Weak-value Metrology and Shot-Noise Limited...

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).


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)


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) =1√µ0

∑m

√~ωm

2[a†m(t) + am(t)]um(kmz), (1.10)

and

By(z, t) =√µ0

∑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


. (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


. (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

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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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