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Weak-value Metrology and Shot-NoiseLimited Measurements
by
Gerardo Ivan Viza
Submitted in Partial Fulfillment of the
Requirements for the Degree
Doctor of Philosophy
Supervised by
Professor John C. Howell
Department of Physics and AstronomyArts, Sciences and Engineering
School of Arts and Sciences
University of RochesterRochester, New York
2016
ii
Dedicated to my family and friends.
Dedicado a mi querida familia y amistades.
iii
Biographical Sketch
Gerardo Ivan Viza was born in Lima, Peru on November 4, 1985. He grew up
in Miami, Florida where he was inspired to study physics through lectures on
Electricity and Magnetism. In May of 2008, he graduated from Georgia Institute
of Technology with a Bachelor of Science degree in Physics and Applied Mathe-
matics. He was admitted to the University of Rochester in the fall of 2008 and
received a Master of Arts degree in Physics in 2010. He then joined Professor John
Howell’s quantum optics group and started his doctoral research in experimental
quantum optics.
iv
Publications
The following publications were a result of work conducted during doctoral study:
[1] Gerardo I. Viza, Julian Martınez-Rincon, Gregory A. Howland, Hadas Frost-
ing, Itay Shomroni, Barak Dayan, and John C. Howell, “Weak-values technique
for velocity measurements,” Opt. Lett. 38, 2949-2952 (2013).
[2] Gerardo I. Viza, Julian Martınez-Rincon, Gabriel B. Alves, Andrew N. Jordan
and John C. Howell, ”Experimentally quantifying the advantages of weak-value-
based metrology,” Phy. Rev. A 92, 032127 (2015).
[3] Gerardo I. Viza, Julian Martınez-Rincon, Wei Tao Liu and John C. Howell,
“Concatenated postselection for weak-value amplification”, in production (2015).
[4] Julian Martınez-Rincon, Wei Tao Liu, Gerardo I. Viza, and John C. How-
ell, “Can anomalous amplification be attained without postselection?”, arXiv
1509.04810 (2015).
v
Conference Proceedings
[1] Gerardo I. Viza, Julian Martınez-Rincon, Wei Tao Liu and John C. Howell,
“Concatenated Weak-values”, Conference on Quantum Information and Quan-
tum Control-VI, Toronto, Ontario (August 2015)
[2] Gerardo I. Viza, Julian Martınez-Rincon, Gabriel B. Alves, Andrew N. Jor-
dan and John C. Howell, “Experimentally Quantifying the Advantages of Weak-
Value-Based Metrology”, Conference on Lasers and Electro-Optics (CLEO, San
Jose, CA (May 2015).
[3] Gerardo I. Viza, Julian Martınez-Rincon, Gabriel B. Alves, Andrew N. Jordan
and John C. Howell, “Quantifying the Technical Advantages of Weak-Value Am-
plification”, Center for Coherence and Quantum Optics, Rochester, NY (Septem-
ber 2014).
[4] Gerardo I. Viza, Julian Martınez-Rincon, Gregory A. Howland, Hadas Frost-
ing, Itay Shomroni, Barak Dayan, and John C. Howell, ”Weak-value technique
for Velocity Measurements”, Conference on Quantum Information and Quantum
Control-V, Toronto, Ontario (August 2013)
[5] Gerardo I. Viza, Julian Martınez-Rincon, Gregory A. Howland, Hadas Frost-
ing, Itay Shomroni, Barak Dayan, and John C. Howell, “Precision Measurement
vi
of Doppler Shifts Inspired by Weak Values”, Frontiers in Optics, Rochester, NY
(October 2012)
vii
Acknowledgments
Nothing in this thesis could have been accomplished without the support of nu-
merous people. Firstly, I wish to praise God for taking me on this journey of
much refining. I would like to thank my family members, in the United States
and abroad, for much support and love, and, in particular, my parents and sister
who always pushed me academically.
I also am greatly thankful for my adviser, as there is probably no one more
optimistic and understanding than John C. Howell. His words of wisdom inside
and outside the laboratory have been inspiring. I would also like to thank Andrew
N. Jordan for challenging me in the fine details of research and writing.
Next I would like to thank my laboratory mates, most of whom have gradu-
ated: P. Ben Dixon, Curtis J. Broadbent, David J. Starling, Praveen Vudyasetu,
Gregory A. Howland and James Schneeloch have always challenged me and have
always been willing to help during experiments or discussion. I would also like
to thank Steve Bloch for his time with me and teaching me the beginnings of
experimental physics. In particular, I would like to thank my colleague Julian
Martınez-Rincon who has often worked with me on different projects. I am also
thankful for many collaborators from China, Brazil, and Israel, especially includ-
ing Wei-Tao Liu who visited from China and Gabriel B. Alves who visited from
Brazil.
I would like to thank the rest of our group for helping me during many phases
of projects, and discussions, and for making the work place enjoyable. I would
viii
like to thank Bethany Little, Daniel J. Lum, Christopher A. Mullarkey, Joseph
Choi, Samuel H. Knarr, and Justin M. Winkler.
I would also like to thank my family in Christ Jesus who has always been there
for me. Lastly, I want to thank my friends for careful editing of this thesis.
ix
Agradecimientos
Nada en esta tesis podrıa haberse completado sin la ayuda de numerosas per-
sonas. Primeramente quiero agradecer a Dios por tenerme en este lugar donde
he podido refinar mis conocimiento. Quisiera agradecer a mi familia por su gran
apoyo y amor, especialmente a mis padres y hermana quienes siempre me moti-
varon avanzar academicamente.
Tambien estoy muy agradecido de mi asesor, probablemente nadie mas opti-
mista y entendible que John C. Howell, sus palabras de sabidura dentro y fuera
del laboratorio han sido inspiradores. Tambien quisiera agradecer a Andrew N.
Jordan por estimularme en mejorar detalles de la investigacion y la redaccin.
Despues quisiera agradecer a mis companeros del laboratorio, la mayorıa de
ellos graduados: P. Ben Dixon, Curtis J. Broadbent, David J. Starling, Praveen
Vudyasetu, Gregory A. Howland y James Schneeloch quienes siempre han estado
dispuestos a ayudarme durante los experimentos o discusiones. Tambin quisiera
agradecer a Julian Martınez-Rincon quien frecuentemente ha estado trabajando
conmigo en diferentes trabajos.
Tambien estoy muy agradecido por muchos colaboradores de la China, Brazil
e Israel, incluyendo a Wei-Tao Liu quien visito de la China y Gabriel B. Alves
quien visito del Brazil. Quisiera agradecer al resto del grupo por su ayuda durante
muchas etapas de los projectos, discusiones y por hacer el lugar de trabajo mas
agradable entre ellos Bethany Little, Daniel J. Lum, Christopher A. Mullarkey,
Joseph Choi, Samuel H. Knarr, y Justin M. Winkler.
x
Quisiera agradecer a mi familia en Cristo Jesus quienes han siempre estado
presentes y finalmente gracias a mis amigos por editar cuidadosamente esta tesis.
xi
Abstract
This thesis contains a subset of the research in which I have participated in during
my studies at the University of Rochester. It contains three projects and one over-
arching theme of weak-value metrology. We start with chapter 1 where we cover
the historical background leading up to quantum optics, which we use for preci-
sion metrology. We also introduce the weak-value formulation and give examples
of metrological implementations for parameter estimation. Chapter 2 introduces
two experiments to measure a longitudinal velocity and a transverse momentum
kick. We show that weak-value based techniques are shot-noise limited because
we saturate the Cramer-Rao bound for the estimator used, and efficient because
we experimentally demonstrate there is virtually no loss of Fisher information of
the parameter of interest from the discarded events. In Chapter 3, we present
a comparison of two experiments that measure a beam deflection. One experi-
ment is a weak-value based technique, while the other is the standard focusing
technique. We set up the two experiments in the presence of simulated techni-
cal noise sources and show how the weak-value based technique out performs the
standard technique in both visibility and in deviation of the transverse momentum
kick. Chapter 4 contains work of the exploration of concatenated postselection
for weak-value amplification. We demonstrate an optimization and conditions
where postselecting on two degrees of freedom can be beneficial to enhance the
weak-value amplification.
xii
Contributors and Funding Sources
This work was supervised by a dissertation committee consisting of Professors
John C. Howell [my research advisor], Andrew N. Jordan, and Stephen Teitel of
the Department of Physics and Astronomy, as well as Professors Nick Vamivakas
and Miguel Alonzo of the Institute of Optics.
I am very grateful that Professor John Howell was able to secure funds for me
and the rest of the group. I am also grateful for the support the University gave me
in teaching assistantship while I did not have an advisor. The following are grants
that funded me and the coworkers that led to the fulfillment of these papers. I
have been funded by the Army Research Office, Grant No. W911NF-12-1-0263
and No. W911NF-09-0-01417. I would also like to thank the funding sources
for the visiting colleagues on the papers starting with the CAPES Foundation,
Process No. BEX 8257/13-2, the National Natural Science Foundation of China
Grant No. 11374368 and the China Scholarship Council.
Julian Martinez-Rincon and I are the main authors of the OSA letter titled
“Weak-values technique for velocity measurements” (see Ref. [1]). Gregory A.
Howland was very helpful in the understanding of the theoretical aspect and the
implications of the results. Gregory A. Howland also helped us in programming
the Pico Harp to use the APDs for data acquisition. Julian Martinez-Rincon
analyzed all the data which both of us collected together. I created all the figures
and wrote the paper with their help. On his sabbatical collaboratively, our advisor
xiii
John Howell, together with Hadas Frostig, Itay Shomroni, and Barak Dayan, came
up with the idea for the experiment.
The PRA article titled “Experimentally quantifying the advantages of weak-
value-based metrology” (see Ref. [2]) was proposed by John Howell and Andrew
Jordan as they and Julian Martinez-Rincon wrote the PRX article titled, “Tech-
nical Advantages for Weak-Value Amplification: When Less Is More.” The PRA
article was a collaborative effort between Julian Martinez-Rincon, Gabriel B. Alves
and myself. Julian Martinez-Rincon and I started the project but we incorporated
Gabriel B. Alves midway. All three of us did data collection. Andrew Jordan was
a strong proponent of the naturally occurring laser beam jitter section, the Fisher
information formalism and the efficiency study of the last section. I wrote the
data acquisition programs and created all the figures. Analyzing the data and
writing the paper was a collaborative effort between all of us.
The article in progress titled, “Concatenated postselection for weak-value am-
plification,” (see Ref. [3]) is a collaborative effort between Julian Martinez-Rincon,
Wei-Tao Liu and myself. The idea was also collaborative as it occurred as we ex-
plored the possibility of photon recycling. Wei-Tao Liu and I collected the data
with the programs I wrote. I analyzed the data and created the figures. Wei-Tao
and Julian Martinez-Rincon proposed to incorporate and study the back ground of
the first postselection. The writing of the paper was a collaborative effort between
all authors. This article lead to Julian’s paper “Can anomalous amplification be
attained without postselection?” (see Ref. [4]).
xiv
Table of Contents
Dedication ii
Biographical Sketch iii
Acknowledgments vii
Agradecimientos ix
Abstract xi
Contributors and Funding Sources xii
List of Tables xvii
List of Figures xviii
List of Acronyms and Abbreviations xx
1 Introduction 1
1.1 The Photon and Quantum Optics . . . . . . . . . . . . . . . . . . 1
1.2 Metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Weak-value: Real and Imaginary . . . . . . . . . . . . . . . . . . 15
xv
1.4 Signal-to-Noise Ratio . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5 The Cramer-Rao Bound and the Fisher Information . . . . . . . . 25
1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.7 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 Weak-value Techniques: Shot-Noise Limited and Efficient 33
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Measuring a Longitudinal Velocity . . . . . . . . . . . . . . . . . 34
2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4 Measuring a Transverse Momentum Kick . . . . . . . . . . . . . . 44
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3 The Technical Advantage of Weak-value Based Metrology 53
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.4 Results: Comparison of WVT and ST . . . . . . . . . . . . . . . . 63
3.5 Results: Laser Beam Jitter . . . . . . . . . . . . . . . . . . . . . . 72
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4 Concatenated Postselection for Weak-value Amplification 76
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.5 Efficiency: Single, Concatenated, and Standard Focusing . . . . . 86
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
xvi
5 Concluding Remarks 94
Bibliography 97
A Velocity Experiment: Bright Port Analysis 110
B Concatenated Postselection 112
B.1 Quarter-Half-Quarter: Pancharatnam-Berry phase . . . . . . . . . 112
B.2 Weak-value Quantum Description . . . . . . . . . . . . . . . . . . 113
B.3 Deviation of First Order in k from the All Order in k Theory . . . 115
xvii
List of Tables
2.1 Table of velocity measurements. . . . . . . . . . . . . . . . . . . . 42
3.1 Table of beam shifts due to the signal and external modulations. . 59
4.1 Results of concatenated postselection for weak-value amplification. 87
xviii
List of Figures
1.1 Split detector caricature to measure beam shifts. . . . . . . . . . . 13
1.2 An imaginary weak-value experiment to measure beam deflections. 19
1.3 A real weak-value experiment to measure beam deflections. . . . . 21
2.1 Michelson setup for velocity measurements. . . . . . . . . . . . . . 37
2.2 Results for the velocity experiment. . . . . . . . . . . . . . . . . . 40
2.3 Error in the measurements of the velocity measurements. . . . . . 41
2.4 Setup of the imaginary weak-value beam deflection experiment. . 46
2.5 Efficiency results of the beam deflection experiment. . . . . . . . . 49
3.1 Beam deflection weak-value vs. standard focusing technique with
technical noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Beam deflection signal and the external modulations for both the
WVT and the ST. . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3 Signal-to-External-Modulation ratio geometric dependence. . . . . 65
3.4 Deviation in k as a function of external modulation comparison
between WVT and ST. . . . . . . . . . . . . . . . . . . . . . . . . 68
3.5 A plot of the geometric factor to optimizing the ST. . . . . . . . 71
3.6 Fourier spectrum of naturally occurring laser beam jitter for WVT
and ST. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
xix
4.1 Experimental setup of concatenated postselection experiment. . . 78
4.2 Single and concatenated data. . . . . . . . . . . . . . . . . . . . . 84
B.1 Percentage of deviation from the first order in k approximation . . 116
xx
List of Acronyms and Abbreviations
APD Avalance Photodiode
BS Beam Splitter
CCD Charged Coupled Device
CRB Cramer-Rao bound
LIGO Laser Interferometer Gravitational-Wave Observatory
PBS Polarizing Beam Splitter
POVM Positive Operator Valued Measure
PVM Projective Value Masurement
SD Split Detector
SNR Signal-to-Noise ratio
ST Standard Technique
WVT Weak-Value Technique
1
1 Introduction
1.1 The Photon and Quantum Optics
Light is one of the first things God created: “And God said, “ ‘Let there be light,’
and there was light (Genesis 1:3).” By one way or another, the vast majority of liv-
ing creatures use light for sustenance or to gather information. As we understand
it today, light is made up of wave packets called photons. Photons are subtle in
the sense that they can be described as both particles and waves. Einstein dis-
covered that photons travel at a constant speed regardless of the reference frame
of the observer. Photons are long lasting as they travel extraordinary distances to
reach our eyes from neighboring galaxies. We start with a brief overview of more
than 100 years of science leading up to quantum optics.
In the pre-quantum days the photon was studied and shown to have extraordi-
nary properties such as in the Young’s double slit experiment performed by Taylor
with a low light level in 1901 [5]. In that time, Planck theorized the energy pack-
ets which we call the photon in his black-body radiation theory of 1901. But it
was not until Einstein explained the photoelectric experiment that the foundation
for quantum thinking was laid [5]. This led to many papers verifying Einstein’s
work, and in 1927 Dirac’s paper on quantum theory of radiation formed the basis
2
of quantum optics [6, 7]. With the invention of the laser in the 1960s, quantum
optics became its own field [8]. Quantum optics is the study of how light in-
teracts with matter: the interaction between photons and atoms. Dirac’s work
led to a more fundamental approach to study electromagnetic waves, where the
photon was treated as electromagnetic radiation. Quantum theory led to many
paradoxes [9–12] as it brought a entirely new way of looking at the world where
particles do not have a well defined position and momentum at the same time. In
the quantum regime, a wave function describes the state of a photon. Today, there
is much debate on the conceptual meaning of the wave function and its relation
to reality [13–15]. The quantum mechanical description of a photon has even led
to the direct measurement of the wave function [16, 17]. In this thesis, we start
with the photon and use both the quantum formalism and the classical formalism
to describe our experiments and results.
Before we begin with Maxwell’s equations, we briefly describe the utility of
photons in our everyday world. The properties of photons have been harnessed
to bring about great technological advances in modern daily life. These utilities
include optical lithography to produce computer processors, and efficient LED
lighting for smart televisions [18]. In addition, photons are used in everyday
stores to scan merchandise and in hospitals for surgical procedures. Photons are
even used in security devices such as retinal scanners.
We understand photons to fundamentally be electromagnetic radiation. Thus
we link the classical electromagnetic fields to the quantum harmonic oscillator
to give us an understanding of the origin of photons. We begin with a classical
electromagnetic field through a medium with a finite volume boundary in vacuum.
Maxwell’s equations in vacuum are as follows:
∇ · E = 0, (1.1a)
∇ ·B = 0, (1.1b)
3
∇× E = −∂B
∂t, (1.1c)
∇×B = µ0ε0∂E
∂t, (1.1d)
where the constants ε0 and µ0 are the permittivity and permeability of vacuum,
respectively. These two fundamental constants are what give rise to the con-
stant speed of light c = 1/√ε0µ0. E and B are the electric and magnetic fields,
respectively.
Next we combine Maxwell’s equations with the identity ∇× (∇×E) = ∇(∇ ·
E)−∇2E to arrive with the wave equations for the two fields. We fix the boundary
conditions to a unit volume cavity and can write the electric field as a sum of modes
as in
E(~r, t) = ε−1/20
∑m
fm(t)um(~r). (1.2)
Assuming the boundary condition of a perfectly conducting surface we have
∇2um = −k2mum, (1.3a)
d2fmdt2
= −ω2mfm, (1.3b)
with ωm = ckm. Then we use the same orthonormal spatial basis of un(r) for
the magnetic field. We then write the solution of the magnetic field in a similar
fashion:
B(~r, t) = µ1/20
∑m
hm(t)(∇× um(~r)). (1.4)
Taking the curl of Eq. (1.1c) we find dhmdt
= −cfm and arrive with the following
relation:d2
dt2hm = −ω2
mhm. (1.5)
4
Now we use the description of both fields in the Hamiltonian that represents the
total energy of the fields:
H =ε02
∫V
E2 + c2B2dr. (1.6)
Using the orthogonality of the spatial basis um(r) we arrive at the Hamiltonian
H =1
2
∑m
(f 2m + ω2
mh2m). (1.7)
From the equality of dfmdt
= ck2mhm we can rewrite the Hamiltonian as
H =1
2
∑m
(f 2m +
1
ω2m
(dfmdt
)2). (1.8)
We can associate this Hamiltonian to the energy in each mode of a harmonic
oscillator and relate fm to the parameter of position qm such that qm = fm/ωm.
We also introduce pm = qm as part of the set of canonical conjugate variables of
position and momentum.
The next step is to make both position and momentum in terms of creation
and annihilation operators as
qm =
√~
2ωm(a†m + am), (1.9a)
and
pm = i
√~ωm
2(a†m − am). (1.9b)
These operators obey the algebra with commutation relationships [am, a†m′ ] =
δm,m′ , [am, am] = 0 and [a†m, a†m] = 0. We recall the operator commutation relation
of position and momentum [q, p] = i~. From this we can write both the electric
and magnetic fields in terms of the ladder operators:
Ex(z, t) =1õ0
∑m
√~ωm
2[a†m(t) + am(t)]um(kmz), (1.10)
and
By(z, t) =õ0
∑m
ic
√~
2ωm[a†m(t)− am(t)](∇× um(kmz)), (1.11)
5
respectively. In Eq. (1.10) and Eq. (1.11) the ladder operators can be written as
parameter functions of time as in the Heisenberg picture. The solution to um is
the oscillating functions of sines and cosines.
This formulation relates the classical electromagnetic fields to the quantum
regime of a single photon in the quantum harmonic oscillator operator formalism.
The electromagnetic waves have well defined amplitudes, but in this formulation,
the amplitude and phase are probabilistic and follow a Heisenberg uncertainty
principle between photon number and phase [19].
We can solve for the ladder operators a and a† in terms of position and mo-
mentum operators. From this we can formulate the following relationship:
a†mam =1
2~ωm(ω2
mq2m + p2m) +
i
2~[qm, pm] (1.12)
Let N = a†mam be the number operator and solve for the Hamiltonian in operator
form
H =∑m
~ωn(a†mam +
1
2
). (1.13)
Now we go to the simplest case where the first mode m = 1 since this is a sum of
independent degree of freedoms:
H = ~ω(N +
1
2
). (1.14)
We denote the eigenvectors of the Hamiltonian as |n〉 where n ∈ {0, 1, 2...}, where
the states are called the Fock number states. We understand the meaning of these
ladder operators as creating, a†, or annihilating, a, a photon from a mode as in
a†|n〉 =√n+ 1|n+ 1〉 and a|n〉 =
√n|n− 1〉.
The number operator, N , corresponds to the quantum of energy ~ω. The
Hamiltonian has two parts. In the first term with the number operator N , the
mode energy can vary only in discrete increments of ~ω. The second part has
a more ellusive meaning. The second term says that even in the ground state
|0〉, there is still energy. The energy ~ω/2 is called the zero-point energy of the
6
vacuum state. The problem arises because the sum of modes is infinite, and this
intuitively means infinite energy, which is not a realistic model. The condition
is also necessary because the Heisenberg uncertainty principle says even at the
ground state, the quantum harmonic oscillator will not be at rest. In research
this subtlety is avoided because we are interested in differences of energies so the
zero-point energy contribution is eliminated [20]. Dirac was one of the first to
study the issue in re-normalization theory [21]. One interesting consequence from
the topic is the Casimir effect [22–25].
1.2 Metrology
Metrology is the study of measurement. Measurements are used to extract param-
eters of interest as precisely as possible. The precision available to an observer is
fundamentally bounded by the resource that is used. In typical experiments the
fundamental bound is not reachable without the use of post-processing devices
such as filters or lock-in-amplifiers. To reach the fundamental bounds, stabilizing
methods of both active and/or passive types are generally used. Active methods
entail the incorporation of a feedback mechanism to the system, while passive
methods entail the use of materials that minimize or isolate the environmental
interactions with the system. The process of extracting a parameter of interest
can involve many steps. If noise is not dealt with in those steps, then extracting
a parameter of interest can prove to be difficult or impractical. Here we present
weak-value-based techniques that can be optimally used for metrology [1, 26].
Weak-value-based techniques both amplify a signal and reduce external modula-
tions and certain types of noise [2, 27, 28].
Post-processing such as amplification of a signal in conjunction with filtering
is used to clear the signals from the noise for measurements. Averaging is an-
other technique for noise reduction. For example, if a signal is embedded with
7
uncorrelated Gaussian white noise, then a simple average makes the measurement
possible by eliminating the noise. The lock-in-amplifier uses a heterodyne tech-
nique combined with frequency filtering and averaging to extract both amplitude
and phase from the signal with a known frequency. For example, one can look
at the electronics in the laser Doppler vibrometry experiment in Ref. [29]. These
devices and techniques quickly become sophisticated depending on how much
noise relative to a signal is present in the system, as in the Laser Interferometer
Gravitational-Wave Observatory (LIGO) [30].
We can generalize stabilization methods into two types: active and passive.
The difference is that the active stabilization type incorporates a feedback mecha-
nism to the system. One of the biggest accomplishments is LIGO, where the 4 km
long Fabry-Perot cavities are actively stabilized arms of a Michelson interferom-
eter. The scientists and engineers from LIGO have stabilized the interferometer
arm length to a strain sensitivity of ∆L/L = 10−23, where L is the length of the in-
terferometer arm [30–34]. The mirrors receive a feedback signal to compensate for
any vibration caused by seismic noise. Examples of passive stabilization include
using materials that have low sensitivity to noise such as zerodur mirrors with
a very low thermal expansion coefficient. Another example of passive stabiliza-
tion, also in the LIGO system, is where the interferometer is in a colossal vacuum
chamber underground in order to be isolated from the outside environment [30].
In an optical example, photons are sent out of a fiber launcher, from which
they traverse a medium where frequency information is imparted. After that,
the photons are spatially separated by an atomic prism [35] according to their
frequency. Then the detection is done on an array of charge coupled devices
(CCDs). The photons are converted into electrical signals which are recorded
by the computer. Throughout the transfer of information, the measurement can
be clouded with noise, and which makes it difficult for measurements to have
shot-noise limited precision.
8
Here we present a compilation of weak-value metrological experiments that are
optimal for parameter estimation. We discuss how a weak-value-based technique
can perform like some of the devices outlined above and enhance the standard
method of performing precision measurements. The weak value as presented in the
seminal work by Aharonov, Albert and Vaidman shows how to amplify signals [27].
The procedure starts with a well defined initial state of a system, |ϕi〉. Then there
is a weak coupling between a meter state and the system through an ancillary
observable, A. Then a postselection is performed on the system state, |ϕf〉, that
is nearly orthogonal to the input state of the system. This leads to a weak-value
of the observable given by,
Awv =〈ϕf |A|ϕi〉〈ϕf |ϕi〉
. (1.15)
The weak-value is different from an expected value in that it can lie outside the
range of allowable eigenvalues. For this reason, this amplification is the anomalous
amplification in the limit of nearly orthogonal postselection. The postselected
meter state has a mean value that will be shifted proportionally to the weak
value, which effectively increases the resolution of detectors.
This has been shown to aid researchers in measuring tiny phenomena such
as the spin hall effect of light [36] using a low-power laser in a table top optical
experiment. Parameters such as frequency [37], temperature [38], velocities [1],
transverse momentum [26, 39], and phase [40, 41] have been measured using weak-
value based techniques. It has been debated whether or not using weak-value
techniques is of any benefit at all [42–46], but like any good technique, it has
been shown useful under certain conditions [2, 28, 47], in many studies [48–55]. It
has been shown both theoretically [28, 56, 57] and experimentally [2], that weak-
value based techniques can outperform standard optimal techniques under the
condition of technical noise. The study of noise is very extensive in many fields,
and in this compilation we will discuss only a small subset of noise sources. Here
we focus on using weak-value techniques optimally and where these are helpful
9
for metrology. We first start with a discussion about a basic measurement in
the noiseless scenario, and in subsequent chapters we will build up to include
deviations from the noiseless case.
Measurements
Starting with a simple example we introduce a measurement of polarization. In
quantum mechanics we can write the state of a photon as |ψ〉. We use projective
measurements to determine the properties of the state. A typical example would
be to determine the polarization of a photon with a polarizer. We write the
state in a superposition of horizontal and vertical polarized light as |H〉 and |V 〉,
respectively. The state in question takes the form
|ψ〉 = a|H〉+ b|V 〉. (1.16)
We do a projective measurement with a polarizer to determine the probability of
finding the state to be horizontally or vertically polarized. We write the projec-
tors as ΠH = |H〉〈H| or ΠV = |V 〉〈V | to make the required measurement. We
reformulate the state into a density matrix ρ = |ψ〉〈ψ| to introduce a more general
formulation of a measurement. The measurement is performed when the photon
reaches the detectors. We write the result of the measurement with the projectors
as
|a|2 = Tr(ΠHρ) = 〈ψ|ΠH |ψ〉, (1.17)
|b|2 = Tr(ΠV ρ) = 〈ψ|ΠV |ψ〉. (1.18)
All measurements in this regime are by nature probabilistic, and since the detec-
tors collect intensity, we lose the phase information in this measurement. The
projector follows certain rules: the projectors must be Hermitian, and the projec-
tor squared must be the projector itself as in
Π†j = Πj, (1.19a)
10
and
Π2j = Πj. (1.19b)
We also assume the projectors to be orthogonal ΠjΠk = δjkΠj and that the pro-
jectors form a complete set,∑
j Πj = 1. We write the state after the measurement
as
|ψ〉f =Πj|ψ〉i√〈ψ|Π†jΠj|ψ〉
. (1.20)
We can also say that the wave function is destroyed after the projector projects to
the detector with the eigenvalue in question. This type of measurement is called
a projective-value-measurement (PVM).
For the weak-value-type measurement (as the term suggests) we make weak
measurements where the measurements do not collapse the wave function. This
measurement type is more general than PVM and is called positive-operator val-
ued measure (POVM). In this formalism the POVM elements are not required
to be orthogonal. The most interesting aspect of this measurement is that the
wave function remains practically undisturbed after the measurement. We will
see examples of this type of measurement in the experiments that follow.
1.2.1 The Shot-Noise Limit
The shot-noise limit is the absolute limit in uncertainty governed by the statistics
of a classical resource. When measuring an observable, we use a large number of
independent events N where every event has the same uncertainty, and because of
the central limit theorem the uncertainty of the measurement has the statistical
scaling of 1/√N . This scaling of uncertainty is known as the standard quantum
limit or the shot-noise limit. This limit is not to be confused with Heisenberg
scaling of uncertainty, 1/N . The Heisenberg scaling is the fundamental bound of
uncertainty of a quantum source, such as a NOON state. The quantum measure-
ment exploits the correlation of the quantum source to go beyond the standard
11
quantum limit to the Heisenberg limit [58]. In this thesis, we use the coherent
state that is statistically bounded by the shot-noise limit and the Heisenberg un-
certainty relation. Before we discuss the uncertainty relation, we will start with
the statistical nature of our resource, that is, the photons out of a laser, or the
coherent state.
When using photons, there is an intrinsic variance or uncertainty in every
measurement. From the formulation of the Harmonic oscillator we will solve for
the uncertainty of measuring position and momentum in the state |n〉. We note
that the expected value of both position in Eq. (1.9a) and momentum in Eq. (1.9b)
are given by 〈n|q|n〉 = 0 and 〈n|p|n〉 = 0. Then the uncertainty relation of the
two conjugate variables is given by
〈n|(∆x)2|n〉〈n|(∆p)2|n〉 =~2
4(2n+ 1)2. (1.21)
The uncertainty of measuring position and momentum of the ground state gives
the minimum uncertainty of ~/2, Heisenberg uncertainty relation. States that are
of minimum uncertainty by the Heisenberg uncertainty relation we call coherent.
According to the Heisenberg uncertainty relation, the ground state of the harmonic
oscillator is the only stationary state that is coherent [7]. Now we will introduce
a general coherent state made up of a superposition of stationary states.
The Coherent State
Another minimal uncertainty state is the coherent state, also known as the Glauber
state. For more properties of the coherent state, please see Ref. [5, 24]. The source
of photons from a laser is of the coherent type where the photons are in phase and
identical within a certain coherence window. This state is produced from a laser
lasing above threshold with stable transmission and temperature. Without going
into the technical details of a laser we start with the coherent state defined as
a|α〉 = α|α〉. (1.22)
12
The state is written as a superposition of Fock states |α〉 =∑
n cn|n〉. We write
|n〉 = (a†)n√n!|0〉 and project to the coherent state |α〉 to determine the coefficient
cn = e−12|α|2 . We then apply these results and arrive with the coherent state
|α〉 = e−12|α|2
∞∑n=0
(αa†)n
n!|0〉. (1.23)
It follows that the coherent state is also equivalent to
|α〉 = e−12|α|2+αa†|0〉. (1.24)
With this state we can determine the expected value of the position and mo-
mentum operators. Then from the expected values we determine the uncertainty
relation that will set the minimum bound set by the shot-noise:
〈α|x|α〉 =
√~
2ωn〈α|a†n + an|α〉 =
√~
2ωn(α∗ + α),
〈α|p|α〉 =1
i
√~ωn
2〈α|a†n − an|α〉 =
1
i
√~ωn
2(α∗ − α).
(1.25)
Hence the uncertainty between the conjugate variables is given by
〈(∆x)2〉α〈(∆p)2〉α =
~2
4. (1.26)
This is the state we will be using in our experiments so we can calculate the
uncertainty of measuring a photon with operator N
〈(∆N)2〉 = 〈N2〉 − 〈N〉2 = |α|2 = N. (1.27)
The exponential operator eiNφ0 adds a phase to the field |α〉. This suggests the
phase is a canonical conjugate to the photon number. This line of thinking has
led to a non-unitary and non-unique description of a phase operator [19, 59].
For a more intuitive understanding of the coherent state, we can determine the
statistics of the state by projecting it to the |n〉 basis to arrive with the intensity
profile
|〈n|α〉|2 = e−〈N〉〈N〉n
n!. (1.28)
13
Figure 1.1: Schematic of a SD. This detector reads out the intensity of the left quadrant minus
the intensity of the right quadrant. The detector also reads out the intensity of the sum of both
quadrants. The drawing is not drawn to scale and it is exaggerating the dead space between
the two pixels. In the experiments the gap is 200 µm. The 2σ is the beam radius at the 1/e2
location. The beam shift is exaggerated; for the calculations we assume δx to be small compared
to σ.
This distribution is a Poisson distribution with mean and variance equal to 〈N〉.
The photon statistics will define the precision of a measurement. In theory we can
use sub-Poisson resources such as squeezed light or other exotic sources of light
for better precision. Since the coherent state satisfies the minimum uncertainty
relation in Eq. (1.26) and the coordinate representation, 〈q|α〉 or 〈p|α〉, is the wave
packet, we can say that the coherent state is the closest representation of the state
of the photons out of a laser cavity [8, 60]. Throughout these experiments we will
use the coherent state as our resource for investigation.
In the experiments we use two types of detection: a time of arrival detection
and a spatial split detection. Here we discuss the split detector (SD) and how we
resolve the minimum shift of a Gaussian beam.
14
Split Detector
The SD is a two pixel spatial detector. We use this detector to determine the
average shift of a beam. As shown in Fig. 1.1, there are two beams. The beam
centered in between the left and right quadrants is unshifted and the beam off to
the left quadrant is the shifted. To determine the signal-to-noise ratio (SNR) of
this system, we can define two Gaussians operators: one even and one odd [61]
such as
aR =1
2
1√2πσ2
e−x2
4σ2 (1 + sign(x)), (1.29a)
aL =1
2
1√2πσ2
e−x2
4σ2 (1− sign(x)). (1.29b)
We observe the aR is non-zero for x > 0 while the aL is non-zero for x < 0. We
use these operators to define the signal which assumes the beam has shifted by
δx in one direction:
〈a†RaR〉 − 〈a†LaL〉 =
2N√2πσ2
∫ δx
0
e−x2
2σ2 dx ≈√
2
π
δx
σN. (1.30)
For the last integration step we assume δx � 1 to approximate the exponential
to first order in x. Using the commutation relation [a, a†] = 1 we calculate the
variance of the signal
〈∆(a†RaR − a†LaL)2〉 = 〈a†RaR + a†LaL〉 = N
∫ ∞−∞
1√2πσ2
e−x2
2σ2 dx = N. (1.31)
We integrate from all of space even though the detector is finite because we assume
our plane wave approximation leads to virtually zero intensity, or zero probability
beyond the detection window. With these two quantities we arrive at the SNR,
S, given as
S =
√2
π
δx
σ
√N (1.32)
The factor√
2/π is there due to the configuration of the detector with two pixels.
Here, N is interpreted as the number of resources or photons arriving on the
15
detector that was used for the measurement. We will continue the discussion of
the SNR in section 1.4.
1.3 Weak-value: Real and Imaginary
In this section we lay out how a weak-values experiment is used for metrology. In
particular we explore three regimes of weak-values: real, imaginary, and inverse.
We show three examples of how they can be used for parameter estimation. We
will refer to these examples throughout the following chapters.
The utility of weak-values for metrology relies heavily on the creativity of the
experimental configuration. The weak-value is complex and in some scenarios
more difficult to measure than a standard protocol. We present the weak-value
theory in the position and momentum domains. For a given observable A with a
set of strong measurements we can extract the mean value of that observable,
〈A〉 = 〈ϕi|A|ϕi〉. (1.33)
This is the standard way of acquiring information of an observable A. We note
that observables are Hermitian and thus they have real eigenvalues. From a set of
strong measurements we have the probabilities for each eigenvalue 〈A〉 =∑
i Piai
for eigenvalue ai and the Pi are the probabilities of the physical system. In 1988,
Aharonov, Albert and Vaidman [27] introduced the weak value
Awv =〈ϕf |A|ϕi〉〈ϕf |ϕi〉
. (1.34)
The formulation of the weak value of an observable A exceeds the range of its
eigenvalues. In the limit where the pre- and postselections are nearly orthogonal
we approach the weak-value amplification regime. This is the regime that we focus
on for the rest of the thesis.
We now introduce a more general method of measurement. We prepare a
state of the meter which is then entangled with a system through the interaction
16
Hamiltonian given by g(t)Aq. We also assume the interaction given by∫ t00g(t)dt =
g, where t0 is the instantaneous time of interaction when the measurement is
performed. We then write the state as
|Ψ〉 = e−igAq|ϕi〉|ψ〉. (1.35)
The final step is to postselect with state |ϕf〉, a near orthogonal state to the
preselected state of |ϕi〉. We will call the final postselected state |Ψ′〉, written as
|Ψ′〉 = 〈ϕf |e−igAq|ϕi〉|ψ〉. (1.36)
The key assumption is that the coupling between the system and the meter is
weak, such that we can expand the interaction to first order in g. The idea is
that the interaction is so small that the meter state remains nearly undisturbed.
After the expansion to first order in g, we postselect to a state that is nearly
orthogonal to the initial state of the system. The last step of the protocol is to
re-exponentiate with the weak-value Awv to arrive with
|Ψ′〉 ≈ 〈ϕf |(1− igAq)|ϕi〉|ψ〉,
|Ψ′〉 ≈∫ (〈ϕf |ϕi〉 − igq〈ϕf |A|ϕi〉
)|q〉〈q|ψ〉dq,
≈ 〈ϕf |ϕi〉∫e−igqAwv |q〉〈q|ψ〉dq,
〈q|Ψ′〉 ≈ 〈ϕf |ϕi〉e−igqAwvψ(q).
(1.37)
In general, the weak-value Awv defined in Eq. (1.34) is complex. From a practical
perspective, the observer building the experiment must identify the parameter of
interest g and make sure that the interaction is weak. Then in order to extract
the information from state |Ψ′〉, the observer can choose to perform a measure-
ment either in the bases p or in its conjugate q. From the parameter of interest
perspective, in this example, g is related to the conjugate variable of q, so g ∝ p.
In this scenario, if Awv is purely imaginary say ib where b ∈ <, the state in
the q basis becomes
〈q|Ψ′〉 ≈ 〈ϕf |ϕi〉ψ(q + 2gbσ2). (1.38)
17
If the weak value were only real, such as when Awv = a where a ∈ <, the state in
the momentum basis becomes
〈p|Ψ′〉 ≈ 〈ϕf |ϕi〉ψ(p+ ga). (1.39)
Depending on whether the weak-value of the experimental setup is real or
imaginary we can project our state to whichever basis p or q is needed for the
measurement. Using the theory by Jozsa [62], the mean values of the meter of the
final postselected state given that Awv = a+ ib, where {a, b} ∈ <, are given by
〈p〉f = 〈p〉i + ga, (1.40a)
and
〈q〉f = 〈q〉i + 2gb(Varq). (1.40b)
Note that the interaction is e−igAp in Ref [62], which differs from our interaction
given by Eq (1.35).
For the experiments explained here, we use an ideal Gaussian pulse in time
and an ideal continuous wave with a transverse Gaussian profile. As an exam-
ple of weak-value-based techniques, we use a Sagnac interferometer to measure
transverse beam deflections.
Before we proceed, we briefly discuss the postselection in the weak-value pro-
tocol and how this postselection differs from ordinary background filtering. The
postselection process is no ordinary background filtering but rather a destructive
interference process with an ancillary system. This critical step is overlooked
and sometimes wrongly thought of as throwing away useful data. This has been
a recent stumbling block in understanding weak-values. There have been toy
models trying to refer to postselection in weak-values as a random selection filter-
ing process arguing that classical conditional probabilities lead to the weak-value
amplification [63]. However, we cannot stress enough that when there exists a
classical model to exhibit a quantum-like effect is not proof that the protocol in
18
question is only classical. We do not address this topic in this thesis but we refer
the reader to Refs. [64, 65].
In a typical example of down conversion, the spontaneously generated photon
pairs are clouded with background light of different characteristics. To isolate the
photons pairs, we separate them according to frequency, polarization, or angular
spread, to name a few. In these cases researchers have devised clever ways to
remove the background with edge filters, polarizers, or lenses. We can all agree
that this type of postselection is of the filtering type and can be applied without
any ancillary system.
It has been recently shown how an anomalous amplification can be observed
away from the weak-value amplification regime without discarding any photons
[4]. This new technique is a new way of thinking for parameter estimation tech-
niques by having an ancillary system with an almost balanced interferometer.
1.3.1 Optical Beam Deflection: Imaginary
To demonstrate an imaginary weak-value example, we present an experiment to
measure a transverse momentum kick with a Sagnac interferometer. We send a
beam of light into a Sagnac interferometer through a 50:50 beam splitter (BS).
The BS is on a piezoactuated mount to give the transverse momentum kick k to
the photons on the reflected port. When the beams recombine in the 50:50 BS,
they destructively interfere at the dark port. We monitor the beam shift of the
light that exits the dark port with a SD. The phase difference between each path
is controlled by a vertical misalignment of the piezoactuated 50:50 BS.
We start with a continuous wave with a transverse Gaussian profile. The beam
is collimated with an objective after it is spatially cleaned out of a pinhole. The
beam has 1/e2 beam radius σ, and the transverse beam profile is written as
〈x|ψ〉 = ψ(x) = E0e− x2
4σ2 . (1.41)
19
k
5x5x
50:50 BS
Split Detector
Pol 1Laser
Half-wave
Figure 1.2: We use a c.w. Gaussian beam exiting a spatial filter pinhole. We use a Sagnac
interferometer to measure transverse beam deflections. The piezoactuated 50:50 BS imparts a
momentum kick k in the horizontal plane. We monitor the photons exiting the dark port with a
SD. We also control the interference by a slight vertical misalignment on the 50:50 BS. The Pol
1 is used for polarization purity and the half-wave plate is used to control the linear polarization
of the input photons.
The experimental configuration can be found in the schematic drawing in Fig. 1.2.
The meter state is coupled to the system via the interaction e−ikAx. We no-
tice the ancillary operator A couples the transverse momentum kick k to beam
position x through the which-path degree of freedom. This interaction breaks the
symmetry of the system and imparts the information of the beam deflection to
the photons. We write the state before postselection as
|Ψ〉 = e−ikAx|ϕi〉|ψ〉. (1.42)
The pre- and postselected states are given by |ϕi〉 = 1√2(| �〉eiφ/2+i| 〉e−iφ/2) and
|ϕf〉 = 1√2(i| �〉+ | 〉), respectively. We note the states |ϕi〉 and |ϕf〉 are nearly
orthogonal for small φ. Then, assuming a weak interaction, the final postselected
20
state of the system-meter is given by
|Ψ′〉 = 〈ϕf |ϕi〉∫e−ikxAwv |x〉〈x|ψ〉dx. (1.43)
Now we calculate the weak-value in this experimental design, where the system
ancillary operator is given by A = | �〉〈� | − | 〉〈 |. In this scenario, the weak-
value is computed as in Eq. (1.34) to be purely imaginary with Awv = −i cot(φ/2).
If we choose to use the imaginary weak-value, we can remain in the position basis
for the measurements.
〈x|Ψ′〉 ≈ 〈ϕf |ϕi〉e−igxAwvψ(x),
Ψ′(x)Im ≈ sin(φ/2)e−kx cot(φ/2)e−x2
4σ2
≈ sin(φ/2) exp
[− 1
4σ2
(x+ 2kσ2 cot(φ/2)
)2].
(1.44)
In the last line of Eq. (1.44), we assume the small interaction approximation given
by k2σ2 cot2(φ/2)� 1. Taking the magnitude square of the last line of Eq. (1.44),
we arrive with an intensity profile
I(x)Im ≈ I0 sin2(φ/2) exp
[− 1
2σ2
(x+ 2kσ2 cot(φ/2)
)2]. (1.45)
We can only read intensities with our detectors, and this profile will describe the
photon collection on the detector. The final intensity reveals an amplified beam
shift of 2kσ2 cot(φ/2) at the cost of I0 cos2(φ/2) photons. If we chose to measure
the momentum space, then our beam shift would be zero because the weak-value
is entirely imaginary, <{Awv} = 0.
1.3.2 Optical Beam Deflection: Real
In this section we expand on the weak-value theory and devise a similar experiment
using a purely real weak-value instead. To acquire a real weak-value, we rework the
ancillary system to one with polarization to measure a transverse beam deflection.
We use a Sagnac interferometer with a polarizing beam splitter (PBS) instead of a
21
k
5x5x
f
PBS
Split Detector
Pol 2Pol 1
Laser
Half-wave
Figure 1.3: We use a c.w. Gaussian beam exiting a spatial filter pinhole. We use a Sagnac
interferometer to measure transverse beam deflections. The piezoactuated polarizing BS im-
parts a momentum kick k on the vertically polarized light. We note all the photons will exit
the interferometer through the exit port of the detector. We postselect with a polarizer near
orthogonal to the input polarization. The photons that survive the polarisation postselection
process are then focused down with a lens with focusing length f . We monitor the beam shift
of the focused beam on a SD. The Pol 1 is used for polarization purity and the half-wave plate
is used to control the linear polarization of the input photons.
50:50 BS as in Fig. 1.3. We include a lens to project our system in the momentum
basis for the measurement. The last step is to perform the postselection with a
polarizer. In this scenario, we note that there is no spatial interference and only
a polarization interference.
We start with a transverse Guassian beam profile with 1/e2 beam radius σ as in
Eq. (1.41). The experimental schematic is shown in Fig. 1.3 which has a PBS and
the postselection is done by the polarizer Pol 2. We use a focusing lens to perform a
Fourier transform on the beam of light. Therefore, the measurement is performed
in the momentum space. The state before the postselection is given by Eq. (1.42)
but with an ancillary system operator A in the polarization basis and a different
22
preselected state |ϕi〉. The ancillary system operator that couples the transverse
momentum kick k to the position degree of freedom is in the polarization basis as
A = |H〉〈H| − |V 〉〈V |. Then our pre- and postselected states are given by |ϕi〉 =
1√2(|H〉+ |V 〉) and |ϕf〉 = 1√
2(cos θ − sin θ)|H〉 − (cos θ + sin θ)|V 〉), respectively.
We note the postselecting angle θ in the postselected state because of the near
orthogonal orientation of Pol 2 from the input polarization of |ψi〉 as depicted in
Fig. 1.3. Now we calculate the weak-value to be Awv = − cot θ from Eq. (1.34).
The real weak-value will modify the postselected state as follows:
|Ψ′〉 = 〈ϕf |ϕi〉∫e−ikx cot θ|x〉ψ(x)dx. (1.46)
This interaction produces a beam shift in the conjugate variable, which in this case
is a shift of k cot θ. We note the small interaction approximation, kσ cot θ � 1,
is used to expand the exponential and re-exponentiate after postselection. We
introduce a lens with focal length f as in Fig. 1.3 to project to the momentum
basis [66]. The postselected wave function and the the postselected state in the
position basis is given by
|Ψ′〉 = 〈ϕf |ϕi〉∫ei2π
xλfx′e−ikx
′ cot θ|x′〉ψ(x′)dx′, (1.47)
and
Ψ′(x)Re = 〈ϕf |ϕi〉 exp
[−1
4σ2f
(x+ f
k
k0cot(θ)
)2], (1.48)
respectively. The lens introduces a quadratic phase that results in a change of
basis. The parameters λ and σf are the wavelength of the photons and 1/e2 beam
radius of the beam in the focal plane, respectively. Then the intensity profile is
attained by squaring the state in the position basis. The intensity profile
I(x)Re = I0 sin2(θ) exp
[−1
2σ2f
(x+ f
k
k0cot(θ)
)2], (1.49)
is related to the probability distribution of the photons arriving on the detector.
The final intensity field shows an amplified beam shift fk cot(θ)/k0 at the cost
23
of I0 cos2(θ) photons. In this scenario, if we were to measure in the conjugate
domain of q instead of p, we would measure zero beam shift because the weak-
value ={Awv} = 0.
1.3.3 Inverse Weak-values
In this section we look closely at the weak-values formalism and reverse the weak-
value amplification approximation to come up with a different parameter estima-
tion technique. In the previous section, for maximum amplification or for the
largest weak-value we used the small angle approximation φ/2 � 1 along with
the small interaction approximation k2σ2 cot2(φ/2) � 1. We consolidate these
approximations to kσ � φ/2 � 1. The inverse weak-value technique starts with
Eq. (1.44), and then we apply the inverse approximation φ/2 � kσ � 1. This
changes the parameter of estimation from k to φ as in
Ψ′(x)Im ≈ sin(φ/2)e−kx cot(φ/2)e−x2
4σ2
≈ sin(φ/2) (1− kx cot(φ/2)) e−x2
4σ2
≈ cos(φ/2)|kσ|(
tan(φ/2)
kσ− x
σ
)e−
x2
4σ2 .
(1.50)
The intensity profile is produced with a slight misalignment k much larger than
φ/2, which will have a bi-modal distribution where the dark fringe shift is directly
proportional to the phase φ. Thus the intensity profile of the bi-modal distribution
is given by the norm of the postselected state as in
I(x)inv = cos2(φ/2)|kσ|2(
tan(φ/2)
kσ− x
σ
)2
e−x2
2σ2 . (1.51)
We then calculate the average shift of this bi-modal distribution and arrive with
〈x〉inv = I0 cos2(φ/2)|kσ|2φk. (1.52)
We note that the postselection is done through a misalignment of k.
24
In summary, these regimes offer different ways to estimate parameters addi-
tional to the standard metrology methods with strong projective measurements
without an ancillary system. Here, we present ways to measure momentum and
phase with weak-values and inverse weak-values, respectively. The standard meth-
ods would be using a focusing lens to measure a transverse momentum and using
a balanced homodyne technique for phase measurements.
1.4 Signal-to-Noise Ratio
The SNR, S, metric is one of the simplest to implement in the laboratory. S can
also be interpreted as the visibility of a signal. This metric is good to identify
where there is substantial noise in the system, so that measures can be taken such
as post-processing to reduce the noise in an experiment. This idea is fundamen-
tally different from the more rigorous metric of Fisher information which we will
discuss the next section 1.5.
We will start with the standard strong projective measurement without an
ancillary system such as a beam deflection experiment with a focusing lens. The
intensity profile of the beam on the detector has the form
Ist(x) = I0 exp
[−1
2σ2f
(x+ f
k
k0
)2]. (1.53)
The lens has focal length f and 1/e2 beam radius at the focus of σf . We are
interested in estimating the momentum kick k, and since the beam is focused, the
standard deviation of the beam gives the uncertainty of σf . We then define the
SNR S as in Eq. (1.32) to be given by
S =
√2
π
fk/k0σf
√N. (1.54)
25
From the Fourier transform and the lens, we know σf = f/2k0σ, and we rewrite
the SNR with parameters of the beam before the lens
S =
√2
π2kσ√N. (1.55)
We will come back to this result as comparison to the weak-value based techniques.
The SNR metric is a quick way to determine quality of a measurement. In
the cases where one manages to isolate the system from noise either by some
type of stabilization or by fast measurements, the S = 1 leads to the shot-noise
limit. From Eq. (1.55) we see that by increasing the intensity of the laser, we will
increase the SNR and hence be able to measure smaller parameters. The catch
to this metric is that the noise term (when measured) is the background of all
noise sources. All noise sources include everything from thermal fluctuations to
mechanical vibrations. This metric is intuitive for experiments; however, we move
on to a more fundamental way of understanding the shot-noise limit through the
Fisher information and the CRB metric in the next section.
1.5 The Cramer-Rao Bound and the Fisher In-
formation
In this section we review an important statistical inference technique that starts
with the maximum likelihood principle. The principle states that a likelihood
function will be large where a parameter of interest is located. Given the likelihood
function that is related to the probability distribution of measurements, we can
infer the parameter of interest with a given uncertainty. From the curvature of
the likelihood function we can determine the plausibility of the estimate [67].
We start with an observation xi with a probability density p(xi; g) for i ∈ [1, N ]
statistically independent measurements. The measurement comes from a proba-
26
bility density p(xi; g) that is a function of the parameter of interest g. The like-
lihood function of parameter g given the data sets of N statistically independent
measurements is given by
L(x; g) =N∏i=1
p(xi; g). (1.56)
The likelihood is a function of the parameter of interest g and is not necessarily
a probability density. The variable x is the set of data points {x1, ..., xN}. The
function L(x; g) is required to satisfy∫L(x; g)dx > 0. To locate the maximum
of the distribution it is mathematically useful to define the log-likelihood. Since
the logarithmic function is monotonic, it will not change the location of the maxi-
mum or the behavior around the maximum. The log-likelihood function gives the
relative likelihood of the parameter of interest, which in this case is g. We define
the log-likelihood function as
logL(x; g) =N∑i=1
log p(xi; g). (1.57)
Next, we define the score to be a measure of the sensitivity of the estimate
as a function of the parameter of interest. We assume that L(x; g) is twice dif-
ferentiable and the score is given as the first derivative of the likelihood function.
We also assume the data is statistically independent from a random sampling and
write the score function as
V (g; x) =N∑i=1
∂
∂glog p(xi; g) =
N∑i=1
V (g;xi) = NV (g;xi), (1.58)
where V (g;xi) is the score associated with the ith component of the data set x.
The likelihood function is maximum where the first derivative is zero so we take
the expected value of the score to be zero. We expected value of the score is given
27
by
E[V (g; x)] = NE[V (g;xi)]
= N
∫p(xi; g)
∂
∂glog p(xi; g)dxi
= N
∫p(xi; g)
1
p(xi; g)
∂
∂gp(xi; g)dxi
= N∂
∂g
∫p(xi; g)dxi = 0.
(1.59)
The next definition is an estimator. An estimator is a rule to determine a
parameter of interest g from a sample set of measurements xi for i ∈ [1, N ]. The
rule we will follow is the mean of the beam shift; that is, we will assume the
information to be in the mean value of the probability distribution.
The Fisher information is then given by the variance of the score. The variance
of the score is a measure of the curvature of the log-likelihood function. Since the
mean of the score is zero, the variance of the score is given by
I(g) = E[V 2(g; x)] = E
[(∂
∂glogL(x; g)
)2]
= −E[∂2
∂g2logL(x; g)
]. (1.60)
Proof of the equality in Eq. (1.60) for L(xi; g):
∂2
∂g2logL(xi; g) =
∂
∂g
(1
L(xi; g)
∂
∂gL(xi; g)
)=
1
Li(x; g)
∂2
∂g2L(xi; g)− 1
L2(xi; g)
(∂
∂gL(xi; g)
)2
,
E
[∂2
∂g2logL(xi; g)
]= 0− E
[(∂2
∂g2logL(xi; g)
)2]. �
(1.61)
From this formulation we can quantify how much information is available in
a system of a parameter of interest, given that we know the distribution of how
the photons arrive at our detector. We can also take one step further and come
to the Cramer-Roa bound (CRB). The CRB is equal to the inverse of the Fisher
information and gives the smallest possible variance using an unbiased estimator.
28
If the CRB is reached, the estimator is said to be efficient, and no other unbiased
estimator will produce a smaller variance. The CRB and Fisher information
relationship [67–69] is given by
1
I(g)= 〈(∆g)2〉CRB ≤ 〈(∆g)2〉. (1.62)
As a reminder, for all the experiments we consider only the efficient estimator
T (~x) = 〈x〉, (1.63)
where the operation is the expected value, or the mean value, from the data set
~x. The parameter x could also be replaced with t time.
Now we provide a proof of Cramer-Rao bound in Eq. (1.62) when L(x; g) =
p(x; g). First we assume the estimator is unbiased such as
E[T (x)] =
∫T (x)L(x; g)dx = ℵ(g), (1.64)
where the parameter of interest is given by g, and the unbiased estimator function
is ℵ(g). We differentiate both sides:∫T (x)
∂
∂gL(x; g)dx =
∂
∂gℵ(g). (1.65)
Note the equality of the product of V (g)L(x; g) from
V (g;x) =∂
∂glogL(x; g) =
1
L(x; g)
∂
∂gL(x; g), (1.66a)
∂
∂gL(x; g) = V (g;x)L(x; g). (1.66b)
We use this equality to substitute it back to Eq. (1.65) as in∫T (x)V (g;x)L(x; g)dx =
∂
∂gℵ(g), (1.67a)
E[T (x), V (g;x)] =∂
∂gℵ(g). (1.67b)
29
Then we note that if the expected value of the score is zero as in Eq. (1.59), then
the covariance between the estimator T (x) and the score V (g;x) is
Cov[T (x), V (g;x)] = E[T (x), V (g;x)]− E[T (x)]E[V (g;x)] = E[T (x), V (g;x)].
(1.68)
We use the fact that for any two random variables, the covariance ρ2 ≤ 1, and it
follows thatCov2[T (x), V (g;x)]
Var[T (x)] Var[V (g;x)]≤ 1, (1.69a)
and
Var[T (x)] ≥( ∂∂gℵ(g))2
I(g)=
1
I(g). � (1.69b)
For the last line we assume our case that ℵ(g) = g as the parameter of interest.
1.6 Summary
For the following experiments, we use optimal systems that will reach the CRB
in the noiseless case and in our cases reach the shot-noise limit for the parameter
of interest. Both the CRB and the shot-noise set the fundamental bound for
a measurement. Practically speaking, measuring the SNR is like measuring the
visibility of a signal in an experiment. To measure the Fisher information one
initially needs to be at the shot-noise limit, which makes the metric hard to use in
experiments. We note that when all noise is subdued, both the Fisher information
and the SNR will result in the ultimate bound of the smallest uncertainty bounded
by the shot-noise limit.
Whichever metric used or whichever way we make a measurement, the resource
sets the limit on the precision of the measurement. The CRB is more fundamental
than the SNR metric to attain the ultimate precision because the CRB is based
on the statistics of a measurement and on the properties of the resource used.
30
We will use the two metrics to quantify the quality of a measurement in the ideal
noiseless case and the realistic case where technical noise is present.
1.7 Thesis Outline
The work presented in this thesis is a subset of six years of Ph.D. research in the
John Howell Group of the University of Rochester. I present the experimental
results of the works of my contribution and elaborate on the utility that weak-
value-based techniques brings to metrology.
In this chapter, we review the historical background of the study of photons and
quantum optics. Then we review the basics of noise mitigation for experiments.
In particular, we highlight the colossal achievement of LIGO and the stabilization
work of that system. We review the coherent state as the resource and the SD
as the primary detector for the measurements. We also demonstrate how the
shot-noise limit is attained with a coherent state and a SD. We then introduce
weak values as a tool for metrology. We introduce the weak-value technique with
three examples: the imaginary, the real, and the inverse. Lastly, we discuss the
signal-to-noise metric and the Fisher information metric to evaluate the benefits
of a parameter estimation technique.
In chapter 2, we highlight the weak-value-based technique as an optimal and
efficient technique. We demonstrate a weak-value based technique with a time-
domain analysis to measure longitudinal velocities. The technique employs the
near-destructive interference of non-Fourier-limited pulses, one of which is Doppler
shifted due to a moving mirror in a Michelson interferometer. We present a veloc-
ity measurement of 400 fm/s and show our estimator to be efficient by reaching its
CRB. Since the weak-value technique reached the CRB and in our case reached
the shot-noise limit for velocity measurements, the technique is optimal. In the
second experiment, we measure a transverse momentum kick with a Sagnac inter-
31
ferometer by monitoring the dark and bright ports. We use two SDs to measure
the Fisher information from both the bright and the dark ports, and show that by
collecting only 1% of the photons in the system we measure 99% of the available
Fisher information. This result highlights the efficiency of monitoring the dark
port without any loss due to discarding data from the bright port.
In chapter 3, we introduce technical noise to a weak-value-based technique
to quantify the noise mitigation properties of the technique when comparing it
to standard methods. We measure small optical beam deflections both using a
Sagnac interferometer with a monitored dark port, the weak-value-based tech-
nique (WVT), and by focusing the entire beam to a SD, the standard technique
(ST). We introduce controlled external transverse detector modulations and trans-
verse beam deflection momentum modulations to quantify the mitigation of these
sources in the WVT versus the ST experiments. We also compare the naturally
occurring beam jitter in both techniques. We show how the WVT exploits the ge-
ometrical configuration of the system, and how in all cases the WVT outperforms
the ST by up to two orders of magnitude in precision for our parameters.
In chapter 4, we present concatenated postselections for weak-value amplifica-
tion. We explore the complementary amplification of postselecting on two degrees
of freedom to measure a beam deflection. We use a Sagnac interferometer to mea-
sure a beam deflection by monitoring the dark port. The first postselection is with
spatial interference and the second postselection is with polarization interference.
We show that when the first degree of freedom has a low contrast such as spatial
interference, adding a second degree of freedom with a larger contrast can pro-
vide an enhancement to the overall effective postselection angle. The optimized
region leads to a smaller postselection angle and a greater amplification. We also
include a theoretical study of the efficiency of the technique and show that the
loss of Fisher information is negligible under specific circumstances in the small
postselection angle limit.
32
In chapter 5, we conclude with some remarks and a summary of the results.
We also include suggestions for future work in the field.
In the Appendices, we present some theoretical calculations that were omit-
ted in the chapters. Appendix A contains a calculation of the efficiency of the
velocimetry article from Chapter 2. Appendix B is derived from the concate-
nated postselection for weak-value amplification experiment in chapter 4. The
Appendix B includes an explicit description of the Berry phase in the experi-
mental setup, the quantum description of the weak-value, and the all order in k
theory.
33
2 Weak-value Techniques:
Shot-Noise Limited and
Efficient
2.1 Introduction
In this chapter, we present a weak-value-based technique that measures the ve-
locity v of a mirror and saturates the shot-noise limit for a velocity measurement.
We demonstrate an interferometric scheme combined with a time-domain analy-
sis to measure longitudinal velocities. The technique employs the near-destructive
interference of non-Fourier limited pulses, one of which is Doppler-shifted due to
a moving mirror in a Michelson interferometer. We monitor the dark port of the
interferometer which amounts to 15% of the total available photons. We achieve
a velocity measurement of 400 fm/s and show our estimator to be efficient by
reaching its CRB [1]. Next, we present the efficiency of a weak-values-based tech-
nique by measuring a transverse beam deflection using a Sagnac interferometer.
We monitor the beam shift of the dark and bright output ports, and recover 99%
of the available Fisher information from the dark port with 1% of the photons
that entered the system [2]. In both experiments, we firstly present the theo-
retical predictions and then explain the experimental realization. Thus we show
34
how weak-values-based techniques are optimal strategies and CRB bounded for
metrology.
We call a technique optimal if there exists an estimator that can reach the
CRB of a parameter of interest. We call a technique suboptimal when there does
not exist an estimator reaching the CRB for a given parameter of interest. There
are claims that weak-values based techniques are suboptimal because of the loss
of resources in post-selection. However, it has been shown that in the weak-value
amplification limit, the technique can be arbitrarily close to optimal conditions
for reasonable experimental values.
This chapter is organized as follows. In Sec 2.2, we introduce the velocity ex-
periment. The theoretical description is explained, Sec. 2.2.1 and the experimental
description in Sec. 2.2.2. The results in the velocity experiment are presented in
Sec. 2.2.3 and a summary in Sec. 2.3. In Sec. 2.4 and Sec. 2.4.1, we introduce the
beam deflection experiment and present the theoretical description, respectively.
Then we present the experimental setup and results in Sec. 2.4.2 and Sec. 2.4.3,
respectively. In Sec. 2.5, we summarize the results of the chapter.
2.2 Measuring a Longitudinal Velocity
In information theory, the CRB [70] is the fundamental limit in the minimum
uncertainty for parameter estimation. Measurements of phase [40, 47, 71], beam
deflection [39, 72], pulse arrival time [73], Doppler shift [29, 74, 75] and veloc-
ity [76–79] are all fundamentally bounded by a CRB. If a measurement technique
reaches the CRB, its estimator is said to be efficient, and no other estimator will
produce a smaller variance.
An important aspect of the weak-values framework is that it provides a method-
ology for mitigating technical noise and amplifying an effect in one domain that
35
is technologically difficult to observe in the conjugate domain. We will address
several types of technical noise found in the laboratory in chapter 3.
In the work by Simon and Brunner [47], they show that an imaginary weak-
value-based technique allows an observer to see a large spectral shift caused by
a small temporal shift. They also show that the measurement of longitudinal
phase shifts with the weak-value-based technique outperforms that of standard
interferometry [47].
Here we consider the opposite regime where a small spectral shift causes a
large temporal shift. It is experimentally easier to redesign the experiment in
order to make measurements in the conjugate domain. The spectral shift in our
experiment is a Doppler frequency shift produced by a moving mirror. Using
established interferometry to measure velocities arriving at the CRB is difficult
but achievable as seen in Ref [70]. Our technique is comparable to standard
interferometry, but allows us to reach its CRB with a relatively simple method.
In this section, we show a weak-value technique to measure sub pm/s velocities.
The protocol reaches the predicted CRB in the low frequency regime.
2.2.1 Theory
The protocol shown in Fig. 2.1 uses a non-Fourier limited Gaussian pulse (i.e.,
cτ � coherence length of the laser, where τ is the length of the pulse). The pulse,
with initial electric field profile
Ein(t) = E0 exp(−t2/4τ 2
)1
0
, (2.1)
enters the Michelson interferometer through one of the input ports. Then the BS
splits the pulse into two, and travels to a slowly moving mirror with velocity v.
The interferometer is tuned slightly off destructive interference by an amount 2φ.
36
The interaction matrix Mv and the BS matrix B are written as
Mv =
ei(φ+k0vt) 0
0 e−i(φ+k0vt)
, (2.2)
and
B =1√2
0 i
i 0
, (2.3)
respectively. The output electric field profile is given by the product of the ma-
trices as in
Eout(t) = BMvBEin(t) = iE0 exp(−t2/4τ 2
)sin(φ+ k0vt)
cos(φ+ k0vt)
. (2.4)
The intensity profile is given by the modulus square of the electric field. We study
the dark port of the interferometer and ignore the bright port. We factor sin2(φ/2)
out of the bright port intensity profile as in
Iout(t) ∝ I0 exp(−t2/2τ 2
)sin2 φ |cos(k0vt) + cot(φ) sin(k0vt)|2 , (2.5)
where k0 = 2π/λ and λ is the center wavelength of the light. Assuming k0vτ �
φ, making a small angle approximation of φ and re-exponentiating the output
intensity, we obtain
Iout(t) ≈ I0 exp (−t2/2τ 2) sin2 φ |1 + cot(φ)k0vt|2 ,
≈ I0 sin2 φ exp
[− 1
2τ2
(t− 2kvτ2
φ
)2]. (2.6)
Near-destructive interference reduces the peak intensity of the pulse by a factor of
sin2 φ, which is the probability for a single photon passing through the interferom-
eter to reach the detector. Importantly, a time shift in the peak output intensity,
δt = 2kvτ 2/φ, has been induced with respect to the input. The velocity v can
be obtained from measurements of the time shift δt. The theory works because
cτ � [coherence length of the laser] � φ. In other words, there is a point by
point interference of the non-Fourier limited pulses.
37
780 nmDiode Laser
OpticalIsolator
AOModulator
Computer
Figure 2.1: An optical modulator generates a non-Fourier limited Gaussian shaped pulse.
We couple the pulse to a fiber and launch it to a Michelson interferometer where one mirror
is moving with constant speed v. The interference is controlled by inducing a phase offset 2φ
with the piezoactuated mirror. Photons exiting the interferometer are coupled into a fiber (not
shown) and the arrival time of single photons are measured with an APD and a photon counting
module.
We can rewrite the time shift in Eq. (2.6) in terms of the spectral shift
δt = 2kvτ 2/φ = 2πfdτ2/φ, where the spectral shift, fd = 2v/λ, of the pulse
is proportional to velocity v. Instead of a direct spectral measurement, we obtain
the velocity by measuring the induced time shift of the non-Fourier limited pulses.
The time shift is amplified in the measurement of v which is accompanied by a
decrease in the measured intensity. These two results are well-known properties of
the weak-value amplification technique. In our case, the use of non-Fourier limited
pulses allows us to produce large time shifts regardless of the laser linewidth.
We now consider the fundamental limitations of our velocity measurement set
38
by the CRB. The CRB is equal to the inverse of the Fisher information, which is
the amount of information a random variable (arrival time of photons) provides
about a parameter of interest (velocity). Assume that N photons are sent through
the interferometer. We want to determine the shift δt from the set of N sin2 φ
independent measurements of photon arrival times. Such measurements follow the
distribution P (t; δt) = (2πτ 2)−1/2
exp [−(t− δt)2/2τ 2]. The Fisher information is
I(δt) = N sin2 φ
∫dt P (t; δt)
[d
d δtlnP (t; δt)
]2≈ Nφ2
τ 2. (2.7)
The CRB, I−1, is the minimum variance [68, 69] of an unbiased estimation of
δt. For our experiment, the estimator is the expected value or the mean of the
probability distribution. The sensitivity in the determination of δt is therefore
bounded by ∆ (δt) ≥ τ/φ√N . The error in the estimation of v is then bounded
by
∆vCRB =∆ (δt) φ
2 k τ 2=
1
2kτ√N. (2.8)
Note that this minimum uncertainty is independent of the actual value of v mea-
sured. This also determines the smallest resolvable velocity, when the SNRo is
unity. The SNR is
S =δt
τφ√N =
v
∆v=
fd∆fd
. (2.9)
In this experiment, we focus on the velocity because the velocity of the mirror is
independent of the laser source, i.e. λ. This calculation is assuming the noiseless
case of the experiment.
2.2.2 Experiment
We use a grating feedback laser with λ ≈ 780 nm. An acoustic optical modulator
creates Gaussian pulses of length τ , which we couple into a fiber. We launch
them through the 50:50 BS of the interferometer. The piezoactuated mirror is
driven by a triangle function with frequency fm and peak-to-peak voltage Vpp.
39
The pulse length is smaller than half the oscillating mirror period, so that a pulse
experiences a single constant velocity. An opposite constant velocity is observed
for each sequential pulse because the sign depends on whether the mirror moves
toward or away from the BS (see Fig. 2.1). The piezoresponse α is calibrated by
varying the voltage to change the dark port to a bright port. The piezoresponse
was calibrated to be α ≈ 27 pm/mV for a low frequency-voltage product. The
arm lengths (BS-mirror distances) are approximately 1 mm (not including the BS
size) to ensure long term phase stability. Photon arrival times are recorded with
an avalanche photon diode (APD) and a photon counting module (PicoQuant
PicoHarp 300). The detector collects arrival times with 350 ps resolution.
To calibrate the experiment, we record the number of photons entering the
interferometer, N . Then, the piezodriven mirror is biased near destructive inter-
ference and fed a triangle signal. We calculate the mean and error of the arrival
time of the Nφ detected photons for each set of pulses. The mean of the Gaussian
determines the time shift δt from which the velocity is extracted, and the angle
φ ≈√Nφ/N is calculated. Lastly, to reach the CRB we attenuate the peak of
the pulses to about a million photons per second.
2.2.3 Results
We present velocity measurements v as a function of the pulse width τ for different
amplitudes on the moving mirror in Fig. 2.2. The lines are the theoretical predic-
tions, v = 2fmVppα, where 2fm = 1/6τ . The mirror voltages are Vpp = {105, 52.5,
26.25, 10.5} mV, and the angle is φ = 0.31 ± 0.02 rad. The results agree with
the theoretical predictions. The smallest measurement of velocity in Fig. 2.2 is
v = 60± 11 pm/s. The angle φ = 0.31 might seem large; however, comparing the
exact form in Eq (2.5), | sin(φ+ kvt)/ sin(φ)|, to the approximation | exp(kvt/φ)|
shows a discrepancy of less than 1% for the experimental parameters.
40
Figure 2.2: The velocity of the mirror v, is plotted as a function of τ = {1.67, 4.17, 16.7,
417 and 833} ms. The phase offset angle is φ = 0.31 radians. The points are the experimental
results and the lines are the theoretical predictions for different voltages. Signal-to-noise ratios
are 54, 27.4, 14.7, and 5.7 for V pp ={105, 52.5, 26.25, 10.5} mV respectively.
The uncertainties of the measurements in Fig. 2.2 are plotted separately in
Fig. 2.3 and compared to the CRB Eq. (2.8). The error matches the CRB, thus
the estimator is efficient, and no other estimator can produce smaller uncertain-
ties. This technique did not require noise filters or frequency locking to reach the
fundamental uncertainty in the mean arrival time of the photons. In addition, the
fluctuations in the post selection angle φ are negligible. Therefore, our velocity
measurement is fundamentally bounded only by its CRB. We note that Fig. 2.3
is a log-log plot with a linear behavior and a negative slope due to the 1/τ√N
shot-noise limit behavior.
It is important to note that our CRB is scaled by the maximum number of
detected photons N . The collection-detection efficiency is about 20% due to the
50:50 BS (not shown in Fig.2.1) located before the APD used for alignment of the
41
Figure 2.3: Experimental error in Fig. 2.2 as a function of τ√N . The solid line is the CRB
as in Eq. (2.8) for N ≈ 54 × 106 photons. Note that there are no error bars because these are
the errors from the data in Fig. 2.2. Shaded green region is the above CRB (or above shot-noise
limit) and the unshaded region is the better than shot-noise region which is forbidden for our
measurements.
dark port, the efficiencies of the APD and the fiber coupling. Our calculations do
not take the collection-detection efficiency into account.
The results show precise and accurate detection of velocity measurements in
the pm/s range. Results from Fig. 2.2 show that smaller velocities can be measured
with longer pulses.
Now we seek to achieve the smallest velocities without the concern of reaching
the CRB. Consider the temporal shift, advance or delay, of the pulse exiting
the interferometer. Since the peak of the pulse is sufficient to detect a shift, we
require a small region around the peak to determine the shift. This allows the use
of effectively large values of τ without requiring long term interferometric stability.
Since the pulses are non-Fourier transform limited, it is not necessary to use an
42
entire Gaussian pulse. We truncate the Gaussian pulse to a width of τ , that is
2fm = 1/τ . In other words, the light intensity into the interferometer never drops
below 88% of the peak intensity, and there is 12% peak to peak intensity variation
following the peak of the Gaussian profile.
Table 2.1: Results of the cut Gaussian profile with τ = 50 s and N ≈ 66×109. The collection-
detection efficiency is about 20%. The error is from the statistics of numerically fitting each
run. Integration time was about two hours worth of data.
Vpp [mV] φ [±0.002 rad] fd [µHz] v [pm/s]
2.0 0.275 3.6± 1.2 1.4± 0.5
1.0 0.276 1.6± 1.1 0.6± 0.4
0.5 0.279 1± 1 0.4± 0.4
We show velocities in the sub pm/s range using truncated pulses in Table 4.1.
The mirror frequency was set to 10 mHz, which corresponds to τ = 50 s, and data
was taken for voltages peak to peak, Vpp = {2, 1, 0.5} mV, for the piezo driving
the mirror. Data was collected in intervals of 10 minutes (due to drift instability in
intensity), and 13 sets of data were taken for each voltage. We did a Gaussian fit
for each 10 minute interval. The time shift and its error were found as the mean
and standard deviation, respectively, of the 13 time shifts obtained. The time
shift was in the tens of milliseconds and corresponds to small Doppler shifts in
the mircoHertz range. This leads to the best technical noise limited measurement
of v = (400 ± 400) fm/s. Nevertheless, both accuracy and precision are lost due
to numerically fitting the truncated distributions. Note that the measurements
are all relative velocities because of the oscillating mirror. In one period, there
are two pulses, each with opposite but equal speeds.
The results remain consistent with the full Gaussian picture theory in Eq. (2.6),
but not with the CRB theory in Eq. (2.8). Calculating the mean arrival time of
the photons is not a good estimator of the time shift because we lack the full
43
Gaussian pulse profile. Therefore, we numerically fit the data to an unnormalized
function A exp [−(t− δt)2/2τ 2], where the shift δt is extracted and the velocity,
v, is backed out.
2.3 Summary
In this experiment, we show that by using non-Fourier limited pulses and standard
interferometry inspired by weak values, sub pm/s velocities can be measured.
Using a Michelson interferometer tuned near a dark port, we measure velocities
as low as 400± 400 fm/s. The uncertainty of the phase measurement is negligible
when compared to the uncertainty of v for our parameter values, which is why our
measured uncertainty reaches the fundamental limit. Additionally, the error in
our measurement of v matches the predicted CRB, making this estimator efficient
and the ultimate limit in uncertainty for velocity measurements. We note that we
only measured the dark port with about 15% of the total available photons in the
system and saturated the CRB for velocity measurements.
There is a difficulty in understanding how discarding data can result in an
optimal measurement. Because of this there has been great criticism of weak-
values based techniques [42–44, 46]. These techniques are counterintuitive in that
even though we would gain more “information” by measuring the bright port of
the interferometer, but the small angle would render the effort fruitless. From
our results, measuring the dark port alone was enough to saturate the CRB for
velocity measurements with a small angle. Thus there is negligible benefit from
observing the bright port.
To address the criticism of monitoring only the dark port we use the Fisher
information formalism and measure the relative available Fisher information out
of the two ports of a weak-value experiment. In this next section, we measure
both the dark and bright ports of a weak-value experiment to demonstrate the
44
efficiency of a weak-values based technique in the small angle regime.
2.4 Measuring a Transverse Momentum Kick
In this section, we present an imaginary weak-value technique to measure a beam
deflection. Using the mean as our estimator, we collect the Fisher information of
the transverse momentum kick k of both bright and dark ports. We show that
observing the bright port of the interferometer provides no Fisher information
because of the deamplification of the beam shift in bright port. Thus there is no
benefit in spending resources to measure these un-informative photons.
It is counterintuitive that the use of a tiny subset of resources can produce a
measurement as precise as with the use of all available resources. We challenged
this intuition through the following experimental demonstration. We present the
weak-value amplified beam shift of the dark-port and the deamplified beam shift
of the bright port. We monitor the beam shift of both dark and bright ports
on separate SDs. By measuring the percentage of Fisher information in each
respective port, we can see that virtually no information is lost by performing
the post-selection. By analogy, we can do the same procedure with the velocity
experiment to arrive with similar results (see App. A).
2.4.1 Theory
We present the theory of the beam deflection with imaginary weak-values from
the introduction. We use classical matrix algebra as in Ref. [72]. We start with a
continuous wave transverse Gaussian profile out of a fiber launcher and collimated
with an objective. The transverse electric field is
Ein(x) = E0e− x2
4σ2
1
0
. (2.10)
45
The beam radius σ is defined where the intensity field falls to 1/e2 of the maximum
intensity. The input intensity field is defined to be the square modulus of the
electric field of Eq. (2.10). The electric field enters the Sagnac interferometer
through the piezoactuated 50:50 BS. The BS matrix is defined as
B =1√2
1 i
i 1
, (2.11)
where the reflected port receives a phase of i.
The BS imparts a momentum kick k and a phase mismatch between the two
paths. The momentum kick k and the phase are imparted on the reflected port
according to Fig. 2.4. Since we measure relative differences, we can write as if
each path gets half the momentum kick k and half of the added phase. We write
the interaction matrix as
Mk =
ei(kx+φ/2) 0
0 e−i(kx+φ/2)
, (2.12)
where k is the momentum kick and φ/2 is the phase imparted by the vertical
misalignment of the 50:50 BS. We perform the matrix multiplication to arrive
with exit port electric field, one directed to Split Detector 1 and the other to Split
Detector 2 as in Fig. 2.4. Ignoring the global phase of the output electric field
from the matrix multiplication gives
Eout(x) = BMkBEin(x) = E0e− x2
4σ2
sin(kx+ φ/2)
cos(kx+ φ/2)
. (2.13)
As of now, there is no weak-value amplification because no approximations
have been made. We will first expand the trigonometric functions and expand the
first order k, that is, sin(kx) ≈ kx and cos(kx) ≈ 1 as in
Eout(x) ≈ E0e− x2
4σ2
sin(φ/2) (1 + kx cot(φ/2))
cos(φ/2) (1− kx tan(φ/2))
. (2.14)
46
PBS
k
PBS
PBS
5x
780nmDiode Laser
Split Detector 1
Split Detector 2
50:50
50:50
PBS
Figure 2.4: We use a c.w. Gaussian beam exiting a fiber. We monitor both the dark port
and the bright port to determine a beam deflection using Split Detector 1 and 2 (SD1 and
2), respectively. The WVT uses a Sagnac interferometer with a piezoactuated 50:50 BS that
imparts a momentum kick k, which we determine from the beam shift on Split Detector 1 and
2. PBS are in orange and 50:50 BS are in blue. The PBS work as mirrors given that the light is
vertically polarized. The phase control between paths is controlled by a vertical misalignment
of the piezoactuated 50:50 BS.
Then we assume the weak-interaction approximation k2σ2 cot2(φ/2) � 1 to re-
exponentiate and complete the square. Then the final step is to square the electric
field profile to arrive with the intensity profile as shown through
Eout(x) ≈ E0 exp
[− x2
4σ2
] sin(φ/2) exp [kx cot(φ/2)]
cos(φ/2) exp [−kx tan(φ/2)]
, (2.15a)
47
and
Iout(x) ≈ I0
sin2(φ/2) exp[−12σ2 (x− 2kσ2 cot(φ/2))
2]
cos2(φ/2) exp[−12σ2 (x+ 2kσ2 tan(φ/2))
2] . (2.15b)
The intensity profile shows that the dark port receives a beam shift amplification
of cot(φ/2) due to the weak-value, and the bright port receives a beam shift
deamplification of tan(φ/2).
The intensity profiles can be written to describe the probability of a photon
arrival on the SD. Using the probability density functions for the dark and bright
ports, we can write the likelihood functions. The Fisher information is then given
by
I(k) = −NE[∂2
∂g2logL1(g;x)
]= N
∫dxP (x; k)
[∂
∂klnP (x; k)
]2, (2.16)
where N is the number of independent events. It follows that the Fisher informa-
tion for the momentum kick k for the dark port (D) and bright port (B) is given
by
ID(k) = 4Nσ2 cos2(φ/2), (2.17a)
and
IB(k) = 4Nσ2 sin2(φ/2), (2.17b)
respectively. Measuring the Fisher information directly can be difficult, so we
measure the relative amounts of Fisher information based on the probability dis-
tribution of each port. Hence, the total amount of Fisher information is the sum
of Fisher information from both ports, 4Nσ2. This leads to the curves of the per-
centage of Fisher information in the dark port and bright ports being cos2(φ/2)
and sin2(φ/2), respectively.
2.4.2 Experiment
We use a grating feedback laser with λ ≈ 780 nm coupled into a polarization-
maintaining single mode fiber. The Gaussian mode exits the fiber, reflects through
48
a PBS for polarization purity, and reflects off a mirror that sends the beam
into the Sagnac interferometer (see Fig. 2.4). The beam propagates through the
piezomounted 50:50 BS and reflects off the three PBS acting as mirrors for the
vertically polarized input light. The beam recombines back in the piezomounted
50:50 BS and exits through the dark and bright ports. The photons exiting the
dark port are sent to SD1. To collect the bright port photons, we add an extra
50:50 BS before the interferometer to direct them to SD2 as in Fig. 2.4.
We calibrated the piezoresponse independently by reflecting the beam from
the actuated device to the Split Detector 1. The piezoresponse of the actuated
50:50 BS was calibrated to be α1 ≈ 68.6 pm/mV.
2.4.3 Results
Here, we study the efficiency of the estimator by using the Fisher information. The
data collected was at best a factor of 7 away from the CRB for momentum kick
k. The time bin for one measurement was 8 µs which is limited by the detector
bandwidth. To extract the Fisher information behavior predicted in Eqs. (2.17a),
we collected the photons from both bright and dark ports. As pointed out in
Refs. [28, 42, 43], the bright port in general also contains information about the
parameter. Here we used a 7 Hz sine wave to drive the momentum kick k and
varied the postselection angle φ. Then, we measured momentum kick k and the
deviation, ∆k, from both the dark and bright ports with Split Detector 1 and
Split Detector 2, respectively (see Fig. 2.4). Averaging the Fourier transform of
the signal allowed us to extract the SNR, S for the relative Fisher information
measurement.
In this scenario, we assume there is no noise. This, however, is an idealization,
so we briefly discuss the detector noise. The detector noise is white-Gaussian-
power-dependent electronic noise, denoted it as J. By taking Fourier transform
49
of the signal we spread out the noise in the frequency domain. We only extract
the frequency component of the signal k needed for the measurement. For further
details of the noise J refer to Chapter 3.
We acquire data from the Fourier transform of the signal and note that the
procedure is only affected by the component of J of the same frequency as the
signal k. For this uncorrelated temporal Gaussian noise, the Fisher information
is related to the SNR as
S2 =
(k
∆kB
)2
= k2I(k). (2.18)
Since both bright and dark ports are measuring the same k, we arrive at the
percentage of Fisher information from each port given the total Fisher information
available,
I%D,B =S2D,B
S2D + S2
B
× 100% =ID,B
ID + IB× 100%. (2.19)
0.3 0.4 0.5 0.6 0.7 0.8 0.90
20
40
60
80
100
Post−selection Angle: φ [rad]
Fish
er In
fo. [
%] Bright Port
sin2(φ/2)
Dark Port
cos2(φ/2)
95% conf bound
Figure 2.5: Fisher information vs post-selection angle φ. Data is taken from the Fourier
transform and averaged over equal numbers of samples. Angle φ ranges from 0.22 to 0.9 rad.
The confidence interval is 95%, and we see the fit break down as φ becomes large. Most of the
information is found to be in the dark port even for large φ. Both dark and bright ports follow
cos2(φ/2) and sin2(φ/2) behavior, respectively, as in Eqs. (2.17).
50
We calculate the percentage of Fisher information by inputting Eqs. (2.17) into
Eq. (2.19). We define I%D,B as the percentage of Fisher information in dark (D) or
bright (B) ports given by,
I%D(k) = cos2(φ/2)× 100%, (2.20a)
and
I%B(k) = sin2(φ/2)× 100%, (2.20b)
respectively.
In Fig. 2.5, the percentages of Fisher information from each port are shown
as a function of postselection angle. We observe the corresponding behavior of
the weak-value regime of Eqs. (2.17) and note that most of the information is
recovered from the dark port with a small post-selection angle φ/2. We fit the
data with about 100 points for both dark and bright ports. The Fisher information
is a near-perfect match to the theoretical prediction in Eq. (2.20). The nonlinear
fit gives a goodness measure r2 = 0.99, and the red lines are the 95% confidence
interval bounds (2σerror). Note that the results deviate from the approximation as
φ increases out of the weak interaction approximation of Eq. (2.15b). In addition,
we find 99 ± 2% of the Fisher information in the dark port and 1 ± 2% of the
Fisher information in the bright port for a postselection angle of φ ≈ 0.22 rad
(1% of the photons). Even though we only measure 1% of the photons, we extract
99% of the Fisher information. From the results we conclude that weak-value
amplification with strong postselection (dark port) extracts almost 100% of the
Fisher information about the momentum kick k, while the Fisher information in
weak-value amplification with failed post-selection (bright port) is negligible for
practical purposes. As predicted in Eqs. (2.17a) [28], the weak-value amplification
technique provides an efficient estimation for this experiment. We note that using
an estimator that also incorporates the bright port will make this technique even
better, albeit only slightly.
51
Although we have extracted 99% of the Fisher information from 1% of the
photons, we wish to stress that this is in no way a limit on the efficiency of the
technique, but rather a proof-of-principle result. We can quantify this point in
the following manner: suppose we wish to demonstrate the efficiency of the weak-
value estimator explored in this chapter to be some fixed fraction of the total
Fisher information, 1− ε, where ε is a small but finite number. This is equivalent
to showing I%D > 1− ε. We can demonstrate the efficiency of the technique to this
level by fixing the post-selection angle to be
φ/2 <√ε, (2.21)
where we recall the fractional Fisher information Eqs. (2.19) and Eqs. (2.17) in
this experiment [28, 51]. This assumes that the condition k2σ2 cot(φ/2) � 1
(controlling the weakness of the interaction) is suitably reduced as well while still
measuring a sufficiently large number of photons. Amazingly, Eq. (2.21) indicates
that the technique increases in efficiency as the number of photons measured in the
dark port decreases. Since ε can be made small, we conclude that the technique
can be made as efficient as desired. The important practical limitations are the
fidelity of the optics, getting a good dark port, and any other deviations from the
theory.
2.5 Summary
From the results presented in Fig. 2.5, no Fisher information about the parameter
of interest k is lost in the small angle approximation. This indicates that the
fewer the photons we measure, the greater the precision we will extract due to
the weak-value technique. Since the data was taken near the shot-noise limit,
we can conclude that there is strong evidence showing that if we were to send
single photons, the results would be the same. In an article by Jordan et al. [28]
52
the theoretical predictions using the classical Fisher information are the same to
those using the quantum Fisher information. From a metrological perspective,
the precision of N independent measurements is recovered from only a fraction
of measurements. With 1% of the photons we recovered 99% of the available
Fisher information of k in the system through the dark port. Even from the
classical perspective as we throw away 99% of the intensity in the WVT, we
recover the precision associated with using all the available measurements as in
the ST. Thus the WVT in a sense imparts all the available Fisher information to
the few surviving postselected photons.
53
3 The Technical Advantage of
Weak-value Based Metrology
3.1 Introduction
Weak-value amplification is a metrological technique intended to precisely mea-
sure many small parameters such as the spin Hall effect of light [36] and optical
beam deflections [39], velocities [1], and many others [37, 38, 40, 41, 54, 55, 80].
These tiny parameters are measured on table top experiments with low-powered
lasers. The ability to perform these measurements precisely naturally leads to the
question of how technical noise behaves in the protocol. Weak-value amplification
has also been shown by the Steinberg group in Ref. [57] to improve the SNR rela-
tive to the non-post-selected case in the presence of additive correlated technical
noise.
Weak-value-based techniques has been proven very useful for parameter esti-
mation and the articles outlined above demonstrate the utility of the technique.
However, there is a large number of theoretical articles claiming that weak-value
amplification shows no advantages in comparison with techniques that use all the
photons when optimal statistical estimators are used [42–46]. Hence we experi-
mentally quantify the weak-value advantage under technical noise with the Fisher
information formalism.
54
Quantum mechanically, the weak-value protocol consists of a weak interaction
that couples a system and a meter, separated in time by nearly orthogonal pre- and
post-selected states of the system [27]. We can define our pre- and postselected
state as |ϕi〉 and |ϕf〉. In this technique, the parameter of interest k controls the
weakness of the coupling between the system and meter through the interaction
eikAx, where A is the system observable. The weak interaction with the almost
orthogonal postselection leads to the weak-value defined as
Awv =〈ϕf |A|ϕi〉〈ϕf |ϕi〉
. (3.1)
The weak-value allows for values outside the range of eigenvalues as experimentally
demonstrated for the first time in 1991 by Ritchie et al. [81]. As such, a small shift
in the value of the parameter corresponds to a large shift in the meter. Then we
measure the meter either in the position basis x for observation of the imaginary
part of the weak-value, or in the momentum basis p for the real part of the weak-
value [62]. Both parts of the weak-value have been extensively studied in both
experiments and theory [82]; we will discuss only the imaginary weak-value in this
chapter.
A well-designed weak-value experiment concentrates almost all available in-
formation about the parameter of interest into the small fraction of events that
survive the post-selection process [28, 83–85], except for a negligibly small amount
that can in principle be extracted from the non-post-selected events. Existing ex-
periments of this kind also have a wave optics interpretation so long as we focus
on intensities and not photon counts [72].
In our experiment, a momentum shift in the interaction results in a transverse
beam displacement. Similarly, in the standard technique, after the momentum
kick, a focusing lens effects a Fourier transform on the beam which results in a
transverse beam displacement.
In making the comparison between the weak-value technique (WVT) and the
55
standard technique (ST), we pay special attention to the statistical estimators
used. For the ST, we use an estimator that can achieve the lowest possible variance
for unbiased estimators. Figure 3.1 contains diagrams of the experiments carried
out. We begin with a Sagnac interferometer to measure a beam deflection as
in [26, 39] and add two external modulating sources, which simulate noise sources
at a given frequency: a transverse momentum modulation, q, and a transverse
detector modulation, d (see Fig. 3.1). We define a measure of sensitivity of the
experiment to these modulations to be the ratio, R, of the signal to the external
modulation amplitudes. Using single-frequency external modulations, we show
that the WVT performs as well as or better than the ST, and the amount of
advantage is governed by the geometry and choice of parameters of the experiment.
We note that the Signal-to-External-Modulation ratio should not be confused
with the Signal-to-Noise ratio. We make the point of creating a new variable
for the metric because our simulated external source is deterministic and does
not have a variance but rather a fixed amplitude root-mean-squared value. This
is justified because a true Gaussian white noise source can be decomposed into
frequency components. Here we study a single tone frequency component of a
Gaussian white noise source. The behavior of a superposition of different frequen-
cies with random amplitude and phase should be similar to that of a Gaussian
white noise source. We show then that modulations and noise sources outside
the interferometer of the WVT are un-amplified and thus suppressed compared to
the signal, while the ST responds similarly to all modulations and noise sources.
In the experiments we perform, all modulating sources are independent of the
parameter of interest. This holds true even for naturally occurring laser-beam-
jitter noise. In demonstrating these effects, we report the following results: (i) the
ratio R of the WVT indicates that the transverse momentum, q, and transverse
detection, d, modulations are suppressed over the ST; (ii) when comparing the
deviation in measurements of transverse momentum k to the smallest predicted
56
Figure 3.1: We use a c.w. Gaussian beam exiting a fiber. We compare the different experiments
WVT (upper box) and the ST (lower box) to determine a beam deflection using split detector
1 (SD1). The WVT uses a Sagnac interferometer, and the ST focuses the deflected beam with
focal length f . The piezoactuated 50:50 BS imparts a momentum kick k, which we determine
from the beam shift on split detector 1. The two external modulations are labeled as q and d.
The d refers to the transverse detector modulation and q refers to the transverse momentum
modulation. PBS are in orange and 50:50 BS are in blue. The PBS work as mirrors given that
the light is vertically polarized. The split detector 2 (SD2) is used to collect the bright port
beam shift from the WVT.
error, the WVT offers improvement for both transverse momentum and transverse
detection modulations over the ST; (iii) the WVT suppresses naturally occurring
laser-beam-jitter noise over the ST. For all our results, we use the same acquisition
time for both the WVT and the ST.
57
This chapter is organized in the following sections. In Sec. 3.2, we review
the theory of beam deflection metrology based on the WVT and the ST. We
also review the concepts of Fisher information and the CRB applied to these
experiments. In Sec. 3.3, we describe the experimental setups. In Sec. 3.4, we
present a comparison of the WVT and the ST based on the accuracy and deviation
of beam deflection measurements. In Sec. 3.5, we compare the WVT and the ST
with naturally occurring intrinsic laser beam jitter. Lastly, in Sec. 3.6, we present
the conclusions we draw from the results.
3.2 Theory
We consider the experimental setup shown in Fig. 3.1, hereafter referred to as
the WVT (upper box) or the ST (lower box). The WVT theoretical description
is described in Chater 2 with a difference in sign of the signal of interest k. In
the weak-values protocol, a Gaussian beam of radius σ with initial electric field
transverse profile,
Ein(x) = E0 exp
[− x2
4σ2
]1
0
, (3.2)
is sent through a Sagnac interferometer. The beam enters the interferometer
through a piezoactuated 50:50 BS, which imparts a momentum kick, k, and phase,
φ, to the reflected beam. The BS matrix is given in Eq. (2.11), and the interaction
matrix is given by
M =
ei(−kx+φ/2) 0
0 e−i(−kx+φ/2)
. (3.3)
The phase, φ, is given by a constant deflection in the vertical y axis. This slight
misalignment is enough to provide control over the lengths between clockwise
and counterclockwise propagating beams. The beams recombine and interfere
back at the BS. The recombination of the beams entangles the which-path degree
58
of freedom to the position-momentum degree of freedom [39]. The final output
electric field is then given by the matrix multiplication
Eout(x) = BMBEin(x). (3.4)
Then the two output fields exit the BS as in Howell et al. [72] is given as Eq. (2.13).1.
We assume the momentum kick k is small for the weak interaction approximation,
k2σ2 cot2(φ/2)� 1. We expand the trigonometric functions up to first order in k
as in Eq. (2.14)2. After expanding, we re-exponentiate and complete the square
to arrive with the dark and bright port beam shifts as in
Eout(x) ≈ E0
sin(φ/2) exp[−14σ2 (x+ 2σ2k cot(φ/2))
2]
cos(φ/2) exp[−14σ2 (x− 2σ2k tan(φ/2))
2] . (3.5)
The weak interaction approximation is sufficient for both dark port and bright
port as the bright port requires a weaker constraint of k2σ2 tan2(φ/2)� 1. Then,
the intensity profile takes the form:
Iwvout(x) ≈ I0
sin2(φ2) exp [−(x+ δd)
2/2σ2]
cos2(φ2) exp [−(x− δb)2/2σ2]
, (3.6)
where the dark and bright port shifts are given by δd = 2σ2k cot(φ/2) and δb =
2σ2k tan(φ/2) respectively. The superscripts wv and st refer to the WVT and the
ST respectively. The actual weak-value in the quantum description can be found
in Chapter 1. We keep the approximation sign, but the equation is accurate in the
small angle approximation. The small angle approximation places this technique
in the weak-value amplification regime, which has been shown to be efficient and
optimal for metrology [1, 26]. The weak-value amplification regime, φ/2 → 0, is
where we conduct all of our measurements.
In the ST protocol, we consider a lens in order to optimize this technique for
deflection measurements. As shown in Fig. 3.1, a lens with focal length f focuses
1The value k is replaced with −k.2The value k is replaced with −k.
59
the beam on Split Detector 1. The SD then measures the transverse displacement
fk/k0. The intensity profile of the ST is written as
Istout(x) = I0 exp
[−1
2σ2f
(x− f k
k0
)2], (3.7)
where the beam radius at the focus is σf = f/2k0σ, and k0 = 2π/λ is the wave
number defined by the center wavelength of the laser λ. For further detail on the
lens, refer back to Chapter 1.
Table 3.1: A summary of the detection of different signals according to the WVT and the ST
following the theory described in Eq. (3.6) or Eq. (3.7), respectively. The three different signals
include k, d, and q are the momentum kick of interest, the transverse detector modulation and
the momentum kick from transverse momentum modulation, respectively. The beam shift is
given by δx, and the distance from the external modulating mirror, q, to the detector SD1 is
given by L.
Sources Weak-values tech. Standard tech.
k δxk = 2kσ2 cot(φ/2) δxk = fk/k0
d δxd = d δxd = d
q δxq = Lq/k0 δxq = fq/k0
For both techniques, we use single-frequency external modulations of two con-
jugate domains of our experiments: a deflecting mirror with transverse momentum
modulation q, and the SD1 on a stage with a transverse detector modulation d.
Since Gaussian white noise can be modeled by randomly changing the size of
the modulation, one can add each Fourier component independently and expect
similar results to Ref. [28].
We now compare the WVT and the ST when measuring a momentum kick k
in the presence of external modulations. The estimator we use for both techniques
is the sample mean
T : ~x→ 〈x〉. (3.8)
60
The sample mean rule refers to a data set, ~x, that has a probability distribution
where 〈x〉 is the expected value or mean value of the data set. We also assume
this estimator is unbiased.
We quantify the size of the signal in comparison to the background modulation
with ratio R. The ratio R is the beam shift at the detector δx due to the signal
k, divided by the modulation q or d (values from Tab. 4.1), Rq,d = δxk/δxq,d. R is
not to be confused with the SNR, S, because the external modulations are single
toned sources that do not have a variance but a fixed amplitude root-mean-squared
value. For the two modulations, we find,
Rwvd =
δxwvkδxwvd
=2k0σ
2
fcot(φ/2)Rst
d , (3.9a)
Rwvq =
δxwvkδxwvq
=2k0σ
2
Lcot(φ/2)Rst
q , (3.9b)
where the superscripts wv and st refer to their respective techniques. From
Eqs. (3.9), Rst � Rwv holds true for reasonable values of σ, L, f , and φ. We
will show this explicitly in Sec. 3.4. We note that the analysis here uses the dark
port of the WVT in Eq. (3.6).
We also compare the WVT to the ST using the Fisher information [68, 69].
Knowing the transverse probability distribution in the presence of random fluctu-
ations that arrive on the SD1 allows us to calculate the Fisher information, I(k),
with respect to the momentum kick k. The CRB sets the minimum possible sta-
tistical variance using unbiased estimators, I−1. (For more on the theory of Fisher
information see Ref. [67–69] or refer back to Chapter 1). The Fisher information
can be written as
I(k) =
∫dxP (x; k)
[∂
∂klnP (x; k)
]2, (3.10)
61
where P (x; k) is of the normalized form of Eqs. (3.6) or (3.7). P (x; k) is the
probability distribution of a photon arriving on the detector with transverse mo-
mentum k. The Fisher information assumes discrete events, even though the light
intensity was derived in Eq (3.6) or (3.7). With Eq. (3.10), we arrive at the
Fisher information with respect to the momentum kick k and number of photons
N (independent trials) for our two techniques:
IwvD (k) = 4Nσ2 cos2(φ/2), (3.11a)
IwvB (k) = 4Nσ2 sin2(φ/2), (3.11b)
Ist(k) = 4Nσ2, (3.12)
where in Eqs. (3.11) the Fisher information, I, of the dark and bright ports of the
WVT are denoted by the subscripts D and B, respectively.
We omit the factor 2/π from the Fisher information that is derived in chap-
ter 1.2.1 in the section on the SD. In that section, we derive this coefficient for
the two pixel split detector. For most of the analysis, this constant factor drops
out, but we include it for the CRB calculations or when needed.
The two Fisher informations for the WVT arise because of the two exit ports
of the BS as in Eq. (3.6). Adding the Fisher information from each port leads us
to the total Fisher information found in the ST. We note that both the ST and
the WVT transform deflections into displacements in conjugate bases with the
Fisher information proportional to the beam radius σ before the transformation.
In addition, note that Eq. (3.12) can also be found from the quantum Fisher
information [44, 85] derived from the transverse wave-function, giving the same
result. Lastly, we note that the Fisher information results in Eqs. (3.11) are only
valid for the weak-interaction approximation, k2σ2 cot2(φ/2)� 1.
62
3.3 Experiment
We use a grating feedback laser with λ ≈ 780 nm coupled into a polarization-
maintaining single mode fiber. The Gaussian mode exits the fiber, reflects through
a PBS for polarization purity, and reflects off a piezoactuated mirror q (see
Fig. 3.1).
In the WVT, the beam propagates through the piezomounted 50:50 BS and
enters a Sagnac interferometer of three PBS acting as mirrors for the vertically
polarized light. The beam recombines back in the piezomounted 50:50 BS and
exits through the dark and bright ports of the interferometer. The photons exiting
the dark port are sent to SD1 on a piezoactuated translation stage. To collect the
bright port photons, we add an extra 50:50 BS before the interferometer to direct
them to SD2 as in Fig. 3.1.
For the ST, the Gaussian beam is reflected from the 50:50 BS. The beam is
then focused onto the SD1 on a piezoactuated translation stage as in Fig. 3.1.
The power is adjusted so that the detector will not saturate due to the focused
beam.
We calibrated the piezoresponses independently by reflecting the beam from
the actuated devices to the SD1 by focusing it with a lens of focal length f .
The piezoresponses of the piezoactuated 50:50 BS, the piezoactuated mirror, and
the piezoactuated translation stage were calibrated to be α1 ≈ 68.6 pm/mV,
α2 ≈ 31.6 pm/mV, and α3 ≈ 75.8 pm/mV, respectively. The piezocalibrations
differ because of different materials and loads3. All the calibration was done before
the experiment using sinusoidal waves with fixed frequencies.
3The different loads refer to the different moments of inertia that each piezoactuator expe-
rienced due to different mounts. The piezoactuated translation stage had a different restoring
force than the piezoactuated mirror because of the different spring coefficients. The mechanics
of the modulations can be written out, but we ignored them, as it does not pertain to the
experiment, and calibrated the piezoactuated responses independently.
63
0 50 100 150 200
−100
−80
−60
−40
−20
0
Frequency [Hz]
20 lo
g(V
pp/V
tota
l) [d
BV
] Weak−valuesStandardMomentum
Modulation
Detector Modulation
Signal
Figure 3.2: A dBV spectrum comparison of the WVT (blue) and ST (green) with both external
modulations, where dBV= 20 log10(V/Vtotal) is plotted as a function of frequency. The peak-to-
peak signals are normalized to the detected power, Vtotal, in their respective experiment, either
the WVT or the ST. We see the “Signal,” have a deflection corresponding to an angle of 48
nrad peak-to-peak at 7 Hz. The external transverse “Momentum Modulation” corresponds to
an angle modulation of 2.5 µrad peak-to-peak at 56 Hz. The second external modulation is the
“Detector Modulation” which corresponds to a displacement of 230 nm peak-to-peak at 28 Hz.
We note that the suppression of the external modulations from the signal, Vpp, collected from
SD1 are not direct deflection measurements (see Eqs. (3.13)).
3.4 Results: Comparison of WVT and ST
First, we show that modulating sources external to a weak values amplifying
system are not amplified and can thus be suppressed. It is important to note that
in the WVT, the modulations external to the interferometer arrive at the detector
without amplification and with a reduced number of photons. On the other hand,
the ST uses a lens to focus the beam with every modulation (external and the
source) to the detector with all the photons.
We now discuss measurements of k in the presence of external modulations.
For the WVT, we have a post-selection angle of φ ≈ 0.38 rad and a distance from
modulating mirror to SD1 of L ≈ 34 cm. The beam size is a constant σ = 1.075
mm out of the fiber. For the WVT, the input power is Pwvin ≈ 1.45 mW. In the
64
ST, we use a focusing lens of f = 1 m and an input power of P stin ≈ 400 µW.
The power is lower for the ST to avoid saturating the detector. The reduction of
power is accounted for by comparing the deviation to the respective lower bound
such that the resulting ratio is independent of the total power. Because of this,
we see that the WVT allows the use of more input power without saturating the
detector and avoids a nonlinear response from the detector. In this comparison,
we also require equal acquisition time for both techniques. Here we normalize
each technique to its respective CRB for a fair comparison.
Figure 3.2 shows the average Fourier transform of the signal measured by the
SD1. We normalize the WVT and the ST Fourier transforms by dividing by
Vtotal, the total voltage corresponding to the power of all detected photons in the
respective technique. The figure can be interpreted as the visibility of the signal
from each respective technique. The voltage Vpp is the raw signal of the detector
read by the oscilloscope. The signal from the SD1 on the oscilloscope is given by
Vpp/Vtotal = δx/2σαcal, where αcal is a calibration constant of the detector and δx
is the beam displacement. We note here that the units of the Fourier transform
are such that 20 dBV is a factor of 10 in volts i.e. dBV= 20 log10 V/Vtotal. Here
we have three single toned modulations at three different frequencies in a side-
by-side comparison of the weak-value amplification. The signal of interest is the
beam deflection labeled as “Signal,” corresponding to an angle of 48 nrad peak-
to-peak at 7 Hz. The transverse “Momentum Modulation” corresponds to an
angle of 2.5 µrad peak-to-peak at 56 Hz. The transverse “Detector Modulation”
corresponds to a displacement of 230 nm peak-to-peak at 28 Hz. The transverse
“Momentum Modulation” is a piezoactuated mirror before the momentum signal
k. The “Detector Modulation” is the SD1 on a piezoactuated translation stage.
The green line is the ST with the higher harmonics of the external sources. The
blue line shows the WVT with signal higher than in the ST because of the weak-
value amplification.
65
0.02 0.04 0.06 0.08 0.11
4
10
40
RSTq
RWVq
Momentum Modulation slope = 258±8
(b)
0.1 0.2 0.3 0.6 1 2
10
20
40
100
RSTd
RWVd
Detector Modulationslope =51±2
(a)
Figure 3.3: A log-log plot of the ratio of the signal voltage to external modulation of the WVT,
Rwv, as a function of the ratio of the ST, Rst, with varying external modulation strengths. In
plot (a), external transverse detector modulation d is applied. In plot (b), external transverse
momentum modulation q is applied. We use 12 points to demonstrate the constant-slope be-
havior of Eqs. (3.9). The postselected angle is 0.38 rad and L ≈ 34 cm. The dotted red lines
are the linear fits of the data.
The spectrum analysis in Fig. 3.2 shows that the WVT mitigates the external
modulation signals at the detector: the transverse detector modulation in volts
is mitigated by 11 times (21 dBV from Fig. 3.2), and the transverse momentum
modulation in volts is mitigated 28 times over (29 dBV from Fig. 3.2) the ST.
We also observe a suppression of the modulations at harmonics of the driving
66
frequencies found in the ST. The “Signal,” however, is amplified by a factor of 3.2
(10 dBV from Fig. 3.2) in the WVT over the ST.
The signal benefits of the WVT over the ST from Fig. 3.2 is predicted in the
following (see Table 4.1):δxwvk σfδxstk σ
= cot(φ/2), (3.13a)
δxwvq σf
δxstq σ=L
f
σfσ
=L
2k0σ2, (3.13b)
δxwvd σfδxstd σ
=σfσ
=f
2k0σ2. (3.13c)
We note that in the ratios in Eqs. (3.13) we divide out the beam radius, thus
the amplification or improvement is not in accuracy, but strictly in the raw signal
given by the detector. The theory prediction of Eq. (3.13a) predicts the WVT
amplification to be 5 over the ST (14 dBV for the signal k) for φ = 0.38. Likewise,
the external modulations of d and q in the ST are 24 and 34 dBV, respectively,
greater than the WVT.
In Fig. 3.3, we plot R of the WVT versus the ST. The data comes from
using two different k values that give 48 and 16 nrad peak-to-peak deflections of
frequency 7 Hz. We set both external modulation sources to 28 Hz and study
them one at a time. By fitting the data, we arrive with the geometric factors
in Eqs. (3.9). From these results, the WVT outperforms the ST by a factor of
258 for transverse momentum modulations and by a factor of 51 for transverse
detector modulations for our parameters. Note the constant slope as predicted by
the theory in Eqs. (3.9). However, there is a discrepancy between the predicted
geometric slope values of 285 and 100 for transverse momentum modulation and
transverse detector modulation, respectively. This discrepancy is consistent with
67
previous experiments [26, 39] and attributed to the quality of the dark port and
imperfections of the optical elements.
After verifying the theoretical behavior, we study how the deviation of k,
∆k, is affected by the external modulations q or d. We use a trapezoid function
at frequency 10 Hz with a rise time of 10 ms to drive the piezoactuated BS. The
trapezoid function gives a constant momentum kick for about 40 ms. The external
modulation is a sine wave with frequency 250 Hz, and our collection window is 4
ms. We collect data with a sample time of T = 8 µs due to the bandwidth limit
of our split detector. This measurement protocol gives us 500 raw data points of
the momentum kick.
We note that the SDs have variable gain settings with white-Gaussian-power-
dependent electronic noise, J, equally present in both techniques.
Jwv =σJ√T
αcal2σ
V wvtotal
tan(φ/2)
2σ2. (3.14a)
Jst =σJ√T
αcal2σfV sttotal
k0f. (3.14b)
In Eqs. (3.14), σJ is the deviation of the intrinsic electrical noise (with laser off),
and T is the sample time. The factor αcal2σ/Vtotal converts the electrical detector
noise to a displacement in meters. The beam radius at the detector is defined to
be 2σ; Vtotal is the voltage proportional to the total power on the detector, and
αcal ≈ 0.66 is a calibration constant from the SD. The last term converts the noise
to momentum units given the technique in use.
The CRB for estimating k is given by I−10 in the absence of technical noise.
We modify the CRB to include the uncorrelated J noise [28, 42] by
∆k2B = 1/I0 + J2. (3.15)
Each technique is compared to its respective lower bound in uncertainty defined
by the CRB in Eq. (3.15). In Fig. 3.4, we plot ∆kB, divided by the deviation
68
0 0.5 1 1.5 2 2.510−3
10−2
10−1
100
qrms
/k0
[µrad]
Weak−Values
Standard
0 50 100 150 200 25010−2
10−1
100
−
2 drms
[nm]
ΔkBΔk
ϑsig
= 48 nrad
ΔkBΔk
ϑsig
= 16 nrad
Standard
Weak−Values(a)
(b)−
Figure 3.4: A plot of the theoretical minimum deviation given by the CRB, ∆kB , divided by
the deviation of the measurements of k, ∆kB/∆k = 1/√
1 + ξ2rms/∆k2B as a function of external
modulation strength ξrms ∈ {drms, qrms}. Data comes from a signal k of 16 and 48 nrad
deflection with variable external modulation at a frequency of 28 Hz. Plot (a) is for transverse
detector modulation and plot (b) is for transverse momentum modulation. The blue lines are
the WVT theory and the green lines are the ST theory. We stress that ξrms is not a noise source,
but rather models one frequency component of a general noise source.
of measurements of k, ∆k =√
∆k2B + ξ2rms, as a function of the external mod-
ulation strength ξrms ∈ {drms, qrms}, where ξrms is the root-mean-square value
of the sinusoidal external modulation. When both techniques have no external
modulation (∆k = ∆kB), ∆k is at best a factor of 7 away from the I−1/20 or the
shot-noise limit for momentum kick k. All of the post-selection was done with
φ ≈ 0.38 rad. Fig. 3.4 shows that the WVT is insensitive to external modu-
lations (1 ≥ ∆kB/∆k ≥ 0.5), while the ST is sensitive. From Figure 3.4, the
69
WVT outperforms the ST in deviation up to a factor of 7 for large transverse de-
tector modulation (230 nm = 2√
2drms) and 145 for large transverse momentum
modulation (2.5 µm =√
2qrms/k0).
We note that we acquire data when the signal k has shifted the beam by δd
to a steady value that remains constant for the integration time. Extracting the
deviation of k includes the electrical detector noise J and external modulation
ξrms. We add ∆k2B and ξ2rms in quadrature to describe the deviation of the mea-
surement of k because both sources are uncorrelated. This is not a general result
since one can devise a single tone modulation that will be correlated with the
detector power-dependent noise, and as a consequence will not be able to add
the modulation in quadrature. However, we want to stress that Fig. 3.4 is for a
single toned external modulation uncorrelated to the power-dependent noise from
the SD. In addition, if one were to superimpose many of these external modula-
tions with random frequency, phase, and amplitude one would expect behavior
following the description in [28] and not as in Fig. 3.4.
To higlight these results, we compare the results of the Fisher information
analysis in Ref. [28]. We start by reviewing the WVT and the ST in the presence
of a Gaussian-distributed angular jitter with the Fisher information formalism [28].
The final probability distribution in position is Gaussian distributed, and the ST
has a mean kf/k0 and variance σ2f + f 2Q2/k20, where Q2 is the angular-jitter
variance. The probability distribution for the WVT has mean 2kσ2 cot(φ/2) and
variance σ2 + (L/2k0σ)2(1 + (2σQ)2). The Fisher information for both techniques
is given by
IwvQ (k) =4Nσ2
1 + ( L2k0σ2 )2[1 + (2σQ)2]
, (3.16a)
IstQ(k) =4Nσ2
1 + (2σQ)2, (3.16b)
70
where the subscript Q denotes the angular-jitter analysis.
The Fisher information for the WVT in Eq. (3.16a) shows suppression of the
angular jitter with larger σ and with shorter L, the distance from the source Q to
detector. However, the Fisher information for the ST in Eq. (3.16b) degrades as
σf decreases (σf ∝ 1/σ). From the Fisher information perspective, it is better to
use a long focal length to acquire more of the available Fisher information in the
ST, but this will introduce turbulence effects. In this infinitely long focal length
approximation, the lens will no longer impart a quadratic phase to the beam. In
the approximation the lens will essentially be a transparent medium.
We also discuss the effect of a Gaussian-distributed detector-displacement jit-
ter on the Fisher information of the WVT and the ST as outlined in Ref. [28]. If
the detector-displacement jitter has a variance of J2, then the ST variance at the
detector becomes σ2f + J2, and the WVT variance becomes σ2 + J2 such that the
Fisher information for both techniques is given by
IwvJ (k) =4Nσ4
σ2 + J2, (3.17a)
IstJ (k) =N(f/k0)
2
σ2f + J2
=4Nσ2
1 + (2k0σJf
)2, (3.17b)
where the subscript J denotes the detector-displacement jitter analysis. This
symbol is not to be confused with J, the electrical noise on the detector, from
Eqs. (3.14).
Similarly, the Fisher information in Eq. (3.17a) shows suppression of the de-
tector jitter with large σ such that σ � J , while the ST Fisher information
in Eq. (3.17b) is shown to be optimal with detector-displacement jitter only for
large values of focal length such that f � 2k0σJ . This is the same as having
71
01 2 20 40 60 80 100
0.05
0.1
0.15
0.2
Figure 3.5: A plot of the geometric factor f ′/2σ2k0 cot(φ/2) from Eq. (3.9) as a function of
beam radius σ and focal length f ′ = f or distance f ′ = L. The plot never surpasses 1, thus the
ST will not outperform the WVT. The plot uses a postselection angle of φ = 0.4 rad.
a larger displacement at the detector (σf � J), which will introduce turbulence
effects [28]. Thus, the WVT with detector-displacement jitter will outperform the
ST under the Fisher information metric.
The Fisher information from both angular jitter Eqs. (3.16) and beam-displacement
jitter Eqs. (3.17) lead to two CRBs. The two CRBs predict a similarl behavior
to our external modulation results in Figs. 3.4. This analysis for the Gaussian-
distributed noises from Ref. [28] reveals that the WVT is superior over the ST in
obtaining Fisher information with technical noise. Our experimental results only
encompass one frequency component in the theory but validate the behavior.
We now search through the possible parameter space to see whether there exist
parameter values to give the ST an advantage over the WVT. When comparing
each technique with external modulations as in Eqs. (3.9) the WVT always out-
performs the ST. We explore possible parameter space to reoptimize the ST with
the following assumptions: (i) that both f and L are no greater than one meter
to avoid turbulence (as discussed in Ref. [28]), and (ii) that we σ is greater than
250 µm and smaller than 2 mm (to avoid saturation of the detector).
72
In Fig. 3.5, we plot the geometric factor of Eqs. (3.9) as f ′/2σ2k0 cot(φ/2) for
experimentally possible parameters of f ′ and σ, where f ′ can be either f or L. The
postselection angle for the plot is φ = 0.4 rad. The geometric value never exceeds
1. Thus in this comparison, the WVT always outperforms the ST. From Eqs. (3.9)
and Fig. 3.5, the WVT advantage over external modulations increases for smaller
postselection angles and larger beam widths. If we consider the bright port case,
the cot(φ/2) in Eqs. (3.9) will change to tan(φ/2). From this, we conclude that
when considering technical noise, only the dark port of the WVT outperforms the
ST. Thus, the technical advantage of the WVT is controlled by the geometry and
parameter selection for the experiment.
3.5 Results: Laser Beam Jitter
In Fig. 3.2, the amplitude of the angular modulation outside the interferometer is
suppressed in the WVT, relative to the ST. This behavior was predicted theoret-
ically regardless of the frequency of the oscillation [28]. We will now see how this
effect can be put to use for more general noise sources. To accomplish this, we
remove the connecting fiber that stabilizes the laser, and direct the light into one
of the two experiments in Fig. 3.1. The signal on the detector then registers noise
consisting of a combination of electronic noise and intrinsic laser jitter. We note
that the statistics of the jitter is neither white nor Gaussian, nor is it stationary.
The angular jitter originates from the physics of the laser, and exists up to around
300 Hz in this experiment. It has strong frequency components at around 50 and
100 Hz. Its constantly changing statistical nature makes any kind of improved
statistical estimation strategy extremely challenging. Nevertheless, the fact that
the weak-value experiment globally suppresses the amplitude of all angular jitter
from outside the interferometer makes the WVT very convenient as a noise re-
duction strategy. Indeed, we see from Fig. 3.6 that the contribution of the laser
73
0 200 400 600 800−60
−50
−40
−30
−20
−10
0
Frequency [Hz]
Weak−valuesStandard
1000 2000 3000 4000 5000−60−50−40−30−20
20 lo
g(V pp
/Vto
tal) [
dBV]
Figure 3.6: (Color online) A spectrum voltage comparison of the WVT (blue) and ST (green)
with naturally occurring laser-beam-jitter noise, where Vpp is the signal from the detector in
volts. Without the fiber, the beam is shaped to σ = 1.12 mm and is sent to the experiments.
The beam jitter is found in the low frequency regime (under 1 kHz). We see about 20 dBV
improvement in the WVT for low frequencies (under 300 Hz). We note similarly to Fig. 3.2,
the comparison is not of beam shift, but of voltages from the laser-beam-jitter noise. Both data
sets were taken by averaging 128 samples. The plots are normalized to the detected power,
Vtotal (either the WVT or the ST in their respective experiments). Note that this is a voltage
comparison and not a deflection comparison of WVT and the ST.
jitter to the noise spectrum is essentially eliminated entirely, being reduced below
the electronic noise floor.
In Fig. 3.6, the Fourier transforms of both the WVT (blue) and the ST (green)
signals as a function of frequency. The Fourier transforms shown are the average
of 128 samples, and the WVT postselection angle is φ ≈ 0.46 rad. We note that
while the ST uses 400 µW and the WVT uses 1.45 mW of power, the Fourier
transform of both signals are renormalized given the total detected power used in
each technique for a fair comparison.
Next, we perform the measurements in the time domain with a sample time
of T = 4 ms and compare the relative error of k in both techniques. The relative
74
error is the deviation of the measurements of k, ∆k, divided by its respective lower
bound, ∆kB from Eq. (3.15). The relative errors of the ST and the WVT are 144
and 5, respectively. Therefore, the WVT suppresses intrinsic beam-jitter noise at
best 29 times over the ST. Most importantly, it can be seen in Fig. 3.6, that the
WVT completely suppresses this laser-beam-jitter noise, showing only electronic
noise from the detector.
We independently verified the intrinsic laser beam jitter to be about 0.3 µrad
peak to peak using the full width at half maximum and twice the deviation of the
data collected from Fig. 3.6. The WVT has a total propagation length of 205 cm
from the laser to detector. The ST used a focal length of 1 m.
We can verify the claim that the WVT globally suppresses laser-beam-jitter
noise by comparing the suppression of the intrinsic (stochastic) beam jitter to the
single frequency modulation at an amplitude chosen to be the typical wander.
From the data in Fig. 3.4(b) where one single frequency is modulating an external
mirror before the interferometer (see Fig. 3.1), we can predict the mitigation factor
for a single tone of deflection angle 0.3 µrad. According to Fig. 3.6, the suppression
factor for the intrinsic beam jitter is at best 29, and the suppression for the single-
frequency tone from Fig. 3.4b is 44 (at 0.3 µrad peak-to-peak), giving comparable
results.
3.6 Summary
In this chapter, we compared two experimental techniques, the ST and the WVT.
The ST is a standard focusing technique to measure a deflection off a tilted mirror,
while the WVT includes a BS, making the system an interferometer that may
be interpreted as a realization of the Aharonov, Albert, Vaidman, weak-value
amplification effect if one output port is monitored [27]. We presented two types of
modulations in the two experiments: an external modulating beam deflection and
75
an external modulating detector modulation. The WVT used the experimental
geometry to give a mitigation effect of these sources over the ST. We understand
this behavior because the weak-value amplifies a signal of interest, while and all
other modulations external to the system are left un-amplified 4. Then we removed
the fiber to both experiments and measured a transverse momentum kick in the
presence of naturally occurring laser beam jitter. The WVT also fared better than
the ST with this real noise source found in the laboratory. We showed a greater
visibility and a smaller deviation in measurements of k in the WVT compared to
the ST. Lastly, we searched parameter space to re-optimize the ST, but practical
parameter values limited the space and left the ST always underperforming when
compared to the WVT in both deviation and visibility.
It is important to stress that in the absence of any technical limitations, both
systems are bounded by the same fundamental CRB or the shot-noise limit for
transverse momentum [see Eqs. (3.11) and Eq. (3.12)]. Therefore, the “weak-
value amplification” alone gives no metrological advantage, unless it is combined
with the other effects we have identified. This point has been studied some time
ago [26], though some authors have recently rediscovered it using the Fisher in-
formation metric [42–45]. However, under realistic conditions such as a detector
that saturates (responds nonlinearly), the presence of vibrational detector noise,
or the presence of angular jitter, it was shown that the WVT can perform orders
of magnitude better than the ST [28]. This is consistent with independent inves-
tigations using variations of this experiment, claiming record precision [80, 86].
We have reported experimental results quantifying this effect under the presence
of transverse detector modulations, transverse momentum modulations and nat-
urally occurring laser beam jitter. All these sources have been mitigated in the
WVT comparison to the ST as theoretically explained by Jordan et al. [28].
4In addition, the bright port signal is de-amplified.
76
4 Concatenated Postselection
for Weak-value Amplification
4.1 Introduction
From the formulation of the first order approximation of the weak-value amplifi-
cation in Chapter 1, it seems that the amplification is without bound. We have
shown that weak-value amplification as a technique saturates the CRB for a pa-
rameter of interest (see chapter 2). Weak-value amplification can be very close
to the orthogonal case between the pre and postselected states with the polar-
ization degree of freedom [87]. High quality optics produces a high polarization
extinction ratio, and weak-value amplification has been studied in that regime.
On the other hand, spatial interference experiments have difficulty attaining the
orthogonality condition. In the following work, we focus on improving effective
interference quality by introducing a concatenated postselection. In this chapter,
we discuss two postselections, that is, a concatenated postselection to overcome
spatial imperfections.
The main practical benefit of weak-values comes from the anomalous amplifica-
tion in the limit of nearly orthogonal postselection [2, 4]. Weak-value amplification
is theoretically bounded [45, 88–92], and the upper bound for this experiment (see
Eq. (B.9b) in App. B.3) is relatively difficult to achieve because it is reliant on
77
spatial interference. The Fisher information metric was used to study the theoret-
ical limitation of precision [84, 85], and orthogonal postselection [87, 88, 90, 93].
These studies have pointed out that the benefits of weak-value amplification de-
pend critically on the postselection angle.
All optical weak-value amplification experiments depend on the quality of the
dark port for greatest amplification. In optical experiments where the interaction
is polarization dependent [36, 81], the experiment is performed with a single prop-
agating beam which is pre- and postselected. In these experiments, the dark port
contrast can be as large as the polarization contrast of the lowest quality polarizing
optic. Other experiments measuring polarization independent effects [1, 37–39]
generally have lower interference contrast when compared to the higher polariz-
ing contrast from polarization dependent effects. The inability to postselect on
small angles in spatial interference experiments limits the benefits that weak-value
amplification has to offer.
In this chapter, we concatenate two postselections: the first with spatial inter-
ference and the second with polarization interference to measure a beam deflection
(see Fig. 4.1). We study the complementary amplification behavior between the
spatial contrast and the polarization contrast of the weak-values. We optimize
the concatenated postselection and arrive with an enhanced amplification.
This chapter is organized as follows. In Sec. 4.2, we start with the theory
for single and concatenated postselection for weak-value amplification to measure
beam deflection with a model that includes spatial interference background. Then,
in Sec. 4.3, we describe the experimental setup. In Sec. 4.4, we present the result
of the weak-value techniques. After that, in Sec. 4.5, we compare the theoreti-
cal efficiency of the weak-value techniques and the standard focusing technique.
Lastly, we discuss the results and conclude in Sec. 4.6.
78
k
5x5x
50:50 BS
Split Detector
Pol 2
Pol 1
QHQ
Laser
Half-wave
Figure 4.1: We send anti-diagonal polarized light (Pol 1) through a Sagnac interferometer.
Inside the interferometer are three wave plates arranged quarter-half-quarter (QHQ) which is
to control the phase for clockwise and counterclockwise propagation (see App. B.1). The coun-
terclockwise propagating beam receives a transverse momentum kick k from the piezocontrolled
50:50 beam splitter. When the beam recombines, it destructively interferes at the dark port.
The beam is then postselected a second time with polarizer 2 (Pol 2).
4.2 Theory
A laser beam with a TEM00 mode and 1/e2 beam radius σ enters a Sagnac in-
terferometer through a piezoactuated 50:50 BS (see Fig. 4.1). The reflected beam
receives a transverse momentum kick k upon both entering and exiting the interfer-
ometer. We monitor the spatial beam shift of the beam exiting the dark port. The
quarter-half-quarter (QHQ) wave plates combination [94] gives a Pancharatnam-
Berry phase [95] of ±φ/2 to each counter-propagating (| �〉, | 〉) beam (details
are shown in App. B.1). The interaction with the system is given by exp(−ikAx),
where the ancillary system operator is A = | �〉〈� | − | 〉〈 |.
For the remainder of the paper, we will describe the experiment in the classical
matrix formalism [72]. For input anti-diagonal polarized light, the output electric
79
field takes the form
Eout(x; β) = E0e−x24σ2
sin(kx+ φ/2) + β
sin(kx− φ/2) + β
, (4.1)
in the horizontal, |H〉, and vertical, |V 〉, polarization basis. We note this equation
is not to be confused with Eq. (2.13), where the output is from the dark and bright
ports of the interferometer. We introduce a spatial interference background, β.
The constant β describes the background that limits the spatial contrast from
reaching the perfect zero output from the dark port and the perfect input power
from the bright port. The difference in sign in Eq. (4.1) for H or V polariza-
tion comes from the asymmetric response of the QHQ combination inside the
interferometer (see App. B.1).
4.2.1 Theory: Single Postselection
First we assume the ideal case of β = 0. Using the modulus square of one compo-
nent of the electric field from Eq. (4.1), we arrive at the intensity profile. With the
intensity profile, we assume that the momentum kick is small for the weak interac-
tion approximation, k2σ2 cot2(φ/2) � 1. We expand the trigonometric functions
up to first order in k and re-exponentiate the quantity. Then we combine the two
exponentials by completing the square to arrive at the dark port intensity profile,
Is(x) = I0 sin2
(φ
2
)exp
[− 1
2σ2(x− δxs)2
]. (4.2)
The subscript s of Eq. (4.2) refers to the single postselection where δxs =
±2kσ2 cot(φ/2) is the beam shift from a horizontally or vertically polarized input
light. This is the standard result from the beam deflection experiment [39], with
a weak-value of Awv = ±i cot(φ/2) (see App. B.2).
For a realistic experiment implementations when φ is small, we assume the
case of β 6= 0 for Eq. (4.1). Assuming β << sin(φ/2) and integrating the square
80
modulus of the electric field of either |H〉 or |V 〉 of Eq. (4.1) yields the normal-
ization factor Ns = (sin(φ/2)± β)2. The mean beam shift on the detector is then
given by
〈x〉s =1
Ns
∫x|Eout
H,V (x; β)|2dx
= δxs
(1∓ β
sin(φ/2)− 2β2
sin2(φ/2)
).
(4.3)
We note that we only keep correction terms up to second order in β. We also note
that the mean shift in Eq. (4.3) has two solutions that depend on the different
components of Eq. (4.1).
4.2.2 Theory: Concatenated Postselection
The second part of the theory is to take advantage of the polarization sensitive
phase φ/2 by inputting anti-diagonal polarized light as in Eq. (4.1), with β = 0.
The orthogonal components of polarization will spatially separate at the dark
port by 2|δxs| since the horizontal and vertical components have opposite weak-
values (see App. B.2). The electric field exits the Sagnac interferometer and passes
through a polarizer with angle θ given by
P(θ) =1
2
1 + sin(2θ) cos(2θ)
cos(2θ) 1− sin(2θ)
. (4.4)
The polarizer angle θ is aligned to be nearly orthogonal to the polarization of the
exit beam from the interferometer.
We assume that the momentum kick is small for the weak interaction approxi-
mation, k2σ2 cot2(φ/2) cot2(θ)� 1. We expand the trigonometric functions up to
first order in k and re-exponentiate the result. We then combine the exponentials
by completing the square to arrive at the dark port intensity profile,
Ic(x) = I0 sin2
(φ
2
)sin2(θ) exp
[−(x− δxc)2
2σ2
]. (4.5)
81
The beam shift after the concatenated postselection is given by δxc = 2kσ2 cot(φ/2) cot(θ) =
|δxs| cot θ. The subscript c refers to the concatenated postselection case.
We now assume the case where the spatial interference background β is not
equal to zero. After the polarizer, we have a new normalization term Nc =
(sin(φ/2) sin(θ) + cos(θ)β)2. From the normalized intensity field, we expand the
trigonometric term and assume β << sin(θ) sin(φ/2) up to second order in β.
Then we re-exponentiate and arrive at the intensity profile of the dark port. The
mean shift of the beam on the detector is then given by
〈x〉c =1
Nc
∫x|P(θ)Eout(x; β)|2dx
= δxc
(1− β cot(θ)
sin(φ/2)− 2β2 cot2(θ)
sin2(φ/2)
).
(4.6)
We note that as θ → 45◦, the polarizer Eq. (4.4) will be aligned to the horizontal
polarization, and hence Eq. (4.6) reduces back to Eq. (4.3) for the horizontally
polarized case.
4.3 Experiment
The experimental setup shown in Fig. 4.1 starts with a grating feedback laser with
a 780 nm center wavelength. Two objectives and a 50 µm pinhole are used to
create a collimated Gaussian beam with radius σ, and a polarizer selects a linear
polarization for the experiment. The beam propagates and enters the Sagnac
interferometer through a 50:50 BS on a piezoactuated mount that provides a
transverse momentum kick k at each reflection. The interferometer has three
wave plates, QHQ, that create a phase difference between paths. The quarter-
wave-plates are set to +45◦ and −45◦. The half-wave plate sets the added phase of
±φ/2 to each path (see App. B.1). When the beams recombine, they destructively
interfere at the dark port. We monitor the beam shift of the light exiting the dark
82
port with a split detector. In the second part of the experiment, we add a polarizer
before the detector for the concatenated postselection.
We use a beam radius of σ = 550 µm with a polarization extinction ratio of
25000:1 measured by putting two crossed Glan Taylor polarizers in the beam path.
The polarization quality of the interferometer is limited to 5000:1 by the wave plate
combination inside the interferometer. The traverse momentum kick k is driven by
a piezostack calibrated separately for 100 Hz with a response of α ≈ 63.9 nm/V.
For all the measurements, we apply a 100 mV sinusoidal wave to the piezostack
which corresponds to a momentum kick k = 2.74 m−1. The first postselection
angle φ/2 is determined by the ratio of the measured power at the dark port,
Pφ/2, to the power of the bright port, Pbright,1, given by Pφ/2 = sin2(φ/2)Pbright,1.
The second interference postselection angle θ is determined by the ratio of the
power after the output polarizer dark port, Pθ, (Pol 2 in Fig. 4.1) to the power of
the polarization interference bright port, Pbright,2, as in Pθ = sin2(θ)Pbright,2.
4.4 Results
One of the advantages of weak-value amplification is the suppression of experi-
mental error such as technical noise, external modulations and naturally occurring
laser beam jitter [2, 28]. All these benefits hinge on the need to postselect with
small angles. In this section, we compare the weak-value amplification with single
and concatenated postselection.
4.4.1 Single Postselection
In Figure 4.2 the single postselection weak-value data (red circles) is labeled as
Single PS. We plot the absolute value of the mean beam shift of Eq. (4.2) versus
the generalized postselection angle Θ. The generalized postselection angle for the
83
single postselection case is given by Θ = φ/2. The data consists of both horizontal
and vertical polarized input light with dark port contrast of 1400:1. The fit to
the single postselection data is labeled as Fit:SPS (solid teal line) and takes on
the positive values of δxs as in the horizontal polarized case of Eq. (4.3). From
Fit:SPS, we extract β and the optimal postselection angle Θopt, where the weak-
value amplification shows the largest signal before it is lost due to the background.
From Fit:SPS, we observe that the largest signal is found with a postselection angle
of φ/2 = 2.6◦. We also see that the constant background parameter is given by
β = (12.4 ± 0.1) × 10−3. The data differs slightly from the theory of Eq. (4.2)
(dotted blue line) because of a systematic error in calibration of the piezoactuated
BS. This theory of Eq. (4.2) is the beam shift δxs = 2kσ2 cot(φ/2) without spatial
interference background consideration.
For each postselection angle, we compare the first order in k weak-value to the
all order in k theory to see if there is a need to include nonlinear effects or consider
loss in accuracy. The all order theory is computed in App. B.2 for experimental
parameter values. The first order in k weak-value approximation is valid for this
angle as it deviates only 0.18% from the all order in k theory (see App. B.3).
We note that as the alignment improved, the quality of the dark port also
improved and the optimal angle for greatest amplification decreased. The results
from the single postselection case of Fig. (4.2) show that the background β limits
the weak-value amplification from the theoretical upper bound [90, 93] and limits
the benefit over technical noise [2, 28].
4.4.2 Complementary Behavior Between Postselections
We note the complementary behavior of the two degrees of freedom, which-path
and polarization. For example, if one postselects the spatial interference to re-
solve maximum amplification of the single postselection (the peak of Single PS
84
0 2 4 6 8 10
10
20
30
40
50
Θ (deg)
Concatenated PSSingle PSFit: SPSFit: CPSδ x s
µm
)⟨
(⟨ x
Figure 4.2: The average beam shift as a function of postselection angle Θ. The variable Θ is
a generalized postselection angle; for the single postselection case, Θ = φ/2 (red circles), but
for the concatenated postselection case it is the product of both postselection angles Θ = θφ/2
(green squares). The label PS refers to postselection. The theory (dotted blue line) is labeled
as δxs = 4kσ2/Θ as in the mean beam shift of Eq. (4.2) with generalized postselection angle Θ.
Fit:SPS (solid teal line) is the fit of the single postselection data, and Fit:CPS (solid purple line)
is the fit to the concatenated postselection data, both of which have corrections terms up to
second order in β as in Eqs. (4.3) and (4.6), respectively. Fit: CPS gives β = (7.1± 0.1)× 10−3
for the concatenated case. This concatenated data set is the largest experimental signal from
Table 4.1 found in row three.
in Fig. 4.2), then there cannot be any polarization improvement because we ob-
serve a polarization contrast close to 10:1 for polarization. To understand this
limitation, we first note that if the first postselection output power is Pφ/2, the
contrast ratio is Pφ/2:Pin. For our case the maximum amplification in the Single
PS is Pφ/2:Pin ≈ 500:1. Then we observe the second postselection to have a con-
trast at best of about 10:1 for a total effective contrast of 5000:1. The effective
contrast of both postselections cannot exceed the contrast of either the spatial or
polarization contrast. In this particular scenario, we are not in the small angle
(because of 10:1 in polarization) regime so the configuration is suboptimal. We
note with a maximum polarization contrast of 10:1 we can expect the location of
the peak slightly to be close to θ ≈ 0.5 rad ≈ 29◦. This angle will not provide an
enhanced beam shift of δc, as the angle is outside the small angle approximation
and will not follow the optimal theory of Eq. (4.5). For our experiment, the spatial
85
interference contrast is 1400:1 and the polarization contrast is 5000:1. Therefore,
we present the results of the optimized case in the next section.
4.4.3 Concatenated Postselection
When compared to the beam shift of the single postselected weak-value, we see
from Eq. (4.5) that the beam shift is further amplified by cot(θ) at the cost of a
fraction cos2(θ) of the input photons. The enhancement of the beam shift increases
the effective resolution of the detector. Thus the use of the extra degree of freedom
of polarization can enhance the weak-value amplification.
We point out that the theory of Eq. (4.5) does not assume any limitations to
the contrast for either spatial or polarization interference. In the case of infinite
contrast there is no benefit in adding a second degree of freedom. Since this is
an idealization, therefore we explore the case of having one degree of freedom
with a higher contrast than the other. In this experiment, the spatial interference
contrast is 1400:1 and polarization contrast is 5000:1, thus there exists an optimal
configuration for greatest amplification.
Now we focus on the concatenated postselection data (green squares) in Fig. 4.2.
We plot the absolute value of the mean beam shift, |〈x〉|, from Eq. (4.5) as a
function of postselection angle Θ. The variable Θ is a generalized postselection
angle given by Θ = θφ/2 for the concatenated postselection and Θ = φ/2 for the
single postselection (red circles). The product of postselection angles is a valid
approximation for the small angle regime. The plot shows the benefit of intro-
ducing the polarization degree of freedom to the experiment which allows us to
achieve smaller effective postselection angles. From Fit:CPS (solid purple line)
from Eq. (4.6), the optimal postselection angle for the concatenated postselection
is about 1.4◦. This angle deviates only about 0.44% from the all order in k theory
(see App. B.3). The improved beam shift with the concatenated postselection
86
has increased by a factor of approximately 1.4 over the single postselection beam
shift. The fit of the concatenated weak-value case gives a background interference
parameter β ≈ (7.1± 0.1)× 10−3. We remind the reader that the data presented
for the concatenated case is the optimized case where there is an enhancement in
the signal.
Now we compare the spatial interference background parameter β of the single
and concatenated cases. The fitting of Eq. (4.3) and Eq. (4.6) to the data reveals
β ≈ (12.4± 0.1)× 10−3 and β ≈ (7.1± 0.1)× 10−3, respectively. The error σerror
is from the fit of the data with 95% confidence. Thus the background has been
mitigated by a factor of 1.7 and the overall effective contrast has improved.
4.5 Efficiency: Single, Concatenated, and Stan-
dard Focusing
In this section, we theoretically compare the efficiency of the concatenated weak-
value technique to the standard focusing technique in the ideal noiseless case.
To compare the two techniques, we use the Fisher information formalism of the
parameter of interest k given by
I(k) =
∫dxP (x; k)
[∂
∂klnP (x; k)
]2, (4.7)
where P (x; k) is the probability distribution of the photons arriving on the detec-
tor. Like the weak-value technique, the standard focusing technique is optimal for
reaching the shot-noise limit of a transverse momentum kick k. The standard fo-
cusing technique uses a lens to Fourier transform the momentum kick to a spatial
beam shift in the Fourier plane. The weak-value technique does not use a lens,
but instead the imaginary part of the weak-value transforms the momentum kick
to a beam shift [28].
87
Table 4.1: Results of optimized concatenated postselection for weak-value amplification. The
results of the last four columns come from numerically fitting the data where the nonlinear fit is
with a 95% confidence and a goodness measure r2 > 0.8. The first column is the output power
of the first postselection. The second column is the first postselection angle, φ/2. Θopt. is the
product θφ/2 with highest signal from the fit. The fourth column is the spatial interference
background parameter β with an uncertainty of σerror from the 95% confidence fit. Ic(k)frac
is the fraction of Fisher information after postselection in the dark port as in Eq. (4.14b) for
rows two through six. We note that the last two columns are theoretical and not measured
results. The first row in column five displays Is(k)frac that corresponds to the fraction of Fisher
information after postselection from Eq. (4.14a). The proximity to one in Is(k)frac ≈ 1 means
that there is no loss of Fisher information when monitoring the dark port. This is not to be
confused with shot-noise limited measurements because this is fractional Fisher information. The
last column is a comparison of the weak-value techniques to the standard focusing technique of
the density of Fisher information per measurement given by the ratio of Eq. (4.11) by Eq. (4.9).
Pφ/2(µW ) φ/2 Θopt. β(10−3) I(k)frac ρc/ρst
– – 2.6◦ 15.1±0.1 ≈ 1.0 480
30 4.3◦ 2.0◦ 10.2±0.1 0.80 720
50 5.7◦ 1.4◦ 7.1± 0.1 0.92 1500
52 5.6◦ 1.3◦ 6.4± 0.1 0.94 1900
100 7.9◦ 1.4◦ 6.6± 0.1 0.95 1700
200 11◦ 1.7◦ 8.3± 0.1 0.94 1100
88
The standard focusing technique focuses the beam on the split detector. The
beam shift on the detector is of fk/k0, where f is the focal length of the lens, k is
the transverse momentum kick and k0 is the wave number of the light. The beam
radius at the focus is given by σf = f/2k0σ. The standard focusing technique has
a probability function given by
Pst(x; k) =N∏i=1
1√2πσ2
f
exp
[−(xi − fk/k0)2
2σ2f
], (4.8)
where N is the the number of independent measurements in the standard focusing
technique. In this study, we will use N to equal the number of photons entering
the system. From the probability function we can define the Fisher information
for parameter k from Eq. (4.7). Then we define the density of Fisher information
per number of independent measurements as
ρst =Ist(k)
N= 4σ2. (4.9)
Next, we consider the probability function of the optimized concatenated weak-
value technique
Pc(x; k) =Nc∏i=1
1√2πσ2
exp
[−(xi − δxc)2
2σ2
], (4.10)
where δxc = 2kσ2 cot2(φ/2) cot2 θ as in Eq. (4.5). The probability function of
the concatenated case has Nc independent measurements which is less than the
number of measurements from the standard focusing technique, Nc � N . We
relate the number of measurements between the concatenated and the standard
focusing cases by Nc = N sin2(φ/2) sin2 θ. The concatenated weak-value technique
has a reduced number of photons that can be accounted for by the bright port
statistics, which we will not address in this paper. Then we define the density of
Fisher information per number of independent measurements as
ρc =Ic(k)
N sin2(φ/2) sin2 θ= 4σ2 cot2(φ/2) cot2 θ. (4.11)
89
We compare the optimized concatenated technique to the standard focusing tech-
nique by the ratio of the Eq. (4.11) and Eq. (4.9), ρc/ρst.
We also study the amount of Fisher information that is collected out of the
dark port of each technique. We note that the total available Fisher information is
the sum of the Fisher information from the dark port (D) and the bright port (B)
of the single postselection case, Is,D + Is,B = 4Nσ2. The Fisher information from
the single postselection, concatenated postselection and the standard focusing
technique is written as
Is(k) = 4Nσ2 cos2(φ/2), (4.12a)
Ic(k) = 4Nσ2 cos2(φ/2) cos2(θ), (4.12b)
and
Ist(k) = 4Nσ2, (4.13)
respectively. Eq. (4.12a) and Eq. (4.12b) are obtained from the dark port of the
single and concatenated techniques, respectively. Each experiment performing ei-
ther the single, concatenated, or standard focusing technique requires N sin2(φ/2),
N sin2(φ/2) sin2 θ, or N , measurements respectively.
4.5.1 Results of the Comparison
In Table 4.1, we present the data from the single and the concatenated postse-
lections. The first column is the output power of the spatial interference post-
selection. The first row is the single postselected data. The single postselected
case has no first or second column because it only uses spatial interference, where
Θ = φ/2. The concatenated results are bottom five rows. The second column
is the angle φ/2 of the first postselection, which is the spatial interference. The
third column is the postselection angle, Θopt., of the largest beam shift, δc, from
90
the fits. The fourth column is the spatial interference background parameter β
from the fits.
From Table 4.1, the optimized concatenated case is in the region of the third
and fourth row. In Fig. 4.2 we plot the experimental run with the largest signal
from row three in Table. 4.1, The optimized case shows not only a large ampli-
fication for the smallest postselection angle but also a lower amount of spatial
interference background, given by β in column four. From column four, β dips to
a minimum around the optimal region. We note that the last two columns are the-
oretical results. The fifth column is the fractional Fisher information of the posts-
elected events from the total Fisher information in the system, Is,D+Is,B = 4Nσ2.
The subscript s refers to the single postselected case. The fractional Fisher in-
formation for the single postselection and concatenated postselection is given by
Is(k)frac =Is,D
Is,D + Is,B= cos2(φ/2), (4.14a)
Ic(k)frac =Ic,D
Is,D + Is,B= cos2(φ/2) cos2(θ), (4.14b)
respectively.
The first row of Table 4.1 has fractional Fisher information given by Eq. (4.14a).
The proximity to 1 of the fractional Fisher information in the first row means that
there is no loss in Fisher information due to monitoring only the dark port. This
is not to be confused with a shot-noise limited measurement because this is a
fractional description of the Fisher information meant to describe efficiency of the
postselected events. For the rest of the rows, the fifth column refers to Eq. (4.14b).
The last column of Table 4.1 is a comparison of the density of Fisher infor-
mation per photon of the weak-value technique against the standard focusing
technique. In the standard focusing technique we assume that every measure-
ment or photon carries an equal amount of Fisher information. In comparison,
91
the weak-value type measurement imparts the Fisher information into a small
subset of events after postselection. The comparison in column six allows us to
identify the optimal region for the concatenated postselection technique.
Looking at the fourth row of Table 4.1, the largest beam shift is found with
a postselection angle of Θopt = 1.3◦. The single postselected angle Θopt = 2.6◦
has therefore improved with the optimized concatenated case. We note that there
exists an optimized case because the polarization degree of freedom has a greater
extinction efficiency than the spatial degree of freedom in this experiment. Rows
three and four have different values of φ/2 because of slightly different measured
input powers. The concatenated postselection could not be optimized any further
because of limitations on our polarization extinction ratio of 5000:1. We note
that the postselection angle for the optimized case is only 1.5% away from the full
theory as seen in App. B.3, so we do not consider any approximation corrections.
The non-optimized cases of the concatenated postselection will be penalized in
the Fisher information because of the product of the two cosines as in Eq. (4.14b),
but the penalty with the small angle approximation can be minimized and extin-
guished. The loss of the available Fisher information is less than 8% for the
optimized region (see fifth column of Table 4.1). The optimized case can only
exist when one interference contrast is higher than the other. Polarization is an
example of a large extinction efficiency where polarizers can have extinction ratios
of 106:1.
The concatenated configuration makes the postselected photons 103 times more
effective than the standard focusing technique. The following experiments perform
one postselection with spatial interference. We also note in the recent technical
noise paper [2] that the smallest postselection angle for deflection measurements
was 10.9◦ which makes the photons close to 30 times more effective than the
standard focusing technique. In papers such as Ref. [1, 39], the postselection
angles are close to 10◦ and in Ref. [26] the postselection angle was 25◦. In cases
92
where spatial interference contrast is needed, adding a second degree of freedom
such as polarization can enhance the effective resolution of the detectors.
In this experiment, we postselect with spatial interference to produce two op-
posite weak-values, each of which carries half of the Fisher information. Then we
use the higher extinction contrast degree of freedom of polarization to postselect
a second time to explore the complementary behavior between the two postselec-
tions. We find an optimized region of parameter space such that the concatenated
postselection provides some benefit with a smaller effective postselection angle Θ,
a reduction of the spatial interference background parameter β, and an increase
of Fisher information per measurement when compared to the standard focusing
technique.
It is worth pointing out that the optics used in our experiment limited the
polarization contrast to 5000:1. This is consistent with our measurements of the
concatenated postselection in Table 4.1. We note that with higher performing
optics, we would amplify the signal with higher visibility and circumvent spatial
interference background. We are in a regime where the weak-values first order
approximation is less than half a percent away from the all order in k theory and
thus we do not account for deformation of the wave function or loss in the accuracy
of the results [87]. We also note that this work is not to be confused with Ref. [51],
where they propose an entangled ancillary system to improve the precision of a
measurement. In our experiment, the best possible precision is bounded by the
standard quantum limit.
4.6 Summary
In this chapter, we have explored concatenated post-selection for weak-values am-
plification to measure a beam deflection. We used a Sagnac interferometer with
spatial interference to measure a transverse beam deflection. We then introduced
93
a second postselection to the system with polarization. The concatenated post-
selection angle, θ, and the first spatial interference postselection angle, φ/2, are
complementary bounded by the highest interference contrast. Only when one of
the two interference contrasts is larger than the other can there be an optimized
regime in parameter space.
In general, it is better to do one postselection, but in the case of low contrast
spatial interference, we can incorporate a higher contrast degree of freedom such
as polarization for improvement. Thus as we have observed, the optimized values
of the postselected angles reduced the spatial interference background, modeled by
parameter β, by a factor of 1.7. The optimized technique also increased the signal,
governed by the beam shift, by a factor of cot θ. Lastly, there was an increase in
the available Fisher information per photon up to 103 over the standard focusing
technique for our parameter values, showing the efficiency of the concatenated
postselection.
With higher quality optics, we could have greater discrepancy between spatial
and polarization extinction ratios and further increase postselection contrast in
an optimized case. This condition would lead to greater reduction in technical
noise [2, 28] which would help reach the shot-noise limit with greater ease. It is
worth noting that a concatenated postselection for weak-values is beneficial only
when the additional degree of freedom has a higher interference contrast than
the first interference. In our optimized case, we lose a small amount of Fisher
information, but in the small angle approximation, this is negligible and therefore
still an optimal technique. This can aid in exploring the orthogonal case of weak-
values and in reaching the shot-noise limit for parameter estimation with non-ideal
detectors.
94
5 Concluding Remarks
In this thesis, we have presented weak-value based techniques as a metrological
tool to measure velocities and beam deflections. We used efficient estimators in all
the experiments and showed how weak-value based techniques can be practically
implemented in the laboratory. We saw that the weak-value-based techniques, be-
cause they are shot-noise limited, result in little to no loss in Fisher information
from the discarded bright port. Then we introduced external deterministic and
stochastic representations of noise to compare the performance of the weak-value
technique (WVT) versus the standard technique (ST) for measuring a transverse
momentum kick. We showed how both optimal techniques experienced different
deviations from the shot-noise and concluded that the WVT outperformed the ST.
Then we explored the idea of concatenated postselection for weak-value amplifi-
cation. We introduced an extra degree of freedom to find an optimized region of
parameter space where the use of two postselections can provide an enhancement
to the amplification. We summarize the results presented in the thesis.
In Chapter 2, we presented two experiments: one to measure the longitudinal
velocity of a mirror, and the other to measure the transverse momentum kick of
a mirror. In the first experiment, we were able to measure a velocity of 400 fm/s
that was shot-noise limited. We used the efficient estimator of the mean to reach
the Cramer-Rao bound for velocity measurements. In the second experiment,
95
we measured a transverse momentum kick k using both the dark port and the
bright port of the interferometer. This allowed us to measure the relative Fisher
information of k out of the system. The results revealed that as expected, there
was little to no Fisher information out of the bright port, and that there is no
benefit in collecting the statistics of the un-postselected photons.
In Chapter 3, we have explored the usefulness of weak-value based techniques
in the presence of technical noise. We compared the standard focusing technique
and the weak-values technique to measure a transverse momentum kick k in the
presence of deterministic and stochastic sources. The WVT outperforms the ST
by orders of magnitude in deviation of the parameter of interest, as predicted in a
more general case in Jordan et al. [28]. We also showed that weak-values exploit
the geometrical configuration of the experimental setup to mitigate the noise.
The mitigation of the external sources occurs because the weak-value amplifies
the signal of interest through a reduction of photons, while the external sources
are not amplified.
In Chapter 4, we explored the complementary behavior of performing two
postselections to measure a transverse momentum kick. The first postselection
uses the spatial degree of freedom through spatial interference. The second inter-
ference deals with the polarization degree of freedom. In our case, the different
degrees of freedom brought two different interference contrasts for postselection.
We showed that the addition of a higher contrast second degree of freedom can
be beneficial by reducing the effective postselection angle. We also explored the
theoretical efficiency of the technique. The optimized case of the technique under
the small angle approximation shows negligible loss of Fisher information through
the dark port. This shows promise of further mitigation of technical noise for
spatial interference weak-value based techniques.
In conclusion, weak-value-based techniques show great promise to improve cur-
rent parameter estimation techniques in the realistic scenario of including external
96
noise sources. The amplification of this optimal technique has been demonstrated
to increase the effective resolution of detectors and to amplify parameters of in-
terest. The idea of measuring the system weakly is also of great interest because
the reduced number of measurements allows for the use of low power detectors.
These topics outlined above demonstrate properties of devices used in metrolog-
ical experiments described in chapter 1. Weak-value-based techniques have been
shown to mitigation noise and amplify signal, therefore it is a suitable tool for
metrology. While a direct application to measure gravitational waves with weak-
value amplification seems infeasible, the technique is still helpful for the detection
of other small parameters. Weak-value techniques alongside with the technology
of filtering, stabilization and amplification (see chapter 1), is a promising combi-
nation to allow researchers to probe tiny phenomena such as universal constants.
There are further proposals to explore recycling [48, 96] and weak-values in other
fields [97, 98], that, tied with the ideas presented here, can continue to explore
small effects in nature.
97
Bibliography
[1] G. I. Viza, J. Martınez-Rincon, G. A. Howland, H. Frostig, I. Shomroni,
B. Dayan, and J. C. Howell, “Weak-values technique for velocity measure-
ments,” Opt. Lett. 38, 2949 (2013), URL http://ol.osa.org/abstract.
cfm?URI=ol-38-16-2949.
[2] G. I. Viza, J. Martınez-Rincon, G. B. Alves, A. N. Jordan, and J. C. Howell,
“Experimentally quantifying the advantages of weak-value-based metrology,”
Phys. Rev. A 92, 032127 (2015), URL http://link.aps.org/doi/10.1103/
PhysRevA.92.032127.
[3] G. I. Viza, J. Martınez-Rincon, W.-T. Liu, and J. C. Howell, “Concatenated
postselection for weak-value amplification,” in preparation (2016).
[4] J. Martınez-Rincon, W.-T. Liu, G. I. Viza, and J. C. Howell, “Can
anomalous amplification be attained without postselection?,” arXiv preprint
arXiv:1509.04810 (2015).
[5] M. Fox, Quantum Optics (Oxford University Press, 2006).
[6] M. Born and E. Wolf, Principles of Optics (Cambridge University Press,
1999).
[7] C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge Univer-
sity Press, 2005).
98
[8] P. W. Milonni and J. H. Eberly, Lasers (John Wiley & Sons, Inc., 1991).
[9] L. Hardy, “Quantum mechanics, local realistic theories, and lorentz-invariant
realistic theories,” Phys. Rev. Lett. 68, 2981 (1992), URL http://link.
aps.org/doi/10.1103/PhysRevLett.68.2981.
[10] J. S. Lundeen and A. M. Steinberg, “Experimental joint weak measurement
on a photon pair as a probe of hardy’s paradox,” Phys. Rev. Lett. 102,
020404 (2009), URL http://link.aps.org/doi/10.1103/PhysRevLett.
102.020404.
[11] M. J. W. Hall, D. Deckert, and H. M. Wiseman, “Quantum phenomena mod-
eled by interactions between many classical worlds,” Phys. Rev. X 4, 041013
(2014), URL http://link.aps.org/doi/10.1103/PhysRevX.4.041013.
[12] P. Shadbolt, J. C. Mathews, A. Laing, and J. L. O’Brien, “Testing founda-
tions of quantum mechanics with photons,” Nature Physics 10, 278 (2014).
[13] M. F. Pusey, J. Barrett, and T. Rudolph, “On the reality of the quantum
state,” Nature Physics 8, 475 (2012), URL http://dx.doi.org/10.1038/
nphys2309.
[14] R. Colbeck and R. Renner, “Is a system’s wave function in one-to-one corre-
spondence with its elements of reality?,” Phys. Rev. Lett. 108, 150402 (2012),
URL http://link.aps.org/doi/10.1103/PhysRevLett.108.150402.
[15] M. Ringbauer, B. Duffus, C. Branciard, E. G. Cavalcanti, A. G. White, and
A. Fedrizzi, “Measurements on the reality of the wavefunction,” Nat Phys
11, 249 (2015), URL http://dx.doi.org/10.1038/nphys3233.
[16] J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber,
“Direct measurement of the quantum wavefunction,” Nature 474, 249
99
(2011), URL http://www.nature.com/nature/journal/v474/n7350/abs/
10.1038-nature10120-unlocked.html#supplementary-information.
[17] G. A. Howland, D. J. Lum, and J. C. Howell, “Compressive wavefront
sensing with weak values,” Opt. Express 22, 18870 (2014), URL http:
//www.opticsexpress.org/abstract.cfm?URI=oe-22-16-18870.
[18] Y. Nanishi, “Nobel prize in physics: The birth of the blue led,” Nat Photon
8, 884 (2014), URL http://dx.doi.org/10.1038/nphoton.2014.291.
[19] S. Barnett and D. Pegg, “On the hermitian optical phase oper-
ator,” Journal of Modern Optics 36, 7 (1989), http://dx.doi.
org/10.1080/09500348914550021, URL http://dx.doi.org/10.1080/
09500348914550021.
[20] L. E. Ballentine, Quantum Mechanics: A Modern Development (World Sci-
entific, 1998).
[21] A. Kar, Ph.D. thesis, University of Rochester (2014).
[22] H. B. G. Casimir and D. Polder, “The influence of retardation on the london-
van der waals forces,” Phys. Rev. 73, 360 (1948), URL http://link.aps.
org/doi/10.1103/PhysRev.73.360.
[23] C. Wilson, G. Johansson, A. Pourkabirian, M. Simoen, J. Johansson, T. Duty,
F. Nori, and P. Delsing, “Observation of the dynamical casimir effect in a
superconducting circuit,” Nature 479, 376 (2011).
[24] J. J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley, 1967).
[25] A. Sushkov, W. Kim, D. Dalvit, and S. Lamoreaux, “Observation of the
thermal casimir force,” Nature Physics 7, 230 (2011).
100
[26] D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, “Optimizing the
signal-to-noise ratio of a beam-deflection measurement with interferometric
weak values,” Phys. Rev. A 80, 041803 (2009), URL http://link.aps.org/
doi/10.1103/PhysRevA.80.041803.
[27] Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measure-
ment of a component of the spin of a spin- 1/2 particle can turn out to be
100,” Phys. Rev. Lett. 60, 1351 (1988), URL http://link.aps.org/doi/
10.1103/PhysRevLett.60.1351.
[28] A. N. Jordan, J. Martınez-Rincon, and J. C. Howell, “Technical advantages
for weak-value amplification: When less is more,” Phys. Rev. X 4, 011031
(2014), URL http://link.aps.org/doi/10.1103/PhysRevX.4.011031.
[29] K. P R, R. T, A. A J, and A. K M, “Laser doppler vibrometry with acous-
tooptic frequency shift,” Opt. Appl. 34, 373 (2004).
[30] F. Matichard, B. Lantz, R. Mittleman, K. Mason, J. Kissel, B. Abbott,
S. Biscans, J. McIver, R. Abbott, S. Abbott, et al., “Seismic isolation of
advanced ligo: Review of strategy, instrumentation and performance,” Clas-
sical and Quantum Gravity 32, 185003 (2015), URL http://stacks.iop.
org/0264-9381/32/i=18/a=185003.
[31] T. L. S. Collaboration, “Enhanced sensitivity of the ligo gravitational
wave detector by using squeezed states of light,” Nat Photon 7,
613 (2013), URL http://www.nature.com/nphoton/journal/v7/n8/abs/
nphoton.2013.177.html#supplementary-information.
[32] U. L. Andersen, “Quantum optics: Squeezing more out of ligo,” Nat Photon
7, 589 (2013), URL http://dx.doi.org/10.1038/nphoton.2013.182.
101
[33] T. L. S. Collaboration, “Ligo: the laser interferometer gravitational-wave
observatory,” Reports on Progress in Physics 72, 076901 (2009), URL http:
//stacks.iop.org/0034-4885/72/i=7/a=076901.
[34] T. L. S. Collaboration, “A gravitational wave observatory operating
beyond the quantum shot-noise limit,” Nat Phys 7, 962 (2011), URL
http://www.nature.com/nphys/journal/v7/n12/abs/nphys2083.html#
supplementary-information.
[35] D. J. Starling, S. M. Bloch, P. K. Vudyasetu, J. S. Choi, B. Little, and J. C.
Howell, “Double lorentzian atomic prism,” Phys. Rev. A 86, 023826 (2012),
URL http://link.aps.org/doi/10.1103/PhysRevA.86.023826.
[36] O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via
weak measurements,” Science 319, 787 (2008), http://www.sciencemag.
org/content/319/5864/787.full.pdf, URL http://www.sciencemag.
org/content/319/5864/787.abstract.
[37] D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, “Precision
frequency measurements with interferometric weak values,” Phys. Rev. A
82, 063822 (2010), URL http://link.aps.org/doi/10.1103/PhysRevA.
82.063822.
[38] P. Egan and J. A. Stone, “Weak-value thermostat with 0.2 mk pre-
cision,” Opt. Lett. 37, 4991 (2012), URL http://ol.osa.org/abstract.
cfm?URI=ol-37-23-4991.
[39] P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive
beam deflection measurement via interferometric weak value amplification,”
Phys. Rev. Lett. 102, 173601 (2009), URL http://link.aps.org/doi/10.
1103/PhysRevLett.102.173601.
102
[40] D. J. Starling, P. B. Dixon, N. S. Williams, A. N. Jordan, and J. C. Howell,
“Continuous phase amplification with a sagnac interferometer,” Phys. Rev.
A 82, 011802 (2010), URL http://link.aps.org/doi/10.1103/PhysRevA.
82.011802.
[41] X.-Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.-F. Li, and G.-C. Guo,
“Phase estimation with weak measurement using a white light source,” Phys.
Rev. Lett. 111, 033604 (2013), URL http://link.aps.org/doi/10.1103/
PhysRevLett.111.033604.
[42] G. C. Knee and E. M. Gauger, “When amplification with weak values fails
to suppress technical noise,” Phys. Rev. X 4, 011032 (2014), URL http:
//link.aps.org/doi/10.1103/PhysRevX.4.011032.
[43] C. Ferrie and J. Combes, “Weak value amplification is suboptimal for es-
timation and detection,” Phys. Rev. Lett. 112, 040406 (2014), URL http:
//link.aps.org/doi/10.1103/PhysRevLett.112.040406.
[44] S. Tanaka and N. Yamamoto, “Information amplification via postselection:
A parameter-estimation perspective,” Phys. Rev. A 88, 042116 (2013), URL
http://link.aps.org/doi/10.1103/PhysRevA.88.042116.
[45] X. Zhu, Y. Zhang, S. Pang, C. Qiao, Q. Liu, and S. Wu, “Quantum measure-
ments with preselection and postselection,” Phys. Rev. A 84, 052111 (2011),
URL http://link.aps.org/doi/10.1103/PhysRevA.84.052111.
[46] L. Zhang, A. Datta, and I. A. Walmsley, “Precision metrology using weak
measurements,” Phys. Rev. Lett. 114, 210801 (2015), URL http://link.
aps.org/doi/10.1103/PhysRevLett.114.210801.
[47] N. Brunner and C. Simon, “Measuring small longitudinal phase shifts:
Weak measurements or standard interferometry?,” Phys. Rev. Lett. 105,
103
010405 (2010), URL http://link.aps.org/doi/10.1103/PhysRevLett.
105.010405.
[48] J. Dressel, K. Lyons, A. N. Jordan, T. M. Graham, and P. G. Kwiat,
“Strengthening weak-value amplification with recycled photons,” Phys. Rev.
A 88, 023821 (2013), URL http://link.aps.org/doi/10.1103/PhysRevA.
88.023821.
[49] O. S. Magana Loaiza, M. Mirhosseini, B. Rodenburg, and R. W. Boyd,
“Amplification of angular rotations using weak measurements,” Phys.
Rev. Lett. 112, 200401 (2014), URL http://link.aps.org/doi/10.1103/
PhysRevLett.112.200401.
[50] Y. Kedem, “Using technical noise to increase the signal-to-noise ratio of mea-
surements via imaginary weak values,” Phys. Rev. A 85, 060102 (2012), URL
http://link.aps.org/doi/10.1103/PhysRevA.85.060102.
[51] S. Pang and T. A. Brun, “Improving the precision of weak measurements
by postselection measurement,” Phys. Rev. Lett. 115, 120401 (2015), URL
http://link.aps.org/doi/10.1103/PhysRevLett.115.120401.
[52] X. Qiu, X. Zhou, D. Hu, J. Du, F. Gao, Z. Zhang, and H. Luo, “Deter-
mination of magneto-optical constant of fe films with weak measurements,”
Applied Physics Letters 105, 131111 (2014), URL http://scitation.aip.
org/content/aip/journal/apl/105/13/10.1063/1.4897195.
[53] G. Jayaswal, G. Mistura, and M. Merano, “Observation of the Imbert-
Fedorov effect via weak value amplification,” Opt. Lett. 39, 2266 (2014),
URL http://ol.osa.org/abstract.cfm?URI=ol-39-8-2266.
[54] G. Jayaswal, G. Mistura, and M. Merano, “Observing angular deviations in
light-beam reflection via weak measurements,” Opt. Lett. 39, 6257 (2014),
URL http://ol.osa.org/abstract.cfm?URI=ol-39-21-6257.
104
[55] A. Feizpour, M. Hallaji, G. Dmochowski, and A. M. Steinberg, “Observation
of the nonlinear phase shift due to single post-selected photons,” Nature
Physics 11, 905 (2015).
[56] G. Strubi and C. Bruder, “Measuring ultrasmall time delays of light by joint
weak measurements,” Phys. Rev. Lett. 110, 083605 (2013), URL http://
link.aps.org/doi/10.1103/PhysRevLett.110.083605.
[57] A. Feizpour, X. Xing, and A. M. Steinberg, “Amplifying single-photon non-
linearity using weak measurements,” Phys. Rev. Lett. 107, 133603 (2011),
URL http://link.aps.org/doi/10.1103/PhysRevLett.107.133603.
[58] V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,”
Nat Photon 5, 222 (2011), URL http://dx.doi.org/10.1038/nphoton.
2011.35.
[59] M. M. Nieto, “Quantum phase and quantum phase operators: some physics
and some history,” Physica Scripta 1993, 5 (1993), URL http://stacks.
iop.org/1402-4896/1993/i=T48/a=001.
[60] M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University
Press, 1997).
[61] S. Barnett, C. Fabre, and A. Matre, “Ultimate quantum limits for resolution
of beam displacements,” The European Physical Journal D - Atomic, Molec-
ular, Optical and Plasma Physics 22, 513 (2003), ISSN 1434-6060, URL
http://dx.doi.org/10.1140/epjd/e2003-00003-3.
[62] R. Jozsa, “Complex weak values in quantum measurement,” Phys. Rev. A
76, 044103 (2007), URL http://link.aps.org/doi/10.1103/PhysRevA.
76.044103.
105
[63] C. Ferrie and J. Combes, “How the result of a single coin toss can turn
out to be 100 heads,” Phys. Rev. Lett. 113, 120404 (2014), URL http:
//link.aps.org/doi/10.1103/PhysRevLett.113.120404.
[64] M. F. Pusey, “Anomalous weak values are proofs of contextuality,” Phys.
Rev. Lett. 113, 200401 (2014), URL http://link.aps.org/doi/10.1103/
PhysRevLett.113.200401.
[65] A. Romito, A. N. Jordan, Y. Aharonov, and Y. Gefen, “Weak values are
quantum: you can bet on it,” arXiv preprint arXiv:1508.06304 (2015).
[66] J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Pub-
lishers, 2005).
[67] H. Liero and S. Zwanzig, Introduction to the Theory of Statistical Inference
(CRC Press Taylor & Francis Group, 2012).
[68] P. Refregier, Noise Theory and Application to Physics (Springer, 2004).
[69] S. M. Kay, Fundamentals of Statistical Signal Processing (Prentics-Hall,
1993).
[70] T. Pfister, A. Fischer, and J. Czarske, “Cramrrao lower bound of laser doppler
measurements at moving rough surfaces,” Measurement Science and Technol-
ogy 22, 055301 (2011), URL http://stacks.iop.org/0957-0233/22/i=5/
a=055301.
[71] C.-F. Li, X.-Y. Xu, J.-S. Tang, J.-S. Xu, and G.-C. Guo, “Ultrasensitive
phase estimation with white light,” Phys. Rev. A 83, 044102 (2011), URL
http://link.aps.org/doi/10.1103/PhysRevA.83.044102.
[72] J. C. Howell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan,
“Interferometric weak value deflections: Quantum and classical treatments,”
106
Phys. Rev. A 81, 033813 (2010), URL http://link.aps.org/doi/10.1103/
PhysRevA.81.033813.
[73] V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced positioning
and clock synchronization,” Nature 412, 417 (2001).
[74] J. Guena, R. Li, K. Gibble, S. Bize, and A. Clairon, “Evaluation of doppler
shifts to improve the accuracy of primary atomic fountain clocks,” Phys.
Rev. Lett. 106, 130801 (2011), URL http://link.aps.org/doi/10.1103/
PhysRevLett.106.130801.
[75] B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof,
X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with
accuracy and stability at the 10-18 level,” Nature 506, 7175 (2014), URL
http://dx.doi.org/10.1038/nature12941.
[76] A. Meier and T. Roesgen, “Imaging laser doppler velocimetry,” Experiments
in Fluids 52, 1017 (2012), ISSN 0723-4864, URL http://dx.doi.org/10.
1007/s00348-011-1192-1.
[77] T. O. H. Charrett, S. W. James, and R. P. Tatam, “Optical fibre laser
velocimetry: a review,” Measurement Science and Technology 23, 032001
(2012), URL http://stacks.iop.org/0957-0233/23/i=3/a=032001.
[78] J. Czarske and H. Muller, “Two-dimensional directional fiver-optic laser
doppler anemometer based on heterodyning by means of a chirp frequency
modulated nd: {YAG} miniature ring laser,” Optics Communications
132, 421 (1996), ISSN 0030-4018, URL http://www.sciencedirect.com/
science/article/pii/0030401896004282.
[79] L. Scalise, Y. Yu, G. Giuliani, G. Plantier, and T. Bosch, “Self-mixing
laser diode velocimetry: Application to vibration and velocity measurement,”
IEEE Trans. Instrum Measur. 53, 223 (2004).
107
[80] J. M. Hogan, J. Hammer, S.-W. Chiow, S. Dickerson, D. M. S. Johnson,
T. Kovachy, A. Sugarbaker, and M. A. Kasevich, “Precision angle sensor
using an optical lever inside a sagnac interferometer,” Opt. Lett. 36, 1698
(2011), URL http://ol.osa.org/abstract.cfm?URI=ol-36-9-1698.
[81] N. W. M. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a mea-
surement of a “weak value”,” Phys. Rev. Lett. 66, 1107 (1991), URL
http://link.aps.org/doi/10.1103/PhysRevLett.66.1107.
[82] J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Col-
loquium : Understanding quantum weak values: Basics and applications,”
Rev. Mod. Phys. 86, 307 (2014), URL http://link.aps.org/doi/10.1103/
RevModPhys.86.307.
[83] H. F. Hofmann, M. E. Goggin, M. P. Almeida, and M. Barbieri, “Estimation
of a quantum interaction parameter using weak measurements: Theory and
experiment,” Phys. Rev. A 86, 040102 (2012), URL http://link.aps.org/
doi/10.1103/PhysRevA.86.040102.
[84] G. C. Knee, G. A. D. Briggs, S. C. Benjamin, and E. M. Gauger, “Quantum
sensors based on weak-value amplification cannot overcome decoherence,”
Phys. Rev. A 87, 012115 (2013), URL http://link.aps.org/doi/10.1103/
PhysRevA.87.012115.
[85] G. B. Alves, B. M. Escher, R. L. de Matos Filho, N. Zagury, and L. Davi-
dovich, “Weak-value amplification as an optimal metrological protocol,”
Phys. Rev. A 91, 062107 (2015), URL http://link.aps.org/doi/10.1103/
PhysRevA.91.062107.
[86] M. D. Turner, C. A. Hagedorn, S. Schlamminger, and J. H. Gundlach, “Pico-
radian deflection measurement with an interferometric quasi-autocollimator
108
using weak value amplification,” Opt. Lett. 36, 1479 (2011), URL http:
//ol.osa.org/abstract.cfm?URI=ol-36-8-1479.
[87] S. Chen, X. Zhou, C. M. Hailu Luo, and S. Wen, “Modified weak mea-
surements for detecting photonic spin hall effect,” arXiv preprint 1505.04239
(2015).
[88] A. Nishizawa, K. Nakamura, and M.-K. Fujimoto, “Weak-value amplification
in a shot-noise-limited interferometer,” Phys. Rev. A 85, 062108 (2012), URL
http://link.aps.org/doi/10.1103/PhysRevA.85.062108.
[89] K. Nakamura, A. Nishizawa, and M.-K. Fujimoto, “Evaluation of weak mea-
surements to all orders,” Phys. Rev. A 85, 012113 (2012), URL http:
//link.aps.org/doi/10.1103/PhysRevA.85.012113.
[90] S. Pang, S. Wu, and Z.-B. Chen, “Weak measurement with orthogonal
preselection and postselection,” Phys. Rev. A 86, 022112 (2012), URL
http://link.aps.org/doi/10.1103/PhysRevA.86.022112.
[91] T. Koike and S. Tanaka, “Limits on amplification by aharonov-albert-
vaidman weak measurement,” Phys. Rev. A 84, 062106 (2011), URL http:
//link.aps.org/doi/10.1103/PhysRevA.84.062106.
[92] A. D. Lorenzo, “Weak values and weak coupling maximizing the out-
put of weak measurements,” Annals of Physics 345, 178 (2014), ISSN
0003-4916, URL http://www.sciencedirect.com/science/article/pii/
S0003491614000621.
[93] S. Wu and Y. Li, “Weak measurements beyond the aharonov-albert-vaidman
formalism,” Phys. Rev. A 83, 052106 (2011), URL http://link.aps.org/
doi/10.1103/PhysRevA.83.052106.
109
[94] P. Hariharan and M. Roy, “A geometric-phase interferometer,” Journal of
Modern Optics 39, 1811 (1992).
[95] J. C. Loredo, O. Ortız, R. Weingartner, and F. De Zela, “Measurement of
pancharatnam’s phase by robust interferometric and polarimetric methods,”
Phys. Rev. A 80, 012113 (2009), URL http://link.aps.org/doi/10.1103/
PhysRevA.80.012113.
[96] K. Lyons, J. Dressel, A. N. Jordan, J. C. Howell, and P. G. Kwiat, “Power-
recycled weak-value-based metrology,” Phys. Rev. Lett. 114, 170801 (2015),
URL http://link.aps.org/doi/10.1103/PhysRevLett.114.170801.
[97] J. P. Groen, D. Riste, L. Tornberg, J. Cramer, P. C. de Groot, T. Picot,
G. Johansson, and L. DiCarlo, “Partial-measurement backaction and non-
classical weak values in a superconducting circuit,” Phys. Rev. Lett. 111,
090506 (2013), URL http://link.aps.org/doi/10.1103/PhysRevLett.
111.090506.
[98] M. Blok, C. Bonato, M. Markham, D. Twitchen, V. Dobrovitski, and R. Han-
son, “Manipulating a qubit through the backaction of sequential partial mea-
surements and real-time feedback,” Nature Physics 10, 189 (2014).
110
A Velocity Experiment: Bright
Port Analysis
We study the relative Fisher information of the velocity experiment. We begin
with the intensity equations of the exit beam out of the Michelson interferometer
as in
Iout(t) = I0 exp(−t2/2τ 2
)sin2(φ+ k0vt)
cos2(φ+ k0vt)
. (A.1)
We factor out a sin2(φ) and a cos2(φ) from the dark and bright port, respectively.
Then expand to first order in v. Then using the weak-value approximation kτ �
φ� 1 we arrive with output intensity profile
Iout(t) = I0 exp(−t2/2τ 2
) sin2(φ)(1 + cot(φ)k0vt)2
cos2(φ)(1− tan(φ)k0vt)2
= I0
sin2(φ) exp [−(t− δd)2/2τ 2]
cos2(φ) exp [−(t+ δb)2/2τ 2]
, (A.2)
where δtd = 2k0vτ2 cotφ and δtb = 2k0vτ
2 tanφ. Now we write the intensity as
two separate probability distributions. We calculate the Fisher information with
respect to the parameter of interest of velocity v for the dark and bright port
sing the Fisher information equation from chapter 1 Eq. (1.60). For our case the
Fisher information is
I(v) = −E[∂2
∂v2logP (t; v)
], (A.3)
111
where the probability function P for the dark and bright port
PD(x; v) =
ND∏i=1
1√2πτ 2
exp
[(ti − δtd)2
2τ 2
], (A.4a)
and
PB(x; v) =
NB∏i=1
1√2πτ 2
exp
[(ti − δtb)2
2τ 2
], (A.4b)
respectively. Here NB = N sin2 φ and ND = N cos2 φ, where N is the number of
independent events or a single photons as in chapter 2. This leads to the Fisher
information of the dark port and the bright port,
ID(v) = 4k2τ 2N cos2(φ), (A.5a)
and
IB(v) = 4k2τ 2N sin2(φ), (A.5b)
respectively. We note we can only arrive with an efficient estimator in the dark
port when we use the small angle approximation of the weak-value amplification
technique. Now we see the relative Fisher information between the dark and
bright ports have the same behavior as in Eq. (2.20).
112
B Concatenated Postselection
B.1 Quarter-Half-Quarter: Pancharatnam-Berry
phase
Inside the Sagnac interferometer the polarization dependent phase is controlled
by the wave plate combination of quarter, half, and quarter wave plates. The
quarter wave plates are set to ±45◦ denoted as the Q matrices and the half wave
plate is denoted by the H matrix. We denote the product of the three wave plate
combination as the C matrix. We will represent the wave plate matrices in the
Jones matrix formalism in the polarization basis of H and V polarization.
C|�〉,|〉
(±φ
4
)= Q (45◦) H
(±φ
4
)Q (−45◦)
=1
2
1 −i
−i 1
cos(φ/2) ± sin(φ/2)
± sin(φ/2) − cos(φ/2)
1 i
i 1
=
0 −e∓iφ/2
e±iφ/2 0
(B.1)
We note, that this configuration of wave plates give a geometric phase that de-
pends on the polarization of the beam.
113
We note the symmetry in C(±φ/4) is broken by the beam propagation direc-
tion either clockwise (| �〉) with +φ/4 or counter clockwise (| 〉) with−φ/4 as the
half-wave plate angle. The wave plate combination C(±φ/4) changes the state as
follows: C(φ/4)| �〉⊗|H〉 = eiφ/2| �〉⊗|V 〉, C(−φ/4)| 〉⊗|H〉 = e−iφ/2| 〉⊗|V 〉,
C(φ/4)| �〉⊗|V 〉 = −e−iφ/2| �〉⊗|H〉 and C(−φ/4)| 〉⊗|V 〉 = −eiφ/2| 〉⊗|H〉.
B.2 Weak-value Quantum Description
The preparation of state for this experiment deals with joint space between the
which-path and polarization degree of freedom. The input state is first linearly
polarized to the anti-diagonal state 1√2(|H〉 − |V 〉). Then the state enters the
beam splitter of the Sagnac interferometer. We define the state after the beam
splitter as
|ξ〉 =1
2(| �〉+ i| 〉)⊗ (|H〉 − |V 〉)). (B.2)
To write the input state before the interaction we include the polarization depen-
dent phase φ/2 described in Appendix B.1.
|ϕ〉1 = C (±φ/4) |ξ〉
=1
2
((| �〉eiφ/2 + i| 〉e−iφ/2)⊗ |V 〉+ (| �〉e−iφ/2 + i| 〉eiφ/2)⊗ |H〉
).
(B.3)
The interferometer has an interaction given by U = exp(ikAx), where the trans-
verse momentum kick k is coupled through the ancillary system operator A to
the meter x. The ancillary system operator A is given in the which path basis by
A = | �〉〈� | − | 〉〈 |. The parameter of interest is the transverse momentum
kick k that the beam of light receives from the reflected port of the BS. Then the
114
first postselection is conducted with spatial interference. The postselected state
is nearly orthogonal to the input state and is given by
|ϕ〉2 =1√2
(| �〉 − i| 〉). (B.4)
We assume the interaction is weak such that we can expand to O(k1) and can
define a weak-value for both horizontal and vertical polarizations. The postselec-
tion is only for the spatial degree of freedom so we have two weak-values given by
AHwv =〈H|〈ϕ2|A|ϕ1〉〈H|〈ϕ2|ϕ1〉
= i cot(φ/2), (B.5a)
and
AVwv =〈V |〈ϕ2|A|ϕ1〉〈V |〈ϕ2|ϕ1〉
= −i cot(φ/2). (B.5b)
We have two weak-values of opposite signs, thus the separation between the two
polarization components becomes 2|δxs| = 4|kσ2 cot(φ/2)| and the signal on the
detector is null. Then we introduce a second postselection in the polarization
basis.
The second postselection is through polarization interference, and the postse-
lected state is given by
|ϕ〉3 =1√2
[(sin θ − cos θ)|H〉 − (sin θ + cos θ)|V 〉] . (B.6)
The angle θ is a small angle bias that determines the orthogonality between the
pre- and postselections in the polarization basis. With the concatenated postse-
lection we have a total effective weak-value given by
ACwv =〈ϕ3|〈ϕ2|A|ϕ1〉〈ϕ3|〈ϕ2|ϕ1〉
= i cot(φ/2) cot(θ). (B.7)
With this concatenated configuration we amplify the visibility of the weak-value
and improve the contrast of the spatial interference by adding the polarization
degree of freedom.
115
B.3 Deviation of First Order in k from the All
Order in k Theory
The concatenated post-selection allows the contrast of the interference to be higher
than one postselection by achieving a relative smaller post-selection angle. There-
fore, it is important to track the deviation of the first order approximation in k
from the full theory to see when the enhancement ceases to be accurate or when
other effects start to dominate.
The probability distribution of the weak-value O(k1) and all order approxima-
tion in k are given by
P1(x; k) =1√
2πσ2e−
(x−2kσ2 cot(φ/2))2
2σ2 , (B.8a)
Pall(x; k) =2 sin2(kx+ φ
2)
(1− e−2k2σ2 cos(φ))√
2πσ2e−
x2
2σ2 . (B.8b)
From the probability distributions we determine the mean shift of the meter state
in position space
〈x〉1 =
∫xP1(x; k)dx = 2kσ2 cot(φ/2). (B.9a)
and
〈x〉all =
∫xPall(x; k)dx =
2kσ2 sin(φ)
e2k2σ2 − cos(φ). (B.9b)
With these two predictions of the mean beam shift, we study deviation of the first
order in k approximation to the full theory. We write the deviation as a function
of postselection angle as
D(φ) =
∣∣∣∣〈x〉all − 〈x〉1〈x〉all
∣∣∣∣ =
∣∣∣∣∣ 1− e2k2σ2
cos(φ)− 1
∣∣∣∣∣ . (B.10)
With this function we determine all our measurements to be well within the first
order approximation. The first order in k approximation is sufficient with less
116
0 0.5 1 1.5 2 2.5 30
1
2
3
4
φ/ 2 (deg)D
evia
tion
(%)
Figure B.1: (Color online) The Percentage of deviation between the first and all order in k
theory of the mean shift of the beam in Eq. (B.10).
than half a percent deviation from the full theory for all of our measurements.
We plot the percentage of Eq. (B.10) in Fig. B.1 as a function of postselecting
angle φ/2 with parameter values of k = 2.74 m−1 and σ = 550 µm.
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