WMITECHNISCHEUNIVERSITÄTMÜNCHEN

WALTHER -MEISSNER -INSTITUT FÜR TIEF -

TEMPERATURFORSCHUNG

BAYERISCHEAKADEMIE DER

WISSENSCHAFTEN

Thesis for a Master degreein

Applied and Engineering Physics

Characterization of hysteretic

flux-driven Josephson parametric amplifier

Martin Betzenbichler

Garching, 22. October 2015

Academic Supervisor: Prof. Dr. Rudolf Gross

Advisor: Dr. Kirill G. Fedorov

Erklärung / Declaration of Originality

Mit der Abgabe der Masterarbeit versichere ich, dass ich die Arbeit selbstständigt verfasst undkeine anderen als die angegebenen Quellen und Hilfsmittel benutzt habe.

By submission of this thesis I hereby certify, that this diploma thesis is my own work and nosources other than the ones given have been used.

Garching, August 2015Ort,Datum Martin Betzenbichler

Contents

1. Theory 31.1. Josephson Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1. The RCSJ model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2. Flux Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3. dc-SQUID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.1. Non hysteretic dc-SQUID - βL ! 1 . . . . . . . . . . . . . . . . . . . . . . 111.3.2. Hysteretic dc-SQUID - βL Á 1 . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4. Quantum states in cirQED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.5. Flux-driven Josephson Parametric Amplifier . . . . . . . . . . . . . . . . . . . . . 18

1.5.1. Description of a reflection type resonator . . . . . . . . . . . . . . . . . . 191.5.2. Flux modulation of a coplanar waveguide cavity . . . . . . . . . . . . . . 221.5.3. Parametric amplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2. Experimental Setup 272.1. JPA samples and sample holder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1.1. JPA samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.1.2. Sample holder assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2. Cryogenic Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2.1. Dilution refrigerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2.2. Input lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2.3. Output lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.2.4. Circulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3. Room temperature Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3. Experimental Results 373.1. Resonator characterization of the JPA . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.1. Flux dependence of the resonant frequency . . . . . . . . . . . . . . . . . 373.1.2. Determination of internal and external quality factors . . . . . . . . . . . 483.1.3. Fitting of the resonant frequency . . . . . . . . . . . . . . . . . . . . . . . 52

3.2. Gain characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2.1. Q200 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2.2. Q600 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4. Conclusion 65

Appendix A Spectra 67Appendix A.1 Q1100-epr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Appendix A.2 Q200 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Appendix A.3 Q600 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Appendix B Hysteretic currents and frequencies 77

i

Bibliography 79

ii

List of Figures

1.1. Schematic of a S-I-S tunneling barrier Josephson Junction . . . . . . . . . . . . . 31.2. Current-Voltage characteristics and equivalent circuit of a Josephson junction . . 61.3. The washboard potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4. Different possibilities for closed contours within a superconductor . . . . . . . . . 81.5. Schematic of a dc-SQUID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6. The magnetic flux dependence of the transport current of a dc-SQUID with neg-

ligible screening parameter βL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.7. Equivalent circuit of a dc-SQUID . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.8. The potential of a symmetric dc-SQUID . . . . . . . . . . . . . . . . . . . . . . 131.9. Evolution of the dc-SQUID for different applied fluxes Φext and different screening

parameter βL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.10. The magnetic flux dependence of the switching current of a dc-SQUID at different

internal flux values Φ and for different values of the screening parameter βL. . . . 161.11. Schematic for the generation of coherent states and coherent squeezed states . . 171.12. JPA equivalent circuit, flux-dependent resonant frequency and schematic for the

input-output relations for signal amplification . . . . . . . . . . . . . . . . . . . . 181.13. Schematic of the lumped element model of a reflection type resonator . . . . . . 201.14. The reflection coefficient Γ of a reflection type resonator as a function of frequency 221.15. Signal and intermodulation gains of a JPA as a function of the signal frequency . 25

2.1. Micrographs of the Q200 Josephson Parametric Amplifier . . . . . . . . . . . . . 282.2. Image of the prepared sample holder . . . . . . . . . . . . . . . . . . . . . . . . . 302.3. Schematic of the cryogenic setup for samples Q200 and Q600 . . . . . . . . . . . 312.4. The sample rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.5. Schematic and image of the utilized dilution refrigerator . . . . . . . . . . . . . . 332.6. Schematics of circulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.7. Schematic of a room temperature setup for samples Q200 and Q600 . . . . . . . 36

3.1. Schematic of the setup for sample Q1100-epr . . . . . . . . . . . . . . . . . . . . 383.2. Processed Magnitude of the resonator reflectance of sample Q1100-epr for different

sweep directions of the applied magnetic field . . . . . . . . . . . . . . . . . . . . 403.3. Overlay of the reflectance magnitude of sample Q1100-epr for both sweep direc-

tions of the magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4. Resonator reflectance of sample Q200 for increasing the magnetic field . . . . . . 433.5. Resonator reflectance of sample Q200 for decreasing the magnetic field . . . . . . 443.6. Resonator reflectance of sample Q600 for increasing the magnetic field . . . . . . 463.7. Resonator reflectance of sample Q600 for decreasing the magnetic field . . . . . . 473.8. Reflection magnitude and phase of samples Q200, Q600 and Q1100-epr . . . . . . 493.9. Quality factor dependence on the external flux for sample Q200 . . . . . . . . . . 503.10. Quality factor dependence on the external flux for sample Q600 . . . . . . . . . . 503.11. Quality factor dependence on the external flux for sample Q1100-epr . . . . . . . 51

iii

3.12. The hysteretic switching current curves of a dc-SQUID used for fitting both sweepdirections of the applied magnetic field . . . . . . . . . . . . . . . . . . . . . . . . 53

3.13. Flux-dependence of the resonant frequency of sample Q1100-epr . . . . . . . . . 533.14. Flux-dependence of the resonant frequency of sample Q200 . . . . . . . . . . . . 543.15. Flux-dependence of the resonant frequency of sample Q600 . . . . . . . . . . . . 553.16. Gain characterization of Q200 at 5.802 GHz . . . . . . . . . . . . . . . . . . . . . 573.17. Gain characterization of Q200 at different flux biases . . . . . . . . . . . . . . . . 583.18. Flux sweep of sample Q200 with a constant pump power of 7.8 dBm . . . . . . . 593.19. Flux sweep of sample Q200 with a constant pump power of 10.0 dBm . . . . . . . 613.20. Gain characterization of Q600 at different flux biases . . . . . . . . . . . . . . . . 63

A.1. Raw data of Q1100-epr for increasing coil current . . . . . . . . . . . . . . . . . . 67A.2. Raw data of Q1100-epr for deqcreasing coil current . . . . . . . . . . . . . . . . . 68A.3. Processed phase data of Q1100-epr . . . . . . . . . . . . . . . . . . . . . . . . . . 69A.4. Frequency response of Q1100-epr at different flux biases . . . . . . . . . . . . . . 70A.5. Raw data of Q200 for increasing coil current . . . . . . . . . . . . . . . . . . . . . 71A.6. Raw data of Q200 for decreasing coil current . . . . . . . . . . . . . . . . . . . . 72A.7. Overlay of the reflectance magnitude of sample Q200 for both sweep directions of

the magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73A.8. Raw data of Q600 for increasing coil current . . . . . . . . . . . . . . . . . . . . . 74A.9. Raw data of Q600 for decreasing coil current . . . . . . . . . . . . . . . . . . . . 75A.10.Overlay of the reflectance magnitude of sample Q600 for both sweep directions of

the magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

B.1. Hysteretic switching currents and frequencies . . . . . . . . . . . . . . . . . . . . 78

iv

List of Tables

2.1. Design parameters of different JPA samples . . . . . . . . . . . . . . . . . . . . . 27

3.1. Extracted quality factors of the samples . . . . . . . . . . . . . . . . . . . . . . . 483.2. Summary of the samples’ determined resonator parameters . . . . . . . . . . . . 54

v

IntroductionSince the formulation of quantum mechanics, researchers have sought systems in which it

is possible to observe and verify the limitations and often strange implications, predicted bythis theory. Their discoveries often led to a new understanding of nature and enabled theconstruction of novel devices, capable of feats previously unheard of. Through the advances insolid state physics and fabrication methods of the last 50 years, quantum systems have beenmade ever more accessible. The field which seeks ways of making quantum mechanical effectsactively accessible to communication and computation systems is quantum information science.Among the most prestigious examples of its applications are quantum computers and bug-proofcommunication lines.An important ingredient for the realization of several applications in quantum information is

entanglement. The state a quantum system occupies prior to any measurement, is described by asuperposition of all possible quantum states. This description is also valid for a quantum systemconsisting of multiple particles. An ensemble of two or more particles is entangled if it is welldescribed by a quantum state, but an individual description for each element can not be given.Another remarkable observation is that the system can still be described as being entangled,even if its constituents are spatially separated. As a consequence for an entangled photon pair,if a measurement determines the polarisation of one photon to be clockwise oriented, the waveform of the second photon will instantaneously collapse with a counter-clockwise polarisation.This prediction led to the famous paradox from Einstein, Podolsky and Rosen [1] and to theformulation of the Bell inequalities [2].Entanglement is a crucial ingredient for quantum information, as it allows among others

superdense coding [3] and is argued to be necessary for quantum computers[4, 5, 6]. Anotherprominent process, which makes use of entangled photon pairs, is quantum teleportation. Unlikea magical instant transport of mass, commonly known from various science fiction sources,quantum teleportation describes the transmission of quantum information, i.e. a quantum state,between two spatially separated points. In a typical quantum teleportation setup [7], a sender,called Alice, and a receiver, named Bob, share an Einstein-Podolsky-Rosen (EPR) pair, providedto them by an additional station responsible for the generation and distribution of entangledquantum states. A quantum state can be teleported from Alice to Bob by exploiting non-localcorrelations between EPR pairs. In order to teleport a quantum state, Alice needs to perform aBell measurement on it and her part of the EPR pair. This results in the collapse of the otherEPR photon into the initial quantum state plus a phase offset. By obtaining Alice’s measurementresult, Bob can reconstruct the original quantum state with a unitary transformation.Proof of principle of quantum teleportation has been shown in the field of cavity quantum

electrodynamics (cQED)[8]. A full realization of unconditional quantum teleportation was shownin Ref. [9] in which various optical coherent states were teleported.As cQED employs atoms and photons to study light-matter interaction, experiments are lim-

ited by the small coupling strength between an atom and a cavity. A way to get rid of theseissues is to transfer the experiments into a different environment. One approach utilizes su-perconducting circuits[10] which triggered the development of circuit quantum electrodynamics(cirQED). This gives researchers a new type of control, in that they can design their artificial

1

atoms, control the coupling strengths or other system properties by making use of state of the artmicrofabrication technologies[11]. This flexibility of designing one’s experiment is one of the keyfeatures of cirQED and is traded off by having to cool the samples to cryogenic temperatures.This is done to achieve the superconducting state of the sample material and to reduce signalinterference by thermal noise. The employed probe signal consists of a few photons in the GHzregime, which compares to a temperature of a few mK. By cooling the sample to temperaturesbelow that, thermal noise contributions can be frozen out. Another drawback is because of thelow photon power there exist no single photon detectors. For detection the microwave signalshave to be amplified additionally.Nevertheless, many observations known from cQED have been demonstrated in cirQED during

the last decade, including teleportation of discrete quantum states[12]. Proof of unconditionalquantum teleportation of continuous states has yet to be shown. The most crucial part ofa functioning teleportation scheme probably is the EPR beam generator. A way of generatingentangled photon pairs is superposing the squeezed states from a Josephson parametric amplifier(JPA) at a 50/50 beam splitter, similar to Ref. [9].Based on the Josephson effect observable in superconductors, Josephson parametric amplifiers

are cryogenic amplifiers capable of signal amplification while adding very low amounts of noiseto an input signal [13, 14, 15]. This makes them especially attractive in cirQED. While operatinga JPA in the phase insensitive mode, the added noise approaches the quantum limit set by theHeisenberg uncertainty principle. In the phase sensitive mode a JPA generates squeezed states asa result of amplifying and deamplifying the input signal’s quadrature moments depending on therelative phase between pump and signal tone[16]. A type of JPA, which is of particular interestfor us, is the flux-driven Josephson parametric amplifier. This type of JPA was developed byT. Yamamoto et al. [17] based on a proposal of T. Ojanen and J. Salo[18]. It consists of aquarter-wavelength resonator shunted to ground with a dc-SQUID.Our long term goal is to implement quantum teleportation with continuous quantum mi-

crowave states utilizing JPAs. We plan to use the squeezed microwave fields produced by theJPAs as a source for entangled microwave photons in future teleportation experiments. To guar-antee a successful outcome of our experiment, a significant portion of our experimental efforts isdedicated to the thorough characterization of our flux-driven Josephson parametric amplifiers.In this thesis we will present our results on characterising flux-driven Josephson parametric

amplifiers. In Chapter 1 a theoretical description of the working principles of JPAs and theirrespective components is given. We start by introducing the Josephson junction including theresistively and capacitively shunted junction model (RCSJ) and the phenomenon of flux quanti-zation in a superconducting ring. We treat direct-current superconducting quantum interferencedevices (dc-SQUID) and give a quantum state representation of the signals used in circuit quan-tum electrodynamics. We conclude Ch. 1 with the flux-driven Josephson parametric amplifier(JPA). Following a general description of the working principals of this device, we describe theJPA resonator and its flux-tunability before giving expressions for the phase-insensitive andphase-sensitive gain.Chapter 2 is dedicated to our experimental setup. We describe the different samples, which

were fabricated at NEC(Japan), the cryogenic and the room temperature part of our measure-ment setup.Our obtained measurement results are displayed in Chapter 3. We present a thorough char-

acterisation of the JPA resonator for three different JPAs. We conclude this thesis by giving ourresults for measuring signal amplification with two of our samples.

2

1. Theory

In order to achieve our goal of continuous variable quantum teleportation in the microwaveregime entangled photon pairs are required. Entangled photon pairs can be effectively producedby superimposing two orthogonally squeezed states. The squeezed states themselves are typicallygenerated by Josephson Parametric Amplifiers (JPA). Therefore, a detailed characterization ofJPAs is an essential part for successful implementation of quantum information protocols withsqueezed microwaves.In the following we present a theoretical description of the flux-driven JPA. We start with

the Josephson equations and flux quantization, followed by a description of the direct currentSuperconducting QUantum Interference Device (dc-SQUID). We proceed by describing the JPAas a combination of a quarter-wavelength resonator and a dc-SQUID. We show how amplificationof microwave signals can be achieved with such a device.

1.1. Josephson Junctions

A Josephson junction can be created by a weak coupling between two pieces of supercon-ductors. Technical implementations range from point contacts, normal metal or semiconductorlayers serving as a tunneling barrier, microconstrictions and others [19]. The Josephson effectis based on a coherent tunneling of Cooper pairs through the tunneling barrier. Cooper pairsare formed by two electrons which give rise to supercurrents in superconductors. A theoreticalexplanation of the Josephson effect has been given in 1962 by Brian D. Josephson [20] and sub-sequently experimentally verified in 1963 by Philip W. Anderson and John M. Rowell [21]. In

S1 Ψ

1S

2 Ψ

2I

Figure 1.1.: Schematic of a S-I-S tunneling barrier Josephson Junction. The evanescent wavefunctions Ψ1 and Ψ2, corresponding to superconductors S1 and S2, and their corre-lated overlap is symbolically displayed into the insulating barrier I.

Fig. 1.1 we depict a tunneling barrier Josephson junction, consisting of two superconducting elec-trodes separated by an insulating barrier. The ensemble of Cooper pairs in each superconductorcan be described by macroscopic wave functions Ψi

Ψ1 ?ns,1e

iθ1 ,

Ψ2 ?ns,2e

iθ2 ,(1.1)

3

CHAPTER 1. THEORY

where θi are the phases and ns,i the Cooper pair densities in the superconducting regions 1/2of the Josephson junction. These macroscopic wave functions are exponentially decaying in theinsulating barrier. For a sufficiently thin barrier both superconductors will be weakly coupledthrough a small overlap of the evanescent wave functions. The insulating barrier serves thereforeas a mediator, coupling both superconductors weakly together and allowing them to exchangeCooper pairs. The resulting supercurrent density through the insulator is expected to be verysmall and will have no effect on the respective Cooper pair densities ni. The supercurrent densitytherefore will depend on the phase difference between the superconductors. The supercurrentdensity Js pr, tq in the bulk of a superconductor is given by [22] as a function of the gauge-invariant phase gradient γ pr, tq, where

ΛJs pr, tq 2πΦ0

r∇θ pr, tqs A pr, tq 2πΦ0γ pr, tq , (1.2)

Λ m

ns q2 is the London coefficient, q 2e is the charge of a Cooper pair, e is the electroncharge, h is the reduced Planck constant, m is the mass of a Cooper pair, ∇θ is the phasegradient in the superconductor, A is a vector potential and Φ0 is the flux quantum, which isdefined as

Φ0 h

2e 2.067833636p86q 1015V s, (1.3)

where h is the Planck constant. We apply Eq. 1.2 to the Josephson junction by assumingthat the supercurrent across the junction is homogeneous and that the gauge-invariant phasegradient γ varies negligible in the superconducting electrodes. Since Js is constant for the entirejunction, the gauge-invariant phase gradient is negligibly small compared to the insulating area.We therefore substitute the phase gradient γ by the value of its integral across the junction indirection of the supercurrent:

ϕ pr, tq 2»

1

γ pr, tq dl θ2 pr, tq θ1 pr, tq 2πΦ0

2»1

A pr, tq dl, (1.4)

where ϕ is a phase difference between the superconducting electrodes. For simplicity, we drophere the vector notation, reverting to the simple one-dimensional case. In order to give Isa final form, we now take some analogies into account from the previous analysis. First, incorrespondence to Eq. (1.1), Is has to be a 2π periodic function. Second, in the absence ofsupercurrents, Is has to be zero and the value of the electrodes’ phases θi have to match by amodulo of 2π: θ1 2πnθ2. In first order approximation we write the supercurrent

Is Ic sin pϕq , (1.5)

where Ic is the critical current across the Josephson junction. Eq. (1.5) is the first Josephsonequation, also called the current-phase relation. The second Josephson equation is derived bylooking at the time evolution of ϕ

BϕBt

Bθ2Bt

Bθ1Bt

2πΦ0

BBt

2»1

A pr, tq dl. (1.6)

The right hand side we can bring into a different form, making use of the energy phase-relationin [22]

BϕBt

2πΦ0

2»1

∇φ BA

Bt dl, (1.7)

4

1.1. JOSEPHSON JUNCTIONS

where φ is an electric potential. From electrodynamics we can relate E ∇φ BABt with an

electric field. Taking also into account that³21Epr, tq dl corresponds to a voltage drop V across

the junction, we can substitute the integral by a voltage V across the Josephson junction. Thissimplified version of the second Josephson equation is called the voltage-phase relation

BϕBt

2πΦ0V. (1.8)

From this differential equation we see, that the phase difference grows linearly in time, propor-tional to the applied dc-voltage to the Josephson junction. Substituting this result into Eq. (1.5)we see that this results in oscillations of the supercurrent with the Josephson frequency, definedas

ν

V 2πω

V 1

Φ0 483.597898p19qMHz

µV . (1.9)

The overlap of the superconducting wave functions in the vicinity of the barrier can be comparedto a binding energy between both superconductors, similar to the electronic wave functions ina molecular bond. This binding energy is known as the Josephson coupling energy and is givenby

EJ Φ0Ic2π r1 cos pϕqs EJ0 r1 cos pϕqs . (1.10)

Another interesting property related to the energy stored in this coupling, is the non-linearinductance of Josephson junctions. Making use of the expression V ptq LBI

Bt , which links thevoltage V to the current I of a circuit through its self inductance, we substitute Eqs. (1.5)and (1.8) and get an expression for the inductance of a Josephson junction as follows

LJ Φ02πIc cos pϕq . (1.11)

1.1.1. The RCSJ modelHere we briefly address current-voltage characteristics of Josephson junctions. For applying

an external current, a voltage drop across the junction will only be observed, when the appliedcurrent exceeds the critical current Ic of the Josephson junction. Below Ic the applied current iscarried by the tunneling Cooper pairs. As soon as we exceed the critical Josephson current Ic,an additional normal current starts to flow through the junction. This normal current consistsof quasiparticle excitations. Thus, there are two operation modes for Josephson junctions: thezero voltage state pI Icq and the voltage state pI ¡ Icq.This rather complicated dynamics of the Josephson junction can be well described by the

Resistively and Capacitively Shunted Junction model (RCSJ) [23, 24], in which the Josephsonjunction is modelled by its equivalent circuit. This circuit is displayed in Fig. 1.2. It consists ofthe Josephson junction shunted in parallel by a resistor, a capacitor and a noise source. Eachelement contributes to a net current flowing across the Josephson junction.The ideal Josephson junction carries the supercurrent Is according to (1.5) and (1.8). The

resistor carries the normal current IN VRN

, where RN is the normal state resistance of theJosephson junction. The capacitor contributes a displacement current, which is represented byID C dV

dt , where C is the junctions capacitance. The noise source is included into the equivalentcircuit to represent a fluctuating current IF. The net current is then given by Kirchhoff’s law

I Is IN ID IF

Ic sin pϕq V

RN C

dV

dt IF.

(1.12)

5

CHAPTER 1. THEORY

V S

D

N

F

b)I

<V>

I

c

R

R

c

a)

I

I

I

II II I

Figure 1.2.: a) Current-Voltage characteristics of a Josephson junction, displaying the Josephsonjunctions hysteretic behaviour by switching between the zero voltage and the voltagestate. Increasing the bias current from zero, a voltage drop will only be observedwhen exceeding the junction critical current Ic. Decreasing the current from thispoint, results in the Josephson junction switching back into the zero voltage stateat a current value IR smaller than the critical current Ic, called the return current.b) equivalent circuit of a Josephson junction consisting of a capacitor, a resistor,a noise source and an element, representing the Josephson junction supercurrent.Each element contributes a current Ii to the total current flowing across the junctionfor the different possible operation modes.

By using Eqs.(1.5), (1.8) we obtain a differential equation for the gauge-invariant phase dif-ference

h

2e

2Cd2ϕ

dt2h

2e

2 1RN

dϕ

dth

2e

Ic

sin pϕq I

Ic IFIc

0. (1.13)

We can interpret Eq.(1.13) by comparing it to the differential equation of a particle moving ina potential Upot with a mass M , subject to a damping η

Md2x

dt2 η

dx

dt∇Upot 0. (1.14)

In this picture ϕ represents the spatial coordinate x of the phase-particle,h2e2C represents its

mass M andh2e2 1

RNis the damping coefficient η. The potential Upot is given by

Upot EJ0 r1 cos pϕq iϕ iFϕs , (1.15)

where we normalized the supercurrent i IIc

and the fluctuation current iF IFIc

by the criticalcurrent Ic, respectively. Eq.(1.15) is the well known washboard potential. In Fig. 1.3 we displaya washboard potential for different bias currents i. The cosine term in Eq.(1.15) leads to thedevelopment of minima and maxima, whose positions are given by

1EJ0

BUpotBϕ sin pϕq i 0. (1.16)

Increasing the bias current results in an increased tilt of the washboard potential. For biascurrents below the critical current, I Ic, the washboard potential still exhibits minima inwhich the phase particle resides, separated from each other by neighbouring potential barriers

6

1.1. JOSEPHSON JUNCTIONS

Ubar

0 2 4 6 8 10 12ϕ

-15

-10

-5

0

Epot

(

IcΦ

0

2π

)

i = 0

i = 0.4

i = 1

i = 1.3

Figure 1.3.: Washboard potential for different applied currents i. In the case of i 1 the phaseparticle is trapped in a minimum, which vanishes at i 1. The energy necessary toescape this minimum is determined by the barrier height Ubar. For values of i ¡ 1the phase particle experiences no more potential barriers and moves down the slopeof the tilted washboard-potential.

Ubar. For applied currents exceeding the critical current I ¥ Ic, the washboard potential willno longer display any minima and the phase particle can move down the slope. Its velocity isdetermined by its mass M and damping η.These two cases of a stationary and moving phase particle correlate to the previously men-

tioned regimes of the Josephson junction. In the stationary state the phase difference ϕ doesnot change in time, which corresponds to the zero voltage state according to Eq. (1.8). Likewisein the dynamical case, the phase difference ϕ changes in time, which corresponds to the voltagestate.We want to conclude the RCSJ-model by briefly addressing two additional properties of

Josephson junctions the RCSJ-model helps to explain. Firstly, we consider the case of a movingphase particle in a washboard potential with applied bias current I ¡ Ic. Upon lowering the biascurrent below the critical current I Ic, the washboards potential will again exhibit minimafor the particle to be trapped in. Due to the particles inertia, the particle will not come toreside in the first minimum encountered but will first have to loose kinetic energy before it isretrapped. This however depends on its mass M , damping coefficient η as well as the loweringrate dI

dt of the applied current. The junction switches back into the zero voltage state at a returncurrent IR Ic lower than the junction critical current. In effect a Josephson junction displaysa hysteretic current-voltage curve, depending on its parameters, which we displayed in Fig. 1.2a.Second we want to mention two escape mechanisms possible for the gauge-invariant phase

difference in the washboard potential for bias currents i close to Ic. In this case the potentialbarrier to the energetically lower minimum is significantly decreased. By the inclusion of noise,the phase particle can escpe its potential well and start moving, by picking up enough kinetic

7

CHAPTER 1. THEORY

energy to overcome subsequent minima. Another way of escape is called macroscopic quantumtunneling, by which the phase particle can tunnel through the potential well.

1.2. Flux QuantizationQuantization of magnetic flux is an important property of superconductors arising from the

macroscopic quantum nature of superconductivity. Let us consider a supercurrent flowing ina homogeneous superconductor along the closed contour C1, as depicted in Fig. 1.4. Taking acontour integral of the supercurrent density (1.2) along C1, we obtain the following expression¾

C1

ΛJs dl»S1

B ds 2πΦ0

¾C1

∇θ dl. (1.17)

Here, we used Stokes theorem, substituting the vector potential A by the magnetic field Baccording to ¾

C1

A dl »S1

p∇Aq ds »S1

B ds, (1.18)

where S1 is a surface enclosed by the contour C1, and B is the magnetic field associated withthe vector potential A. We first evaluate the right hand side of Eq. (1.17). The argument of

S1

S2

C1

C2

Ф

Figure 1.4.: Different possibilities for contours Ci in a superconductor. C1 is in a simply-connected and C2 in a multiply-connected region.

the integral is the gradient of a scalar function. From vector calculus we know that its solutionis given by the function’s values at the limits of the integral

r2»r1

∇θ dl θ pr2, tq θ pr1, tq . (1.19)

For a closed path this means that both end points coincide, r2 Ñ r1, and θ pr2, tq θ pr1, tq.However, this does in general not mean that the integral is zero. As we mentioned in Sec. 1.1,the superconducting state is described by the macroscopic wave function of the form Ψ pr, tq ?nse

iθ (1.1). Although the value of the wave-function Ψ prq at position r is well defined, this is

8

1.3. DC-SQUID

not the case for the value of its phase θ. There exist an infinite amount of possible values for θof the form: θn θ0 2πn, for which Ψ will have the same value. This means that the phaseis only specified within a modulo of 2π of its principal value θ0, ranging in the interval rπ, πs.As θ0 is single-valued, the value of the right hand side of Eq. (1.17) is therefore¾

C1

∇θ dl 2πn. (1.20)

Rewriting Eq. (1.17) with this result, we get the equation for the fluxoid quantization¾C

ΛJs dl»S

B ds nΦ0, (1.21)

where the left hand side expression is denoted as the fluxoid. We now consider different ge-ometries of the superconducting region. For simply-connected areas, similar to S1 in Fig. 1.4,Eq. (1.21) has to be fulfilled everywhere regardless of the choice of the contour line. This willalso be true, when the length of C1 approaches zero. For a simply-connected region, this resultsin the trivial case of n 0 and corresponds to the well known Meißner-Ochsenfeld effect [25].In a multiply-connected region like surface S2 depicted in Fig. 1.4, C2 contains a superconduct-

ing and a non-superconducting area. The contour is therefore substantial and from Eq. (1.21)we see, that the magnetic flux inside the non-superconducting region is quantised and can onlyobtain integer values of the flux quantum Φ nΦ0. We note here, that this is also true forexternally applied magnetic fields, resulting in a non-integral external flux Φext. We have totake into account the flux induced by the supercurrent ΦL and the internal flux Φ, given by thesum of external flux Φext and induced flux ΦL must have discrete values. This is valid for anymultiply-connected superconducting regions e.g. superconducting ring structures.In addition, by choosing an integration contour deep in the superconductor, we can neglect

the supercurrent density Js in Eq. (1.21). The fluxoid quantization relaxes thereby to the fluxquantization, which was independently proven experimentally by Doll and Näbauer [26], as wellas Deaver and Fairbank [27].

1.3. dc-SQUIDA Superconducting QUantum Interference Device displays Josephson physics as well as fluxoid

quantization in a superconducting loop. Of the two known species, only the direct currentSQUID will be discussed here. It consists of two Josephson junctions connected in parallel ina superconducting loop, a schematic of which can be seen in Fig. 1.5. For us, the current-flux-characteristic Is pΦextq of the dc-SQUID in the zero voltage state is of particular interest. Thisis also known as the switching current, because applying currents larger than Is pΦextq results inthe dc-SQUID switching into the voltage state.The transport current Is is defined as the current flowing across the SQUID. For a symmetrical

dc-SQUID, both Josephson junctions are assumed to be identical and their critical currents tobe equal, Ic,1 Ic,2. By making use of Kirchhoff’s law and Eq. (1.5) we get an expression for Is

Is Is,1 Is,2 2Ic cosϕ1 ϕ2

2

sin

ϕ1 ϕ2

2

. (1.22)

Applying an external flux Φext, we know from Sec. 1.2 that the effective flux Φ in the dc-SQUIDwill be quantized. In the case of non-integer applied flux Φext, a circulating current J will start

9

CHAPTER 1. THEORY

to flow in the ring. We can define J in a similar fashion to the transport current Is

J Is,1 Is,2 2Ic sinϕ1 ϕ2

2

cos

ϕ1 ϕ2

2

, (1.23)

where we assumed a circulating current flowing counter-clockwise. From the definitions ofEqs. (1.22) and (1.23), we see that maximizing Is results into a minimized J and vice versa.By taking the integral of the gauge-invariant phase gradient along the closed contour C of the

s

J

ФФ

L

Фext

s,1

1

s,2

2

I

!

I

!

I

C

Figure 1.5.: Schematic of a dc-SQUID. The transport current Is flows across both Josephsonjunctions in the ring. An applied external flux Φext induces a circulating J , whichcontributes with its generated flux ΦL to the flux Φ inside the SQUID loop. Thisinternal flux Φ locks the phases ϕ1 and ϕ2 to each other, cf. Eq. (1.24).

dc-SQUID in Fig. 1.5, similar to the derivation of Eq. (1.4), we gain a relation for the gauge-invariant phase differences ϕ1 and ϕ2 in the SQUID

ϕ2 ϕ1 2πΦΦ0

. (1.24)

This expression is called the flux-phase relation. We see here, that the phase differences of theJosephson junctions are not independent, but that they are coupled to each other by the effectiveflux Φ in the dc-SQUID.With this result we can simplify Eq. (1.22) accordingly

Is 2Ic cosπΦΦ0

sin

ϕ1 πΦ

Φ0

. (1.25)

The effective flux Φ is divided between the external flux Φext, coupled in e.g. by an externalcoil, and the inductive flux ΦL, generated by the circulating current J in conjunction with theinductance Ls of the SQUID

Φ Φext ΦL Φext LsJ. (1.26)

10

1.3. DC-SQUID

Similar to our discussion in Sec. 1.1 by making use of Eq. (1.25) and (1.8), we get a generalexpression for the inductance of the dc-SQUID

Ls Φ02πIs

Φ0

4πIc cosπΦΦ0

cos

ϕ1 πΦ

Φ0

. (1.27)

We see that Eqs. (1.25) and (1.26) are interdependent and can not be solved analytically. Inour further discussion to get an expression for the switching current Is pΦextq of the dc-SQUID,we introduce the screening parameter βL, defined as

βL 2LsIcΦ0

. (1.28)

The screening parameter βL represents the amount of flux which can be screened from the insideof the ring by the circulating supercurrent J in combination with the SQUID’s inductance.Furthermore it is an indicator, whether the current-flux-curve of a SQUID displays a hystereticbehaviour.

1.3.1. Non hysteretic dc-SQUID - βL ! 1

In the case of negligible screening, βL ! 1 flux quantisation inside the dc-SQUID plays a lessdominant role. The inductively generated flux ΦL is very small and can therefore be neglected.The flux inside the SQUID equals the externally applied flux

Φ Φext. (1.29)

For a given external flux Φext the maximum transport current Is is found by maximizing (1.25)with respect to ϕ and taking the absolute value of the sin-term. We therefore find

Ims 2Ic

∣∣∣∣cosπΦext

Φ0

∣∣∣∣. (1.30)

We display the switching current in Fig. 1.6. The SQUID’s inductance Ls reduces accordingly

Ls Φ0

4πIc∣∣∣cos

πΦΦ0

∣∣∣ (1.31)

1.3.2. Hysteretic dc-SQUID - βL Á 1

In the case of non-negligible screening, βL Á 1, the situation becomes more complicated.Fluxoid quantization in the loop of the dc-SQUID plays a more dominant role as the inductiveflux ΦL LJ is of considerable magnitude and can therefore no longer be neglected. A wayof working around Eqs. (1.25) and (1.26) is by employing the RCSJ-model for the dc-SQUID.From its equivalent circuit, which is displayed in Fig. 1.7, we obtain an equation of motion forthe phase difference ϕi of each Josephson junction [28]

h

2e

2 C

Ic

d2ϕ1dt2

h

2e

2 1RIc

dϕ1dt

h

2e

rsin pϕ1q i iFs 1

πβLpϕ1 ϕ2 2πφextq 0,

h

2e

2 C

Ic

d2ϕ2dt2

h

2e

2 1RIc

dϕ2dt

h

2e

rsin pϕ2q i iFs 1

πβLpϕ1 ϕ2 2πφextq 0,

(1.32)

11

CHAPTER 1. THEORY

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2Φext/Φ0

0

0.2

0.4

0.6

0.8

1

Im s/2I c

Figure 1.6.: Dependence of the maximal possible supercurrent Ims on the external magnetic field

Φext for negligible inductive flux ΦL.

1

2

V S

s

S,2

D,2

N,2

F,2

I II I S,1

D,1

N,1

F,1

I II IV V

L

I

Figure 1.7.: Equivalent circuit of a dc-SQUID, consisting of the equivalent circuits of two Joseph-son junctions shunted in parallel by an inductor.

where the additional term introduced in Eq. (1.32), compared to Eq. (1.15), represents thecoupling between both phases analogous to the flux-phase relation Eq. (1.24). The strengthof the coupling depends on the value of 1

πβLand φext Φext

Φ0represents the externally applied

flux. For low values of the screening parameter βL, both phases couple strongly and the SQUIDbehaves like a single Josephson junction. For large values of βL, the flux-phase relation relaxesand the SQUID behaves similar to two uncoupled Josephson junctions [22, 29].Similar to our discussion in Subsec.1.1.1, we can apply the analogue of a phase particle with

coordinates pϕ1, ϕ2q, moving in a two dimensional potential. The associated potential USQUIDfor a fully symmetric dc-SQUID, similar to (1.15), is given by[29]

USQUID EJS

cos pyq cos pxq ix 1

πβLpy πφextq2

, (1.33)

12

1.3. DC-SQUID

where a coordinate transformation x ϕ1ϕ22 , y ϕ1ϕ2

2 was applied and EJS is the SQUIDJosephson energy of the complete system, cf. Eq.(1.10) [29, 30]. We note here, that for acompletely symmetric dc-SQUID, EJS is given by twice the value of EJ0 of its constituentjunctions.We have displayed the potential surface described by (1.33) in Figs. 1.8 and 1.9. In the

following we give a qualitative description on the potentials shape and discuss how changes inparameters βL and φext affect it. Concluding, we provide an analysis on the stability of a phaseparticle initiated in one of the potentials many minima, which we use to calculate the switchingcurrent Is pΦextq in the hysteretic regime.

Φ0

5

Φ−1

Φ1

0 x

-50

-6 -4 -2 0 2 4 6

5

y

10

USQUID

(

2E

JΦ

0

2π

)

15

Figure 1.8.: The potential USQUID of a dc-SQUID for parameters I 0 Ic, Φext 0 and βL 1.5.The red lines are guides for the eyes and indicate different flux quantum states ofthe dc-SQUID. Note the periodic arrangement of minima and maxima along thex-axis connected by saddle points offset by either side. For better visibility, we havemarked existent minima, saddle points and maxima in the potential with green,yellow and red dots, respectively.

The overall shape of this 2-dimensional potential resembles that of a parabolic, which is bentalong the y-axis. In the case of no applied current i, its deepest points reside at the value ofapplied flux y πφext. Applying a current i results in a slope along the x-axis, similar to thewashboard potential. In addition, the surface is periodically textured by hills and valleys dueto the double cosine-term in Eq. (1.33). In combination with the parabolic shape this resultsin the formation of periodic minima and maxima in proximity to y πφext along the x-axis,interconnected by saddle points. For a given parameter set pβL, i, φextq the position of theseextrema is given by the Jacobi vector JSQUID

12EJ0

JSQUID i sin pxq cos pyq

cos pxq sin pyq 2πβL

py πφextq 0. (1.34)

13

CHAPTER 1. THEORY

Minima, maxima and saddle points are identified by a positive-definite, negative-definite orindefinite Hessian HSQUID, respectively

12EJ0

HSQUID

cos pxq cos pyq sin pxq sin pyq sin pxq sin pyq cos pxq cos pyq 2

πβLpy πφextq

. (1.35)

We have marked in Fig. 1.8 minima with green points, saddle-points with yellow points andmaxima with red points. Increasing the screening parameter βL for such a potential surfaceincreases the impact of the double-cosine term, as can be seen by comparing Figs.1.9a) and1.9e) with values of βL 0.5 and βL 1.5, respectively. This leads to the formation of new setsof next-nearest extrema, cf. Figs.1.8 and 1.9e), parallel to the fundamental set at the y πφextline. Minima residing at the same y-value form a set corresponding to the same flux quantumstate Φ nΦ0 in the dc-SQUID, indicated by the red lines in Fig. 1.8 and Figs. 1.9.The stability of a phase particle initialized in a certain potential well depends on the applied

external flux φext and the current i, in contrast to only the current i for the washboard potential.A phase particle will stay trapped in its corresponding well, as long as condition (1.34) is fulfilledand the Hermitian H is positive-definite. Due to the periodicity of the double cosine-term, fora parameter change of only φext, new minima are generated, cf. Fig. 1.9b) and in the case ofuntrapping, cf. Fig. 1.9c) the phase particle will always be retrapped. For increasing values ofi this is generally not the case and for high enough i the phase particle will start to move downthe surface without coming to rest in another potential minimum. Classical escape of a phaseparticle will happen in the vicinity of a saddle point, because the presented potential barrier hasits lowest value here.Similar to a Josephson junction, a moving phase particle corresponds to the voltage state,

while a trapped phase particle represents the zero voltage state. For the transition of the phaseparticle between adjacent minima along the y-axis, the dc-SQUID switches for a short amountof time into the voltage state. While in the voltage state, the SQUID changes the flux inside itsloop and occupies a different flux quantum state Φ nΦ0. The amount of time in which thedc-SQUID is in the voltage state depends on the screening parameter βL, the sweeping rate ofthe applied flux, as well as the mass M and damping coefficient η of the phase particle [19].With this qualitative description of the SQUID potential USQUID, we can turn to our quantity

of interest: the switching current Is pΦextq of a hysteretic dc-SQUID. For initializing a phaseparticle in a potential well corresponding to an internal flux value of Φ, we identify the switch-ing current Is for an applied external flux Φext with the current i necessary to destabilize thephase particle. I.e. when the minimum it resides in turns into a saddle point, cf. Eq. (1.35). Wehave displayed the results of this analysis in Fig. 1.10a for different internal flux values Φ witha screening parameter of βL 1.5. Due to the high screening parameter βL multiple crossingsof current curves corresponding to different internal flux values can be observed. We observeadditionally, that the commonly given current-flux characteristics of hysteretic dc-SQUIDs, dis-played e.g. in Refs. [19, 31], equates to the composed envelope of all Is-curves above the firstcrossing point with next-nearest Is-curves, shown by the red dotted line in Fig. 1.10a. Fig. 1.10bdisplays Is pΦextq for different values of the screening parameter βL. From the general definitionsof the switching current Is and the circulating current J , we see that a decreasing switchingcurrent Is is related to an increasing circulating current J . The higher βL is for a correspondingIs pΦextq, the more externally applied flux can be screened by the circulating current J of thedc-SQUID. This is in good agreement with the definition of βL of Eq. (1.28). In this interpre-tation, switching occurs when the externally applied flux Φext exceeds the screening capabilityof the internally generated flux ΦL and forces the dc-SQUID to occupy a different internal flux

14

1.3. DC-SQUID

Φ1

Φ0= ΦextΦ

−1

-3 -2 -1 0 1 2 3y

-5

0

5x

0

2

4

6

8

USQUID

(

2E

JΦ

0

2π

)

Φ1

Φ0

Φ−1

Φext

-3 -2 -1 0 1 2 3y

-5

0

5

x

0

5

10

15

USQUID

(

2E

JΦ

0

2π

)

Φ1

Φ0

Φ−1

Φext

-3 -2 -1 0 1 2 3y

-5

0

5

x

0

5

10

15

20

USQUID

(

2E

JΦ

0

2π

)

Φ1= ΦextΦ

0Φ

−1

-3 -2 -1 0 1 2 3y

-5

0

5

x

0

5

10

15

20

25

USQUID

(

2E

JΦ

0

2π

)

Φ1

Φ0= ΦextΦ

−1

-3 -2 -1 0 1 2 3y

-5

0

5

x

-1

0

1

2

3

USQUID

(

2E

JΦ

0

2π

)

a) b)

c) d)

e)

Figure 1.9.: Top view of the SQUID potential USQUID with a screening parameter of βL 0.5for graphs a) d) and βL 1.5 for graph e). The values of the applied flux area) Φext 0,b) Φext 0.41 Φ0, c) Φext 0.63 Φ0,d) Φext Φ0 and e) Φext 0.Equipotential levels are given by the grey lines. The red lines are guides for theeyes and indicate different flux quantum states of the dc-SQUID. The green linedisplays the value of the applied external flux Φext for each graph. The generationof new minima due to the double-cosine term in USQUID by increasing Φext is shownin b). Switching between two different flux quantum states Φ is displayed in c).The different color scale for d) results from the potential increase in the observed x-y-region. Graph e) displays additional metastable states for an increased screeningparameter βL, compared to graph a).

15

CHAPTER 1. THEORY

Φ1

Φ0

Φ−1

Φ−2

Φ2

a)

-2 -1 0 1 2Φext (Φ0)

0

0.2

0.4

0.6

0.8

1

Is(2Ic)

b)

0 0.25 0.5 0.75 1 1.25 1.5Φext (Φ0)

0

0.2

0.4

0.6

0.8

1

Is(2Ic)

βL = 0.1

βL = 0.5

βL = 1.0

βL = 1.5

βL = 2.5

Figure 1.10.: a) Switching current Is pΦextq-curves for different internal flux values Φ with ascreening parameter of βL 1.5. The internal flux values are given atop thepeaks. b) Switching currents for different screening parameters βL. Unlike the caseof negligible screening there still exists a persistent switching current Is attributedto a flux quantum state outside the typical interval of rΦ02,Φ02s.

quantum state Φ.

1.4. Quantum states in cirQED

We consider a time varying electromagnetic signal a ptq, which can be decomposed into quadra-tures of the field, I ptq and Q ptq respectively [32]

a ptq I ptq cos p2πftq Q ptq sin p2πftq , (1.36)

where f is the signal frequency.As we are interested in quantum effects observable in our system we give in the following a

short overview on possible representations of microwave signals in the quantum limit. Part of itwill later be needed to give a more detailed explanation on the parametric amplification processof our JPA.Suitable sets to represent photonic fields are coherent states [33]. Therefore we introduce X1

and X2 as dimensionless quadrature operators

X1 12a: a

,

X2 i

2a: a

,

(1.37)

where a is an annihilation operator, and a: is a creation operator.The coherent states form an overcomplete, non-orthogonal, set of eigenstates fulfilling the

lower bound of the Heisenberg uncertainty relation with the same amount of noise in eachquadrature moment [32, 34]

∆X1 ∆X2 14. (1.38)

16

1.4. QUANTUM STATES IN CIRQED

Coherent states are eigenstates of the annihilation operator a

a |αy α |αy , (1.39)

which gives rise to their complex eigenvalues, as a is a non-hermitian operator. Coherent statesare generated by applying a displacement operator D on a vacuum state |0y,

D pαq |0y expαa: αa

|αy . (1.40)

A graphical representation of this process is illustrated in Fig. 1.11 a. The radius of the errorcircle is given by Eq. (1.38) to r ∆X1 ∆X2 12.An important aspect of coherent states is that they approximate the sinusoidal behaviour of

classical light, according to the correspondence principle [35].

1

2

X

X

1

2

a) b)

φ

e -r

er

= "⁄$

X

X

r

Figure 1.11.: Generation of a) a coherent state by a displacement of a vacuum state b) genera-tion of a coherent squeezed state by displacing the vacuum field and then squeezingit by the amount r and angle φ

We plan to generate squeezed vacuum states with JPAs and use them further for quantumcommunication. According to [34] such states can be generated by a variety of non-linearprocesses. Squeezed states are a general class of minimum-uncertainty states which fulfil theHeisenberg uncertainty relation on the lower bound, just like coherent states, but in additionhave less noise in one quadrature amplitude compared to the other. Correspondingly the otherquadrature amplitude is amplified to fulfil Eq. (1.38) [32, 34, 35]. Squeezed vacuum states aregenerated by operating with the squeezing operator S on a vacuum state.Squeezed coherent states are generated by a succession of squeezing and displacement operator|α, εy DS |0y. The squeezing operator S is given by

S pεq exp12εa2 12εa:2 , (1.41)

with a complex squeezing parameter ε rei2φ, r a squeezing factor and φ a squeezing angle.A representation of a displaced squeezed state is displayed in Fig. 1.11 b). We see that thesqueezing operator S decreases the variance of one quadrature while increasing the other oneby the squeezing factor r. The attenuated and amplified quadrature moments are also calledsqueezed and anti-squeezed moments, respectively.

17

CHAPTER 1. THEORY

1.5. Flux-driven Josephson Parametric AmplifierA crucial point while dealing with microwave signals at the quantum level is that they need

to be sufficiently amplified before they can be detected. As such, the amplification process mustnot add too much noise. A suitable device for this is a Josephson Parametric Amplifier, which iscapable of quantum-limited amplification of single-photon microwave signals. Theoretically, ina phase-insensitive, or non-degenerate, operation mode, input microwave signals are amplifiedwhile adding only one half of a noise photon. In a phase-sensitive, or degenerate, mode it istheoretically possible to amplify one quadrature of the input signal without adding any noiseat all [13, 14]. The phase-sensitive amplification is an important requirement for generatingsqueezed states [32, 36].The effect of parametric amplification can also be observed classically. An archetypical exam-

ple of parametric amplification is a child standing on a swing. Modulating the center of massof the swing at certain frequencies leads to the parametric amplification (excitation). The childdoes this most efficiently by modulating the swing at twice the oscillation frequency.

a)

b)

c)

f0f f0-

A

input signal

f0 f +f0

A

f f0-

M·AG A·

output signal

f02·f0

pump signal

sig

na

l p

ort

pu

mp

po

rt

JPA

resonator

pump

dc SQUID

C csignal

Φ +dc rfΦ

Fre

qu

en

cy (

GH

z)

6.0

5.5

5.0

4.5-0.5 0 0.5

Flux

Figure 1.12.: a) circuit diagram of a flux-driven Josephson parametric amplifier. A chain ofLC oscillators represents the coplanar waveguide resonator. A coupling capacitorat the left end enables signal exchange. A dc-SQUID is included in the last os-cillator block, controlling the resonator’s coupling to ground. The pump line isportrayed by an inductance, by which pump signals can couple to the dc-SQUID.b) The flux-tunable resonant frequency of a Josephson parametric amplifier withan approximate internal quality factor Qint 30, which was previously measuredin house. For a set working point f0dc (blue point), the resulting frequency mod-ulation by applying a pump tone at 2f0dc is indicated by the black arrows. c)Black box representation of a JPA for subjected input and output fields in thephase insensitive operation mode. For an applied pump tone of 2f0, an appliedsignal f0 f detuned from the working point f0 will be amplified by a gain fac-tor G. Additionally, an idler mode f0 f will be generated, characterised by anintermodulation gain M [37].

Our flux-driven Josephson Parametric Amplifier [17] consists of a quarter wavelength res-onator, which is shunted to ground by a dc-SQUID at one end. A pump line couples fluxinductively to the SQUID. The equivalent circuit of our JPA can be seen in Fig. 1.12a. Bycontrolling the flux Φext threading the dc-SQUID, the resonant frequency f0 of the JPA can be

18

1.5. FLUX-DRIVEN JOSEPHSON PARAMETRIC AMPLIFIER

tuned. An external field coil supplies the magnetic field necessary for that. For a set workingpoint f0, parametric amplification for an input signal fs f0 f at the resonator is achievedby applying a pump signal at double the resonant frequency fp 2f0 via the pump line. Thefast oscillating pump field adds to the flux Φext in the dc-SQUID and effectively modulates theresonant frequency f0. This leads to signal amplification as the energy of the pump is convertedinto the signal field. Parametric amplification can be seen as a three-wave mixing process, withthe JPA serving as a non-linear medium. The third wave involved is the generated idler modefi. Its frequency is given by energy conservation

fp fs 2f0 pf0 fq f0 f fi (1.42)

From Eq. (1.42) we see, that in the case of a detuned input signal fs f0 from the workingpoint, the generated idler mode deviates from all other modes in frequency. This corresponds tothe phase-insensitive amplification process mentioned earlier. A schematic representation of theinput and output fields during phase-insensitive amplification is displayed in Fig. 1.12c). For aninput signal equal to the working point fs f0, one can no longer talk about idler generationas idler and signal occupy the same mode. Momentum conservation poses a strict boundary onall phases involved in the mixing process and signal amplification or even de-amplification canbe observed, depending on the phase difference between signal and pump mode. This is thephase-sensitive operation mode. Fig. 1.12b) [13, 32, 37] illustrates the effect of an applied RFsignal on the set working point.In the following we give a more detailed description on the JPA’s working principles. As

we have previously explained the dc-SQUID, we continue with the remaining constituent, thestripline resonator. We give a brief summary on the basic properties, which help us to describeour resonator. Further we explain, how incorporating a dc-SQUID into a stripline cavity resultsin a flux-tunability of its resonant frequency. As a last point, a more detailed analysis of theparametric amplification process will be given in Subsec. 1.5.3.

1.5.1. Description of a reflection type resonator

For applying microwave probe tones the quarter-wavelength resonator is the main componentof our Josephson parametric amplifier. It serves as a microwave cavity, into which we can couplein signal photons. We therefore are interested in the characteristic properties describing thequality of our resonator. These are its resonant frequency, bandwidth and quality factors. Thequantity which allows us to determine all three properties is the reflection coefficient Γ.Like any oscillator, a coplanar waveguide resonator is described by the equation of motion for

a harmonic oscillatord2q

dt2 ω2

0q 0, (1.43)

where q is a generalized coordinate and ω0 is the resonant frequency of the oscillator, defined byits physical dimensions. For a lumped element resonator, the resonant frequency is in generalgiven by

ω0 1?LC

(1.44)

where L is the inductance and C is the capacitance of the resonator.To allow for signal application and detection, our JPA has to be connected to an external

device, which will lead to a shift in the resonant frequency ω0. The representation of a re-flection type resonator as a lumped element is sufficient to explain this coupling between the

19

CHAPTER 1. THEORY

measurement device and our JPA. In this model, our resonator is described by a resistor R, aninductance L and a capacitance C connected in parallel. Via the coupling capacitance Cc it isconnected to the measurement device, which can be portrayed as a current source I with loadimpedance Z0. A diagram of this lumped element can be seen in Fig. 1.13. The current source

CL 0

I

R

in

Z

Z c

C

Figure 1.13.: Schematic of the lumped element model of a reflection type resonator. The mea-surement device, symbolized by a current source, is directly connected with itsload impedance Z0 to the resonator. The input impedance Zin with its compo-nents R,L,C,Cc is marked by the yellow box.

will treat our resonator as an additional input impedance Zin, to which it is connected. Fromdiagram 1.13, we can calculate the resonator’s input impedance Zin to

Zin 1iωCc

11R

1iωL ωC

1iωCc

11R iωC

1 ω2

0ω2

1iωCc

11R 2i pω ω0q

,

(1.45)

where in Eq.1.44 was used, and Zin was approximated in first order for applied frequenciesω ω0. We further transform Eq.(1.45) by dividing it into its real and imaginary components,yielding

Zin R

1 4C2R2 pω ω0q i

1ωCc

2 pω ω0qCR2

1 4C2R2 pω ω0q2. (1.46)

When applying an alternating signal Ain with the current source, we can measure the back re-flected signal Aout from the resonator. Both quantities are connected by the reflection coefficientΓ by Aout ΓAin. An expression for the reflection coefficient Γ from the lumped element modelis given to

Γ Zin Z0Zin Z0

. (1.47)

By applying a spectrum of frequencies to the cavity, we will detect a deviation from the bareresonant frequency ω0 in Aout due to the coupling to the measurement device. This measuredresonance is called the loaded resonant frequency ωL and can be obtained from Eq. (1.46)by searching for frequencies, for which the imaginary part of Zin vanishes. In the case of

20

1.5. FLUX-DRIVEN JOSEPHSON PARAMETRIC AMPLIFIER

R2ω20C

2cC " r4C, 2Ccs, the two solutions obtained from this condition take the form

ωL,1 ω0,

ωL,2 ω0

1 2Cc2C

.(1.48)

As the first solution is the unperturbed resonant frequency ω0, only ωL,2 has real physicalsignificance, which we refer to as ωL for the remainder.A further characteristic observation from the backscattered signal Γ is, that the loaded resonantfrequency ωL is not a single sharp line, but resembles a dip with finite width and deepness, similarto the spectral lines observed in atoms. The full width at half maximum of this Lorentzian shapeddip is described as the resonator bandwidth ∆ωFWHM and connected to the resonator’s qualityfactor Q by [38]

∆ωFWHM ωLQ. (1.49)

The quality factor Q is a figure of merit describing the quality of a resonator, i.e. how well itcan store energy over time. A general definition of the quality factor is given by [39]

Q ωL time average energy storedenergy loss per second (1.50)

As energy storage is always connected with losses, we can divide the quality factor into anexternal Qext and internal quality factor Qint, corresponding to the two possible loss mechanismfor a resonator

1Q 1Qint

1Qext

. (1.51)

The external quality factor Qext is a measure, how long an in-coupled photon will stay insidethe cavity, before it gets emitted. The internal quality factor indicates, when an in-coupledphoton will decay inside the JPA due to internal losses, respectively. In the case of Qext ¡ Qintone speaks of an undercoupled resonator, where in the case of Qext Qint the resonator isovercoupled.A more accessible form of the reflection coefficient Γ is obtained using input-output formal-

ism [40]

Γ pω ωLq2 iκ2 pω ωLq κ21κ

22

4ω ωL iκ1κ2

22 , (1.52)

where κ1 and κ2 are coupling constants, which are related to the JPA’s quality factors by theloaded resonant frequency ωL

κ1 ωLQext

, (1.53)

κ2 ωLQint

. (1.54)

With (1.52) we can fully describe the resonator by recording magnitude and phase of the backreflected signal Aout over a broad frequency range around the loaded resonant frequency ωL. InFig. 1.14 we display the real and imaginary part of the reflection coefficient for an under-coupledand over-coupled resonator, as displayed in [40].

21

CHAPTER 1. THEORY

Frequency (GHz)

0

0.2

0.4

0.6

0.8

1Magnitude(arb.units)

undercoupled

overcoupled

5.5 5.75 6 6.25 6.5Frequency (GHz)

0

60

120

180

240

300

360

420

Phase

(deg.)

undercoupled

overcoupled

a)

b)

Figure 1.14.: a) Magnitude and b) phase of the reflection coefficient Γ of a reflection type res-onator as a function of frequency. The parameters used were fL 6.00 GHz,Qin 700, Qext 1100 for the under-coupled case, and fL 6.00 GHz, Qin 700,Qext 200 for the over-coupled case.

1.5.2. Flux modulation of a coplanar waveguide cavity

The discussion of the lumped element model resulted in the properties by which a purereflection resonator can be described. However, the coplanar waveguide resonator of our JPAhas a dc-SQUID incorporated in it. In the following we want to explain how this leads to aflux dependence of the resonant frequency, which is a key necessity for a flux-driven Josephsonparametric amplifier.Contrary to the lumped element model, a physical transmission line resonator possesses has

spatial extensions comparable to the wavelengths of the relevant signals. By viewing it as adistributed element resonator as in Fig. 1.12a, the resonator can be described by as a chainof LC oscillators, each with superconducting phases φi, inductances Li and capacitances Ci,as displayed by the circuit diagram of Fig. 1.12. Summing up all elements will result in theinductance L and capacitance C of the lumped element model, which we refer to as Lcav andCcavfor the remainder.The resonant frequency of a distributed one dimensional coplanar transmission line cavity is

determined by the boundary conditions for the superconducting phases φi. In our case, theseboundaries for our resonator are given by the coupling capacitance Cc on the left end and thedc-SQUID on the right end, which itself is shunted to ground, cf. Fig 1.12.

22

1.5. FLUX-DRIVEN JOSEPHSON PARAMETRIC AMPLIFIER

The boundary conditions given by coupling one end to ground while leaving the other endopen results in a maximum for the spatial distribution of the superconducting phase at theopen end and a node at the grounded end. This configuration corresponds to a λ4-resonatorwith eigenmode frequencies ωn

π?LcavCcavd

pn 12q. Leaving both resonator ends openresults in a λ2-resonator with eigenmode frequencies ωn

π?LcavCcavd

n.

Coming back to our system, the coupling capacitance at the left end equates to an openend for our resonator. Incorporating a dc-SQUID at the right end renders the state of theboundary condition tunable by the applied flux to the SQUID, similar to a stepless switch. Aconnection to ground corresponds to a formally infinite SQUID Josephson energy EJS, wherean open end corresponds to a fully suppressed SQUID Josephson energy. In effect, by changingthe applied flux in an interval of half a flux quantum 0 Φ Φ02, the resonator configurationcan be swept between open-to-ground and open-to-open, i.e. a λ4 and λ2 element. Theeigenmode frequencies related to this tunability are in the interval

π?LcavCcavd

n ωn

π?LcavCcavd pn 12q.

The accessible frequency band associated with the SQUID Josephson energy is given by thedispersion relation of the resonator wave equation[30]

pkndq tan pkndq

2eh

2LcavEJS pΦextq Cs

Ccavpkndq2 , (1.55)

where d is the length of the resonator, Cs is the SQUID capacitance and Lcav and Ccav are thecavities bare inductance and capacitance without the SQUID, respectively. kn is the associatedwave vector of an eigenmode ωn, defined by kn

?LcavCcavωn.

The SQUID Josephson energy EJS pΦextq represents the flux dependent Josephson energy ofthe complete dc-SQUID, whose values theoretically . In order to compare the SQUID Josephsonenergy with the cavity inductance Lcav, we express it in the form of an inductance. The totalinductance Ltot of a real SQUID has two contributions. The first is the kinetic inductance Ls,which is given by Eq.(1.27) and represents the inertial energy of all Cooper pairs. The secondpart is the geometric inductance Lgeom and is determined by the geometry of our SQUID. Withboth contributions, the Josephson coupling energy can be expressed as:

EJS Φ20

4π2 pLs pΦextq Lgeomq . (1.56)

As the capacitive term in Eq. 1.55 can be neglected for SQUID configurations of our interest,inserting Eq. (1.56) into Eq. (1.55), we can express the dispersion relation by the involvedinductance terms

pkndq tan pkndq LcavLs pΦq Lgeom

. (1.57)

In practice, SQUIDs always display an asymmetry in the critical currents of its Josephsonjunctions. This implies a non-zero value for the SQUID Josephson energy EJS even when fullysuppressed by the external flux. In addition, the maximum EJS has a finite value. The accessiblemodulation interval of a real SQUID terminated coplanar waveguide resonator is narrower - thanthe range given above. In order to achieve a broad modulation interval, a high value of LcavEJSshould ideally be chosen, while SQUID asymmetries should be avoided during fabrication [30].

1.5.3. Parametric amplificationIn order to understand the parametric amplification process in flux driven JPAs, we give a

quantum description based on input-output formalism. We are interested in expressions for the

23

CHAPTER 1. THEORY

phase insensitive and phase sensitive signal gain of the JPA. We start by classifying the JPA asa resonant structure exchanging photons with its environment.As mentioned earlier, a flux driven JPA consists of a λ4 resonator connected to ground with

a dc-SQUID. This connection allows to parametrically excite the resonator by applying a pumpsignal. The quantum description of the parametrically-modulated harmonic oscillator is given,in first order approximation, by [41]

Hpar hω0

a:a ε cos pαω0tq

a a:

2, (1.58)

where the first term represents the harmonic oscillator and the second term is owed to thepresence of the SQUID. α is a scalar factor, relating the applied pump frequency fp αω0 tothe resonant frequency ω0. a and a: are the annihilation and creation operator for photons inthe JPA, and ε is the amplitude of the modulation term.The complete Hamiltonian H is given by adding terms Hin for exchanging photons with an

external photon source and Hloss describing for internal losses

H Hpar Hin Hloss, (1.59)

withHint

»dω

hωb pωq: b pωq ih

cκ12π

a:b pωq b pωq: a

, (1.60)

Hloss »dω

hωc pωq: c pωq ih

cκ22π

a:c pωq c pωq: a

. (1.61)

In this picture Hin and Hloss are considered as heat baths exchanging photons of frequency ωwith the JPA. Signal photons of frequency ω are represented by b pωq and noise photons aregiven by c pωq. The first term in each Hamiltonian portrays the ensemble of photons present ineach bath. The second terms constitute coupling between the corresponding heat bath to theJPA. The strength of each coupling is represented by the coupling constants κ1 and κ2, relatedto Eqs. (1.53) and (1.54), respectively.From Eq. (1.59) an expression for the parametrically amplified signal gain Gs can be calculated

from the JPA’s steady state response. We therefore consider a classical signal, which implies:a xay,

AbE Eeiβω0t, where E is the input signal power, and xcy 0. A differential equation

describing the JPA’s steady state response, subject to input signals and applied pump power, isgiven by [41]

d2 xaydt2

rκ1 κ2 iαω0s d xaydt

pκ1 κ2q2

4 Ω201 ε2 α

iαω0κ1 κ2

2

xay

?κ1E

κ1 κ2

2 i pα β 1qω0

eiβω0t iεω0

?κ1E

eipαβqω0t,

(1.62)

where β is a scalar factor, relating the signal frequency fs βω0 to the JPA’s resonant frequencyω0. The left-hand-side of this equation represents the resonator states. By applying a sufficientlyhigh pump power ε, an in-coupled signal is parametrically amplified and resonator modes areexcited. As the latter is not desirable, an analysis of the solutions for the resonator eigenstatesxayr in Eq. (1.62) gives an expression for the values of the pump power, where ε resonator modesare excited. For α 2, when the frequency of the pump tone fp is twice the JPA’s resonantfrequency ω0, we define the critical pump power εc as

εc κ1 κ22ω0

12

1

Qext 1Qint

, (1.63)

24

1.5. FLUX-DRIVEN JOSEPHSON PARAMETRIC AMPLIFIER

which is the threshold value, above which resonator excitation occurs.Expressions for the input signal gain Gs and idler gain Gi are obtained by an analysis of the

solutions for the input signals, represented by the right hand side of Eq. (1.62). In the case ofnon-degenrate gain, β 2α, the signal Gs and intermodulation gain Gi is given by [41]

Gs ∣∣∣∣E ?

κ1AsE

∣∣∣∣2, (1.64)

Gi ∣∣∣∣?κ1AiE

∣∣∣∣2, (1.65)

where As and Ai are field operators for the signal and idler states, respectively, given by

As ?κ1E

κ1κ2

2 i pα β 1qω0

κ1κ22 i p1 βqω0

κ1κ2

2 i pα β 1qω0 ε2ω2

0, (1.66)

Ai iεω0?κ1E

κ1κ2

2 i p1 βqω0

κ1κ22 i pα β 1qω0

ε2ω20. (1.67)

Figure 1.15.: a) Signal and b) intermodulation gains as a function of the signal frequency inunits of βω0 for different pump powers below the threshold εc from Ref. [41]. Theparameters used for the calculation are ω0

2π 10.41 GHz, Qext 240, Qint 2200and α 2.

We plot Eqs. (1.64) and (1.65) as a function of signal frequency βω0 for different values ofpump power ε εc below threshold in Fig. 1.15, as given in [41].If the input frequency is half the pump frequency β 2α, we also have to consider the phase

of the input signal E ∣∣∣E∣∣∣eiθ. An expression for the resulting degenerate gain is given by

Gd ∣∣∣∣E ?

κ1As ?κ1Ai

E

∣∣∣∣2 ∣∣∣∣∣∣1 κ1

κ1κ22 i pβ 1qω0 iεω0e

2iθ

pκ1κ2q2

4 p1 βq2 ω20 ε2ω2

0

∣∣∣∣∣∣2

. (1.68)

In this operation mode, the minimum gain Gmind is obtained by setting the phase to θ π4nπ

Gmind

εω0 κ1κ2

2εω0 κ1κ2

2

2

, (1.69)

25

CHAPTER 1. THEORY

and the maximum gain Gmaxd by setting θ 3π4 nπ

Gmaxd

εω0 κ1κ2

2εω0 κ1κ2

2

2

(1.70)

26

2. Experimental Setup

In this chapter we describe the experimental setup we used for characterizing our JPA. Westart by discussing the JPA samples. Then we proceed with an overview of a sample holder,which allows us to apply microwave signal and pump tones to our sample. The cryogenic setupis described next - we present our cryogen-free dilution refrigerator and the components of ourmicrowave lines. At the end, we present room temperature electronics responsible for controllingand measuring of our JPAs.

2.1. JPA samples and sample holder

The Josephson parametric amplifiers were designed and provided by T. Yamamoto and K.Inamoto at NEC, similar to the ones previously characterized in [37]. We received 6 differenttypes of JPAs and 32 samples in total. In Tab. 2.1 we give the designed parameter values of thedifferent JPA types. These are: the external quality factor Qext, the coupling capacitance Cc,the resonator length lres, the critical current Ic of the SQUID’s Josephson junctions, and theSQUID area.We differentiate between the different samples in the following by referring to the designed

external quality factor. The suffix "epr" for the JPAs Q1100-epr and Q5000-epr, states thatthese samples are intended to be used for Einstein-Podolsky-Rosen pair production.

sample Qext Cc lres Ic/junction SQUID area(fF) (mm) (µA) (µm2)

Q80 85 54 4.32 1.1 4.22.4Q200 214 34 4.80 2.0 4.22.4Q600 680 19 5.10 4.0 4.22.4Q1100 1091 15 5.19 5.6 4.22.4Q1100-epr 1090 15 5.06 3.0 4.22.4Q5000-epr 5002 7 5.18 4.0 4.22.4

Table 2.1.: Design parameters of different JPA samples.

2.1.1. JPA samples

In this thesis we present results for JPA samples Q200, Q600 and Q1100-epr. A micrograph ofthe Q200 sample is displayed in Fig. 2.1. An overview of all JPA parts is given in Fig. 2.1a. Thesubstrate, on which the JPA structure is deposited, is a silicon wafer. The ground planes, centerconductor of the transmission line resonator and the pump line were fabricated from niobiumwith an approximate layer thickness of 50 nm. Fig. 2.1b) displays the coupling capacitor. Thedc-SQUID terminating the JPA’s resonator is made of aluminum deposited on top of the niobiumlayer by shadow evaporation[42]. A magnification of the SQUID can be seen in Fig. 2.1c).

27

CHAPTER 2. EXPERIMENTAL SETUP

Figure 2.1.: Micrographs of a Q200 Josephson Parametric Amplifier. The complete structureof the JPA is displayed in a). The coplanar waveguide resonator in the center ofthe image is the JPA’s resonator. The coupling capacitor is marked by a red box.The green box marks the position of the dc-SQUID. The pump line is the coplanarwaveguide resonator in the right third of the image. b) magnified image of thecoupling capacitor. c) magnified image of the dc-SQUID. The bright color depictsaluminum, while the darker one stands for niobium.

28

2.2. CRYOGENIC SETUP

2.1.2. Sample holder assembly

We install our samples for measurements in a cryostat, which we will describe in Sec. 2.2.1.Electrical contact between the measurement equipment and all other microwave components isestablished with microwave coaxial cables. This poses two demands on our sample: First, wehave to establish a connection between standard SMA coaxial cables and a coplanar waveguideresonator to the JPA. Second, we have to mount our sample in a mechanically secure way inour cryostat. These are met by our sample holder assembly which consists of:

• sample holder

• V-type microwave connectors

• glass beads

• printed circuit board (PCB)

The sample holder serves as a housing for a JPA sample, the PCBs, glass beads and microwaveconnectors. It will then be installed on the sample rod in our cryostat, see Fig. 2.4. The sampleholder provides shielding from stray electromagnetic fields. The sample holders are fabricatedby our in-house workshop and were gold plated subsequently. Electrical contacts between thecoaxial cables, the JPA’s signal, and the pump lines were established by a chain of a V-typeconnector, a glass bead and a printed circuit board. The V-connectors and the glass beadsare attached to the sample holder through its designated recesses. The aluminum oxide PCBfeatures a gold-plated coplanar waveguide resonator on top, connected to a ground plane by vias.The connection between the PCB and the sample is achieved by Al-wire bonds. A preparedsample is shown in 2.2. The additional wire bonds between the sample holder, the sample andthe PCBs are installed to provide a broadband impedance matching.

2.2. Cryogenic SetupExperiments in circuit quantum electrodynamics(cirQED) require low temperatures. On the

one hand, superconducting circuits require low temperatures for the materials to enter thesuperconducting state, e.g. Tc pAlq 1.20 K or Tc pNbq 9.26 K. On the other hand, we areworking with microwave signals in the GHz regime. Here, the thermal population of photonstates should be negligible. This requires the thermal energy to be much lower than the energyof a microwave photon, which requires the samples to be cooled down to cryogenic temperaturesin the mK regime1.We use a dilution refrigerator to cool our samples. A schematic of a cryogenic setup used

for characterizing our samples is displayed in Fig. 2.3. Attenuated input lines are used fortransmission of the signal and the pump tones to our samples. Output lines with HEMTamplifiers are used to transfer the signal to a detection setup at room temperature. A magneticfield necessary for flux-characterization of our samples is provided by superconducting coilsmounted on top of the sample holders. Circulators are employed for measurement and shieldingpurposes. In this scheme, the JPA sample marked as Q200 was selected for further experimentsincorporating a dual path setup [32], for which a heatable attenuator, a directional coupler,and a hybrid ring have been additionally installed. Since the dilution refrigerator provides

1To be more precise, as we want to employ signal frequencies close to 6 GHz we need to operate our cryostatbelow 280 mK in order for our signals not to be dominated by thermal noise

29

CHAPTER 2. EXPERIMENTAL SETUP

Figure 2.2.: Prepared sample holder for sample of type Q200 before its installation into thecryostat. 1 - the JPA sample 2 - the PCBs 3 - the glass bead center conductor pin.The sample holder used is of the top-mount design, which renders the V-connectorsin this image not visible. The marks S and P mark the connection to the signaland the pump lines of the JPA, respectively.

large experimental space at base temperature other experiments are mounted in parallel. Themicrowave components at base temperature are mounted onto a silver rod, which is attached tothe mixing chamber plate of our cryostat. The assembled sample rod can be seen in Fig.2.4.

2.2.1. Dilution refrigerator

Dilution refrigerators (DR) have become standard work-horses in low temperature experi-ments, as they can be run for long periods of time by simultaneously offering a relatively highamount of cooling power. In our dry dilution refrigerator precooling the system is achieved bya pulse tube refrigerator (PTR).The cryostat in which we conduct our experiments is a dry dilution refrigerator with a separate

1K-stage[43, 44]. A photograph of our cryostat alongside a scheme with the correspondingtemperature stages is shown on Fig. 2.5.Pre-cooling is facilitated by a CRYOMECH PT410-RM. The PTR works similar to a stirling

engine with regenerator, where the pulse tube replaces mechanical components in the low tem-perature part of the device. A compressor unit expands and compresses He gas through thepulse tube, which leads to heat intake at the cold head and release at the room temperatureheat exchanger. For a more detailed description we refer to [45]. Our PTR operates at twostages and achieves 50 K at the first and 3 K at the second stage, respectively.Dilution refrigerators make use of the heat of mixing of 3He and 4He isotopes. In the mixing

chamber liquid He undergoes a phase separation at very low temperatures into a 3He-rich phaseand a 3He-sparse phase, where 3He is diluted in superfluid 4He. In the mixing process, 3Hefrom the rich phase is transferred into the dilute phase. The required energy is provided by themixing chamber and its environment, and represents an effective cooling power of the DR. A

30

2.2. CRYOGENIC SETUP

10 dB 10 dB 10 dB 10 dB

10 dB 10 dB 10 dB 10 dB

10 dB 10 dB 10 dB 10 dB

10 dB 10 dB 10 dB 10 dB

50 Ω

50 Ω

50 Ω

50 Ω

50 Ω

JPA Q600

topmount

JPA Q200

topmount

30 dB 50 Ω 50 Ω

S

S

P

P

Output lines

Input

port 1

Signal lines Pump lines

10 dB

50 Ω

1 2 3 4 1 2 5 RT

50 K

3 K

1 K

15 mK

Heater

+

temp. Sensor

magnet coil

matched load

a"enuator

male-male

adapter

microwave

feedthrough

HEMT amplifier

power divider

hybrid ring

circulator

direc$onal coupler

Figure 2.3.: Schematic of the cryogenic setup installed in our dilution refrigerator for measuringJPA samples Q200 and Q600. The different temperature stages are represented bythe dashed boxes, surrounding the corresponding equipment installed at them. Sig-nal and pump tones pass through the attenuation chain before being applied to thesamples. Two different multiport flanges were used to couple in signals to the exper-iment. The heatable attenuator was installed to later perform power calibration onthe input signals. Circulators are used for signal separation (red box) and noise at-tenuation (yellow box). The output signals are amplified at cryogenic temperaturesbefore exiting the cryostat through single port flanges. The directional coupler, thedirectional coupler and the hybrid ring were installed for later measurements.

31

CHAPTER 2. EXPERIMENTAL SETUP

Figure 2.4.: a) Front and b) back views of our sample rod installed at the mixing chamber plate.1 - sample rod 2 - Q600 sample box 3 - Q200 sample box 4 - superconducting coils5 - circulators 6 - directional coupler 7 - heatable attenuator 8 - hybrid ring.

precise description of the mixing process is given by [46].In addition radiation shields are installed at different temperature levels of the cryostat, which

can also be seen in Fig.2.5. These allow us to shield our samples to some extent from magneticstray fields from the fridge’s environment.

2.2.2. Input lines

We supply the signal and pump tone, which are generated by microwave sources at the roomtemperature setup, with input microwave coaxial cables to our JPAs. Signals generated at roomtemperature are coupled into the evacuated cryostat through vacuum tight flanges. For our inputlines inside the cryostat we use a chain of rigid steel coaxial cables, which are impedance matchedat 50 Ω, suitable for operation at cryogenic temperatures. The cables are connected between thedifferent temperature stages of the cryostat by microwave feedthroughs, which cool down theinner conductors of the coaxial cables. A direct application of microwave signal would exposeour samples to room temperature noise. Therefore, we use attenuators to mitigate the thermalnoise transmitted down to a level, which poses no threat to our experiments. The attenuatordampens and thermalises the incoming power by its specified value in dB. Similar to a blackbody, the attenuator adds white noise to the input signal, corresponding to its temperature.Ideally all attenuation would be done at the lowest temperature of our cryostat, in order tokeep the signal to noise ratio at its optimum. However this is not possible, as the signal powerdissipated by an attenuator can exceed the cooling power of the mixing chamber. We therefore

32

2.2. CRYOGENIC SETUP

PTR

PT405

RM

Sample

room

temperature

plate

1st PTR stage

50 K

2nd PTR stage

3 K

1 K stage

S!ll plate

0.6 K

mixing chamber

plate

15-50 mK

heat

exchanger

mixing

chamber

separate 4He

stage

3He/4He

mixing cycle

s!ll

circulators

HEMT

outer

vacuum can

radia!on

shields

a) b)

Sample

PTR

1K He bath

Figure 2.5.: a) Schematic drawing of our dilution refrigerator. We display the components ofthe 3He 4He mixing cycle, the 1 K refrigerator as well as the stages of thepulse tube refrigerator at the corresponding temperature stages of our cryostats.b) A photograph of our dilution cryostat. We marked the positions of the HEMTamplifiers, circulators and the sample.

33

CHAPTER 2. EXPERIMENTAL SETUP

resort to a chain of attenuators installed at various temperature stages of our cryostat. Note thatthe heat dissipated at each temperature stage has still to be taken into account. To safeguardthe overall operation of our refrigerator, an integral heating power dissipated by the attenuatorsshould not surpass the effective cooling power at the corresponding temperature stage of thecryostat.

2.2.3. Output linesWith output lines we direct backscattered signals from the JPAs to the detector. These

signals are weak and need additional amplification - which is why we use superconducting NbTicables from the JPA to a cryogenic amplifier and rigid silver-plated stainless steel cables fromthe amplifier to the output flanges. We use HEMT amplifiers from lownoisefactory, capable ofoperating at cryogenic temperatures in a frequency band of 4-8 GHz, with an average signalgain of +40 dB. During the amplification process 7 noise photons on average are added to oursignal. Thermalisation of output microwave lines is achieved by clamping copper braids to theouter conductor of the output lines and anchoring them at the corresponding temperature stagesof the cryostat.

2.2.4. CirculatorAn important component of our setup is a microwave circulator, which allows to separate

signals with respect to their direction of propagation. An ideal circulator is a passive, non-reciprocal three port device. Signals entering one port are transmitted to the next port in aspecific rotation direction of the circulator, as can be seen in Fig. 2.6. In the opposite rotationdirection, the circulator ports are isolated from each other. I.e. for a counter-clockwise circulator,as shown in Fig. 2.6a, a signal applied to port 1 will be transferred to port 2, while no signalis passed to port 3. From network analysis this process is described by the following scatteringmatrix:

bout1bout2bout3

0 0 1

1 0 00 1 0

bin1bin2bin3

, (2.1)

where bouti

bin

irepresent outgoing(incoming) signals at the i-th circulator port.

1

1

2

2

3

3

a) b)

Figure 2.6.: Schematic of a a) counter-clockwise and b) clockwise circulator. The different signaltransmission between the circulator’s ports is indicated by arrows.

The circulators we used were from QuinStar, designed for a frequency range of 4-8 GHz andspecified for use at cryogenic temperatures [47]. As every physical system is not ideal, there

34

2.3. ROOM TEMPERATURE SETUP

will be signal transmission between isolated ports, which will be attenuated by 18 dB for ourcirculators. In our setup, the circulators allow to implement two important tasks:

• Separation of JPA signals: Because of the design of our JPA, backscattered signalsleave the resonator through the same port through which the original signals enter. Asthe signals are weak and need further amplification before detection, reusing the inputline is not an option. The circulators marked by red box in Fig. 2.3 are used to separateinput and output signals of our JPAs. The incoming signals from the input line are directedtowards the JPA and the outgoing signals are fed into the output line toward the cryogenicamplifiers.

• Isolation of the sample stage from noise: No attenuators are built into the outputline to allow for a minimal amount of signal losses along the detection chain. This howeverposes a significant threat for the sample stage, as noise from higher temperature stages canpropagate unmitigated along the output line towards the sample stage. The circulatorsmarked by the yellow box in Fig. 2.3 address this issue. The port transmitting the incom-ing noise is terminated with a 50 Ω matched load, which absorbs all input noise power.Similar to the attenuators, it will emit white-noise of a black body of its correspondingtemperature.

2.3. Room temperature SetupIn this section we present the room temperature setup and describe devices we use. In order

to characterize our sample we use signal generators to send microwave signals to our samplesvia our input lines. Backscattered signals from the output lines are then detected by detectors.Although the signals have been pre-amplified in the cryostat by our HEMT amplifiers, thesignals are still too weak for some room-temperature devices. We therefore need an additionalamplification step.We create the input signals with a Rohde & Schwarz Network Analyzer (ZVA), capable of

signal generation in a range of 10 MHz up to 24 GHz and output powers up to 15 dBm. Weadditionally use the ZVA as a detector in order to determine the complex S-parameters of oursample. The pump signals we need to amplify our input signals are generated by a Rohde& Schwarz Microwave Signal Generator (MSG). The frequency range of our MSG is between100 kHz to 43.5 GHz with output powers of -130 up to 25 dBm .For additional signal amplification, additional microwave amplifiers were installed at the DR’s

frame. We used Agile AMT-A0033 rf-amplifiers, which yield a signal gain of up to 28 dB forsignals between 2 to 8 GHz. To ensure their phase and gain stability, they were temperaturestabilized by a Peltier cooler plus controller. In order to assure frequency and phase stabilitybetween our microwave sources and detectors we used a 10 MHz reference signal from a Rubidiumclock generator Stanford FS725.In Fig. 2.7 we display the room temperature setup used for the characterization of samples

Q200 and Q600. A 50 dB attenuator was installed at the signal port of the ZVA, to achieve apower level -139 dBm at the sample stage, adding to the attenuation chain of the input signalline. In addition to the already mentioned measurement equipment, the dual path receiverwas installed for later measurements [13, 32]. The coaxial switches are installed to allow forswitching different measurement methods. The switches used were Agilent N1810 TL, providinghigh isolation between uncoupled ports. Control and automation of the setup was faciliated bya LabView program.

35

CHAPTER 2. EXPERIMENTAL SETUP

ZVA

Port 2 Port 1

MSG

Pump

50 dB

Frequency

standard

signal

port

pump

port

output

port

coaxial

switch

Figure 2.7.: Schematic of a room temperature setup for the characterization of JPA samples.We use the ZVA to measure the complex scattering parameters of our sample. TheMSG serves as a source for the pump tone. The rubidium frequency standard assuresphase stabilization between separate microwave devices.

36

3. Experimental Results

In this chapter we summarize the experimental results with different flux-driven Josephsonparametric amplifiers. For our first measurements, we chose to characterise one sample of typeQ1100-epr, Q200 and Q600, respectively.As a first characterization step we probed the flux tunability of our JPAs by measuring the

response of the JPAs’ resonant frequency over a broad range of external magnetic flux. Fromthe recorded data we extracted the samples’ parameters and determined, how well they matchtheir respective design values. The second part of the JPA characterization contained measuringthe non-degenerate gain with samples of type Q200 and Q600.

3.1. Resonator characterization of the JPAAs we described in Sec. 1.5, the core element of a Josephson parametric amplifier is a quarter

wavelength resonator, shunted to ground by a dc-SQUID. This renders the resonator’s elec-trical length controllable by a magnetic flux. This sensitivity is utilised in flux-driven JPAsto parametrically amplify input signals. Important characteristics of the JPA are its internaland external quality factors, which influence the amplification properties. In our first measure-ments we record the flux dependence of the JPA frequency and determine the quality factorsfor different flux values.

3.1.1. Flux dependence of the resonant frequency

We studied the JPA response to an external magnetic field by measuring the back-reflectedsignal Aout Γ Ain for an applied input signal Ain at different flux values. As described inSubsec. 1.5.1, the back-reflected signal will display a Lorentzian dip in the magnitude and a phaseshift of 2π in the phase around the JPA resonance frequency. We vary the external magneticfield by changing the electrical current through a superconducting coil mounted on top of thesample holder. The microwave input tone was generated by a ZVA network analyser. Port 2was used as a source and port 1 as a receiver, to measure the back-reflected signal separated bythe measurement circulator. The ZVA calculates complex scattering parameters of the deviceunder test by comparing the input and output signals. The S-parameters determined in ourmeasurements include an accumulated attenuation from the complete setup. We performeda calibration measurement for all our samples by recording a frequency trace, for which theresonant frequency of our JPA was tuned outside of the observed frequency band.

Sample Q1100-epr

The first characterized sample is Q1100-epr. Fig. 3.1 displays the setup during characteriza-tion. Port 2 of the ZVA was connected to the JPA signal port and port 1 to the JPA’s pumpport. In this configuration, we measure the transmission between the JPA ports. This results ina Lorentzian peak at resonance in the signal band. In our measurements, the ZVA sweeps theprobe microwave tone between 5.450 GHz and 6.150 GHz with 1001 steps. The output power

37

CHAPTER 3. EXPERIMENTAL RESULTS

10 dB 10 dB

10 dB 10 dB

10 dB 10 dB

10 dB 10 dB

50 Ω

JPA Q1100

topmount

30 dB

S P

Output

line

Input

lines 3 4 1

RT

50 K

3 K

1 K

30 mK

RT Amplifier

ZVA

Port 2 Port 1

Frequency

standard

10 dB

50 Ω

Heater

+

temp. Sensor

magnet coil

matched load

a#enuator

male-male

adapter

microwave

feedthrough

HEMT amplifier

circulator

Figure 3.1.: Schematic of the measurement setup for characterizing sample Q1100-epr. The ZVAserves as a signal source as well as a detector. In the transmission measurement, theZVA port 2 was attached to the JPA signal port and port 1 to the JPA pump port.The frequency standard provides a reference signal for the ZVA. Circulators areused for signal separation (red box) and noise attenuation (yellow box). The outputsignal is amplified by a HEMT amplifier at the 1 K stage and by room temperatureamplifiers, before detection by the ZVA.

38

3.1. RESONATOR CHARACTERIZATION OF THE JPA

was set to 20 dBm. Taking the attenuation chain into account, we estimate a signal power of90 dBm at the JPA input port. Between recordings, the magnetic flux was varied by sweepingthe coil current from 500 µA to 500 µA in steps of 1 µA. We display the raw data of thismeasurement in App. A.The recorded data were processed with an evaluation software. From the magnitude traces an

average was subtracted. For the phase traces a linear phase fit was calculated and subtracted. InFig. 3.2 we portray the magnitude data for a narrower frequency span. We show the processedphase in App. A, because they are very noisy. A reason could be the large isolation between thesignal and pump ports of the JPA. Fig. 3.2a resulted for increasing and Fig. 3.2b for decreasingthe coil current.We observe the modulation of the resonant frequency of Q1100-epr in a frequency interval of

approximately 5.90 GHz to 6.12 GHz. We clearly see the Lorentzian peak at resonance in bothcolor plots. The resonance is better visible in Fig. 3.2 at higher frequencies. This is a plottingartefact due to the high sample rate in the coil current. In coil current intervals, in which theresonant frequency modulates stronger, it is less visible. Single frequency traces are given asexample in Fig. A.4 of App. A.Theoretical predictions suggest, that the frequency modulates from low values to maximum

to low values, in a current interval corresponding to one flux quantum Φ0, similar to Fig. 1.12b). For sample Q1100-epr we observe, that these arch-like features are incomplete. Dependingon the sweep direction of the coil current, the frequency modulates continuously to its locallylowest value and jumps to a higher frequency point discontinuously. For increasing coil currentsthe left flank of the arch-like frequency response is missing, while for decreasing coil currents theright flank is cut off. The coil current intervals of continuous frequency response vary between60 µA to 155 µA. The peak positions of each continuous block are subject to an additional, slowflux modulation, by which peak positions vary between 6.08 GHz to 6.13 GHz.For higher coil currents additional irregularities appear. The first irregularity can be observed

at coil current values of 300 µA for both sweep directions, 80 µA in Fig. 3.2a and at 300 µA inFig. 3.2b. Here, the resonant frequency displays a local minimum instead of modulating to lowervalues. At 300 µA in Fig. 3.2b, the curve progression to the minimum is rather smooth, whilein the other cases it is very sharp. The second irregularity appears at a coil current of 400 µAfor Figs. 3.2a and 3.2b. The frequency modulates to a low value here, where a comparison withthe rest of the spectrum would suggest a peak position. In addition, the continuous frequencyresponse curves in the proximity of this irregularity display an asymmetry, which can be bestobserved in Fig. 3.2a.Based on the discrepancy between the different sweep directions of the coil current, we see

that our sample is hysteretic with respect to the externally applied flux. To clearly observethe hysteresis, we added up the processed magnitude data of both sweep directions and plottedthem in 3.3. We see that in certain current intervals, the frequency response of our sampleis independent of the sweep direction of the coil current. Between those intervals, we observea dependence on the sweep direction of the coil current. We note here, that the increasedmagnitude values of the resonant frequency in Fig. 3.3 are due to the simple summation of bothdata sets. The magnitude is increase, when sample Q1100-epr displays a frequency responseindependent of the sweep direction of the coil current.We find that our spectrum is offset in the coil current, because no maximum is centred at

zero applied flux. We attribute this shift to a stray magnetic field. Alternatively, the reason forthis shift might be Abrikosov vortices, trapped during cooldown.

39

CHAPTER 3. EXPERIMENTAL RESULTS

-400 -200 0 200 400Coil current (µA)

5.9

5.95

6

6.05

6.1

6.15

Frequency

(GHz)

-60

-40

-20

0

20

Magnitude(dB)

-400 -200 0 200 400Coil current (µA)

5.9

5.95

6

6.05

6.1

6.15

Frequency

(GHz)

-60

-40

-20

0

20

Magnitude(dB)

Figure 3.2.: Processed magnitude of the resonator reflectance of sample Q1100-epr for sweepingthe externally applied magnetic field from low-to-high (top figure) and from high-to-low values (bottom figure). The resonance is clearly visible compared to thenoise background. Sample Q1100-epr displays a hysteretic behaviour in the fluxtunable resonant frequency, dependent on the sweep direction of the magnetic fieldin addition to a frequency modulation over a broad range of the applied magneticfield.

40

3.1. RESONATOR CHARACTERIZATION OF THE JPA

-400 -200 0 200 400Coil current (µA)

5.9

5.95

6

6.05

6.1

6.15

Frequency

(GHz)

-60

-40

-20

0

20

40

Magnitude(dB)

Figure 3.3.: A false color spectrum obtained from adding up the magnitude data of the differentsweep directions of the magnetic field. We can clearly observe the hysteresis ofour sample. Mutually shared frequency ranges give rise to an increased amplitude,which is responsible for the deviations in the colour code values and has no realphysically meaning.

Sample Q200

The next JPAs we characterized were of type Q200 and Q600. The cryogenic setup is given byFig. 2.3 and the room temperature setup is displayed in Fig. 2.7. For both measurements Port 2of the ZVA was connected to the respective input lines and port 1 to the respective output lines.Through the installed circulators we record the back reflected signal of the JPA. This results ina Lorentzian dip at resonance in the signal band. For the characterization of both samples, thecoil current was swept from 400 µA to 400 µA at a step size of 4 µA.For sample Q200 the ZVA swept the probe microwave tone between 5.70 GHz and 6.00 GHz

with 301 steps. The applied signal power was 25 dBm and we estimate 145 dBm at the JPAsignal port. The raw data were processed in a similar fashion to Q1100-epr. The unprocesseddata are given in App. A. In Fig. 3.4 we display the magnitude and phase data for increasingand in Fig. 3.5 for decreasing coil current, respectively. A summation of the magnitude andphase data, similar to Fig. 3.3, is given in App. A.For both sweeps we can identify the JPA’s resonant frequency in the magnitude and phase

data. For coil currents exceeding 280 µA we do not observe a resonance in the recorded signalbandwidth. For coil currents between 292 µA and 8 µA sample Q200 displays a regular be-haviour. One modulation period of the resonant frequency corresponds to a coil current intervalof 150 µA. The frequency maximum is at 5.98 GHz.At coil currents exceeding the aforementioned current interval, Q200 displays hysteretic and

irregular behaviour with respect to the applied flux. In Fig. 3.4 we observe a hysteretic responsein a coil current interval from 368 µA to 292 µA. For coil currents between 8 µA to 188 µAwe observe a modulation to low frequency values at 80 µA, resembling the irregularity of sampleQ1100-epr observed at a coil current of 400 µA. For coil currents of 188 µA to 288 µA we observe a

41

CHAPTER 3. EXPERIMENTAL RESULTS

non-hysteretic frequency response of Q200. With a coil current interval of 100 µA correspondingto one modulation period and a frequency maximum of 5.91 GHz, this response differs from thepreviously stated normal current region. Although we observe a resonance from 400 µA to368 µA we cannot attribute it to the previously described responses, because the majority ofthe response curve is outside of the recorded measurement window.In Fig. 3.5 we also observe a hysteretic and irregular behaviour of sample Q200. Between

coil currents from 400 µA to 292 µA we observe a modulation to low frequencies at 392 µA.Because we lack information about the sample’s response for lower coil currents in this sweepcase, we cannot assign this feature to the type of irregularity observed for Q1100-epr at 400 µA.For coil currents of 8 µA to 188 µA the spectrum deviates from Fig. 3.4. Instead of the irregularitythe sample displays a hysteretic response between 8 µA and 56 µA and a complete modulationperiod between 56 µA and 188 µA. With a current interval of 132 µA and a peak frequency of5.94 GHz this response also differs from the normal current region. Unlike Fig. 3.4, sample Q200displays a hysteretic response for coil currents of 192 µA to 276 µA in the case of decreasing coilcurrent.In summary, the coil current intervals of continuous frequency response in the hysteretic

regimes vary between 48 µA and 84 µA. Similar to Q1100-epr, the peak position of each con-tinuous and discontinuous resonant curve is subject to a slow flux modulation. The maximumfrequency value modulates between 5.91 GHz and 5.98 GHz.

42

3.1. RESONATOR CHARACTERIZATION OF THE JPA

-400 -300 -200 -100 0 100 200 300 400Coil current (µA)

5.7

5.75

5.8

5.85

5.9

5.95

6

Frequency

(GHz)

-8

-6

-4

-2

0

2

Magnitude(dB)

-400 -300 -200 -100 0 100 200 300 400Coil current (µA)

5.7

5.75

5.8

5.85

5.9

5.95

6

Frequency

(GHz)

-200

-100

0

100

200

Phase

(deg)

Figure 3.4.: Processed magnitude (top figure) and phase (bottom figure) data of the resonatorreflectance of sample Q200 for increasing the applied magnetic field. The dip inmagnitude as well as the 2π phase shift of the resonator are clearly visible. SampleQ200 displays coil intervals with normal and hysteretic response.

43

CHAPTER 3. EXPERIMENTAL RESULTS

-400 -300 -200 -100 0 100 200 300 400Coil current (µA)

5.7

5.75

5.8

5.85

5.9

5.95

6

Frequency

(GHz)

-8

-6

-4

-2

0

2

Magnitude(dB)

-400 -300 -200 -100 0 100 200 300 400Coil current (µA)

5.7

5.75

5.8

5.85

5.9

5.95

6

Frequency

(GHz)

-200

-100

0

100

200Phase

(deg)

Figure 3.5.: Processed magnitude (top figure) and phase (bottom figure) data of the resonatorreflectance of sample Q200 for decreasing the applied magnetic field.

44

3.1. RESONATOR CHARACTERIZATION OF THE JPA

Sample Q600

For the characterization of sample Q600 the probe microwave tone of the ZVA varied between5.70 GHz and 6.1 GHz in 801 steps. The signal power was set to 40 dBm and we estimate130 dBm at the JPA input port. The raw data are given in App. A. We display the processedmagnitude and phase data for increasing the coil current in Fig. 3.6 and for decreasing the coilcurrent in Fig. 3.7. A summation of the magnitude and phase data is given in App. A.We clearly see the Lorentzian dip at resonance for both sweep cases in magnitude and phase

data. Sample Q600 displays a hysteretic frequency response, similar to Q1100-epr. We observe acontinuous frequency response over 144 µA on average for coil currents from 250 µA to 400 µA.In both spectra we also observe smaller hysteretic responses, which vary from 28 µA to 48 µAin coil current. We find these centered at 225 µA and 40 µA in Fig. 3.6 and at 270 µA,90 µA and 370 µA in Fig. 3.7. We cannot make a statement about the resonance observed forcoil currents of 380 µA to 400 µA, because we lack information about the samples response forhigher coil currents.For coil currents from 400 µA to 250 µA in Fig. 3.6 we observe at 336 µA a modulation to

a low frequency value, similar to the irregularity at 400 µA for sample Q1100-epr. We do notsee a similar behaviour for decreasing coil currents in Fig. 3.7. We observe a modulation from288 µA to 340 µA to low frequency values and a small response from 372 µA to 384 µA.The peak frequency of the longer current intervals varies between 6.04 GHz and 6.07 GHz. In

the shorter current intervals, the peaks range from 6.00 GHz and 6.02 GHz.We repeated the flux characterization several times for sample Q1100-epr after heating our

sample above the critical temperature Tc of Niobium. The hysteretic dependence of the JPAfrequency on external flux remained exactly the same after multiple cooldowns, suggesting thatrandom trapping of Abrikosov vortices does not take influence on the behavior of our samples.One has to mention that vortex trapping may be not of random nature at all, in case, if thereare certain "weak spots" in the JPA, defined by design or fabrication flaws. In this case, wecould have trapped vortices always at the same spot, resulting in similar JPA frequency-fluxdependencies with reproducible irregularities.

45

CHAPTER 3. EXPERIMENTAL RESULTS

Coil current (µA)-400 -300 -200 -100 0 100 200 300 400

Frequency(G

Hz)

5.7

5.75

5.8

5.85

5.9

5.95

6

6.05

6.1

Magnitude(dB)

-43

-42.5

-42

-41.5

-41

-40.5

-40

Coil current (µA)

-400 -300 -200 -100 0 100 200 300 400

Phase

(deg)

5.7

5.8

5.9

6

6.1

Phase

(deg)

-400

-200

0

200

400

Figure 3.6.: Magnitude (top figure) and phase (bottom figure) of the resonator reflectance ofsample Q600 in the case of increasing externally applied field. The dip in magnitudeas well as the 2π phase shift of the resonator are clearly visible. Similar to sampleQ1100-epr sample Q600 displays a hysteretic behaviour in the flux tunable resonantfrequency.

46

3.1. RESONATOR CHARACTERIZATION OF THE JPA

Coil current (µA)-400 -300 -200 -100 0 100 200 300 400

Frequency(G

Hz)

5.7

5.75

5.8

5.85

5.9

5.95

6

6.05

6.1

Magnitude(dB)

-43

-42.5

-42

-41.5

-41

-40.5

-40

Coil current (µA)

-400 -300 -200 -100 0 100 200 300 400

Phase

(deg)

5.7

5.8

5.9

6

6.1

Phase

(deg)

-400

-200

0

200

400

Figure 3.7.: Magnitude (top figure) and phase (bottom figure) of the resonator reflectance forsample Q600 in the case of decreasing externally applied field.

47

CHAPTER 3. EXPERIMENTAL RESULTS

3.1.2. Determination of internal and external quality factorsWe now turn to the quantitative analysis of our data. As it is explained in Subsec.1.5.1, we

can fully describe a loaded λ4 resonator by a respective complex reflection coefficient Γ. Wecan extract the quality factors of the resonator by a least squares fit of Γ defined in Eq. (1.52)simultaneously to a measured reflectance magnitude and phase. Fig. 3.8 displays some particularresults of this fitting procedure for JPA samples Q200, Q600 and Q1100-epr.We applied this fitting procedure to all recorded traces of our samples. The determined

quality factors of samples Q200, Q600 and Q1100-epr are plotted versus the applied coil currentin Figs. 3.9, 3.10, 3.11, respectively. For sample Q600 Qext is increased by a factor of 2 incomparison with the designed values. The internal quality factors Qint of our samples scatterover 2 orders of magnitude. We observe that Qint of Q600 corresponds to a typical value of10000, while Qint of Q200 is decreased by a factor of 10 and Qint of Q1100-epr is increased by afactor of 7. Tab. 3.1 summarizes the retrieved quality factors.

Q200 Q600 Q1100-eprQext 282 1094 1162Qint 1624 12229 72695

Table 3.1.: The extracted quality factors obtained by fitting.

48

3.1. RESONATOR CHARACTERIZATION OF THE JPA

Frequency (GHz)

5.8 5.85 5.9 5.95 6

Magnitude(arb.units)

0.6

0.8

1

1.2

Frequency (GHz)

5.8 5.85 5.9 5.95 6

Phase

(deg)

-400

-300

-200

-100

0

Frequency (GHz)

5.95 6 6.05

Magnitude(arb.units)

0.8

0.85

0.9

0.95

1

1.05

Frequency (GHz)

5.95 6 6.05

Phase

(deg)

-400

-200

0

200

400

Frequency (GHz)

6.05 6.1

Magnitude(arb.units)

0.94

0.96

0.98

1

1.02

Frequency (GHz)

6.05 6.1

Phase

(deg)

-100

-50

0

50

100

a) b)

c) d)

e) f)

Q200

Q600

Q1100-epr

Figure 3.8.: Magnitude and phase traces of samples Q200, Q600 and Q1100-epr recorded forzero coil current. The red line is a fit of Eq. (1.52) to the data. Obtained fit resultswere: fL 5.83 GHz, Qext 292 and Qint 1596 for a) and b). fL 6.07 GHz,Qext 1148 and Qint 12142 for c) and d). fL 6.11 GHz, Qext 1190 andQint 67843 for e) and f).

49

CHAPTER 3. EXPERIMENTAL RESULTS

Qext

0

200

400

600

coil current (µA)

-200 -150 -100 -50 0 50 100 150

Qint

0

500

1000

1500

2000

Figure 3.9.: The determined external Qext (top) and internal Qint (bottom) quality factors ofsample Q200 versus the applied coil current.

coil current (µA)

-400 -300 -200 -100 0 100

Qint

0

5000

10000

15000

Qext

0

500

1000

1500

Figure 3.10.: The determined external Qext (top) and internal Qint (bottom) quality factors ofsample Q600 plotted against the coil current.

50

3.1. RESONATOR CHARACTERIZATION OF THE JPA

coil current (µA)

-100 -50 0 50 100

Qint

(

×104)

×104

0

2

4

6

8

10

Qext

0

500

1000

1500

Figure 3.11.: External Qext (top) and internal Qint (bottom) quality factors of sample Q1100versus the coil current.

51

CHAPTER 3. EXPERIMENTAL RESULTS

3.1.3. Fitting of the resonant frequency

We then fitted the retrieved JPA resonant frequency with the modified dispersion relationof the JPA’s resonator, given by Eq. (1.57). An expression for the flux dependent SQUIDinductance is given by Eq. (1.27). Considering only the base mode (n 0) of the quarterwavelength resonator, the resulting dispersion relation is given by:

ωLω0

tanωLω0

Lcav

Φ02πIspΦextq

Lgeom, (3.1)

where k0 ωL?LCd and the bare resonator frequency ω0 1?LC were used.

In the fitting procedure, both sides of Eq. (3.1) are calculated independently and the fittingparameters are determined by minimizing the fit’s deviation from the data using a least squarealgorithm.Since the ZVA records the loaded resonant frequency fL we transform the corresponding

angular frequency ωL of a quarter wave length resonator by fLωL 2πfL to match our data.

In our model, the flux dependent JPA frequency is related to the switching current of theincorporated dc-SQUID. For JPAs with non-hysteretic dc-SQUIDs, the magnetic field is cali-brated to a flux quantum by a modulation period of the JPA frequency. From Sec. 1.3.2 weknow, that hysteretic dc-SQUIDs display a continuous switching current Is pΦextq for appliedflux exceeding a flux quantum, as can be seen in Fig. 1.10b.In the case of integer values of the screening parameter βL, the flux intervals of Is pΦextq

correspond to multiples of the flux quantum nΦ0 and continuous Is pΦextq curves can be con-structed with no current overlap, resembling the regular case. In the case of non-integer valuesof the screening parameter βL, the flux interval of the switching current curves Is pΦextq alwayshave an overlapping region and display hysteresis. In order to calibrate the magnetic field forfitting the JPA frequency, we limit our analysis to screening parameters βL between 0 and 1. Acomparison between hysteretic switching currents Is and frequencies for different values of βL isgiven in App. B.To fit the hysteretic JPA frequency we made use of our results of the switching current Is pΦextq

for a given βL. In the case of a continuously driven external magnetic flux, we calculated thecorresponding Is pΦextq curves for different screening parameters. Fig. 3.12 depicts the resultingIs pΦextq curve for a screening parameter of βL 0.5 for both sweep directions of the externallyapplied flux.We display the retrieved resonant frequency of sample Q1100-epr for coil currents between

250 µA and 50 µA in Fig. 3.13. The continuous JPA frequency responses outside this magneticfield interval have a different length in the coil current and differ in their maximum frequencyand are therefore excluded from the fitting. We depict both sweep cases in the graph to betterdisplay the hysteresis, similar to Fig.3.3. The x-axis has been calibrated to correspond to valuesof the flux quantum Φ0 with a current-to-flux conversion factor of 127µAΦ0.From the performed fit we extract the critical current Ic, the geometric resonator inductance

Lcav, the SQUID geometrical inductance Lgeom and the screening parameter βL. The respectiveresults are given in Tab. 3.2. From the fitting result, we find that the critical current Ic ofQ1100-epr’s Josephson junctions is 13 times higher than their designed value.In Fig. 3.14 we display the retrieved resonant frequency of sample Q200 for coil currents of

r292, 8s µA. We focus on this interval because for these magnetic fluxes sample Q200 displaysnormal frequency response. We made use of (1.30) to generate the switching current. Wefound a current-to-flux conversion factor of 150µAΦ0. The extracted fit parameters are given

52

3.1. RESONATOR CHARACTERIZATION OF THE JPA

-2 -1 0 1 2Φext (Φ0)

0

0.2

0.4

0.6

0.8

1

1.2

I s/2I c

increasing Φext

decreasing Φext

Figure 3.12.: Hysteretic current curves calculated for a value of the screening parameter β0 0.5for increasing and decreasing external flux Φ0.

Φext/Φ0

-2 -1.5 -1 -0.5 0 0.5 1

Frequency(G

Hz)

5.85

5.9

5.95

6

6.05

6.1

6.15

6.2

data

fit

Figure 3.13.: The determined resonant frequencies of sample Q1100-epr for both sweep direc-tions of the externally applied flux Φext. The solid black lines correspond to thecalculated fitting curve resulting form theory.

in Tab.3.2. The critical current Ic was determined to be 4 times higher than the designed valuefor sample Q200.The retrieved JPA frequency of sample Q600 for coil currents between 20 µA and 360 µA

is shown in Fig. 3.15. We display both sweep cases, similar to Fig. 3.13. The current-to-flux

53

CHAPTER 3. EXPERIMENTAL RESULTS

βL Ic Lgeom f0 Lcav design-Ic(µA) (pH) (GHz) (nH) (µA)

Q200 0.15 7.65 19.7 6.08 2.43 2.00Q600 0.58 35.1 17.1 6.15 1.57 4.00Q1100-epr 0.8 40.4 20.4 6.22 1.45 3.00

Table 3.2.: Summary of the resulting device parameters from fitting the flux dependent reso-nant frequency. The designed critical currents for each JPA are recapitulated forcomparison.

-1.5 -1 -0.5 0 0.5Φext/Φ0

5.6

5.7

5.8

5.9

6

6.1

Frequency

(GHz)

data

fit

Figure 3.14.: The determined resonant frequency of sample Q200 versus the external flux Φext.The solid black line corresponds to the fitting curve from (3.1).

conversion factor was 144µAΦ0. The fit parameters are summarized in Tab.3.2. The determinedcritical current is about 9 times higher the design value.

54

3.1. RESONATOR CHARACTERIZATION OF THE JPA

Φext/Φ0

-1.5 -1 -0.5 0 0.5 1 1.5

Frequency(G

Hz)

5.85

5.9

5.95

6

6.05

6.1

6.15

data

fit

Figure 3.15.: The determined resonant frequencies of sample Q600 for both sweep directions ofthe externally applied flux Φext. The solid black lines correspond to the calculatedfitting curve.

55

CHAPTER 3. EXPERIMENTAL RESULTS

3.2. Gain characterizationThe next step of JPA characterization is to measure gain in a non-degenerate regime for

various pump powers.Similar to the resonator characterisation, we determined the gain of our sample by measuring

the back reflected signal for an applied pump tone with our ZVA. With the applied pump tonewe expose our dc-SQUID to an alternating magnetic field, by which we modulate the resonantfrequency. The magnitude of this frequency modulation ∆ω is determined by the slope of thefrequency-flux curve for a chosen resonant frequency ωL and the applied flux modulation. Wecopy from [37]:

∆ω dω

dΦext

∣∣∣∣ωL

∆Φext. (3.2)

As high gains are expected for high frequency modulations, we set the working point ωL of ourJPA to a value with a reasonable slope in the frequency response of the JPA. We want to notehere, that choosing a working point with too high slope will render our JPA susceptible to fluxnoise.

3.2.1. Q200For a first measurement, the JPA’s resonant frequency was determined to be 5.797 GHz for

an applied coil current of 148 µA. The slope in the JPA response at this working point isdωdΦext 1.945 GHzΦ0. The probe microwave tone of the ZVA varied in a band of 80 MHzcentred at 5.802 GHz in 101 steps for various pump powers. The ZVA output power was set to35 dBm and we estimate the signal power to 155 dBm at the signal port of the JPA. Thepump frequency was set to 11.604 GHz. The pump tone was generated with the MSG. The pumppower was increased between each frequency sweep from 6 dBm to 8 dBm in steps of 0.1 dBm.The pump tone was attenuated by 40 dB by the attenuation chain. Due to the change in existingstray fields, the initial flux-characterisation from Sec. 3.1 cannot serve as an absolute calibrationof our JPAs’ resonant frequency.We determined the JPA gain by comparing the output signals for an applied pump power

to a calibration curve with pump power off. The results of this measurement are portrayedin Fig. 3.16. We observed signal amplification, which can be seen in Fig. 3.16a for differentpump powers. The maximum achieved gain was 24.7 dB in a bandwidth of 1.11 MHz at a signalfrequency of 5.802 GHz and an applied pump power of 7.6 dBm. At higher pump powers, theresulting maximum gain decreased, as can be seen in Fig. 3.16b.As the non-degenerate gain is expected to be working point dependent, we repeated our

initial measurement at different flux biases. The chosen coil currents were in an interval ofr155,145sµA. At each working point the pump frequency was set to be 10 kHz less thandouble the working point. The pump power was increased from 6 dBm to 8.6 dBm in steps of0.2 dBm between each recorded signal band. The ZVA applied a signal band of 100 MHz centredat 5.8 GHz, recording 101 samples. The signal powers stayed the same as in the previous case.We present the results of this measurement for coil currents of 153 µA, 152 µA, 150 µA

and 149 µA in Fig. in Fig. 3.17a, c, e and g, respectively. During varying the flux bias weobserve the formation of a bifurcation in the amplified signal frequency. Because we also haveno calibration measurements for our measurements, we reference the signal magnitude to thepower average at frequencies, for which no amplification occurs.We observe the formation of a bifurcation at certain flux biases of our resonator. In Figs. 3.17b,

d, f and h we show the recorded signal band at different pump powers to better display this

56

3.2. GAIN CHARACTERIZATION

Signal frequency (GHz)

5.79 5.8 5.81 5.82

Gain(dB)

0

5

10

15

20

25 6.1 dBm6.7 dBm7.2 dBm7.6 dBm

Pump power (dBm)

6 6.5 7 7.5 8

Gain

max(dB)

10

15

20

25

a) b)

Figure 3.16.: Gain characterization of Q200. a) calibrated frequency traces at different pumppowers. b) determined maximum gain value for given pump power

formation. In Fig. 3.17a, where the coil current was set to 153 µA, we have no bifurcationin the recorded spectrum. We observe a steady signal gain with a maximum achieved gain of13.7 dB in a bandwidth of 6.72 MHz. By changing the coil current to lower values, a bifurcationdevelops above a certain pump power in the signal gain, which can be seen in Figs. 3.17c, e andg. This bifurcation threshold shifts to lower pump powers for decreasing coil current. Below thebifurcation threshold, a steady signal gain can be observed. Above the bifurcation threshold thesignal gain starts to split into a low gain and high gain component.The maximum gain for the low gain component stays constant in our measurements at ap-

proximately 13.2 dB to 14 dB for bandwidths of approximately 5.40 MHz for increasing pumppowers. The maximum gain of the high gain component ranges from 28 dB to 32 dB in a sig-nal bandwidth of approx. 2.60 MHz in our measurements. In Figs. 3.17e and g, the high gaincomponent can no longer be observed for pump powers exceeding 8 dBm.We investigated the bifurcation further and repeated the resonator characterisation, similar

to Subsec. 3.1.1 but with an applied constant pump power to our JPA. The pump powersinvestigated were 7.8 dBm and 10.0 dBm. The pump frequency was set 10 kHz lower than twicethe resonant frequency. The ZVA output signal power was set to 35 dBm, recording 151samples in a signal bandwidth of 60 MHz. In the case of low pump power, the center frequencywas changed between 5.827 GHz and 5.730 GHz, while for high pump power the center frequencywas changed from 5.876 GHz to 5.767 GHz, in order to keep the resonant frequency in the focusof the recorded signal band. Similar to Fig. 3.17 we normalized the detected magnitude to anaverage of the back reflected signal powers with no gain.For a pump power of 7.8 dBm we changed the coil current from 147 µA to 155 µA in steps of

0.1 µA. We portray the result in Fig. 3.18. We see from Fig. 3.18a, that the high gain componentof the bifurcation exists for coil currents from 154 to 151 µA. It reaches its maximum gainof 33.1 dB in a bandwidth of 1.81 MHz at a coil current of 153 µA and stays constant fromthere to higher coil currents. The signal gain of the low gain component ranged from 10 dB forlow and 16 dB for high coil currents. In Fig. 3.18b we display frequency traces for which theformation and separation of the feature can be seen for different coil currents.Fig. 3.19 gives our results for a similar measurement with the pump power of 10.0 dBm. The

coil current was increased by 0.1 µA from 160 µA to 155 µA. Contrary to Fig. 3.18, the high

57

CHAPTER 3. EXPERIMENTAL RESULTS

6 6.5 7 7.5 8 8.5Ppump (dBm)

5.75

5.8

5.85

Frequency

(GHz)

-15

-10

-5

0

5

10

5.8 5.81 5.82 5.83 5.84Frequency (GHz)

0

20

40

60

Magnitude(dB)

6 dBm6.8 dBm7.6 dBm8.4 dBm

6 6.5 7 7.5 8 8.5Ppump (dBm)

5.8

5.81

5.82

5.83

5.84

Frequency(G

Hz)

-10

0

10

20

30

5.79 5.8 5.81 5.82 5.83 5.84Frequency (GHz)

0

20

40

60

Magnitude(dB)

6 dBm6.8 dBm7.6 dBm8.4 dBm

6 6.5 7 7.5 8 8.5Ppump (dBm)

5.79

5.8

5.81

5.82

5.83

Frequency(G

Hz)

-10

0

10

20

5.78 5.79 5.8 5.81 5.82 5.83Frequency (GHz)

0

20

40

60

Magnitude(dB)

6 dBm6.8 dBm7.6 dBm8.4 dBm

6 6.5 7 7.5 8 8.5Ppump (dBm)

5.78

5.8

Frequency(G

Hz)

-10

0

10

20

5.77 5.78 5.79 5.8 5.81 5.82Frequency (GHz)

0

20

40

60

Magnitude(dB)

6 dBm6.8 dBm7.6 dBm8.4 dBm

a) b)

c) d)

e) f)

g) h)

Icoil = − 153 µA

Icoil = − 152 µA

Icoil = − 151 µA

Icoil = − 149 µA

Figure 3.17.: Gain characterization at different flux biases. Spectra a), c), e) and g) correspondto coil currents of 153 µA, 152 µA, 151 µA, and 149 µA. Related frequencytraces at different pump powers are given by b), d), f) and h). To increasevisibility, they are offset by 10 dB. The magnitude label relates to the color bar aswell as the y-axis of graph pairs a)-b), c)-d), e)-f) and g)-h).

58

3.2. GAIN CHARACTERIZATION

-155 -154 -153 -152 -151 -150Coil current (µA)

5.76

5.78

5.8

5.82

5.84Frequency

(GHz)

0

10

20

30

Magnitude(dB)

a)

5.76 5.77 5.78 5.79 5.8 5.81 5.82 5.83 5.84Frequency (GHz)

0

10

20

30

40

50

60

70

Magnitude(dB)

-155 µA

-154 µA

-153.5 µA

-152.5 µA

-151 µA

-150 µA

b)

Figure 3.18.: Flux sweep for a constant applied pump power of 7.8 dBm. The resulting spectrumis given in a). The evolution and splitting of the high gain feature is portrayed inb). The frequency traces are offset by 10 dB.

59

CHAPTER 3. EXPERIMENTAL RESULTS

gain component persists in a smaller current span of approx. 0.6 µA, starting at 157.5 µA andreaching a maximum gain of 33.0 dB relative to the noise floor. The maximum gain of the lowgain component varies from 4.7 dB at 162 µA to 12.0 dB at 149 µA. Similar to 3.18b we showthe separation of the feature over different coil currents in Fig. 3.19b.

60

3.2. GAIN CHARACTERIZATION

-160 -159 -158 -157 -156 -155Coil current (µA)

5.82

5.83

5.84

5.85

5.86

5.87

5.88Frequency

(GHz)

0

10

20

30

Magnitude(dB)

a)

5.83 5.84 5.85 5.86 5.87Frequency (GHz)

-10

0

10

20

30

40

50

60

70

Magnitude(dB)

-159 µA

-158 µA

-157.5 µA

-157.2 µA

-156.5 µA

-155.5 µA

b)

Figure 3.19.: Flux sweep for a constant applied pump power of 10.0 dBm. a) spectrum of theback reflected signal magnitude. b) frequency traces at different coil currents. Thetraces are offset by 10 dB for better visibility.

61

CHAPTER 3. EXPERIMENTAL RESULTS

3.2.2. Q600For sample Q600 we measured the non-degenerate gain by setting the resonator’s working

point to 5.900 GHz at the coil current of 350 µA. The pump frequency was detuned by 10 kHzfrom twice the resonant frequency. We increased the pump power from 2 dBm to 10 dBm insteps of 0.5 dBm. The pump tone was attenuated by 40 dB to the pump port by the attenuationchain. We measured the signal with the ZVA in a band of 80 MHz centred at the working pointfor various pump powers. The ZVA power was set to 35 dBm and we estimate a signal powerof 125 dBm at the JPA signal port. We recorded 301 samples with an IF-frequency of 10 Hz.The results of this measurement can be seen in Fig. 3.20a.We reference our measurements to the back reflected signal power, where we observe no gain.

We observe a response of our device in a bandwidth of approx. 0.60 MHz. The recorded gainincreases to 10 dB for a pump power of 3 dBm. For higher pump powers the gain diminishes.In addition, a dip develops in the vicinity of the gain peak at a lower frequency value. The

maximum attenuation of this dip reaches 16 dB at a pump power of 7 dBm. In Fig. 3.20b) wedepicted the single traces, recorded at 2, 3.5, 7 and 10 dBm. At pump powers of 2 dBm and3.5 dBm we observe an additional dip at lower frequencies, which is less pronounced at higherpump powers. We consider these as parasitic losses for our device.We repeated the measurement for a different working point. We set the resonant frequency

to 6.025 GHz with a coil current of 220 µA. The pump frequency was set to 12.04 GHz andwe increased the pump power from 13 dBm to 17 dBm in steps of 0.2 dBm. The ZVA measuredthe signal in a band of 70 MHz at a power of 35 dBm centred at the resonant frequency. Werecorded 401 samples. The results of this measurement is displayed in Fig. 3.20c.Similar to the previous result, the gain peak increases in frequency over the course of the

measurement. For low applied gain powers we see parasitic losses in addition at a lower frequencyin each measurement trace, which is shifted to lower frequencies for increasing pump powers,which can also be seen in Fig. 3.20d. The value of the gain increases at a pump power of14.4 dBm to 17 dB and fluctuates between 1 dB to 20 dB for higher pump values.We repeated this measurement again at a coil current of 22 µA. The pump power was

increased from 12 dBm to 17 dBm in steps of 1 dBm for a pump frequency of 12.12 GHz. TheZVA measured the signal in a band of 20 MHz with a signal power of 35 dBm centred at theworking point at 6.065 GHz. We recorded 101 samples. Figs. 3.20e and 3.20f portray our results.We see a frequency dip, which is constant in frequency for increasing pump powers. The dip

decreases in magnitude from 1.7 dB for a pump power of 12 dBm to 1.9 dB for a pump powerof 17 dBm.

62

3.2. GAIN CHARACTERIZATION

2 4 6 8 10Ppump (dBm)

5.905

5.91

5.915

5.92Frequency

(GHz)

-15

-10

-5

0

5

10

5.905 5.91 5.915 5.92Frequency (GHz)

0

10

20

30

40

50

Magnitude(dB)

2 dBm3.5 dBm7 dBm10 dBm

13 14 15 16 17Ppump (dBm)

6.048

6.05

6.052

6.054

6.056

Frequency(G

Hz)

-10

0

10

20

6.048 6.05 6.052 6.054 6.056Frequency (GHz)

0

10

20

30

40

50

Magnitude(dB)

13 dBm14.6 dBm15.8 dBm17 dBm

12 14 16Ppump (dBm)

6.055

6.06

6.065

6.07

6.075

Frequency(G

Hz)

-1.5

-1

-0.5

0

6.06 6.065 6.07 6.075Frequency (GHz)

-4

-2

0

2

4

6

Magnitude(dB)

12 dBm14 dBm17 dBm

a) b)

c) d)

e) f)

Icoil = 350µA

Icoil = −220µA

Icoil = −22µA

Figure 3.20.: Gain characterization of sample Q600 at different flux biases. Spectra a), c), ande) correspond to coil currents of 350 µA, 220 µA and 22 µA. Related frequencytraces at different pump powers are given by b), d) and f). To increase visibility,they are offset by 10 dB. For spectra a) and c) we observe a divergent feature inthe applied signal band, which increases in frequency for increasing pump powers.In e) we observe no gain for increasing pump powers. The magnitude label relatesto the color bar as well as the y-axis of graph pairs a)-b), c)-d) and e)-f).

63

4. Conclusion

For this thesis we have characterised three samples from the newly received batch of flux-driven Josephson parametric amplifiers in order to include them into our setup for quantumteleportation. We measured the JPAs’ response to input signals over a broad range of flux biasesfor no applied pump power. Based on these measurements we determined the JPAs’ resonatorscharacteristics. As a last step we recorded the achievable gain with one of our samples.For our measurements we chose samples, which differed in the designed value of the exter-

nal quality factor. Upon characterizing the flux dependent resonant frequency of our JPAs’resonator, we discovered that all of our samples deviate strongly from the behaviour predictedby theory. We observed that the magnitude of the applied coil current, in which the resonantfrequency modulates by one period, varies in the investigated flux range. Furthermore the max-imum of the resonant frequency for each supposed period also varies in the order of 100 MHz. Inaddition to other irregularities observed for high fluxes, two of our samples, Q600 and Q1100-eprrespectively, displayed a hysteretic response in respect to the sweep direction of the externallyapplied magnetic field. This also stands in strong contrast to the measurement results obtainedfrom the initial batch of JPAs in [37]. A considerable effort has been invested in the localizationof the hysteretic behaviour of our samples. We attribute the hysteresis to faulty dc-SQUIDs inour JPA’s resonator with a screening parameter βL higher than the design value. In the theorysection we show, how dc-SQUIDs with a too high screening parameter can display a hystereticcurrent flux relation, when operated in the zero voltage state. We find that the external qualityfactor of all our samples are slightly higher than the design vale, with the deviation for Q600 be-ing almost double. The internal quality factor of Q1100-epr is also higher compared to samplesQ200 and Q600, which have a lower internal quality factor. We have recorded a maximum signalgain of 24.7 dB for an applied pump power of 7.6 dBm. For pump powers exceeding this value,the maximal observed gain decreased in magnitude. In consecutive measurements, in which wetried to optimize our samples output gain, we also observed the formation of an erratic featurein the back reflected signal spectrum at certain resonant frequencies, which is characterized bya high gain in a small bandwidth.For sample Q600 we could not produce a gain for three different working points. In two of

our measurements we observed the formation of a divergent frequency response of our device,characterised by a dip followed by a peak in the reflectance of the applied signal bandwidth.Further investigation into this feature revealed its increasing bandwidth in pump frequency forincreasing pump powers. For constant applied pump powers we observed multiple formations ofthis feature for applied pump and signal frequencies.In summary, we documented the behaviour of three flux driven Josephson parametric ampli-

fiers for potential use in the experimental realization of quantum teleportation with continuousvariable quantum states.

65

A. Spectra

A.1. Q1100-epr

-400 -200 0 200 400Coil current (µA)

5.9

5.95

6

6.05

6.1

6.15

Frequency

(GHz)

-60

-40

-20

0

20

40

Magnitude(dB)

-400 -200 0 200 400Coil current (µA)

5.9

5.95

6

6.05

6.1

6.15

Frequency

(GHz)

-5000

0

5000

10000

Phase

(deg)

Figure A.1.: Raw magnitude (top figure) and phase (bottom figure) data of sample Q1100-eprfor increasing coil current.

67

APPENDIX A. SPECTRA

-400 -200 0 200 400Coil current (µA)

5.9

5.95

6

6.05

6.1

6.15

Frequency

(GHz)

-60

-40

-20

0

20

40

Magnitude(dB)

-400 -200 0 200 400Coil current (µA)

5.9

5.95

6

6.05

6.1

6.15

Frequency

(GHz)

-5000

0

5000

10000

Phase

(deg)

Figure A.2.: Raw magnitude (top figure) and phase (bottom figure) data of sample Q1100-eprfor decreasing coil current.

68

A.1. Q1100-EPR

-400 -200 0 200 400Coil current (µA)

5.9

5.95

6

6.05

6.1

6.15

Frequency

(GHz)

-200

-100

0

100

200

Phase

(deg)

-400 -200 0 200 400Coil current (µA)

5.9

5.95

6

6.05

6.1

6.15

Frequency

(GHz)

-200

-100

0

100

200

Phase

(deg)

Figure A.3.: Processed phase data of sample Q1100-epr after subtracting a linear phase fit withthe data evaluation program. Top figure - increasing current. Bottom figure -decreasing coil current.

69

APPENDIX A. SPECTRA

5.7 5.75 5.8 5.85 5.9 5.95 6 6.05 6.1 6.15Frequency (GHz)

-20

0

20

Magnitude(dB)

5.7 5.75 5.8 5.85 5.9 5.95 6 6.05 6.1 6.15Frequency (GHz)

-20

0

20

Magnitude(dB)

5.7 5.75 5.8 5.85 5.9 5.95 6 6.05 6.1 6.15Frequency (GHz)

-20

0

20

Magnitude(dB)

5.7 5.75 5.8 5.85 5.9 5.95 6 6.05 6.1 6.15Frequency (GHz)

-20

0

20

Magnitude(dB)

a)

b)

c)

d)

Icoil = 311µA

Icoil = 297µA

Icoil = 289µA

Icoil = 287µA

Figure A.4.: Frequency response of Q1100-epr at different flux biases during decreasing the coilcurrent.

70

A.2. Q200

A.2. Q200

-400 -300 -200 -100 0 100 200 300 400Coil current (µA)

5.7

5.75

5.8

5.85

5.9

5.95

6

Frequency

(GHz)

-38

-36

-34

-32

-30

-28

-26

Magnitude(dB)

-400 -300 -200 -100 0 100 200 300 400Coil current (µA)

5.7

5.75

5.8

5.85

5.9

5.95

6

Frequency

(GHz)

-2500

-2000

-1500

-1000

-500

0

Phase

(deg)

Figure A.5.: Raw magnitude (top figure) and phase (bottom figure) data of sample Q200 forincreasing coil current.

A.3. Q600

71

APPENDIX A. SPECTRA

-400 -300 -200 -100 0 100 200 300 400Coil current (µA)

5.7

5.75

5.8

5.85

5.9

5.95

6

Frequency

(GHz)

-38

-36

-34

-32

-30

-28

-26

Magnitude(dB)

-400 -300 -200 -100 0 100 200 300 400Coil current (µA)

5.7

5.75

5.8

5.85

5.9

5.95

6

Frequency

(GHz)

-2500

-2000

-1500

-1000

-500

0

Phase

(deg)

Figure A.6.: Raw magnitude (top figure) and phase (bottom figure) data of sample Q200 fordecreasing coil current.

72

A.3. Q600

-400 -300 -200 -100 0 100 200 300 400Coil current (µA)

5.7

5.75

5.8

5.85

5.9

5.95

6

Frequency

(GHz)

-8

-6

-4

-2

0

2

Magnitude(dB)

-400 -300 -200 -100 0 100 200 300 400Coil current (µA)

5.7

5.75

5.8

5.85

5.9

5.95

6

Frequency

(GHz)

-500

0

500

Phase

(deg)

Figure A.7.: False colour plots from adding up the magnitude (top figure) and phase (bottomfigure) data of sample Q200 for the different sweep directions of the magnetic field.

73

APPENDIX A. SPECTRA

-400 -300 -200 -100 0 100 200 300 400Coil current (µA)

5.7

5.8

5.9

6

6.1

Frequency

(GHz)

-43

-42.5

-42

-41.5

-41

-40.5

-40

Magnitude(dB)

-400 -300 -200 -100 0 100 200 300 400Coil current (µA)

5.7

5.8

5.9

6

6.1

Frequency

(GHz)

-16

-14

-12

-10

-8

-6

-4

-2Phase

(deg)

×104

Figure A.8.: Raw magnitude (top figure) and phase (bottom figure) data of sample Q600 forincreasing coil current.

74

A.3. Q600

-400 -300 -200 -100 0 100 200 300 400Coil current (µA)

5.7

5.8

5.9

6

6.1

Frequency

(GHz)

-43

-42.5

-42

-41.5

-41

-40.5

-40

Magnitude(dB)

-400 -300 -200 -100 0 100 200 300 400Coil current (µA)

5.7

5.8

5.9

6

6.1

Frequency

(GHz)

-16

-14

-12

-10

-8

-6

-4

-2

Phase

(deg)

×104

Figure A.9.: Raw magnitude (top figure) and phase (bottom figure) data of sample Q600 fordecreasing coil current.

75

APPENDIX A. SPECTRA

-400 -300 -200 -100 0 100 200 300 400Coil current (µA)

5.7

5.8

5.9

6

6.1

Frequency

(GHz)

-3

-2

-1

0

1

Magnitude(dB)

-400 -300 -200 -100 0 100 200 300 400Coil current (µA)

5.7

5.8

5.9

6

6.1

Frequency

(GHz)

-500

0

500Phase

(deg)

Figure A.10.: False colour plots from adding up the magnitude (top figure) and phase (bottomfigure) data of sample Q200 for the different sweep directions of the magnetic field.

76

B. Hysteretic currents and frequencies

In order to fit the hysteretic frequency response of our JPAs we calculate the switching currentof dc-SQUIDs according to Subsec.1.3.2. For different values of the screening parameter βLwe can calculate different switching currents Is pΦextq as shown for both sweep directions inFigs. B.1a, B.1b. As already stated, the flux range for each continuous Is pΦextq increases withthe screening parameter.During characterisation of regular JPAs, the coil current interval is calibrated to a flux quan-

tum Φ0 for a period in the JPA frequency response. For hysteretic JPAs we only observe acut off frequency response in the recorded spectrum, corresponding to the continuous switchingcurrent Is pΦextq. This poses an issue for calibrating the coil current to flux, due to the differ-ent calculable Is pΦextq curves with increasing screening parameter βL. The magnitude of thescreening parameter βL prior to fitting cannot be guessed from the recorded spectra.Because the cut off frequency resembles the switching current, a brute force approach, cali-

brating the various Is pΦextq from Figs. B.1a, B.1b to a flux quantum is given in Figs. B.1c, B.1d.The resulting frequency curves from these calibrated Is pΦextq curves with the fitting algorithmis given in Figs. B.1e, B.1f. Note the similarity between the different current and frequencycurves, especially for screening parameters βL 0.5 and 2.5Due to the presented ambiguity in respect to the fitting parameter βL, we restrict the analysis

and fitting of the JPA resonant frequency to values βL between 0 and 1 in this thesis.

77

APPENDIX B. HYSTERETIC CURRENTS AND FREQUENCIES

0 1 2 3 4Φ/Φ0

0

0.2

0.4

0.6

0.8

1

I s/2I c

βL = 0.5βL = 1.5βL = 2.5

0 1 2 3 4Φ/Φ0

0

0.2

0.4

0.6

0.8

1

I s/2I c

βL = 0.5βL = 1.5βL = 2.5

0 1 2 3 4Φ/Φ0

0

0.2

0.4

0.6

0.8

1

I s/2I c

βL = 0.5

βL = 1.5

βL = 2.5

0 1 2 3 4Φ/Φ0

0

0.2

0.4

0.6

0.8

1

I s/2I c

βL = 0.5

βL = 1.5

βL = 2.5

0 1 2 3 4Φ/Φ0

5

5.5

6

Frequ

ency

(GHz)

βL = 0.5

βL = 1.5

βL = 2.5

0 1 2 3 4Φ/Φ0

5

5.5

6

Frequ

ency

(GHz)

βL = 0.5

βL = 1.5

βL = 2.5

a) b)

c) d)

e) f)

Figure B.1.: Comparison beween hysteretic switching currents and frequencies for screening pa-rameters βL r0.5, 1.5, 2.5s. Switching currents for a), c) increasing and b), d)decreasing magnetic field. In a) and b) the increasing screening capability for in-creasing βL can be good observed. In c and d the magnetic field of each switchingcurrent is normalized to a flux quantum. These normalized currents were used withthe fitting results for sample Q1100-epr to calculate JPA frequency curves. Asidethe shift in the magnetic field, the resulting JPA frequencies do not deviate strongly.

78

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