Novel precision control techniques in a trapped Yb+ ionimplementation

Jarrah SastrawanQuantum Control Laboratory

School of PhysicsFaculty of Science

University of Sydney

2015

A thesis submitted in fulfilment of the requirements for the degree of Master of Science

AbstractPrecise control of quantum systems is vital to scientific and technological progress, including therealisation of quantum computation and simulation, record-breaking timekeeping and positioningapplications. Control of quantum systems is hampered by the effects of random environmental orhardware noise, which leads to unknown deviations from the system’s desired evolution. This the-sis presents a set of interaction-focussed methods for improving precision control, tailored to theproblems of quantum error suppression and stabilisation of oscillators, which share a common basicstructure. These methods are based on a theoretical framework called the filter-transfer function for-malism, which expresses the convolution of user-applied control and random noise in the languageof spectral filtering. This powerful and accessible approach is experimentally verified in this thesis,and is used to formulate novel control techniques and improve on existing ones. This thesis ex-perimentally demonstrates the effectiveness of novel composite pulse schemes for suppressing errorin quantum bits. Furthermore, the thesis derives and demonstrates a novel predictive technique forstabilising oscillators by means of combining multiple frequency measurements against a quantumreference. The thesis therefore advances the theoretical understanding of a frequency-domain for-malism for noise-affected quantum systems, on which basis it presents and demonstrates novel andimproved techniques for mitigating the effects of such noise on the user’s precision control over thesystem.

Statement of ContributionI wrote this thesis. The conception, argumentation, wording and appearance of the thesis is my

work and responsibility. I played a leading role in the theoretical and experimental work concerningfrequency standards, presented in Chapters 6 and 7. On the qubit control project, I took the datacomparing traditional composite pulses to uncompensated pulses, as well as developing the single-tone scanning technique for obtaining spectral information about an arbitrary control operation. Mydescription of the experimental setup in Chapter 2 is based on my own laboratory notes, communica-tion with my colleagues, precedents in the literature and equipment manuals provided by the variousmanufacturers. The mathematical derivations and proofs presented in Chapter 3 are my own synthesisof the existing literature.

What I have presented in this thesis has also relied on the work of my colleagues. AssociateProfessor Michael Biercuk guided and directed this project. He charted the main courses of the re-search of which this thesis is part, including the development of a filter function approach to universalqubit noise, the noise engineering techniques used for both experiments, and the predictive schemefor stabilising oscillators. His close oversight of the drafting of this thesis has given it a narrativeconsistency and a crafted purpose that would have been impossible to achieve otherwise.

MC Jarratt contributed to the optical setup for these experiments. She built the RF circuitry todrive the electro-optical modulators (EOM) at the required frequencies to produce sidebands on thelaser beams, as well as writing software for a control panel to easily manipulate these frequencies.The laser frequency stabilisation setup in its current form, including the free-space UV switch drivenby a custom circuit that consists of three flip-flop elements, was built by her.

Alexander Soare played a central role in building the microwave delivery system, writing theIGOR Pro software modules for coordinating the ion trap hardware and the qubit control experiment,developing the noise engineering platform using IQ modulation, as well as contributing to the opticalsetup. He built an early version of the laser frequency stabilisation setup using a UV switch driven bya single flip-flop element. The design and results of the qubit control experiment, in particular, owe agreat debt to his involvement.

Harrison Ball did much of the legwork for the theoretical aspects of this thesis, as well as playinga leading role in conceiving of the qubit control experiment. He contributed to the development of thefidelity metric for qubit coherence and the tricky task of establishing the fit between theoretical pre-diction and experimental data. He played a principal role in the analysis of composite pulse schemesand comprehending them within the Walsh basis framework.

Todd Green laid the theoretical groundwork for the general approach taken in this thesis. His workon generalising the filter function to universal noise provided the mathematical tools needed to buildthese novel techniques for quantum control. Cody Jones worked with me on the frequency standardproject in its earliest days, and wrote some important derivations for the predictive approach. StephenDona collected several crucial pieces of data for presentation in this thesis, assisted by Nitin Nand.David Hayes, James McLoughlin and Xinglong Zhen were involved in conceiving and building thequbit control experiment and in collecting the data for it.

Parts of the work presented here have been published or submitted elsewhere, as the products ofmy group’s collective research effort. Content pertaining to the qubit control experiment appeared inthe Nature Physics publication [69] reproduced in Appendix A, and content pertaining to the noise

engineering technique developed by the group appeared in the Physical Review A publication [70] re-produced in Appendix B. I am a co-author on both of these publications and my specific contributionsare indicated in each of them.

Content pertaining to the frequency standard experiment appears in two submitted works of whichI am the lead author. The concept and definition of a predictive correction for oscillator stabilisationappears in my Honours thesis (submitted October 2013). More advanced stages of this work, in-cluding numerical simulations of a frequency standard undergoing predictive correction, appears amanuscript submitted to Physical Review Letters which is on the pre-print server [68] and is repro-duced in Appendix C.

Acknowledgements

When doing science, you depend on others constantly. Each bit of progress that you make dependson those who support you: people who guide you, people who work with you, people who precededyou and people who listen to you. Many people have supported me in various aspects of this projectand the thesis came out of it. I want to acknowledge each of them in turn.

Mike’s guidance and engagement with my scientific work has been extremely dedicated. Hishands-on involvement in all the projects has been most valuable to my understanding of them. Hisexperience and knowledge of every part of the experimental apparatus has allowed me to rapidlybecome familiar and comfortable in using it, while his teaching of the theoretical foundations ofthis work has been exceptionally clear and accessible. His passion for physics and his desire tocontribute to the scientific community have made for a rewarding three years for me. I thank him forthe opportunities he has given me, and for his support in the conclusion of my work, which culminatesin this thesis.

MC is my dearest friend in physics. She mentored me as I first learned how to work in the lab, withpatience and good humour. Her sharp instinct for troubleshooting and the rigour of her experimentaltechnique set an example that I admired and strove to follow. Her presence in the lab eased the mostdifficult tasks and made fun the most tedious days. Her trust and companionship have been immenselyimportant to me, and I would be glad to have had anywhere near as positive an impact on her as shehas had on me.

Alex has a breathtaking understanding of the experimental design and its operation. I relied onhim a great deal to teach me how to run, fix and design the experiments described in this thesis.His ability to communicate his experience and his memory for detail were indispensable; withoutthem I would not have been able to so quickly acquire the knowledge and skills needed to run theseexperiments. I have no doubt that he will establish himself as an authority in anything he turns hisattention to.

Harrison has been a steadfast and dependable source of advice and solidarity. His disciplinedattitude to work and commitment to thoroughness has impressed me greatly. He helped me throughthe challenges of theoretical physics and of working as a physicist. I wish him every good fortune forthe rest of his studies and hope that they will serve him well.

Terry McRae’s humour and clear-eyed view on life has buoyed me during difficult periods, and isdeeply appreciated. Based their long ion-trapping experiences, Karsten Pyka and James McLoughlingave me advice and filled in gaps in my knowledge, thereby significantly improving my understanding

of my work. I thank Claire Edmunds for her friendship and for helping me to complete the descriptionof the experimental setup that appears here.

Our collaborators in South Africa, Hermann Uys and Ismail Akhalwaya, gave me many usefulinsights into the field. Their work, which complements my own, has enriched my understanding ofthe possibilities and relevance of what I have presented here. Peter Fisk and Bruce Warrington, whonow lead the National Measurement Institute, once ran the ion trap that is used in these experiments.I was most grateful to have the opportunity not only to discuss with them the technical details andhistory of the trap, but also to seek their expert opinions on the techniques I have developed to improveoscillator stability. David Reilly and Stephen Bartlett gave me important advice and support in mydegree, which I very much appreciate.

Of the many friends who have lent me a sympathetic ear, I would like to single out Steph, Ero-manga and Clare, for their kindness and understanding.

Veronica has witnessed the side-effects of the tough aspects of physicist life. I thank her for herperseverance and dedication to taking care of me when I needed it, assuring me when I was concernedand calming me down when I was anxious. Her love and care have been vitalising.

Harmy’s wisdom and depth of experience have been crucial. Many moments of uncertainty andworry were resolved by her clear thinking. My written expression owes much to her careful reading,and my enthusiasm persists in no small part thanks to her encouragement.

My mum and dad gave me everything I needed to do this. At every point they have helped meto take control of events and work towards the best outcomes. Without them always behind me, ofcourse, this thesis would be impossible.

Contents

1 Introduction 11.1 Introduction to precision quantum control . . . . . . . . . . . . . . . . . . . . . . . 11.2 Unified approach to precision quantum control . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Abstract features of precision control . . . . . . . . . . . . . . . . . . . . . 21.2.2 Theoretical and experimental methods . . . . . . . . . . . . . . . . . . . . . 3

1.3 Specifics of the control problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.1 Qubit control problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.2 Frequency standard problem . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Contributions of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Structure of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Experimental Platform: Ytterbium Ions in a Linear Paul Trap 62.1 Trap design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Yb+ ion energy level structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Driving of ion optical transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Two-stage photoionisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.2 Doppler cooling of ion cloud . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.3 Depopulation of undesired states . . . . . . . . . . . . . . . . . . . . . . . . 112.3.4 State initialisation and detection . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4.1 External cavity diode lasers . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4.2 Laser frequency stabilisation . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5 Measurement system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.6 Coherent control of the qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.6.1 Derivation of qubit Rabi oscillation . . . . . . . . . . . . . . . . . . . . . . 152.6.2 Microwave setup for driven qubit control . . . . . . . . . . . . . . . . . . . 18

2.7 Improved photon detection accuracy by Bayesian normalisation . . . . . . . . . . . 20

3 Semiclassical stochastic noise in quantum systems 213.1 Spectral character of noise statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Power spectral density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3 The Wiener-Khinchin theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Characterising the Fidelity of Qubit Control 264.1 Introduction to quantum control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2 Fidelity metric for effect of noise on qubit control . . . . . . . . . . . . . . . . . . . 26

4.2.1 Single-qubit Hamiltonian picture . . . . . . . . . . . . . . . . . . . . . . . . 274.2.2 Magnus expansion of the Hamiltonian . . . . . . . . . . . . . . . . . . . . . 28

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4.3 Fidelity metric for control accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3.1 Magnus expansion of fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3.2 Fidelity approximations beyond the quadratic power . . . . . . . . . . . . . 304.3.3 Fidelity in the frequency domain . . . . . . . . . . . . . . . . . . . . . . . . 31

5 The Filter Function Approach to Characterising Qubit Control 345.1 Introduction to composite pulse methods of quantum control . . . . . . . . . . . . . 345.2 Quantum control experiment setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.2.1 Noise engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2.2 Experimental implementation via IQ modulation . . . . . . . . . . . . . . . 36

5.3 Experimental validation of filter function predictions . . . . . . . . . . . . . . . . . 375.4 Composite pulse sequences for noise suppression . . . . . . . . . . . . . . . . . . . 39

5.4.1 Traditional composite pulses . . . . . . . . . . . . . . . . . . . . . . . . . . 395.4.2 Single-tone engineered noise scans for BB1 and SK1 . . . . . . . . . . . . . 39

5.5 Search for optimised Walsh amplitude modulated filters . . . . . . . . . . . . . . . . 415.5.1 Dynamically corrected gates as composite pulses . . . . . . . . . . . . . . . 415.5.2 Walsh synthesis of composite pulses . . . . . . . . . . . . . . . . . . . . . . 415.5.3 Walsh synthesis software . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.5.4 Randomised benchmarking . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6 Characterising Frequency Standard Instabilities 466.1 Introduction to frequency standard stabilisation . . . . . . . . . . . . . . . . . . . . 466.2 Filter function approach to frequency standards . . . . . . . . . . . . . . . . . . . . 47

6.2.1 Time-average sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.2.2 Variances in the frequency domain . . . . . . . . . . . . . . . . . . . . . . . 48

6.3 Covariance in the frequency domain . . . . . . . . . . . . . . . . . . . . . . . . . . 506.4 Extending analytic variance expressions to locked local oscillators . . . . . . . . . . 51

7 Transfer Functions for Predictive Stabilisation of Frequency Standards 557.1 Introduction to predictive protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . 557.2 Deriving the linear predictor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.2.1 An alternative formulation: Kalman filtering . . . . . . . . . . . . . . . . . 577.3 Numerical simulation of frequency standard protocols . . . . . . . . . . . . . . . . . 59

7.3.1 Correction accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597.3.2 Long-term stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.4 Experimental emulator for frequency standards . . . . . . . . . . . . . . . . . . . . 617.4.1 Frequency measurement technique . . . . . . . . . . . . . . . . . . . . . . . 61

7.5 Experimental verification of stability improvement . . . . . . . . . . . . . . . . . . 647.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

8 Conclusion 668.1 Summary of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668.2 Contributions of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668.3 Directions for future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

A Nature Physics publication 69

B Physical Review A publication 87

C arXiv pre-print 100

D MATLAB code for Frequency Standard Simulation 110D.1 queue.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110D.2 timedomain.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115D.3 intnoisecov.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129D.4 colourednoise.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135D.5 showbins.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

References 144

Chapter 1

Introduction

1.1 Introduction to precision quantum controlThe ability to precisely control quantum systems is a vital requirement for the advancement of scienceand technology. Improvements in humans’ power to control the behaviour of quantum systems offerconcrete benefits to physics and engineering communities. Speaking about atomic clocks, which arethe highest-precision quantum devices in existence [8], Nobel Laureate David Wineland states thatthroughout history "when a better clock was built, use was immediately found for it" [1]. High-precision control of individual quantum systems is the basis on which quantum computing may berealised [58], is necessary for quantum simulation [13] and is central to time-keeping and positioningapplications [3, 48, 40]. Precision control is also necessary for fundamental science: advances inthe precision of frequency standards beyond the 10−19 level could allow for laboratory-scale testsof general relativity [12] and improved bounds on the measurement of the time-variation of the finestructure constant, an empirical parameter in the Standard Model [7].

Improvement in precision control over quantum systems is, to a large extent, achieved by sup-pressing inaccuracies and instabilities, generically labelled as ‘error’, that result from physical noiseprocesses. A fruitful area of research to achieve such improvements is quantum control theory, whichfocusses user-end techniques in order to make quantum systems more resistant to the effects of noise.The precision and stability of a quantum device, such as a quantum memory [42], may be improvedby tailored error-suppressing operations performed by the experimentalist, without the need for fur-ther hardware modifications. The application of such interaction-focussed techniques to quantumsystems has become prominent only since the early 2000s [77, 46], and there remains much scope forthe investigation of experimentally-useful control techniques for improving the precision of realisticquantum devices, such as quantum information processors and atomic clocks [69, 68].

This thesis makes theoretical and experimental contributions to the field of precision control ofquantum systems. The theoretical contributions involve the development of a unifying mathematicalformalism, known as the ‘filter transfer function’ formalism, to describe the spectral response of aquantum system to an arbitrary control protocol. This formalism allows for straightforward and accu-rate prediction of the system’s dynamics, by capturing the sensitivity of the system to environmentaland control noise. The thesis provides a comprehensive treatment of the filter transfer function for-malism. On the basis of this formalism, it generalises and improves upon existing techniques for errorsuppression.

The experimental contributions of this thesis are: (i) the validation of the accuracy of the filterfunction approach in real physical systems, (ii) the development of a method for engineering artificialnoise that allows the experimental apparatus to emulate a broad range of realistic noise scenarios, and(iii) demonstrating a novel predictive stabilisation scheme for oscillators that may have applicationsin the improvement of frequency standards, as well as general predictive control methods for sup-

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pressing decoherence in quantum systems. The experimental results demonstrate the usefulness ofthe theory and provide a platform for optimising error-suppression protocols for other physical sys-tems. Therefore the relevance of these results is not limited only to a particular quantum system, e.g.trapped ions, but may have direct implications for any quantum system in which precision control isa priority.

This introductory chapter is organised as follows. In Section 1.2, an explanation of precisionquantum control is presented in general terms, which forms the common basis for a unified treatmentof the two control problems examined in this thesis. The usefulness of a unified theoretical formal-ism and experimental platform is discussed. In Section 1.3 the specific features of the two controlproblems are introduced, to show how they focus on different parts of the same composite system.Section 1.4 is a description of the contributions made by the thesis to the scientific community, whilein section 1.5 the structure of the whole thesis is outlined.

1.2 Unified approach to precision quantum control

1.2.1 Abstract features of precision controlThis general description presents a unified treatment of two precision control problems that share anunderlying commonality: (i) maintaining the coherence of a quantum state (the ‘quantum bit con-trol’ problem), and (ii) stabilising the frequency of an oscillator for use as a standard (the ‘frequencystandard’ problem). The two problems are associated with distinct research communities and tra-ditions, though there has been recognition of their similarities [19]. This thesis treats the quantumbit (‘qubit’) control and frequency standard problems as having a common basic structure, showinghow the same theoretical formalism (the filter transfer function approach) and the same experimentalapparatus (microwave-driven trapped ions) may be used to find insights into both.

In abstract terms, the precision quantum control problems considered in this thesis involve ran-dom deviations over time from the system’s desired behaviour (‘error’), caused by instabilities in thehardware or coupling to fluctuating parameters of the environment (‘noise’). The archetypal systemof interest is a two-energy-level quantum system (the quantum bit) driven by an oscillator, e.g. anatomic transition in an ion being driven by a coherent microwave source [20]. The primary source oferror in the systems of interest to this thesis are fluctuations in time; spatial inhomogeneities acrossensemble systems, such as are familiar from nuclear magnetic resonance (NMR) spectroscopy, arenot directly targeted by the techniques developed in this thesis, although it can be shown that thesetechniques indirectly suppress error due to quasistatic spatial inhomogeneities.

In order to accurately control the quantum system’s behaviour, an experimentalist may use a hard-ware approach that involves isolating the system from environmental perturbations (e.g. shieldingagainst magnetic field fluctuations), as well as optimising the performance of the driving oscillator.Nevertheless, improving the hardware alone is insufficient for many applications. This fact has lead tothe development of interaction-focussed approaches that suppress the effects of noise by manipulatingthe system in such a way that the noise is cancelled. Both the controller (e.g. an oscillator) and the sys-tem (e.g. atomic transition) are susceptible to error caused by noise. An interaction-focussed approachto error-suppression involves the experimentalist modifying only the controller-system interaction, inorder to make the combined setup more robust against noise. By contrast, a hardware-only approachwould involve improving the controller and system individually to reduce their sensitivity to noise.

The qubit control problem and the frequency standard problem share a fundamental physical sim-ilarity. The basic architecture of both problems, which consists of a driving oscillator coupled to aquantum system, motivates a unified treatment of characterisation and methods for suppressing theeffects of noise. The specific noise processes present in both cases lead to the same kinds of error inthe qubit; e.g. both dephasing noise acting on a qubit from the environment and frequency noise ina local oscillator that interrogates that qubit produce the same undesired phase evolution. The type

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of noise studied in this thesis is semiclassical, meaning that it produces unitary randomisation of thequantum state, rather than e.g. entanglement of the quantum system with the environment.

1.2.2 Theoretical and experimental methodsThe fact that the qubit control and the frequency standard problems have the same physical struc-ture means that the same theoretical and experimental platforms may be used to investigate them.In line with classical control theory approaches to suppressing random noise processes, the theoret-ical framework used in this thesis is a frequency-domain, statistical framework. The noise affectingthe controller and/or system is characterised in terms of its frequency-domain power spectral den-sity, while the action of the control protocol is quantified as a ‘filter transfer function’, also in thefrequency domain. The use of frequency-domain analysis emerged independently in the quantumcontrol [54, 47, 76] and frequency standard [14, 2, 66] communities. This thesis recognises that thesame mathematics underpins both approaches, and thus develops a unified theoretical formalism inChapter 3.

The same experimental setup is used to realise these two control problems. A high-quality os-cillator that performs coherent driving on a qubit may be used either as a qubit control setup or as afrequency standard, depending on which element is used to stabilise by the other. The typical quan-tum control experiment involves a qubit, undergoing noisy evolution due to the ambient environment,that is subjected to an open-loop control protocol by a ‘clean’ oscillator, which cancels the noise tosome extent and thereby improves the experimentalist’s precision control over the qubit state. Con-versely, the typical frequency standard involves an oscillator, whose frequency fluctuates randomly,being locked in a closed feedback loop to match the stable measured frequency of a qubit.

A key benefit offered by the physical commonality of the two problems is that noise in one of theelements may be used to emulate noise in the other. The noise processes in the two cases produceidentical physical effects: dephasing noise in the qubit produces the same phase randomisation asfrequency noise in the oscillator, while amplitude noise in the oscillator mimics the effect of relaxationof the qubit excited state. The commonality between qubit and oscillator noise allows noise to beengineered by deliberately degrading the oscillator signal according to some desired prescription [70].If the engineered noise dominates the underlying system noise (‘intrinsic’ noise), the experimentalistmay use this apparatus to emulate the kinds of noise found in other systems. The experimentalapparatus used in this thesis may therefore used to test the effectiveness of various error-suppressingsoftware protocols in a range of realistic noise scenarios.

1.3 Specifics of the control problems

1.3.1 Qubit control problemThe qubit control problem consists of a qubit that experiences noise, either due to interaction withthe environment or by instabilities in the driving oscillator. The noise is described as ‘semiclassical’because it can be modelled as a random perturbation of a classical (many-photon) field interactingwith a quantum system, removing the need for a fully quantum-mechanical treatment of the oscillator-qubit interaction, e.g. that found in [76]. The primary noise mechanisms are called dephasing, whichacts to randomise the relative phase between the two components of the quantum state, and drivingnoise, which leads to inaccuracy in the state-space trajectory of the driven quantum state duringinteraction with the oscillator.

The effect of such semiclassical noise is to produce ‘decoherence’ of the quantum state, whichin this model refers to randomisation of the state that leads to a loss of quantum behaviour, averagedover an ensemble of identical systems [69]. So-called ‘open-loop’ techniques for mitigating the effectof noise on coherent quantum systems make use of an early insight from nuclear magnetic resonance

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(NMR) spectroscopy [30], which finds that a tailored sequence of driven control operations can can-cel out slowly-varying qubit noise. These techniques, which involve only the interaction betweencontroller and system, result in a ‘recovery’ of qubit coherence at the endpoint of the sequence. Thecontrol operations therefore act as a spectral filter for the noise, since the extent of error-suppressiondepends on the frequency of the noise fluctuations in terms of the timescale of the control operations.

1.3.2 Frequency standard problemThe frequency standard problem consists of a ‘local oscillator’ (LO) that experiences frequency noisedue to hardware instabilities or environmental perturbation. The LO frequency is compared to thecharacteristic frequency associated with the qubit splitting energy, which is in this problem is as-sumed to be far more stable, by means of a spectroscopic interrogation. The resulting frequencydifference information is used to correct the LO frequency in a feedback loop, thereby improving thefrequency stability of the locked local oscillator (LLO). This problem belongs to the general categoryof ‘oscillator stabilisation’ control problems; however, due to the importance of frequency standardsas the most significant example of oscillator stabilisation to a stable qubit reference, this thesis refersto this problem as the ‘frequency standard problem’.

The LO instability is quantified by metrics of variance of the measured frequency differencesbetween the LO and qubit. The characterisation of oscillator stability is a field in its own right,with several types of variance quantity (e.g. sample variance [66], Allan variance [2], frequencyvariance [14, 51]) in use. The frequency standard community has developed an analytic methodfor calculating these variances, based on transfer functions [66]. A transfer function captures thesensitivity of its associated variance metric to the spectral content of the LO noise, thereby allowingan analytic prediction of the instability of a free-running oscillator, given spectral information aboutthe noise and time-domain sensitivity information about the measurement sequence.

1.4 Contributions of this thesisFor the qubit control problem, this thesis seeks to develop useful methods of characterising the effectof noise on quantum coherence, and to find new control schemes with improved filtering power. Interms of characterisation methods, the thesis presents the first experimental validation of the accuracyof a filter function approach to calculating trace fidelity (a widespread metric of coherence for drivenevolution). In terms of finding improved error-suppression schemes, the thesis develops a unifyingtheoretical framework that expresses known composite pulses in terms of the Walsh basis functions.The spectral features of these Walsh functions provide a recipe for synthesising novel composite pulsesequences with improved noise-filtering capability, hence better maintaining quantum coherence.

For the frequency standard problem, this thesis extends the transfer function method of charac-terising instability to apply to locked local oscillators (LLO). The contribution to characterisationinvolves the derivation of a recursive analytic form that may be simply expressed in the frequencydomain, allowing for a direct transfer function calculation of variance for an LLO, entirely analogousto traditional calculations of LO variance. This extension to characterisation of the LLO is based onthe derivation of a frequency-domain covariance function. This covariance transfer function is used,in combination with insights from optimal control theory, to develop a novel predictive technique thatis presented in the thesis, which is demonstrated to improve the overall stability of the LLO. The con-tribution to the problem of oscillator stabilisation is the derivation and demonstration, by numericalsimulation and experiment, of this predictive technique, which exploits correlations between multiplesequential measurements in order to predict the trajectory of LO noise at the moment of correction.The use of both pre-characterised spectral information and real-time measurements makes this tech-nique a hybrid of feedback and feedforward, resulting in improved overall stability in the lockedoscillator in comparison to traditional feedback methods.

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1.5 Structure of thesisThe thesis is organised as follows. Chapter 2 provides a detailed description of the apparatus usedto perform the experiments for the qubit control and frequency standard problems. The descriptionincludes the physics of the linear ion trap in which the experiments are conducted, the atomic physicsof the ytterbium ion ensemble which constitutes the qubit, the optical system used to control andmanipulate the ions, the measurement system used detect the state of the qubit and the precisionmicrowave setup that acts as the local oscillator of the system. Particular emphasis is placed on theimplementation of engineered noise to the oscillator, which is a novel technique for the emulation ofquantum systems.

Chapter 3 provides theoretical background to the frequency-domain treatment of the types ofsemiclassical noise that are pertinent to this thesis. A rigorous definition of the power spectral densityis provided for the canonical case of stochastic, Gaussian-distributed noise. The Wiener-Khinchintheorem is proven, on which the conversion from time-domain to frequency-domain calculations isbased.

Chapter 4 introduces the application of the filter function formalism to universal qubit noise, viaan average Hamiltonian theory. An experimentally-accessible metric for the accuracy of an arbitrarycontrol operation, viz. the average fidelity of the operation, is derived in the convenient form of anoverlap integral of the noise spectral density and the filter function.

Chapter 5 presents experimental results on the qubit control problem. The chapter verifies theaccuracy of the filter function formalism in making predictions of the fidelity of qubit operations. Thetrapped-ion qubit is used as an experimental testing platform for the filter function approach. Theusefulness of the filter function approach is further demonstrated by the improved error-suppressionperformance of composite pulses, as predicted by the features of their respective filter functions.The relationship between the shape of the filter function and the manner in which the pulses aresynthesised in the Walsh basis allows for the search and experimental testing of novel high-ordercomposite pulses, which improve on existing protocols.

Chapter 6 introduces methods for characterising the stability of frequency standards, expressedin the vernacular of filter transfer functions. The traditional analytic approach to frequency standardvariance calculations is extended to the case of locked local oscillators. The key tool for perform-ing this extension, the novel ‘pair covariance transfer function’ that captures statistical correlationsbetween two time-separated measurements, is derived.

Chapter 7 presents results on the stabilisation of frequency standards. The covariance transferfunction derived in the previous chapter is used to develop a scheme for linearly combining multiplemeasurement outcomes in order to predict the trajectory of noise. This scheme produces improvedfeedback correction accuracy and hence improved LLO stability. Since this predictive scheme relieson spectral information about the noise, it is tested in a range of parameter regimes by numericalsimulation. The success of the scheme in simulation motivates the development of the trapped-ionqubit system as an experimental emulator of frequency standards, which may be tuned within a widerange of control and noise parameter settings. The improved stability afforded by the predictivescheme is verified by experimental results.

Chapter 8 summarises the main contributions made to the field of precision quantum control inthis thesis. On the basis of these contributions, it suggests fruitful directions for future research andapplications of the results of the thesis.

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Chapter 2

Experimental Platform: Ytterbium Ions in aLinear Paul TrapTrapped ions are a proven architecture for extremely high-precision experiments in precision quantumcontrol [7]. The primary advantages of trapped ions in this context are high-accuracy control overthe electronic degrees of freedom, as well as the exceptionally long ion lifetimes and internal statecoherence times [61]. Trapped ions have been a leading technology in time and frequency metrologyfrom the 1990s to the present [19, 3, 52], and are an important platform for the development ofquantum information processing [49, 60, 62] and quantum simulation [45, 10].

The trapped ions used in this thesis are 171Yb+ ions in a linear Paul trap kept under ultrahighvacuum (UHV) around 2 × 10−10 Torr. A ‘cloud’ of ions is produced by the photoionisation of abeam of neutral Yb, and is Doppler-cooled in the centre of the trap by an ultraviolet laser at 369.5 nm.The multilevel electronic structure of Yb+ is reduced in effect to a two-level system (the hyperfineground state doublet) by using visible and infrared repump lasers. The cloud therefore forms anensemble of identical two-level quantum systems (qubits). This hyperfine qubit with a transition at12.6 GHz is coherently driven by unidirectional free space microwaves [71] and the occupancy of theexcited F = 1 state is detected by a fluorescence shelving technique [17]. Fluorescence is measuredwith a photomultiplier tube and photon counter, and the hardware is centrally coordinated by a PCrunning WaveMetrics’ IGOR Pro.

This chapter is organised as follows. Section 2.1 outlines the design of the ion trap used in theseexperiments, including the basic derivation of the trapping pseudopotential. Section 2.2 describes therelevant electronic energy level structure of Yb+, including the relationship between radiation polar-isation and the spatial orientation of the ions’ magnetic dipole. Section 2.3 describes the optically-driven transitions used for ionisation, cooling, state preparation and detection, while section 2.4 de-scribes the optical system that delivers the laser beams to the ion cloud. Section 2.5 describes the ionstate measurement apparatus. Section 2.6 describes the setup used to deliver coherent microwaves forprecision control of the qubit state. Section 2.7 presents a control-based normalisation technique formitigating photon detection noise.

2.1 Trap designThe type of ion trap used for these experiments is a linear radiofrequency (RF) trap, also knownas a Paul trap [79]. It was proven in the nineteenth century by Earnshaw [18] that a point chargecannot be confined by electrostatic forces alone, so the Paul trap uses a combination of static andoscillating electric fields to confine ions. The following summary of Fisk [19] explains how such anarrangement of fields produces confinement. Consider in one dimension an ion of mass m and chargeq, in the presence of a sinusoidally oscillating electric field with frequency Ω and position-dependentamplitude E0(y). The ion experiences a force

6

F (t) = qE0(y) cos (Ωt) (2.1)

and so its position, given the initial condition y(0) = 0, is

y(t) =−qE0(y)

mΩ2cos (Ωt) (2.2)

For sufficiently large Ω, the deviation |y − y| of the ion from its average position is small, and thepseudopotential approximation can be made. In this approximation, the electric force is written

F (t) = qE0(y) cos (Ωt) + qdE0(y)

dy(y − y) cos (Ωt) (2.3)

= qE0(y) cos (Ωt)− q2E0(y)

mΩ2

dE0(y)

dycos2 (Ωt) (2.4)

where the second line is obtained by substituting in (2.2). Averaging (2.4) over a single cycle of theelectric field oscillation gives

F (y) =−q2E0(y)

2mΩ2

dE(y)

dy(2.5)

Extending the problem to three spatial dimensions ~r = (x, y, z) and defining the ponderomotivepseudopotential F (~r) = q∇Ψ(~r) gives

Ψ(r) =qE2

0(r)

4mΩ2(2.6)

Any electric field configuration that at some position satisfies∇Ψ = 0 and∇2Ψ > 0 can trap chargedparticles. Paul traps use a combination of static (DC) and oscillating (RF) potentials to satisfy theseconditions, resulting in a quadrupole potential at the centre of the trap.

Prestage et al. [64] first demonstrated the use of a linear Paul trap, consisting of four parallel rodsacting as RF electrodes with the DC potentials applied by a pair of end caps. The advantage conferredby a linear geometry over the conventional hyperbolic geometry is that the pseudopoential minimumis extended along the trap axis, allowing more ions to be trapped. The resulting ion cloud has anellipsoid shape whose axial extent can be tuned from a few mm up to 1 cm by the DC potentials onthe end caps, as opposed to the spherical cloud produced in a hyperbolic trap.

In the linear Paul trap used for these experiments, the trapping conditions are fulfilled with RFapplied only to one diagonal pair of rods, shown schematically in Figure 2.1. For general operation,the RF frequency is set to 425 kHz and the RF amplitude to 150 Vpp. The other pair of rods haveDC voltages applied to them for the purpose of shimming, whereby spontaneous asymmetries inthe trapping potential are compensated for by applying a DC bias perpendicular to the RF electricfield. Such asymmetries are usually produced by the accumulation of unwanted charge on conductingsurfaces inside the vacuum chamber or trap mounting. Shimming is done by regularly optimising theDC rod voltages in order to centre the ion cloud in the trap, and can be enhanced by applying a DCbias to the RF potential.

2.2 Yb+ ion energy level structureThe ytterbium ion is an attractive choice for quantum control and frequency standard experimentsbecause the relevant optical transitions can be accessed with commercial lasers, allowing for fastand efficient cooling of the ion cloud and preparation and detection of its electronic state [61]. The

7

Figure 2.1 : A schematic diagram of the linear Paul ion trap setup used in these experiments. The end caps haveholes in the centre to allow laser beams to be incident on the cloud in the axial (Y) direction. The free-spacemicrowaves from the horn and lens combination propagate radially (X), while the aligning magnetic field isoriented 45 in the XZ plane.

hyperfine qubit is defined to be the F = 0 → 1,mF = 0 → 0 transition of the ground state 2S1/2,which is known as a ‘clock transition’ because it is first-order insensitive to magnetic fields, makingit useful for frequency standard applications [21, 61]. Borrowing notation from quantum information,the qubit states are defined |0〉 ≡ |2S1/2(F = 0,mF = 0)〉 and |1〉 ≡ |2S1/2(F = 1,mF = 0)〉.

Figure 2.2 shows the energy level structure of the Yb+ ion. The 2S1/2 and 2P1/2 states do notform a closed subsystem of the Yb+ electronic manifold, as the 2P1/2(F = 1) state decays to the2D3/2(F = 1, 2) states with 0.5% probability. Decay out of the 2P1/2 manifold causes full populationof the undesired 2D3/2 state in less than a second, given the ion cloud sizes used in this experiment.Furthermore, collision of the ions with background gas drives the 2D3/2 → 2D5/2 transition, fromwhich the ions decay to the extremely long-lived 2F7/2 state [55, 61]. In order to recover these ionsinto the desired manifold, repump lasers are used, as described in Section 2.3.3.

The detailed S − P level diagram in Figure 2.4 shows the hyperfine splitting of 2S1/2 and 2P1/2

at 12.643 GHz and 2.105 GHz respectively, with each F = 1 level further Zeeman split into a mF =−1, 0, 1 triplet. As seen in the next section, the S − P transition is required for both state detectionand cooling of the ion cloud.

The qubit is defined by the magnetic dipole transition |0〉 → |1〉 (F = 0 → 1) that exists inthe ground state manifold. In order to drive the qubit transition uniformly over the whole cloud, themagnetic dipole must be spatially oriented. This requires the application of a static, uniform magneticfield across the cloud. This field is achieved by three pairs of Helmholtz coils experiencing fixed DCto produce a homogeneous field across the ions of strength 1.8 Gauss, as depicted in Figure 2.3. Theapplication of this field produces a small Zeeman triplet splitting of the |1〉 state by some tens ofMHz. The entire trap and vacuum chamber is magnetically insulated from ambient magnetic fieldsby a Ni-Co shield. The induced spatial alignment allows linearly polarised optical and microwaveradiation to preferentially target all three Zeeman-split states of |1〉 (optical cooling beam), or onlythe |1〉 (mF = 0) (microwave driving beam), as outlined in section below.

8

Figure 2.2 : Energy level structure, showing transitions in the UV (purple arrows), visible (red arrows) and IR(green arrows) regions of the spectrum. Thin grey lines indicate spontaneous decay paths between levels. Thedashed black lines show transitions due to collisions with background gas that provide a path to the undesired2F7/2 states. The term symbols of the 3[3/2]1/2 and 1[5/2][5/2] states indicate that two elections are excited andcouple to each other as well as to the atomic core, a departure from the usual LS-coupling [61].

2.3 Driving of ion optical transitions

2.3.1 Two-stage photoionisationTo trap an ion cloud, a filament of isotopically-enriched Yb is heated, producing a neutral beam ofpredominantly 171Yb through the centre of the trap. During this loading stage, two overlapping UVbeams at 398.9 nm and 369.5 nm cross the Yb beam and via a two-stage photoionisation processproduce Yb+. The 398.9 nm photon excites a bound electron from the ground state to the 1P1 excitedstate and the 369.5 nm photon excites the electron into the continuum of unbound states [61]. Thenumber of ions in a trapped cloud depends on the density of the neutral Yb beam, which is tuned viathe current applied to the filament. In this manner, large ion numbers between 103 and 104 can beobtained, which results in an excellent measured signal to projection noise ratio [36].

2.3.2 Doppler cooling of ion cloudDoppler cooling of the ion cloud is achieved by a beam of 369.5 nm light with approximately 100µW of optical power which is red-detuned from the 2S1/2(F = 1)→ 2P1/2(F = 0) optical transitionby 10 MHz, half its natural linewidth. The ions experience stimulated photon absorption preferen-tially, since they are more likely to absorb a photon if the ions and photons are moving in oppositedirections. Since the ions re-emit isotropically, the momentum component in the opposing directionis reduced, and after many absorption and emission events, the overall ion kinetic energy is absorbedby the restoring force of the trap. This process cools the ions down to a ‘Doppler limit’ set by thephoton recoil energy [82]. In the experiment, the cooling is achieved by scanning the 369.5 nm laserfrequency across 200 MHz on the red side of the optical transition in a sawtooth pattern with a cy-cle time of 10 s. The UV cooling beam is equally split into a ‘straight’ and ‘oblique’ beam before

9

Figure 2.3 : Photographs of Helmholtz coils around the ion trap vacuum chamber. Upper: side view of threesets of Helmholtz coils around the vacuum chamber allow for flexible and precise modification of the appliedmagnetic field on the ions. The white microwave horn is visible in the foreground. Lower: view from above ofthe Helmholtz coils and a magnetometer that allows for precise measurements of the ambient magnetic field.This provides a method for correctly compensating for the Earth’s magnetic field, thereby ensuring the desiredfield strength and stability at the ions.

entering the trap and recombining at an angle in the trap centre, in order to increase Doppler coolingefficiency by targeting a larger section of the ions’ velocity space. It is sufficient for these experi-ments to work near the Doppler cooling limit, and so additional cooling techniques such as resolvedsideband cooling are not necessary [37].

Since there is a non-negligible probability of off-resonant driving and decay to 2S1/2(F = 0)and 2P1/2(F = 1) levels, sidebands are added to the cooling beam at 14.75 GHz using the second-order modulation of a 7.374 GHz electro-optical modulator (a New Focus free-space EOM, modelnumber 4851-02). The combination of 369.5 nm carrier and the 14.75 GHz sideband allows Dopplercooling of all states in the S and P manifolds except for the 2S1/2(F = 1,mF = ±1) Zeeman splitstates. To couple these Zeeman split states to the F = 0 states and hence allow them to be cooled, it isnecessary to tailor the polarisation of the cooling beam. In order to drive both the ∆mF = 0 transition,which requires linearly polarised photons, and the ∆mF = ±1 transitions, which requires left andright circularly polarised photons, the cooling beam’s polarisation must be chosen with respect tothe alignment of the qubit dipole axis. The cooling beam is linearly polarised at 45 to the dipoleaxis, so that it is decomposed into parallel and perpendicular components with respect to that axis.The parallel component drives the ∆mF = 0 transition while the perpendicular component is furtherdecomposed into left and right circular components, which drive the ∆mF = ±1 transitions [55].Thus the mF = ±1 states are coupled to relevant mF = 0 states and can be Doppler cooled. An

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2S1/2

2P1/2

F = 0

F = 1 mF = 0

mF = -1

mF = 1

Cooling

2.105 GHz

12.643 GHz

369.5 nm

F = 0

F = 1

mF = 0

mF = -1

mF = 1

mF = 0

mF = -1

mF = 1

mF = 0

mF = -1

mF = 1m

F = 0

mF = -1

mF = 1m

F = 0

mF = -1

mF = 1

Initialisation Detection

Figure 2.4 : S − P transitions in Yb+. Thick solid lines indicate resonant transitions driven by the carrierfrequencies while thin solid lines indicate transitions driven by sideband frequencies. Doppler cooling; purplearrows show the 369.5 nm stimulated transitions: the three ∆mF = −1, 0, 1 transitions at resonance and theone ∆mF = 0 on the 14.75 GHz sideband. Light grey arrows show spontaneous decay paths; all P → Stransitions are permitted by selection rules except the (F = 0) → (F = 0) and (F = 0,mF = 0) → (F =1,mF = ±1) decay transitions. The use of the 14.75 GHz sideband and polarisation alignment allows Dopplercooling of all the hyperfine and Zeeman states in the 2S1/2 and 2P1/2 manifolds. State initialisation; 369.5nm beam is modulated by 2.105 GHz sidebands, which excite the (F = 1) → (F = 1) transitions. The2S1/2(F = 1)→ 2P1/2(F = 0) resonant transitions and 2P1/2(F = 0)→ 2S1/2(F = 1) spontaneous decayare present without contributing to the initialisation cycle, and so for clarity they are not shown in this middlesub-figure. After excitation to 2P1/2(F = 0) by the sideband transition, the qubit decays into the |0〉 hyperfineground state without being excited further, producing state initialisation in |0〉. State detection; no sidebands areapplied to the 369.5 nm, resulting in driving only of the 2S1/2(F = 1)→2 P1/2(F = 0) transition. This formsa closed subspace that prevents change in the relative population of |0〉 and |1〉, resulting in an unambiguousqubit state measurement. Transitions are not shown to scale.

energy level diagram depicting the cooling transitions appears in Figure 2.4.

2.3.3 Depopulation of undesired states

To maintain the ion state in the desired subspace, two ‘repump’ lasers are applied. An IR laser at935.2 nm with approximately 4 mW of optical power drives the 2D3/2(F = 2) → 2[3/2]7/2(F = 1)transition, with an EOM-generated (a fibre EOM from EOspace) sideband at 3.067 GHz driving the2D3/2(F = 1) → 3[3/2]7/2(F = 0) transition. From the 2[3/2]7/2 state the ions rapidly decay tothe ground state 2S1/2 and are returned to the cooling cycle. To depopulate 2F7/2 state, a visiblered laser alternating between 638.610 nm and 638.616 nm is overlapped with the repump IR beam.The rate of decay into the 2F7/2 state is sufficiently slow that the hyperfine-split levels can be de-populated by alternating between the two transition frequencies every 30 s. The red beam drives the2F7/2 → 1[5/2]5/2 transition (details of the hyperfine splittings are shown in the figure), from whichthere is rapid decay back into the 2D3/2 states.

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2.3.4 State initialisation and detectionThe ion state is initialised by inducing decay to the 2S1/2(F = 0) state (i.e. |0〉), which is achievedby introducing 2.105 GHz EOM-generated sidebands (a free-space EOM from New Focus, customAR-coated model number 4431-01) onto the cooling beam and applying the two repump beams. The7.374 GHz sidebands are absent from the cooling beam during initialisation, so that there is no drivencoupling from |0〉 to any other state. This configuration results in initialisation of all the ions to |0〉.Detection of the qubit state is achieved by coupling the |1〉 state to 2P1/2(F = 0) using 369.5 nm lightwith no sidebands. This transition is closed, resulting in a large number of photons scattered froma repeated absorption and emission process and hence an improvement in the fluorescence signal incontrast to driving to 2P1/2(F = 1) [20]. An energy level diagram depicting the state initialisationand detections transitions appears in Figure 2.4.

Since the frequencies of these sidebands must be very precisely determined, RF signals in the GHzregime are synthesised from a 10 MHz RF reference using a pair of Hittite micro synthesisers (modelHMC-C070 for the 7.374 GHz signal and HMC-C083 for the 2.105 GHz signal) and a Mini-Circuitsvoltage-controlled oscillator (VCO, for the 3.067 GHz signal). The frequencies of these synthesisersare set via a USB interface from National Instruments, and monitored on a Hewlett-Packard frequencycounter.

2.4 Laser systemIn order to achieve the efficient and accurate driving of the optical transitions covered in the previoussection, the optical system must fulfil the following criteria:

• narrow linewidth < 5 MHz

• long-term frequency stability

• precisely tunable sidebands

The sources for all four laser frequencies are external cavity diode lasers (ECDL) from MOGLabs,which are frequency-locked using a commercial wavemeter from High-Finesse/Ångstrom runningin multi-channel mode. The sidebands are added to the relevant beams by electro-optical modulators(EOM) driven by synthesised RF power at the required GHz frequencies. The application and removalof sidebands and beamlines on the sub-millisecond time scales of the experiments are performed byTTL-triggered switching of the optical modulators. A more detailed description of each part of theoptical system follows.

2.4.1 External cavity diode lasersAll the frequencies used in this optical system: UV at 369.5 nm and 398.9 nm, IR at 935.2 nm and vis-ible at 638.6 nm, are produced by semiconductor diode lasers coupled to external cavities in a Littrowconfiguration [38]. The ECDLs are from MOGLabs with model number ECD-003, with controllerboxes from MOGLabs with model number DCL-202. For the generation of 369.5 nm, the ECDLsare a low-cost, low-power alternative to previous techniques involving a high-power Ti:sapphire laserand lossy frequency doubling techniques [22]. The diode consists of a semiconductor heterostructurewhich acts both as a light-emitting part and a gain medium. Emitted photons pass through a collima-tor and are incident on a diffraction grating whose angle is piezoelectrically controlled; this forms theexternal cavity.

The arrangement of internal and external cavities produces three overlapping mode structures: thegrating modes, the internal cavity modes and the external cavity modes. The output frequency of theECDL is determined by the combined mode with the highest power. Since the three mode structuresare shifted by separate physical mechanisms (grating modes by the piezo voltage, internal modes by

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Figure 2.5 : A photograph of ECDL lasers of the type used in this experiment. The semiconductor diodes andexternal cavities are housed in MOGLabs boxes, which are controlled via external software link.

the current through the diode, external modes by an overall temperature control of the cavity), theexperimentalist is able to select the desired mode and tune it in a range of ∼ 5 GHz. [38]

2.4.2 Laser frequency stabilisationThe frequencies of the four lasers are cyclically measured using a wavelength meter, commonly re-ferred to as a wavemeter, from High-Finesse/Ångstrom with model number WSU U-10, which can op-erate in an multi-channel switching mode to monitor up to eight frequencies in tandem. The waveme-ter is a relative frequency reference that is calibrated to an absolute Helium-Neon (HeNe) laser refer-ence, achieving 1 MHz relative and 10 MHz absolute frequency accuracy [38]. The wavemeter itselfhas only a front and back fibre optical port, so two external four-channel switches are triggered bythe wavemeter to multiplex the four input beams into the wavemeter. The IR and visible beams aresent through a commercial fibre optical switch that TTL-triggered by the wavemeter. The two UVbeams are switched using a custom-built free-space switch: a pair of AOMs pass the two beams onan eight-channel cycle coordinated by a flip-flop-based circuit and TTL-triggered by the wavemeter,as shown in Figure 2.6.

By measuring the laser frequencies in real time, the wavemeter can be used to lock them to thedesired set points. The frequency measurements are delivered to a PC, which acts as a softwarePID controller that sends a command back to the wavemeter to output a control voltage. The controlvoltages are routed to each laser’s piezoelectric control, closing the feedback loop and allowing stabil-isation of the laser’s output frequency. In switching mode the wavemeter applies corrections to eachlaser frequency approximately once a second, which results in deviation at the sub-MHz level thatis acceptable for these experiments, without the need for more sophisticated stabilisation at higherbandwidths such as Pound-Drever-Hall locking or polarisation spectroscopy [38, 50]. This is pos-sible because intrinsic current noise in the controller is so low that additional linewidth narrowing

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Figure 2.6 : Switch and wavemeter setup. In eight-channel switch mode, the wavemeter accepts light fromeither the front or back port, sending the measurement outcome for each channel to the PC via USB (solidblack arrow) and trigger the switch and flip-flop circuit by TTL pulses (dotted arrows). The fibre-optic switchoperates on a four-stage cycle, sending in IR, red and no light in series; the fibre components are shown asbrown solid lines. The flip-flop circuit operates on an eight-stage cycle, sending TTL pulses to two RF switchesat the second and third out of eight triggers. The RF switches regulate the 70 MHz driving of two free-spaceAOMs, which act as ‘shutters’, rapidly producing and removing first-order diffraction beams that continue onto a combiner and are overlapped into a fibre-coupler. The effect of this switch-AOM setup is allow the twoUV beams to enter the front wavemeter port in series, as triggered by the flip-flop circuit. Since light enteringthe front port overrides measurements taken at the back port, the net effect of this two-port setup is that eachof the four beams (369.5 nm, 398.9 nm, 935.2 nm and 638.6 nm) are measured on their own channels. On thebasis of the measurements, the PC calculates a PID control value which is sent back to the wavemeter, in orderto piezo-control each laser.

14

via current feedback and an external Fabry-Perot optical cavity is unnecessary [50]. These experi-ments represent the first Doppler-limited cooling using the output of a bare ECDL without additionallinewidth narrowing.

2.5 Measurement systemWhen sufficiently cooled, the ion cloud fluoresces at 369.5 nm. During the loading and initial coolingstages, a CCD video camera (Andor Solis) with a 369.5 nm UV filter is used to precisely determine the2S1/2 → 2P1/2 resonance frequency and to estimate the size of the cloud. The fluorescence intensityof the cloud is directly proportional to the probability of observing the qubit in the |1〉 state, so themeasured intensity of scattered photons is used as a proxy for a projective Z-basis measurement ofthe qubit state. A photomultiplier tube (PMT) from Products for Research and a gated photon counter(from Stanford Research Systems with model number SR400) linked to the experiment PC via an RS-232 serial connection, is used to measure ion fluorescence. The photons are counted for a fixed gateperiod, generally 1 ms, giving a more accurate measurement of fluorescence intensity than possiblewith the CCD camera. The accuracy of the counter photon number is enhanced by a Semrock filterFF01-370/6 placed on the aperture of the PMT that transmits light at 369.5 nm, within a bandwidthof 10 nm, reducing the effect of background photon noise. In order to switch between CCD cameraand photon counter imaging of the cloud, the two devices are placed on a pair of precision translationstages (from ThorLabs with model number NRT100E). By alternating the translation stages betweentwo sets of optimised settings, the user can choose to align either the CCD camera or the photoncounter with the static ion trap. The image is focussed using a 100mm diameter Knight optical lenswith 160mm focal length (model number LPV160100) followed by an 80mm diameter lens with100mm focal length.

2.6 Coherent control of the qubitA useful method of visualising the qubit state is the image of the Bloch sphere, where a pure state

|ψ〉 = cos (θ/2) |0〉+ sin (θ/2)eiφ |1〉 (2.7)

is represented as a vector from the centre of the sphere to its surface, where θ and φ are the polarand azimuthal angles on the sphere. This geometrical interpretation is possible due to the homomor-phism between the SU(2) group symmetry of the qubit state space and the SO(3) symmetry of the 3Dgeometrical space [24].

Coherent driving of the qubit state, say from |0〉 to |1〉, may be represented as rotations on theBloch sphere. For example, phase-coherent driving of the qubit by the microwave oscillator results inrotation of the Bloch vector about some axis in the XY plane, while dephasing perturbations of thequbit state result in random rotations about the Z axis. A visualisation of such a driven rotation on theBloch sphere is shown in Figure 2.7. The following section derives the relationship between constantuniform driving of the qubit state and the resulting rotation of the qubit state, which is known as ‘Rabioscillation’. The term ‘π time’ refers to the time taken to drive a π-rotation on the Bloch sphere, andas such is an inverse measure of Rabi rate.

2.6.1 Derivation of qubit Rabi oscillationSince the quantum system targeted by these experiments is the hyperfine qubit that resides in theground state 2S1/2 manifold, the problem of microwave field-ion state interaction can be formulatedas the ‘two-level Rabi problem’. The solution is found by a time-dependent perturbation theoryapproach to the response of a quantum dipole system to a coherent classical field. The field-qubitinteraction produces a coherent oscillation of the qubit state on a time scale set by the amplitude of

15

Figure 2.7 : Driven rotation on the Bloch sphere. The phase of the driving oscillator determines the axis ofrotation n, while the amplitude of the driving field determines the rate of rotation (the ‘Rabi rate’). For a givendriving time, the rate determines the arc length of the Bloch vector’s trajectory θ(t). This figure is based on thatfound in [24].

the field, known as ‘Rabi oscillation’ or ‘Rabi flopping’.The following derivation is based on the summary in [71]. Starting from a discrete multi-level

quantum system, an arbitrary state |Ψ〉 may be written as a linear combination of energy eigenstates|n〉, defined as

Hq |n〉 = En |n〉 (2.8)

where Hq is the quantum system’s time independent Hamiltonian. The effect of interaction with atime-varying field is most simply written by expressing the total Hamiltonian as the sum

H(t) = Hq + Hi(t) (2.9)

where H(t) is the total Hamiltonian and Hi(t) is the time-dependent interaction Hamiltonian. Theperturbative approximation lies in assuming that the interaction is sufficiently small that the state canbe expressed as a normalised superposition of the eigenstates of Hq

|Ψ(t)〉 =∑

n

cn(t)e−iEnt/~ |n〉 (2.10)

where the coefficients cn(t) are in general complex. Substituting this ansatz into the time-dependentSchrödinger equation

i~∂

∂t|Ψ(t)〉 = H |Ψ(t)〉 (2.11)

and left-multiplying by the eigenstate 〈m| gives, after some manipulation

i~cm(t)e−iEmt/~ =∑

n

cn(t)e−iEnt/~ 〈m| Hi |n〉 (2.12)

16

The matrix element Hmn of the interaction Hamiltonian is defined: Hmn = 〈m| Hi |n〉, and thefurther restriction is applied that since Hi captures interaction with a classical field, the matrix Hmn

is Hermitian so that Hmn = H∗nm. Reducing the dimension of Hmn to the two-level system (a qubit)relevant to this physical system produces a pair of coupled ordinary differential equations (ODE)describing the time evolution of the qubit state

i~c1(t) = c0(t)H10eiω10t (2.13)

i~c0(t) = c1(t)H∗10e−iω10t (2.14)

where ω10 = (E1 − E0)/~ and |0〉 and |1〉 label the ground and excited energy states.Assuming a coherent dipole interaction with a magnetic field linearly polarised and aligned with

the quantisation axis of the qubit, the interaction Hamiltonian can be written

Hi = −~µB cos (ωt+ φ) (2.15)

where ~µ is the qubit magnetic dipole moment and B and φ are the amplitude and phase of the drivingmagnetic field. Making the magnetic dipole approximation, which consists of assuming that varia-tions in the magnetic field strength are insignificant on the scale of the qubit’s spatial wavefunction,is justified in this experimental system [71]. Substituting this dipole interaction Hamiltonian into thecoupled ODEs gives

c1(t) = c0(t)iΩ

2

(ei(ω+ω10)teiφ + e−i(ω−ω10)te−iφ

)(2.16)

c0(t) = c1(t)iΩ∗

2

(ei(ω−ω10)teiφ + e−i(ω+ω10)te−iφ

)(2.17)

where the complex resonant Rabi rate is defined Ω ≡ 〈1| ~µ |0〉B/~. To solve this coupled ODEsystem, a further approximation is required. The rotating wave approximation assumes that the fre-quency detuning ∆ ≡ ω − ω10 is negligible compared to the nominal frequency ∆ ω so that thefrequency sum components ω + ω10 can be considered to average to zero on the time scale of thefrequency detuning components ∆. Given the rotating wave approximation, the coupled ODEs canbe rewritten

c1(t) = c0(t)iΩ

2e−i∆te−iφ (2.18)

c0(t) = c1(t)iΩ∗

2ei∆teiφ (2.19)

These ODEs can be combined into a single second-order ODE in c1(t)

c1(t) + i∆c1(t) +

∣∣∣∣Ω

2

∣∣∣∣2

c1(t) = 0 (2.20)

whose general solution is

c1(t) = e−i∆t(ae−iΩRt/2 + beiΩRt/2) (2.21)

17

where a and b are coefficients that depend on the initial state c1(0) and ΩR =√|Ω|2 + ∆2 is called

the ‘generalised’ Rabi rate, which for zero detuning reduces to the resonant Rabi rate |Ω|.Since a solution is required for an arbitrary initial state, the time evolution of c0(t) and c1(t) is

written as a linear transformation by a ‘rotation matrix’ R(t)

[c1(t)c0(t)

]=

[R11(t) R10(t)R01(t) R00(t)

] [c1(0)c0(0)

](2.22)

This matrix equation can be solved for an arbitrary initial state by applying the general solution (2.21)and setting the initial state to be two basis states [1 0] and [0 1]. The rotation matrix is therefore

R(t) =

[e−i∆t/2(cos (ΩRt/2) + i∆

ΩRsin (ΩRt/2)) iΩ∗

ΩRei∆t/2eiφ sin (ΩRt/2)

iΩΩRe−i∆t/2e−iφ sin (ΩRt/2) ei∆t/2(cos (ΩRt/2)− i∆

ΩRsin (ΩRt/2))

](2.23)

which in the case of a resonant driving field ∆ = 0 reduces to

R(t) =

[cos (ΩRt/2) ieiφ sin (ΩRt/2)

ie−iφ sin (ΩRt/2) cos (ΩRt/2)

](2.24)

with ΩR = |Ω|. This form of the rotation matrix completely captures the effect of coherently drivinga qubit. From the definition of the Rabi rate it can be seen that the rate of driven transition betweenthe two states is proportional to the amplitude B of the driving field, while ‘axis of rotation’ in theBloch sphere picture is given by the phase of the driving field φ.

2.6.2 Microwave setup for driven qubit controlThe qubit state transition |0〉 ↔ |1〉 is coherently driven by microwaves at 12.6 GHz, whose linearpolarisation is parallel aligned to the qubit dipole axis. In order to accurately and precisely controlthe ion state, there are several criteria that the microwaves must fulfil:

• high power for rapid driven Rabi rate on the order of Ω ∼ 100 kHz

• spatial homogeneity over the ion cloud to maintain the coherence of the ensemble

• precise and arbitrary modulation of phase and amplitude

• excellent frequency, power and polarisation stability

The microwaves are generated by an Agilent vector signal generator (VSG) with model numberE8267D and the UNY option for enhanced ultra-low phase noise, amplified by an 18 dB RF amplifier(from Microsemi with model number AML618P1802) and delivered as directed free-space wavesvia a conical microwave horn and dielectric lens (from Flann Microwave with model number CL320-4901) [71]. The overall microwave setup is schematically depicted in Figure 2.8. A primary benefit ofusing the VSG is that the continuous-wave output may be arbitrarily phase and amplitude modulated,either by a software link or by an external arbitrary waveform generator (AWG). The experimentPC uses a 16-bit Pulseblaster (from Spincore with model number DDS-IV 1000) to produce 5 V TTLpulses that trigger the VSG, producing sharply rectangular gated pulses, and in the case of FM, triggerthe AWG to start applying the modulating waveform. The Pulseblaster can be preprogrammed by thePC to execute a complex series of triggers that manage the whole apparatus including microwaveand optical components, and these triggers are synchronised to the 10 MHz Cs reference, which issupplied by the National Measurement Institute and cleaned using a Wenzel crystal.

The VSG produces a carrier RF signal with mHz precision and is also frequency-stabilised to the10 MHz Cs reference. The phase noise specification of the VSG output, which can be quantified in

18

Figure 2.8 : A schematic diagram of the microwave setup. RF signals are depicted in blue, DC signals in solidgreen, transistor-to-transistor logic (TTL) voltage signals in dashed green, free-space microwaves in red anddigital I/O links in black (the type of link is labelled in italics).

terms of power at a fixed offset from the carrier, is given by Agilent as -138 dBc/Hz at 10 kHz and-141 dBc/Hz at 100 kHz offset. After amplification, the maximum achievable microwave power is33 dBm which corresponds to a measured minimum π time τπ = 4 µs, though most experimentsare operated at τπ > 25 µs. The use of an external gate trigger supplied by a TTL signal fromthe Pulseblaster, which achieved via the pulse modulation setting on the VSG, allows these rapid µspulses to be precisely timed. The TTL pulse jitter and shape can be examined using high-bandwidthoscilloscope and the fidelity of the output microwave pulses can be checked using an Agilent MXAVector Analyser N9020A.

Several methods could have been used to deliver microwaves to trapped ions, including applyingmicrowaves to the trap rods themselves, open-ended waveguides or microwave horn antennae [71].The benefit of the horn and lens combination used in this setup is that it produces a highly directedbeam, which maximises the power directly incident on the ions and minimises undesired reflectionsfrom the vacuum chamber walls. Thus the criteria of high power and spatial homogeneity are well metby the horn and lens. The microwave horn acts as a waveguide whose diverging geometry reduces thereflection at the free-space interface, while the dielectric lens focusses the microwaves into a narrowwaist near the trapped ions.

19

2.7 Improved photon detection accuracy by Bayesian normalisa-tion

Fluctuations in the 369.5 nm laser amplitude and detector shot noise lead to errors in the qubit statemeasurement. To mitigate the effect of amplitude fluctuations slower than the measurement time,a normalisation procedure is employed. Before the measurement of the desired state |ψ〉, the |1〉‘bright’ and |0〉 ‘dark’ states are prepared and measured. A linear interpolation is performed betweenthe bright state count B and dark state count D in order to cancel the effect of slow laser amplitudedrifts on the measured fluorescence of |ψ〉.

To mitigate the high-frequency shot noise, an iterative Bayesian estimation scheme is employed.Identical sets of normalisation and state measurements are performed up to a desired number ofiterations I . A modified form of Bayes’ rule is used:

P (θ|ci) =P (c|θ)P (θ|ci−1)∫ π

0P (c|θ)P (θ|ci−1)dθ

(2.25)

where ci is the measured photon count on the ith iteration and θ is the Bloch vector inclination from|0〉 to |1〉. The prior state probability density P (θ), initially assumed to be uniform, is iterativelyrefined I times by applying the rule. The likelihood function P (c|θ), which captures the detectornoise probability, is experimentally verified to be Gaussian-distributed within the bounds of the brightand dark state normalisation counts [70]:

P (c|θ) = exp[−(

D + B−Dπθ

σD + σB−σDπ

θ

)2](2.26)

where B, D are the mean bright and dark state counts, while σB, σD are the standard deviations overthe set of iterations. The final estimate for θ is found by taking the expected value of the posteriorafter all the iterations are complete, which gives the result of the projective measurement with greaterthan 98% accuracy [70].

20

Chapter 3

Semiclassical stochastic noise in quantumsystemsIn order to achieve precision control of the systems of interest, it is necessary to achieve a detailedcharacterisation of the dominant sources of noise. The quantum control community has a particularinterest in stochastic, non-Markovian, universal qubit noise models [70], which are semiclassicalmodels that capture many features of real noise environments [24]. The resulting qubit dynamicsare non-dissipative, non-entangling and non-depolarising, which means that they can be modelledas unitary evolution of a single qubit. It is desirable to quantify this noisy evolution by quantitiesaccessible in experiment. A convenient measure to use in this regard is the power spectral density(PSD), which is a frequency-domain function that captures the autocorrelation in the noise (alsoreferred to as its ‘colour’). The PSD of a noisy signal can be easily inferred from spectral diagramsmeasured on a signal analyser.

A key task in quantum control is to produce techniques for error-robust control, taking into ac-count information such as the PSD. The action of arbitrary control operations on the qubit may berepresented in the frequency domain by filter transfer functions, well known in classical control the-ory and frequency metrology [14, 5, 66]. An important advantage of working with filter transferfunctions is ease of use: the time-domain convolution of noise trajectories and response functionsis transformed via the Wiener-Khinchin theorem, proven in this chapter, into a product of frequencydomain quantities that may be straightforwardly evaluated. In this framework, the accumulation oferror during a control protocol in a known noise environment is captured by metrics that are definedas the overlap integral of the transfer function of the control and the PSD of the noise. This transferfunction framework allows for convenient calculation of error due to noise and for straightforwardanalysis of the error-suppressing properties of different control protocols.

The purpose of this chapter is to provide the relevant mathematical background to the controlprotocols used in later chapters. The chapter is organised as follows. Section 3.1 outlines the statisticalbasis for the frequency-domain approach of this thesis, rigorously defining concepts of ensemble,autocorrelation and ergodicity. Section 3.2 gives a first-principles derivation of the PSD that is akey quantity that captures the spectral features and correlations in the noise processes consideredhere. Section 3.3 gives a proof of the Wiener-Khinchin theorem that states that the PSD and thetime-domain autocorrelation function form a Fourier pair.

3.1 Spectral character of noise statisticsIn the quantum systems of interest, it is the statistical properties of an ensemble that are the mostrelevant and easily accessible [19, 6]. A fundamental reason for this is that projective measurementof a quantum system causes the wavefunction to collapse randomly into one of its eigenstates [73],so to achieve a reliable estimate of the state it is necessary to take an ensemble of measurements.Furthermore, since the relevant noise processes are stochastic, any robust characterisation of them

21

must be found by extracting features that are invariant over an ensemble of realisations, effectively‘averaging over’ the stochasticity. Such statistical measures are of central importance to precisionmetrology and quantum measurement [66, 83]. The following derivation elaborates on the treatmentof the PSD given in [56].

The main control problems of interest to this thesis involve stochastic noise processes that degradethe accuracy of driven manipulation of the qubit state. A generic time-domain continuous noise vari-able y(t) is used in the following derivation; its specific definition depends on the type of experiment.Since y(t) is a random function of time, it is different for every realisation of an experiment. Such arealisation is referred to in this thesis as a run and a set of runs is referred to as an ensemble; generallythe former is indexed by r and the latter denoted by E , where the ensemble’s cardinality (numberof runs) is written |E|. Often the ensemble has infinite cardinality and the characteristics of y(t) av-eraged for this infinite E are assumed to reproduce the statistics of a single run of y(t), consistentwith the frequentist interpretation of probability [29]. The aim is to describe the spectral characterof the statistics of y(t), since such a description is more robust and informative than time-domainmeasurements of particular realisations.

The notation used for a mean of a stochastic function f(a) over an infinite ensemble is E[F ]where F is a random variable whose realisations are the values of f(a) for each run r. The mean ofa particular run of that function fr(a) over its domain is written 〈fr〉a. The definitions of these twotypes of mean are

E[F ] ≡∫

Ω

Fp(dF ) (3.1)

〈fr〉a ≡ limb→∞

1

b

∫ b

0

fr(a)da (3.2)

where Ω is the sample space of F , p(dF ) is the probability of obtaining the outcome in the range[F, F + dF ] (since it is assumed that the range of f(a) is a subset of R), f(a) has domain a ∈ [0,∞)and fr(a) refers to the realisation of f(a) on the rth run. The subscript indexing the run is oftenomitted in general statements about the characteristics of f(a) not peculiar to individual runs. E[F ]is generally written E[f(a)] for convenience.

Both E[f(a)] and 〈fr〉a are ideal quantities that cannot be measured directly. Therefore it isassumed that they can be accurately estimated by E[f(a)] and fr where

E[f(a)] ≡ 1

|E|∑

r∈Efr(a) (3.3)

fr ≡1

b

∫ b

0

fr(a)da (3.4)

such that the estimators converge to the ideal means in the infinite limits:

E[f(a)] = lim|E|→∞

E[f(a)] (3.5)

〈fr〉a = limb→∞

fr (3.6)

The autocorrelation function RTSyy (∆t)

22

RTSyy (∆t) ≡ E[y(t)y(t+ ∆t)] (3.7)

defined on the domain ∆t ∈ R and is symmetric about ∆t = 0 due to the assumption of wide-sensestationarity, explained below. The same information can be captured in a one-sided autocorrelationfunction ROS

yy (∆t), defined

ROSyy (∆t) ≡ 2RTS

yy (∆t), for ∆t ≥ 0 (3.8)

In general, one-sided quantities are preferred, so Ryy(∆t) is used to refer to ROSyy (∆t).

Ryy(∆t) captures the time-correlations of y(t) over the whole ensemble E . By contrast, the time-average autocorrelation function is defined for a single run r

Ryy(∆t, r) ≡ 2 〈yr(t)yr(t+ ∆t)〉 t , for ∆t ≥ 0 (3.9)

where Ryy(∆t, r) is directly defined to be one-sided. Care should be taken to distinguish these twoautocorrelation quantities, as Ryy(∆t, r) is often used to estimate Ryy(∆t), via the assumption ofergodicity, which is defined below.

The following assumptions are made about y(t):

• that it is a continuous-time real-valued stochastic process

• that it is defined on t ∈ [0,∞)

• that its ensemble mean is zero at every instant: E[y(t)] = 0, ∀t and for |E| → ∞

• that its probability density function (PDF) is Gaussian, i.e. that ∀t, y(t) is distributed in aGaussian manner over E

• that it is wide-sense stationary (WSS), i.e. that its two-sided ensemble autocorrelation functionRTSyy (∆t) is invariant under time-translations of ∆t

An important concept concerning the relationship between time and ensemble properties of a processf(a) is ergodicity. Broadly speaking, an ergodic process is one whose ensemble averages can befound via the time averages of a particular realisation of that process. The two relevant cases ofergodicity are: ergodicity in the mean and ergodicity in the autocorrelation:

〈fr〉a = E[f(a)] (3.10)Rff (∆t, r) = Rff (∆t) (3.11)

for all r. For practical purposes, ergodicity may be assumed in order to justify the use of a time-average quantity as an estimator of an ensemble average, which may be difficult to calculate inde-pendently. In the context of real experiments, it is not usually possible to simultaneously measure anensemble of identical systems. Instead, the measurements are repeated and their outcomes combinedto provide an estimate of ensemble quantities. In these instances, the validity of the estimate dependson whether the process is properly ergodic.

23

3.2 Power spectral densityMany experimental devices, such as spectrum analysers, work natively in the frequency domain. It istherefore useful to describe characteristics of the noise as a power spectral density (PSD). The PSDof y(t) is found via its one-sided truncated Fourier transform, denoted YT (ω), which is defined

YT (ω) ≡∫ T

0

y(t)e−iωtdt (3.12)

using the angular frequency, non-unitary convention for Fourier transform that is common in physics.The truncated Fourier transform is used here because there is no guarantee that the integral of yr(t)for a single run will converge over t ∈ [0,∞). The tactic is to place a finite upper bound T on theintegral, take the expectation value and then take the limit T →∞.

The PSD of y(t), written Syy(ω), is thus defined

STSyy (ω) ≡ limT→∞

E[|YT (ω)|2]

T(3.13)

By this definition, Syy(ω) is a non-negative even function whose domain is all real ω, and is henceknown as a two-sided PSD. The corresponding one-sided PSD can be found in a straightforward way:

SOSyy (ω) = 2STSyy (ω), for ω ≥ 0 (3.14)

Since one-sided PSDs are more convenient and more commonly used, PSDs without the superscriptsOS or TS are understood to refer to the one-sided version, by analogy with the autocorrelation func-tion, hence

Syy(ω) ≡ 2 limT→∞

E[|YT (ω)|2]

T, for ω ≥ 0 (3.15)

The definitions for autocorrelation functions and PSD can be generalised to involve two separateWSS processes x(t) and y(t):

Rxy(∆t) ≡ 2 E[x(t)y(t+ ∆t)] (3.16)Rxy(∆t) ≡ 2 〈x(t)y(t+ ∆t)〉 t (3.17)

Sxy(ω) ≡ 2 limT→∞

E[X∗T (ω)YT (ω)]

T(3.18)

all of which are defined to be one-sided, i.e. over ∆t ∈ [0,∞) and ω ∈ [0,∞). These quantities arereferred to as cross-correlation functions and cross-power spectral densites.

3.3 The Wiener-Khinchin theoremThis theorem is the mathematical basis on which it is possible to describe in the frequency domain theeffect of a measurement and control protocol, in the presence of noise. This theorem states that, underspecial conditions, the two-sided PSD STSff (ω) and the two-sided autocorrelation function RTS

ff (∆t)

24

of the function f(a) form a Fourier pair [80]. Some of these special conditions are:

• that f(a) is WSS

• that RTSff (∆t) exists and is finite for all ∆t

• that the integrated spectrum F (ω) ≡∫ ω

0STSff (ω′)dω′ is continuous and differentiable every-

where

• that STSff (ω) and RTSff (∆t) satisfy the conditions for Fourier inversion to be valid

This section proves the theorem for the case of y(t) as defined here. Substituting the definition ofYT (ω) (3.12) into the definition of the PSD (3.15) gives

STSyy (ω) = limT→∞

1

TE

[∫ T

0

y∗(t)eiωt dt

∫ T

0

y(t′)e−iωt′dt′]

(3.19)

= limT→∞

1

T

∫ T

0

∫ T

0

E[y(t)y(t′)]eiω(t−t′) dt dt′ (3.20)

= limT→∞

1

T

∫ T

0

∫ T

0

E[y(t)y(t+ ∆t)]e−iω∆t dt dt′ (3.21)

= limT→∞

1

T

∫ T

0

∫ T

0

RTSyy (∆t)e−iω∆t dt dt′ (3.22)

where y∗(t) denotes the complex conjugate of y(t) (since y(t) is real, y∗(t) = y(t)) and the substitu-tion t′ = t+ ∆t is made . In order to convert the double integral in dt and dt′ into a single integral ind∆t, the geometrical argument in [56] is used to write

Syy(ω) = limT→∞

1

T

∫ T

−T(T −∆t)RTS

yy (∆t)e−iω∆t d∆t (3.23)

=

∫ ∞

−∞RTSyy (∆t)e−iω∆td∆t (3.24)

= FRTSyy (∆t) (3.25)

where Fx(t) denotes the two-sided Fourier transform of x(t).By substituting the relation between the one- and two-sided PSD (3.14), it is possible to derive the

same result for the one-sided quantities, i.e. SOSyy (ω) = FROSyy (∆t). This result is valid provided

∫ ∞

−∞∆t Ryy(∆t)e

−iω∆td∆t <∞ (3.26)

This Fourier pair relationship also holds between the cross-correlation function and the cross-PSD fortwo different signals x(t) and y(t):

Rxy(∆t)FT−−−−IFT

Sxy(ω) (3.27)

For non-WSS processes, a more general proof is required than is given here. Some sources use thisresult in order to define the PSD in terms of the autocorrelation function [2, 14, 5, 66].

25

Chapter 4

Characterising the Fidelity of Qubit Control4.1 Introduction to quantum controlA primary requirement for precision quantum control is the suppression of qubit error due to envi-ronmental noise, which allows the coherence of the quantum state to be maintained [24]. In the fieldof quantum control, dynamical error suppression (DES) is an open-loop control technique that allowsfor the possibility long-term quantum memories [42] and high-accuracy single-qubit logic gates [25].The filter transfer function approach, which frames the task of DES as a filter-design problem [6], isapplied in this chapter to arbitrary, single-qubit control. Analytic expressions are derived for controlfidelity in the presence of universal noise.

This chapter and the following chapter presents the work published as [69] and reproduced inAppendix A. This chapter focusses on the theoretical underpinning of qubit control, and is organisedas follows. Section 4.2 presents an average Hamiltonian theory for deriving expressions for fidelity,which quantifies the effect of stochastic noise on a qubit under universal control. Section 4.3 presentsa Magnus expansion of the fidelity in order to express it in the convenient overlap integral form thatis easily utilised in experiment.

4.2 Fidelity metric for effect of noise on qubit controlWith a spectral model for the noise affecting the qubit, it is necessary to determine the effect of thatnoise on the accuracy of the qubit control operation. Uncompensated stochastic noise during controloperations, e.g. quantum logic gates, results in errors in the qubit evolution. The suppression of suchnoise is vital to applications such as quantum error correction (QEC), which require very low errorrates and detailed understanding of the noise processes causing those errors [59].

The metric used to quantify the deleterious effect of noise is the average fidelity of the desiredoperation on the qubit state, i.e. a measure of how accurately that operation is performed in thepresence of noise. This fidelity is a proxy for the coherence of the quantum state at the end of thepulse. To capture the evolution of the qubit state in the presence of universal (i.e. both dephasing andamplitude) stochastic noise, as well as coherently driven control, an average Hamiltonian theory isemployed. The qubit evolution can be modelled as unitary in this experiment because: (i) there areno projective measurements made during this open-loop control protocol, and (ii) the dynamics arenon-dissipative on the timescale of the experiment [69].

The main theoretical challenge is that the universality of the noise and control means that thequbit’s Hamiltonian does not necessarily commute with itself at different times, which severely com-plicates the analytic expression. The problem is simpified by using a Magnus expansion of the quan-tum state propagator U(τ), where the nested commutators appear only in high-order terms that maybe truncated in the weak noise limit [24]. The work of Green et al. [25, 24] forms the core theoreticalframework of this chapter.

26

4.2.1 Single-qubit Hamiltonian pictureThe derivation begins with the single qubit state, which is used to represent the state of an ensembleof identical non-interacting qubits such as an ion cloud

|ψ〉 = cos (θ/2) |0〉+ sin (θ/2)eiφ |1〉 (4.1)

where θ and φ are the polar and azimuthal angles of the Bloch vector representing the state. Sincethe qubit experiences only unitary evolution due to both noise and control, it obeys the Schrödingerequation

id

dtU(t, 0) = H(t)U(t, 0) (4.2)

where ~ = 1, H(t) is the time-dependent Hamiltonian experienced by the qubit and U(t, 0) is thepropagator that captures the time evolution of the state: |ψ(t)〉 = U(t, 0) |ψ(0)〉 for t ≥ 0. Forconvenience the initial time argument may be assumed to be 0 and dropped: U(t)

The overall Hamiltonian can be separated into parts pertaining to the noise and to the control

H(t) = HN(t) + HC(t) (4.3)

where

HN(t) = b(t)σ (4.4)

HC(t) = h(t)σ (4.5)

where b(t) = [bx(t) by(t) bz(t)] and h(t) = [bx(t) by(t) bz(t)] are 1 × 3 matrices of the noise andcontrol fields respectively, while σ = [σx σy σz]

T is a 3 × 1 matrix of the three Pauli spin operators.Rather than attempting to solve directly for the overall Hamiltonian H(t), a ‘control propagator’ isdefined as the solution of the Schrödinger equation under HC(t) alone

id

dtUC(t) = HC(t)UC(t) (4.6)

In general, this equation cannot be integrated to determine the state evolution, due to non-commutingterms. It is therefore necessary to adopt another reference frame that makes calculation of the statefidelity more tractable.

To this end, an ‘error propagator’ U(t) is defined that captures any deviation from the desired statetrajectory due to noise

U(t) = UC(t)U(t) (4.7)

and a ‘toggling frame Hamiltonian’ is defined

HN(t) = U †C(t)HN(t)UC(t) (4.8)

This toggling frame is interpreted as the rest frame of the noiseless control at all times t. By substitu-tion it is found that the error propagator satisfied the Schrödinger equation governed by the togglingframe Hamiltonian

27

id

dtU(t) = HN(t)U(t) (4.9)

The error propagator U(t) therefore describes the time evolution under the noise Hamiltonian alone.In the cases where there is no noise, or the noise is effectively suppressed by some control protocol,the error propagator is the identity U(t) = I .

By substituting the definition of the noise Hamiltonian (4.4) into that of the toggling frame Hamil-tonian (4.8), it is possible to define a 3× 3 ‘control matrix’ T [24] that captures the reorienting effectof coherent control

HN(t) =∑

i=x,y,z

βi(t)U†C(t)σiUC(t) (4.10)

=∑

i,j=x,y,z

βi(t)Tijσj (4.11)

= β(t)T(t)σ (4.12)

Since this expression is the product of a matrix with the Pauli matrices, the term β(t)T(t) is identifiedwith a 1× 3 ‘error matrix’ a(t) that captures the evolution of the Bloch vector due to noise alone (i.e.in the toggling frame), giving

a(t) = β(t)T(t) (4.13)

4.2.2 Magnus expansion of the Hamiltonian

The presence of noncommuting operators in the Hamiltonian makes calculation of the propagatorintractable, due to the exponentiation required to find the propagator U ∝ e−iH . It is thereforenecessary to approximate U by employing a Magnus expansion and truncating at a useful point [53].The Magnus expansion of the toggling frame Hamiltonian is written

HN(t) =∞∑

µ=1

Φµ(t) (4.14)

where the first few terms in the expansion are [24]

Φ1 =

∫ t

0

HN(t1)dt1 (4.15)

Φ2 = − i2

∫ t

0

∫ t1

0

[HN(t1), HN(t2)] dt2 dt1 (4.16)

Φ3 = −1

6

∫ t

0

∫ t1

0

∫ t2

0

[HN(t1), [HN(t2), HN(t3)]] + [HN(t3), [HN(t2), HN(t1)]] dt3 dt2 dt1 (4.17)

By substituting the definition of the control matrix T(t) (4.10) into the Magnus expansion, it is pos-sible to expand the error matrix a(t):

28

a(t) =∞∑

µ=1

aµ = β(t)T(t) (4.18)

(4.19)

where the row matrix Ti ≡ [Tix Tiy Tiz] is defined in order to write

a1(t) =∑

i=x,y,z

∫ t

0

βi(t1)Ti(t1) dt1 (4.20)

a2(t) =∑

i,j=x,y,z

∫ t

0

∫ t1

0

βi(t1)βj(t2)(Ti(t1)× Tj(t2)) dt2 dt1 (4.21)

a3(t) =2

3

∑

i,j,k=x,y,z

∫ t

0

∫ t1

0

∫ t2

0

βi(t1)βj(t2)βj(t3) (4.22)

(Ti(t1)× (Tj(t2)× Tk(t3)) + (Tk(t3)× Tj(t2))× Ti(t1)) dt3 dt2 dt1

The commutators in the above are converted into cross products using the identity

[uσ,vσ] = 2i(u× v)σ (4.23)

for all u,v with real elements [24].The Magnus expansion has traditionally been used as a perturbative tool to deal with static errors,

such as those that arise from spatial inhomogeneities in NMR ensembles [81]. This treatment appliesthe Magnus expansion to the control matrix T, which captures the time variations of the noise, pro-ducing two distinct sets of orders of terms: the Magnus order and the filter order. This distinction doesnot arise in the traditional NMR case because the time invariance of the inhomogeneities cancels allthe high filter order terms, leaving Magnus order as the only meaningful expansion. In the next chap-ter, the distinction between Magnus and filter order is elaborated and demonstrated experimentally. Arecent mathematical proof of the distinction between Magnus and filter order is given in [63].

4.3 Fidelity metric for control accuracy

4.3.1 Magnus expansion of fidelity

A useful metric for quantifying the extent to which a propagator U1 replicates a desired propagatorU2, hence acting as a proxy for the accuracy of an operator in the presence of noise, is the trace fidelity

F(t) ≡ 1

4

∣∣∣Tr[UT2 U1]

∣∣∣2

(4.24)

In experimental settings, the figure of merit is the ensemble average of the fidelity E[F(t)]. In theabove formalism, the desired propagator is always I in the toggling frame, while the actual propagatorenacted is U(t), so the average fidelity is given by

E[F (t)] =1

4E

[∣∣∣Tr[U(t)]∣∣∣2]

(4.25)

29

Expressing the error propagator in terms of the error vector, the average fidelity may be written

E[F ] =1

2(E[cos (2a)] + 1) (4.26)

where a = |a| is the magnitude of the error vector and the explicit time-domain dependence of F hasbeen suppressed. Expanding the cosine term in a power series gives

E[F ] =

(1− E[a2] +

2

3E[a4]− · · ·

)(4.27)

and then Magnus expanding E[a2]

E[a2] =∞∑

µ,ν=1

E[aµaTν ] (4.28)

= E[a1]2 + E[a2]2 + · · ·+ 2(E[~a1~aT2 ] + E[~a1~a

T3 ] + E[~a2~a

T3 ] + · · · ) (4.29)

as well as the more complicated quartic term E[a4] gives, with rearrangement in terms of Magnusorder,

E[F ] = 1− E[a21]− 2E[~a1~a

T2 ]−

(E[a2

2] + 2E[~a1~aT3 ]− a4

1

3

)+ · · · (4.30)

These terms can be expressed in terms of auto- and cross-correlation functions as described in section3.2: the a1 terms contain two-point correlation functions, the a2 terms contain four-point correlationfunctions, and so on. For sufficiently weak noise, the quadratic term of a1 is the dominant contributorto fidelity and all higher powers of a1 may be neglected, as well as the higher Magnus orders a2, a3 · · · .[24]. The truncated first-order average fidelity is thus written

E[F1] = 1− E[a21] (4.31)

4.3.2 Fidelity approximations beyond the quadratic powerThe criterion for ‘sufficiently weak noise’ used to justify truncating the power series (4.30) at thequadratic term is given by a smallness parameter

ξ ≡ ∆yτ/2 (4.32)

where ∆y =√

E[y2] is the root-mean-square (RMS) of the noise and τ is the total time durationof the control protocol [25]. The quadratic truncation is used in situations where ξ2 1. In caseswhere the noise too strong for this criterion to be fulfilled, the first-order average fidelity E[F1] ceasesto be a useful metric [69]. It becomes necessary to include higher powers of the first-order Magnusorder term E[a2m

1 ] for m > 1. The similarity between the higher powers of E[a21] and the power series

expansion of the exponential e2E[a21] allows for the definition of a new fidelity expression E[Fχ] where

E[Fχ] =1

2

(1 + e−χ(t)

)(4.33)

= 1− E[a21] + E[a4

1]− 2E[a61]

3+ · · · (4.34)

≈ E[F ] (4.35)

30

where the error is χ(t) = E[a21], as above. The distinctions between the three average fidelity quanti-

ties are:

• E[F ] includes all powers of all Magnus terms, valid for all ξ

• E[F1] includes the quadratic power of first Magnus term, valid for ξ2 1

• E[Fχ] includes all powers of first Magnus term, valid for ξ2 . 1 (though agreement withexperiment is shown even for ξ2 = 1 [69])

Since E[Fχ] is useful for a wider range of noise strengths but still has a simple exponential form thatis expressible in terms of a1 alone, it is the metric of choice in the following experiments. The higherpowers E[Fχ] continue to dominate the higher Magnus order terms for sufficiently weak noise, whichjustifies disregarding contributions by a2, a3, · · · [69].

4.3.3 Fidelity in the frequency domainAll of the average fidelity quantities so far have been expressed in the time domain. Since it is moreexperimentally convenient to deal with the spectral characteristics of stochastic noise processes, it isdesirable to translate the problem into the frequency domain. The derivation begins by finding theoverlap integral for E[F1], which contains only two-point correlation functions of the formRij(t1, t2).

The higher Magnus terms aµ for µ > 1 that contribute to E[F ] contain n-point functionsRij(t1, · · · , tn)(for even n only), but since y(t) is assumed to be Gaussian-distributed, the Gaussian moment theoremcan be applied to express the n-point correlation functions in terms of two-point correlation functions[25, 24]. Hence the substitution E[F1] → E[F ] is done after the conversion to the frequency do-main by defining higher-power filter functions F (n)(ω, · · · ) to include contributions from the n-pointcorrelations.

The time-domain expression for the truncated first-order average error is written

E[a21] =

∑

i,j=x,y,z

∫ t

0

∫ t1

0

E[βi(t1)βj(t2)]Ti(t1)TTj (t2) dt2 dt1 (4.36)

=∑

i,j=x,y,z

∫ t

0

∫ t1

0

RTSij (t1, t2)Ti(t1)TT

j (t2) dt2 dt1 (4.37)

=∑

i,j=x,y,z

∫ t

0

∫ t1

0

RTSij (t2 − t1)Ti(t1)TT

j (t2) dt2 dt1 (4.38)

where RTSij (t1, t2) is the two-sided cross-correlation function derived in the previous section, and the

third line is valid assuming all the noise processes are WSS.Using the Wiener-Khinchin theorem derived in section 3.3, it is possible to use the PSD of the

noise

Rij(t2 − t1) =1

2π

∫ ∞

−∞Sij(ω)eiω(t2−t1) (4.39)

as well as Fourier transforming the control matrix elements

Tij(ω) =

∫ t

0

Tij(t1)e−iωt1 dt1 (4.40)

to obtain the overlap integral for the first-order average fidelity

31

E[F1] = 1− 1

2π

∑

i,j=x,y,z

∫ ∞

−∞Sij(ω)Ti(ω)TT

j (ω)dω (4.41)

To exemplify the form of T(ω) in experimentally-relevant situations, it can be shown [24] that fora driven rotation on the Bloch sphere around the X axis of magnitude π, in the presence of purelydephasing noise (i.e. where the noise vector ~β(t) has only aZ-component), the control matrix containsonly two nonzero elements:

Tzz(ω) =ω2

ω2 − Ω2(eiωτπ + 1) (4.42)

Tzy(ω) =iωΩ

ω2 − Ω2(eiωτπ + 1) (4.43)

where Ω = π/τπ is the Rabi rate of the coherent driving of the qubit.The frequency-domain control matrix T(ω) provides the elements for calculating the filter transfer

functions known in the quantum dynamical error suppression (DES) literature [6, 24]. The primarydifference between the above derivation and those found in the DES literature is in the definition of(4.40), where the literature includes a factor of−iω in the righthand side of the definition. By omittingthis factor, the thesis uses a consistent definition that is valid for both the ‘filter functions’ known tothe DES community and the ‘transfer functions’ known to the frequency metrology community. Thetwo functions are mathematically identical and in this treatment are defined in the same way.

A simple and experimentally-useful example of deriving a filter function from the control matrixis the case of pure dephasing noise [25]. The error due to dephasing χz is given to first Magnus orderby

χz =1

2π

∫ ∞

−∞Sz(ω)F (1)

z (ω)dω (4.44)

where the fidelity is reduced by dephasing alone E[F1] ' 1−χz and the first-Magnus-order, dephasingfilter function is defined

F (1)z (ω) ≡

∑

i=x,y,z

|Tzi(ω)|2 (4.45)

Another important source of noise in experimental systems is amplitude noise, which is coaxial withthe driving field. Since the axis of noise varies as the direction of driven control of the qubit, itis necessary to define the time-dependent axis of rotation σΩ = cos (φ(t))σx + sin (φ(t))σy of thedriving field. The error due to amplitude noise has the same form

χΩ =1

2π

∫ ∞

−∞SΩ(ω)F

(1)Ω (ω)dω (4.46)

where the first-Magnus-order amplitude filter function is defined

F(1)Ω (ω) ≡

∑

i=x,y,z

cos2 (φ(t)) |Txi(ω)|2 + sin2 (φ(t)) |Tyi(ω)|2 (4.47)

In the presence of universal noise, the contributions from dephasing and amplitude noise are simplysummed

32

E[F1] = 1− (χz + χΩ) (4.48)

= 1− 1

2π

∫ ∞

−∞(Sz(ω)F (1)

z (ω) + SΩ(ω)F(1)Ω (ω)) dω (4.49)

A replacement of the truncated first-order average fidelity E[F1] with the preferred E[Fχ] requiresonly a substitution of the term E[a2

1] by the exponential e−2E[a21], as indicated by the definition (4.33)

E[Fχ] =1

2

(1 + e−χ(t)

)(4.50)

=1

2

(1 + exp

[1

2π

∑

i=z,Ω

∫ ∞

−∞Si(ω)F

(1)i (ω) dω

])(4.51)

It is important to note that this expression contains only first-order Magnus contributions.To calculate the full unapproximated average fidelity E[F ] requires the inclusion of high-order

Magnus terms, which therefore involve the use of n-order filter functions:

E[F ] = 1− 1

2π

∑

i=z,Ω

∫ ∞

−∞Si(ω)F

(1)i (ω) dω +

1

(2π)2

∑

i=z,Ω

∫ ∞

−∞

∫ ∞

−∞Si(ω)Si(ω

′)F (2)i (ω, ω′) dω + · · ·

(4.52)

where the Gaussian moment theorem has allowed the n-point noise correlations Rij(t1, t2, · · · , tn)to be Fourier transformed into powers of the two-point quantity S(ω) [25]. Hence the only newquantities required are the higher order filter functions F (n)(ω, · · · ). A more generalised approach forcalculating the higher order filter functions in terms of a set of fundamental filter functions has beenrecently derived in [63].

The next chapter will test and verify the accuracy of this overlap integral expression for E[Fχ]as a metric for experimentally measured qubit fidelity. Using insights from the spectral analysis ofthis framework, existing composite pulse schemes for error suppression are extended and improved,leading to a new understanding of the practical benefits of using the filter function in this way.

33

Chapter 5

The Filter Function Approach toCharacterising Qubit Control5.1 Introduction to composite pulse methods of quantum controlThis chapter presents the first experimental validation of generalised filter-transfer functions castingarbitrary quantum control operations as noise spectral filters [25, 24]. Using this validated theoreti-cal framework, composite pulses known from the magnetic resonance community are analysed andexperimentally tested. The filter function framework plays a central role in both characterising andimproving the performance of traditional and novel composite pulse schemes.

The use of composite pulses for noise suppression in quantum systems has received significant at-tention [11, 43, 4]. Many open-loop quantum control techniques make use of ideas from nuclear mag-netic resonance (NMR) due to the structural similarities between the qubit architecture and the spin-ensemble system [6, 4], while others have been developed as Hamiltonian-engineering approachesfor quantum computation [43, 41]. Using a framework known as Walsh basis synthesis to systema-tise and unify the methods for constructing composite pulses, this chapter demonstrates these pulses’enhanced error-suppressing properties in comparison to non-compensated ‘primitive’ pulses. The ex-perimental results also contribute to the characterisation of these pulses, by demonstrating the utilityof the filter function description in realistic, time-varying noise environments.

This chapter and the previous chapter presents the work published as [69] and is organised asfollows. Section 5.2 outlines the experimental setup specific to the qubit control experiment. Section5.3 presents results showing the predictive power of fidelity calculations based on filter functions, inthe case of ‘primitive’ uncorrected pulses. Section 5.4 introduces the composite pulses sequences ofinterest: BB1 (broadband) and SK1 (Solovay-Kitaev). It also demonstrates the conceptual distinctionbetween Magnus order and filter order in determining fidelity, by means of an experimental single-tone spectral scan. Section 5.5 presents results demonstrating the improved noise filtering capabilityof amplitude-modulated pulses synthesised using higher-order Walsh functions, which systematisesprevious work done on dynamically-corrected gates (DCG) [43].

5.2 Quantum control experiment setupThe experiment involves the application of composite microwave pulses to a trapped ion cloud, witha user-defined noise spectrum engineered on the microwave carrier, and the measurement of the pulsefidelity via photon counts. The trapping and optical setup described in Chapter 2 is common to thisexperiment and the frequency standard experiment. The only features specific to the composite pulseexperiment are: (i) the IQ modulation for engineering noise on a continuously-driven pulse, and (ii)the software for synthesising pulses in the Walsh basis.

34

5.2.1 Noise engineeringThis thesis is concerned with demonstrating techniques for precision control in the presence of ar-bitrary noise power spectral densities (PSD). It is therefore necessary to emulate a variety of noiseenvironments by deliberately degrading the microwave signal according to a user-determined noisePSD. This deliberate error is scaled to completely dominate the effect of ‘intrinsic’ phase and ampli-tude noise in the microwaves, so that the ‘engineered’ noise PSD of the output microwaves accuratelyreflects the desired noise character.

While Syy(ω) can in general be any positive semi-definite function of ω, the standard model fornoise PSDs relevant to both the composite pulse and frequency standard experiments is the powerlaw model. This model expresses the PSD as the sum of powers of ω such that particular power-dependences dominate in particular regions. In these experiments, a power-law model defined onthree regions is used, whose boundaries are defined by the low roll-off frequency ωb and the highroll-off frequency ωc:

Syy(ω) =

rb(ω) , for ω < ωb∑+2α=−2 hαω

α , for ω ∈ [ωb, ωc]

rc(ω) , for ω > ωc

(5.1)

where rb(ω) and rc(ω) are the two roll-off functions that describe the PSD behaviour for very lowand very high frequencies, and the coefficients are set to ensure Syy(ω) is continuous over the wholedomain. In these experiments rb(ω) is often set to a constant (‘white ceiling’) and rc(ω) to 0 (‘hardcutoff’), although the model gives extensive scope for customising the PSD to mimic realistic noisesettings.

In the relevant physical settings, phase and amplitude errors arise by different noise processes, andso the engineered noise in each of these quadratures is synthesised separately. As seen in the previouschapter, a range of noise processes including qubit dephasing due to environmental interactions, aswell as phase and amplitude errors in the driving microwave field, may be emulated by performingamplitude and phase modulation (AM and PM) on the driving field only [70]. Since the engineerednoise PSD is expressed in the frequency domain, a Fourier transform is required to obtain the time-domain AM and PM waveforms. The noisy control field may be written

B(t) = Ω(1 + βΩ(t)) cos (ωt+ βΦ(t))n (5.2)

where βΩ,Φ(t) are the AM and PM waveforms, Ω and ω are the carrier amplitude and frequency, andn is the spatial direction of the magnetic part of the control field. The modulating waveforms areexpanded as a discrete truncated inverse Fourier transform

βΩ(t) = αΩ

jc∑

j=1

FΩ(j) cos (ωjt+ ψj) (5.3)

βΦ(t) = αΦ

kc∑

k=1

FΦ(k) cos (ωkt+ ψk) (5.4)

where FΩ,Φ are the envelope functions of the discrete PSDs, ωj,k are the component frequencies, αis the overall noise scaling, the subscript c denotes the maximum index before the high-frequencytruncation, and ψj,k ∈ [0, 2π) are uniformly-distributed pseudorandom phases. The desired time-correlations in the noise are captured by the envelope functions while the stochastic time-domain

35

S (ω

)

Angular Frequency, ω!0

J!0

!0

F (j)

Figure 5.1 : A logarithmic plot of an engineered comb spectrum PSD. The envelope function F (j) sets thepower law behaviour, up to a hard cutoff at Jω0. Since the comb spacing is linear, it appears to become denseras frequency increases on this logarithmic plot.

behaviour is provided by the random phases between each frequency component. The spacing of ωj =jω0 is linear with minimum frequency spacing ω0. The characteristic time scale of the modulatingwaveforms βΩ(t) and βΦ(t) is generally much longer than the carrier period, so jcω0 ω. Whenthere is more than one component frequency, the discrete spectrum is referred to as a ‘comb’ spectrum,and in the limit of a single component frequency, it is referred to as a ‘tone’ spectrum. A schematicdiagram of a comb spectrum is shown in Figure 5.1.

For phase errors, the frequency noise PSD Sz(ω) is more commonly quoted than the phase noisePSD, in which case the appropriate envelope functions are

F (j) =

jp2 for amplitude noisejp2−1 for phase noise

(5.5)

By setting the parameters and envelope functions in equations (5.3) and (5.4), the experimentalist isable to engineer analytic functions of stochastic noise with the desired PSD [70]. During a run of theexperiment, a noise trace is generated and discretised according to the cycle structure of the particularexperiment. The resulting noise samples are imposed on the microwave carrier signal, by IQM in thequbit control experiment and by FM in the frequency standard experiment.

5.2.2 Experimental implementation via IQ modulation

In order to drive arbitrary qubit rotations and to apply engineered noise, precise time-dependent con-trol over the microwave phase and amplitude is necessary. The composite pulse experiment usesinternal IQ modulation (IQM), according to a software-defined waveform downloaded from the ex-periment PC. The principle of IQM is a decomposition of the signal into in-phase (I) and quadrature(Q) components, which effectively converts combined phase and amplitude modulation of the carrierinto amplitude-only modulation of the components. This can be understood as a change of basis of thesignal phasor diagram from polar coordinates (amplitude and phase) to rectilinear coordinates (I andQ). Considering time-dependent amplitude modulation A(t) and phase modulation δ(t) of a signalwith carrier angular frequency ωc

S(t) = A(t) cos (ωct+ δ(t)) (5.6)

the required decomposition can be found by trigonometric identities

36

S(t) = I(t) cos (ωct) +Q(t) cos (ωct+ π/2) (5.7)

where the modulating waveforms are defined

I(t) = A(t) cos (δ(t)) (5.8)Q(t) = A(t) sin (δ(t)) (5.9)

Figure 5.2 : IQ modulation circuit diagram.

Figure 5.2 depicts a circuit diagram of IQM, represented as a circuit of mixers. The VSG downloadsthe modulating waveforms not as continuous functions, but as a pair of discrete vectors of ‘waveformsamples’. The time interval between each sample is user-defined and the VSG’s interpolation algo-rithm converts the downloaded waveform file into continuous functions I(t) and Q(t) for mixing.Thus the experimentalist, by generating discrete waveform files and uploading them to the VSG, canenact arbitrary rotations of the qubit for any duration longer than the inverse of the sample rate.

5.3 Experimental validation of filter function predictions

In order to demonstrate the usefulness of this filter function approach to quantum control, it is neces-sary to show that the theoretical expressions for fidelity accurately predict measured fidelity. Such atest is done by performing a ‘primitive’ π pulse, resulting in the qubit transition |0〉 → |1〉 in the pres-ence of engineered noise. The key parameters, namely the PSD scale factor α and cutoff frequencyωc, are scanned to demonstrate the frequency-dependence of fidelity. A schematic of the cutoff scanused in this section and the single-tone scan used in section 5.4.2 is shown in Figure 5.3.

37

0.01 0.1 1

ωcωt

Frequency ωc (2π/τπ)S

(ω)

(a.u

.)

α2

Figure 5.3 : Engineered noise PSDs. The white noise spectrum (blue) takes the form of a step function witha hard cutoff at ωc and is used for the validation experiment (section 5.3). The single-tone spectrum (black)consists of a delta function whose centre frequency ωt is scanned for the experimental testing BB1 and SK1composite pulse performance (section 5.4.2). The overall noise strength is expressed in terms of the noiseamplitude α.

1.0

0.5

0.0

0.01 0.1 1

Cutoff Frequency, ωc (2π/τπ)

0.01

1

100

1.0

0.5

0.0

1.0

0.5

0.0

1.0

0.5

0.0

0.01

1

100

0.01

1

100

0.01

1

100 Sm

alln

ess P

ara

mete

r, ξ

2

α = 10

α = 20

α = 50

α = 200

Pro

ba

bili

ty B

rig

ht

Figure 5.4 : Fidelity of a primitive π pulse in the presence of a white noise PSD of varying strength α. Theexperimentally observed bright state probability (circles) match closely the theoretically calculated fidelityE[Fχ] (solid black curves) when the smallness parameter (red line) ξ2 ≤ 1. The smallness parameter, asdefined in the previous chapter, is proportional to the total power in the noise, and therefore determines thebreakdown point (dashed vertical lines) of the first-order fidelity approximation. Each experimental data pointis the average of 50 different noise realisations. The dotted line at P (|1〉) = 0.5 bright state probability indicateszero fidelity, i.e. total decoherence of the qubit state.

The measurements of single-gate fidelity in the presence of a white engineered noise spectrum

38

are shown in Figure 5.4, overlain with filter-function-based predictions of fidelity. Measurementsare conducted for a simple πX enacted while varying the high-frequency cutoff, ωc, of a flat-topengineered noise PSD. As the high-frequency cutoff of the noise is increased and fluctuations that arefast compared to the control (τπ) are added to the noise power spectrum, Sz(ω), errors accumulatereducing the measured fidelity. The fidelity predictions are based on a direct calculation of Fχ =1−exp [

∫∞0Sz(ω)Fz(ω)dω]. These predictions fit the experimentally observed values closely, with no

free parameters. The agreement breaks down only in the strongest noise environment α = 200, wherethe weak-noise assumption underpinning the first-Magnus-order approximation no longer holds.

5.4 Composite pulse sequences for noise suppression

5.4.1 Traditional composite pulses

Quantum logic requires the enactment of arbitrary single-qubit rotations Uφ(θ), with high fidelity evenin the presence of qubit or field noise. A huge variety of composite schemes have been invented in thefield of NMR, which are designed to compensate for inhomogeneities in the orienting magnetic field[75, 81]. Composite pulses work by geometrically manipulating the qubit state, such that first-ordersensitivity of the state to environment noise is transformed into high-Magnus-order contributions. Asthese high-order contributions scale with the strength of the noise, they become insignificant in theweak-noise quasistatic limit. It has been shown that such pulses also provide significant benefits inthe presence of time-varying noise, which motivates their investigation in this quantum system [11].

Two composite pulse schemes for enacting arbitrary rotations, BB1 and SK1, are constructed byperforming a rotation of arbitrary angle θ on the Bloch sphere and then a series of ‘compensating’pulses that sum to the identity operator. The net result of this composite pulse is a driven rotation of θwith higher fidelity than if the same rotation had been performed ‘primitively’, i.e. without additionalerror-compensating pulses. The pulses can be defined in terms of time evolution operators Uφ(θ)where φ is the axis of rotation on the Bloch sphere (φ = 0 corresponding to σx) and θ is the angle ofdriven rotation. The composite pulses of interest are defined

U(BB1)φ (θ) = Uφ+φBB1

(π)Uφ+3φBB1(2π)Uφ+φBB1

(π)Uφ(θ) (5.10)

U(SK1)φ (θ) = Uφ−φSK1

(2π)Uφ+φSK1(2π)Uφ(θ) (5.11)

where φBB1 = φSK1 = cos−1 (−θ/4π) [81, 11].In order to quantify the error-suppressing capabilities of these two composite pulse schemes, the

average fidelity E[F ] is used. It can be shown [69] that the Magnus expansion of E[F ] for BB1 has asecond-order squared leading term E[a2

2] while for SK1 the leading term is first-order squared E[a21].

For static amplitude noise, the leading Magnus order determines the overall error suppression. Acomparison of the fidelity of experimentally enacted BB1 and SK1 pulses, for varying strength oftime-varying noise, allows a determination of the threshold where the first-Magnus-order approxima-tion E[Fχ] ceases to be a good approximation for E[F ].

5.4.2 Single-tone engineered noise scans for BB1 and SK1The usefulness of the Magnus-term expression of pulse fidelity and the filter function may be illus-trated by comparing the performance of different composite pulses in the presence of the same noiseenvironment. The PSD used here is an ‘amplitude tone spectrum’, which consists of a single sinu-soid component at frequency ωt of random initial phase modulating the microwave amplitude. ωt isswept through a range, and since the tone spectrum PSD is the delta function S(ω) = δ(ω − ωt), thefirst-order fidelity depends directly on the filter function

39

Figure 5.5 : Swept single-tone scan of first-order filter function and measured fidelity for three π pulses:primitive, SK1 and BB1. The tone angular frequency ωt is in units of inverse τπ, which is the time requiredto enact a primitive π rotation. (a) The solid lines indicate the first-order amplitude filter function FΩ(ωt) forthe three pulses. The dotted lines mark the frequencies at which the filter functions cross, indicating pointsof inversion in the relative performance of the pulses. (b) (weak noise) and (c) (strong noise) The solid linesindicate analytically-calculated error (χΩ = exp [FΩ(ωt)]) and markers indicate experimental measurementsof error, averaged over 20 realisations of engineered noise. The physical observable used as a proxy for error is1−P (|1〉), since an ideal π pulse results in P (|1〉) = 1. The noise amplitude α is given in terms of the drivingamplitude Ω. The measurement resolution limit is depicted as a grey shade.

E[Fχ] = 1− exp

[ ∫ ∞

0

F (ω)δ(ω − ωt)dω]

(5.12)

= 1− exp

[F (ωt)

](5.13)

Single-tone scans are useful for ‘tracing out’ the shape of the filter function, indirectly observing thespectral sensitivity of a particular composite pulse sequence to noise.

A single-tone scan with varying noise strength α to compare a primitive π pulse, a BB1 π pulseand an SK1 π pulse may be used to test the improved performance of BB1 and SK1 in contrast tothe primitive pulse for time-varying noise, as well as in the static limit for which they were originallydesigned. Figure 5.5 shows the result of such a single-tone scan.

The vertical dotted lines indicating the filter function crossovers closely match the crossovers ofexperimentally measured fidelities. Hence the filter function approach accurately predicts the compar-ative performance of primitive, SK1 and BB1 pulses across the spectrum. In panel (b), the calculatederror fits the experimental data well over three decades of frequency, down to the measurement res-olution limit of χ < 1.5%, verifying the predictive power of the filter function approach in the weaknoise regime. SK1 and BB1 composite pulses outperform primitive pulses in the low ωt regime be-cause their filter order in the quasistatic limit is quadratic, in contrast to the constant filter order of the

40

primitive pulse, i.e. F (ω) ∝ ω2 for BB1 and SK1 and ∝ ω0 for primitive, for ω → 0.In panel (c), the calculated error fits the experimental data only for frequencies higher than the

primitive-SK1 crossover. In the low frequency regime, the second-order Magnus term contributessignificantly to the overall fidelity and the first-order filter function ceases to accurately predict theerror. The experimental fidelity E[F ] diverges from the calculated first-order approximation E[F ]differently for SK1 and BB1. As noted above, in the static limit SK1 has a first-order leading Magnusterm while BB1 has a second-order leading Magnus term, i.e. there is no static, first-Magnus-ordercontribution to the error for BB1 as there is for SK1. This difference between the two composite pulsesmeans that in the quasistatic regime for strong noise, the BB1 is dominated by the second order term,resulting in higher overall fidelity E[F ]. The comparison between (b) and (c) therefore illustrates thenoise strength threshold at which the filter function approach ceases to accurately predict fidelity.

The different performance of BB1 and SK1 in (c) illustrates an important theoretical result: thatMagnus order and filter order are distinct and dominate the overall fidelity in different regimes. In thelimit of strong, low frequency noise, the leading Magnus order provides the dominant contributionto fidelity, leading to the superior performance of BB1 over SK1 despite having the same filter orderFΩ(ω) ∝ ω2. In the weak noise regime, the contribution of the first-order filter function dominates thefidelity. The distinction between Magnus and filter order that is shown experimentally in this thesishas been given a rigorous theoretical proof [63].

5.5 Search for optimised Walsh amplitude modulated filters

5.5.1 Dynamically corrected gates as composite pulsesA generalisation of the NMR approach to time-varying noise environments was proposed by Khod-jasteh et al., known as ‘dynamically (error-)corrected gates’ (DCG) [43]. Such DCGs produce thesame state net unitary evolution as the primitive pulses with lower error per gate, by enacting a com-posite pulse sequence consisting of piecewise-constant rotations about various axes. The advantagesof the original dynamical correction approach are: that DCGs reduce error in a universal noise en-vironment and that they are effective in the presence of time-varying noise governed by a stochasticHamiltonian [41]. These features make the DCG class of composite pulses attractive for implemen-tations in realistic physical systems, especially for the purposes of quantum logical and computation[42].

The original formulation of the DCG involves a group theoretic approach [43], but more recentlyit has been shown that the same pulses may be understood in terms of Walsh functions, as Walshamplitude modulated filters [4]. The advantage of applying Walsh synthesis to the notion of DCGsis the relative ease of use of the Walsh functions, and the direct relationship between Paley order andfilter order, which is demonstrated in this chapter.

5.5.2 Walsh synthesis of composite pulsesThe Walsh functions, which are the square wave analogues of sine and cosine functions, providea useful method for synthesising novel composite pulses, since they are naturally compatible withdiscrete-time clocking and digital logic [4]. The infinite set of Walsh functions Wk(x) are definedon the interval 0 ≤ x ≤ 1 and take only values −1 and 1. The set forms an orthonormal basis andhence any continuous function may be expressed in terms of Walsh functions [78]. The number andlocation of zero-crossings of any Walsh function depends on its Paley order k, which is the orderingconvention of the Walsh set used in this work. Figure 5.6 shows plots of the first 32 Paley-orderedWalsh functions.

The formal definition of the Walsh functions is

41

Figure 5.6 : A graphical depiction of the first 32 Paley-ordered Walsh functions. The functions of Paley orderk = 1, 3, 7, 15, 31 are highlighted in red, as these Walsh functions have the best amplitude noise suppressingproperties in their classes. A class of Walsh functions is defined as a set of Walsh functions with the sameminimum inter-crossing interval. In the Paley order, each class is equivalent to a logarithmic octave in k.

Wk(x) =m∏

j=1

sgn(sin (2jπx))bj (5.14)

where (bm, bm−1, · · · , b1)2 is the binary representation of k. The Walsh functions or superpositionsof them are applied as modulation to the driving field in order to produce Walsh synthesised gates. Amodulating time-domain function M(t) is produced from a sum of the relevant Walsh functions

M(t) =∑

k∈SnkWk(t/τ) (5.15)

where S is the set containing the synthesised basis functions, nk is the relative scaling factor for eachbasis function and τ is the total duration of the whole Walsh-modulated pulse.

In the Walsh amplitude modulated filter (WAMF) pulses examined here, the value of M(t) de-termine the strength of the applied driving field, producing piecewise-constant amplitude modulationof the driving field. It can be seen that the first-order DCG NOT gate [42] is the result of amplitudemodulation by a synthesis of the zeroth and third Paley order Walsh functions

MDCG(t) = 1.5W0(t/τ) + 0.5W3(t/τ) (5.16)

with amplitude modulation depth ΩR = π/τπ, where ΩR is the Rabi rate. The extension of the DCGconcept to higher order is thus straightforwardly done using the Walsh functions [4].

It can be shown that other composite pulses familiar to NMR, such as BB1 and SK1, may bewritten in terms of Walsh-synthesised phase modulation. These two pulses consist of the initial‘primitive’ part followed by the ‘compensating’ part, and the compensating part may be expressed inthe Walsh basis:

MBB1(t) = 2W0(t/τ) +W3(t/τ) (5.17)MSK1(t) = W1(t/τ) (5.18)

with phase modulation depth φBB1 = φSK1 = cos−1 (−θ/4π) [4].

42

5.5.3 Walsh synthesis softwareThe experiment PC runs the hardware for the composite pulse experiment using software written inthe IGOR Pro language. The custom-written software module has a graphical user interface (GUI)that synthesises modulating functions M(t) out of arbitrary weightings of Walsh functions, as wellas standard NMR pulses such as BB1 and SK1. The engineered noise, which is expressed as a dis-crete ‘comb spectrum’ that approximates a continuous PSD [70], is applied by adding the engineerednoisy time trace y(t) to the function M(t). The computer discretises these modulating functions andsends the resulting waveform vectors via GPIB link to the VSG, which then applies IQ modulationinternally.

The ability to synthesise composite pulses in the Walsh basis, either by phase or amplitude mod-ulation, makes it desirable to attempt a search for novel high-performance pulses. The Walsh basisprovides a convenient restriction on the dimensionality of the search space, making the process highlyefficient. The filter function formalism makes this approach particularly attractive, since the searchcost function may be derived from a specific noise model or experimentally-observed PSD, whichturns the search into an optimisation problem tailored to a particular need.

The cost function Ai used to evaluate the performance is the integral of the filter function overa spectral band of interest ω ∈ [ωl, ωc], thereby capturing the bandlimited performance of a controloperation:

Ai =

∫ ωc

ωl

Fi(ω)dω (5.19)

where i ∈ [z,Ω].A convenient search space is the two-dimensional space spanned by the Walsh basis functions

W0 and W3, to which the amplitude modulating profile for the traditional first-order DCG belongs(equation 5.16). A colour plot of the the cost function in this search space is given in Figure 5.7.

The blue regions of the colour plot correspond to areas where Az is minimised, resulting in su-perior error suppression. Since Wk is a mean-zero function for all k > 0, the overall net rotationperformed is given entirely by the amplitude of W0. By targeting the blue regions, it is possible toeasily derive an error-suppressing composite pulse for an arbitrary desired rotation θ = X0 (mod 2π).

The numerical search results may be verified experimentally by scanning over X3 for a fixed X0,shown in Figure 5.8(a)-(c). The regions of high error-suppression (blue in the colour plot), appearsas infidelity troughs in the experimental data. Direct integration of the Schödinger equation in thepresence of simulated noise fits the experimental data well. This data confirms that Walsh synthesisof composite pulses results in improved error suppressing for an arbitrary desired rotation θ.

5.5.4 Randomised benchmarkingRandomised benchmarking [44] is used as a tool for resolving small gate errors below the measure-ment resolution limit. The technique works by repeating the desired pulse about a randomised axis,leading to an amplified overall error that is more easily measured, allowing the mean ‘error per gate’to be determined. Our randomised benchmarking sequence consists of interleaved ‘reorienting’ π/2pulses and ‘under test’ π pulses, each applied along axes randomly selected from ±X and ±Y . Theπ/2 pulses enact the random change of rotation axis, while the π is the pulse under test, which may besubstituted for any composite pulse for the purpose of comparison. A given randomised benchmark-ing sequence consists of l computational gates followed by a final reorienting pulse whose directionis chosen such that the the final qubit state may be measured by a Z projective measurement.

It is desirable to understand the behaviour of composite pulses in the presence of weak noise,where resolving measurements of fidelity very close to 100% becomes difficult. Using the techniqueof randomised benchmarking to amplify the error, it can be seen in Figure 5.8(d) that even in the limit

43

M(t

)

-12

-8

-4

0

4

8

12

PA

L0 A

mp

litu

de

, X

0 (

π)

-12 -8 -4 0 4 8 12 PAL3 Amplitude, X3 (π)

-8 -7 -6 -5 -4

X0

2X3

Normalised Time

log(Az)

a

b

Figure 5.7 : Numerical search over the two-dimensional space spanned by W0,W3. (a) Schematic of thetime-domain profile of a Walsh-synthesised amplitude-modulated filter (WAMF), where X0 and X3 are theweighting coefficients of the two basis functions such that M(t) = X0W0(t) +X3W3(t). (b) Colour plot overthe search space in terms of the logarithm of the dephasing noise cost function Az , calculated over the low-frequency band ω ∈ [10−9, 10−1]τ−1, where τ is the total duration of the pulse. The dotted white lines representoverall rotations X0 = 2.25π, 2.5π, 3π corresponding to error-suppressed net rotations of θ = π/4, π/2, πrespectively.

of weak noise, the WAMF composite pulse consistently outperforms the primitive.

5.6 ConclusionThe focus of this chapter has been on validating the filter function framework for the task of predictingquantum dynamics in realistic environments and demonstrating the relevant physics through construc-tion of error-suppressing composite pulses. The Walsh-modulated pulses presented here complementand systematise the existing formulations. The determination of a restricted search space for error-suppressing pulses in terms of Walsh basis functions provides an efficient method of tailoring com-posite pulse sequences optimised for known noise environments. The chapter therefore provides botha validation of the general approach, namely the filter function approach to qubit evolution, and anew technique for the specific goal of synthesising error-resistant composite pulses in the presenceof time-varying, stochastic qubit noise. Furthermore, the demonstration that filter and Magnus orderare distinct has an important consequence for quantum information and control experiments. Thisconsequence is that historic insights into how to tailor control protocol on the basis of knowledgefrom the field of NMR is insufficient, and a more complete dynamical picture is required, to whichthe filter function formalism is well suited.

44

1511219161310

Gate Number

1.0

0.8

0.6

0.4

0.2

0.0

Pro

babili

ty B

right

WAMF Primitive

0.3

0.2

0.1

0.0 10-8

10

-5 10

-2

0.4

0.3

0.2

0.1

0.0 10-8

10

-5 10

-2

0.6

0.4

0.2

0.0

121086420

X3 (π)

10-8

10

-5 10

-2

Lo

g F

F A

rea

Infidelit

y

c

b

a

d

Experiment Control Simulation A (Γ4)

Figure 5.8 : (a)-(c) Experimental measurement of single-gate infidelity (left axis) for rotations constructedfrom various Walsh coefficients in the presence of engineered noise (white dephasing with ωc/2π = 20 Hz).Black dots and line represented calculated fidelity by Schrödinger equation integration (raw and smoothedrespectively). All values ofX3 for a givenX0 implement the same net rotation, indicated by control experimentwith no noise. Total rotation time is scaled with X3 to preserve a maximum Rabi rate. Black dashed line(right axis) corresponds to the cost function Az . (d) Randomized benchmarking results (50 randomisations)demonstrating superior performance of modulated gate in the small-error limit (infidelity < 0.5% per gate).The dots represent each randomisation, the open circles represents the mean bright state probability, and thesolid lines are smoothed fits.

45

Chapter 6

Characterising Frequency StandardInstabilities6.1 Introduction to frequency standard stabilisationThe problem of stabilisation of a frequency standard may be considered to have the same basic struc-ture as the problem of qubit control. In frequency standards, the noisy dynamics of a classical devicesuch as a laser is controlled by locking to some characteristic frequency of a quantum system, e.g. thequbit splitting frequency. Frequency stability is a chief criterion for evaluating a frequency standard[19], and the vast majority of research in the field focus on breaking stability records by hardware im-provements, recently reaching frequency deviations at the 10−18 level [8, 34]. This chapter presents aplatform-independent, software technique for improving the stability of frequency standards, in whichstatistical correlations between measurements are used to predict the time-domain noise trajectoriesof frequency. An important insight of this part of the thesis is that the standard operating protocols offrequency standards, such as Ramsey interrogation and feedback correction, may be comprehendedusing the same descriptive framework as that of the measurement of the stability of frequency stan-dards, such as variance metrics and transfer functions. The stability improvements shown in thischapter, which are informed by real-world experimental parameters, are compatible with a wide rangeof existing frequency standard technologies.

The vast majority of cutting-edge frequency standards are classified as passive, which means thatthey consist of a local oscillator (LO) that produces a signal e.g. as a laser, and a passive reference towhich the frequency of the signal is locked e.g. an atomic transition. Traditional techniques of lockinginvolve a simple feedback loop, in which a measurement of the frequency deviation of the signal isused to immediately correct the LO frequency, producing a more stable ‘locked local oscillator’ (LLO)signal. Such LO frequency deviations, quantified in terms of variance metrics, may be naturally ex-pressed in the same filter transfer function language as the qubit noise problem. The transfer functionapproach may be used to calculate the time-correlations between multiple measurement outcomes,allowing for future trajectories of the LO frequency noise to be predicted. Transfer-function-basedprediction may be used to improve the accuracy of the feedback lock, leading to a novel method offrequency control known as predictive hybrid feedforward.

This chapter and the next chapter presents work that is available as a pre-print [68] which is re-produced in Appendix C. This chapter is organised as follows. Section 6.2 provides the mathematicalbackground to the method of sampling frequency noise in the time domain and expressing the vari-ance of those measurement samples. Section 6.3 derives a novel transfer function form that capturesthe covariance between two measurements, allowing for the extension of the transfer function formsto include locked local oscillator variance quantities, which are presented for the first time in Section6.4.

46

6.2 Filter function approach to frequency standardsThe type of noise of interest to the frequency standard community is frequency noise of the localoscillator (LO), which for this purpose is assumed to dominate the noise of the qubit, which acts asa ‘clean’ frequency reference for the LO. The type of metric used to quantify the effect of noise onthe LO frequency stability is variance, which is found by successive measurement of the frequencyfluctuations. In contrast to the unitary qubit dynamics described in the previous section, the frequencystandard is a closed loop control system. There are several variance metrics used by the communitythat are naturally expressed in the same filter transfer function language as the universal qubit noiseproblem.

6.2.1 Time-average samplingThe standard method of measuring frequency deviations between an LO and a qubit reference isRamsey interrogation [65] (originally referred to as the ‘method of separated fields’), which involvesperforming two π/2 rotations of the Bloch vector separated by a long interrogation period of durationT (R). The frequency fluctuations during the interrogation period result in an axis shift of φ betweenthe two π/2 rotations, and hence a difference in the final state Z-projection of

∆P↑ ∝ sin2 (φ) (6.1)

which may be directly measured.Ramsey interrogation provides the template for a more abstract ‘time average sample’ that may be

applied to a broader range of physical systems than the molecular beams and trapped ions to whichRamsey interrogation is native. In this model, an LO frequency noise trace y(t) is sampled in overdiscrete time bins labelled k: [tsk, t

ek], where a modulating sensitivity function g(t) ∈ [0, 1] captures

the contribution of y(t) to the final measured average over the whole interval of yk. The definition ofthe kth sample is thus

yk =1

T(R)k

∫ tek

tsk

y(t)g(t− tsk)dt (6.2)

where the kth Ramsey time is T (R)k ≡ tek − tsk. All variance metrics for LO instability are calculated

from sequences of such samples.For ideal Ramsey interrogation with infinitely fast π/2 pulses, g(t) has a simple rectangular func-

tional form

g(t) =

1 for 0 < t < T

(R)k

0 otherwise(6.3)

in which case yk reduces to a simple time-average over [tsk, tek].

For non-ideal cases where the π time τπ > 0, the sensitivity function is given by [67]

g(t) =

sin (tπ/2τπ) for 0 < t < τπ

1 for τπ < t < T(R)k − τπ

1− sin ((t− T (R)k + τπ)π/2τπ) for T (R)

k − τπ < t < T(R)k

0 otherwise

(6.4)

More complicated sensitivity functions, such as the Hadamard window [66] and triangular windows[15], may be used to find probe specific parts of the noise spectrum.

In these experiments τπ T(R)k , so the rectangular sensitivity function (6.3) is used exclusively.

47

6.2.2 Variances in the frequency domainIt is useful to derive expressions for different variance metrics in terms of the known frequency noisePSD and the sensitivity functions of the interrogation scheme used. In a similar way to the derivationof the frequency-domain fidelity expression for universal qubit noise, the Wiener-Khinchin theoremcan be used to derive filter transfer functions that capture the spectral effect of particular interrogationschemes. Some of these variance metrics pertain to a particular measurement in the sequence andare found by deviations over the ensemble, e.g. true and Allan variances, while others are found bycalculating the deviations among a sequence of measurements belonging to a single run, e.g. samplevariance [68].

An ideal variance metric that cannot be realised experimentally is the frequency variance, whichis the variance over the ensemble E of y(t), corresponding to an instantaneous sample at time t. Thefrequency variance is written Var[y(t)]:

Var[y(t)] = E[y(t)2]− E[y(t)]2 (6.5)= E[y(t)2] (6.6)

due to the zero-mean assumption E[y(t)] = 0.The simplest experimentally-useful variance metric is called the true variance σ2

y(k) [66], defined

σ2y(k) = E[y2

k] (6.7)

= E

[(1

T(R)k

∫ tek

tsk

y(t)g(t− tsk)2dt

)](6.8)

where the zero-mean assumption on y(t) results in E[yk]2 = 0. Defining a normalised, time-reversed

sensitivity function on the assumption that g(t) is time-reversal symmetric about tmk , the centre pointof [tsk, t

ek]

g(tmk ) =g(t− tsk)T

(R)k

(6.9)

which allows the expansion of (6.7)

σ2y(k) = E

[(∫ tek

tsk

y(t)g(tmk − t)dt)2]

(6.10)

=

∫ ∞

−∞

∫ ∞

−∞E[y(t)y(t′)]g(tmk − t)g(tmk − t′)dt′dt (6.11)

=

∫ ∞

−∞

∫ ∞

−∞RTSyy (t′ − t)g(tmk − t)g(tmk − t′)dt′dt (6.12)

Applying the Wiener-Khinchin result to the PSD and Fourier transforming the sensitivity function,similarly to section 4.3.3, gives

σ2y(k) =

1

2π

∫ ∞

0

Sy(ω) |Gk(ω)|2 dω (6.13)

where |Gk(ω)|2 is the transfer function for the kth sample. For the case of rectangular g(t), the

48

transfer function has the simple form

|Gk(ω)|2 ≡∣∣∣∣∫ ∞

−∞g(tmk − t)eiωtdt

∣∣∣∣2

(6.14)

=sin2 (ωT

(R)k /2)

(ωT(R)k /2)2

(6.15)

Another important metric for frequency instability is the Allan variance Aσ2y(k) [2], which found by

a two-piece rectangular function that covers two adjacent equal-duration interrogation bins with zerodead time between them

g(t) =

−1 for 0 < t < T (R)

1 for T (R) < t < 2T (R)

0 otherwise(6.16)

where T (R)k = T

(R)k+1 for all k by the uniformity requirement of the Allan variance definition. The

resulting transfer function is

∣∣AG(ω)∣∣2 =

2 sin4 (ωT (R)/2)

(ωT (R)/2)2(6.17)

and hence the Allan variance is given by the overlap integral

Aσ2y(k) =

1

2π

∫ ∞

0

Sy(ω)∣∣AG(ω)

∣∣2 dω (6.18)

This thesis relies on measures of frequency stability such as the true variance and sample variance,rather than the more commonly employed Allan variance. This selection has been deliberate as theform of the Allan variance specifically masks the effect of LO noise components with long correlationtimes. In fact the Allan variance, taking the form Aσ2

y(k) = 12E[(yk+1 − yk)2] is employed by the

community in part because it does not diverge at long integration times τ due to LO drifts, as wouldthe sample or true variance [74, 5, 66, 28]. The Allan variance is most appropriate when comparingthe stability, over a range of time scales, of frequency standards where both LO and reference (i.e.qubit) contribute to overall instability [23]. In this system, where LO frequency noise dominates therelatively stable qubit, alternative metrics that assume a fixed reference frequency e.g. true and samplevariance more useful.

The third metric of interest is the sample variance, which is defined as the variance of a sequenceof samples

σ2y[N ] ≡ 1

N − 1

N∑

k=1

(yk −

1

N

∑

l=1

yl

)(6.19)

where the use of brackets for the sample index indicates that the variance is calculated over the firstN samples. Since this variance is based on data from only a single run, a more robust measure is theexpected value of the sample variance over the ensemble. The expected value of the sample variancecan be expressed in terms of the true variance σ2(k) and a novel quantity called the covariance σ(k, l):

49

Figure 6.1 : The sum and difference functions for two time-separated samples yi and yj . These are time-domainfunctions that capture the sensitivity of a pair of measurements, modulated by the zeroth-order Walsh function(sum sensitivity) and the first-order Walsh function (difference sensitivity). Expressing measured samples interms of these modulated functions constitutes a basis rotation of 45, which allows for direct calculation of thecorrelation and anticorrelation between the two original samples.

E[σ2[N ]] =1

N − 1

N∑

k=1

(σ2(k)− 1

N

∑

l=1

σ(k, l)

)(6.20)

Even though expressed as a nested sum over measurement indices k, l, this expression for the expectedvalue of the sample variance is fundamentally a frequency domain quantity, since the true varianceand covariance are written in terms of the PSD and relevant transfer functions.

6.3 Covariance in the frequency domainA vital quantity that has appeared only tangentially in the literature [5] is the covariance betweentwo measurements in a sequence. The covariance is required for calculating transfer function-basedexpressions for quantities involving time-correlations, e.g. sample variance, and for the predictiveprotocols introduced in subsequent chapters, which exploit those time correlations.

The covariance σ(yi, yj) of two measurement results indexed (i, j) is found via a pair of sum anddifference sensitivity functions. This formalism compares the variance of sum and difference of thetwo measurements in order to extract their covariance, using the identity

σ2(A+B) = σ2(A) + σ2(B)− 2σ(A,B) (6.21)

We defineG+ij(ω) andG−ij(ω) as the Fourier transforms of the sum and difference sensitivity functions

g+ij(t) and g−ij(t). These time-domain functions are plotted in Figure 6.1. The expressions for the sum

and difference sensitivity functions for the flat-top case are

g±ij(t) =

1, for t ∈ [tsi , tei ]

±1, for t ∈ [tsj , tej ]

0, otherwise(6.22)

from which the sum and difference transfer functions are obtained by Fourier transform:

50

G±ij(ω) =i

ω

(e−iωt

ei − e−iωtsitei − tsi

± e−iωtej − e−iωtsjtej − tsj

)(6.23)

By substituting (6.13) into the identity (6.21) and rearranging terms, it is possible to obtain an expres-sion for the covariance of the two measurements:

σ(yi, yj) =1

4

∫ ∞

0

Sy(ω)(|G+ij(ω)|2 − |G−ij(ω)|2)dω (6.24)

≡∫ ∞

0

Sy(ω)G2ij(ω)dω (6.25)

This covariance expression introduces the concept of the pair covariance transfer function G2ij(ω),

which captures covariance in the frequency domain in an analogous manner to the traditional transferfunction |G(ω)|2, which captures variance. This expression is a crucial theoretical result of this thesis.It takes the same overlap integral form as variance quantities of the previous section, capturing insteadthe covariance between two time-separated measurement windows. Subsequent results on lockedlocal oscillators, as well as the predictive stabilisation of the next chapter, rely on the correlationsencapsulated in Gij(ω).

In the flat-top case of a sharply rectangular sensitivity function, a closed-form expression for thistransfer function can be obtained:

G2ij(ω) =

1

ω2Tmi Tmj

(cos (ω(tsj − tsi )) + cos (ω(tej − tei ))− cos (ω(tej − tsi ))− cos (ω(tsj − tei ))

)

(6.26)This new result permits the calculation of the covariance of any two measurements using (6.25) in atime-domain sequence with arbitrary start and end times for each measurement.

6.4 Extending analytic variance expressions to locked local oscil-lators

The conventional measures for oscillator performance consider only a free-running LO, excluding theeffects of feedback locking. In this section explicit analytic forms are presented for variance quantitiesfor locked local oscillators (LLO), i.e. variance in the presence of feedback.

The link between LO and LLO variances is made via a time-domain treatment. Consider thetrajectory of the same frequency noise realisation y(t) in the cases of no correction, yLO(t) and ofcorrection, yLLO(t). The relation between these two cases of y(t) is

yLLO(t) = yLO(t) +n∑

k=1

Ck (6.27)

where Ck refers to the value of the kth frequency correction applied to the LLO, n of which haveoccurred before time t.

Under traditional feedback, each correction is directly proportional to the immediately precedingmeasurement outcome: Ck = wky

LLOk , where wk is correction gain. Since yLLOk is calculated by

51

convolving yLLO(t) with a sensitivity function pertaining to the measurement parameters, (6.27) is arecursive equation in general. It is possible to cancel all but one of the recursive terms by setting thecorrection gain equal to the inverse of the average sensitivity gk ≡

∫ Tk0g(t)/Tkdt of the preceding

measurement, i.e. wk = −g−1k , where the minus sign indicates negative feedback. With this constraint

we can write

yLLOk = yLOk −gkgk−1

yLOk−1 (6.28)

where gk = 1 for an ideal Ramsey interrogation and measurement with negligibly short pulses.

The frequency variance of an LLO can be found straightforwardly, by substituting (6.27) into thedefinition of frequency variance, with the additional substitution of (6.28) for the particular case oftraditional feedback:

Var[yLLO(t)] = E[yLLO(t)2]− E[yLLO(t)]2 (6.29)

= E

[(yLO(t) +

n∑

k=1

Ck

)2](6.30)

= E

[(yLO(t)− yLOn

gn

)2](6.31)

= E[yLO(t)2] +1

g2n

E[(yLOn )2]− 2

gnE[yLO(t)yLOn ] (6.32)

= E[yLO(t)2] +σ2yLO(n)

g2n

− 2

gnσ(yLO(t), yLOn ) (6.33)

where the progression from (6.30) to (6.31) is valid for traditional feedback only, and E[yLLO(t)] = 0by assumption and n indexes the last measurement before t.

The true variance for an LLO can be found in a similar way by substituting (6.28) into the defini-tion of true variance:

σ2yLLO(k) = Var[yLLOk ] (6.34)

= Var

[yLOk −

gkgk−1

yLOk−1

](6.35)

= σ2yLO(k) +

(gkgk−1

)2

σ2yLO(k − 1)− 2gk

gk−1

σ(yLOk−1, yLOk ) (6.36)

where the appropriate forms of the measurement transfer function (6.13) and the pair covariancetransfer function can be substituted in to express σ2

yLLO(k) in terms of Sy(ω).

The expected value of the LLO sample variance can be found by substituting (6.27) into thedefinition of the sample variance:

52

E[σ2yLLO[N ]] =

1

N − 1

N∑

k′=1

σ2yLLO(k′) +

1

N2

N∑

p′=1

N∑

q′=1

σ(yLLOp′ , yLLOq′ )− 2

N

N∑

l′=1

σ(yLLOk′ , yLLOl′ )

(6.37)

=1

N − 1

N∑

k′=1

(σ2yLO(k′) + g2

k′

bk′/nc∑

r=1

bk′/nc∑

s=1

σ(Cr, Cs)− 2gk′

bk′/nc∑

u=1

σ(yLOk′ , Cu)

)

+1

N2

N∑

p′=1

N∑

q′=1

σ(yLOp′ + gp′

bp′/nc∑

p=1

Cp, yLOn + gq′

bq′/nc∑

q=1

Cq)

− 2

N

N∑

l′=1

σ(yLOk′ + gk′

bk′/nc∑

u=1

Cu, yLOl′ + gl′

bl′/nc∑

v=1

Cv)

(6.38)

=1

N − 1

N∑

k′=1

(σ2yLO(k′) + g2

k′

bk′/nc∑

r=1

bk′/nc∑

s=1

σ(Cr, Cs)− 2gk′

bk′/nc∑

u=1

σ(yLOk′ , Cu)

)

+1

N2

N∑

p′=1

N∑

q′=1

(σ(yLOp′ , y

LOq′ ) + gp′ gq′

bp′/nc∑

p=1

bq′/nc∑

q=1

σ(Cp, Cq)

)

− 2

N

N∑

l′=1

(σ(yLOk′ , y

LOl′ ) + gk′ gl′

bk′/nc∑

k=1

bl′/nc∑

l=1

σ(Ck, Cl)

)(6.39)

where N is defined as the total number of measurements and n as the number of measurements percycle. The summation signs with unprimed indices are sums over whole cycles (of which there arebN/nc) and the primed indices are sums over all N measurements. In general, E[σ2

yLLO[N ]] containsrecursive terms that cannot be concisely expressed in terms of the LO PSD Sy(ω) and covariancetransfer function G2(ω).

This expression is valid both for the case of traditional feedback where each correction cyclecontains only a single measurement (see Figure 7.1b) or novel forms of correction where correctioncycles may be interleaved with multiple measurements (see Figure 7.1c).

In the case of traditional feedback, the distinction between primed and unprimed indices disap-pears and the expression reduces to:

E[σ2yLLO[N ]] =

1

N − 1

N∑

k=1

(σ2yLO(k) + g2

k

k−1∑

r=1

k−1∑

s=1

σ(Cr, Cs)− 2gk

k−1∑

u=1

σ(yLOk , Cu)

)

+1

N2

N∑

p=1

N∑

q=1

(σ(yLOp , yLOq ) + gpgq

p−1∑

w=1

q−1∑

x=1

σ(Cx, Cy)

)

− 2

N

N∑

l=1

(σ(yLOk , yLOl ) + gkgl

k−1∑

y=1

l−1∑

z=1

σ(Cy, Cz)

)(6.40)

where each term can be expressed in terms of Sy(ω) and G2(ω).The LLO Allan variance can be found by substituting (6.28) into the definition of the Allan vari-

ance (6.18):

53

Aσ2yLLO(k) =

1

2E[(yLLOk+1 − yLLOk )2] (6.41)

=1

2E

[(yLOk+1 −

gk+1

gkyLOk − yLOk +

gkgk−1

yLOk−1

)2](6.42)

=1

2

(σ2yLO(k + 1) +

(1 +

gk+1

gk

)2

σ2yLO(k) +

(gkgk−1

)2

σ2yLO(k − 1)

+2gkgk−1

σ(yLOk+1, yLOk−1)− 2

(1 +

gk+1

gk

)σ(yLOk , yLOk+1)− 2(gk + gk+1)

gk−1

σ(yLOk , yLOk−1)

)

(6.43)

The next chapter takes a key concept from this chapter, the pair covariance transfer function, anduses it to derive a novel predictive method of correction. By contrast to the traditional feedbacklocking presented in this chapter, the predictive scheme combines multiple measurements to calculatean optimised predictor of the noise trajectory at a future time.

54

Chapter 7

Transfer Functions for PredictiveStabilisation of Frequency Standards7.1 Introduction to predictive protocolsThe transfer function approach is powerful because it may also be employed to craft new measure-ment feedback protocols based on predicted noise trajectories. In particular, the aim is to mitigatethe deleterious effect of noise on LO frequency stability by combining information from multiplemeasurements with statistical information contained in the PSD. The key insight is that the non-Markovianity of dominant noise processes in typical LOs – captured through the low-frequency biasin the PSD – implies the presence of temporal correlations in y(t) that may be exploited to improvefeedback stabilisation.

The formal basis of the analytic approach is a frequency-domain measure of correlation betweentime-separated measurements, using the pair covariance transfer function (Equation 6.26). In sum-mary, a covariance matrix is calculated in the frequency domain via transfer functions to capture therelative correlations between sequential measurement outcomes of an LLO, and this matrix is used toderive a linear predictor of the noise at the moment of correction. This predictor provides a correctionwith higher accuracy than that derived from a single measurement for experimentally-relevant noisespectra, allowing us to improve the performance of the LLO. Since the predictor is found using in-formation from previous measurements (feedback) and a priori statistical knowledge of the LO noise(feedforward), this scheme is called hybrid feedforward. Effectively, the ability to predict the evo-lution of y(t) by exploiting correlations captured statistically in Sy(ω) allows feedback stabilisationwith increased accuracy and reduced sensitivity to dead time.

This chapter is organised as follows. Section 7.2 derives the linear predictor in terms of the covari-ance between measurements, via the transfer function formalism presented in the previous chapter.The relation between the linear predictor and the technique of Kalman filtering is discussed. Sec-tion 7.3 presents data from numerical simulations of a frequency standard, which demonstrate thesuperior stabilisation performance of the predictive scheme by contrast to traditional feedback sta-bilisation. The source MATLAB code for this numerical simulation appears in Appendix D. Section7.4 presents the experimental setup that allows the trapped-ion system to be used as an emulator offrequency standards via LO noise engineering, and experimental data verifying the effectiveness ofthe predictive scheme.

7.2 Deriving the linear predictorIn hybrid feedforward, results from a set of n past measurements are linearly combined with weightingcoefficients ck optimised such that the kth correction, Ck, provides maximum correlation to y(tck) atthe instant of correction tck (Figure 7.1c). Assuming that the LO noise is Gaussian, the linear least

55

1

0

g (t)

-50

0

50

y(t)

(a.u

.)

543210Measurement number

1

0

g (t)

t (a.u.)

C(n)k

C(n)k+1

Tc

TR TD

ts1 ts2 ts3 ts4 ts5 te5te4te3te2te1

C(n)k+2

y(LO)k y

(LLO)k

y(LO)(t) y(LLO)(t)

Feedback

Hybrid Feedforward

(a)

(b)

(c)

Figure 7.1 : (a) Simulated traces for a noisy LO, unlocked (black) and locked with traditional feedback (red),with correction period Tc. The dotted horizontal bars indicate the measurement outcomes (samples) over eachcycle, which are applied as correction at the end of the cycle (assuming zero dead time). The arrows on thefar right schematically indicate how locking reduces the variance of y(t) though it does not eliminate it. (b)Schematic timing diagram of a standard feedback protocol with Ramsey measurement time, T (R) and dead timeT (D), where T (D) + T (R) = Tc. (c) Schematic diagram of hybrid feedforward with an example protocol withn = 3, with an alternative option for non-uniform duration measurements instead of the uniform measurementsthat are illustrated here. Corrections C(n=3)

k (depicted as down arrows occurring at instants tck) may be appliedeither after every n measurements only, or after every measurement. The latter option is depicted here, and iscalled the interleaved option because each measurement outcome is reused n times as the correction advances.Dashed red arrows indicate the first corrections performed without full calculation of the covariance matrix;this effect vanishes for k > n.

minimum mean squares estimator (LMMSE) is optimal, and the optimal value of the correction isgiven by Ck = ck · yk. The dot product of a set of correlation coefficients ck derived from knowledgeof Sy(ω) and a set of n past measured samples, yk = yk−n+1, · · · , yk. An (n + 1) × (n + 1)covariance matrix is defined where the (n + 1)th term represents an ideal zero-duration sample at tckand in the second line the covariance matrix is written in block form:

Σk ≡

σ(yk−n+1, yk−n+1) · · · σ(yk−n+1, y(tck))

σ(yk−n+2, yk−n+1) · · · σ(yk−n+2, y(tck))

· · · · · · · · ·

σ(y(tck), yk−n+1) · · · σ(y(tck), y(tck))

(7.1)

≡[Mk Fk

FTk σ(y(tck), y(tck))

](7.2)

where each covariance matrix element σ(yi, yj) is calculated as a spectral overlap

σ(yi, yj) =1

2π

∫ ∞

0

Sy(ω)G2ij(ω)dω (7.3)

This novel overlap integral form for covariance, introduced in the previous chapter, comes directly

56

from the transfer function approach to the problem of covariance, thus linking the predictive correc-tion concept back to the general filter-transfer function formalism that is central to this thesis.

This pair covariance transfer function captures the physics of combining any two Ramsey inter-rogations of different durations (hence sampling different regions of the LO noise spectrum). TheLMMSE optimality condition is then fulfilled for

ck =FkwkN√FTk MkFk

(7.4)

where wk is an overall correction gain and N = 12π

∫∞0Sy(ω)dω is a normalisation factor related to

the overall noise strength in Sy(ω).

7.2.1 An alternative formulation: Kalman filteringThe field of optimal estimation yields a technique known as Kalman filter or linear quadratic estima-tion, which has close affinity with the predictive scheme presented above, and developed indepen-dently. Optimal estimation seeks to estimate the evolution of a system given a physical model (in thiscase, the statistics captured by the PSD) and a series of measurements. Both the physical model andthe measurement may contain uncertainty. The Kalman filter aims to optimally reconstruct the systemevolution given that information, and may be extended to predict future evolution. The goals of opti-mal estimation thus overlap with the aim of the transfer-function-based predictive hybrid feedforwardtechnique.

This derivation of the Kalman gain vector, which plays the same role as the weighting vector ck

above, follows [39]. Working with in the discretised time domain consisting of discrete-time samples,the kth measurement outcome is represented as an m×1 vector yk based on an underlying state n×1vector xk and an p× 1 applied control vector uk applied to the state, where k indexes the time-step:

yk = Hkxk + vk (7.5)xk = Jkxk−1 + Gkuk (7.6)

where Hk is a m× n matrix capturing the measurement transformation, Jk a n× n matrix capturinga model of the dynamical process affecting the state, Gk a n× p matrix capturing the action of noiseon the state and vk a m × 1 measurement noise vector. The control vector uk in the conventionalnomenclature of Kalman filtering refers to applied control, but since in this case the only perturbationto the system frequency is LO noise, the thesis treats uk as the noisy term in the equation.

To specify this system to the frequency standard problem examined in this chapter, the followingassumptions would be made: m = 1 (single degree of freedom) and vk = 0 (negligible measurementnoise). The derivation is presented here in the more general form without these assumptions.

The estimator for the measured outcome is denoted yk|k−1, which may be interpreted as ‘theestimate for yk given the set of measured outcomes y1, · · · ,yk−1’, and is defined

yk|k−1 = Hkxk|k−1 + vk|k−1 (7.7)

where the estimator xk|k−1 is a prediction of the actual state xk based only on the dynamical model.vk|k−1 is interpreted similarly, as an estimate of the effect of measurement noise. The residual betweenthe estimated measurement outcome and the actual measured outcome, called the ‘innovation’ in thiscontext, is given by

57

zk = yk − yk|k−1 (7.8)= yk −Hkxk|k−1 (7.9)

since it is assumed that measurement noise is uncorrelated with previous measurements, so E[vlyk] =0 for all l < k.

From the relation of the state estimate from step to step that follows directly from (7.6)

xk|k−1 = Jkxk−1|k−1 + Gkuk (7.10)

and from the definition of the covariance matrix of the state residual Pk|k ≡ Cov[xk − xk|k], it isfound

Pk|k−1 = JkPk−1|k−1JTk (7.11)

It is important to note that the term ‘covariance matrix’ pertaining to Pk does not explicitly capturecovariance over multiple values of k, as does the covariance matrix Σk over the n previous measure-ments, in the predictive hybrid feedforward calculation of the previous section. Rather, the covariancematrix in the Kalman filter derivation Pk captures correlations across the elements of xk, i.e. the mul-tiple dimensions of the state space. The time-correlations appear in the Kalman filter in its nature asa recursive filter, such that the dependence of Pk|k on Pk|k−1 reflects the cumulative effect of corre-lations between measurements. To emphasise this distinction between the explicit correlations acrossthe state space captured by the covariance matrix and the implicit time-correlations captures in therecursive filter, the notation σ(xk, xl) is used to refer to covariance between points in time labelledk, l, while Cov[x] is used to refer to covariance between the elements of vector x.

The covariance of the innovation is defined Sk ≡ Cov[ek], and so may be expanded to

Sk = HkPk|k−1HTk + Rk (7.12)

where Rk = Cov[v].The crux of the Kalman filtering approach is that the update to the state estimate xk|k, i.e. the

optimal estimate given the latest information, is expressed as a linear combination of the innovationzk and the Kalman gain Kk

xk|k = xk|k−1 + Kkzk (7.13)

where the Kalman gain is defined

Kk = Pk|k−1HTk S−1

k (7.14)

where given certain optimality conditions, the Kalman gain minimises the innovation, i.e. the errorbetween the estimated and measured value for yk.

The predictive hybrid scheme may be understood as a specialised Kalman filter, where yk is1 × 1, and the vector xk represents a string of previous measurements. Hence the elements of thestate-space vector xk are chosen to be a ‘recent history’ of the system. This modification allows theKalman filter to capture time-correlations more explicitly than in the usual formulation, since the statespace explicitly includes past measurements.

The Kalman filter and the predictive hybrid scheme are compatible. The frequency-domain ap-proach of the predictive hybrid scheme is preferred because of its accessibility to experimental charac-terisation (e.g. the PSD) and its mathematical consistency with the transfer functions already known

58

in the frequency standard community. This formal correspondence allows for the combination ofinsights from several different research communities: frequency standards, optimal control and quan-tum control, to produce new results. Kalman filtering has already been applied to an ensemble offrequency standards, viz. the weighted combination of multiple clock signals into a more stable com-posite signal [27]. The single-clock application of the technique may work well in conjunction withmultiple-clock time standard schemes.

7.3 Numerical simulation of frequency standard protocolsIn the practical setting of a frequency standard experiment, it is necessary to improve both the ac-curacy of each correction, by maximising the correlation between Ck and y(tck), and the long-termstability of the LLO output, captured by the variance metrics. In order to test the general performanceof hybrid feedforward in different regimes, numerical simulations are performed of noisy LOs withuser-defined statistical properties, characterised by Sy(ω). An ensemble E of realisations of y(t) ispseudorandomly generated with statistics given by Sy(ω). The simulations are performed using asuite of custom-designed MATLAB scripts. The simulated ‘noisy traces’, representing trajectories ofy(t) in an ensemble of realisations, are used to calculate measures such as the sample variance over asequence of simulated measurement samples with user-defined Ramsey measurement times and deadtimes. In these calculations it may assumed that the LO is either free running, experiencing standardfeedback, or employing hybrid feedforward. The calculations include various noise power spectra,with tunable high-frequency cutoffs, including ‘flicker frequency’ (Sy(ω) ∝ 1/ω), and ‘random walkfrequency’ (Sy(ω) ∝ 1/ω2) noise, as appropriate for experiments incorporating realistic LOs.

7.3.1 Correction accuracyThe first performance criterion is the correction accuracy associated with a single correction cycle,defined as the extent to which a correction brings yLLO(t) → 0 at the instant of correction, t = tck.The metric for correction accuracy is defined as the inverse of frequency variance at tck relative to thefree-running LO

Ak ≡E[yLO(tck)

2]

E[yLLO(tck)2]

(7.15)

Closed form analytic expressions for correction accuracy may be calculated in terms of elementsof the covariance matrix. A consideration of a single cycle can provide a value for the frequencyvariance E[yLLO(tck)

2] in terms of covariance matrix elements, which in turn provides a metric for thecorrection accuracy of hybrid feedforward:

Ak ≡〈yLO(tck)

2〉〈yLLO(tck)

2〉 (7.16)

=

(1 + w2

k − wk|Fk|2√

FTk MkFk

)−1

(7.17)

Tunability in the hybrid feedforward protocol comes from the number of measurements to becombined, n, in determining Ck as well as the selected Ramsey periods, permitting an operator tosample different parts of Sy(ω). As an example, the predictor is fixed to consider n = 2 sequentialmeasurements and permit the Ramsey durations to be varied as optimization parameters. A Nelder-Mead simplex optimization over the measurement durations finds that a hybrid feedforward protocolconsisting of a long measurement period followed by a short period maximises correction accuracy(Figure 7.2a). This structure ensures that low-frequency components of Sy(ω) are sampled but the

59

measurement sampling the highest frequency noise contributions are maximally correlated with y(tck).With Sy(ω) ∝ 1/ω and Sy(ω) ∝ 1/ω2, correction accuracy is improved under hybrid feedforwardwhile the rapid fluctuations in y(t) arising from a white power spectrum mitigate the benefits ofhybrid feedforward, as expected. In the parameter ranges studied numerically, correction accuracy ismaximised for n = 2 to 3, with diminishing performance for larger n.

10 100

1.20

1.15

1.10

1.05

0.01 0.1 1

1.2

1.0

0.8

0.6

0.4

0.2

1.5

1.0

0.5

0.0

12 3 4 5 6 7 8

102 3 4 5 6 7 8

100

1

0

g (t)

Number of Samples, N Duty Cycle (TR / Tc)

LO

-Nor

m. S

amp.

Var

ianc

e

Nor

mal

ized

Acc

urac

y

Ratio of Ramsey Measurements

1/ω

a)

b) c)

White noise 1/ω noise 1/ω2 noise

1/ω 1/ω

1/2

FB, 1/ω HFF, 1/ω FB, 1/ω + spurs HFF, 1/ω + spurs

Sam

ple

Varia

nce

Impr

ovem

ent

〈 σ2 y [N]

〉(FB) / 〈

σ2 y [N]

〉(HF

F)

〈 σ2 y [20

] 〉( HFF

/FB)

/ 〈 σ

2 y [20

] 〉(LO)

A k=

2(HFF

) / A

k=1(F

B)

Figure 7.2 : (a) Calculated correction accuracy of the first correction for hybrid feedforward normalised tofeedback (A(FB) = 1), under different forms of Sy(ω) as a function of the ratio of Ramsey periods between thetwo measurements employed in constructing C(2)

k . Correction accuracy for feedback is calculated assumingthe minimum Ramsey time; thus for the ratio of Ramsey measurements taking value unity on Sy(ω) the Xaxis, the hybrid feedforward scheme takes twice as long as feedback. Inset: depiction of the non-interleavedoption for hybrid feedforward correction. (b) Calculated sample variance for interleaved hybrid feedforward,as a function of measurement number N . The duty cycle of the measurements is fixed at 10%. Data presentedas the normalised ratio of E[σ2

y [N ]](FB)/E[σ2y [N ]](HFF ) in order to demonstrate improvement due to hybrid

feedforward (larger numbers indicate smaller sample variance under hybrid feedforward). Calculations assumeSy(ω) ∝ 1/ω, with a high-frequency cutoff ωc/2π = 100/Tc and Sy(ω) ∝ 1/ω1/2 with a cutoff frequencyωc/2π = 1/Tc, demonstrating the importance of high-f noise near ω/2π = T−1

c . PSDs with different ω-dependences are normalised to have the same value at ωlow = 1/100Tc. (c) Calculated E[σ2

y [N ]] for N = 20as a function of duty cycle, normalised to the sample variance for the free-running LO. Data above red dashedline indicate that the standard feedback approach produces instability larger than that for the free-runningoscillator. Both data sets assume Sy(ω) ∝ 1/ω, with ωc/2π = 100/Tc. Crosses represent data with ten noisespurs superimposed on Sy(ω), starting at ω/2π = 1.15T−1

c , and increasing linearly with step size 0.15T−1c .

7.3.2 Long-term stabilityIn all locked frequency standards, repeated measurements and corrections provide long-term stabil-ity. Instability is a measure of how the output frequency of the LLO deviates from its mean value

60

over time, and the chosen metric for instability is the sample variance of a time-sequence of measure-ment outcomes averaged over an ensemble of noise realizations, E[σ2

y[N ]]. The interleaved style ofhybrid feedforward provides improved long-term stability, as the correction Ck will depend on theset of measurement outcomes yk = yk−n+1, · · · , yk, among which previous corrections have beeninterleaved, as illustrated in Figure 7.1c.

Figure 7.2b shows the resulting normalised improvement in E[σ2y [N ]] up to N = 100 measure-

ments, calculated using feedback and hybrid feedforward with n = 2, and assuming uniform TR.Clear improvement (reduction) in E[σ2

y[N ]] is observed when using the hybrid feedforward approach,with benefits of order 5− 25% of 〈σ2

y[N ]〉 relative performance improvement over standard measure-ment feedback. Data is presented for different functional forms of Sy(ω), including low-frequencydominated flicker noise (∝ 1/ω), and power spectra (∝ 1/ω1/2) with more significant noise near T−1

c .The benefits of our approach are most significant in the long term when high-frequency noise reducesthe efficacy of standard feedback.

The improvement provided by hybrid feedforward is most marked for low duty cycle d, defined asthe ratio of the interrogation time to total cycle time: d ≡ T (R)/Tc. As d→ 1 the feedback and hybridfeedforward approaches converge, as standard feedback corrections become most effective when deadtime is shortest. However as the dead time increases, feedback efficacy diminishes until 〈σ2

y[N ]〉 forthe feedback-locked LO approaches that for the free-running LO (value unity in Figure 7.2c). In thislimit the utility of the measurement-feedback diminishes as the LO noise evolves substantially duringthe dead time, but even here knowledge of correlations in the noise allows hybrid feedforward toprovide significant gains in stability. The benefit afforded by hybrid feedforward is reduced in casesof high duty cycle because the loss of information due to dead time, which is what hybrid feedforwardspecifically targets, contributes proportionally less to the overall error. Other sources of error such asmeasurement noise begin to dominate the instability in high duty cycle cases, thereby setting a limitof the effectiveness of hybrid feedforward. Figure 7.2c illustrates that in the presence of a typical1/ω power spectrum, the presence of noise spurs near ω/2π = T−1

c results in certain regimes wherestandard feedback makes long-term stability worse, while feedforward provides useful stabilization.Exact performance depends sensitively on the form and magnitude of Sy(ω), but results demonstratethat systems with high-frequency noise content around ω/2π ≈ T−1

c benefit significantly from hybridfeedforward.

7.4 Experimental emulator for frequency standards

7.4.1 Frequency measurement techniqueThe development of noise engineering in this system allows it to be used as an emulator for a widevariety of frequency standard systems. The engineered noise PSD may be tailored to match thosereported in real experimental systems, e.g. optical lattice clocks [34, 8], trapped single-ion clocks[16], atomic fountain clocks [73], crystal sapphire oscillators [31, 57], among other technologies.The timing of the Ramsey interrogation intervals may be set arbitrarily to mimic the real duty cyclelimitations experienced by such systems. The trapped Yb+ microwave frequency standard describedin this thesis may therefore be used to test software protocols, including the predictive hybrid feedfor-ward scheme described above, in the presence of noise environments that are engineered to emulateany realistic frequency standard system.

The ideal method for sampling a noisy LO frequency y(t) involves a single Ramsey interrogationwith τπ = 0 resulting in a perfectly rectangular sensitivity profile g(t). These experiments work inthe regime where τπ = 50 µs while T (R) ≥ 100 ms, so the rectangular profile approximation is valid.However, in the context of an experimental frequency standard, performing a Ramsey interrogationin the close vicinity of resonance results in a symmetrical brightness fringe P|↑〉(ω), so that only themagnitude and not the sign of yk can be determined from a single Ramsey interrogation.

61

ȳk

+ȳk

–

ȳk

+

ȳk

– ȳk

+

ȳk

–

Frequencyȳk

= 0 ȳk

> 0 ȳk

< 0

Brightness

Γ

Frequency Frequency

Figure 7.3 : Spectral diagram of the double Ramsey measurement technique. The LO frequency (solid blackline) fluctuates relative to the stable resonance frequency of the qubit (dashed line), with the fractional differencebetween them defined as yk. Due to the symmetry of the brightness peak, the same detuning on either side ofqubit resonance gives the same brightness, so a single-Ramsey measurement is ambiguous with respect to thesign of yk. By taking two measurements detuned Γ/2 above and below the nominal LO frequency (black dots),both sign and magnitude of yk may be determined as a function of y+

k − y−k .

In order to determine the sign of yk, which is required to lock to the centre frequency of thefringe, the Ramsey interrogation is performed twice: once with the microwave carrier detuned Γ/2above the fringe centre and a second time detuned by Γ/2 below the centre, where Γ is the full widthat half-maximum of the fringe. The sign and magnitude of yk is determined by difference between themeasured outcomes y+

k and y−k obtained from the two interrogation periods. The frequency modulatedapproach gives sign information and is also more sensitive to ω-fluctuations than interrogating thefringe at the peak [19]. This technique of frequency measurement is depicted in Figure 7.3. In orderto perform normalisation to counter photon detector fluctuations, a single normalisation bright statepulse is required, by contrast to the additional dark state normalisation required in the qubit controlexperiment.

-1

0

1

Fre

quency e

rror

(Hz)

16x10312840

Sample Number

102

103

104

105

106

107

PS

D (

a.u

.)

0.001 0.1 Frequency (Hz)

Measured 1/f Fit

10-13

10-12

10-11

10-10

10-9

Alla

n D

evia

tio

n

0.1 1 10 100

Ramsey Averaging Time (s)

a b c

Figure 7.4 : Measurements of intrinsic frequency noise in the trapped-ion standard. (a) Time-domain measure-ments of LO fluctuations relative to the qubit frequency. The Ramsey interrogation period is T (R) = 100 ms,with average dead time of T (D) = 600 ms between each measurement. (b) Power spectral density calculatedfrom the time-domain data, showing 1/f statistics. The source of this characteristic flicker frequency noisesignature is believed to be fluctuations in the LO carrier frequency that are produced by instabilities in theMicrosemi amplifier, since the bare VSG’s manufacturer-quoted noise spectrum is white. (c) Allan deviationfor the LO noise, exhibiting the independence of Ramsey time that is consistent with the 1/f noise statistics ofthe measured PSD in (b).

The frequency standard experiment relies on rectangular-wave FM of the carrier to perform Ram-sey spectroscopy of the qubit frequency. This is achieved by using an external arbitrary waveform

62

generator (AWG, with model number DG4602 from RIGOL), which produces a rectangular voltagewaveform that is sent to the VSG’s external FM port. The frequency and amplitude of the modu-lating waveform is sent to the generator from the experiment PC, and is triggered by the SpinCorePulseblaster. Unlike the case of IQM, no discretisation of the FM waveform is necessary.

It is necessary to have an understanding of the intrinsic stability of the trapped-ion system beforeusing it as an emulator with engineered noise. Since frequency stability is the criterion of interest,the frequency deviations between the free-running LO and the qubit are measured over a long period(16384 measurements ≈ 3 hours), shown in Figure 7.4. From these time-domain measurements, thepower spectral density is calculated and found to fit well to a 1/f flicker frequency noise profile. Thecalculated Allan variance is independent of the timescale τ , as expected for flicker noise [66], withbest performance at Aσ2

y(τ) = 5× 10−12.

tNormalisation Red-detuned Ramsey Blue-detuned Ramsey

T(R) T(R)T(D) T(D)

ty(t)

τπ

τπ/2 τ

π/2 τ

π/2 τ

π/2

Figure 7.5 : Schematic diagram of the noise engineering protocol for the frequency standard experiment. Thetop panel shows the engineered noise trace applied to the LO (red trace), with the user-defined dead timesindicated by grey shading to show that the noise trajectory in that period is not measured. The bottom panelshows the control sequence, with microwave pulses in red cross hatch and photon detection windows in greydiagonal hatch.

In order to use the apparatus as an emulator, engineered noise is applied to the LO. For a givenuser-determined noise PSD, a continuous analytic function representing the time-domain noise tra-jectory is calculated in IGOR Pro. This continuous function is divided piecewise according to user-determined parameters of the measurement protocol: number of measurements, Ramsey interrogationtime and dead time. From this piecewise division, engineered samples are produced with the appro-priate time spacings, and applied to the LO as frequency offsets, imitating the effect of measuredfrequency noise. The samples are found by integrating the frequency noise trace over the appropriatetime interval, but for the sake of computational efficiency, they are calculated by deriving the cor-responding phase noise trace and then finding the average gradient across the interval. The benefitof ‘pre-calculating’ the noise trajectory in this way is that the measurement protocol parameters are

63

highly flexible. Tests of the emulator show that applying engineered noise with a substantially dif-ferent character to the LO’s intrinsic 1/f signature produces measured Allan deviations consistentwith the character of the engineered noise. This demonstrates the effectiveness of the emulator, sincethe engineered noise dominates the intrinsic fluctuations to such an extent that the latter cannot bediscerned during noise injection.

-6

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eere

d F

requency O

ffset (H

z)

6005004003002001000

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Figure 7.6 : A demonstration of the engineered noise technique. The red trace is the trajectory of the PC-generated engineered noise trace, which is then imposed on the VSG using the AWG as a source external FM.The black dots are the measured frequency offset of the VSG relative to the qubit frequency, showing that themeasured samples match the trajectory of the engineered noise.

In Figure 7.5, the engineered noise trace (solid red) is calculated as a continuous time-domainfunction that conforms to user-defined PSD parameters. The noise trace is applied directly to theLO in order to mimic the effect of intrinsic LO frequency noise, with the advantage that since theapplication is software-based, it may be triggered and repeated arbitrarily. This allows the user tofreely set the ‘engineered’ dead time T (D), and the noise trace is restarted at the point after a desireddead time period, rather than the actual dead time period enforced by the experimental hardware. Thedouble Ramsey measurement is performed by repeating the noise waveform identically, apart froman FM offset (dashed red lines). This repetition of the waveform ensures that the same engineerednoise is being measured both times, once with the LO red-detuned by Γ/2 and a second time withthe LO blue-detuned by Γ/2, as depicted in Figure 7.3. In combination with the initial bright statenormalisation pulse, this control sequence produces an unambiguous measurement of yk that is robustagainst photon detection noise. No engineered noise is applied during the normalisation pulse, sinceits purpose is simply to compensate for intrinsic detection noise. The accuracy of this engineerednoise technique is shown in Figure 7.6, where the measured frequency offsets are seen to follow thetrajectory of the injected noise.

7.5 Experimental verification of stability improvementAn experimental scan of sample variance as a function of the number of samples is shown in Figure7.7, comparing the stability of the frequency standard emulator under traditional feedback (FB) andthe novel predictive hybrid feedforward scheme (HFF). The approximately 10% improvement of HFFover FB in the high N regime (representing long-term stability) is falls within the 5% − 15% rangepredicted by the numerical simulation shown in Figure 7.2b. This successful implementation of apredictive HFF correction scheme, and verification of the accuracy of previous simulations, opensthe door to an exploration of the scheme in a wide range of realistic noise environments, emulatingreal-world frequency standards.

64

0.45

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ple

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10008006004002000

Number of Samples, N

FB HFF

Figure 7.7 : Experimental demonstration of predictive hybrid feedforward locking. This figure shows measuredexperimental data for a fixed set of parameters, where the HFF predictor weightings have been optimised to theinjected engineered noise spectrum. The engineered noise PSD obeys a 1/ω power law with a high-frequencycutoff ωc = 10 Hz and a low-frequency cutoff ωb = 1 mHz with a comb tooth spacing of ω0 = 1 mHz. The dutycycle of this measurement scheme is d = 10% with Tc = 1s. The optimised weights for the HFF in this scan are[-0.0397, -0.3466, 0.864]. The sample variance as a function of N is lower under hybrid feedforward locking(solid blue) than traditional feedback locking (solid red). The number of samples N is to be understood as thenumber of samples that contribute to the calculation of sample variance, rather than the number of samples thatcontribute to the calculation of the hybrid feedforward predictor. Each data point is an average of individualnoise realisations (pale red and blue).

It is important to note that even on a single run of the experiment, the predictive hybrid feedfor-ward scheme produces improved correction accuracy, and so improved sample variance over the longterm, as indicated by the pale traces in Figure 7.7. Averaging over the ensemble removes the largefluctuations in the low N (short term) regime, showing the superior performance of predictive hybridfeedforward on all timescales (solid lines).

7.6 ConclusionThis thesis seeks to make two primary contributions to the study of frequency standards. First, anextension of the conventional transfer function approach to frequency standards to the novel purposesof (a) characterising locked local oscillators (presented in the previous chapter), and (b) predictionof noise trajectories by exploiting correlations, leading to the novel predictive hybrid feedforwardtechnique. Second, a demonstration of both a numerical simulator and an experimental emulator offrequency standards. Using these tools, the efficacy of the hybrid feedforward scheme is tested ina range of realistic noise settings. The results of these tests suggest that the technique may be ofparticular use in real frequency standards that are limited by long dead times, such as cutting-edgeoptical lattice clocks [8].

65

Chapter 8

Conclusion8.1 Summary of the thesisThis thesis presents significant theoretical and experimental contributions to the field of precisionquantum control. The thesis begins with a unifying abstract treatment of two control problems thatshare an underlying commonality, those of qubit control and of frequency standards, and proceedsto demonstrate new analysis and improved techniques that are specific to each problem. Chapter 1introduces the field of precision quantum control and describes, in general terms, the physical systemthat is investigated in this thesis. Chapter 2 provides a detailed description of the experimental as-pects of the project. Chapter 3 derives a mathematical formalism for characterising noise as a spectraldensity, which is applied to both control problems. Chapter 4 presents the theoretical underpinningsof the filter function approach to qubit control, via an average Hamiltonian derivation. Chaper 5presents experimental results showing both verification of the filter function as a characterisation tooland its use in constructing improved error-suppressing composite pulses. Chapter 6 introduces thetransfer function approach to the characterisation of frequency standards, presenting novel extensionsof the approach to calculate variances for feedback-stabilised local oscillators. A key new concept isdeveloped, the pair covariance transfer function, which captures frequency-domain information per-taining to correlations between time-separated measurements. Chapter 7 presents a novel predictivemeasurement and correction scheme based on this covariance transfer function, along with numericalsimulations and experimental verification of its superior stabilisation performance in comparison totraditional feedback.

8.2 Contributions of the thesisThe fields of quantum control and frequency standards are distinct, though both acknowledge the sim-ilarity of their systems and the influence of resonance spectroscopy disciplines, such as NMR and ESR[77, 19]. This thesis seeks to unite the two control problems by showing that they represent differentways of looking at the same physical system: a classical oscillator coherently driving a two-levelquantum system (qubit). In the interaction-focussed approach taken here, the experimentalist im-proves the noise-resistance of the overall system by modifying only the interaction between oscillatorand qubit. The thesis demonstrates the unity of the qubit control and frequency standard problemsby treating them using the same theoretical formalism, the ‘filter-transfer function formalism’, andperforming the experiments on the same apparatus using a novel noise engineering technique thatallows the system to emulate either qubit or oscillator noise.

The primary contribution to qubit control presented in Chapter 4 is the application of the filterfunction approach in semiclassical universal noise environments. This theoretical chapter uses anaverage Hamiltonian theory to derive novel frequency-domain expressions for the qubit trace fidelity(i.e. the accuracy with which an intended operation is achieved in the presence of noise).

Chapter 5 finds that the fidelity predictions accurately match those measured in experiment,

66

demonstrating for the first time that the filter function formalism is an effective framework in whichto cast universal qubit control in the presence of realistic noise. Furthermore, the chapter systematisesa range of known error-suppressing composite pulses in terms of Walsh basis modulation. This sys-temisation contributes to the understanding of the structure and behaviour of such composite pulses,particularly the conceptual distinction between Magnus and filter order. This distinction becomesimportant in noise environments that fluctuate more rapidly than the quasistatic case for which thepulses were traditionally designed. The generality of the Walsh-modulation prescription allows thedevelopment of novel high-filter-order composite pulses, which are experimentally shown to haveimproved resistance to noise compared to known pulses.

The contributions in Chapter 6 are the novel extensions of the transfer function approach to char-acterising noise in local oscillators. The extensions to noise characterisation are a set of new analyticforms for the instability of locked local oscillators, where the literature has previously only providedsuch forms for free-running oscillators. The basis of these novel theoretical results is the derivationof the ‘pair covariance transfer function’, which is frequency-domain function that captures statisticalcorrelations between a pair of measurements.

Chapter 7 presents the derivation and implementation of a novel method of predictive hybridfeedforward to improve the stability of locked local oscillators. The technique allows multiple mea-surements to be weighted, given a prior characterisation of the oscillator noise statistics, to produce anoptimal estimate of the noise trajectory at future times. Since the technique is flexible with respect tonoise spectra and measurement parameters (e.g. dead time), it is platform-independent. The numer-ical simulations and experimental emulations done in this chapter verify the superior performanceof the novel predictive technique in parameter regimes of interest. Given its flexibility, predictivefeedforward may provide tangible benefits to existing frequency standard technologies, such as therecord-breaking optical lattice clocks, since this technique is a modification to the control protocolonly. The affinity of this technique with Kalman filtering techniques suggests the possibility of usefulsynthesis of insights from optimal control theory beyond what has been presented in this thesis.

8.3 Directions for future researchBoth sets of experimental results, those in Chapters 5 and 7, represent starting points for furtherinvestigation. As noted above, Chapter 5 presents the first experimental validation of the filter functionapproach as a means of calculating the fidelity of quantum control operations in the presence of noise.Filter-function-based calculations may be used in other noisy systems to predict the performanceof error-suppressing composite pulses in a range of different noise environments. Since the powerspectral density is a characterisation tool that is familiar to experimentalists, it is expected that thefilter function approach will be a convenient method of predicting and assessing the performance ofreal-world quantum systems.

Similarly, the experimental result of Chapter 7 is the first demonstration of the improved per-formance of the novel predictive technique, using concepts from optimal estimation for a single fre-quency standard. This result, in conjunction with the numerical simulations that give a more extensivepicture of the regimes where the technique is effective, provides a starting point for further applica-tion and exploration of the predictive approach. Further investigation would involve the testing of thepredictive technique for a broad range of experimental parameters, including noise spectra with real-istic features such as spurs, a wide range of duty cycles to include the different dead time limitationsof different systems. The flexibility of the novel technique in combining measurements of differentdurations could be integrated with the emulator’s double Ramsey technique to provide further im-provements in accuracy. A long-term aim of an exhaustive parameter survey of this kind would be thetesting of the predictive technique in other advanced frequency standard systems, via this trapped-ionemulator, to demonstrate the benefits of this control technique to the community at large. Further

67

exploration of the predictive technique beyond frequency standard applications may lead to broaderuse for the techniques in such fields as quantum state estimation.

68

Appendix A

Nature Physics publication

69

LETTERSPUBLISHED ONLINE: 19 OCTOBER 2014 | DOI: 10.1038/NPHYS3115

Experimental noise filtering by quantum controlA. Soare1,2†, H. Ball1,2†, D. Hayes1,2‡, J. Sastrawan1,2, M. C. Jarratt1,2, J. J. McLoughlin1,2, X. Zhen3,T. J. Green1,2 and M. J. Biercuk1,2*Extrinsic interference is routinely faced in systemsengineering, and a common solution is to rely on a broadclass of filtering techniques to aord stability to intrinsicallyunstable systems or isolate particular signals from a noisybackground. Experimentalists leading the development of anew generation of quantum-enabled technologies similarlyencounter time-varying noise in realistic laboratory settings.They face substantial challenges in either suppressing suchnoise for high-fidelity quantum operations1 or controllablyexploiting it in quantum-enhanced sensing2–4 or systemidentification tasks5,6, due to a lack of ecient, validatedapproaches to understanding and predicting quantumdynamics in the presence of realistic time-varying noise. Inthis work we use the theory of quantum control engineering7,8

and experiments with trapped 171Yb+ ions to study thedynamics of controlled quantum systems. Our results providethe first experimental validation of generalized filter-transferfunctions casting arbitrary quantum control operations onqubits as noise spectral filters9,10. We demonstrate the utilityof these constructs for directly predicting the evolution ofa quantum state in a realistic noisy environment as well asfor developing novel robust control and sensing protocols.These experiments provide a significant advance in ourunderstanding of the physics underlying controlled quantumdynamics, and unlock new capabilities for the emerging fieldof quantum systems engineering.

Time-varying noise coupled to quantum systems—typicallyqubits—generically results in decoherence, or a loss of‘quantumness’ of the system. Broadly, one may think of thestate of the quantum system becoming randomized throughuncontrolled (and often uncontrollable) interactions with theenvironment during both idle periods and active control operations(Fig. 1a). Despite the ubiquity of this phenomenon, it is achallenging problem to predict the average evolution of a qubitstate undergoing a specific, but arbitrary operation in the presenceof realistic time-dependent noise—how much randomization doesone expect and how well can one perform the target operation?Making such predictions accurately is precisely the capability thatexperimentalists require in realistic laboratory settings. Moreover,this capability is fundamental to the development of novel controltechniques designed to modify or suppress decoherence asresearchers attempt to build quantum-enabled technologies forapplications such as quantum information and quantum sensing.

These considerations motivate the development of novelengineering-inspired analytic tools enabling a user to accuratelypredict the behaviour of a controlled quantum system in realistic

laboratory environments. Recent work has demonstrated that theaverage dynamics of a controlled qubit state evolution may becaptured using filter-transfer functions (FFs) characterizing thecontrol. The fidelity of an arbitrary operation over duration τ ,Fχ (τ ) ∝ e−

∫∞

0 dωS(ω)F(ω), is degraded owing to frequency-domainspectral overlap between noise in the environment given by apower spectrum S(ω), and the filter-transfer functions denotedF(ω) (Methods)11–14.

The FF description of ensemble-average quantum dynamicstremendously simplifies the task of analysing the expectedperformance of a control protocol in a noisy environment as itpermits consideration of control as noise spectral filtering. TheFFs themselves may be described using familiar concepts suchas frequency passbands, stopbands and filter order, enabling asimple graphical representation of otherwise complex conceptsin the dynamics of controlled quantum systems (Fig. 1b). Noisefiltering, in practice, is achieved through construction of a controlprotocol (Fig. 1a) which modifies the controllability of the quantumsystem by the noisy environment over a defined frequency band.Adjusting F(ω) and changing its overlap with the noise spectrumthus allows a user to change the average dynamics of the system ina predictable way.

To see the importance of this capability we may consider thevarious tasks that might be of interest in experimental quantumengineering and the role of noise spectral filtering in theseapplications. In quantum information an experimentalist mayaim to suppress broadband low-frequency noise to maximize thefidelity of a bounded-strength quantum logic operation (Fig. 1b,upper trace), and then calculate the residual error. Alternatively, inquantum-enabled sensing or system identification he or she mayperform narrowband spectral characterization of a given operation(Fig. 1b, lower trace), where any change in the measured fidelityunder filter application represents the signal of interest4,6.

The intuitive nature of this framework is belied by thechallenge of calculating FFs for arbitrary control protocols,generally involving time-domain modulation of control parameterssuch as the frequency and amplitude of a driving field. Thenature of quantum dynamics means that the implementedcontrol framework is generally nonlinear; for instance, one findscomplex dynamics in circumstances where the noise and controloperations do not commute, such as a driven operation (∝ σx)in the presence of dephasing noise (∝ σz ). Recent theoreticaleffort has allowed calculation of FFs for arbitrary single-qubitcontrol and arbitrary universal classical noise9,10, expandingsignificantly beyond previous demonstrations restricted tothe identity operator in pure-dephasing environments15. It

1ARC Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, NSW 2006, Australia, 2National Measurement Institute,West Lindfield, NSW 2070, Australia, 3State Key Laboratory of Low-dimensional Quantum Physics and Department of Physics, Tsinghua University,Beijing 100084, China. †These authors contributed equally to this work. ‡Present address: Lockheed Martin Corporation, USA.*e-mail: michael.biercuk@sydney.edu.au

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© 2014 Macmillan Publishers Limited. All rights reserved

LETTERS NATURE PHYSICS DOI: 10.1038/NPHYS3115

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xx

Figure 1 | Noise filters and experimental validation of the predictive power of the filter-transfer function. a, Time-varying noise during an operation(a rotation on the Bloch sphere, here θ=π/2, φ=0) produces a broad range of outcomes (red uncertainty cone, left) and may yield an oset of the averagefinal state from the target state, measured as operational infidelity. Schematic filtered state evolution, depicted as a user-defined modulation pattern on thecontrol (coloured segments), changes the measured fidelity by reducing the uncertainty due to noise in a specified band. b, Schematic representation ofnoise filters of interest—shaded areas represent filter stopbands—crafted by control modulation as indicated above. a.u., arbitrary units. c, Measurementsof operational fidelity with engineered dephasing noise for primitive π rotation, |0〉→|1〉, as a function of dimensionless noise cuto frequency overlaidwith FF-based calculations of Fχ (τ ). Decay to value 0.5 corresponds to full decoherence. Each data point is the result of averaging over 50 dierent noiserealizations. d, Schematic representation of the quasi-white noise power spectrum with cuto ωc employed in c and single-tone power spectrum withfrequency ωt employed in f and g. Noise strength parameterized by α. e, Calculated F(ω), for primitive and compensating π pulses (ref. 18). Vertical linesindicate frequencies where F(ω) for SK1 (red) and BB1 (black) cross primitive (blue), indicating an expected inversion of performance. f,g, Swept-tonemultiplicative amplitude-noise measurements, S(ω)∝δ(ωt−ω), for various π rotations (averaged over 20 noise realizations). Vertical axis is a proxy formeasured operational infidelity. Solid lines indicate 1−Fχ (τ ). f, Good agreement is revealed in the weak-noise limit across three decades of frequency,down to the measurement fidelity limit,∼98.5%, indicated by grey shading. Measured gate-error crossover points correspond well with crossovers in theFFs for these gates (vertical dashed lines). Detailed performance dierences between protocols in the low-error limit can be revealed through randomizedbenchmarking, as performed later (Fig. 3). g, In the strong-error limit, first-order approximations are violated and contributions from higher-order Magnusterms contribute to the measured error in the low-frequency limit, yielding (expected) dierences between SK1 and BB1 due to Magnus order cancellationand not captured by the FF.

is this more general case where the impact of noise filteringand the FFs may have the most significant impact on thequantum engineering community, and where experimental testsare vital.

In our experimental system, based on the 12.6GHz qubittransition in 171Yb+ (Supplementary Methods), we are able toperform quantitative tests of operational fidelity for arbitrarycontrol operations; thesemay then be compared against calculationsof Fχ (τ ) as a fundamental test of FF validity. A key tool in ourstudies is bath engineering16, in which we add noise with user-defined spectral characteristics to the control system, producingwell-controlled unitary dephasing or depolarization.

As a first example (Fig. 1c), experimental measurements ofoperational fidelity for a πx-pulse driving qubit population from thedark state to the bright state, |0〉→|1〉, in the presence of engineeredtime-dependent dephasing noise give good agreement with analyticcalculation of Fχ (τ ) using the noise power spectrum and analyticFFs (ref. 10) with no free parameters (Methods). This approachtherefore immediately demonstrates the predictive power of theFF formalism.

The FFs for much more complex control, such as compensatingcomposite pulses17,18, can be calculated and experimentally validatedas well (Fig. 1e). These protocols are commonly used in nuclearmagnetic resonance and electron spin resonance in attempting tosuppress static offsets in control parameters such as the frequencyof the drive inducing spin rotations. Calculating the FFs forthese protocols now reveals their sensitivity to time-dependent

noise—an important characteristic for deployment in realisticquantum information settings19. We experimentally demonstratea form of quantum system identification (Methods), effectivelyreconstructing the amplitude-noise filter functions, F(ω), fortwo well-known compensating pulse sequences known by theshorthand designations SK1 and BB1 (Supplementary Methods).Again, calculations of Fχ (τ ) match data well over the entire bandin the weak-noise limit (Fig. 1f) with no free parameters.

Our choice of characterizing these compensating pulse sequenceshighlights an important issue in the prediction of ensemble-average dynamics of controlled quantum systems. Ultimately, theunderlying physical principles giving rise to the analytic form ofF(ω) are based on the well-tested average Hamiltonian theory20exploited in crafting these pulses. Despite this shared theoreticalfoundation, the calculation of spectral filtering properties is quitedistinct from calculation of quasi-static error terms in a Magnusexpansion, with important consequences for average quantumdynamics in realistic time-varying noise environments10.

Accordingly, our tests of the FF formalism reveal thatcompensating pulses designed to suppress errors to high order in aMagnus-expansion framework need not be efficient noise spectralfilters (Supplementary Methods and ref. 19). Despite significantdifferences in their construction—the BB1 protocol is designed toprovide higher-order cancellation of Magnus terms than SK1—both of the selected composite pulses provide similar filtering oftime-dependent noise, given by the filter order (slope of the FF inFig. 1e). In the weak-noise limit frequency-domain characteristics

826 NATURE PHYSICS | VOL 10 | NOVEMBER 2014 | www.nature.com/naturephysics

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NATURE PHYSICS DOI: 10.1038/NPHYS3115 LETTERS

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ΓΓ T8

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= M τ HM X

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Figure 2 | Synthesis of high-pass amplitude-modulated filters from theWalsh functions. a, The first eight Walsh functions used in filter synthesis,PAL0,PAL7, with maximum-Hamming-weight-indexed functionshighlighted in red. b, Representative amplitude profiles for filterconstructions found via numerical search over the Walsh basis with four(red, denoted W1) and eight (blue, denoted W2) time steps. Vertical axisrepresentsΩ , the Rabi rate per time step; negative values indicate π-phaseshifts. Synthesis may be performed over square (flat-top) pulse segmentsor Gaussian-shaped pulse segments, with results diering only in theresulting Walsh coecients. The matrix representing filter characteristicsover eight segments is superimposed on the amplitude profiles (for n=4,neighbouring segments between red dashed lines are combined). The firstrow (the angles of rotation in each segment of the filter) is determined viaWalsh synthesis, indicated by the vectors XT

∣∣n0, containing the spectral

weights over PAL0→PALn. In the case of Gaussian pulse envelopes Walshsynthesis sets the first line, θl. The symbol X indicates reordering forHadamard synthesis, with listed coecients appropriate for square pulseenvelopes. c, The filter-transfer function for a primitive πx rotation and forsynthesized noise filters. Performance improvement over the desiredstopband of the filter captured in cost function A(0W1(W2)

4(8) ) and itsdierence relative to that for the primitive operation, A(0Prim

1 ). Filter W1gives improvement indicated by the red shading, with additionalimprovement in the cost function given by W2 indicated by blue shading.

are captured accurately through the FF across frequencies rangingfrom quasi-static to rapidly fluctuating on the timescale of thepulse (Fig. 1f). Performance deviations between the pulses ariseand the FF approximation breaks down as the noise strength isincreased and higher-order terms in the Magnus series becomeimportant, but only at low frequencies (Fig. 1g). At frequenciesfast relative to the control the FF again accurately predicts therelevant quantum dynamics even in the strong-noise limit. This isthe first direct manifestation of the difference between studyingquantum dynamics in terms of frequency-domain noise filteringand calculation of error contributions in a Magnus expansion as isappropriate in the quasi-static limit.

These simple but powerful validations of the predictive powerof the generalized FF formalism now open the possibility ofdemonstrating the construction of noise filters with a specifiedspectral response, employing the filter-transfer functions as keyanalytic tools. Filters may take a wide variety of forms asneeded by users—including high-pass filters for broadband noisesuppression and band-stop filters useful for narrowband noisecharacterization (Fig. 1b).

In the discussion that follows, we focus on a common settingin which we aim to improve operational fidelity by reducingthe influence of broadband non-Markovian noise on a targetstate transformation. Filters are realized as n-step sequences oftime-domain control operations with tunable pulse amplitude andphase, similar in spirit to compensating composite pulses in NMR(refs 17–19), dynamically corrected gates (DCGs) in quantuminformation21,22, and open-loop modulated pulses in quantumcontrol23,24. However, recalling the difference between Magnuscancellation order and filtering order described above, in this settingwe wish to synthesize a filter with arbitrary, user-defined spectralcharacteristics captured by a cost function, A(0n), to be minimizedfor a filter represented by 0n(θl ,τl ,φl) (Fig. 2b,c and Methods).

To provide efficient solutions to filter design we restrict ourcontrol space and focus on constructions synthesized using conceptsfrom functional analysis in the basis set ofWalsh functions—square-wave analogues of the sines and cosines4,25,26 (Fig. 2a). This approachprovides significant benefits for our problem26, but is by no meansthe only basis set for composite filter construction27,28.

As an example we synthesize noise filters via weighted linearcombinations of Walsh functions, PALk(x) denoted by the Paley-ordered index, k. These filters are designed to suppress time-varyingdephasing noise over a low-frequency stopbandwhile implementinga bounded-strength driven rotation about the x axis on theBloch sphere (Supplementary Methods). In this case the Walsh-synthesized waveform dictates an amplitude modulation patternfor the control field over discrete time segments. Importantly,Walsh filter synthesis is compatible with pulse segments possessingarbitrary pulse envelopes, including sequences of, for example,square (used here) or Gaussian pulse segments (Fig. 2b).

Analytic design rules provide simple insights into how one maycraft effective modulation protocols, and a Nelder–Mead simplexoptimization is used to find high-performing operations as definedby our cost function. Relative to an unfiltered primitive gate,the dephasing filter function, Fz(ω), for the simplest four-pulseconstructionW1 shows increased steepness in the stopband (Fig. 2c,red), reducing A(04) (here the gate performs θ=π). This measureof filter order may be further increased via construction W2, inturn reducing the cost function for optimization (blue shaded areain Fig. 2c). Relating back to earlier demonstrations of filter orderin compensating pulses, W2 presents an interesting case of a high-order noise filter over the target band which provides only first-order Magnus cancellation.

Filters W1 and W2 are representative, rather than uniquesolutions. In Fig. 3b we show the calculated cost function, A(04),as a function of the Walsh coefficients used in constructingW1, X0andX3, giving themodulation profile indicated in Fig. 3a. Blue areasmeet ourminimized target, indicating useful filters, revealing a widevariety of possible constructions with favourable characteristics.Experimental tests of these protocols reveal that Walsh-modulatedwaveforms minimizing A(04) effectively suppress noise in thedesignated stopband for arbitrary rotation angles (Fig. 3c–e), andoutperform standard pulses in the small-error limit germane toquantum information (Fig. 3f). See Methods.

Our focus has been on providing a validated frameworkfor the vital task of predicting quantum dynamics in realisticenvironments and demonstrating the relevant physics throughconstruction of noise spectral filters. The Walsh-modulated filters

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LETTERS NATURE PHYSICS DOI: 10.1038/NPHYS3115

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X0 = 2.25 , = /4θπ π

π

PAL3 amplitude, X3 ( )π X3 ( )π

A(Γ4)

(a.u

.)Ω

Figure 3 | Construction of the first-orderWalsh amplitude-modulated dephasing-suppressing filter. a, Schematic representation of Walsh synthesis for afour-segment amplitude-modulated filter (WAMF). Walsh synthesis can be used to determine either the modulating envelope of square pulse segments,or the net area of discrete Gaussian pulses with diering amplitudes. b, Two-dimensional representation of the integral metric defining our target costfunction, A(04) integrated over the stopband ω∈[10−9, 10−1]τ−1. Areas in blue minimize A(04), representing eective filter constructions. The X0determines the net rotation enacted in a gate while X3 determines the modulation depth, as represented in a. White lines indicate possible constructionsfor filters implementing rotations of θ=π , θ=π/2 and θ=π/4 from top to bottom. c–e, Experimental measurement of gate infidelity (left axis) forrotations constructed from various Walsh coecients in the presence of engineered noise (ωc/2π=20 Hz). Black dots and line represented calculatedfidelity by Schroedinger equation integration (raw and smoothed respectively). All values of X3 for a given X0 implement the same net rotation, indicatedby a control experiment with no noise. Total rotation time is scaled with X3 to preserve a maximum Rabi rate. Black dashed line (right axis) corresponds toA(04) from a. In experiments we always perform a net π rotation |0〉→|1〉 by sequentially performing identical copies of rotations for θ <π . f, Randomizedbenchmarking results (50 randomizations) demonstrating superior performance of a modulated gate in the small-error limit (infidelity<0.5% per gate),see Supplementary Methods.

presented here—based on the achievable frequency-domain filterorder—complement existing techniques rather than attempting toprovide optimal-performance error-robust gates. Our results onhigh-pass noise filters, for instance, add to existing compensatingpulse sequences designed for quasi-static noise, as well as gateconstructions with interleaved dynamical decoupling that seek toperiodically ‘refocus’ quantum evolution29–32.

Importantly, recent work has demonstrated that the filter-transfer function formalism is applicable to multi-qubit settingswhere dynamics may be considerably more complex than thesingle-qubit case33–35. In addition, ongoing efforts suggest thereexists a path towards further extension of the generalized filter-transfer function and noise filtering formalisms to arbitrary controlsettings involvingmultiple qubits subject to general noise from non-Markovian classical and/or quantummechanical environments. Webelieve thatwith the validations providedhere, this simple extensibleframework with precise predictive power will provide a path forexperimentalists to characterize and suppress the effects of noise ingeneric quantum coherent technologies, ultimately enabling a newgeneration of engineered quantum systems.

MethodsThe fidelity of a control operation for a single qubit in the presence of atime-dependent environment is reduced as Fχ (τ )=1/2(1+e−χ(τ)), whereχ(τ)=1/π

∑i

∫∞

0 dωSi(ω)Fi(ω), and τ is the total duration of the operation. Inthis expression for fidelity, the integral considers contributions from independentnoise processes through their frequency-domain power spectra Si(ω), i∈z ,Ω,capturing dephasing along z and amplitude noise co-rotating with a resonantdrive field (Supplementary Methods). We employ here the so-called modifiedfilter-transfer function, which subsumes a factor of ω−2 into the definition ofFi(ω). See refs 6,14 for details.

Experimental measurements involve state initialization in |0〉 followed by acontrol operation—or series of control operations—designed to drive qubit

population from the dark state to the bright state, |0〉→|1〉. For instance, tests offilters used for rotations θ <π are repeated sequentially such that the net rotationenacts |0〉→|1〉 (Fig. 3d,e). The operational fidelity is measured as the probabilitythat the qubit is in the bright state over an ensemble of measurements. Typicalexperimental uncertainties are limited by measurement fidelity (∼98.5%) andquantum projection noise with maximum value comparable to the measurementinfidelity for qubit states near the equatorial plane of the Bloch sphere. In general,a non-Markovian noise bath is engineered with specific properties of interest (seeSupplementary Methods for full details). Additional measurement uncertainty oforder ∼3–5% is added through finite sampling of the infinite ensemble ofpossible noise realizations. This is visible as fluctuations between neighbouringpoints in, for example, Fig. 3c–f.

Measurements in Fig. 1c are conducted for a simple πx enacted while varyingthe high-frequency cutoff, ωc , of a flat-top engineered non-Markovian dephasingbath (Fig. 1d). As the high-frequency cutoff of the noise is increased andfluctuations fast relative to the control (τπ ) are added to the noise powerspectrum, Sz (ω), errors accumulate, reducing the measured fidelity. Forωc/2π=1 the highest frequency contribution to Sz (ω) undergoes a completecycle of oscillation over τπ , indicating that the noise is time-dependent on thescale of a single experiment even for ωc/2π1. We calculate Fχ (τ ) using theform of the noise and the analytic FF for a driven primitive gate underdephasing10, finding good agreement with experimental measurements using nofree parameters.

Measurements in Fig. 1f,g employ a narrowband ‘delta-function’ noise powerspectrum swept as an experimental variable, ωt . Injected noise takes the form offixed-frequency amplitude modulation of the near-resonant driving field duringapplication of a control pulse, with strength (modulation depth) parameterized interms of Ω , the Rabi rate of the drive. The form of Fχ (τ ) demonstrates that thecalculated fidelity involves an exponentiated value of the FF at frequency ωt ,meaning that fidelity measurements effectively reconstruct the filter functions.Key features in the data, such as performance-crossover frequencies betweenprimitive and compensating gates and deep notches in the filter at highfrequency, are quantitatively reproduced in experimental measurements.

Filter construction presented in Figs 2 and 3 is parameterized as a functionof controllable properties of a near-resonant carrier frequency enacting drivenoperations. An arbitrary n-segment filter is represented over successive timestepsthrough the matrix quantity 0n(θl ,τl ,φl) (Fig. 2b); in each segment of duration τl

828 NATURE PHYSICS | VOL 10 | NOVEMBER 2014 | www.nature.com/naturephysics

© 2014 Macmillan Publishers Limited. All rights reserved

NATURE PHYSICS DOI: 10.1038/NPHYS3115 LETTERSwe perform a driven operation generating a rotation through an angleθl=

∫ tltl−1Ωl(t)dt about the axis rl=(cos(φl), sin(φl), 0), with Ωl(t) the Rabi rate

over the lth pulse segment.The value of n is chosen to be a power of two, compatible with synthesis over

discrete-time Walsh functions. The Walsh functions are piecewise-constant oversegments which are all integer multiples of base period τl . This approach bringsbenefits for the current setting26; for instance, their piecewise-constantconstruction builds intrinsic compatibility with discrete clocking and classicaldigital logic, while the well-characterized mathematical properties of the Walshfunctions provide a basis for establishing simple analytic filter-design rules, andflexibility in realizing a wide variety of filter forms.

For the filters W1 and W2 presented in the main text, Walsh-synthesisdesign rules dictate that we implement our filtered rotation by θx over aminimum of four discrete steps, permitting synthesis over PAL0 to PAL3. Withinthis small set, the coefficient of PAL0, denoted X0, sets the total rotation angleθ mod 2π for the modulated driven evolution, and only non-zero X3 preservessymmetry. We experimentally test the performance of four-segmentamplitude-modulated filters by scanning over X3 for fixed X0 (denoted by whitedotted lines in Fig. 3b). Values of X3 minimizing A(04) (dips in the dashed trace,right axis) also minimize the experimentally measured infidelity in the presenceof engineered low-frequency noise (open circles, left axis). This behaviour isobserved for various target rotation angles of interest (Fig. 3c–e), with predictedshifts in the optimal values of X3 with changes in X0 borne out throughexperiment. Filter W2 is constructed over PAL0 to PAL7, and has twice as manytimesteps as W1. Interestingly, W1 is a special case of an analytically constructeddynamically corrected NOT gate (a π-rotation)21. For details of the Walshfunctions, Walsh synthesis and Walsh-basis analytic design rules seeSupplementary Methods.

Received 1 April 2014; accepted 2 September 2014;published online 19 October 2014

References1. Smith, A. et al. Quantum control in the 6Cs1/2 ground manifold using

radio-frequency and microwave magnetic fields. Phys. Rev. Lett. 111,170502 (2013).

2. Dolde, F. et al. Electric-field sensing using single diamond spins. Nature Phys.7, 459–463 (2011).

3. Kucsko, G. et al. Nanometre-scale thermometry in a living cell. Nature 500,54–58 (2013).

4. Cooper, A., Magesan, E., Yum, H. & Cappellaro, P. Time-resolved magneticsensing with electronic spins in diamond. Nature Commun. 5, 3141 (2014).

5. Álvarez, G. A. & Suter, D. Measuring the spectrum of colored noise bydynamical decoupling. Phys. Rev. Lett. 107, 230501 (2011).

6. Bylander, J. et al. Noise spectroscopy through dynamical decoupling with asuperconducting flux qubit. Nature Phys. 7, 565–570 (2011).

7. Clark, J. W., Lucarelli, D. G. & Tarn, T. J. Control of quantum systems. Int. J.Mod. Phys. B 17, 5397–5411 (2003).

8. Bouten, L., Handel, R. V. & James, M. R. An introduction to quantum filtering.SIAM J. Control Optim. 46, 2199–2241 (2007).

9. Green, T. J., Uys, H. & Biercuk, M. J. High-order noise filtering in nontrivialquantum logic gates. Phys. Rev. Lett. 109, 020501 (2012).

10. Green, T. J., Sastrawan, J., Uys, H. & Biercuk, M. J. Arbitrary quantumcontrol of qubits in the presence of universal noise. New J. Phys. 15,095004 (2013).

11. Kofman, A. & Kurizki, G. Universal dynamical control of quantum mechanicaldecay: Modulation of the coupling to the continuum. Phys. Rev. Lett. 87,270405 (2001).

12. Uhrig, G. Keeping a quantum bit alive by optimized pi-pulse sequences.Phys. Rev. Lett. 98, 100504 (2007).

13. Cywinski, L., Lutchyn, R. M., Nave, C. P. & Sarma, S. D. How to enhancedephasing time in superconducting qubits. Phys. Rev. B 77, 174509 (2008).

14. Biercuk, M. J., Doherty, A. C. & Uys, H. Dynamical decoupling sequenceconstruction as a filter-design problem. J. Phys. B 44, 154002 (2011).

15. Biercuk, M. J. et al. Optimized dynamical decoupling in a model quantummemory. Nature 458, 996–1000 (2009).

16. Soare, A. et al. Experimental bath engineering for quantitative studies ofquantum control. Phys. Rev. A 89, 042329 (2014).

17. Vandersypen, L. M. K. & Chuang, I. NMR techniques for quantum control andcomputation. Rev. Mod. Phys. 76, 1037–1069 (2004).

18. Merrill, J. T. & Brown, K. R. in Quantum Information and Computation forChemistry: Advances in Chemical Physics Vol. 154 (ed. Kais, S.) Ch. 10, 241–294(John Wiley, 2014).

19. Kabytayev, C. et al. Robustness of composite pulses to time-dependent controlnoise. Phys. Rev. A 90, 012316 (2014).

20. Haeberlen, U. &Waugh, J. S. Coherent averaging effects in magnetic resonance.Phys. Rev. 175, 453–467 (1968).

21. Khodjasteh, K. & Viola, L. Dynamically error-corrected gates for universalquantum computation. Phys. Rev. Lett. 102, 80501 (2009).

22. Wang, X. et al. Composite pulses for robust universal control of singlet–tripletqubits. Nature Commun. 3, 997 (2012).

23. Fauseweh, B., Pasini, S. & Uhrig, G. Frequency-modulated pulses for quantumbits coupled to time-dependent baths. Phys. Rev. A 85, 022310 (2012).

24. Ticozzi, F., Nishio, K. & Altafini, C. Stabilization of stochastic quantumdynamics via open and closed loop control. IEEE Trans. Auto. Control 58,74–85 (2013).

25. Beauchamp, K. G.Walsh Functions and their Applications (Academic, 1975).26. Hayes, D., Khodjasteh, K., Viola, L. & Biercuk, M. J. Reducing sequencing

complexity in dynamical quantum error suppression by Walsh modulation.Phys. Rev. A 84, 062323 (2011).

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33. Su, Z-K. & Jiang, S-J. Filter-design perspective applied to dynamical decouplingof a multi-qubit system. J. Phys. B 45, 025502 (2012).

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AcknowledgementsWe thank P. Fisk, M. Lawn, M. Wouters and B. Warrington for technical assistance andK. Brown, J. T. Merrill and L. Viola for useful discussions. This work partially supportedby the US Army Research Office under Contract Number W911NF-11-1-0068, theAustralian Research Council Centre of Excellence for Engineered Quantum SystemsCE110001013, the Office of the Director of National Intelligence (ODNI), IntelligenceAdvanced Research Projects Activity (IARPA), through the Army Research Office, andthe Lockheed Martin Corporation. X.Z. acknowledges useful discussions with G. L. Longand support from the National Natural Science Foundation of China (GrantsNo. 11175094 and No. 91221205) and the National Basic Research Program of China(No. 2011CB9216002).

Author contributionsA.S., H.B., D.H. and M.J.B. conceived and performed the experiments, built experimentalapparatus, contributed to data analysis and wrote the manuscript. T.J.G. conceived therelevant theoretical constructs. J.S., M.C.J. and X.Z. assisted with development of theexperimental system and data collection. J.J.M. assisted with data collection.

Additional informationSupplementary information is available in the online version of the paper. Reprints andpermissions information is available online at www.nature.com/reprints.Correspondence and requests for materials should be addressed to M.J.B.

Competing financial interestsThe authors declare no competing financial interests.

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© 2014 Macmillan Publishers Limited. All rights reserved

Supplementary material for: Experimental noise filtering by quantum control

A. Soare,1, ∗ H. Ball,1, ∗ D. Hayes,1, † J. Sastrawan,1 M. C. Jarratt,1

J. J. McLoughlin,1 X. Zhen,2 T. J. Green,1 and M. J. Biercuk1, ‡

1ARC Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, NSW 2006 AustraliaNational Measurement Institute, West Lindfield, NSW 2070 Australia

2Tsinghua University, Beijing, People’s Republic of China

Theoretical model

We consider a model quantum system consisting of anensemble of identically prepared noninteracting qubitsimmersed in a weakly interacting noise bath and drivenby an external control device. Working in the interactionpicture with respect to the qubit splitting ωa state trans-formations are represented as unitary rotations of theBloch vector. The generalized time-dependent Hamilto-nian is then written

H(t) = Hc(t) + H0(t) (1)

where Hc(t) describes perfect control of the qubit state,e.g. via an ideal external driving field, and thenoise Hamiltonian H0(t) captures undesirable interac-tions with a (universal) noise bath.

The specific forms taken by Hc(t) and H0(t) in thiswork are given in the sections below, where we treatboth dephasing (detuning) and amplitude-damping (co-herent relaxation) noise processes. We will begin withthis model to craft time-dependent noise filters, and de-tail this method in the following sections.

Defining the control space

Representing the qubit state on the Bloch sphere,state manipulation maps to a rotation of the Bloch vec-tor in R3 and described by the unitary U(θ, σr) :=exp

(−iσ·rθ2

), reflecting the homeomorphism between

SU(2) and SO(3). In effect, the spin operator σr := r ·σgenerates a rotation though an angle θ about an axis de-fined by the unit vector r ∈ R3. For our purpose controltakes the form of a composite pulse sequence consisting ofn such unitaries executed over a time period [0, τ ], withthe lth pulse in the sequence written

Pl := U(θl, σφl) = exp

[− i

σφl

2

∫ tl

tl−1

Ωl(t)dt]

(2)

σφl:= cos(φl)σx + sin(φl)σy. (3)

Here Ωl(t) is the Rabi rate with arbitrary amplitude enve-lope in a single pulse, τl = tl − tl−1 is the pulse duration,and the spin operator σφl

, parametrized by φl ∈ [0, 2π],

generates a rotation θl =∫ tl

tl−1Ωl(t)dt of the Bloch vector

about an axis rl ≡ (cos(φl), sin(φl), 0) in the xy-plane1.This sequence of control unitaries implies a natural par-tition of the total sequence duration τ into n subintervalsIl = [tl−1, tl], l ∈ 1, n, such that the lth pulse has du-ration τl = tl − tl−1 with tl−1 and tl the start and endtimes respectively. Here t0 ≡ 0 and tn ≡ τ .

The control Hamiltonian associated with this compos-ite pulse sequence takes the form

Hc(t) =

n∑

l=1

G(l)(t)Ωl(t)

2σφl

(4)

where the function G(l)(t) is 1 if t ∈ Il and zero oth-erwise. The sequence of n triples (θl, τl, φl)n

l=1 com-pletely characterizes the net effect of the applied control(Pl = Pl(θl, Ωl(t), τl, φl)) at the end of successive pulseapplications. We define the n × 3 composite pulse se-quence matrix

Γn =

θl τl φl

P1 θ1 τ1 φ1

P2 θ2 τ2 φ2

......

......

Pn θn τn φn

(5)

to compactly describe any arbitrary n-pulse control se-quence. The entire space of such control forms there-fore corresponds to an infinite set of Γn matrices rangingcontinously over all possible values taken by the controlparameters. We denote this set by Cn and refer to it asthe n-pulse control space. Written formally

Cn :=Γn

∣∣θl, τl > 0, φl ∈ [0, 2π], l ∈ 1, ..., n, Σnl τl = τ

.

Noise bath model

We consider semi-classical time-dependent dephasing(detuning) and amplitude damping (relaxation) pro-cesses, captured respectively through the appearance ofstochastic rotations about σz and σφ := cos(φ)σx +

1 For a resonantly driven qubit φl is the phase of the driving fieldand Ωl is linearly proportional to the driving amplitude.

Supplementary material for: Experimental noise filtering by quantum control

A. Soare,1, ∗ H. Ball,1, ∗ D. Hayes,1, † J. Sastrawan,1 M. C. Jarratt,1

J. J. McLoughlin,1 X. Zhen,2 T. J. Green,1 and M. J. Biercuk1, ‡

1ARC Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, NSW 2006 AustraliaNational Measurement Institute, West Lindfield, NSW 2070 Australia

2Tsinghua University, Beijing, People’s Republic of China

Theoretical model

We consider a model quantum system consisting of anensemble of identically prepared noninteracting qubitsimmersed in a weakly interacting noise bath and drivenby an external control device. Working in the interactionpicture with respect to the qubit splitting ωa state trans-formations are represented as unitary rotations of theBloch vector. The generalized time-dependent Hamilto-nian is then written

H(t) = Hc(t) + H0(t) (1)

where Hc(t) describes perfect control of the qubit state,e.g. via an ideal external driving field, and thenoise Hamiltonian H0(t) captures undesirable interac-tions with a (universal) noise bath.

The specific forms taken by Hc(t) and H0(t) in thiswork are given in the sections below, where we treatboth dephasing (detuning) and amplitude-damping (co-herent relaxation) noise processes. We will begin withthis model to craft time-dependent noise filters, and de-tail this method in the following sections.

Defining the control space

Representing the qubit state on the Bloch sphere,state manipulation maps to a rotation of the Bloch vec-tor in R3 and described by the unitary U(θ, σr) :=exp

(−iσ·rθ2

), reflecting the homeomorphism between

SU(2) and SO(3). In effect, the spin operator σr := r ·σgenerates a rotation though an angle θ about an axis de-fined by the unit vector r ∈ R3. For our purpose controltakes the form of a composite pulse sequence consisting ofn such unitaries executed over a time period [0, τ ], withthe lth pulse in the sequence written

Pl := U(θl, σφl) = exp

[− i

σφl

2

∫ tl

tl−1

Ωl(t)dt]

(2)

σφl:= cos(φl)σx + sin(φl)σy. (3)

Here Ωl(t) is the Rabi rate with arbitrary amplitude enve-lope in a single pulse, τl = tl − tl−1 is the pulse duration,and the spin operator σφl

, parametrized by φl ∈ [0, 2π],

generates a rotation θl =∫ tl

tl−1Ωl(t)dt of the Bloch vector

about an axis rl ≡ (cos(φl), sin(φl), 0) in the xy-plane1.This sequence of control unitaries implies a natural par-tition of the total sequence duration τ into n subintervalsIl = [tl−1, tl], l ∈ 1, n, such that the lth pulse has du-ration τl = tl − tl−1 with tl−1 and tl the start and endtimes respectively. Here t0 ≡ 0 and tn ≡ τ .

The control Hamiltonian associated with this compos-ite pulse sequence takes the form

Hc(t) =

n∑

l=1

G(l)(t)Ωl(t)

2σφl

(4)

where the function G(l)(t) is 1 if t ∈ Il and zero oth-erwise. The sequence of n triples (θl, τl, φl)n

l=1 com-pletely characterizes the net effect of the applied control(Pl = Pl(θl, Ωl(t), τl, φl)) at the end of successive pulseapplications. We define the n × 3 composite pulse se-quence matrix

Γn =

θl τl φl

P1 θ1 τ1 φ1

P2 θ2 τ2 φ2

......

......

Pn θn τn φn

(5)

to compactly describe any arbitrary n-pulse control se-quence. The entire space of such control forms there-fore corresponds to an infinite set of Γn matrices rangingcontinously over all possible values taken by the controlparameters. We denote this set by Cn and refer to it asthe n-pulse control space. Written formally

Cn :=Γn

∣∣θl, τl > 0, φl ∈ [0, 2π], l ∈ 1, ..., n, Σnl τl = τ

.

Noise bath model

We consider semi-classical time-dependent dephasing(detuning) and amplitude damping (relaxation) pro-cesses, captured respectively through the appearance ofstochastic rotations about σz and σφ := cos(φ)σx +

1 For a resonantly driven qubit φl is the phase of the driving fieldand Ωl is linearly proportional to the driving amplitude.

Experimental noise filtering by quantum control

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHYS3115

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© 2014 Macmillan Publishers Limited. All rights reserved.

2

sin(φ)σy. The universal noise Hamiltonian then takes

the form H0(t) = H(z)0 (t) + H

(Ω)0 (t) where H

(z)0 (t) and

H(Ω)0 (t) denote noise interactions associated with the de-

phasing and amplitude noise quadratures respectively.The dephasing noise Hamiltonian is then given by

H(z)0 (t) = βz(t)σz (6)

where βz(t) is a classical stochastic process. During eachpulse we also make the substitution Ωl(t) −→ (Ωl(t) +

βΩ

(t)Ω(max)l ) where β

Ω(t) describes captures a (multi-

plicative) stochastic noise scaled by the maximum Rabirate in a pulse segment in the amplitude of the drivingfield. Thus the amplitude noise Hamiltonian takes theform

H(Ω)0 = β

Ω(t)

n∑

l=1

G(l)(t)Ω

(max)l

2σφl

(7)

and generates errors in intended rotation angle coaxialwith the target rotation axis σφl

.

Calculating operational fidelity & the first-orderapproximation

In the absence of noise, state evolution is determinedby iUc(t) = Hc(t)Uc(t) with Uc(t) describing a target op-eration. Including the effects of noise, however, the ac-tual evolution operator U(t) satisfies iU(t) = (Hc(t) +H0(t))U(t). A measure of the average gate fidelityFav(τ) = 1

4 〈|Tr(U†c (τ)U(τ))|2〉 is then obtained usingthe Hilber-Schmidt inner product, effectively measuringthe overlap between the intended and realized opera-tors. These evolution dynamics however are challeng-ing to compute due to sequential application of noncom-muting, time-dependent operations giving rise to bothdephasing and depolarization errors. Our approach fol-lows the method developed by Green et al. [11], usingaverage Hamiltonian theory and the generalized filter-transfer function formalism to obtain a first order ap-proximation for gate infidelity.

Magnus Expansion

We write the total evolution operator U(t) =Uc(t)U(t), where the error propagator U(t) satisfies the

Schrodinger equation i ˙U(t) = H0(t)U(t) in a frame co-rotating with the control defined by the Toggling frameHamiltonian H0(t) := U†c (t)H0(t)Uc(t). Thus, in theevent that U(τ) = I, the realized evolution operatorU(τ) approaches the target operation Uc(τ) and errorsdo not affect the gate. This can be systematized by writ-ing U(τ) = exp[−iΦ(τ)] in terms of a time-independent

effective error operator Φ(τ) =∑∞µ=1 Φµ(τ) with Magnus

expansion terms

Φ1(τ) =

∫ τ

0

dtH0(t)

Φ2(τ) = − i2

∫ τ

0

dt1

∫ t1

0

dt2[H0(t1), H0(t2)

]

Φ3(τ) =1

6

∫ τ

0

dt1

∫ t1

0

dt2

∫ t2

0

dt3

[H0(t1),

[H0(t2), H0(t3)

]]

+[H0(t3),

[H0(t2), H0(t1)

]]

...

generally taking the form of time-ordered integrals overnested commutators in H0(t).

One may then use vector identities (given Unitary pro-cesses) to re-express the so-called error vector in the tog-gling frame, again in an infinite power-series expansion

a(τ) =∞∑

µ

aµ(τ). (8)

This is a reexpression of the Magnus expansion errorterms in the language of our control Hamiltonian. Fordetails of this derivation and the definition of all termssee [11].

Our theoretical predictions based on the filter-transferfunction formalism involve an approximation to the traceor gate fidelity defined by

Fav(τ) =1

4〈|Tr(U(τ))|2〉 =

1

4〈|Tr(e−ia(τ)σ|2〉 (9)

where a(τ) =∑∞µ aµ(τ) is the error vector given in

terms of the Magnus expansion. Following the methoddeveloped in Ref. [11] and expanding the exponential inEq. 9 we obtain

Fav(τ) =1

2[〈cos(2a)〉+ 1] (10)

=1

2

[1 +

∞∑

m=0

(−1)m22m

(2m)!〈a2〉m

](11)

〈a2〉 =∑

µν

[〈a21〉+ 〈a2

2〉+ (12)

+ 2(〈a1aT2 〉+ 〈a1a

T3 〉+ 〈a2a

T3 〉+ ...)] (13)

with a2 ≡ a(τ)a(τ)T the norm square of the error vec-tor. The full expansion rapidly becomes too complexto write explicitely, however it is convenient to writeFav =

∑∞k=0O(ξ2k) where ξ is the smallness parameter

quantifying the RMS deviation of the noise integratedover the sequence duration, ξ ≡ ∆βτ/2 [12]. For thisseries to formally converge we require ξ2 < 1 (see maintext and Fig. 1).

Here odd powers of ξ are omitted since these in-volve ensemble averages over odd powers of the noise

© 2014 Macmillan Publishers Limited. All rights reserved.

3

strength and vanish under our assumption of zero-mean,Guassian-distributed random variables. Writing theO(ξ0),O(ξ2),O(ξ4) classes explicitely we have

Fav =1− 〈a21〉 −

[〈a2

2〉+ 2〈a1aT3 〉 −

〈a41〉

3

]+

∞∑

k=3

O(ξ2k)

(14)

Immediately we see that there is a collection of termswith equal magnitude arising from different orders of theMagnus expansion (e.g. a2

2 vs a41). The individual terms

in the series expansion of the fidelity rely on time-domaincorrelation and cross-correlation functions and convolu-tion with a multidimensional control matrix capturingthe effect of the control operations.

The fidelity is thus expressed explicitly in terms of

noise correlations and the control matrix. For instance,

〈a21〉

=∑

i,j=x,y,z

∫ τ

0

dt2

∫ τ

0

dt1〈βi(t1)βj(t2)〉Ri(t1)RTj (t2)

=∑

i,j,k=x,y,z

∫ τ

0

dt2

∫ τ

0

dt1〈βi(t1)βj(t2)〉Rik(t1)Rjk(t2)

(15)

contains all two-point noise cross-correlation functions〈βi(t1)βj(t2)〉, for i, j ∈ x, y, z, Higher-order terms con-tain multipoint correlation functions (this is determinedby the sum of subscript indices, as they indicate theexpansion-order of the error vector).

We rewrite these terms in the frequency domain, defin-ing the Fourier transform Si1...in(ω1, ..., ωn) of an n-pointcross-correlation function via

〈βi1(t1)βi2(t2)...βin(tn)〉 ≡ 1

(2π)n

∫dω1...

∫dωnSi1...in(ω1, ..., ωn)ei(ω1t1+...+ωntn) (16)

The fidelity above can then be rewritten as

Fav = 1−∞∑

n=2

1

(2π)n

∑

i1...in

∫dω1...

∫dωnSi1...in(ω1, ..., ωn)Ri1...in(ω1, ..., ωn)

(17)

where Ri1...in(ω1, ..., ωn) is determined solely by the control matrix and increases in complexity at higher order.Explicit expressions for terms to arbitrary order are found in [11].

First-order fidelity approximation

Here we briefly explain the choice of fidelity metricused in the figures of the main text to produce the theorycurves against which our experimental data is compared.Experimental fidelities are determined by measuring thebrightness of the ion cloud after completing the controlsequence, effectively yielding a projective measurementonto the |↑〉 state. We denote this metric by P↑(τ) ∈ [ 1

2 , 1]and refer to it as the state fidelity, with lower and upperbounds corresponding to complete decoherence and per-fect fidelity respectively.

If the noise is sufficiently weak (ξ2 1) we may trun-cate the series expansion for fidelity after the O(ξ2) termyielding the approximation

FO(1) = 1− 〈a21〉. (18)

Here the term which dominates the measured infidelity is〈a2

1〉 := 〈a1(τ)aT1 (τ)〉, defined as the ensemble averagedmodulus square of the first order error vector a1(τ). As-suming wide sense stationarity, independence and zeromean of both noise fields βz(t) and β

Ω(t) we may derive

a spectral representation of 〈a21〉 of the form

〈a21〉 =

1

2π

∫ ∞

−∞

dω

ω2Sz(ω)Fz(ω) +

1

2π

∫ ∞

−∞

dω′

ω′2SΩ(ω′)FΩ(ω′).

(19)

Here Sz(ω) and SΩ(ω) denote the dephasing and ampli-tude noise PSDs. The dephasing Fz(ω) and amplitudeF

Ω(ω) filter functions, on the other hand, capture the

spectral response of the control sequence and are com-pletely defined as functions of the control sequence.

As the integrated noise content increases, however,higher-order error contributions must be included; ne-glecting to do leads to the unphysical result that FO(1) ≤0 when ξ2 ≥ 1. Although computation of all higher-ordercontributions is challenging we may gain some insightinto the full expansion by considering terms of the form〈a2m

1 〉 ≡ 〈a21〉m in each class O(ξ2m). This collection of

terms is obtained by setting a2 → a21 in Eq. 10, effec-

tively including only the first-order Magnus expansion

© 2014 Macmillan Publishers Limited. All rights reserved.

4

term in the expansion for Fidelity, yielding

F ′O(1)(τ) =1

2

[1 +

∞∑

m=0

(−1)m22m

(2m)!〈a2

1〉m]

(20)

= 1− 〈a21〉+

〈a41〉

3− 2〈a6

1〉45

+ ... (21)

The oscillating sign of these terms is characteristic ofthe higher-order classes in converging to the true express-sion. To overcome the unphysicality of FO(1) as the noisecontent increases we employ a metric Fχ with the phys-ically reasonable properties that

Fav ≈ FO(1) ≈ Fχ, 〈a21〉 1 (22)

FO(1) ≤ F ′O(1) ≤ Fav,Fχ, 〈a21〉 ≈ 1 (23)

Fχ → 1/2→ P↑(τ), 〈a21〉 1 (24)

We may satisfy these conditions by noticing the qualita-tive resemblance between Eqs. 20 and 21 and the expan-sion for a simple exponential

1− 〈a21〉+ 〈a4

1〉 −2〈a6

1〉3

+ ... =1

2

[1 +

∞∑

m=0

(−1)m2m

m!〈a2

1〉m]

(25)

=1

2

[1 +

∞∑

k=0

(−χ(τ))m

m!

]

(26)

where we have defined χ(τ) ≡ 2〈a21〉. Hence we use the

following metric in calculating fidelities to be comparedwith experimental data

Fχ =1

2

1 + exp[−χ(τ)]

(27)

This approximation represents the first-order fidelityapproximation: it ignores higher-order cross correlationsin the noise arising from higher-order Magnus contribu-tions to the error vector, with diminishing overall magni-tude (as given by the smallness parameter), but incorpo-rates an approximation to higher-order terms importantas the total noise-induced infidelity grows. We work inthis limit throughout this manuscript.

Breakdown of the first-order fidelity approximation

As described above, the first-order fidelity ignoreshigher-order terms expressed as nested-integrals overcross-correlations between noise along different direc-tions, assuming weak noise. As these contributions togate infidelity grow in importance (for instance with α)we expect the filter-transfer-function fidelity calculationsto underestimate measured error in cases where the con-trol has filtered the noise to leading order.

We measure the probability that a πx-pulse drivesqubit population from the dark state to the bright state,|0〉 → |1〉, as a function of the high-frequency cutoff,ωc, of a white dephasing bath (Fig. 1b). As the high-frequency cutoff of the noise is increased and fluctua-tions fast compared to the control are added to the noisepower spectrum, Sz(ω), errors accumulate reducing themeasured fidelity. The value of ωc at which the fidelitydrops from near unity decreases as a function of the noisestrength, parametrized by α. In all cases for the primi-tive π pulse the fidelity calculated using the filter transferfunction matches the measured data well with no free pa-rameters.

Performance is notably different when studying thefour-segment WAMF π-pulse, W1, indicated in Fig. 2b-cof the main text. The WAMF construction provides first-order filtering of time-dependent noise (red line in Fig.2c) (effectively cancelling terms proportional to 〈a2

1〉), butdoes not provide suppression of higher-order terms in theMagnus expansion for fidelity which grow in importancewith noise strength. Unlike data for the primitive gate,as the noise strength increases we observe a growing di-vergence between the measured fidelity and the fidelitycalculated using the filter-transfer functions introducedabove assuming a first order approximation (Fig. 1b).

This phenomenon is not a function of total error mag-nitude, but instead occurs for ξ2 ≥ 1 (red lines, rightaxis), a proxy measure indicating that we are not for-mally able to truncate the series expansion for fidelity atfirst order and must consider higher-order error contri-butions [11], including Magnus terms above 〈a2

1〉. Thesemeasurements therefore reveal the efficacy of noise fil-tering and quantitatively demonstrate the bounds of thefirst-order fidelity approximation as breakdown routinelyoccurs near the predicted value ξ2 ≥ 1. Notably, whileformal convergence of this series requires ξ2 1, we findreasonable agreement between experiment and theory upto ξ2 ∼ 5 (Fig. 1c).

Time-domain filter order vs. Magnus order

We may formally indicate the functional dependenceof the filter function on the control sequence by writingF (τω) = F (τω; Γn). Noise filtering (and hence errorsuppression) corresponds to minimizing the area underthe filter transfer function in the spectral region wherethe noise PSD is non-negligible. We therefore define acost function over a user-defined frequency band whichmay take the form

A(Γn) :=

∫ ωc

ωL

dωF (τω; Γn) (28)

to diagnose the filtering effectiveness achieved by the con-trol sequence Γn; the smaller the integral A(Γn), themore effective the noise filtering in this band. Having

© 2014 Macmillan Publishers Limited. All rights reserved.

5

1.0

0.5

0.0

0.01 0.1 1

1.0

0.5

0.0

1.0

0.5

0.0

1.0

0.5

0.0

10-4 10-2 100 102

ω (2π/ τπ)

10-5 10-2 101

S (ω

) (ra

d/s)

ωc

0.01 0.1 1

0.011100

0.011100

0.011100

0.011100

πx Primitive

Cutoff Frequency, ωC (2π / τπ)

c

0.2 1 10ξ

2

1.00.80.60.4

α2White

a b

Sm

alln

ess

Par

amet

er, ξ

2α = 10

α = 20

α = 50

α = 200

d

Pro

babi

lity

Brig

ht

πx W1

FIG. 1. Experimental validation of the breakdown of the first-order approximation to fidelity. a)-b) Measurements of oper-ational fidelity with engineered noise for rotation |0〉 → |1〉as a function of dimensionless noise cutoff frequency. Prim-itive rotation (a) and four-segment WAMF modulated rota-tion, W1, (b). Decay to value 0.5 corresponds to full de-coherence. Each data point is the result of averaging over50 different noise realizations. Maximum Rabi rate is fixed;filter W1 is conducted over time 4τπ. Black dotted lines indi-cate the (smoothed) results of numerical integration of theSchroedinger equation, indicating that divergence betweendata and filter-function calculation is not due to experimentalartefacts. Red lines (right axis) give ξ2 employed in makingthe first-order filter-transfer function approximation. Verti-cal dashed lines indicate values of ωc beyond which ξ2 ≥ 1.c) Measurements of W1 fidelity revealing a growing break-down in agreement between filter-transfer function fidelityprediction and experimental data beyond ξ2 & 1 , taken forωc/2π = 1.7%τ−1

π . d) Schematic representation of the whitenoise power spectrum employed.

defined control sequences as continuous elements of thecorresponding control space, for a given n we may inprinciple construct a variational procedure on Cn to de-rive “values” of Γn satisfying a given cost function.

The filter transfer function may be approximated by apolynomial expression F (ωτ) ∝ (ωτ)2p for some p nearω ≈ 0. As p increases the integral in 28, and hence theinfidelity, decreases: the noise in the time domain is thensaid to be filtered to order p− 1.

Equivalently stated, a control sequence Γn ∈ Cn filterstime-dependent noise to order p− 1 if Γn is a concurrentzero of the first p−1 coefficients in the Taylor expansion2

2 This procedure is valid for frequencies sufficiently lower than 1/τ(the inverse of the total sequence duration).

of the filter transfer function about ω = 0.

F (ωτ ; Γn) =∞∑

k=1

C2k(Γn)(ωτ)2k. (29)

The dependence of the expansion coefficients on ourcontrol parameters Γn has been made explicit, and weinclude only even powers of ωτ due to the evennessof the filter transfer function. In this case A(Γn) ≈C2p(Γn) (τωc)2p+1

2p+1 and the condition that

A(Γn)

C2p(Γn)= O

( (τωc)2p+1

2p+ 1

)(30)

therefore implies the control sequence Γn filters noise toorder p− 1. This effect is visualized through the slope ofthe filter transfer function in the stopband on a log-logplot (Fig. 2c, main text). A high-order filter has a higherslope in this region, indicating improved suppression oftime-dependent noise.

General filter design focuses on a band of interest, per-mitting spectral response to diverge outside of the spec-tral region of interest - for instance electrical filters inthe microwave may appear transparent in the THz or Hz.Therefore, in addition to the asymptotic, zero-frequencyfilter order (p − 1), we introduce a more general metriccapable of describing filter performance over an arbitraryspectral band. The local filter order (p∗ − 1) establishesthat the filter-transfer function is well approximated byFi ∝ (ωτ)2p∗ over the band [ωL, ωc]. It is this morenarrowly defined metric that is used in most practicalfilter-design tasks, including those undertaken above.

The performance of filter-order (p − 1) or local filterorder (p∗ − 1) for time-dependent noise described abovemust be distinguished from the order of error suppres-sion for quasistatic errors in the Magnus expansion. Thelatter measure is typically used in NMR literature to thepulse sequences designed to compensate for quasistaticerrors. In this regime the time dependence of the dephas-ing (amplitude) noise fields reduces to constants βz (βΩ)

and the Magnus expansion terms Φ(DC)µ are evaluated

strictly as time integrals over ideal control operationsscaled by powers of the offset magnitude βz

µ (βΩµ). A

pulse sequence for which Φ(DC)1 = ... = Φ

(DC)µ−1 = 0 is then

said to compensate offset errors to order µ−1. In this case

the total error operator satisfies Φ(DC)(τ) = O(Φ(DC)µ )

and is dominated by the residual error proportional tothe µth power in the offset magnitude.

High-order error suppression in the Magnus expansiondoes not imply high-order time-domain noise filtering.Table I reveals the importance of not conflating thesetwo measures when assessing the performance of a controlsequence against static vs stochastic errors. The upperpanel compares the two performance measures for somewell-known phase-modulated NMR sequences, the nam-ing conventions for which are consistent with the review

© 2014 Macmillan Publishers Limited. All rights reserved.

6

Amplitude Errors Dephasing Errorsµ− 1 p− 1 µ− 1 p− 1

SK1 1 1 0 0P2 2 1 0 0B2 2 1 0 0C1 0 0 1 1

C2 (π) 0 0 2 0

W1 0 0 1 1W2 0 0 1 2

UWMG1,SK1 1 1 1 1

TABLE I. Comparision between suppressing stochastic errorsto order p− 1 (filter order) and compensating for static offseterrors to order µ−1 (Magnus order). Naming conventions forNMR sequences in top panel are consistent with the reviewarticle by Merrill and Brown [13]. The final entry correspondsto a concatenated construction described below.

by Merrill and Brown[13]. For completeness, in the lowerpanel we also make the comparison for the novel controlsequences derived in this paper.

Later we will return to the question of time-domainfilter order and introduce a set of analytic design rulesfor filter construction based on the characteristics of ourselected basis functions - the Walsh functions.

Walsh basis functions

We impose physically motivated constraints on theform of Γn in order to reduce the search to a manageablesubspace of Cn, and elect to synthesize control sequencesfrom the Walsh basis functions. The set of Walsh func-tions wk : [0, 1] → ±1, k ∈ N form an orthonormal-complete family of binary-valued square waves defined onthe unit interval and are the digital analogues of the sinesand cosines in Fourier analysis. Since their formulationin the first half of the twentieth century, Walsh functionshave played an important role in scientific and engineer-ing applications. Their development and utilzation hasbeen strongly influenced by parallel developments in digi-tal electronics and computer science since the 1960s, withWalsh-type transforms replacing Fourier transforms in arange of engineering applications such as communication,signal processing, image processing, pattern recognition,noise filtering and so forth[14].

We summarize the relevant mathematical details ofthe Walsh basis, outlining two equivalent representations,Paley ordering and the Hadamard representation, both ofwhich are useful to understand the Walsh control space.

Paley ordering

The Walsh functions are aperiodic and hence do notadmit to the unique ordering according to increasing fre-quency characteristic of the sinusoids in the Fourier basis.

302826242220181614121086420

Pal

ey O

rder

ed W

alsh

Fun

ctio

ns

0 5 10 15 20 25 30

Time Bins

312927252321191715131197531

FIG. 2. The first 32 Walsh functions in Paley ordering.

A number of differnt orderings exist with associated def-initions of the basis elements. We employ the Paley or-dering in which basis functions are generated from prod-ucts of Rademacher functions, a family of square wavesdefined by

Rj(x) := sgn[

sin(2jπx)], x ∈ [0, 1], j ≥ 0,

(31)

switching between ±1 at rate 2j . The Walsh function ofPaley order k, denoted PALk(x), is then defined by

PALk(x) =m∏

j=1

Rj(x)bj (32)

where (bm, bm−1, ..., b1)2 is the binary representation ofk. That is, k = bm2m−1 + bm−12m−2 + ...+ b120, wherebm ≡ 1 is defines the most significant binary digit. HenceRj(x) is a factor of PALk(x) whenever bj is a nonzerobinary digits of k. The total number of nonzero bj ’s in kdefine the Hamming wieght, denoted by r. We write m(k)and r(k) when it is desirable to emphasize that both mand r are functions of k. For illustration, the first 32Walsh functions in the Paley ordering are shown in Fig.2.

Hadamard representation

For our purposes we require an expression for thepiecewise-constant structure of an arbitrary superposi-tion of Walsh functions, and therefore desire a prioriknowledge of the locations of their various zero cross-ings. A general expression, however, is difficult dueto the aperiodicity of the Walsh functions. It is con-venient instead to use the Hadamard representation in

© 2014 Macmillan Publishers Limited. All rights reserved.

7

which any continuously defined basis member PALk(x)projects completely onto a digital vector in R2n

providedm(k) ≤ n, which is true for the 2n Paley orders in theset k ∈ 0, 1, ..., 2n − 1. Since these vectors have di-mension 2n and inherit the orthogonality of the PALk(x)they therefore form a discrete Walsh basis spanning R2n

.Such a projection is clearly possible since the fastest mod-ulation rate in PALk(x) derives from the periodicity ofRm(k)(x), which switches sign 2m(k) times over x ∈ [0, 1].The projection then involves partitioning the domain into2n bins and asociating the value of PALk(x) in the jth

bin to the jth element P(k)j ∈ ±1 of the discrete digital

vector

P(k)2n =

[P

(k)1 , P

(k)2 , ... P

(k)2n

]. (33)

Using the so-called Sylvester construction [15], the 2n-dimensional Hadamard matrix H2n is generated recur-sively by

H2n =

[H2n−1 H2n−1

H2n−1 −H2n−1

]= S⊗n (34)

S =

[1 11 −1

], H1 = 1 (35)

where S is the Sylvester matrix, and ⊗n denotes n ≥ 1applications of the Kronecker product. In this construc-

tion P(k)2n defines the i(k) = 1 +

∑m(k)j=1 bj2

n−j column(row) of H2n . The orthogonality of the Walsh basis isthereby reflected in the property that H2nHT

2n = 2nI,implying the orthogonality of the Hadamard matrices.

This representation is particularly useful for efficientlyconstructing Walsh-synthesized waveforms. Consider anarbitrary function f(x) =

∑Nk=0XkPALk(x) synthesized

in the Walsh basis where N sets the highest (Paley) or-dered function in the construction. Then, from the abovediscussion, all Walsh functions in this synthesis projectedonto a Hadamard matrix of dimension ≥ 2m(N), withM = 2m(N) giving the minimal sufficient dimension. Adiscrete representation of the function f(x) therefore ex-ists as a projection onto the column space of HM bywriting

f = HMX. (36)

The column vector X =[X1, X2, ... XM

]Tcontains

the reordered Paley amplitudes Xk reoardered under thechange of basis map i(k) specified b

Xi(k) =

Xk for 0 ≤ k ≤ N0 for N < k <M . (37)

The vector f =[f1, f2, ... fM

]Tso generated then

represents the piecewise constant structure of f(x), withfj giving the value taken by f(x) on the jth of M equalsubintervals partitioning x ∈ [0, 1].

Walsh synthesis for amplitude modulated filters

Any square integrable function f(x) on the interval[0, 1] has a unique spectral decomposition in the Walshbasis

f(x) =

∞∑

k=0

Xkwk(x) ⇐⇒ Xk :=

∫ 1

0

f(x)wk(x)dx.

We consider a control regime referred to as single-axisamplitude-modulation defined by

Γn ∼= (τl, θl)nl=1, φl = φ0 ∀l ∈ 1, ..., n (38)

where the waveform structuring the total angle sweptout in each pulse segment, θl, is based on a linear su-perposition of well-defined square waves known as Walshfunctions. We refer to these sequences as Walsh ampli-tude modulated filters (WAMFs), and speak of searchingover the Walsh-modulated control subspace.

Using the above framework, we may efficiently con-struct arbitrary WAMFs for arbitrary pulse-segment en-velopes. We define a time-varying Rabi rate Ωl(t), l ∈1, ...,M over the time period t ∈ [tl−1, tl] of each seg-ment, l ∈ 1, ...,M, t ∈ [tl−1, tl]. Each segment hasduration τl = τ/M and generates a total rotation anglefor each segment, θl given by the integral

θl =

∫ tl

tl−1

Ωl(t)dt (39)

We now treat the rotation angles θl as parameters bywhich to optimize filter performance. For efficient filterconstruction, however, it is convenient to instead trans-form this optimization over θl to an optimization overa Walsh spectrum. This is achieved by writingθl =θl(X0, X1, ..., XN ) with the dependence on the Walshspectra defined by the Hadamard-matrix equation

~θ =(θ1, , θ2, ..., θM

)T= (τ/M)HMX. (40)

Defined in this way, the M-segment arbitrary-envelopeconstruction achieves total gate-rotation angle Θ =∑nl=1 θl = X0τ , completely determined by the spectral

amplitude of PAL0. All symmetry-based design rulescarry over, regardless of the modulation envelope for anindividual pulse segment.

Square pulses

The special case of square pulse segments is treatedhere as it allows a reduction in synthesis complexity andis compatible with many experimental systems. We mayreplace the time-dependent Rabi rate Ω(t) over a sin-gle segment with a piecewise-constant (over a single-segment) construction used in Walsh synthesis over a

© 2014 Macmillan Publishers Limited. All rights reserved.

8

complete pulse sequence

Ω(t) =

N∑

k=0

XkPALk(t/τ), t ∈ [0, τ ]. (41)

This permits synthesis over the Rabi rate per segmentrather than the total rotation angle, which is often sim-pler in experimental settings. Substituting Ω(t/τ) forf(x) in Eq. 36 we obtain Ω = HMX. The vector

Ω =[

Ω1, Ω2, ... ΩM]T

thus defines a sequence ofmodulated Rabi rates each functionally dependent on theWalsh amplitudes

Ωl = Ωl(X0, X1, ..., XN ), l ∈ 1, ...,M. (42)

The WAMF is then defined by explicitly writing Ω as anadditional column leading the representation of Eq. 5,yielding the form

ΓM =

Ωl θl τl φl

P1 Ω1Ω1τM

τM φ0

P2 Ω2Ω2τM

τM φ0

......

......

...

PM ΩMΩMτM

τM φ0

(43)

Here the degree of freedom associated with τl has ap-parently been removed. This reflects the fact that thechoice of τl has been transformed into the choice of Walshbasis functions in the synthesis, each contributing itscharacteristic temporal profile. The remaining degreesof freedom reside in functional dependence of Ωl on theWalsh spectrum and our variational search is thus limitedto the subspace of CM effectively spanned by X.

Negative Walsh spectral amplitudes may produce neg-ative valued Rabi rates under a linear superposition. Fora pulse of the form Pl = exp

[− iΩlτlσφl

/2]

the nega-tive sign of Ωl may be absorbed into the spin operatorφl corresponds physically to the application of a π phaseshift int he diriving field. This follows from the fact thatσφ+π = −σφ, which is clear from the definition of ourspin operator in Eq. 3. Thus including negative-valuedWalsh spectral amplitudes generally produces single axiscontrol only up to a sign change.

Gaussian pulse envelopes

The square-envelope pulses studied experimentally inthe main text are easy to generate in our experimentalsystem but may prove difficult in other settings whereabrupt amplitude shifts at timestep-edges produce sig-nificant pulse distortion. Here we show that achiev-ing Walsh synthesized filters using a common Gaussianpulse envelope yields comparable results with a simplere-optimization of Walsh-synthesis coefficients.

The square amplitude-modulated waveform is here re-placed with a smoothly varying pulse envelope in eachsegment, each associated with a specific rotation angleθl subject to optimization. We assume a Gaussian pro-file Gl(t;µl, σl) defined on t ∈ [tl−1, tl] with mean µl andstandard deviation σl. Specifically, we construct

Gl(t;µl, σl) =θl

Clσl√

2πexp

[− (t− µl)2

2σ2l

](44)

µl =tl−1 + tl

2(45)

σl = gτ/M (46)

with µl the segment midpoint and σl expressed as a mul-tiple g of the segment duration. The normalizing factor

Cl :=

∫ tl

tl−1

1

σl√

2πexp

[− (t− µl)2

2σ2l

]dt (47)

is included to ensure the total rotation implementedby the Gaussian pulse in the lth segment is given by∫ tl−1

tlGl(t;µl, σl)dt = θl. We now impose the same struc-

ture on the segment rotations θl as presented above inEq. 40. Defined in this way, the M-segment Gaussian-pulse sequence shares with the square WAMF construc-tion the property that the total gate rotation angleΘ =

∑nl=1 θl = X0τ is completely determined by the

spectral amplitude of PAL0.We may therefore construct a Gaussian-pulse varia-

tion on any candidate WAMF such that, having setg to some value relevant to the control hardware, thesmooth pulse sequence remains strictly parametrized inthe Walsh spectrum X. In particular, filter optimiza-tion proceeds in the same manner as for ordinary Walsh-modulated control by minimizing the cost function withrespect to the Walsh spectrum.

Analytic design rules

An advantage of Walsh synthesis is that the well-defined spectral properties and symmetries of the Walshfunctions may be employed to further restrict the searchspace available for filter construction.

First, in practice the achievable filter order over theentire stopband is limited by the number of constituentcontrol operations; one may achieve higher p at the costof higher n. The maximum achievable value of p for agiven filter is set by the power-law expansion of the fil-ter for the single Walsh function with the highest Paleyorder for a given n. As has been shown previously, allWalsh functions with given Hamming weight of the Pa-ley order have the same power-law expansion near zerofrequency [16]. Therefore, in principle, every doubling ofn increases the maximum achievable time-domain filterorder by one. The Walsh functions highlighted in red in

© 2014 Macmillan Publishers Limited. All rights reserved.

9

Fig. 2 represent those with the highest Paley-order Ham-ming weight for a given n. Nonetheless we find that ingeneral we are able to construct filters with higher orderthan prescribed over narrow regions in the stopband, asa result of Walsh synthesis (see the multiple slopes forthe blue line in Fig. 2c).

In filter construction we may further constrain the formof a candidate pulse sequence by imposing required phys-ical properties on the sequence, such as fixing the totalrotation angle of the Bloch vector in order to implementa target logic operation. In order to proceed we then par-tition the Walsh spectrum X ≡ (Xν ,Xρ) into spectralamplitude classes Xν and Xρ to be treated as varia-tional and fixed parameters respectively. Fixed parame-ters set the physical state transformation of interest whilethe remaining unconstrained components in Xν serve astuning parameters by which to minimize A(Xν ;Xρ).

The primary constraint in WAMF constructions is thatthe total rotation angle executed depends only on thevalue of X0, the zeroth order spectral component; it setsthe effective average Rabi rate for the WAMF. This canbe seen as follows. First observe all Walsh functions ofhigher than zeroth order are balanced in the sense that∫ 1

0PALk(x)dx = δ0k. For the control field defined by Eq.

41 the total gate rotation angle Θ =∫ τ

0Ω(t)dt then takes

the form

Θ =

∫ τ

0

N∑

k=1

XkPALk(t/τ)dt

= τN∑

k=1

Xk

∫ 1

0

PALk(x)dx

= τN∑

k=1

Xkδ0k = X0τ.

In this case the net gate rotation θ = Θ mod 2π is givenby

θ = X0τ mod 2π (48)

implying the necessary constraint on X0 in order toachieve a desired θ.

Next, we observe that the Walsh functions have dis-tinct parity, but that filter constructions mandate sym-metric constructions in order to enact a target operationand provide effective noise cancellation. The result isthat odd-parity Walsh functions may generally be ex-cluded from the variational search. While this is notnecessarily strictly required (multiple odd parity Walshfunctions may in principle be added with opposite signsto produce net symmetric constructions), it is convenientand effective to restrict the synthesis space to the so-called CAL subset of the Walsh functions.

Our reduced search problem may then be representedformally by replacing ΓM → (Xν ,Xρ) in Eq. 28 to

obtain

A(Xν ;Xρ) :=

∫ ωc

0

dωF (τω;Xν ,Xρ) (49)

with the variational search now restricted to the subspacespanned by Xν with the Xρ held constant.

First order WAMFs

As a first application of the above result, we derive afamily of nontrivial gates decoupled to first order againstdephasing noise by constructing a pulse sequence from

the synthesis θ(t) = τ4

(X0PAL0(t/τ) + X3PAL3(t/τ)

).

Note that θ(t) is only formally defined at the end ofpulse segments. That is, we set Xρ ≡ X0 and Xν ≡X3. In this case N = 3 and M = 4, so the min-imal pulse duration is τ/4. Using Eq. 37 we ob-

tain X =[X1, X2, X3, X4

]T=[X0, 0, 0, X3

]T,

yielding the minimal Hadamard representation

θ =τ

4

1 1 1 11 −1 1 −11 1 −1 −11 −1 −1 1

X0

00X3

=

τ

4

X0 +X3

X0 −X3

X0 −X3

X0 +X3

(50)

These sequences therefore span the control subspaceparametrized by (τ = 1)

WAMF O(1) =

Ωl θl τl φl

P1 X+X+

414 0

P2 X−X−4

14 0

P3 X−X−4

14 0

P4 X+X+

414 0

(51)

where we have defined X± = X0±X3. This choice is mo-tivated by the fact that Paley order k = 3 corresponds tothe lowest order non-constant Walsh function with evensymmetry about τ/2 (Fig. 2). Hence 50 is the sim-plest Walsh modulated form which includes the zerothorder Walsh function in the synthesis and which pos-sesses time-reversal symmetry about the sequence mid-point. The former property ensures a nontrivial gateangle is executed. The latter is chosen due to the ob-servation in dynamic decoupling literature that usingtime-symmetric building blocks often improves the per-formance of the sequence compared to sequences formedby time-asymmetric building blocks[17][18]. In this con-struction we have maintained a leading column of rabirates, Ωl, as would be appropriate for the square-pulseforms used in the main text.

© 2014 Macmillan Publishers Limited. All rights reserved.

10

Gaussian pulse construction

Adding to the results presented in the main text, con-structing the 4-segment filter W1 using square pulse seg-ments, we examine the Gaussian-pulse variation here.The cost function Az(X3;X0) =

∫ ωc

ωLdωFz(ωτ ;X3;X0)

may be computed by partitioning the time domain intoa large number Ns of subintervals on which the con-tinuous Gaussian envelope is treated as approximatelyconstant. Fig. 3 shows a two-dimensional represen-tation of Az(X3;X0) integrated over the interval ω ∈[10−9, 10−6]τ−1, with g = 1/6 and Ns = 100. The valueof Log10

[Az(X3;X0)

]is indicated by the color scale. To-

tal sequence length is normalized to τ = 1 in this data,so the total gate rotation angle Θ ≡ X0 is given directlyby the X0-axis. Regions in blue represent effective (first-order) filter constructions, where the cost function is min-imized.

We conclude useful filter construction using Gaussianpulses is a simple matter of re-optimization in the Walsh-synthesis framework. This is readily achieved using a

Ω (a

.u.)

-6

-4

-2

0

2

4

6

PA

L 0 A

mpl

itude

, X

0 (π)

-6 -4 -2 0 2 4 6 PAL3 Amplitude, X3 (π)

-22 -20

X0

2X3

Normalized Time

A(4)

1 2 3 4

FIG. 3. Construction of the first-order Walsh amplitude mod-ulated dephasing-suppressing filter using Gaussian-shapedpulse segments. a) Schematic representation of Walsh syn-thesis for a four segment gate of discrete Gaussian segments.Walsh synthesis determines the overall amplitude of Gaussianpulses with fixed duration and standard deviation, setting theeffective pulse area in each segment. b) Two-dimensional rep-resentation of the integral metric defining our target cost func-tion, A(Γ4) integrated over the stopband ω ∈ [10−9, 10−6]τ−1.Areas in blue minimize A(Γ4), representing effective filter con-structions. The X0 determines the net rotation enacted in agate while X3 determines the modulation depth, as repre-sented in a).

Nelder-Mead optimization of Az(X3;X0) for any partic-ular choice of g, ωL, ωc, X0 or Ns, in a manner preciselythe same as for square envelopes.

Universal Filters by Concatenation

Phase-modulated sequences robust against amplitudenoise may also be found in the Walsh basis, yieldingWalsh phase-modulated filters (WPMFs) analogous toWAMFs, and implementing arbitrary target rotationsθ. There are a variety of techniques to construct suchWPMFs, but we use analytic design rules in which a tar-get rotation is performed (with some error due to noise),and phase-modulated segments are added in order to pro-duce the net filtering effect. The simplest WPMF addstwo segments phase modulated according to Walsh func-tion PAL1 with coefficient X1, subject to the constraintthat one enacts the desired driven rotation by θ.

The result of this approach yields a WPMF that isidentical to the NMR sequence SK1, with value X1 =cos−1(−θ/4π) ≡ φ

SK1(θ) [13], as represented

Γ(SK1)3 =

Ωl θl τl φl

P1 Ω0 θ τθ 0

P2 Ω0 2π τ2π φSK1

P3 Ω0 2π τ2π −φSK1

(52)

Ω0 =θ + 4π

τ, τθ =

θ

Ω0, φ

SK1(θ) := cos−1

(− θ

4π

).

(53)

Note that the Walsh timing construction only holds inthe two correction steps represented above. Followinga similar route allows one to construct a sequence withmodulation given by PAL3 which is formally identicalto the three-segment (four-timestep) phase modulationgiven by gate P2 [13].

These WPMF sequences perform as first-order time-dependent noise filters, captured in the form of FΩ(τω),and noted in Table I (column 2). For example, filterfunctions for the WPMF that is equivalent to SK1 areshown in Fig. 4b, revealing first-order filtering of ampli-tude noise, but not dephasing noise.

We may now concatenate WAMFs and WPMFs in or-der to simultaneously filter universal noise. We focuson an explicit example providing first-order amplitudeand dephasing noise filtering. The basic procedure is toimplement each constant-amplitude segment of a four-segment WAMF, W1, using a constant-amplitude phase-modulated sequence robust against amplitude noise. As areminder, the noise-filtering performance of W1 is shownin Fig. 4c. Here we use the WPMF≡SK1 sequence for

© 2014 Macmillan Publishers Limited. All rights reserved.

11

0

Ω SK1(1)

Ω SK1(2)

τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 τ 7 τ 8 τ 9

100

10-5

10-10

10-15 F

ilter

Fun

ctio

n

10-6 10

-2 102

FΩ Fz

100

10-5

10-10

10-15

10-6 10

-2 102 10

-6 10-2 10

2 ω (τ −1

)

a

dcb

0

Ω SK1(1)

Ω SK1(2)

τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 τ 7 τ 8 τ 9

0

ΩSK1

τ 1 τ 2 τ 3

0

X -

X +

1.000.750.500.250.00 Normalized Time

WPMF WAMF Concatenated

FIG. 4. Concatenation scheme for universal noise suppres-sion. a) Concatenation of WPMF≡SK1 within first orderWAMF sequence yielding UWMF1,SK1. White fill indicatesrotations enacted with φ = 0; orientation of hatching de-notes SK1 phase flips φ = ±φSK1 . b) Filter functions forWPMF≡SK1 sequence. c) Filter functions for four-segmentWAMF sequence, W1. d) Filter functions for concatenatedsequence.

the phase modulation. We refer to the concatenated gateas a Universal Walsh Modulated Gate, UWMF1,SK1.

Referring to Eq. 51, the WAMF filter is similarly writ-ten P3(X+/4, 0)P2(X−/2, 0)P1(X+/4, 0). Concatenationthen involves the operator substitutions

P1(X+/4, 0)→ SK1(1)(X+/4) (54)

P2(X−/2, 0)→ SK1(2)(X−/2) (55)

P3(X+/4, 0)→ SK1(3)(X+/4). (56)

The composite structure for UWMF1,SK1 is shown inFig. 4a. Here the SK1 phase flips φ = ±φ

SK1within

each segment of the WAMF profile are indicated by theoppositely oriented hatching; φ = 0 is indicated by whitefill. The dephasing and amplitude filter functions for theconcatenated sequence are shown in Fig. 4d, indicatingeffective filtering of both amplitude and dephasing noise.

Ytterbium Ion Trapping

We use trapped 171Yb+ ions as our experimental plat-form; a detailed description of related experimental ap-proaches appears in [19, 20]. A linear Paul trap enclosedin an ultra-high vacuum (UHV) chamber is used to trapseveral hundred 171Yb+ ions as a small homogeneous en-semble (in magnetic field and microwave field amplitude).Doppler cooling of the ions is achieved using 369 nm laserlight, slightly red-detuned from the 2S1/2 to 2P1/2 tran-sition. Additional lasers near 935 nm and 638 nm areemployed to depopulate metastable states.

Our qubit is the 12.6 GHz hyperfine splitting betweenthe 2S1/2 |F = 0,mf = 0〉 and 2S1/2 |F = 1,mf = 0〉

states. For notational simplicity we will designate |0〉and |1〉 to these states respectively. Addition of a 2.1GHz sideband to the 369nm laser using an electro-opticmodulator permits high-fidelity state preparation in |0〉.For details of ion loading, laser cooling, state preparation,and state detection see [20]. While we typically employa small ensemble of ions, the system behaves similarly tosingle-ion experiments in our lab, and benefits from bothhigh-fidelity state initialization and projective measure-ment - the system does not bear similarity to NMR-styleensembles.

State detection is achieved by counting 369 nm photonsscattered from the ions and converting to a probabilitythat the Bloch vector lies at a particular location along ameridian of the Bloch sphere. This measurement is sus-ceptible ion loss in the ensemble and both laser ampli-tude and frequency drifts over long timescales, resultingin variable maximum and dark count rates over time. Wetherefore employ a normalization and Bayesian estima-tion procedure for state detection, see [20].

An important advantage of this system is that the se-lected qubit transition is first order insensitive to mag-netic field fluctuations; the measured free-evolution inour setup is T2 ≈ 4 s, limited by coherence betweenthe qubit and the master oscillator [20]. Coherent ro-tations between the measurement basis states are drivenby using the magnetic field component of resonant mi-crowave radiation. The Rabi rate for driven oscillationsreaches ∼ 14 µs in our system, with typical operationnear ∼ 50 µs. Rotations are implemented about an axis~r lying on the xy-plane of the Bloch sphere and set bythe phase of the microwaves as ~r = (cosφ(t), sinφ(t), 0).Driven operations, characterized by randomized bench-marking, exhibit a mean fidelity in excess of 99.99%.

Noise Engineering

In the laboratory we rely on engineering noise in ourcontrol system to provide a method to accurately repro-duce decoherence processes of interest. We begin with adesired noise power spectral density in either the ampli-tude or detuning quadrature (or both), assuming theyare statistically independent. From this power spec-trum, defined by the noise strength α, the exponent ofthe power-law scaling p, the comb spacing ω0, and thehigh-frequency cutoff ωc ≥ Jω0, we numerically generatetime-domain vectors for amplitude and frequency errors.Noise is injected into the system by adding these modu-lation patterns on top of the control sequence being im-plemented (e.g. a pulse of radiation for implementing aπ-pulse)using IQ modulation in our vector signal gener-ator [20].

© 2014 Macmillan Publishers Limited. All rights reserved.

12

Randomized Benchmarking

We use randomized benchmarking as a tool for resolv-ing small gate errors which cannot be resolved in the ap-plication of a single gate. Our randomized benchmarkingsequence consists of interleaved π/2 and π pulses eachapplied along axes randomly selected from ±x and ±y.Each sequential pair of π/2 and π rotations is referred toa computational gate. A given randomized benchmark-ing sequence consists of l computational gates followedby a final correcting gate which is selected such that theaggregate Unitary operation applied is a π rotation. Foreach l we measure 50 randomizations (dots in Fig. 3f ofthe main text), and in each randomization average over20 different realizations of a white dephasing noise bath.Each realization, in turn, employs 20 measurements inour Bayesian state-detection algorithm, in addition toassociated normalization experiments.

Comparisons of W1 to primitive π rotation perfor-mance in randomized benchmarking is conducted via re-placement of all π pulses with W1 constructions, againabout randomly selected axes. In either case the π/2rotation is achieved using a primitive gate, although wehave also validated that replacement of the π/2 gateswith WAMF constructions yields net improvement ingate fidelity.

∗ These two authors contributed equally to this work.† Present address: Lockheed Martin Corporation‡ michael.biercuk@sydney.edu.au

[1] Green, T. J., Sastrawan, J., Uys, H. & Biercuk, M. J.Arbitrary quantum control of qubits in the presence ofuniversal noise. New J. Phys. 15, 095004 (2013).

[2] Green, T. J., Uys, H. & Biercuk, M. J. High-order noisefiltering in nontrivial quantum logic gates. Phys. Rev.Lett. 109, 020501 (2012).

[3] Merrill, J. T. & Brown, K. R. Progress in com-pensating pulse sequences for quantum computation.arXiv:1203.6392 (2012).

[4] Beauchamp, K. G. Walsh Functions and their Applica-

tions (Academic Press, London, 1975).[5] Horadam, K. J. Hadamard Matrices and their Applica-

tions (Princeton University Press, Princeton, 2007).[6] Hayes, D., Khodjasteh, K., Viola, L. & Biercuk, M. J.

Reducing sequencing complexity in dynamical quantumerror suppression by Walsh modulation. Phys. Rev. A84, 062323 (2011).

[7] Lidar, D. A. & Brun, T. A. Quantum Error Correction(Cambridge University Press, New York, 2013).

[8] Souza, A. M., Alvarez, G. A. & Suter, D. Experimen-tal protection of quantum gates against decoherence andcontrol errors. Phys. Rev. A 86, 050301 (2012).

[9] Olmschenk, S. et al. Manipulation and detection of atrapped yb+ hyperfine qubit. Phys. Rev. A 76, 052314(2007).

[10] Soare, A. et al. Experimental bath engineering for quan-

titative studies of quantum control. Phys. Rev. A 89,042329 (2014).

[11] Green, T. J., Sastrawan, J., Uys, H. & Biercuk, M. J.Arbitrary quantum control of qubits in the presence ofuniversal noise. New J. Phys. 15, 095004 (2013).

[12] Green, T. J., Uys, H. & Biercuk, M. J. High-order noisefiltering in nontrivial quantum logic gates. Phys. Rev.Lett. 109, 020501 (2012).

[13] Merrill, J. T. & Brown, K. R. Progress in com-pensating pulse sequences for quantum computation.arXiv:1203.6392 (2012).

[14] Beauchamp, K. G. Walsh Functions and their Applica-tions (Academic Press, London, 1975).

[15] Horadam, K. J. Hadamard Matrices and their Applica-tions (Princeton University Press, Princeton, 2007).

[16] Hayes, D., Khodjasteh, K., Viola, L. & Biercuk, M. J.Reducing sequencing complexity in dynamical quantumerror suppression by Walsh modulation. Phys. Rev. A84, 062323 (2011).

[17] Lidar, D. A. & Brun, T. A. Quantum Error Correction(Cambridge University Press, New York, 2013).

[18] Souza, A. M., Alvarez, G. A. & Suter, D. Experimen-tal protection of quantum gates against decoherence andcontrol errors. Phys. Rev. A 86, 050301 (2012).

[19] Olmschenk, S. et al. Manipulation and detection of atrapped yb+ hyperfine qubit. Phys. Rev. A 76, 052314(2007).

[20] Soare, A. et al. Experimental bath engineering for quan-titative studies of quantum control. Phys. Rev. A 89,042329 (2014).

© 2014 Macmillan Publishers Limited. All rights reserved.

Appendix B

Physical Review A publication

87

PHYSICAL REVIEW A 89, 042329 (2014)

Experimental bath engineering for quantitative studies of quantum control

A. Soare,1 H. Ball,1 D. Hayes,1 X. Zhen,1,2 M. C. Jarratt,1 J. Sastrawan,1 H. Uys,3 and M. J. Biercuk1,*

1ARC Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, NSW 2006 Australiaand National Measurement Institute, West Lindfield, NSW 2070 Australia

2Tsinghua University, Beijing, People’s Republic of China3Council for Scientific and Industrial Research, National Laser Centre, Pretoria, South Africa

(Received 18 March 2014; published 28 April 2014)

We develop and demonstrate a technique to engineer universal unitary baths in quantum systems. Using thecorrespondence between unitary decoherence due to ambient environmental noise and errors in a control systemfor quantum bits, we show how a wide variety of relevant classical error models may be realized through in-phaseor in-quadrature modulation on a vector signal generator producing a resonant carrier signal. We demonstrateour approach through high-bandwidth modulation of the 12.6-GHz carrier appropriate for trapped 171Yb+ ions.Experiments demonstrate the reduction of coherent lifetime in the system in the presence of both engineereddephasing noise during free evolution and engineered amplitude noise during driven operations. In both cases,the observed reduction of coherent lifetimes matches well with quantitative models described herein. Thesetechniques form the basis of a toolkit for quantitative tests of quantum control protocols, helping experimentalistscharacterize the performance of their quantum coherent systems.

DOI: 10.1103/PhysRevA.89.042329 PACS number(s): 03.67.−a, 37.10.Ty

I. INTRODUCTION

The discipline of quantum control engineering [1–4] isaddressing pressing challenges in the fields of quantumphysics, quantum information, and quantum engineering,attempting to provide the community with a broad range ofnovel capabilities in the precise manipulation of quantumsystems [5–9]. For instance, protocols derived from open-loopcontrol employing sequences of SU(2) operations, knowncollectively as dynamical decoupling, have proven usefulin extending the coherent lifetime of qubits in quantummemories [10–13] and in producing effective noise filters forquantum sensors [14–18].

Beginning with the work of Kurizki et al. [19,20], therehas been a substantial effort in the field towards incorporat-ing filter-transfer functions into the vernacular of quantumcontrol [21,22]. This has extended from trivial application ofthe identity in dynamical decoupling [10,23–27] to arbitrarysingle- [28–31] and two-qubit operations [32,33]. In thisframework, a metric of interest, generally an ensemble-averaged operational fidelity, may be simply calculated fromthe product of the environmental noise power spectrum and afilter transfer function capturing the effects of the control in theFourier domain. This approach has been shown to be a generaland efficient approach capturing arbitrary control and arbitraryuniversal noise in quantum systems [30] and is a powerful toolfor understanding the influence of realistic colored classicalnoise power spectra on quantum systems.

These advances are providing a means for theoreticalresearchers to move away from the unphysical Markovianassumptions for stochastic, uncorrelated error models se-lected for convenience in quantum error correction and thelike [34,35], and has provided a simple platform for thedevelopment of novel protocols aimed at improving controlfidelity in quantum systems. As these protocols transitionfrom theoretical concepts into the laboratory, experimentalists

*michael.biercuk@sydney.edu.au

require techniques to quantitatively verify the predicted perfor-mance in different noise environments and compare outcomesin a manner that is insensitive to underlying imperfectionsin their hardware. Such precise validations are necessary forresearchers to confidently develop quantum control techniquesusing substantiated methodologies and subroutines.

In this paper, we describe a technique to engineer arbitraryunitary baths consisting of dephasing and amplitude-dampingprocesses for quantitative tests of experimental quantum con-trol. We present a simple theoretical model for approximatingarbitrary classical power spectra via discrete frequency combswith user-selected envelopes (e.g., 1/f ). We describe howthis model permits simple and verifiable creation of time-dependent noise realizations in both dephasing and amplitude-damping quadratures, compatible with experimental systems.Through demonstration of the isomorphism of unitary controlerrors and environmental decoherence, we map these noiserealizations to modulation of a carrier signal in an experimentalcontrol system, e.g., for a single quantum bit. Using trapped171Yb+ ions with splitting ∼12.6 GHz, we demonstrateour bath engineering approaches via IQ modulation onthe microwave carrier. Ramsey spectroscopy measurementsquantitatively verify the predicted influence of engineereddephasing noise on the coherent lifetime of our qubits.

The remainder of this paper is structured as follows. InSec. 2, we provide a detailed theoretical derivation of ourselected method of unitary noise engineering for both de-phasing (detuning) and amplitude-damping Hamiltonians, anddescribe how these noise spectra may be translated to widelyavailable time-domain IQ-modulation waveforms applied toa carrier signal. We then move on to describe our experimentalsystem and its capabilities in Sec. 4 A. This is followed inSec. 4 by a characterization of an experimentally implementednoise-engineered bath through direct examination of the carrierand a demonstration of engineered dephasing environments viameasurements of coherent lifetimes for 171Yb+ ion qubits. Thepaper concludes with a discussion and outlook towards futureexperiments.

1050-2947/2014/89(4)/042329(12) 042329-1 ©2014 American Physical Society

A. SOARE et al. PHYSICAL REVIEW A 89, 042329 (2014)

II. PHYSICAL SETTING

In many quantum systems of interest we may consider twogeneral classes of unitary time-dependent errors. Dephasingprocesses are associated with rotations about σz induced bya stochastic relative detuning between a qubit’s transition(angular) frequency ωa and the experiment’s master clock,defined by a local oscillator (LO). Dephasing is frequentlydominated by instabilities in the qubit splitting caused byenvironmental (e.g., magnetic field) fluctuations. However,in the limit of very stable qubits (e.g., clock transitions inatomic systems [36,37]), observed dephasing may be causedby frequency instabilities in the experimental LO. Similarly,one may consider coherent amplitude-damping processes,causing unwanted rotations along meridians of the Blochsphere, and arising either through ambient environmentalfluctuations (e.g., microwave leakage from nearby systems)or from imperfections in the amplitude of the applied controlfield.

Together, these two classes of error capture so-called uni-versal (multiaxis) rotations of the Bloch sphere. Importantly,the consideration of time-dependent errors in both dephasingand amplitude quadratures allows us to capture the dominantforms of non-Markovian noise processes characterized by thepresence of long-time correlations; realistic laboratory settingsare typically dominated by such noise terms. Dissipative errorpathways with Markovian characteristics may be capturedthrough linearly independent error terms that we ignorethrough this treatment, as quantum control generally providesno relevant benefits in error resistance for these effects.

We consider a model quantum system consisting of an en-semble of identically prepared noninteracting qubits immersedin a weakly interacting noise bath and driven by an externalcontrol device. Working in the interaction picture with respectto the qubit splitting, state transformations are represented asunitary rotations of the Bloch vector. Including both controland noisy interactions, we may therefore write the generalizedtime-dependent Hamiltonian

H (t) = Hc(t) + H0(t). (2.1)

The term Hc(t) = h(t)σ represents perfect control over thequbit state via the application of an external field, while thegeneralized noise term H0(t) = η(t)σ captures all interactionsdue to the noise bath. Here, σ denotes a column vector ofPauli matrices and the row vectors h(t),η(t) ∈ R3 denote,respectively, the Cartesian components of the control andnoise fields in the basis of Pauli operators [29,30,38]. Thestochastic noise fields ηi(t), i ∈ x,y,z, model semiclassicaltime-dependent error processes in each of the three spa-tial directions. In this formulation, dephasing processes arecaptured through the appearance of stochastic terms alongσz. General coherent amplitude-damping terms on the otherhand are captured by terms proportional to the spin operatorσφ := cos(φ)σx + sin(φ)σy parametrized by the driving phaseφ ∈ [0,2π ].

Our choice to write separate control and noise terms inthe Hamiltonian in Eq. (2.1) belies the fact that, when ex-pressed in an appropriate interaction picture, time-dependentfluctuations in either term are effectively indistinguishable.This observation permits a formulation in which the noise

terms are all incorporated into the control Hamiltonian, andone assumes the presence of a perfectly stable qubit (i.e., thereis no ambient decoherence). This is a good approximation inthe case of a sufficiently stable qubit so long as native errorrates and ambient noise susceptibilities are small compared torelevant scales under study. We therefore proceed by providinga model for quantum dynamics that permits us to captureunitary decoherence through the control.

With these generalized notions in mind, we proceed inlaying out the detailed Hamiltonian framework relevant to ourstudy. The system considered in this paper consists of a qubitwith transition (angular) frequency ωa , driven by a LO withmagnetic field component aligned with σz taking the form

B(t) = (t) cos[ωμt + φ(t)]z, (2.2)

(t) = C(t) + N (t), (2.3)

φ(t) = φC(t) + φN (t) (2.4)

with ωμ the carrier frequency. In this formulation, thetime dependence of the phase φ(t) and amplitude (t) hasbeen formally partitioned into components denoted by thesubscripts C and N , capturing the desired control and noisyinteractions, respectively. Using standard approximations, andworking in an interaction picture (see Appendix A), the systemHamiltonian ( = 1) may be expressed as

HI = − φN (t)

2σz + 1

2(t)cos[φC(t)]σx + sin[φC(t)]σy.

(2.5)

The noise component φN (t) of the engineered phase φ(t)produces net rotations about σz through its time derivativeφN (t), and the resonant carrier field drives coherent Rabiflopping between the qubit states |1〉 and |0〉. The instantaneousRabi rate in this case is proportional to (t) and rotations,generated by the spin operator σφC (t), are driven about theaxis r = [cos φC(t), sin φC(t),0] in the xy plane of the Blochsphere.

Given sufficient control over both the phase and amplitudeof our driving field, Eq. (2.5) therefore indicates we mayengineer a variety of effective control Hamiltonians withdephasing and amplitude-damping terms of interest. Forinstance, setting φC(t) = 0, we may generate

H (t) ∝

⎧⎪⎨⎪⎩

hx(t) [1 + ηx(t)] σx (Mult. Amp. Noise),

[hx(t) + ηx(t)] σx (Add. Amp. Noise),

hx(t)σx + ηz(t)σz (Add. Deph. Noise),

(2.6)

where the control field hx(t) is proportional to C(t) and thenoise fields ηx(t) or ηz(t) may be switched on, with desiredspectral properties, by an appropriate choice of N (t) andφN (t), respectively.

The Hamiltonians in Eq. (2.6) correspond to familiar errormodels from NMR and quantum information [39,40], but nowexplicitly incorporate non-Markovian time-dependent effectsthrough the power spectra of the relevant terms in η(t). The firstnoise model may be produced in the absence of Hamiltonian

042329-2

EXPERIMENTAL BATH ENGINEERING FOR . . . PHYSICAL REVIEW A 89, 042329 (2014)

terms that look like hx(t)ηx(t) by virtue of the ability toarbitrarily parametrize ηxhx(t),t.

Following previous work, we express the first-order aver-aged fidelity of an arbitrary unitary control operation on SU(2)in the presence of noise as [30]

Fav(τ ) ≈ 1

21 + exp[−χ (τ )], (2.7)

χ (τ ) = 2

π

[ ∫ ∞

0

dω

ω2Sz(ω)Fz(ω) +

∫ ∞

0

dω′

ω′2 S(ω′)F

(ω′)

].

(2.8)

Here, we have defined independent noise power spectraSz(ω) and S

(ω), with angular frequency ω, for fluctuations

in φN (t) and N (t), respectively, while the quantities Fz(ω)and F

(ω) represent the spectral characteristics of the control

under study. While we will not focus on the particular form ofthis so-called filter transfer function expression for operationalfidelity [28,30], we can clearly see the importance of thesenoise power spectra in determining the performance of anarbitrary control operation. Consequently, in the followingsection, we derive their forms.

III. ENGINEERING NOISE IN THE CONTROL SYSTEM

In the laboratory we rely on engineering noise in our controlsystem to provide a method to accurately reproduce deco-herence processes of interest. This approach has significantbenefits over, e.g., noise injection in ambient magnetic fieldcoils, as it minimizes potential nonlinearities and frequency-dependent responses in hardware elements, exploiting insteadthe modulation capabilities of a carrier synthesis system [41].By engineering noise through a highly accurate control systemwith linear response, we gain the ability to perform quantitativetests of quantum control in the presence of unitary noiseHamiltonians.

We employ the phase- and amplitude-modulation capabil-ities in state-of-the-art quantum control systems in order toprovide access to the error models of interest. In the remainderof this section, we present a mathematical formalism linkingour error model in the geometric picture of unitary dynamics tothe properties of a near-resonant drive field of the form givenin Eq. (2.2).

A. Arbitrary dephasing (detuning) power spectra

We begin with the case of noise proportional to σz.Our method relies on generating stochastic detuning errorsby performing phase modulation on a constant-amplitudecarrier, thereby implementing an effective pure dephasingHamiltonian. Setting N (t) = φC = 0 and C = 0 we write

B(t) = 0 cos[ωμt + φN (t)]z, (3.1)

φN (t) = α

J∑j=1

F (j ) sin(ωj t + ψj ), (3.2)

where ψj is a random number. That is, the driving carrier toneis modulated to include a time-dependent, stochastic error in

the phase constructed as a discrete Fourier series with a basefrequency ω0 = ωj/j , with α being a global scaling factor [42].

The link between this phase modulation and the dephasingpower-spectral density of interest is revealed by definingthe instantaneous phase in terms of the carrier plus a time-dependent detuning βz(t):

(t) = 0 +∫ t

0dt ′[ωμ + βz(t

′)], (3.3)

where βz(t) is a zero-mean time-dependent random variable.This then implies

φN (t) = 0 +∫ t

0dτ βz(τ ) ⇐⇒ βz(t) = d

dtφN (t)

(3.4)

so the time-dependent detuning noise βz(t), explicitly linked tothe phase modulation of the carrier, characterizes the strengthof the dephasing-noise term in Eq. (2.5).

Using the Euler decomposition, we may then write

βz(t) = αω0

2

J∑j=1

jF (j )[ei(ωj t+ψj ) + e−i(ωj t+ψj )]. (3.5)

Assuming wide-sense stationarity, the two-time correlationfunction for βz(t) is then written

〈βz(t + τ )βz(t)〉t = α2ω20

2

J∑j=1

[jF (j )]2 cos(ωjτ ), (3.6)

where 〈. . .〉t denotes averaging over all times t from whichthe relative lag of duration τ is defined. Invoking the Wiener-Khintchine theorem [43] and moving to the Fourier domain,we then obtain the power-spectral density

Sz(ω) = πα2ω20

2

J∑j=1

[jF (j )]2[δ(ω − ωj ) + δ(ω + ωj )].

(3.7)

The detailed derivation of the above expressions appears inAppendix B.

The power-spectral density (PSD) is thus representedas a Dirac comb of discrete frequency components withthe amplitude of the j th tooth determined by the quantity[jF (j )]2. We now have an explicit relationship betweenan effective dephasing- or frequency-detuning -noise powerspectrum and the phase modulation of the carrier frequency inthe control system required to achieve that PSD.

It is then straightforward to specify the construction ofany power-law PSD by writing the amplitude of the j thfrequency component as a power law Sz(ωj ) ∝ (jω0)p. Ittherefore follows that the envelope function for the comb teethin the phase modulation scales as

F (j ) = jp

2 −1. (3.8)

Table I shows the functional form required for F (j ) in orderto achieve dephasing-noise PSDs of interest.

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TABLE I. Functional form of F (j ) for well-known dephasing-and amplitude-noise PSDs.

Dephasing Amplitude

1/f 2 1/f White Ohmic 1/f 2 1/f White Ohmic

p −2 −1 0 1 −2 −1 0 1F (j ) j−2 j−3/2 j−1 j−1/2 j−1 j−1/2 j 0 j 1/2

B. Arbitrary amplitude power spectra

Derivation of the relevant amplitude noise power spectraproceeds in a similar manner. We consider here multiplicativeamplitude noise, although the derivation maintains a similarform in the case of additive noise. Further, in our modelthe amplitude noise is always assumed to be coaxial withthe driving field. While this is not a strict requirement,it greatly simplifies the analysis and broadly represents aninteresting class of time-dependent error models incorporatingdriving-field noise. The relevant modulation capability here is,as expected, amplitude modulation on a carrier signal. Settingφ(t) = 0, C(t) = 0, and N = 0β

(t), Eq. (2.2) reducesto

B(t) = 0[1 + β(t)] cos(ωμt)z. (3.9)

That is, amplitude modulation transforms the control fieldstrength as

0 → 0[1 + β(t)], (3.10)

where, again, β(t) is a zero-mean stochastic random variable

here capturing fluctuations in the drive amplitude. This termis realized directly through the comb of discrete frequencycomponents with randomly selected phase shifts

β(t) = α

J∑j=1

F (j ) cos(ωj t + ψj ) (3.11)

= α

2

J∑j=1

F (j )[ei(ωj t+ψj ) + e−i(ωj t+ψj )]. (3.12)

The form of this expression is similar to that above for dephas-ing noise, except the direct-amplitude modulation removes afactor of j from the expression. We are interested in producinga PSD for the quantity β

(t) as this captures the amplitude

errors pertaining to the second term of Eq. (2.8). Followingthe same method as before (see Appendix B), we obtain

S(ω) = πα2

2

J∑j=1

[F (j )]2[δ(ω − ωj ) + δ(ω + ωj )]. (3.13)

Once again, we may define a relationship between the powerlaw of the target noise power-spectral density and the quantity[F (j )]2, which determines the amplitude of the j th tooth in thefrequency comb PSD above. In this instance, the removal ofa differential relationship between the modulation and desirednoise power spectrum yields the simplified expression F (j ) =jp/2 for S

(ωj ) ∝ (jω0)p.

S (ω

)

Angular Frequency, ωω0

Jω0

ω0

F (j)

FIG. 1. (Color online) Schematic depiction of the frequencycomb generated in our noise-engineering protocol and its relation-ships to key parameters on a logarithmic scale.

C. Summary

With these relationships, we now have explicit linksbetween the quantities we wish to engineer in realizing unitarydephasing or relaxation noise power spectra and the relevantparameters entering into the modulation of a control signal.We will employ these relations to engineer arbitrary unitarynoise baths for quantitative tests of various quantum controlprotocols.

In implementing bath engineering in the laboratory, we areleft with the following free parameters:

(i) z/: the quadrature of noise injection (dephasing vsamplitude);

(ii) ω0: the fundamental frequency of the Dirac comb andthe lower cutoff of the noise power spectrum;

(iii) J : the maximum number of comb teeth in the discretesum, setting the upper frequency cutoff Jω0;

(iv) p: the exponent setting the frequency dependence ofthe effective noise power spectrum.

These parameters provide an experimentalist with a broadset of capabilities for bath engineering (see Table I and Fig. 1).

IV. EXPERIMENTAL BATH ENGINEERING

A. Experimental platform

The approach we have described above is quite generic forthe case of a quantum system controlled by an oscillatorysignal, including most atomic, superconducting, and manysemiconductor-based spin qubits. In this section, we describethe experimental platform we will employ for validation of ourmethod.

We use trapped 171Yb+ ions as our model experimentalplatform; a detailed description of related experimental ap-proaches appears in [37]. Neutral 171Yb is ionized using atwo-photon process whereby 399-nm light excites electronsfrom 1S0 to 1P 1 and 369-nm light is sufficiently energetic tofurther excite electrons to the continuum. A linear Paul trapenclosed in an ultrahigh vacuum (UHV) chamber is used totrap several hundred 171Yb+ ions. Doppler cooling of the ionsis achieved using 369-nm laser light, slightly red-detuned fromthe 2S1/2 to 2P 1/2 transition. Additional lasers near 935 and638 nm are employed to depopulate metastable states. Typicalexperiments employ ensembles of approximately 100–1000ions with high homogeneity in magnetic field and microwavefield over the ensemble.

Our qubit is the 12.6-GHz hyperfine splitting betweenthe 2S1/2|F = 0,mf = 0〉 and 2S1/2|F = 1,mf = 0〉 states.For notational simplicity we denote these states by |0〉

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and |1〉 respectively. The system may be optically pumpedto |0〉 using a 2.1-GHz sideband on the 369-nm coolingbeam, which couples the states 2S1/2|F = 1〉 ↔2P 1/2|F = 1〉following [37].

State detection is achieved using resonant light near 369 nm,which preferentially couples the state |1〉 to the excited P state,resulting in a large probability of detecting scattered photons.These photons are detected using a pair of large-diameterlenses and a photomultiplier tube. State discrimination isconducted by photon counting followed by conversion to aprobability that the Bloch vector lies at a particular locationalong a meridian of the Bloch sphere. This measurement issusceptible to ion loss in the ensemble and both laser amplitudeand frequency drifts over long time scales, resulting in variablemaximum and dark count rates over time. We therefore employa normalization and Bayesian estimation procedure to improvemeasurement fidelity.

Dark-state normalization is achieved by cooling and opti-cally pumping the ions to the F = 0 state of the 12.6-GHzhyperfine manifold and then performing a measurement ofphoton counts, while bright-state normalization includes anadditional πx gate implemented using microwaves beforethe photon count measurement. We have experimentallyascertained that for any angle of declination of the Bloch vectorwith respect to the −z axis, θ , photon counts over a repeatednumber of identical experiments are normally distributed abouta mean value with standard deviation dominated by rapid laserfrequency fluctuations with magnitude ∼ 1 MHz.

We use Bayesian inference to statistically determine thequbits’ z projection given the number of scattered 369-nmphotons we measure, denoted P (θ |c). We write

P (θ |c) = P (c|θ )P (θ )

P (c)(4.1)

= P (c|θ )P (θ )∑i

∫ π

0 dθP (c|θ )P (θ ), (4.2)

where in the second line we have incorporated the factthat P (c) is dependent on our knowledge of the probabilitydistribution function for θ . We have experimentally verifiedthat we may write the probability density function P (c|θ ) =exp

[ − ( D+ B−Dπ

θ

σD+ σB −σDπ

θ

)2]. Here, we have defined a Gaussian with

center defined by linearly interpolating between the meandetected photon counts for the bright state B and mean countsfor the dark state D. The standard deviation of the Gaussianis defined by linearly interpolating between the standarddeviations of the photon-count distributions for the brightstate σB and dark state σD . The quantities B, D, σB , and σD

are found by performing bright- and dark-state normalizationbefore each iteration of the experiment of interest. P (θ ) isinitially assumed to be a uniform probability distribution,and is iteratively refined with subsequent experiments. Wethen calculate the mean and standard deviation and apply thetransform P|1〉 = sin2( θ

2 ). Using this method, we can achievemeasurement fidelity in excess of 98%.

A simpler method of calculating the approximatebright-state probability is to take a simple normalized averageof the form (E − D)/(B − D) where B and D are definedas before and E is the mean detected photon count for the

experiment of interest. This method agrees well with Bayesianestimation for states that are not near the poles of the Blochsphere and provides a simpler and faster measurement methodin cases where maximizing measurement fidelity near thepoles is not required.

To produce our master oscillator signal we use an ultralow-phase-noise vector source referenced to a caesium clock and10-MHz Wenzel cleanup oscillator for long-term stabilityand good short-term phase noise. The output of the signalgenerator is amplified using a low-phase-noise amplifier withmaximum output of approximately +33 dBm. A commerciallyavailable, microwave horn-lens combination is used to producea highly directional free-space linearly polarized microwavefield (+25 dBi directional gain) which can be directed atthe ions, approximately 150 mm from a 150-mm-diameterviewport on the UHV chamber.

Coherent rotations between the measurement basis statesare driven using the magnetic field component of resonantmicrowave radiation. The Rabi rate for driven oscillationsis linearly proportional to the microwave magnetic fieldamplitude, with rotations about an axis r lying on thexy plane of the Bloch sphere and set by the phase ofthe microwaves as r = [cos φ(t), sin φ(t),0]. Rabi floppingexperiments [Fig. 2(c)] demonstrate high-visibility coherentqubit rotations where we can achieve hundreds of flopsbefore seeing appreciable decay. With ∼ +30 dBm nominalmicrowave power (e.g., not accounting for cable losses) weachieve π times as low as ∼ 15 μs, but we typically operatenear 50 μs. We have confirmed that in these experiments ourmeasured Rabi flopping times are limited by small microwavefield amplitude inhomogeneities over the ion ensemble causedby diffraction of the microwave beam at the aperture of theUHV chamber.

A standard technique for characterizing oscillator stabilityis Ramsey spectroscopy [44]. We prepare the ions in |0〉 androtate to | + y〉 using a π/2 pulse applied about x, but slightlydetuned from resonance by +4 Hz. After a free-evolutionperiod, a second π/2 pulse will rotate the qubit to |0〉 or|1〉 depending on the phase accumulated between the masteroscillator and the qubit. Scanning the evolution time τ revealssinusoidal fringes due to the free evolution of the qubit relativeto the control during the delay period. Instabilities of the phaseover time cause the relative phase between the qubit and masteroscillator to become randomized, thus reducing the visibilityof Ramsey fringes.

An important advantage of this system is that the selectedqubit transition is first order insensitive to magnetic fieldfluctuations. As a result, the intrinsic free-evolution coherencetime of this hyperfine qubit has been measured to be at least15 min [45]. A T2 decay time of approximately 4 s, inferredfrom Ramsey experiments [Fig. 2(d)], demonstrates long-termcoherence between the qubit and our LO, ultimately limitedby phase stability of the LO (typically −80 dBc phase noise at100 Hz offset from carrier). These experimental measurementsreveal that this system therefore provides a “clean” baselinefor quantitative tests of bath engineering.

B. Implementation of bath engineering by IQ modulation

The bath engineering technique described above provides ageneric framework allowing noise to be generated for specific

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1.0

0.5

0.0 Pro

b. B

right

3.02.52.01.51.00.50.0

Time (ms)

20

15

10

5 Cou

nts

(k)

3.02.52.01.51.00.50.0

FID Time (s)

(a)

(d)

(b)2P1/2

2S1/2

(c)

Cs Reference

10 MHz

QuartzOscillator

VSG

Digital In

Mod Waveforms

10 MHz

12.6 GHz

+18 dB

To TrapF=1

F=0

12.6 GHz

F=0

F=1

~369.52 nm

FIG. 2. (Color online) Experimental control system. (a) Simplified level structure of 171Yb+ ions with qubit at 12.6-GHz splittinghighlighted. Solid arrows indicate excitation via UV detection laser, dotted arrows indicate spontaneous emission pathways. Repumpingtransitions to mestatable D and F states not shown. (b) Microwave synthesis chain employed for 171Yb+ qubits. Digital programming of thevector signal generator (VSG) conducted via either GPIB or LAN. (c) Measured Rabi flopping at 12.6 GHz on the clock transition in the171Yb+ ground state. In this measurement, we use Bayesian estimation to map raw measured photon counts to bright- or dark-state probability.(d) Free evolution measured via Ramsey interferometry and presented using raw photon counts for simplicity. Interrogating π/2 pulses appliedwith frequency detuning of ∼ 4 Hz to yield interference fringes. After approximately 2 s of free evolution, the fringe frequency appears to shiftabruptly and then become unstable. The overlaid damped oscillation assumes a Gaussian decay and a T2 of 4 s and matches the general decayenvelope well. However, due to the appearance of statistically nonstationary dynamics during the experiment, fitting struggles to provide anaccurate reproduction of the data. Nonetheless, the data clearly indicate coherence between the LO and qubits beyond approximately 3 s.

Hamiltonians of interest. We must now demonstrate howsuch noise may be implemented using the kind of controlhardware typically available for quantum control experiments:IQ modulation on the resonant carrier.

To model a desired control field in the presence of noise, wegenerate a microwave field of the form set out in Eqs. (2.2)–(2.4). In order to engineer the bath in our experimental system,we begin with a desired noise power-spectral density in eitherthe amplitude or detuning quadrature (or both), assumingthey are statistically independent. From this power spectrum,defined by the noise strength α, the exponent of the power-lawscaling p, the comb spacing ω0, and the high-frequency cutoffωc Jω0, we numerically generate time-domain vectors forN (t) and φN (t) using the relationships appearing in Sec. 2,and randomly selecting the phase ψj for each comb toothin the Fourier decomposition. Thus, we may independentlygenerate our control and noise modulation signals for thecarrier amplitude and phase (see Fig. 3).

Independent and arbitrary control over these properties ofthe carrier may be achieved using IQ modulation [46]. I (t)and Q(t) are simply a polar-to-Cartesian coordinate transformof the familiar amplitude (t) and phase φ(t) components ofa modulated signal S(t) = (t) sin[ωμt + φ(t)] as

S(t) = I (t) sin(ωμt) − Q(t) sin(ωμt − π/2), (4.3)

I (t) = (t) cos[φ(t)], Q(t) = (t) sin[φ(t)]. (4.4)

The numerically generated noise and control modulationpatterns are thus converted to the IQ basis and applied asa modulation pattern in time. While our method typicallyrelies on a Fourier decomposition for the generation of the

IQ-modulation patterns, arbitrary time-domain noise may beengineered, such as the influence of random telegraph noise incarrier frequency.

As mentioned above, our carrier frequency for the clocktransition in 171Yb+ is 12.6 GHz, produced by a vectorsignal generator. The key feature of this unit is a digitallyprogrammable baseband generator producing the modulationenvelopes for I and Q. The functions are defined sample-wisewith 16-bit resolution in order to approximate a continuousfunction. In our system, care must be taken to ensure thatdiscontinuities in the waveforms are avoided as the basebandgenerator employs an interpolation algorithm that can produceringing in the applied modulation.

LO RF

IQ Mod

Dual DAC

Control Phase

Phase Noise

Noisy Phase Mod. Noisy Amp. Mod.

Control Amplitude

Amplitude NoiseIQ Conversion

FIG. 3. (Color online) Schematic representation of process flowinvolved in experimental noise engineering. Independent waveformsfor phase and amplitude are determined numerically, shifted to theIQ basis and sent to the dual digital-to-analogue converter (DAC)used for IQ modulation in our vector signal generator.

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EXPERIMENTAL BATH ENGINEERING FOR . . . PHYSICAL REVIEW A 89, 042329 (2014)

C. Direct characterization of the microwave carrier

In order to quantitatively verify the noise engineeringprocess, we begin by measuring the resultant phase noiseon a 12.6-GHz carrier in the presence of bath engineering.Interpreting such data requires a brief quantitative analysis ofthe effect of amplitude and phase modulation as representedin the Fourier domain. In the case of amplitude modulation, asignal consisting of a single-frequency amplitude-modulated(AM) carrier can be expressed as

S(t) = [Aμ + Am sin(ωmt)] sin(ωμt) (4.5)

= Aμ sin(ωμt) + Am

2cos(δ−t) − Am

2cos(δ+t), (4.6)

where Aμ and Am are the amplitudes of the carrier andmodulating sinusoid and ωμ and ωm are their frequencies,respectively. In the second line above we see the signal isrepresented as a weighted sum of the carrier frequencies as wellas two symmetric sideband frequencies δ± = ωμ ± ωm fromthe carrier. Referring back to Eq. (3.11), each comb tooth givesrise to a pair of sidebands with amplitude (power) proportionalto αF (j ) [α2F (j )2]. In this case, the power-law scalings of thecomb teeth, p, and the measured phase-noise power spectrumare identical.

The case of dephasing noise is slightly more complicateddue to the effect of frequency or phase modulation asrepresented in the Fourier domain. Single-frequency phasemodulation with amplitude m at frequency ωm gives a signal

S(t) = Aμ sin (ωμt + m sin(ωmt))

= Aμ

∞∑n=−∞

Jn( m) sin[(ωμ + nωm)t]

≈ Aμ sin(ωμt) + Aμ m

2sin(δ+t) − Aμ m

2sin(δ−t).

(4.7)

Such modulation produces an infinite comb of frequencies,centered around the carrier, spaced by ωm, and weighted byBessel functions [47]. In the last line above we have assumedsmall modulation depth allowing the infinite comb to betruncated beyond first order. As a result of this expression, therelationship between the power law of comb teeth in Eq. (3.7)and the expected form of the phase noise is p ↔ (p − 2).

Phase-noise power spectra are presented in Figs. 4(a)–4(d)using units of dBc/Hz as a function of offset from the carrier,for different forms of bath engineering. These data provide ameasure of the total power at a particular offset referenced tothe carrier in a one-Hertz integration bandwidth. In all cases,we observe a strong increase in the measured phase noise overthe (unmodulated) carrier noise floor up to a cutoff frequencycorresponding to the programed ωc. The form of decay in thephase-noise power law is well described using the expressionsabove, as indicated by guides to the eye superimposed on themeasured data. Agreement is good for both amplitude anddetuning noise. We also observe that as the noise strength (i.e.,modulation depth) increases for dephasing noise, the first-order approximation above fails and the cutoff frequency is no

0.6

0.5

0.4

0.3

0.2

0.1

0.0121086420

Noise strength

1.0

0.5

0.01086420

-90

-75

-60

-45

300200100 300200100

-85

-70

-55

-40

1/

1/2 White

White

-2

-2

-3

c

Am

plitu

deD

epha

sing

Pow

er r

elat

ive

to c

arrie

r (d

Bc/

Hz)

Frequency offset from carrier (Hz)

(a) (b)

(c) (d)

(e)

1.0

0.5

0.01.0

0.5

0.01.0

0.5

0.02.01.51.00.50.0

Time (ms)

1.0

0.5

0.0 (f)

(g)

(h)

(i)

constant

Brig

ht s

tate

pro

babi

lity

Time (ms)

Ram

sey

Dec

ay T

2-1 (

ms-1

)

= .015

= .02

= .0325

= .035

FIG. 4. (Color online) Experimental validation of noise engi-neering. (a)–(d) Phase-noise spectrum of carrier with engineerednoise measured using a vector signal analyzer. Black data representthe unmodulated noise floor at 12.6 GHz, while the red representdata with engineered bath. Blue overlays are guides to the eyedemonstrating the predicted phase-noise scaling associated withparticular power spectra. (e) Scaling of measured T2 as a function ofnoise strength α for white dephasing noise F (j ) ∝ j−1. Error bars aredetermined from the fitting procedure employed in analyzing Ramseydata and blue line represents a quadratic fit, in line with expectations(see text). Inset: representative Ramsey data with overlaid exponentialdecay. Data taken with fixed carrier detuning ∼1 kHz in order to showRamsey fringes. (f)–(i) Rabi oscillations in the presence of amplitudenoise with a white power spectrum measured in experiment (blackmarkers) and calculated via numerical integration of the Schrodingerequation (red markers) for different noise strengths. Line traces showGaussian decay fits.

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A. SOARE et al. PHYSICAL REVIEW A 89, 042329 (2014)

longer clearly visible in these plots due to the infinite comb ofsidebands.

D. Qubits in a noisy bath

We quantitatively demonstrate our bath engineering tech-niques by studying the effect of engineered dephasing noiseon the free evolution of our trapped-ion qubits. We producetime-domain dephasing noise using a discrete comb withfundamental frequency ω0 = 2π × 4 Hz, and a cutoff ωc =2π × 3 kHz. Selecting F (j ) = j−1 yields a white-dephasing-noise power-spectral density.

In our experiments, we perform Ramsey spectroscopy inthe presence of engineered noise. We observe the decay timeconstant of fringe visibility is significantly reduced in thepresence of the engineered bath, as expected. A representativeRamsey experiment is presented in the inset of Fig. 4(e)showing the fringe visibility decays over a time scale oforder milliseconds in the presence of the engineered bath.Since Ramsey spectroscopy measures the relative coherencebetween two oscillators (the LO and the qubit), engineereddephasing in the LO results in net decoherence even duringfree-evolution periods.

The scaling of the measured coherence time withnoise strength is a key validation of our bath engi-neering techniques. We write the ensemble-averaged co-herence for free evolution in the presence of dephas-ing noise as W (t) = exp[−χ (τ )] [48,49] where χ (τ ) =2π

∫ ∞0

Sβ (ω)ω2 sin2(ωτ/2)dω [21]. Incorporating the relevant

form of Sβ(ω) gives

χ (τ ) = 2α2ω20

750∑j=1

sin2(jω0τ/2)

(jω0)2, (4.8)

where the upper limit on the sum is determined from thespecifics of the comb tooth spacing and noise cutoff frequency.The expected decay envelope resulting from the integral aboveand for our value of ω0 is a simple exponential, althoughslight modification yields more complex functional forms (seeAppendix C). As a result, we expect a quadratic scaling of thedecay time constant with α.

We study the scaling of the 1/e coherence time T2 as afunction of the noise strength, parametrized by α. Ramseyfringes are recorded for each value of α and a fit to a sinusoidwith a simple exponential decay envelope is performed in orderto extract T2. These data are plotted as the decay rate T −1

2 ofthe qubit coherence as a function of α in Fig. 4(e). The decayrate is observed to scale as T −1

2 ∝ α2, as expected, with theoverall decay time scales determined by the specifics of thenoise power-spectral density.

Similarly, we study driven evolution such as that presentedin Fig. 2(c), but now in the presence of engineered amplitudenoise [Figs. 4(f)–4(i)]. Increasing the strength of the noiseresults in a Gaussian decay envelope for the Rabi floppingdata with decreasing decay constants. The evolution of themeasured decay constant does not take a convenient analyticform with noise strength and so we perform numericalintegration of the Schrodinger equation considering a drivingfield with the same noise characteristics. The data show good

agreement with the numerical calculations qualitatively andquantitatively.

Other experiments incorporating dephasing noise duringdriven evolution and/or universal noise also reveal behaviorin quantitative agreement with the formulation providedabove, but form the subject of separate papers. Overall,these measurements validate the efficacy of our approach ingenerating a quantitatively useful noise bath in a real quantumsystem.

V. CONCLUSION

In this paper, we have presented a detailed technicalprescription for the quantitative engineering of unitary bathsfor studies of quantum control. We produce a discrete combof noise frequencies possessing an overall scaling chosen toreproduce a noise power spectrum of interest in either thedephasing or amplitude-damping quadrature. We show howthis technique lends itself to simple numerical construction ofcomplex time-dependent noise processes using common IQ-modulated carriers for single-qubit control. We validate ourtechnique through both examination of the modulation form ona vector signal analyzer and through application of engineereddephasing noise to the free evolution of trapped 171Yb+ ions.Our measurements demonstrate that the coherent lifetimeof the qubits probed by a 12.6-GHz carrier incorporatingengineered noise scales as expected based on a simple physicalmodel.

The technique we present is applicable to a wide varietyof experimental systems employing carrier signals in the rfor microwave. It is particularly useful in trying to understandthe spectral sensitivity of various quantum control protocolssuch as dynamical decoupling and dynamically correctedgates. For instance, our group has utilized this techniqueto quantitatively validate the error-suppressing properties ofnovel classes of modulated error-suppressing gates, as will bedescribed in future papers. The incorporation of engineerednoise in such experiments is vital to help elucidate the boundsand performance scaling of such protocols in regimes wherethe measured errors (the signal of interest) are in generalnot sufficiently large to exceed intrinsic state preparation andmeasurement errors. It is also possible to combine this kindof unitary noise engineering with dissipation [50–53], forinstance, through leakage of off-resonant lasers or otherwiseinducing spontaneous emission properties, or to expand thegeneral technique to multilevel manifolds [8]. We hope thatour general technique will prove useful to the quantum controland quantum information communities as they push towardsultrahigh-fidelity gate operations.

ACKNOWLEDGMENTS

This work was partially supported by the US ArmyResearch Office under Contract No. W911NF-11-1-0068, andthe Australian Research Council Centre of Excellence forEngineered Quantum Systems CE110001013, the Office ofthe Director of National Intelligence (ODNI), IntelligenceAdvanced Research Projects Activity (IARPA), through theArmy Research Office, and the Lockheed Martin Corporation.All statements of fact, opinion, or conclusions contained herein

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EXPERIMENTAL BATH ENGINEERING FOR . . . PHYSICAL REVIEW A 89, 042329 (2014)

are those of the authors and should not be construed asrepresenting the official views or policies of IARPA, the ODNI,or the US Government.

APPENDIX A: DERIVATION OF THE SYSTEMHAMILTONIAN

The system considered in this paper consists of a qubit withtransition (angular) frequency ωa , driven by a local oscillatorwith magnetic field component aligned with σz taking the form

B(t) = (t) cos[ωμt + φ(t)]z, (A1)

(t) = C(t) + N (t), (A2)

φ(t) = φC(t) + φN (t) (A3)

with ωμ the carrier frequency. In this formulation, thetime dependence of the phase φ(t) and amplitude (t) hasbeen formally partitioned into components denoted by thesubscripts C and N , capturing the desired control and noisyinteractions, respectively. Using the dipole approximation, thesystem Hamiltonian ( = 1) in the laboratory frame is

HS = ωa

2σz + (t) cos[ωμt + φ(t)]σx . (A4)

The first term corresponds to the Hamiltonian of the qubitunder free evolution, and the second term corresponds tothe qubit-field interaction. Moving to the interaction picturecorotating with the carrier frequency and making the rotating-wave approximation, the dynamics are described by theHamiltonian

H(ωμ)I =

(

2

)σz + 1

4(t)e−iφ(t)σ+ + eiφ(t)σ−, (A5)

where = ωa − ωμ is the carrier detuning from the transitionfrequency, and we define the usual operators σ± = σx ±iσy . Setting = φN (t) = 0, Eq. (A5) reduces to H

(ωμ)I =

12(t)cos[φC(t)]σx + sin[φC(t)]σy, so the resonant carrierfield drives coherent Rabi flopping between the qubit states |1〉and |0〉. The instantaneous Rabi rate in this case is proportionalto (t) and rotations, generated by the spin operator σφC (t), aredriven about the axis r = [cos φC(t), sin φC(t),0] in the xy

plane of the Bloch sphere. For instance, a noise-free π pulseabout the x axis would have N,φN,φC = 0, and C(t) =

for t ∈ [0,τπ ], with τπ = π/.The noise component φN (t) of the engineered phase φ(t),

in the exponentials of Eq. (A5), may be mapped to rotationsabout σz if we move to a second interaction picture defined by

H(ωμ,φN )I := U

†φN

H(ωμ)I UφN

− HφN, (A6)

where UφN(t) = exp[−i

φN (t)2 σz] is the evolution operator under

the the engineered dephasing Hamiltonian HφN:= 1

2 φN (t)σz.Setting the carrier detuning = 0, it is then straightforwardto show

H(ωμ,φN )I = − φN (t)

2σz

+ 1

2(t)cos[φC(t)]σx + sin[φC(t)]σy. (A7)

APPENDIX B: DERIVATION OF NOISEPOWER-SPECTRAL DENSITIES

Let h(t) be any time-dependent function. We use nonunitaryangular frequency notation consistent with the usage inRef. [30] to define a Fourier transform pair

H (ω) = F [h(t)] =∫ ∞

−∞h(t)e−iωtdt, (B1)

h(t) = F−1[H (ω)] = 1

2π

∫ ∞

−∞H (ω)eiωtdω. (B2)

In this case, a time-domain signal β(t) is related to its PSDSβ(ω) by the Wiener-Khintchine theorem [43] which takes theform

〈β(t1)β(t2)〉 = 1

2π

∫ ∞

−∞dω Sβ(ω)eiω(t2−t1). (B3)

In this paper, we assume all noise processes are wide-sensestationary in which case the two-point correlation function〈β(t1)β(t2)〉 depends only on the relative difference τ = t2 − t1between t1 and t2 and reduces to the autocorrelation functionCβ(τ ) := 〈β(t1)β(t1 + τ )〉. The angle brackets now denoteaveraging over all times t1 with respect to which the relative lagτ is defined. Consequently, Cβ(τ ) = 1

2π

∫ ∞−∞ dω Sβ(ω)eiωτ , or

using Eqs. (B1) and (B2),

Sβ(ω) = F[Cβ(τ )

] =∫ ∞

−∞〈β(t)β(t + τ )〉e−iωτ dτ. (B4)

1. Dephasing-noise PSD

We require an expression for the autocorrelation functionof βz(t), the dephasing-noise field, in order to derive thecorresponding PSD Sz(ω) given by Eq. (B4). Using Eq. (3.5),it is straightforward to show

βz(t + τ )βz(t)

= α2ω20

4

J∑j,j ′=1

jj ′F (j )F (j ′)[eiωj τ ei(ωj +ωj ′ )t ei(ψj +ψj ′ )

+ eiωj τ ei(ωj −ωj ′ )t ei(ψj −ψj ′ ) + c.c.]. (B5)

Or, averaging over t ,

〈βz(t + τ )βz(t)〉t

= α2ω20

4

J∑j,j ′=1

jj ′F (j )F (j ′)[eiωj τ ei(ψj +ψj ′ )〈ei(ωj +ωj ′ )t 〉t

+ eiωj τ ei(ψj −ψj ′ )〈ei(ωj −ωj ′ )t 〉t+ e−iωj τ e−i(ψj −ψj ′ )〈e−i(ωj −ωj ′ )t 〉t+ e−iωj τ e−i(ψj +iψj ′ )〈e−i(ωj +ωj ′ )t 〉t ].

Since ωj ,ωj ′ are always positive, we know ωj + ωj ′ is alwayspositive. Consequently, the term e±i(ωj +ωj ′ )t is always anoscillating term with nonzero frequency ±(ωj + ωj ′ ), and

042329-9

A. SOARE et al. PHYSICAL REVIEW A 89, 042329 (2014)

average over t yields

〈e±i(ωj +ωj ′ )t 〉t = 0. (B6)

Similarly, when ±(ωj − ωj ′ ) is nonzero (i.e., when ωj =ωj ′ ⇐⇒ j = j ′), we have

〈e±i(ωj −ωj ′ )t 〉t = 0, j = j ′. (B7)

However, when ωj = ωj ′ (which occurs when j = j ′)

〈e±i(ωj −ωj ′ )t 〉t = 1, j = j ′ (B8)

or more concisely

〈e±i(ωj −ωj ′ )t 〉t = δjj ′ , (B9)

where δij is the Kronecker delta. Thus, the autocorrelationfunction for βz(t) takes the form

〈βz(t + τ )βz(t)〉t

= α2ω20

4

J∑j,j ′=1

jj ′F (j )F (j ′)

× [eiωj τ ei(ψj −ψj ′ )δjj ′ + e−iωj τ e−i(ψj −ψj ′ )δjj ′] (B10)

= α2ω20

4

J∑j=1

j 2[F (j )]2[eiωj τ + e−iωj τ ] (B11)

= α2ω20

2

J∑j=1

[jF (j )]2 cos(ωjτ ). (B12)

Substituting this into Eq. (B4) yields

Sz(ω) =∫ ∞

−∞〈βz(t)βz(t + τ )〉t e−iωτ dτ (B13)

=∫ ∞

−∞

[α2ω2

0

2

J∑j=1

[jF (j )]2 cos(ωjτ )

]e−iωτ dτ (B14)

= α2ω20

2

J∑j=1

[jF (j )]2F [cos(ωjτ )]. (B15)

Using the result from Fourier analysis that

F [cos(ω′τ )] =∫ ∞

−∞cos(ω′τ )e−iωτ dτ

= π [δ(ω − ω′) + δ(ω + ω′)], (B16)

we therefore obtain our result

Sz(ω) = α2ω20

2

J∑j=1

[jF (j )]2[π [δ(ω − ωj ) + δ(ω + ωj )]].

(B17)

2. Amplitude-noise PSD

We require an expression for the autocorrelation functionof β

(t), the amplitude-noise field, in order to derive the

corresponding PSD S(ω) given by Eq. (B4). Using Eq. (3.11),

and following the same procedure used in the above section,it is straightforward to show

〈β(t + τ )β

(t)〉t = α2

2

J∑j=1

[F (j )]2 cos(ωjτ ). (B18)

Substituting this into Eq. (B4) and using Eq. (B16), wetherefore obtain the result

S(ω) = α2

2

J∑j=1

[F (j )]2F [cos(ωjτ )] (B19)

= πα2

2

J∑j=1

[F (j )]2[δ(ω − ωj ) + δ(ω + ωj )]. (B20)

APPENDIX C: DEPENDENCE OF χ ON τ FOR FID

In the case of a pure white-noise-detuning PSD it isrelatively simple to calculate an exact form for χ (τ ). Start-ing with χ (τ ) = 2

π

∫ ∞0

Sβ (ω)ω2 sin2(ωτ/2)dω and incorporating

0.3

0.2

0.1

0.0

(

non.

dim

.)

1086420

8

6

4

2

0

x10

-3

1086420

(ms)

2.0

1.5

1.0

0.5

0.01086420

0 = 4 Hz 0 = 0.1 Hz 0 = 30 Hz

(a) (b) (c)

FIG. 5. (Color online) Dependence of χ (τ ) on τ for various choices of ω0. χ (τ ) is calculated for free evolution in the presence of engineeredwhite detuning noise. The τ axis runs over the time period used in the experiments described in the main text. Without loss of generality, weeliminate α as a free parameter by setting it equal to 1.

042329-10

EXPERIMENTAL BATH ENGINEERING FOR . . . PHYSICAL REVIEW A 89, 042329 (2014)

Sβ(ω) = α2 gives

χ (τ ) = 2α2

π

∫ ∞

0

sin2(ωτ/2)

ω2dω = τα2

2(C1)

giving χ a linear dependence on τ , and hence producing asimple exponential decay in fidelity (fringe visibility). Bycontrast, in the limit of weak low-frequency-dominated noise,use of the small-angle approximation for the sinusoidal termresults in a quadratic dependence χ (τ ) ∝ α2τ 2, yielding aGaussian decay envelope in Ramsey fringe visibility.

The choice of fit function for the Ramsey decay thereforedepends sensitively on the details of the noise model employed.Our engineered white-noise PSD consists of a finite set ofcomb teeth spaced by a finite-frequency interval designed toapproximate a continuous white-noise PSD with a finite-cutofffrequency. In Sec. III D, we defined a noise power spectrum

producing a coherence integral

χ (τ ) = 2α2ω20

750∑j=1

sin2(jω0τ/2)

(jω0)2, (C2)

where ω0 = 4 Hz. In such circumstances, we rely on numericalcalculations to determine the behavior χ (τ ) over the timeinterval relevant to the Ramsey spectroscopy experiments inFig. 5(a). For our choice of ω0 we find good approximationof the integral to a linear function of τ , suggesting the use ofa simple exponential fit to Ramsey decay in free-inductiondecay (FID) experiments with engineered noise. However,we observe that modifying the fundamental frequency of thepower spectrum changes the dependence of χ (τ ) within thesame evolution time interval, requiring careful attention to thenoise model in use when performing quantitative studies ofbath engineering.

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042329-12

Appendix C

arXiv pre-print

100

Improving frequency standard performance by optimized measurement feedback

J. Sastrawan,1 C. Jones,1 I. Akhalwaya,2 H. Uys,2 and M.J. Biercuk1, ∗

1ARC Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, NSW 2006 Australiaand National Measurement Institute, West Lindfield, NSW 2070 Australia

2National Laser Centre, Council for Scientific and Industrial Research, Pretoria, South Africa(Dated: July 16, 2014)

We demonstrate a new approach to precision frequency standard characterization and stabiliza-tion, providing performance enhancements in the presence of non-Markovian noise in the local oscil-lator with “software-only” modifications. We develop a theoretical framework casting various mea-sures for frequency standard variance in terms of frequency-domain transfer functions, incorporatingthe effects of feedback stabilization via a chain of Ramsey measurements. Using this framework weintroduce a novel optimized hybrid feedforward measurement protocol which employs results frommultiple measurements and transfer-function-based calculations of measurement covariance giventhe local oscillator power spectrum. This approach exploits statistical correlations in the local oscil-lator frequency noise to provide high-accuracy corrections in the presence of uncompensated deadtime in the measurement cycle or oscillator-frequency evolution during the Ramsey measurementperiod itself. We present numerical simulations of oscillator performance under competing feedbackschemes and demonstrate benefits in both correction accuracy and long-term oscillator stabilityusing hybrid feedforward.

High-performance frequency standards play a majorrole in technological applications such as network syn-chronization and GPS [1] as well as many fields of physi-cal inquiry, including radioastronomy (very-long-baselineinterferometry) [2], tests of general relativity [3], and par-ticle physics [4]. Atomic clocks exploiting the stability ofCs [5–8] or other atomic references [9–13] are knownas the most precise timekeeping devices available, butconstant performance gains are sought for technical andscientific applications.

A major limit to the performance of passive frequencystandards comes from the quality of the local oscillator(LO) used to interrogate the atomic transition. The LOfrequency evolves randomly in time due to intrinsic in-stabilities from the underlying hardware [10, 11], leadingto deviations of the LO frequency from that of the stableatomic reference. These instabilities are partially com-pensated through use of a feedback protocol designed totransfer the stability of the reference to the LO, but theireffects cannot be mitigated completely. LO fluctuationsultimately produce instabilities in the locked local os-cillator (LLO) due to both uncompensated noise duringinitialization and readout stages of the measurement cy-cle (dead time), as well as phenomena such as aliasing ofnoise at harmonics of the feedback-loop period, known asthe Dick effect [14–16]. Accordingly, significant researchfocus in the frequency standards community has beenplaced on improving LO performance, using e.g. ultra-low-phase-noise cryogenic sapphire oscillators or similar[17, 18], with concomitant increases in hardware infras-tructure requirements and complexity.

In this Letter we demonstrate a method by which boththe accuracy and stability of passive frequency standards

∗ michael.biercuk@sydney.edu.au

can be improved without the need for hardware modifi-cation. We present a new theoretical framework cap-turing the effects of LO noise and feedback protocols inthe frequency domain in order to calculate the expectedcorrelations between multiple sequential measurements.Thus, given statistical knowledge of the LO noise char-acteristics, we are able to produce a new form of hybridfeedforward stabilization incorporating the results of anarbitrary number of past measurements with variable du-ration to calculate an improved correction to the LO. Incases where dead time is significant and there is sub-stantial uncompensated LO evolution, we show that thisapproach allows corrections with improved accuracy tobe applied to the frequency standard. Numerical simu-lations also demonstrate that long-term stability of theLLO is improved through a moving-average correctionscheme, where corrections are made based on weight-ing values determined analytically in the same hybridfeedforward approach. The method described here isa technology-independent software-oriented approach toimproving the performance of frequency standards.

Ramsey spectroscopy provides a means to determinethe average fractional frequency offset over a period, TR;this information is used to determine the correction tobe applied in the standard LO feedback loop (Fig. 1a).We represent the fractional frequency offset of an LOrelative to an ontologically perfect (atomic) referencey(t) ≡ (ν(t)−ν0)/ν0, where ν0 is the reference frequencyand ν(t) is the LO frequency. A Ramsey measurementperformed over the kth time interval [tsk, t

ek], with du-

ration T(k)R ≡ tek − tsk, is characterized by a sensitivity

function g(t) ∈ [0, 1], capturing the extent to which LOfluctuations at some instant t contribute to the measuredoutcome for that interval [19], yielding measurement out-

come yk = 1

T(k)R

∫ tektsky(t)g(t−tsk)dt. In the case of a square

sensitivity function over the Ramsey period, the form of

arX

iv:1

407.

3902

v1 [

quan

t-ph

] 1

5 Ju

l 201

4

2

yk reduces to the time-average of y(t) over the interval[tsk, t

ek].

The performance of the frequency standard is statis-tically characterized, for instance, by the time-domain

sample variance σ2y[N ] = 1

N−1∑Nk=1(yk − 1

N

∑Nl=1 yl)

2

for N sequential finite-duration measurements yk [19].This metric captures the evolution of LO frequency as afunction of time. Similarly we may define the true vari-ance of yk, σ2

y(k) = E[y2k], equal to the expected value

of y2k, since y(t) is assumed to be a zero-mean process.The true variance captures the spread of measurementoutcomes due to different noise realizations in a singletimestep. Applying feedback corrections sequentially af-ter measurements is able to effectively reduce y(t) overmany cycles, improving long-term stability; whether thesample or the true variance is used, larger variance cor-responds to a frequency standard with greater fractionaluncertainty (worse performance).

Despite the power of feedback stabilization, evolutionof y(t) during a measurement period, Tc can cause fre-quency deviations that are uncompensated in the feed-back loop, limiting the accuracy of the corrections andincreasing the statistical variance of the locked frequencystandard. These effects are exacerbated in circumstanceswhere dead time, TD, is substantial, due, for instance,to the need to reinitialize the reference between mea-surements (Fig. 1b). We require an efficient theoreticalframework in which to capture these effects, and hencetransition to the frequency domain, making use of thepower spectral density of the LO, Sy(ω), in order to char-acterize average performance over a hypothetical statisti-cal ensemble. In this description residual LLO instabilitypersists because the feedback is insensitive to LO noiseat high frequencies relative to the inverse measurementtime. Additional instability due to the Dick effect comesfrom aliasing of noise at harmonics of the loop band-width.

We may calculate the effects of measurement, dead-time, and the feedback protocol itself on frequencystandard performance in the frequency domain as fol-lows. Defining a normalised, time-reversed sensi-

tivity function g(tmk − t) = g(t − tsk)/T(k)R , where

g(t) is assumed to be time-reversal symmetric abouttmk , the midpoint of [tsk, t

ek], we can express, for in-

stance, the true variance as a convolution σ2y(k) =

E

[( ∫∞−∞ y(t)g(tmk − t)dt

)2]. Expanding this expression

and using the Wiener-Khinchin theorem gives the truevariance of measurement outcomes as an overlap inte-gral σ2

y(k) = 12π

∫∞0Sy(ω) |Gk(ω)|2 dω, with Gk(ω) ≡∫∞

−∞ g(tmk − t)eiωtdt. Here |Gk(ω)|2 is called the trans-fer function for the kth sample. For measurementsperformed using Ramsey interrogation with π/2 pulsesof negligible duration and zero dead time, the transferfunction has a sinc-squared analytic form |Gk(ω)|2 =

(sin (ωT(k)R /2)/(ωT

(k)R /2))2.

We thus see that this approach allows expression

1

0

g (t)

-50

0

50

y(t)

(a.u

.)

543210Measurement number

1

0

g (t)

t (a.u.)

C(n)k

C(n)k+1

Tc

TR TD

ts1 ts2 ts3 ts4 ts5 te5te4te3te2te1

C(n)k+2

y(LO)k y

(LLO)k

y(LO)(t) y(LLO)(t)

Feedback

Hybrid Feedforward

(a)

(b)

(c)

FIG. 1. (a) Simulated traces for a noisy LO, unlocked (red)and locked with traditional feedback (black), with correctionperiod Tc. The dotted horizontal bars indicate the measure-ment outcomes (samples) over each cycle, which are appliedas correction at the end of the cycle (assuming zero deadtime). The arrows on the far right schematically indicate howlocking reduces the variance of y(t) though it does not elimi-nate it. (b) Schematic timing diagram of a standard feedbackprotocol with Ramsey measurement time, TR and dead timeTD, where TD + TR = Tc. (c) Schematic diagram of hybridfeedforward with an example protocol with n = 3, with theoption of non-uniform duration measurements instead of the

uniform measurements illustrated here. Corrections C(n=3)k

are applied in either non-overlapping blocks of three measure-ments or as a moving average (depicted here). In the lattercase, the covariance matrix must be recalculated to correctlyaccount for any variations in measurement duration. Dashedred arrows indicate the first corrections performed withoutfull calculation of the covariance matrix. This effect vanishesfor k > n.

of time-domain LO variances as overlap integrals be-tween Sy(ω) and the transfer function, |G(ω)|2, captur-ing the effects of the measurement and feedback proto-col. Through this formalism we may calculate analyticexpressions for measures such as the true variance, sam-ple variance and Allan variance for either free-runningLOs or LLOs undergoing feedback stabilization, and itpermits incorporation of arbitrary measurement proto-cols (e.g. arbitrary and dynamic Ramsey periods anddead times) (see Supplementary Material).

This approach is powerful because it may also be em-ployed to craft new measurement feedback protocols de-signed to improve the performance of the LLO. In partic-ular we aim to mitigate the deleterious effect of dead timeon LLO accuracy and stability by combining informationfrom multiple measurements with statistical informationabout LO noise, Sy(ω). Our key insight is that the non-Markovianity of dominant noise processes in typical LOs– captured through the low-frequency bias in Sy(ω) –

3

implies the presence of temporal correlations in y(t) thatmay be exploited to improve feedback stabilization.

The formal basis of our approach is a frequency-domainmeasure of correlation between time-separated measure-ments, using transfer functions derived here. In sum-mary, we calculate a covariance matrix in the frequencydomain via transfer functions to capture the relative cor-relations between sequential measurement outcomes ofan LLO, and use this matrix to derive a linear predictorof the noise at the moment of correction. This predic-tor provides a correction with higher accuracy than thatderived from a single measurement for experimentally-relevant noise spectra, allowing us to improve the per-formance of the LLO. Since the predictor is found usinginformation from previous measurements (feedback) anda priori statistical knowledge of the LO noise (feedfor-ward), we call the scheme hybrid feedforward. Effectively,our ability to predict the evolution of y(t) by exploitingcorrelations captured statistically in Sy(ω) allows feed-back stabilization with increased accuracy and reducedsensitivity to dead time.

In hybrid feedforward, results from a set of n pastmeasurements are linearly combined with weighting co-efficients ck optimized such that the kth correction, Ck,provides maximum correlation to y(tck) at the instant ofcorrection tck (Fig. 1c). Assuming that the LO noise isGaussian, the optimal least minimum mean squares es-timator (MMSE) is linear, and the optimal value of thecorrection is given by Ck = ck · yk: the dot product ofa set of correlation coefficients ck derived from knowl-edge of Sy(ω) and a set of n past measured samples,yk = yk,1, · · · , yk,n. We define an (n+ 1)× (n+ 1) co-variance matrix where the (n + 1)th term represents anideal zero-duration sample at tck and in the second linewe write the covariance matrix in block form:

Σk ≡

σ(yk,1, yk,1) · · · σ(yk,1, y(tck))

σ(yk,2, yk,1) · · · σ(yk,2, y(tck))

· · · · · · · · ·

σ(y(tck), yk,1) · · · σ(y(tck), y(tck))

(1)

≡[Mk FkFTk σ(y(tck), y(tck))

]. (2)

The MMSE optimality condition is then fulfilled for

ck =Fk√

FTk MkFk

wk2π

∫ ∞

0

Sy(ω)dω (3)

where wk is an overall correction gain.The covariance matrix elements are calculated as a

spectral overlap σ(yk, yl) = 12π

∫∞0Sy(ω)G2

k,l(ω)dω usinga pair covariance transfer function

G2k,l(ω) = (ω2T

(k)R T

(l)R )−1

[cos (ω(tsl − tsk)) + cos (ω(tel − tek))

− cos (ω(tel − tsk))− cos (ω(tsl − tek))]

(4)

in the case of arbitrary-length Ramsey interrogationsover the intervals [tsk,l, t

ek,l] (see Supplementary Material).

This is a generalization of the transfer function previouslyderived for the special case of periodic, equal-durationRamsey interrogations [19, 20], and allows effective es-timation of y(t) for any t and for any set of measuredsamples yk.

In the practical setting of a frequency standard exper-iment, we wish to improve both the accuracy of eachcorrection, by maximising the correlation between Ckand y(tck), and the long-term stability of the LLO out-put, captured by the metrics of frequency variance, sam-ple variance, and Allan variance. In order to test thegeneral performance of hybrid feedforward in differentregimes we perform numerical simulations of noisy LOswith user-defined statistical properties, characterized bySy(ω). We produce a fixed number of LO realizations inthe time domain and then use these to calculate measuressuch as the sample variance over a sequence of “measure-ment” outcomes with user-defined Ramsey measurementtimes, dead times, and the like. In these calculationswe may assume that the LO is free running, experienc-ing standard feedback, or employing hybrid feedforward,and then take an ensemble average over LO noise real-izations. Our calculations include various noise powerspectra, with tunable high-frequency cutoffs, includingcommon ‘flicker frequency’ (Sy(ω) ∝ 1/ω), and ‘randomwalk frequency’ (Sy(ω) ∝ 1/ω2) noise, as appropriate forexperiments incorporating realistic LOs,

We begin by studying the improvement in correctionaccuracy associated with a single correction cycle, definedas the extent to which a correction brings yLLO(t) → 0at the instant of correction, t = tck. We define a metricfor accuracy as the inverse of frequency variance at tck

relative to the free-running LO, Ak ≡ 〈yLO(tck)2〉

〈yLLO(tck)2〉 . Closed

form analytic expressions for correction accuracy may becalculated in terms of elements of the covariance matrix(see Supplementary Material).

Tunability in the hybrid feedforward protocol comesfrom the number of measurements to be combined, n, indetermining Ck as well as the selected Ramsey peri-ods, permitting an operator to sample different parts ofSy(ω). As an example, we fix our predictor to considern = 2 sequential measurements and permit the Ram-sey durations to be varied as optimization parameters.A Nelder-Mead simplex optimization over the measure-ment durations finds that a hybrid feedforward protocolconsisting of a long measurement period followed by ashort period maximizes feedforward accuracy (Fig. 2a).This structure ensures that low-frequency components ofSy(ω) are sampled but the measurement sampling thehighest frequency noise contributions are maximally cor-related with y(tck). With Sy(ω) ∝ 1/ω and Sy(ω) ∝ 1/ω2

we observe increased accuracy under hybrid feedforwardwhile the rapid fluctuations in y(t) arising from a whitepower spectrum mitigate the benefits of hybrid feedfor-ward, as expected. In the parameter ranges we have stud-ied numerically we find that correction accuracy is max-

4

imized for n = 2 to 3, with diminishing performance forlarger n.

In all slaved frequency standards we rely on repeatedmeasurements and corrections to provide long-term sta-bility, a measure of how the output frequency of the LLOdeviates from its mean value over time. We study thisby calculating the sample variance of a time-sequenceof measurement outcomes averaged over an ensemble ofnoise realizations, 〈σ2

y[N ]〉. A “moving average” style ofhybrid feedforward provides improved long-term stabil-ity, as the correction Ck will depend on the set of mea-surement outcomes yk = yk−n+1, · · · , yk, among whichprevious corrections have been interleaved, as illustratedin Fig. 1c.

In Fig. 2b we demonstrate the resulting normalizedimprovement in 〈σ2

y[N ]〉 up to N = 100 measurements,calculated using feedback and hybrid feedforward withn = 2, and assuming uniform TR. We observe clearimprovement (reduction) in 〈σ2

y[N ]〉 through the hybridfeedforward approach, with benefits of order 5 − 25%of 〈σ2

y[N ]〉 relative performance improvement over stan-dard measurement feedback. We present data for differ-ent functional forms of Sy(ω), including low-frequencydominated flicker noise (∝ 1/ω), and power spectra(∝ 1/ω1/2) with more significant noise near T−1c . Thebenefits of our approach are most significant in the longterm when high-frequency noise reduces the efficacy ofstandard feedback.

The improvement provided by hybrid feedforward ismost marked for low duty factor d, defined as the ratioof the interrogation time to total cycle time: d ≡ TR/Tc.As d→ 1 we observe that the feedback and hybrid feed-forward approaches converge, as standard feedback cor-rections become most effective when dead time is short-est. However as the dead time increases, feedback ef-ficacy diminishes until 〈σ2

y[N ]〉 for the feedback-lockedLO approaches that for the free-running LO (value unityin Fig. 2c). In this limit the utility of the measurement-feedback diminishes as the LO noise evolves substantiallyduring the dead time, but even here knowledge of corre-lations in the noise allows hybrid feedforward to providesignificant gains in stability. In Fig. 2c we demonstratethat in the presence of a typical 1/ω power spectrum, thepresence of noise spurs near ω/2π = T−1c results in cer-tain regimes where standard feedback makes long-termstability worse, while feedforward provides useful stabi-lization. Exact performance depends sensitively on theform and magnitude of Sy(ω), but results demonstratethat systems with high-frequency noise content aroundω/2π ≈ T−1c benefit significantly from hybrid feedfor-ward.

In this work we have relied on measures of frequencystability such as the true variance and sample variance,rather than the more commonly employed Allan variance,in line with recent experiments [21]. This selection hasbeen deliberate as the form of the Allan variance specifi-cally masks the effect of LO noise components with longcorrelation times. In fact the Allan variance, taking the

10 100

1.20

1.15

1.10

1.05

0.01 0.1 1

1.2

1.0

0.8

0.6

0.4

0.2

1.5

1.0

0.5

0.0

12 3 4 5 6 7 8

102 3 4 5 6 7 8

100

1

0

g (t)

Number of Samples, N Duty Cycle (TR / Tc)

LO

-Nor

m. S

amp.

Var

ianc

e

Nor

mal

ized

Acc

urac

y

Ratio of Ramsey Measurements

1/ω

a)

b) c)

White noise 1/ω noise 1/ω2 noise

1/ω 1/ω

1/2

FB, 1/ω HFF, 1/ω FB, 1/ω + spurs HFF, 1/ω + spurs

Sam

ple

Varia

nce

Impr

ovem

ent

〈 σ2 y [N]

〉(FB) / 〈

σ2 y [N ]

〉(HF

F)

〈 σ2 y [20

] 〉( HFF

/FB)

/ 〈 σ

2 y [20

] 〉(LO)

A k=

2( HFF

) / A

k=1(F

B)

FIG. 2. (a) Calculated accuracy of the first correction for hy-brid feedforward normalized to feedback (accuracy = 1), un-der different forms of Sy(ω) as a function of the ratio of Ram-sey periods between the two measurements employed in con-

structing C(2)k . Accuracy for feedback is calculated assuming

the minimum Ramsey time; thus for the ratio of Ramsey mea-surements taking value unity on the x-axis, the hybrid feedfor-ward scheme takes twice as long as feedback. Inset: depiction

of the form of C(2)k used in hybrid feedforward. (b) Calcu-

lated sample variance for hybrid feedforward, as a function ofmeasurement number N . Data presented as the normalizedratio of 〈σ2

y[N ]〉(FB)/〈σ2y[N ]〉(HFF ) in order to demonstrate

improvement due to hybrid feedforward (larger numbers indi-cate smaller sample variance under hybrid feedforward). Cal-culations assume Sy(ω) ∝ 1/ω, with a high-frequency cutoff

ωc/2π = 100/Tc and Sy(ω) ∝ 1/ω1/2 with a cutoff frequencyωc/2π = 1/Tc, demonstrating the importance of high-f noisenear ω/2π = T−1

c . PSDs with different ω-dependences arenormalised to have the same value at ωlow = 1/100Tc. (c)Calculated 〈σ2

y[N ]〉 for N = 20 as a function of duty factor,normalized to the sample variance for the free-running LO.Data above red dashed line indicate that the standard feed-back approach produces instability larger than that for thefree-running oscillator. Both data sets assume Sy(ω) ∝ 1/ω,with ωc/2π = 100/Tc. Crosses represent data with ten noisespurs superimposed on Sy(ω), starting at ω/2π = 1.15T−1

c ,and increasing linearly with step size 0.15T−1

c .

form Aσ2y(τ) = 1

2 〈(yk+1 − yk)2〉 is employed by the com-munity in part because it does not diverge at long inte-gration times τ due to LO drifts, as would the sample ortrue variance [19, 20, 22, 23]. In numerical calculationswe do not observe significant improvements in Aσ2

y(τ) us-ing hybrid feedforward. However we note that this is onlyan artefact of characterization - the LLO is actually expe-riencing smaller average deviations relative to the stablefrequency reference when using hybrid feedforward.

In summary, we have presented a set of analytical toolsdescribing LLO performance in the frequency domain for

5

arbitrary measurement times, durations, and duty cycles.We have employed these generalized transfer functionsto develop a new software approach to LO feedback sta-bilization in slaved passive frequency standards. Thistechnique leverages a series of past measurements andstatistical knowledge of the noise to improve the accu-racy of feedback corrections and ultimately improve thestability of the slaved LO. We have validated these theo-retical insights using numerical simulations of noisy localoscillators and calculations of relevant stability metrics.

The results we have presented have not by any meansexhausted the space of modifications to clock protocolsavailable using this framework. For instance we havenumerically demonstrated improved correction accuracyusing nonuniform-duration TR over a cycle, as well aslong-term stability improvement using only the simplestcase of uniform TR. These approaches may be com-bined to produce LLOs with improved accuracy at thetime of correction and improved long-term stability. Incases where the penalty associated with increasing TRis modest (lower high-frequency cutoff), such compos-ite schemes can provide substantial benefits as well, im-proving both frequency standard accuracy and stability.Other expansions may leverage the basic analytic formal-ism we have introduced; we have introduced the transferfunctions, |G(ω)|2 and G2

k,l(ω), but have assumed only

the simplest form for the time-domain sensitivity func-tion and fixed overall gain. However, it is possible to crafta measurement protocol to yield |G(ω)|2 that suppressesthe dominant spectral features of the LO noise. We haveobserved that through such an approach one may reducethe impact of aliasing on clock stabilization, providinga path towards reduction of the so-called Dick limit inprecision frequency references.

In the parameter regimes we have studied the relativeperformance benefits of the hybrid feedforward approachare of metrological significance - especially consideringthey may be gained using only “software” modificationwithout the need for wholesale changes to the clock hard-ware. We believe the approach may find special signif-icance in tight-SWAP applications such as space-basedclocks where improving the LO quality is generally im-possible due to system-level limitations. Note: Whilepreparing this manuscript we became aware of relatedwork seeking to employ covariance techniques to improvemeasurements of quantum clocks [24].Acknowledgements: The authors thank H. Ball, D.

Hayes, and J. Bergquist for useful discussions. Thiswork partially supported by Australian Research Coun-cil Discovery Project DP130103823, US Army ResearchOffice under Contract Number W911NF-11-1-0068, andthe Lockheed Martin Corporation.

[1] H. Katori, Nat. Photon. 5, 203 (2011)[2] N. Nand, J. Hartnett, E. N. Ivanov, and G. Santarelli,

IEEE Trans. Microwave Thy and Tech. 59 (2011)[3] C. W. Chou, D. B. Hume, T. Rosenband, and D. J.

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sciencemag.org/content/329/5999/1630.abstract

[4] S. Blatt, A. D. Ludlow, G. K. Campbell, J. W. Thomsen,T. Zelevinsky, M. M. Boyd, J. Ye, X. Baillard, M. Fouche,R. Le Targat, A. Brusch, P. Lemonde, M. Takamoto, F.-L. Hong, H. Katori, and V. V. Flambaum, Phys. Rev.Lett. 100, 140801 (Apr 2008), http://link.aps.org/

doi/10.1103/PhysRevLett.100.140801

[5] J. Guena, M. Abgrall, D. Rovera, P. Laurent, B. Chupin,M. Lours, G. Santarelli, P. Rosenbusch, M. Tobar, R. Li,K. Gibble, A. Clairon, and S. Bize, IEEE Trans. Ultra-son., Ferroelec., and Freq. Control 59 (March 2012)

[6] O. I. de la Convention du Metre, “The international sys-tem of units (si),” (2006)

[7] D. B. Sullivan, J. C. Bergquist, J. J. Bollinger, R. E.Drullinger, W. M. Itano, S. R. Jefferts, W. D. Lee,D. Meekhof, T. E. Parker, F. L. Walls, and D. J.Wineland, J. Res. Natl. Inst. Stand. Technol. 106, 47(January-February 2001)

[8] C. Audoin and B. Guinot, The Measurement of Time,1st ed. (Cambridge University Press, 2001)

[9] P. T. H. Fisk, Rep. Prog. Phys. 60, 761 (1997)[10] N. Hinkley, J. Sherman, N. Phillips, M. Schioppo,

N. Lemke, K. Beloy, M. Pizzocaro, C. Oates, and A. Lud-low, Science 341 (May 2013)

[11] N. Huntemann, M. Okhapkin, B. Lipphardt, S. Weyers,C. Tamm, and E. Peik, phys. rev. lett. 108 (March 2012)

[12] C. W. Chou, D. B. Hume, J. C. J. Koelemeij,D. J. Wineland, and T. Rosenband, Phys. Rev. Lett.104, 070802 (Feb 2010), http://link.aps.org/doi/10.1103/PhysRevLett.104.070802

[13] T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou,A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger,T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C.Swann, N. R. Newbury, W. M. Itano, D. J. Wineland,and J. C. Bergquist, Science 319 (March 2008)

[14] G. Dick, Proc. 19th Annual Precise Time and Time In-terval (PTTI) Appls. and Planning Meeting(1987), http://books.google.com.au/books?id=pX3pSgAACAAJ

[15] Y. Y. Jiang, A. D. Ludlow, N. D. Lemke, R. W. Fox,J. A. Sherman, L.-S. Ma, and C. W. Oates, Nat. Photon.5, 158 (2011), http://dx.doi.org/10.1038/nphoton.

2010.313

[16] M. Takamoto, T. Takano, and H. Katori, Nature Pho-tonics 5, 288 (2011), http://www.nature.com/nphoton/journal/v5/n5/full/nphoton.2011.34.html

[17] J. Hartnett and N. Nand, IEEE Trans. Microwave Thyand Tech. 58, 3580 (2010), ISSN 0018-9480

[18] C. Hagemann, C. Grebing, T. Kessler, S. Falke,N. Lemke, C. Lisdat, H. Schnatz, F. Riehle, and U. Sterr,IEEE Trans. Instr. and Meas. 62, 1556 (2013), ISSN0018-9456

[19] J. Rutman, Proceedings of the IEEE 66, 1048 (1978),ISSN 0018-9219

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6

[21] W. H. Oskay, S. A. Diddams, E. A. Donley, T. M. Fortier,T. P. Heavner, L. Hollberg, W. M. Itano, S. R. Jefferts,M. J. Delaney, K. Kim, F. Levi, T. E. Parker, and J. C.Bergquist, Phys. Rev. Lett. 97, 020801 (Jul 2006), http://link.aps.org/doi/10.1103/PhysRevLett.97.020801

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SUPPLEMENTARY MATERIAL

Frequency-domain variance expressions for generalmeasurement windows

Point-like realisations of the stochastic process y(t) can-not be obtained experimentally. Instead, the LO fre-quency error can be measured by a method that producesintegrated samples, denoted yk and indexed in time byk:

yk ≡1

T(k)R

∫ tek

tsk

y(t)g(t− tsk)dt (5)

where T(k)R ≡ tek − tsk, [tsk, t

ek] is the time interval over

which the kth sample is taken, and g(t) is a sensitivityfunction capturing the extent to which LO fluctuationsat some instant t contribute to the measured outcome forthat sample. The range of g(t) is [0, 1] and its domain

is t ∈ [0, T(k)R ]. The ideal case is the rectangular window

case, where

g(t) =

1 for t ∈ [0, T

(k)R ]

0 otherwise(6)

in which case yk reduces to the time-average of y(t) overthe interval [tsk, t

ek].

The variance of yk, denoted σ2y(k) and often called true

variance [19], is equal to the expected value of y2k, sincey(t) is assumed to be a zero-mean process:

σ2y(k) = E[y2k] (7)

= E

[(1

T(k)R

∫ tek

tsk

y(t)g(t− tsk)dt

)2](8)

Defining a normalised, time-reversed sensitivity function

g(tmk − t) = g(t − tsk)/T(k)R , where g(t) is assumed to

be time-reversal symmetric about tmk , the midpoint of[tsk, t

ek], we can express the integral as a convolution:

σ2y(k) = E

[(∫ ∞

−∞y(t)g(tmk − t)dt

)2](9)

Expanding this expression gives

σ2y(k) = E

[∫ ∞

−∞y(t)g(tmk − t)dt

∫ ∞

−∞y(t′)g(tmk − t′)dt′

]

(10)

=

∫ ∞

−∞

∫ ∞

−∞E[y(t)y(t′)]g(tmk − t)g(tmk − t′)dt′dt

(11)

=

∫ ∞

−∞

∫ ∞

−∞RTSyy (∆t)g(tmk − t)g(tmk − t′)dt′dt

(12)

where RTSyy (∆t) is the two-sided autocorrelation functionand ∆t ≡ t′−t. Substituting the Wiener-Khinchin resultRTSyy (∆t) = F−1STSyy (ω) gives

σ2y(k) =

∫ ∞

−∞

∫ ∞

−∞

(1

2π

∫ ∞

−∞STSyy (ω)eiω(t

′−t)dω

)(13)

× g(tmk − t)g(tmk − t′)dt′dt (14)

Defining the Fourier transform of g(tmk − t):

Gk(ω) ≡∫ ∞

−∞g(tmk − t)eiωtdt (15)

and substituting it into the above expression for σ2y(k)

gives

σ2y(k) =

1

2π

∫ ∞

−∞STSyy (ω)Gk(ω)G∗k(ω)dω (16)

=1

2π

∫ ∞

−∞STSyy (ω) |Gk(ω)|2 dω (17)

=1

2π

∫ ∞

0

Sy(ω) |Gk(ω)|2 dω (18)

where |Gk(ω)|2 is called the transfer function for the kthsample. The substitution of the one-sided PSD Sy(ω) for

the two-sided PSD STSyy (ω) is possible because |Gk(ω)|2is even. This result is similar to the convolution theo-rem, which states that Ff ? g = Ff · Fg, where ?denotes a convolution and f and g are Fourier-invertiblefunctions.

The Allan variance, the conventional measure of fre-quency standard instability, can be expressed analo-gously

Aσ2y(y) =

1

2π

∫ ∞

0

Sy(ω)∣∣AG(ω)

∣∣2 dω (19)

7

where the transfer function, for ideal Ramsey interroga-tion, is

∣∣AG(ω)∣∣2 =

2 sin4 (ωTR/2)

(ωTR/2)2(20)

where TR lacks an index because the definition of the Al-lan variance assumes equal-duration interrogation bins[19]. The Allan variance calculated via this frequency-domain approach can be compared to its value via thetime-domain approach, which consists of finding the vari-ance of the difference between consecutive pairs of mea-surement outcomes:

Aσ2y(y) =

1

2〈(yk+1 − yk)2〉 (21)

where yk is the kth measurement outcome and 〈· · · 〉 mayindicate a time average or an ensemble average, depend-ing on whether y(t) is assumed to be ergodic.

Deriving the pair covariance transfer function

In order to calculate covariances involving multiplemeasurement outcomes and obtaining a useful feedfor-ward predictor, we need a way of capturing the corre-lations between different measurements in the frequencydomain. Using the identity σ2(A±B) = σ2(A)+σ2(B)±2σ(A,B), we define a sum and a difference sensitivityfunction: g+k,l(t) and g−k,l(t), with respect to two mea-

surements indexed k and l. g+k,l(t) and g−k,l(t) are generalfunctions of time with two regions of high sensitivity.

g±k,l(t) ≡

g(t− tsk), for t ∈ [tsk, tek]

±g(t− tsl ), for t ∈ [tsl , tel ]

0, otherwise

(22)

These time-domain sum and difference sensitivity func-tions have their corresponding frequency-domain transferfunctions, defined as their Fourier transforms normalised

by T(k,l)R :

G±k,l(ω) ≡∫ ∞

−∞

(g(tmk − t)T

(k)R

± g(tml − t)T

(l)R

)eiωtdt (23)

Substituting this and the form of the true variance (18)into the variance identity above and rearranging termsgives the covariance of the two measurement outcomes

σ(yk, yl) =1

2π

∫ ∞

0

Sy(ω)

4

( ∣∣∣G+k,l(ω)

∣∣∣2

−∣∣∣G−k,l(ω)

∣∣∣2)dω

(24)

≡ 1

2π

∫ ∞

0

Sy(ω)G2k,l(ω)dω (25)

whereby G2k,l(ω) is defined to be the pair covariance

transfer function. Using (23), it is possible to recoverknown expressions for the true, Allan and sample vari-ances pertaining to the LO, in terms of Sy(ω) and theappropriate transfer functions.

Calculating variances for locked local oscillators

The standard measures for oscillator performance con-sider either a free-running LO or provide a means onlyto statistically characterize measurement outcomes underblack-box conditions. Here we present explicit analyticforms for different measurements of variance in the pres-ence of feedback locking.

We initially make the link between LO and LLO vari-ances via a time-domain treatment. Consider the trajec-tory of the same frequency noise realisation y(t) in thecases of no correction, yLO(t) and correction, yLLO(t).The relation between these two cases of y(t) is

yLLO(t) = yLO(t) +n∑

k=1

Ck (26)

where Ck refers to the value of the kth frequency correc-tion applied to the LO, n of which have occurred beforetime t.

Under traditional feedback, each correction is directlyproportional to the immediately preceding measurementoutcome: Ck = wky

LLOk , where wk is correction gain.

Since yLLOk is calculated by convolving yLLO(t) with asensitivity function pertaining to the measurement pa-rameters, (26) is a recursive equation in general. It ispossible to cancel all but one of the recursive terms bysetting the correction gain equal to the inverse of the av-

erage sensitivity gk ≡∫ T (k)

R

0g(t)/T

(k)R dt of the preceding

measurement, i.e. wk = −g−1k , where the minus sign in-dicates negative feedback. With this constraint we canwrite

yLLOk = yLOk − gkgk−1

yLOk−1 (27)

and for a Ramsey interrogation and measurement withnegligibly short pulses, gk = 1.

The frequency variance of an LLO can be foundstraightforwardly, by substituting (26) into the definitionof frequency variance, with the additional substitution of(27) for the particular case of traditional feedback:

8

Var[yLLO(t)] = 〈yLLO(t)2〉 − 〈yLLO(t)〉2 (28)

=

⟨(yLO(t) +

n∑

k=1

Ck

)2⟩(29)

=

⟨(yLO(t)− yLOn

gn

)2⟩(30)

= 〈yLO(t)2〉+1

g2n〈(yLOn )2〉 − 2

gn〈yLO(t)yLOn 〉

(31)

= 〈yLO(t)2〉+σ2yLO(n)

g2n− 2

gnσ(yLO(t), yLOn )

(32)

where the progression from (29) to (30) is valid for tra-ditional feedback only, and 〈yLLO(t)〉 = 0 by assumptionand n indexes the last measurement before t.

Although the LLO frequency variance under hybridfeedforward for more than a single cycle cannot be ex-pressed in a closed non-recursive form, a considerationof a single cycle can provide a value for 〈yLLO(tck)2〉 interms of covariance matrix elements, which in turn pro-vides a metric for the accuracy of the hybrid feedforwardcorrection:

Ak ≡〈yLO(tck)2〉〈yLLO(tck)2〉 (33)

=

(1 + w2

k − wk|Fk|2√

FTk MkFk

)−1(34)

The true variance for an LLO can be found in a sim-ilar way by substituting (27) into the definition of truevariance:

σ2yLLO(k) = Var[yLLOk ] (35)

= Var

[yLOk − gk

gk−1yLOk−1

](36)

= σ2yLO(k) +

(gkgk−1

)2

σ2yLO(k − 1) (37)

− 2gkgk−1

σ(yLOk−1, yLOk ) (38)

where the appropriate forms of the measurement transferfunction (18) and the pair covariance transfer functioncan be substituted in to express σ2

yLLO(k) in terms of

Sy(ω).The expected value of the LLO sample variance can

be found by substituting (26) into the definition of thesample variance, producing a generic expression for tra-ditional feedback (one measurement per correction cycle)and hybrid feedforward (multiple measurements per cy-cle):

E[σ2yLLO[N ]] =

1

N − 1

N∑

k′=1

σ2yLLO(k′) +

1

N2

N∑

p′=1

N∑

q′=1

σ(yLLOp′ , yLLOq′ )− 2

N

N∑

l′=1

σ(yLLOk′ , yLLOl′ )

(39)

=1

N − 1

N∑

k′=1

(σ2yLO(k′) + g2k′

bk′/nc∑

r=1

bk′/nc∑

s=1

σ(Cr, Cs)− 2gk′

bk′/nc∑

u=1

σ(yLOk′ , Cu)

)

+1

N2

N∑

p′=1

N∑

q′=1

σ(yLOp′ + gp′

bp′/nc∑

p=1

Cp, yLOn + gq′

bq′/nc∑

q=1

Cq)−2

N

N∑

l′=1

σ(yLOk′ + gk′

bk′/nc∑

u=1

Cu, yLOl′ + gl′

bl′/nc∑

v=1

Cv)

(40)

=1

N − 1

N∑

k′=1

(σ2yLO(k′) + g2k′

bk′/nc∑

r=1

bk′/nc∑

s=1

σ(Cr, Cs)− 2gk′

bk′/nc∑

u=1

σ(yLOk′ , Cu)

)

+1

N2

N∑

p′=1

N∑

q′=1

(σ(yLOp′ , y

LOq′ ) + gp′ gq′

bp′/nc∑

p=1

bq′/nc∑

q=1

σ(Cp, Cq)

)

− 2

N

N∑

l′=1

(σ(yLOk′ , y

LOl′ ) + gk′ gl′

bk′/nc∑

k=1

bl′/nc∑

l=1

σ(Ck, Cl)

)(41)

where in the case of hybrid feedback, N is defined to be total number of measurements and n is the number ofmeasurements per cycle. The summation signs with unprimed indices are sums over whole cycles (of which there are

9

bN/nc) and the primed indices are sums over all N measurements. In general, E[σ2yLLO[N ]] contains recursive terms

that cannot be concisely expressed in terms of the LO PSD Sy(ω) and covariance transfer function G2(ω).In the case of traditional feedback, the distinction between primed and unprimed indices disappears and the

expression reduces to:

E[σ2yLLO[N ]] =

1

N − 1

N∑

k=1

(σ2yLO(k) + g2k

k−1∑

r=1

k−1∑

s=1

σ(Cr, Cs)− 2gk

k−1∑

u=1

σ(yLOk , Cu)

)

+1

N2

N∑

p=1

N∑

q=1

(σ(yLOp , yLOq ) + gpgq

p−1∑

w=1

q−1∑

x=1

σ(Cx, Cy)

)

− 2

N

N∑

l=1

(σ(yLOk , yLOl ) + gkgl

k−1∑

y=1

l−1∑

z=1

σ(Cy, Cz)

)(42)

where each term can be expressed in terms of Sy(ω) and G2(ω).The LLO Allan variance can be found by substituting (27) into the definition of the Allan variance (21):

Aσ2yLLO(k) =

1

2E[(yLLOk+1 − yLLOk )2] (43)

=1

2E

[(yLOk+1 −

gk+1

gkyLOk − yLOk +

gkgk−1

yLOk−1

)2](44)

=1

2

(σ2yLO(k + 1) +

(1 +

gk+1

gk

)2

σ2yLO(k) +

(gkgk−1

)2

σ2yLO(k − 1)

+2gkgk−1

σ(yLOk+1, yLOk−1)− 2

(1 +

gk+1

gk

)σ(yLOk , yLOk+1)− 2(gk + gk+1)

gk−1σ(yLOk , yLOk−1)

)(45)

Appendix D

MATLAB code for Frequency StandardSimulationThis appendix includes the key scripts that constitute the numerical model for calculating, simulatingand searching for feedforward protocols.

D.1 queue.m

1 clear all

2 tstart=clock;

3 % =======INITIAL VARIABLES ========

4 % the one that is stepped by q is replaced inside the loop

5 ensemble=100;

6 power=-1; % dominant noise power law

7 b12=1e-3; % low-frequency noise cutoff

8 b23=1e1; % high-frequency noise cutoff

9 Tc=1; % cycle time

10 Tr=0.1; % interrogation (Ramsey) time

11 gran = 0.5e-1; % time-domain granularity

12 duty=Tr/Tc; % measurement duty cycle

13 gain=1; % overall correction gain

14 measnum=10; % number of simulated measurements

15 hff=0; % hybrid feedforward toggle

16 overlap=1; % overlapping hybdrid feedforward toggle

17 averages=1; % averaged measurements for intergration time scan

18 fixed = 1; % the scaling factor for the PSD

19 usecmat=1; % FALSE to override the covariance matrix with own weightings

20 order='mdc'; % Defining cycle structure: mdc = ...

'measure-deadtime-correct', dmc = 'deadtime-measure-correct',dmdc ...

='dead-measure-dead-correct'

21 bins=2; % number of measurement bins in the protocol

22 seq_type='uniform'; % for FF: choices are 'uniform' or 'first'

23 scale=1; % scale factor for the long first bin protocol

24

25 % ======= INITIALISE DATA VECTORS AND MATRICES ======

26

27 qstep=[0.1]; % specify the scan points

28 %qstep=[10/676.9];

29 qmax=numel(qstep); % set the number of steps

110

30 %qmax=1;

31 qvec=[1:qmax]; % vector of steps

32 viewvec=[100]; % select the trace to see in figure 3

33

34 time=zeros(qmax,Tc*measnum/gran);

35 freqvarlo=zeros(qmax,Tc*measnum/gran);

36 freqvarfb=zeros(qmax,Tc*measnum/gran);

37 freqvarff=zeros(qmax,Tc*measnum/gran);

38 sampvarlo=zeros(qmax,measnum);

39 sampvarfb=zeros(qmax,measnum);

40 sampvarff=zeros(qmax,measnum);

41 allanvarlo=zeros(qmax,1);

42 allanvarfb=zeros(qmax,1);

43 allanvarff=zeros(qmax,1);

44 measvarlo=zeros(qmax,measnum);

45 measvarfb=zeros(qmax,measnum);

46 measvarff=zeros(qmax,measnum);

47 meanlo=zeros(qmax,1);

48 meanfb=zeros(qmax,1);

49 meanff=zeros(qmax,1);

50 meanlodev=zeros(qmax,1);

51 meanfbdev=zeros(qmax,1);

52 meanffdev=zeros(qmax,1);

53

54 for q=1:qmax

55 duty=qstep(q); % allocate the stepped variable

56 Tc=Tr/duty;

57 Tr

58

59 hff=0;

60 xcentrein=ones(100,1);

61

62 str=['Starting FB scanpoint ',num2str(q),' of ',num2str(qmax)]

63 [intime,infreqvarlo,infreqvarllo,insampvarlo,insampvarllo,inmeasvarlo,...

64 inmeasvarllo,inallanvarlo,inallanvarllo,inmeanlo,inmeanllo,inylodev,...

65 inyllodev,xcentreout] = timedomain(fixed,power,b12,b23,Tc,gran,...

66 ensemble,measnum,duty,gain,hff,overlap,averages,xcentrein,usecmat,...

67 order,bins,scale,seq_type);

68

69 time(q,:)=intime;

70

71 if size(insampvarlo,2) 6=size(sampvarlo,2)

72 insampvarlo=[zeros(1,size(sampvarlo,2)-size(insampvarlo,2)) insampvarlo];

73 insampvarllo=[zeros(1,size(sampvarlo,2)-size(insampvarllo,2)) insampvarllo];

74 inallanvarlo=[zeros(1,size(allanvarlo,2)-size(inallanvarlo,2)) inallanvarlo];

75 inallanvarllo=[zeros(1,size(allanvarlo,2)-size(inallanvarllo,2)) ...

inallanvarllo];

76 inmeasvarlo=[zeros(1,size(measvarlo,2)-size(inmeasvarlo,2)) inmeasvarlo];

77 inmeasvarllo=[zeros(1,size(allanvarlo,2)-size(inmeasvarllo,2)) inmeasvarllo];

78 end

79

111

80 freqvarlo(q,:)=infreqvarlo;

81 freqvarfb(q,:)=infreqvarllo;

82 sampvarlo(q,:)=insampvarlo;

83 sampvarfb(q,:)=insampvarllo;

84 allanvarlo(q)=inallanvarlo;

85 allanvarfb(q)=inallanvarllo;

86 measvarlo(q,:)=inmeasvarlo;

87 measvarfb(q,:)=inmeasvarllo;

88 meanlo(q)=inmeanlo;

89 meanfb(q)=inmeanllo;

90 meanlodev(q)=inylodev;

91 meanfbdev(q)=inyllodev;

92

93 hff=1;

94 xcentrein=xcentreout;

95

96 clear insampvarlo

97 clear insampvarllo

98 clear inallanvarlo

99 clear inallanvarllo

100 clear inmeasvarlo

101 clear inmeasvarllo

102

103 str=['Starting HFF scanpoint ',num2str(q),' of ',num2str(qmax)]

104 [intime,infreqvarlo,infreqvarllo,insampvarlo,insampvarllo,inmeasvarlo,...

105 inmeasvarllo,inallanvarlo,inallanvarllo,inmeanlo,inmeanllo,inylodev,...

106 inyllodevm,xcentreout] = timedomain(fixed,power,b12,b23,Tc,gran,...

107 ensemble,measnum,duty,gain,hff,overlap,averages,xcentrein,usecmat,...

108 order,bins,scale,seq_type);

109

110 if size(insampvarlo,2) 6=size(sampvarlo,2)

111 insampvarlo=[zeros(1,size(sampvarlo,2)-size(insampvarlo,2)) insampvarlo];

112 insampvarllo=[zeros(1,size(sampvarlo,2)-size(insampvarllo,2)) insampvarllo];

113 inallanvarlo=[zeros(1,size(allanvarlo,2)-size(inallanvarlo,2)) inallanvarlo];

114 inallanvarllo=[zeros(1,size(allanvarlo,2)-size(inallanvarllo,2)) ...

inallanvarllo];

115 inmeasvarlo=[zeros(1,size(measvarlo,2)-size(inmeasvarlo,2)) inmeasvarlo];

116 inmeasvarllo=[zeros(1,size(measvarlo,2)-size(inmeasvarllo,2)) inmeasvarllo];

117 end

118

119

120 freqvarlo(q,:)=(infreqvarlo+freqvarlo(q,:))./2;

121 freqvarff(q,:)=infreqvarllo;

122 sampvarlo(q,:)=(insampvarlo+sampvarlo(q,:))./2;

123 sampvarff(q,:)=insampvarllo;

124 allanvarlo(q)=(inallanvarlo+allanvarlo(q))./2;

125 allanvarff(q)=inallanvarllo;

126 measvarlo(q,:)=(inmeasvarlo+measvarlo(q,:))./2;

127 measvarff(q,:)=inmeasvarllo;

128 % meanlo(q)=(meanlo(q)+inmeanlo)./2;

129 meanff(q)=inmeanllo;

112

130 % meanlodev(q)=(meanlodev(q)+inylodev)./2;

131 meanffdev(q)=inyllodev;

132

133 end

134

135 % PLOT RESULTS %

136

137 figure(1) % Frequency variance vs time

138 hold on

139

140 plot(time',freqvarfb')

141 hold on

142 plot(time',freqvarff','r')

143 plot(time',freqvarlo','k')

144

145 xlabel('Time')

146 ylabel('Instantaneous variance')

147 title(['LLO Instantaneous variance vs. time. FF = ',num2str(hff),', ...

overlap = ',num2str(overlap)])

148

149 figure(2) % Sample variance vs measurement number

150

151 % plot(truevarlo','Color',[0 0.8 0],'LineStyle','-','Marker','*')

152 % hold on

153 % plot(truevarllo','Color',[0.5 0.8 0],'LineStyle','-','Marker','*')

154 plot(sampvarff(:,2:end)','LineStyle','-','Marker','*')

155 hold on

156 plot(sampvarfb(:,2:end)','LineStyle','-','Marker','+')

157 plot(sampvarlo(:,2:end)','LineStyle','-','Marker','o')

158 plot(measvarff(:,2:end)','LineStyle','-','Marker','*','Color','r')

159 hold on

160 plot(measvarfb(:,2:end)','LineStyle','-','Marker','+','Color','r')

161 plot(measvarlo(:,2:end)','LineStyle','-','Marker','o','Color','r')

162 % errorbar(sampvarllo(2:end)',devsigma(2:end),'Color',[0.5 0.8 0])

163 % errorbar(sampvarlo(2:end)',devsigmaLO(2:end),'Color','m')

164 xlabel('Number of measurements')

165 ylabel('Sample & measurement variance')

166 title(['LO and FF LLO sample & measure variances. FF = ',num2str(hff),', ...

overlap = ',num2str(overlap)])

167

168

169 figure(3) % Sample variance vs selected variable

170

171 % plot(truevarlo,'Color',[0 0.8 0],'LineStyle','-','Marker','*')

172 % hold on

173 % plot(truevarllo,'Color',[0.5 0.8 0],'LineStyle','-','Marker','*')

174 plot(qstep,sampvarff(:,viewvec),'LineStyle','-','Marker','*')

175 hold on

176 plot(qstep,sampvarfb(:,viewvec),'LineStyle','-','Marker','+')

177 plot(qstep,sampvarlo(:,viewvec),'LineStyle','-','Marker','o')

178 % errorbar(sampvarllo(2:end),devsigma(2:end),'Color',[0.5 0.8 0])

113

179 % errorbar(sampvarlo(2:end),devsigmaLO(2:end),'Color','m')

180 xlabel('Selected variable')

181 ylabel('Sample variance')

182 title(['LO and FF LLO sample variances. FF = ',num2str(hff),', overlap = ...

',num2str(overlap)])

183

184 figure(4) % Normalised sample variance vs selected variable

185

186 % plot(truevarlo,'Color',[0 0.8 0],'LineStyle','-','Marker','*')

187 % hold on

188 % plot(truevarllo,'Color',[0.5 0.8 0],'LineStyle','-','Marker','*')

189 plot(qstep,sampvarff(:,viewvec)./sampvarlo(:,viewvec),'LineStyle','-','Marker','*')

190 hold on

191 plot(qstep,sampvarfb(:,viewvec)./sampvarlo(:,viewvec),'LineStyle','-','Marker','+')

192 % errorbar(sampvarllo(2:end),devsigma(2:end),'Color',[0.5 0.8 0])

193 % errorbar(sampvarlo(2:end),devsigmaLO(2:end),'Color','m')

194 xlabel('Selected variable')

195 ylabel('Sample variance (normalised to LO)')

196 title(['LO and FF LLO sample variances. FF = ',num2str(hff),', overlap = ...

',num2str(overlap)])

197

198 figure(5)

199 semilogx(qstep,abs(meanlo),'Marker','o')

200 hold on

201 errorbar(qstep,abs(meanlo),meanlodev)

202 semilogx(qstep,abs(meanfb),'Marker','+')

203 errorbar(qstep,abs(meanfb),meanfbdev)

204 semilogx(qstep,abs(meanff),'Marker','*')

205 errorbar(qstep,abs(meanff),meanffdev)

206

207 figure(6) % Allan variance vs selected variable

208 interrogationstep=qstep*duty*Tc

209

210 loglog(interrogationstep,allanvarff,'LineStyle','-','Marker','*')

211 hold on

212 loglog(interrogationstep,allanvarfb,'LineStyle','-','Marker','+')

213 loglog(interrogationstep,allanvarlo,'LineStyle','-','Marker','o')

214 xlabel('Selected variable')

215 ylabel('Allan variance')

216 title(['LO and FF LLO Allan variances. FF = ',num2str(hff),', overlap = ...

',num2str(overlap)])

217

218 figure(7) % Normalised Allan variance vs selected variable

219

220 plot(qstep,allanvarff/allanvarlo,'LineStyle','-','Marker','*')

221 hold on

222 plot(qstep,allanvarfb./allanvarlo,'LineStyle','-','Marker','+')

223 xlabel('Selected variable')

224 ylabel('Allan variance (normalised to LO)')

225 title(['LO and FF LLO Allan variances. FF = ',num2str(hff),', overlap = ...

',num2str(overlap)])

114

226

227 % end timer

228 tend=clock;

229 elapsed=etime(tend,tstart)

D.2 timedomain.m

1 function [time,instlo,instllo,meansigmaLO,meansigma,truevarlo,truevarllo,...

2 aveallanvarlo,aveallanvarllo,yloave,ylloave,ylodev,yllodev,xcentreout] =...

3 timedomain(fixed,power,boundary12,boundary23,T,gran,rep,measnum,duty,gain,...

4 ff,overlap,averages,xcentrein,usecmat,order,bins,scale,seq_type)

5

6

7 %=============== Define the Power Spectral Density ===============

8 % Low cutoff below which PSD has power law 'lowfpower', high cutoff above

9 % which PSD has high-f roll-off function

10

11 locked=1; % apply correction or not, more efficient than just setting ...

gain = 0

12

13 timemax=T*measnum; % length of time series in time units

14 fine=timemax/gran; % Number of grains in the time vector

15

16 %boundary12=10^-5;

17 % boundary23=cutoff;

18 %boundary01=10^-5;

19

20 batch=10000; % putting the ensemble in batches to save RAM

21 maxmatsize=5e7;

22

23 if batch>rep

24 batch=rep;

25 str=['Batch size reduced to ',num2str(batch)]

26 end

27

28 if measnum*batch/gran>maxmatsize

29 reduce=(measnum*batch/gran/maxmatsize);

30 batch=batch/reduce;

31 str=['Batch size reduced by a factor of ',num2str(reduce)]

32 end

33

34 %============ Choose traditional feedback or hybrid feedforward ===========

35

36 %ff=1; % choose hybrid feedforward (1) or traditional feedback (0)

37 if ¬ff38 overlap=0; % traditional feedback does not need a moving average flag

39 bins=1; % traditional feedback only uses one bin

40 scale=1;

41 end

42

115

43

44 analytic=1;% if FB, whether or not to calculate analytic

45

46 if (ff==1&&bins==1)||(ff==0&&bins 6=1)

47 % error('Mismatch between number of bins and feedback/feedforward ...

setting') % prevent inconsistent inputs

48 end

49

50 if (ff 6=1&&overlap==1)

51 % error('You are trying to do a moving predictor with traditional ...

feedback') % prevent inconsistent inputs

52 end

53

54 %============= Define the vectors in frequency and time ===============

55

56

57 if (strcmp(seq_type,'uniform')&&scale>1)

58 error('You are trying to apply a scale factor to a uniform bin ...

protocol') % prevent inconsistent inputs

59 end

60

61 if (¬strcmp(seq_type,'uniform')&&scale==1&&ff==1)62 error('You are trying to apply unity scale factor to a first bin ...

protocol') % prevent inconsistent inputs

63 end

64

65 if (¬ff&&scale>1)66 error('You are trying to apply a scale factor to traditional feedback') ...

% prevent inconsistent inputs

67 end

68

69 %================ Define the sensitivity function =================

70

71 if ff

72 int_times=zeros(bins+1,2);

73 cycle_int_times=zeros(bins+1,2,bins);

74 for j=1:bins

75 if strcmp(seq_type,'uniform')

76 if strcmp(order,'dmc')

77 int_times(j,:)=[T*(j-duty),T*j]; % dmc

78 elseif strcmp(order,'mdc')

79 int_times(j,:)=[T*(j-1),T*(j-1+duty)]; % mdc

80 elseif strcmp(order,'dmdc')

81 int_times(j,:)=[T*(j-(1+duty)/2),T*(j-(1-duty)/2)]; %dmdc

82 else

83 error('Not a valid order of measurement, deadtime and correction')

84 end

85 elseif strcmp(seq_type,'first')

86 if scale>1

87 if overlap

88 loopend=(bins+scale-1);

116

89 else

90 loopend=1;

91 end

92 for w=1:loopend

93 if w≤scale % calculating what the inversion should be

94 v=1;

95 else

96 v=w-scale+1;

97 end

98 if j==v

99 if strcmp(order,'dmc')

100 cycle_int_times(j,:,w)=[T*(j-duty),T*(scale+j-1)];

101 elseif strcmp(order,'mdc')

102 cycle_int_times(j,:,w)=[T*(j-1),T*(scale+j-2+duty)];

103 elseif strcmp(order,'dmdc')

104 cycle_int_times(j,:,w)=[T*(j-(1+duty)/2),T*(scale+j-1-(1-duty)/2)];

105 else

106 error('Not a valid order of measurement, deadtime and correction')

107 end

108 elseif j<v

109 if strcmp(order,'dmc')

110 cycle_int_times(j,:,w)=[T*(j-duty),T*(j)];

111 elseif strcmp(order,'mdc')

112 cycle_int_times(j,:,w)=[T*(j-1),T*(j-1+duty)];

113 elseif strcmp(order,'dmdc')

114 cycle_int_times(j,:,w)=[T*(j-(1+duty)/2),T*(j-(1-duty)/2)];

115 else

116 error('Not a valid order of measurement, deadtime and correction')

117 end

118 elseif j>v

119 if strcmp(order,'dmc')

120 cycle_int_times(j,:,w)=[T*(scale+j-1-duty),T*(scale+j-1)];

121 elseif strcmp(order,'mdc')

122 cycle_int_times(j,:,w)=[T*(scale+j-2),T*(scale+j-2+duty)];

123 elseif strcmp(order,'dmdc')

124 cycle_int_times(j,:,w)=[T*(scale+j-1-(1+duty)/2),T*(scale+j-1-(1-duty)/2)];

125 else

126 error('Not a valid order of measurement, deadtime and correction')

127 end

128 end

129 end

130 else

131 if j==1

132 if strcmp(order,'dmc')

133 int_times(j,:)=[T*(j-duty),T*(scale+j-1)];

134 elseif strcmp(order,'mdc')

135 int_times(j,:)=[T*(j-1),T*(scale+j-2+duty)];

136 elseif strcmp(order,'dmdc')

137 int_times(j,:)=[T*(j-(1+duty)/2),T*(scale+j-1-(1-duty)/2)];

138 else

139 error('Not a valid order of measurement, deadtime and correction')

117

140 end

141 else

142 if strcmp(order,'dmc')

143 int_times(j,:)=[T*(scale+j-1-duty),T*(scale+j-1)];

144 elseif strcmp(order,'mdc')

145 int_times(j,:)=[T*(scale+j-2),T*(scale+j-2+duty)];

146 elseif strcmp(order,'dmdc')

147 int_times(j,:)=[T*(scale+j-1-(1+duty)/2),T*(scale+j-1-(1-duty)/2)];

148 else

149 error('Not a valid order of measurement, deadtime and correction')

150 end

151 end

152 end

153 end

154 end

155

156 if scale>1

157 for w=1:loopend

158 if w≤scale % calculating what the test distance should be, long first bin ...

only

159 y(w)=w-1;

160 else

161 y(w)=0;

162 end

163 cycle_int_times(bins+1,:,w)=[T*(bins+scale-1+y(w)),T*(bins+scale-1+y(w))+1e-12];

164 %show_bins(cycle_int_times(:,:,w),'parameterspace',w)

165 end

166 int_times(:,:)=cycle_int_times(:,:,1);

167 Tc=int_times(bins+1,1);

168 else

169 int_times(bins+1,:)=[T*(bins+scale-1),T*(bins+scale-1)+1e-12];

170 show_bins(int_times,'parameterspace',1)

171 Tc=int_times(bins+1,1);

172 end

173 else

174 if strcmp(order,'dmc')

175 int_times=[T*(1-duty),T;T,T+1e-12];

176 elseif strcmp(order,'mdc')

177 int_times=[0,T*duty;T,T+1e-12];

178 elseif strcmp(order,'dmdc')

179 int_times=[T*(1-duty)/2,T*(1+duty)/2;T,T+1e-12];

180 else

181 error('Not a valid order of measurement, deadtime and correction')

182 end

183 show_bins(int_times,'feedback',1);

184 Tc=T;

185 end

186

187 for b=1:rep/batch

188

118

189 % ============== SIMULATE NOISE, MEASUREMENT AND CONTROL ...

==================

190

191 recmLLO=zeros(batch,floor(timemax/Tc*bins)); % matrix for recorded ...

measurement of LLO

192 recmLO=zeros(batch,floor(timemax/Tc*bins)); % matrix for recorded ...

measurement of LO

193 corrLLO=zeros(batch,floor(timemax/Tc*bins)); % matrix for corrections ...

applied to LLO

194 freq=logspace(-5,5,1000);

195

196 % find the index of freq corresponding to PSD roll-off boundary points

197 f12=1;

198 f23=numel(freq);

199 for f=1:numel(freq)

200 if freq(f)<boundary12

201 f12=f;

202 elseif freq(f)≥boundary12&&freq(f)<boundary23

203 f23=f;

204 end

205 if freq(f)*T<1e-3

206 ft=f;

207 else

208 ft=1;

209 end

210 end

211

212 f_low = freq(f12);

213 f_high=freq(f23);

214

215 str=['Starting batch ',num2str(b),' of ',num2str(rep/batch)]

216

217 [recylo,time,nf1,nf2,nf3,boundary3end,normtype,psdy,psdx]=...

218 colourednoise(fixed,batch,timemax,gran,power,boundary12,boundary23,T,duty,overlap);

219

220 % calculate the modified PSD for overlapping correction

221 figure(90)

222 loglog(freq,nf1(freq),'b')

223 hold on

224 loglog(freq,nf2(freq),'b')

225 loglog(freq,nf3(freq),'b')

226

227

228 if overlap

229 nf1corr = @(x)(2*nf1(x).*( (sin(x.*pi*T*duty)./(x.*pi*T*duty)).^2 ...

230 - (2*cos(2*x.*pi*T) - cos(2.*x.*pi*T*(1-duty)) - cos(2.*x.*pi*T*(1+duty)) ...

)...

231 ./(2*x.*pi.*T.*duty).^2));

232 nf2corr = @(x)(2*nf2(x).*( (sin(x.*pi*T*duty)./(x.*pi*T*duty)).^2 ...

233 - (2*cos(2*x.*pi*T) - cos(2.*x.*pi*T*(1-duty)) - cos(2.*x.*pi*T*(1+duty)) ...

)...

119

234 ./(2*x.*pi.*T.*duty).^2));

235 nf3corr = @(x)(2*nf3(x).*( (sin(x.*pi*T*duty)./(x.*pi*T*duty)).^2 ...

236 - (2*cos(2*x.*pi*T) - cos(2.*x.*pi*T*(1-duty)) - cos(2.*x.*pi*T*(1+duty)) ...

)...

237 ./(2*x.*pi.*T.*duty).^2));

238

239 if strcmp(normtype,'area')

240 invalpha=integral(@(x)(nf1corr(x)),0,boundary12,'ArrayValued',true)+...

241 integral(@(x)(nf2corr(x)),boundary12,boundary23,'ArrayValued',true);

242 % ...

integral(@(x)(nf3corr(x)),boundary23,boundary3end,'ArrayValued',true)./fixed;

243 elseif strcmp(normtype,'lowcutoff')

244 invalpha=nf1corr(f12);

245 elseif strcmp(normtype,'highcutoff')

246 invalpha=nf2corr(f23);

247 else

248 error('Work out the noie type please.')

249 end

250

251

252 nf1=@(x)(nf1corr(x)./invalpha);

253 nf2=@(x)(nf2corr(x)./invalpha);

254 nf3=@(x)(nf3corr(x)./invalpha);

255

256 end

257

258 figure(91)

259 loglog(freq,[nf1(freq(1:f12-1)) nf2(freq(f12:end))],'m')

260

261

262

263 recyllo=recylo;

264 corrylo=zeros(batch,size(recylo,2));

265 gee=zeros(T/gran,1);

266

267 if strcmp(order,'dmc')

268 gap=0;

269 elseif strcmp(order,'mdc')

270 gap=T*(1-duty);

271 elseif strcmp(order,'dmdc')

272 gap=T*(1-duty)/2;

273 end

274

275 freq=linspace(1/timemax,1/gran,fine);

276

277

278

279 % ===== CALCULATE HFF PREDICTOR COEFFICIENTS USING COVARIANCE MATRIX ...

=========

280 if ff

281 if scale>1&&overlap==1

120

282 cyclelim=int_times(bins+1,1)/T; % as many corrections as T_seq/T

283 else

284 cyclelim=1;

285 cycle_int_times(:,:,1)=int_times(:,:);

286 int_times

287 if overlap==1

288 y(1)=1; % what to do for uniform: 1=apply whitened PSD

289 else

290 y(1)=0;

291 end

292 end

293 for w=1:cyclelim

294 cycle_cmat(:,:,w)=intnoisecov(cycle_int_times(:,:,w),nf1,boundary12,nf2,...

295 boundary23,nf3,psdy,psdx)

296

297

298 for i=1:bins

299 for j=1:bins

300 V(i,j)=cycle_cmat(i,j,w);

301 if abs(V(i,j))<1e-10 % getting rid of spurious small values

302 V(i,j)=0;

303 error('The correlations are too low to perform predictive feedforward')

304 end

305 end

306 ET(i)=cycle_cmat(i,bins+1,w);

307 if abs(ET(i))<1e-10 % getting rid of spurious small values

308 ET(i)=0;

309 error('The correlations are too low to perform predictive feedforward')

310 end

311 end

312 F=cycle_cmat(bins+1,bins+1,w)

313

314 if usecmat

315 coeff(:,w)=ET.*sqrt(F./(ET*V*ET')).*gain; % use the ...

covariance matrix to determine weightings

316 % coeff(:,w)=coeff(:,w).*gain/(sum(coeff(:,w)));% further ...

normalisation so the gain sum of coefficients is 1

317 else

318 coeff(:,w)=[-0.308;-0.3498;0.8601].*gain;% use ...

predetermined fixed weightings

319 end

320

321 if ¬(size(coeff,1)==bins)322 error('The number of weighting coefficients are not equal to the number ...

of bins per cycle')

323 end

324

325 if overlap

326 corrnum=bins; % as many corrections as measurements per cycle

327 else

328 corrnum=1; % one correction per cycle

121

329 end

330 end

331 else

332

333 corrnum=1;

334

335 end

336

337 if ff

338 coeff

339 end

340

341 % ========= override coefficients to preset values =============

342 %

343

344 % ================= EXTRACT MEASUREMENT OUTCOMES ====================

345

346 % Build the full time-domain sensitivity function

347

348 for t=1:T/gran

349 for q=1:bins

350 if t≥round((int_times(q,1))/gran)&&t<((int_times(q,2))/gran)

351 gee(t)=1;

352 end

353 end

354 end

355

356 sense=gee;

357 figure(76)

358 plot(gee,'+')

359

360 for m=2:measnum

361 sense=vertcat(sense,gee);

362 end

363 prepmLO=reshape(recylo',T/gran,measnum,batch); % reshape the array ...

for Ramsey averaging

364

365 % Match dimensions of traces and sensitivity function and multiply

366 if measnum>size(sense,2)

367 sense=vertcat(sense,zeros(size(recylo,2)-size(sense,1),1));

368 elseif size(recylo,2)<size(sense,2)

369 sense=sense(1:measnum);

370 end

371

372 sensemat=(sense*ones(size(sense,2),batch));

373 prepsense=reshape(sensemat,T/gran,measnum,batch);

374

375 measylo=prepmLO.*prepsense; % modulating the traces by the ...

sensitivity function

376 recmLO=reshape(mean(measylo),measnum,batch)'./duty; % reshape and ...

take Ramsey average

122

377 recmLLO=recmLO; % initialise LLO measurement outcomes to LO outcomes

378

379 % =========== CALCULATE CORRECTIONS USING FB OR HFF ==================

380

381 if ff

382 for m=1:bins-1 % for k < n (first few measurements)

383 recmLLO(:,m)=recmLLO(:,m)-sum(corrLLO,2);

384 w=1;

385 recmLLO(:,1:m);

386 coeff(bins-m+1:bins,1);

387 corrLLO(:,m)=(recmLLO(:,1:m)*coeff(bins-m+1:bins,1));

388 end

389 for m=bins:measnum % for k ≥ n (rest of the measurements)

390 recmLLO(:,m)=recmLLO(:,m)-sum(corrLLO,2);

391

392 if size(coeff,2)>1

393 w=mod(m,bins)+1;

394 else

395 w=1;

396 end

397 recmLLO(:,1:m);

398 coeff(1:bins,w);

399 corrLLO(:,m)=(recmLLO(:,m-bins+1:m)*coeff(:,w));

400 end

401 else

402 for m=1:measnum

403 recmLLO(:,m)=recmLLO(:,m)-sum(corrLLO,2);

404 corrLLO(:,m)=gain.*recmLLO(:,m);

405 end

406 end

407

408 % apply the correction

409 for m=1:measnum

410 if m<measnum

411 corrylo(:,numel(gee)*m+1:numel(gee)*(m+1))=sum(corrLLO(:,1:m),2)*ones(1,numel(gee));

412 end

413 pastcorr(:,numel(gee)*(m-1)+1:numel(gee)*m)=corrLLO(:,m)*ones(1,numel(gee));

414 end

415

416

417 recyllo=recylo-corrylo; % apply corrections to the LLO

418 meanyllo=mean(mean(recyllo)); % find the mean y_LLO value

419 meanylo=mean(mean(recylo)); % find the mean y_LO value

420

421 % apply averaging over contiguous measurements for variance calcs

422

423 if mod(measnum,averages) 6=0

424 error('All values of the averaging set must be factors of the ...

total number of measurements')

425 end

426

123

427 averecmLLO=zeros(batch,measnum/averages);

428 averecmLO=zeros(batch,measnum/averages);

429

430 for k=1:size(averecmLO,2)

431 averecmLO(:,k)=mean(recmLO(:,(k-1)*averages+1:k*averages),2);

432 averecmLLO(:,k)=mean(recmLLO(:,(k-1)*averages+1:k*averages),2);

433 end

434

435 % calculate sample variance per trace

436

437 for k=2:size(averecmLLO,2)

438 clear meanelementLLO

439 clear meanelementLO

440 clear meanaveLLO

441 clear meanaveLO

442 for j=1:k

443 meanelementLLO(:,j)=averecmLLO(:,j);

444 meanelementLO(:,j)=averecmLO(:,j);

445 end

446 meanaveLLO=sum(meanelementLLO,2)./k;

447 meanaveLO=sum(meanelementLO,2)./k;

448 clear varelementLLO

449 clear varelementLO

450 for i=1:k

451 varelementLLO(:,i)=(recmLLO(:,i)-meanaveLLO).^2;

452 varelementLO(:,i)=(recmLO(:,i)).^2;

453 end

454 sampvarLLO(:,k)=sum(varelementLLO,2)/(k-1);

455 sampvarLO(:,k)=sum(varelementLO,2)/(k-1);

456 end

457

458 % calculate Allan variance per trace

459

460

461 if overlap||¬ff462 for k=2:size(averecmLLO,2)

463 allanLLO(:,k-1)=(averecmLLO(:,k)-averecmLLO(:,k-1));

464 allanLO(:,k-1)=(averecmLO(:,k)-averecmLO(:,k-1));

465 end

466 elseif scale==1

467 for k=1:floor(size(averecmLLO,2)/bins)

468 allanLLO(:,k)=averecmLLO(:,1+k*bins)-averecmLLO(:,1+(k-1)*bins);

469 allanLO(:,k)=averecmLO(:,1+k*bins)-averecmLO(:,1+(k-1)*bins);

470 end

471 end

472

473

474 % calculate average Allan variance over ensemble

475

476 if scale==1

477 allanvarllo=sum((allanLLO.^2),2)/(2*size(allanLLO,2));

124

478 allanvarlo=sum((allanLO.^2),2)/(2*size(allanLO,2));

479 end

480

481

482 % ============= CALCULATE VARIANCE QUANTITIES ==============

483

484 if batch==rep

485 % yloave=mean(recylo); % time-average

486 % ylloave=mean(recyllo); % average numerical instantaneous LLO ...

variance

487

488

489 % find numerical LO instantantaneous var

490 instlo=var(recylo);

491 aveinstlo=mean(instlo);

492

493 % find numerical LLO instantaneous var

494 instllo=var(recyllo);

495 aveinstllo=mean(instllo);

496

497 % find num LLO true var values

498 truevarllo=var(recmLLO);

499 truevarlo=var(recmLO);

500

501

502 % find num Allan var values

503 if scale==1

504 allanvarllo=var(allanLLO)/2;

505 allanvarlo=var(allanLO)/2;

506 end

507 % print the average Allan var values

508 % batchallanvarlo=mean(allanvarlo);

509 % devallanvarlo=sqrt(var(allanvarlo));

510 % batchallanvarllo=mean(allanvarllo);

511 % devallanvarllo=sqrt(var(allanvarllo));

512

513 else

514

515 % yloave=zeros(size(recylo,2));

516 % ylloave=zeros(size(recyllo,2));

517 instlo=zeros(1,size(recylo,2));

518 aveinstlo=0;

519 instllo=zeros(1,size(recyllo,2));

520 aveinstllo=0;

521 truevarllo=zeros(1,size(recmLLO,2));

522 truevarlo=zeros(1,size(recmLO,2));

523

524 % find num Allan var values

525 %

526 allanvarllo=var(allanLLO)/2;

527 allanvarlo=var(allanLO)/2;

125

528 %

529 % % print the average Allan var values

530 % aveallanvarlo=mean(allanvarlo);

531 % devallanvarlo=sqrt(var(allanvarlo));

532 % aveallanvarllo=mean(allanvarllo);

533 % devallanvarllo=sqrt(var(allanvarllo));

534 %

535

536 end

537

538

539 % find numerical sample var LLO values

540 %batchvarsigma(b,:)=var(sampvarLLO);

541 batchmeansigma(b,:)=mean(sampvarLLO);

542 %batchdevsigma()=sqrt(varsigma);

543

544 % find numerical sample var LO values

545 %batchvarsigmaLO=var(sampvarLO);

546 batchmeansigmaLO(b,:)=mean(sampvarLO);

547 %batchdevsigmaLO=sqrt(varsigmaLO);

548

549 batchylo(b,:)=mean(recylo,2);

550 batchyllo(b,:)=mean(recyllo,2);

551

552 batchyloave(b)=mean(batchylo(b,:),2);

553 batchylloave(b)=mean(batchyllo(b,:),2);

554

555 if scale==1

556 batchallanvarlo(b)=mean(allanvarlo,2);

557 batchallanvarllo(b)=mean(allanvarllo,2);

558 end

559 end

560

561 % find numerical sample var LLO values

562 %varsigma=var(sampvarLLO);

563 meansigma=mean(batchmeansigma,1);

564 %devsigma=sqrt(varsigma);

565

566 % find numerical sample var LO values

567 %varsigmaLO=var(sampvarLO);

568 meansigmaLO=mean(batchmeansigmaLO,1);

569 %devsigmaLO=sqrt(varsigmaLO);

570

571 if scale==1

572 aveallanvarlo=mean(batchallanvarlo,2);

573 aveallanvarllo=mean(batchallanvarllo,2);

574 else

575 aveallanvarlo=zeros(size(batchmeansigma,1));

576 aveallanvarllo=mean(size(batchmeansigma,1));

577 end

578

126

579 yloave=mean(batchyloave);

580 ylloave=mean(batchylloave);

581

582 yloprepdev=reshape(batchylo,1,[]);

583 ylloprepdev=reshape(batchyllo,1,[]);

584 ylodev=sqrt(var(yloprepdev,0,2));

585 yllodev=sqrt(var(ylloprepdev,0,2));

586

587 figure(rep+1+ff)

588 hold off

589 hist(ylloprepdev,ceil((rep-20)/100)+20)

590 yllonumel=hist(ylloprepdev,ceil((rep-20)/100)+20);

591 hold on

592 scatter(mean(ylloprepdev),[1],[],'r')

593 scatter(mean(ylloprepdev)-yllodev,[1],[],'r','Marker','+')

594 scatter(mean(ylloprepdev)+yllodev,[1],[],'r','Marker','+')

595 if ¬ff596 title(['Distribution of FB LLO trace means - mean ',num2str(ylloave),' ...

deviation ',num2str(yllodev),'. Total number of elements = ...

',num2str(sum(yllonumel))])

597 else

598 title(['Distribution of Ismail-optimised LLO trace means - mean ...

',num2str(ylloave),' deviation ',num2str(yllodev),'. Total number of ...

elements = ',num2str(sum(yllonumel))])

599 end

600

601 figure(rep+3)

602 if ¬ff603 hist(ylloprepdev,ceil((rep-20)/100)+20)

604 [¬,xcentreout] = hist(ylloprepdev,ceil((rep-20)/100)+20);

605 hold on

606 else

607 hist(ylloprepdev,xcentrein,'Color','r')

608 xcentreout=xcentrein;

609 hold off

610 end

611

612 figure(rep)

613 hold off

614 hist(yloprepdev,ceil((rep-20)/100)+20)

615 ylonumel=hist(yloprepdev,ceil((rep-20)/100)+20);

616 hold on

617 scatter(mean(yloprepdev),[1],[],'r')

618 scatter(mean(yloprepdev)-ylodev,[1],[],'r','Marker','+')

619 scatter(mean(yloprepdev)+ylodev,[1],[],'r','Marker','+')

620 title(['Distribution of LO trace means - mean ',num2str(yloave),' ...

deviation ',num2str(ylodev),'. Total number of elements = ...

',num2str(sum(ylonumel))])

621

622 % ============= PLOT RESULTS =======================

623 %

127

624

625 figure(81)

626 plot(time,recylo(1,:),'k')

627 hold on

628 plot(time,recyllo(1,:),'b')

629 plot(time,corrylo(1,:),'r')

630

631 intimesize=size(time);

632 infreqvarlosize=size(instlo);

633 infreqvarllosize=size(instllo);

634 inmeansigmaLO=size(meansigmaLO);

635 inmeansigmaLLO=size(meansigma);

636 inmeasvarLO=size(truevarlo);

637 inmeasvarLLO=size(truevarllo);

638

639

640

641 %

642 % if size(recmLLO,2)/bins≥2

643 % figure(5)

644 % hold on

645 % if ff

646 % if overlap

647 % plot(allantime,allanvarllo,'Color',[0 0.5 ...

0],'LineStyle','-','Marker','*')

648 % else

649 % plot(allantime,allanvarllo,'Color',[0 1 ...

0],'LineStyle','-','Marker','*')

650 % end

651 % hold on

652 % plot(allantime,allanvarlo,'Color',[0 0 1],'LineStyle','-','Marker','*')

653 % xlabel('Number of measurements')

654 % ylabel('Allan variance')

655 % title('LO and FF LLO Allan variances')

656 % else

657 % plot(allantime,allanvarllo,'Color',[0 0 ...

0],'LineStyle','-','Marker','*')

658 % hold on

659 % plot(allantime,allanvarlo,'Color',[0 0 1],'LineStyle','-','Marker','*')

660 % xlabel('Number of measurements')

661 % ylabel('Allan variance')

662 % title('LO and FB LLO Allan variances')

663 % end

664 % end

665 %

666 %

667 % if ¬locked668 % figure(4)

669 % hold on

670 % plot(time,var(recylo),'Color',[0 1 0])

128

671 % line([time(1) time(end)],[aninstlo ...

aninstlo],'LineStyle','--','Color',[0 1 0],'LineWidth',2)

672 % title('Comparing analytic and numerical instantaneous LO ')

673 % end

674 %

675 % figure(99)

676 % plot(elapsed,portion)

677 %

678 % figure(8)

679 % plot(printvar)

680 % hold on

681 % plot(printnorm,'r')

D.3 intnoisecov.m

1 function [ cov_matrix, allan_cov_matrix, f_ul] = intnoisecov(int_times,...

2 noise_function1,...

3 boundary12,...

4 noise_function2,...

5 boundary23,...

6 noise_function3,...

7 psdy,psdx)

8

9 discrete=1;

10

11 freq1=logspace(-2,2,10000);

12

13 % intnoisecov() returns a covariance matrix of random variables for noise

14 % integrated over the periods specified by [int_times], for a stationary,

15 % zero-mean gaussian noise process specified by noise functions and their

16 % boundaries.

17

18 % ** Note that frequency is always in units of 1/t. **19

20 % [int_times]: (n x 2) matrix, where each row specifies the start and stop

21 % time of an interval over which noise is integrated. Note that this

22 % assumes noise g(t) sensitivity comes in rectangular pulses of unit

23 % height.

24

25 % [noise_function1]: the first noise function, extending from f = 0 to f =

26 % infinity if there are no further arguments, or f = boundary12.

27

28 % [boundary##]: frequency defining the transition between piecewise noise

29 % in segments # and #+1. Optional arguments which are only used if the next

30 % noise function is specified.

31

32 % [noise_function#]: noise function in segment #. Optional for # > 1.

33

34 % Example usage: c_mat = intnoisecov([0,1;1,2], @(f)exp(-f.^2/(2*f_c^2)))

35

129

36 % Example usage: c_mat = intnoisecov([0,1;1,2], @(f)100, 1, @(f)100/f)

37

38 % if nargin > 6

39 % error('intnoisecov:TooManyInputs', ...

40 % 'requires at most 6 optional inputs');

41 % end

42

43 if mod(nargin,2) 6= 0

44 error('intnoisecov:MissingInputs', ...

45 'number of inputs should be even');

46 end

47

48 % Create flags for whether noise functions are used, whether a maximum

49 % frequency for integration is specified.

50 % num_noise_segments = 1;

51 % boundaries = [0;0];

52 %

53 nf1 = @(f)(noise_function1(f));

54 % %nf1 = noise_function1;

55 %

56 % % Several optional argument pairs

57 % switch nargin

58 % case 4

59 % num_noise_segments = 2;

60 % if ¬isscalar(boundary12)61 % error('intnoisecov:NonScalarInput', ...

62 % 'input boundary12 must be scalar');

63 % end

64 % boundaries(1) = boundary12;

65 % nf2 = fcnchk(noise_function2);

66 % %nf2 = noise_function2;

67 % case 6

68 % num_noise_segments = 3;

69 % if ¬isscalar(boundary12)||¬isscalar(boundary23)70 % error('intnoisecov:NonScalarInput', ...

71 % 'inputs boundary12 and boundary23 must be scalar');

72 % end

73 % boundaries(1) = boundary12;

74 % boundaries(2) = boundary23;

75 nf2 = @(f)(noise_function2(f));

76 nf3 = @(f)(noise_function3(f));

77 % end

78

79 num_bins = size(int_times,1)-1;

80

81 % Build the covariance matrix element-by-element. Note that this is a

82 % symmetric matrix, so only need upper triangular elements

83

84 cov_matrix = zeros(num_bins+1);

85 allan_cov_matrix = zeros(num_bins+1);

86

130

87 granularity=1000;

88 fmin=-15;

89 fmax=log10(boundary23);

90 fplotting=logspace(fmin,fmax,granularity*(fmax-fmin));

91

92 figure(90)

93 loglog(freq1,nf1(freq1))

94 hold on

95 loglog(freq1,nf2(freq1))

96 loglog(freq1,nf3(freq1))

97

98 for i = 1:num_bins+1

99 for j = 1:num_bins+1

100

101

102

103 % Create shorthand variables for interval start/stop times

104 t1 = int_times(i,1);

105 t2 = int_times(i,2);

106 t3 = int_times(j,1);

107 t4 = int_times(j,2);

108

109

110 % Define an ultra-low frequency boundary for numerical-integration

111 % stability. Below this frequency, use Taylor approximation for

112 % transfer function

113 max_time_lag = max(abs(t4-t1),abs(t2-t3));

114 % frequency always in units of 1/t

115 % relative error in Taylor series should be < 1e-8.

116 %

117 min_time_lag = min(abs(t4-t2),abs(t3-t1));

118 %

119 %

120 if(min_time_lag == 0)

121 min_time_lag = min(abs(t4-t3),abs(t2-t1));

122 abs(t4-t3);

123 abs(t2-t1);

124

125 end

126

127 if(min_time_lag == 0)

128 Ti=abs(t2-t1)

129 Tj=abs(t4-t3)

130 error('intnoisecov: The minimum timescale is too small for ...

MATLAB')

131 min_time_lag = (1e-10)*max_time_lag;

132

133 end

134 %

135 min_time_lag;

136

131

137 f_ul = 0.01/min_time_lag; % frequency always in units of 1/t

138

139

140 f_ulbigger=0;

141 bottom=0;

142 smaller=f_ul;

143

144 if f_ul > boundary23

145 f_ulbigger=1;

146 smaller=boundary23;

147 end

148

149 f_ulbigger=0;

150

151 % relative error in Taylor series should be < 1e-8.

152 f_ub = 10^(fmax-2); % integration upper bound

153

154 %min_time_lag;

155

156 if f_ul==0

157 error('intnoisecov: The minimum timescale is too small for ...

MATLAB')

158 end

159

160

161

162 limh=floor(granularity*(log10(smaller)-fmin));

163 lims12=floor(granularity*(log10(boundary12)-fmin)); % change this ...

for three-part noise

164 lims23=floor(granularity*(log10(boundary23)-fmin));

165

166 Htaylor_cutoff=fplotting(limh);

167 S_boundary12=fplotting(lims12);

168 S_boundary23=fplotting(lims23);

169

170

171 % Testing for the FID case

172

173

174

175 % Three cases: f_ul is below the first boundary (infinity if only

176 % one segment; f_ul is below the second boundary; f_ul is above the

177 % second boundary.

178

179 % Taylor coefficients

180 c1 = -0.5*((t4-t2)^2+(t3-t1)^2-(t4-t1)^2-(t3-t2)^2);

181 c2 = (1/24)*(2*pi)^2*((t4-t2)^4+(t3-t1)^4-(t4-t1)^4-(t3-t2)^4);

182 H_Taylor = @(f)(((c1 + c2*f.^2))/(t4-t3)/(t2-t1));

183

184 %./(pi*(t4-t3+t2-t1)))

185

132

186 H_exact = @(f)((1./(2*pi.*f).^2).*(1/(t4-t3)/(t2-t1)).*...

187 (cos(2*pi.*f*(t4-t2))+cos(2*pi.*f*(t3-t1))...

188 -cos(2*pi.*f*(t4-t1))-cos(2*pi.*f*(t3-t2))));

189

190

191 H_f =@(f)(1./(2*pi.*f)./(t2-t1)).*(sin(2*pi.*f.*(t3-t1))-...

192 sin(2*pi.*f.*(t3-t2)));

193

194 H_Allan = @(f)(1./(2.*pi.*f).^2).*(1/(t4-t3)/(t2-t1)).*...

195 (cos(2*pi.*f*(t4-t2))+cos(2*pi.*f*(t3-t1))...

196 +cos(2*pi.*f*(t4-t1))+cos(2*pi.*f*(t3-t2))...

197 - 2*cos(2*pi.*f*((t4+t3)/2 - t1)) - 2*cos(2*pi.*f*((t3+t4)/2 ...

- t2))...

198 - 2*cos(2*pi.*f*((t1+t2)/2 - t3)) - 2*cos(2*pi.*f*((t1+t2)/2 ...

- t4))...

199 + 4*cos(2*pi.*f*((t1+t2)/2 - (t3+t4)/2))) ;

200

201 ca1 = -0.5*((t4-t2)^2+(t3-t1)^2+(t4-t1)^2+(t3-t2)^2-...

202 2*((t4+t3)/2-t2)^2-2*((t4+t3)/2-t1)^2-2*((t1+t2)/2-t3)^2-...

203 2*((t1+t2)/2-t4)^2+4*((t1+t2)/2-(t3+t4)/2)^2);

204 ca2 = (1/24)*(2*pi)^2*((t4-t2)^4+(t3-t1)^4+(t4-t1)^4+(t3-t2)^4-...

205 2*((t4+t3)/2-t2)^4-2*((t4+t3)/2-t1)^4-2*((t1+t2)/2-t3)^4-...

206 2*((t1+t2)/2-t4)^4+4*((t1+t2)/2-(t3+t4)/2)^4);

207

208 H_Allan_Taylor = @(f)((ca1 + ca2*f.^2)/(t4-t3)/(t2-t1));

209

210 if discrete

211 if (i==num_bins+1&&j==num_bins+1)

212 cov_matrix(i,j)= H_f(psdx)*psdy';

213 elseif (i==1&&j==2)

214 cov_matrix(i,j)= H_exact(psdx)*psdy';

215 figure(69)

216 loglog(psdx,H_exact(psdx).*psdy,'ro')

217 hold on

218 loglog(psdx,psdy,'bo')

219 loglog(psdx,H_exact(psdx),'go')

220 else

221 cov_matrix(i,j)= H_exact(psdx)*psdy';

222

223 end

224 [i j];

225

226 else

227

228 if ¬(i==num_bins+1)&&¬(j==num_bins+1)229

230

231 if f_ulbigger==0

232

233

234 cov_matrix(i,j)= ...

133

235 integral( @(x)(H_Taylor(x).*nf1(x)),0,boundary12) ...

+...

236 integral( ...

@(x)(H_Taylor(x).*nf2(x)),boundary12,f_ul)+...

237 integral( @(x)(H_exact(x).*nf2(x)),f_ul,boundary23);

238

239 line=216;

240

241 elseif f_ulbigger==1

242

243 cov_matrix(i,j)= ...

244 integral( @(x)(H_Taylor(x).*nf1(x)),0,boundary12) ...

+...

245 integral( ...

@(x)(H_Taylor(x).*nf2(x)),boundary12,boundary23);

246

247 line=224;

248

249 end

250

251

252 allan_cov_matrix(i,j)=...

253 integral(@(x)(H_Allan_Taylor(x).*nf1(x)),0,boundary12) ...

+...

254 integral(@(x)(H_Allan_Taylor(x).*nf2(x)),boundary12,f_ul) ...

+...

255 integral(@(x)(H_Allan(x).*nf2(x)),f_ul,boundary23);

256

257

258

259 elseif (i==num_bins+1&&j==num_bins+1)

260

261 cov_matrix(i,j)= ...

262 integral(@(x)(nf1(x)),bottom,boundary12) +...

263 integral(@(x)(nf2(x)),boundary12,boundary23);

264

265 allan_cov_matrix(i,j)=...

266 integral(nf1,bottom,boundary12) +...

267 integral(nf2,boundary12,boundary23);

268

269 line=243;

270

271 else

272

273 if f_ulbigger==0

274 cov_matrix(i,j)= ...

275 integral( @(x)(nf1(x)),bottom,boundary12) +...

276 integral( @(x)(nf2(x)),boundary12,f_ul)+...

277 integral( @(x)(H_f(x).*nf2(x)),f_ul,boundary23);

278

279 line=253;

134

280

281 elseif f_ulbigger==1

282

283 cov_matrix(i,j)= ...

284 integral( @(x)(nf1(x)),bottom,boundary12) +...

285 integral( @(x)(nf2(x)),boundary12,boundary23);

286

287 line=261;

288

289

290

291 end

292

293 if num_bins==1

294 [t1,t2,t3,t4];

295 end

296

297 end

298 end

299 end

300 end

301

302

303

304 for i=1:num_bins+1

305 for j=1:num_bins+1

306

307

308 % Set other half of off-diagonal entries

309

310

311 if j>i

312

313 cov_matrix(j,i) = cov_matrix(i,j);

314 %cov_error(j,i)=cov_error(i,j);

315 %cov_rel_error(j,i)=cov_rel_error(i,j);

316

317 allan_cov_matrix(j,i) = allan_cov_matrix(i,j);

318 %cov_error(j,i)=cov_error(i,j);

319 %cov_rel_error(j,i)=cov_rel_error(i,j);

320 end

321

322 end

323 end

324 end

D.4 colourednoise.m

1 function [trace,time,nf1,nf2,nf3,b3end,normtype,psdoverlap,psdx] = ...

colourednoise(fixed,ensemble,maxtime,mintime,power2,b12,b23,T,duty,overlap)

135

2

3

4 %========= INTENDED POWER SPECTRAL DENSITY ==========

5 % specifies the power laws and rolloffs for a 3-part PSD

6

7 b01=1e-1; % lowest point of part 1

8 %b12=1e-1; % boundary between parts 1&2

9 %b23=1e1; % boundary between parts 2&3

10 b3end=1e1; % highest point of part 3

11

12 power1=0; % power law slope of part 1

13 %power2=-1; % power law slope of part 2

14 power3=0; % power law slop of part 3

15

16 nf1=@(f)((b12/b23).^(power2).*(f./b12).^(power1)); % Constant chosen to ...

be continuous with 2nd part of PSD

17 nf2=@(f)(f./b23).^(power2); % 2nd part defined so that PSD = 1 at ...

boundary, prior to normalisation

18 nf3=@(f)(0); % high-f roll-off = 0 corresponds to a hard cut-off

19

20

21 spurson=0 % toggles whether to insert spurs symmetrically around 1/T

22

23 %=========== NORMALISE INTENDED PSD ===========

24 normtype='area';

25 if ¬...(strcmp(normtype,'area')||strcmp(normtype,'lowcutoff')||strcmp(normtype,'highcutoff'))

26 error('Error: The type of normalisation you chose has not been coded ...

yet.')

27 end

28

29 if strcmp(normtype,'area')

30 normalpha=integral(@(x)(nf1(x)),b01,b12)+...

31 integral(@(x)(nf2(x)),b12,b23)+...

32 integral(@(x)(nf3(x)),b23,b3end,'ArrayValued',true); % area under ...

the PSD

33 elseif strcmp(normtype,'lowcutoff')

34 normalpha=nf1(b12); % PSD evaluated at b12

35 elseif strcmp(normtype,'highcutoff')

36 normalpha=nf2(b23); % PSD evaluated at b23

37 end

38

39 nf1=@(f)(nf1(f)./normalpha.*fixed);

40 nf2=@(f)(nf2(f)./normalpha.*fixed);

41 nf3=@(f)(nf3(f)./normalpha.*fixed);

42

43 alpha=integral(@(x)(nf1(x)),b01,b12)+...

44 integral(@(x)(nf2(x)),b12,b23)+...

45 integral(@(x)(nf3(x)),b23,b3end,'ArrayValued',true); % with area ...

normalisation, this should be equal to fixed

46

136

47 %=========== TIME AND FREQUENCY UNITS ================

48 %mintime=0.5e-2; % set the shortest realised time interval, the highest ...

PSD element realised will be 1/(2*mintime)

49 %maxtime=1e2; % set the total trace duration

50 nyquist=1/2/mintime; % find the Nyquist rate with given sampling interval

51 tracelength=2*round(maxtime/mintime/2);

52 time=linspace(0,maxtime-mintime,tracelength);

53 freq=logspace(log10(b01),log10(b3end),tracelength); % a frequency vector ...

to plot the intended PSD

54

55 % locating the boundary points in the freq vector

56 f01=1;

57 f12=1;

58 f23=numel(freq);

59 f3end=numel(freq);

60 for f=2:numel(freq)

61 if freq(f)≥b01&&freq(f-1)<b01

62 f01=f;

63 elseif freq(f)≥b12&&freq(f-1)<b12

64 f12=f;

65 elseif freq(f)≥b23&&freq(f-1)<b23

66 f23=f;

67 elseif freq(f)≥b3end&&freq(f-1)<b3end

68 f3end=f;

69 end

70 end

71

72 % work out the amount of aliasing

73 if nyquist<b23

74 alphanyquist=integral(@(x)(nf1(x)),b01,b12)+...

75 integral(@(x)(nf2(x)),b12,nyquist);

76 alias=1+integral(@(x)(nf2(x)),nyquist,b23)/alphanyquist';

77 else

78 alias=1;

79 end

80

81

82 %================ PLOT INTENDED AND REALISED PSD ============

83 psdint=[nf1(freq(f01:f12-1)) nf2(freq(f12:f23-1)) ...

nf3(freq(f23:f3end))]/maxtime;

84 if numel(psdint)<numel(freq)

85 freq=freq(1:numel(psdint));

86 end

87

88 %============= ENSEMBLE OF TIME-DOMAIN REALISATIONS ==========

89 offset='initial'; % options: mean (all traces have mean zero) or initial ...

(all traces start at zero)

90 trace=zeros(ensemble,tracelength);

91

92 % low frequency cutoff

93 fb=2*round(b12*maxtime)-1;

137

94 if fb<1

95 fb=1;

96 end

97

98 % high frequency cutoff

99 fc=2*round(b23*maxtime)-1;

100 if fc>tracelength

101 fc=tracelength;

102 end

103

104 % middle of frequency range

105 fmid=2*round(maxtime)-1;

106

107 % Generate the coefficients.

108

109 hfa = zeros ( 2 * tracelength, 1 );

110 hfa(1) = 1.0;

111

112 for i = 2 : tracelength

113 hfa(i) = hfa(i-1) * ( 0.5 * (-power2) + ( i - 2 ) ) / ( i - 1 );

114 end

115 hfa(tracelength+1:2*tracelength) = 0.0;

116

117

118 fh = fft ( hfa ); % FFT the coefficients

119 fh = vertcat(zeros(fb-1,1).*fh(fb),fh(fb:fc),zeros(tracelength-fc,1)...

120 .*fh(fc+1:tracelength)); % truncate appropriately

121

122 % add spurs around 1/T

123

124 spur=max(abs(fh));

125 spurnum=11;

126 spurgap=100;

127 if spurson

128 fh(fmid-spurgap*(spurnum-1)/2:spurgap:fmid+spurgap*(spurnum-1)/2)=spur;

129 end

130

131 for r=1:ensemble

132

133 % Sample the white noise.

134

135 wfa = sqrt(alpha) * randn ( tracelength, 1 );

136

137 % Pad the array with zeros in anticipation of the FFT process.

138

139 z = zeros ( tracelength, 1 );

140 wfa = [ wfa; z ];

141

142 % Perform the discrete Fourier transforms.

143 fw = fft ( wfa );

144

138

145 % Truncate the vectors appropriately

146 fw = vertcat(fw(1:fc),zeros(tracelength-fc,1));

147

148 if size(fw,1)>size(fh,1)

149 fw=fw(1:size(fh,1));

150 elseif size(fw,1)<size(fh,1)

151 fw=vertcat(fw,zeros(size(fh,1)-size(fw,1),1));

152 end

153

154 size(fw,1);

155 size(fh,1);

156

157 % Multiply the two complex vectors.

158 fw = fh.*fw;

159

160 % Scale to match the conventions of the Numerical Recipes FFT code.

161

162 fw(1) = fw(1) / 2.0;

163 fw(end) = fw(end) / 2.0;

164

165 % Pad the array with zeros in anticipation of the FFT process.

166

167 z = zeros ( tracelength - 1, 1 );

168 fw = [ fw; z ];

169

170 % Take the inverse Fourier transform.

171

172 x = ifft ( fw );

173

174 % Only the first half of the inverse Fourier transform is useful.

175

176 trace(r,:) = 2.0 * real ( x(1:tracelength) );

177

178 if strcmp(offset,'initial')

179 trace(r,:)=trace(r,:)-trace(r,1);

180 elseif strcmp(offset,'mean')

181 trace(r,:)=trace(r,:)-ones(1,tracelength).*mean(trace(r,:));

182 end

183

184 end

185

186 % ================ CALCULATE REALISED PSD =============

187

188 psd=mean(abs(fft(trace')').^2);

189 psd=2*psd(2:numel(psd)/2);

190

191 % preparing an x-vector to plot the realised PSD

192 psdx=[1:numel(psd)]./maxtime;

193

194 % locating the boundary points in the psdx vector

195 p01=1;

139

196 p12=1;

197 p23=numel(psdx);

198 p3end=numel(psdx);

199 for p=2:numel(psdx)

200 if psdx(p)≥b01&&psdx(p-1)<b01

201 p01=p;

202 elseif psdx(p)≥b12&&psdx(p-1)<b12

203 p12=p;

204 elseif psdx(p)≥b23&&psdx(p-1)<b23

205 p23=p;

206 elseif psdx(p)≥b3end&&psdx(p-1)<b3end

207 p3end=p;

208 end

209 end

210

211 % renormalise the trace amplitudes so the area under their realised PSD is

212 % what we want, i.e. the variable 'area'

213 if strcmp(normtype,'area')

214 psdarea=sum(psd);

215 elseif strcmp(normtype,'lowcutoff')

216 psdarea=psd(p12);

217 elseif strcmp(normtype,'highcutoff')

218 psdarea=psd(p23);

219 end

220

221 %fudge=1/sqrt(2);

222

223 trace=trace./sqrt(psdarea).*sqrt(fixed);

224

225 psd=mean(abs(fft(trace')').^2);

226 psd=2*psd(2:numel(psd)/2);

227

228 psdAA = psd./alias;

229

230 psdoverlap=psd.*((sin(psdx.*pi*T*duty)./(psdx.*pi*T*duty)).^2 - ...

231 ((2*cos(2*psdx.*pi*T) - cos(2.*psdx.*pi*T*(1-duty)) - ...

cos(2.*psdx.*pi*T*(1+duty))))...

232 ./(2*psdx.*pi.*T.*duty).^2).^overlap;

233

234 if ensemble>10 % prevents small ensembles having noisy PSDs displayed

235 figure(30)

236 loglog(freq,psdint)

237 hold on

238 loglog(psdx,psd,'r*-')

239 loglog(psdx,psdAA,'ro')

240 % loglog(psdx,psdoverlap,'+')

241 xlabel('Fourier Frequency')

242 ylabel('Power Spectral Density')

243 title('Intended (blue solid) and Realised PSD, aliasing-compensated ...

(red stars) and uncompensated (red circles)')

244 end

140

245

246 % ============= EXPORT DETAILS OF THE PSD ============

247

248 % work out the number of teeth in the f12-f23 band

249 bandvec=psdx(p12:p23);

250 bandteeth=size(bandvec);

251

252 % ============== CALCULATE DICK LIMIT ================

253

254 dead = 0.5;

255 Tc = 2;

256

257 for h=1:1000

258 dickvector(h) = ((h/Tc*0.1).^(-1)).*(sin(2*pi*h*dead)/(2*pi*h)).^2;

259 end

260

261 figure(20)

262 plot(dickvector)

263

264 dick=(2/dead^2)*sum(dickvector)

265

266 %================ PLOT EXAMPLE TRACES =================

267 %showtrace=[1,10]; % a vector containing the indices of the traces to be ...

displayed

268 %figure(2)

269 %plot(time,trace(showtrace,:))

D.5 showbins.m

1 function [bin_points,hugosity] = show_bins(int_times,generate_type,cmatnum)

2

3 fineness=1;

4 fineness2=1;

5

6 square_lim=ceil(sqrt(fineness2));

7

8 int_times;

9

10 for r=[1:fineness]

11 for p=[1:fineness2]

12 clear bin_points

13 hugosity=size(int_times,1);

14

15

16 for i=[1:hugosity]

17 for j=[1:2]

18 int_times_mat(fineness,fineness2,i,j)=int_times(i,j);

19 end

20 end

21 int_times;

141

22

23 % make the vector for the bin-step function

24 width=4*hugosity-2;

25

26 for d=2:width % Four points for each bin

27 if (mod(d,4)==2||mod(d,4)==3)

28 bin_points(d,1)=int_times(ceil(d/4),1);

29 elseif (mod(d,4)==0)

30 bin_points(d,1)=int_times(ceil(d/4),2);

31 elseif mod(d,4)==1

32 bin_points(d,1)=int_times(ceil(d/4)-1,2);

33 end

34 if (mod(d,4)==3||mod(d,4)==0)

35 bin_points(d,2)=1;

36 elseif (mod(d,4)==1||mod(d,4)==2)

37 bin_points(d,2)=0;

38 end

39 end

40

41

42 bin_points(1,1)=0;

43

44 bin_points(1,2)=0;

45

46

47 if p==1

48 fignum=40+cmatnum;

49 figure(fignum)

50 if strcmp(generate_type,'parameterspace')

51 subplot(2,1,1)

52

53

54 hold off

55 plot(bin_points(:,1),bin_points(:,2),'LineWidth',2);

56

57 last_bin=(int_times(hugosity,1)-int_times(hugosity-1,1));

58

59 for k=1:ceil(int_times(hugosity,1)/last_bin)

60 % line([k*last_bin k*last_bin],[0 ...

1],'LineStyle','--','Color','r')

61 end

62

63 plot(bin_points(:,1),bin_points(:,2),'LineWidth',2,'Color','b')

64

65 xlim([-0.1*last_bin int_times(hugosity,1)+0.1*last_bin])

66 ylim([-0.2 1.2])

67

68 ff_length=int_times(hugosity,1);

69

70

71 %set(gca,'XTick',[])

142

72 set(gca,'YTick',[0 1])

73 ylabel('g(t)','FontSize',24)

74 set(gca,'FontSize',16)

75

76 %hold on

77 % ...

plot(int_times(hugosity,1),0,'mx','LineWidth',2,'MarkerSize',10)

78

79

80 elseif strcmp(generate_type,'feedback')

81

82 subplot(2,1,2)

83

84

85 ylabel('g(t)','FontSize',24)

86

87 hold off

88

89 last_bin=int_times(2,1);

90 bin_size=int_times(2,1)-int_times(1,1);

91

92

93

94

95 %plot(bin_points(:,1)-k*bin_size,bin_points(:,2),'LineWidth',2,'Color','r');

96 plot(bin_points(:,1),bin_points(:,2),'LineWidth',2,'Color','r')

97

98

99 xlim([-0.1*bin_size last_bin+0.1*bin_size])

100 ylim([-0.2 1.2])

101

102 %plot(bin_points(width,1),0,'go','MarkerSize',10,'LineWidth',2)

103 %plot(int_times(hugosity,1),0,'mx','MarkerSize',10,'LineWidth',2)

104

105 ylabel('g(t)','FontSize',24)

106 set(gca,'YTick',[0 1])

107 xlabel('t (1/f_c)','FontSize',24)

108 set(gca,'FontSize',16)

109 title(['last_bin=',num2str(last_bin),', ...

bin_size=',num2str(bin_size)])

110

111 end

112

113 end

114 end

115 end

143

References

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[8] B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang,S. L. Bromley, and J. Ye. An optical lattice clock with accuracy and stability at the 10-18 level.Nature, 506(7486):71–75, 02 2014.

[9] B.J. Bloom, T.L. Nicholson, J.R. Williams, S.L. Campbell, M. Bishof, X. Zhang, W. Zhang,S.L. Bromley, and J. Ye. An optical lattice clock with accuracy and stability at the 10−18 level.ArXiv e-prints, 2013.

[10] Joseph W. Britton, Brian C. Sawyer, Adam C. Keith, C. C. Joseph Wang, James K. Freericks,Hermann Uys, Michael J. Biercuk, and John J. Bollinger. Engineered two-dimensional isinginteractions in a trapped-ion quantum simulator with hundreds of spins. Nature, 484(7395):489–492, 04 2012.

[11] Kenneth R. Brown, Aram W. Harrow, and Isaac L. Chuang. Arbitrarily accurate compositepulse sequences. Phys. Rev. A, 70:052318, Nov 2004.

[12] C. W. Chou, D. B. Hume, T. Rosenband, and D. J. Wineland. Optical clocks and relativity.Science, 329(5999):1630–1633, 2010.

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