Quantum Imaging and Information

by

P. Benjamin Dixon

Submittted in Partial Fulfillment

of the

Requirements for the Degree

Doctor of Philosophy

Supervised by

Professor John C. Howell

Department of Physics and Astronomy

Arts, Sciences and Engineering

School of Arts and Sciences

University of Rochester

Rochester, New York

2011

Dedicated to my parents and grandparents.

iii

Curriculum Vitae

P. Ben Dixon was born on October 16, 1981 in Hanover, NH. He attended

the University of Florida as a Florida Academic Scholar from 2000 to 2005 and

graduated with a Bachelor of Science in Mechanical Engineering in 2005. He came

to the University of Rochester in the Summer of 2006 and received a Master

of Arts in Physics in 2008. He pursued his doctoral research in experimental

quantum optics under the supervision of John C. Howell.

CURRICULUM VITAE iv

Publications

1. Quantum Mutual Information Capacity for High Dimensional En-

tangled States, P. Ben Dixon, Gregory A. Howland, James Schneeloch,

and John C. Howell, arXiv:1107.5245v1 [quant-ph], in submission.

2. A theoretical analysis of quantum ghost imaging through turbu-

lence, Kam Wai Clifford Chan, D. S. Simon, A. V. Sergienko, Nicholas

D. Hardy, Jeffrey H. Shapiro, P. Ben Dixon, Gregory A. Howland, John

C. Howell, Joseph H. Eberly, Malcolm N. O’Sullivan, Brandon Rodenburg,

and Robert W. Boyd, Physical Review A 84, 04807 (2011).

3. Photon-Counting Compressive Sensing Lidar for 3D Imaging, Gre-

gory A. Howland, P. Ben Dixon, and John C. Howell, Applied Optics, 50,

5917− 5920 (2011).

4. Quantum ghost imaging through turbulence, P. Ben Dixon, Gre-

gory A. Howland, Kam Wai Clifford Chan, Colin O’Sullivan-Hale, Bran-

don Rodenburg, Nicholas D. Hardy, Jeffrey H. Shapiro, D. S. Simon, A.

V. Sergienko, R. W. Boyd and John C. Howell, Physical Review A 83,

051803(R) (2011).

5. Precision frequency measurements with interferometric weak val-

ues, David J. Starling , P. Ben Dixon, Andrew N. Jordan and John C.

Howell, Physical Review A 82, 063822 (2010).

6. Heralded single-photon partial coherence, P. Ben Dixon, Gregory How-

land, Mehul Malik, David J. Starling, R. W. Boyd, and John C. Howell,

Physical Review A 82, 023801 (2010).

7. Continuous phase amplification with a Sagnac interferometer, David

J. Starling, P. Ben Dixon, Nathan S. Williams, Andrew N. Jordan, and

John C. Howell, Physical Review A 82, 011802(R) (2010).

8. Interferometric weak value deflections: Quantum and classical

treatments, John C. Howell, David J. Starling, P. Ben Dixon, Praveen K.

Vudyasetu, and Andrew N. Jordan, Physical Review A 81, 033813 (2010).

CURRICULUM VITAE v

9. Optimizing the signal-to-noise ratio of a beam-deflection measure-

ment with interferometric weak values, David J. Starling, P. Ben Dixon,

Andrew N. Jordan, and John C. Howell, Physical Review A 80, 041803(R)

(2009).

10. Ultrasensitive Beam Deflection Measurement via Interferometric

Weak Value Amplification, P. Ben Dixon, David J. Starling, Andrew N.

Jordan, and John C. Howell, Physical Review Letters 102, 173601 (2009).

11. Realization of an All-Optical Zero to π Cross-Phase Modulation

Jump, Ryan M. Camacho, P. Ben Dixon, Ryan T. Glasser, Andrew N.

Jordan, and John C. Howell, Physical Review Letters 102, 013902 (2009).

12. On the feasibility of detection and identification of individual

bioaerosols using laser-induced breakdown spectroscopy, P. Ben Dixon

and D. W. Hahn, Analytical Chemistry, 77:631-638 (2005).

CURRICULUM VITAE vi

Conference Proceedings

1. Quantum Ghost Imaging through Turbulence, P. B. Dixon, G. A.

Howland, K. W. C. Chan, C. O’Sullivan-Hale, B. Rodenburg, N. D. Hardy,

J. H. Shapiro, D. S. Simon, A. V. Sergienko, R. W. Boyd and J. C. Howell, in

Advanced Photonics Congress: Optical Sensors (Optical Society of America,

2011), p. SWD3.

2. Weak Values and Beam Deflection Measurements, P. B. Dixon, D.

J. Starling, N. S. Williams, P. K. Vudyasetu, A. N. Jordan, and J. C. Howell,

in Frontiers in Optics (Optical Society of America, 2010), p. FTuE4.

3. Heralded Single Photon Partial Coherence, P. B. Dixon, G. A. How-

land, M. Malik, D. J. Starling, R. W. Boyd, and J. C. Howell, in Confer-

ence on Lasers and Electro-Optics (Optical Society of America, 2010). p.

CMCC2.

4. All Optical Zero to π Cross Phase Modulation, P. B. Dixon, R. M.

Camacho, R. T. Glasser, A. N. Jordan, and J. C. Howell, in Frontiers in

Optics (Optical Society of America, 2008), p. FTuI6.

vii

Acknowledgments

It is a pleasure to thank the people who helped me in the process of my

graduate studies. I thank my advisor, John C. Howell, for giving me help and

support in many aspects of my life, including guiding my research. In addition

to John’s guidance there has been a sense of humor and a fantastic sense of

camaraderie in the laboratory work environment—for this I would like to thank

the graduate students that I have worked closely with including: Irfan Ali Khan,

Curtis J. Broadbent, Ryan M. Camacho, Michael V. Pack, David J. Starling,

Gregory A. Howland, and the visiting Ryan T. Glasser. The physics department

faculty and staff has helped me navigate the University rules and regulations, for

this I thank: Barbara Warren, Sondra Anderson, Janet Fogg-Twichell, Michie

Brown, Connie M. Hendricks, Connie Jones, Ali DeLeon, Patricia T. Sulouff,

Eric Blackman, Dan Watson, Arie Bodek, and Nicholas Bigelow. I thank the

collaborators outside of my lab who have I have worked and who helped my

investigations. These people include, at the University of Rochester: Kam Wai

Cliff Chan, Justin Dressel, Nathan S. Williams, Colin O’Sullivan-Hale, Brandon

Rodenburg, Andrew N. Jordan, Robert W. Boyd, Joseph Eberly, and Emil Wolf,

and at other institutions, Alexander V. Sergienko, David Simon, Jeffrey Shapiro,

and Nicholas Hardy. I would like to thank the entire University community and

larger Rochester community for providing me with a wonderful place in which to

live and study. Finally, I would like to thank Ellie Rose Adair for her support

and care.

viii

Abstract

Quantum optics provides a unique avenue to investigate quantum me-

chanical effects. Typically, it is easier to observe the particle-like behavior of a

physical object than it is to observe wave-like behavior. Optics presents us with

the reverse case, observing the particle-like behavior of light is difficult. I investi-

gate the utility and limitations of two quantum mechanical effects—weak values

and spatial entanglement—in the context of experimental quantum optical com-

munication channels. I show that weak values can be used to increase the signal

power and effectively decrease the noise power in a physical communication chan-

nel, up to the standard quantum limit for signal to noise ratio. I also show show

a method for decreasing the negative environmental effects on a communication

channel using spatial entanglement and show that such a channel can be used to

transmit over 7 bits of information per joint photon detection event.

ix

Table of Contents

Foreword 1

Chapter 1. Introduction 3

1.1 Information Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Discrete Probabilities . . . . . . . . . . . . . . . . . . . . . 5

1.1.2 Continuous Probability Densities . . . . . . . . . . . . . . . 8

1.1.3 Channel Limitations . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Low Dimensional Images and Weak values . . . . . . . . . . . . . 13

1.2.1 Weak Values . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2.2 Deflection Amplification . . . . . . . . . . . . . . . . . . . . 17

1.2.3 Controversy . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.2.4 What Is Classical and What Isn’t . . . . . . . . . . . . . . . 19

1.2.5 Weak Value Investigations . . . . . . . . . . . . . . . . . . . 20

1.3 High Dimensional Images and Entanglement . . . . . . . . . . . . 20

1.3.1 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.3.2 Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.3.3 Bell Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 23

1.3.4 Nonlocality . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.3.4.1 Entropic Uncertainty . . . . . . . . . . . . . . . . . 27

1.3.5 Related Concepts . . . . . . . . . . . . . . . . . . . . . . . 30

1.3.6 Spontaneous Parametric Down-Conversion . . . . . . . . . . 30

1.3.7 Entanglement in High Dimensional Imaging Applications . 35

Chapter 2. Weak Values and Deflection 36

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2 Theoretical Description . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3 Experiment and Results . . . . . . . . . . . . . . . . . . . . . . . 40

2.4 Channel Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 47

x

Chapter 3. Weak Values SNR for Deflections 48

3.1 Theoretical Description . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Technical Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.4 Channel Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 60

Chapter 4. Ghost Imaging Through Turbulence 61

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2 Theoretical Description . . . . . . . . . . . . . . . . . . . . . . . . 64

4.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.4 Channel Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Chapter 5. Mutual Information 76

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2 Theoretical description . . . . . . . . . . . . . . . . . . . . . . . . 78

5.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 89

Chapter 6. Conclusion 90

xi

List of Figures

1.1 Relation between mutual information and marginal entropies . . . 6

1.2 Relation between mutual information and conditional entropies . 7

1.3 Electric field and intensity plots of low order TEM modes . . . . . 13

1.4 Electric field and intensity plots of a sum of low order TEM modesas a beam deflection . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5 Example weak value experiment using beam deflection and polar-ization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.6 Effect on electric fields in weak value experiment using polarizationand beam deflection . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.7 An EPR experiment using photon polarization measurement . . . 23

1.8 A Bell experiment using photon polarization measurements . . . . 24

1.9 A “map” of related concepts in quantum mechanics . . . . . . . . 31

1.10 A conceptual SPDC interaction in a nonlinear crystal . . . . . . . 32

1.11 Probability density for a position-momentum entangled state . . . 33

1.12 Probability density for a separable state . . . . . . . . . . . . . . 34

2.1 Experimental setup for interferometric weak values beam deflec-tion measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.2 Effect of beam radius on interferometric weak values beam deflec-tion measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.3 Angular mirror displacement in interferometric weak values beamdeflection measurement . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1 Experimental setup for interferometric weak values signal to noisemeasurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2 Signal to noise ratio for interferometric weak values metrology andstandard metrology techniques for different deflections . . . . . . . 55

3.3 Signal to noise ratio for interferometric weak values metrology andstandard metrology techniques for different beam sizes . . . . . . 56

4.1 Experimental setup for ghost imaging through turbulence mea-surement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2 Conceptual setup for ghost imaging through turbulence measurement 63

4.3 Conceptual setup for ghost imaging through turbulence measurement 63

4.4 Representative ghost image profiles . . . . . . . . . . . . . . . . . 69

4.5 Ghost image visibilities turbulence near the object . . . . . . . . . 70

xii

4.6 Ghost image visibilities for turbulence near the illumination source 71

4.7 Increased mutual information in the novel configuration . . . . . . 73

5.1 Experimental setup for high dimensional quantum mutual infor-mation characterization . . . . . . . . . . . . . . . . . . . . . . . . 78

5.2 High dimensional mutual information capacity data for positioncorrelation measurements . . . . . . . . . . . . . . . . . . . . . . . 83

5.3 High dimensional mutual information capacity data for momentumcorrelation measurements . . . . . . . . . . . . . . . . . . . . . . . 84

1

Foreword

This dissertation investigates two quantum mechanical effects: weak val-

ues and spatial entanglement. These effects are investigated in the context of

quantum imaging experiments. I introduce the main concepts of images, weak

values, spatial entanglement, and information theory in chapter 1. This broad

topic introduction summarizes the standard understanding of these topics and

introduces no new research or results. New research is described in chapters

2 through 5. Experiments investigating weak value metrology are described in

chapters 2 and 3. Experiments involving ghost imaging relying on spatial en-

tanglement are described in chapters 4 and 5. In all of these chapters, relevant

theoretical descriptions are given along with discussions of the results and their

meaning.

The research is experimental in nature and the light sources, optics, and

detection equipment used in the experiments were commercially available. The

scope of the research was not to create new detectors or sources, but rather

to create or observe new physical effects with the equipment available. The

experiments were small in scale, taking up no more than several square feet of

optical table space, with additional space for supporting detection equipment.

In all of the experiments presented, I built the experiment and took the

data as part of a small team of two to three graduate students. The original ideas

for each experiment came mainly from John C. Howell in discussions with the

graduate student teams. For the results in chapter 2 the experimental team of

graduate students consisted of David J. Starling and myself; Andrew N. Jordan

2

was the main contributor to the theoretical analysis, and I was the main contribu-

tor to the data analysis and manuscript preparation. For the results in chapter 3,

the experimental team of graduate students again consisted of David J. Starling

and myself; John C. Howell, David J. Starling, and Andrew N. Jordan were the

main contributors to the theoretical description, data analysis and manuscript

preparation. For the results in chapter 4 the experimental team of graduate stu-

dents consisted of Gregory A. Howland and myself; I was the main contributor

to the data analysis and manuscript preparation and the theoretical analysis was

performed mainly by Kam Wai Cliff Chan and myself. For the results in chapter

5, the experimental team of graduate students consisted of Gregory A. Howland,

James Schneeloch, and myself; I was the main contributor to the data analysis

and manuscript preparation, the theoretical analysis was performed mainly by

John C. Howell and myself.

Additional helpful contributions were made by Nathan S. Williams, Justin

Dressel, Curtis Broadbent, Joseph Eberly, and Emil Wolf, all from the Depart-

ment of Physics and Astronomy here at the University of Rochester; Robert

W. Boyd, Colin O’Sullivan-Hale, and Brandon Rodenburg from the Institute of

Optics here at the University of Rochester; Alexander V. Sergienko and David

Simon from Boston University; and Jeffrey Shapiro and Nicholas Hardy from the

Massachusetts Institute of Technology.

3

Chapter 1

Introduction

The field of quantum optics—concerned with the quantum mechanical

properties of light—begins with the start of quantum mechanics itself. Max

Planck was investigating electromagnetic radiation when he made his initial quan-

tum conjecture; that light could only be emitted with a quanta of energy E = hν

[1, 2]. Einstein added to the beginnings of quantum mechanics with his work on

the photo-electric effect in which he again discussed the quanta of light energy.

Einstein’s work proposed the idea that the quantized energy of light was a fun-

damental aspect of light itself, not just the emission process [3]. These quanta of

light are what modern physicists call “photons”—a term coined in the 1920’s.

The stellar interferometry experiment of Robert Hanbury Brown and Richard

Q. Twiss, using intensity correlations led to much debate and research that clar-

ified aspects of quantum optics [4]. As Brown and Twiss pointed out in response

to the controversy, the predictions from Maxwell’s equations for classical elec-

trodynamics are identical to the predictions of quantum optics with quantized

photo-detections (i.e. detecting photons) [5]. Their experiment inspired the for-

mulation of a quantum mechanical theory of optical coherence and detection

[6–10], extending the earlier work in classical coherence to the quantum optical

and single photon domain [11].

More recently, the quantum mechanical properties of images and imaging

techniques have been investigated [12–14]. Imaging, a subfield of optics, deals

4

primarily with the transverse spatial degrees of freedom of an optical field. Re-

search has included quantum ghost imaging [15–17], quantum communication

[18, 19], and quantum enhanced sensing and lithography [20–23]. Two interest-

ing features of quantum mechanics are entanglement and weak values. These

concepts are connected to quantum measurement theory and for this reason have

the potential for practical use in metrology or measurement based communication

technology. The research I present aims to answer the question: What are prac-

tical applications of these effects and what are the limits of those applications?

It is often the case when dealing with complex problems that considera-

tion of a related but simplified problem is useful. In imaging applications this

corresponds to considering low dimensional images first, and then progressing to

higher dimensional images. I use the term “dimension” here to refer not to the

spatial dimensions of the image (width and height for example), but rather to

the number of free parameters needed to fully characterize the image.

The concept of a low dimensional image may seem counter-intuitive, but

is actually very useful in target tracking or in scanning [24, 25]; although a full

image is usually collected, the location of a target in that image can be accurately

described by a small number of free parameters. High dimensional images are

more in line with what we think of as an image. These images are in principle

infinite dimensional, however in many cases continuous locations and intensities

are digitized, reducing them to a high but finite dimension.

Both low and high dimensional signals can be thought of as communication

channels. Techniques to improve aspects of these images can then be cast in

terms of improved channel capacity. This approach allows seemingly different

experiments to be treated on the same footing, making their similarities and

connections more transparent.

5

In this chapter I review both low and high dimensional imaging systems

and their characterization in terms of communication channels. I also review two

non-classical concepts—weak values and spatial entanglement—and discuss their

role in these imaging systems.

1.1 Information Theory

Information theory is a relatively young field, beginning in the 1940’s

with the seminal work of Claude Shannon [26, 27]. It concerns itself with the

communication over a noisy channel and uses entropy to quantify information.

Communication or information transfer can be quantified in terms of mutual

information, here I describe the concept of mutual information for continuous

and discrete probabilities.

1.1.1 Discrete Probabilities

Consider random variables X (the sent message) and Y (the received

message) which take the values x and y, characterized by the discrete probabilities

P (x) and P (y). The mutual information between these variables is the sum of

their individual marginal entropies minus their joint entropy

I(X;Y ) = H(X) +H(Y )−H(X, Y ), (1.1)

where H(X) is the marginal entropy. This is shown conceptually in Fig. 1.1. The

marginal entropy of X is given by

H(X) = −∑x∈X

P (x) log(P (x)

), (1.2)

the joint entropy of X and Y is given by

H(X, Y ) = −∑x∈Xy∈Y

P (x, y) log(P (x, y)

), (1.3)

6

Figure 1.1: The concept of mutual information is shown visually. The systemdefined by the variable X is represented by the blue circle on the left, the systemdefined by the variable Y is represented by the pink circle on the right. Marginalentropies for each system H(X) and H(Y ), respectively, and the entropy of thejoint system H(X, Y ) can be calculated. The mutual information is the overlapof the marginal entropies of systems X and Y . The sum of the marginal entropiesfor system X and system Y counts this overlap region twice, and by subtractingoff the joint system entropy we are left only with the mutual information asdescribed by Eq. 1.1.

7

Figure 1.2: An alternative concept of mutual information is shown visually. Thesystem defined by the variable X is again represented by the blue circle on the leftand the system defined by the variable Y is again represented by the pink circleon the right. Non-overlapping conditional entropies for each system H(X|Y ) andH(Y |X), respectively, can be calculated. The mutual information is the overlapof the entropies of systems X and Y . The joint system entropy H(X, Y ) minusthese conditional entropies gives the mutual information as described by Eq. 1.4.

where the function P (x, y) is the joint probability which characterizes the corre-

lation between X and Y .

An alternative formulation of the mutual information is

I(X;Y ) = H(Y )−H(Y |X) = H(X)−H(X|Y ), (1.4)

where H(X|Y ) is the conditional entropy of X given Y :

H(X|Y ) = −∑x∈Xy∈Y

P (x, y) log(P (x|y)

), (1.5)

where P (x|y) is the probability of X = x given that Y = y. This is shown

conceptually in Fig. 1.2. This formulation of the mutual information can be

useful when the correlation between the random variables is known.

8

1.1.2 Continuous Probability Densities

In considering continuous probability densities, it is common to replace the

discrete probabilities in the entropic calculations with the continuous probability

density functions p(x), p(y), and p(x, y), resulting in

Hc(X) = −∫p(x) log

(p(x)

)dx, (1.6)

Hc(X, Y ) = −∫p(x, y) log

(p(x, y)

)dxdy, (1.7)

and

Hc(X|Y ) = −∫p(x, y) log

(p(x|y)

)dx, (1.8)

where the subscript c indicates the quantity uses continuous probability distri-

butions.

These continuous probability densities however, present somewhat of a

problem for several reasons, including the fact that the probability density func-

tions p(x), p(y), and p(x, y) can exceed 1 (over a sufficiently narrow domain).

Additionally, the densities are no longer unitless and one cannot sensibly take

the logarithm of anything with units. The solution to this problem is to intro-

duce a unit magnitude dimension-conversion constant whose units are the same

as that of the probability amplitude functions. These units will depend on the

nature of what is being measured, but examples include probability per unit time

and probability per unit area, and it is common to suppress the writing of this

conversion term.

The fact that the continuous probability distribution can exceed 1 indi-

cates the continuous and discrete entropic formulas may not converge in the limit

of small discretization widths. The probability density function is related to the

9

discrete probabilities in the following manner:

p(x) = lim∆x→0

P (x)

∆x, (1.9)

where the region of x with significant probability is discretized into b bins of

width ∆x. We now compare the discrete formulas to the continuous ones:

lim∆x→0

H(X) =− lim∆x→0

∑x∈X

p(x)∆x log(p(x)∆x

)=− lim

∆x→0

(∑x∈X

p(x) log(p(x)

)∆x

)− lim

∆x→0

(∑x∈X

p(x) log(∆x)∆x

)

=−∫p(x) log

(p(x)

)dx+ lim

∆x→0log

(1

∆x

)=Hc(X) + lim

∆x→0log

(1

∆x

).

(1.10)

The entropy formulas for discrete and continuous probabilities therefore do not

converge to the same value for small discretization width. They are offset by the

logarithm of the number of discretization bins b;

H(X) ≈ Hc(X) + log(b). (1.11)

This is not the case for the mutual information formula however; because

the mutual information involves adding and subtracting entropies I(X;Y ) =

H(X) +H(Y )−H(X, Y ), the divergent offset terms cancel out, resulting in

I(X;Y ) = Ic(X;Y ). (1.12)

The experiments that I describe involve either communication or measure-

ment schemes, both of these types of experiments can be described using mutual

information. For a communication scheme, the variables are the sent message

and the received message. For a measurement scheme, we can think of the mea-

surement apparatus as the communication channel between the system and the

10

observer. The variable X is then the true value of what is being measured, and

the variable Y is the measured value. All of the measurements I make use discrete

probabilities.

1.1.3 Channel Limitations

Physical channels have several limitations, one of them is that the signal

power itself is limited. This manifests itself as a limitation of the possible values

that X can take. If we assume that the form of this channel limitation is that

it has finite variance 〈x2〉 = S, then, using calculus of variations, the p(x) that

maximizes entropy (and thus the channel capacity) satisfies the equation

d

dx

(∫p(x) log

((p(x)

)dx+ λ1

(∫x2p(x)dx− S

)+ λ2

(∫p(x)dx− 1

))= 0.

(1.13)

This simplifies to the requirement that

p(x) log(p(x)

)= −λ1x

2p(x)− λ2p(x). (1.14)

Assuming a nonzero p(x), this requires

p(x) = exp(−λ1x2 − λ2) = A exp(−λ1x

2), (1.15)

where A = exp(λ2). The conditions that 〈x2〉 = S and∫p(x)dx = 1 require

λ1 = 1/(2S) and A = 1/√

2πS. The result is that the channel probability

distribution that maximizes the entropy, subject to an average power limitation,

is a Gaussian distribution:

p(x) =1√2πS

exp

(−x2

2S

), (1.16)

and the corresponding entropy is:

H(X) = −∫

1√2πS

exp

(−x2

2S

)log

(1√2πS

exp

(−x2

2S

))dx

=1

2log(2πeS

).

(1.17)

11

The multiple spatial dimensions in images act as independent channels.

Entropies of independent channels are additive resulting in

H(X) =n

2log(2πeS

)(1.18)

for images in n spatial dimensions.

Another common model for channel limitation, rather than an average

power limitation, is a peak power limitation. This type of limitation means there

is a finite range of values the signal variable can take. The same type of process

shows that a channel with this limitation maximizes its mutual information when

its probability distribution p(x) is flat across the possible range of values. For

this type of channel limitation, a Gaussian distribution is not possible, however

such a channel will still have a variance in signal power. The maximum entropy

from Eq. 1.18 is then an upper bound that cannot be reached.

In addition to signal power limitations, another fundamental limitation

is that noise is present in the channel. Noise in the channel manifests itself in

the joint probability function p(x, y), or alternatively in p(x|y), the conditional

probability function—the sent message is not perfectly correlated to the received

message. We can model this noise as additive and uncorrelated to the sent mes-

sage, such that the received message distribution is the sent message distribution

plus noise Y = X + Z. The noise distribution Z is a random variable taking

values z with a probability distribution p(z). The mutual information is given

by:

I(X, Y ) = H(Y )−H(Y |X) = H(Y )−H(X +Z|X) = H(Y )−H(Z|X). (1.19)

But the noise Z is assumed to be uncorrelated to the signal X so this reduces to

I(X, Y ) = H(Y )−H(Z). (1.20)

12

The mutual information capacity of a noisy channel is the capacity of the mea-

sured variable minus the capacity of the noise. If we assume, in addition to the

sent message variable being Gaussian distributed with variance 〈x2〉 = S, that

the noise is Gaussian distributed (meaning its deleterious effect is maximized)

with variance 〈z2〉 = N , then 〈y2〉 = 〈(x + z)2〉 = 〈x2〉 + 〈z2〉 + 〈x〉〈z〉 = S + N .

The entropy of Y is then

H(Y ) =n

2log(2πe(S +N)

), (1.21)

and the entropy of the noise is

H(Z) =n

2log(2πeN

), (1.22)

resulting in a mutual information capacity of the channel of

I(X;Y ) =n

2log

(1 +

S

N

). (1.23)

The units of a channel’s mutual information is the somewhat nebulous “in-

formation per transmission.” The logarithm base determines what units are used

for information; for base 2 logarithm information is measured in units of bits.

Meanwhile, the concept of “transmission” depends on the nature of the chan-

nel. Commonly, a transmission is considered to be an interval of time—resulting

in units of bits/second, or an amount of received power—resulting in units of

bits/watt. It should be noted that changing what is considered a “transmission”

changes the noise power. It is in this way that this description accounts for the

standard practice of averaging out noise. In this thesis, we concern ourselves with

“transmissions” consisting of detected single photons or joint detections of pairs

of photons—yielding mutual information in units of bits/photon.

13

Figure 1.3: A profile of the TEM00 mode as defined in Eq. 1.24 is shown in (a),representing the electric field (in arbitrary units). Plots in (b) and (c) show aprofile and a density plot of |TEM00|2, representing the intensity of this field (inarbitrary units). A profile of the TEM10 mode as defined in Eq. 1.25 is shown in(d), representing the electric field (in arbitrary units). Plots in (e) and (f) showa profile and a density plot of |TEM10|2, representing the intensity of this field(in arbitrary units).

1.2 Low Dimensional Images and Weak values

Transverse beam deflections are a simple type of a low dimensional image.

The beam profile makes up the image and the moving beam centroid then makes

up a one or possibly two dimensional characterization. A common mathematical

description of this type of image is to expand the field into transverse electro-

magnetic (TEM) modes. An undeflected beam is simply the Gaussian TEM00

mode given by

TEM00 =

(2

πw2

)1/4

exp

(−x2

w2

), (1.24)

where w is the 1/e2 beam radius (in power). Deflections are represented by

combining the TEM00 with nonsymmetric higher order modes such as the TEM10

mode, given by

TEM10 =

(2x

w

)(2

πw2

)1/4

exp

(−x2

w2

). (1.25)

14

Figure 1.4: A sum of TEM modes as a beam deflection is shown. In the functiondisplayed, 85% of the power is from the TEM00 mode and the remaining 15%is from the TEM00 mode. A cross section of the sum of modes, representingthe electric field (in arbitrary units), is displayed in (a). A cross section and adensity plot of the magnitude square of the mode sum, representing the intensity(in arbitrary units), is shown in (b) and (c), respectively. This sum of modesapproximates a deflection of the beam by almost half the beam radius.

These modes are shown in Fig. 1.3. An example of how these modes can be added

to cause a beam shift is shown in Fig. 1.4 displaying a beam with 85% of the

power in the TEM00 mode and 15% of the power in the TEM10 mode.

In addition to target sensing and tracking, beam deflection measurements

have applications in metrology fields as diverse as positioning, microcantilever

cooling, and atomic force microscopy [24, 28, 29]. The physics of beam deflection

metrology and the ultimate measurement sensitivities of such low dimensional

images have been studied extensively [25, 29–31]. I use standard methods of

creating these low dimensional images, namely using a mirror to tilt a beam,

but use a novel technique—weak values—to increase the channel capacity of this

imaging communication channel.

Weak values are a recent and striking development in quantum physics.

The concept was introduced in 1989 in the context of time symmetric quan-

tum mechanics involving both forward and reverse causality [32, 33]. The ba-

sic premise is as follows: A pre-selected quantum state with multiple degrees

of freedom is weakly perturbed such that two degrees of freedom are linked (or

15

entangled)—the resulting state is then post-selected on only one degree of freedom

and the remaining degree of freedom is measured. This pre- and post-selected

(thus time-symmetric) state can exhibit strange behavior when measured. The

strange behavior can aid in aspects of beam deflection metrology [34], where the

beam is the quantum system and the beam centroid is one of the degrees of free-

dom that we use. Before getting into the details of weak values it is perhaps

advantageous to point out that in many situations involving optics, weak values

can be described classically [35, 36], a point that is discussed more fully in section

1.2.4.

1.2.1 Weak Values

A typical example of weak values using light is as follows: A beam of light

is pre-selected on a polarization state by using a polarizer. The polarized beam

then passes through a thin calcite crystal that introduces a slight relative displace-

ment between certain orthogonal polarization (dependent on how the crystal is

oriented). This is the weak perturbation—entangling the polarization degree of

freedom to the position degree of freedom. The beam is then post-selected on a

polarization state using a polarizer and finally the centroid position is measured.

Strange behavior happens when the post-selection state is nearly orthogonal to

the pre-selection state. Because the position degree of freedom has a higher di-

mensionality (infinite dimensional) than the polarization degree of freedom (two

dimensional), there is ambiguity in assigning specific position measurements to

specific polarization states—it can appear that measured polarization value lies

far outside the normal eigenvalue range.

More formally, we preselect an initial quantum state involving two degrees

of freedom with different dimensionalities, here I use a two dimensional discrete

16

variable P and an infinite dimensional continuous variable x for definiteness:

|ψi〉 = |Pi〉 ⊗∫fi(x)|x〉dx, (1.26)

where fi(x) is is the initial wavefunction of the state in the variable x. We then

weakly perturb the system, linking the degrees of freedom with the interaction

Hamiltonian H = kPx where the modulus of k is the strength parameter for

the perturbation, and P describes how the different polarizations are affected

differently. This results in an intermediate state of

|ψ〉 =

∫eikPx/~|Pi〉 ⊗ fi(x)|x〉dx. (1.27)

As long as kP/~ �√〈x2〉, meaning the x shifts induced by this perturbation

are small compared with the initial wavefunction width, we can approximate the

intermediate state as

|ψ〉 =

∫ (1− ikPx

~

)|Pi〉 ⊗ fi(x)|x〉dx. (1.28)

We then post-select on a final polarization state |Pf〉:

|ψ〉 =

∫〈Pf |

(1− ikPx

~

)|Pi〉 ⊗ fi(x)|x〉dx

= 〈Pf |Pi〉∫ (

1− ik~〈Pf |P |Pi〉〈Pf |Pi〉

x

)⊗ fi(x)|x〉dx.

(1.29)

The overlap term 〈Pf |Pi〉 describes attenuation and is eliminated by renormalizing

the state. This results in the final state:

|ψf〉 =

∫exp

(−ikPwx

h

)⊗ fi(x)|x〉dx, (1.30)

where Pw = 〈Pf |P |Pi〉/〈Pf |Pi〉 is the weak value. This final form is valid assuming

kPwx/~ �√〈x2〉. In this final form we can see that the weak, shifting pertur-

bation is modulated by the weak value, which can in principle be larger than 1.

17

Figure 1.5: An example weak value experiment, described in section 1.2.2, isshown. An unpolarized beam of light is incident on a pre-selection polarizer ori-ented to pass only vertically polarized light. The beam then passes through acalcite crystal oriented to slightly displace diagonal and anti-diagonal polariza-tions. The pre-selected and perturbed beam then passes through a post-selectionpolarizer oriented to pass a polarization slightly off of horizontal. A measurementof the beam deflection is then made.

Viewing the weak value in this way—of modulating a shift of the wavefunction—

allows us to see that the assumption used in the final simplification means simply

that we cannot post-select on a portion of the wavefunction that was not there

to begin with.

1.2.2 Deflection Amplification

For beam deflections we use an initial state involving optical beam polar-

ization and spatial beam profile, pre-selecting on vertical polarization |Pi〉 = |−〉

and post-selecting on slightly off of horizontal polarization |Pf〉 = |+〉 + δ|−〉,

where δ � 1. This experimental setup is shown in Fig. 1.5. We use a calcite

crystal oriented to weakly displace diagonal |D〉 = |+〉 + |−〉 and antidiagonal

|A〉 = |+〉−|−〉 polarizations, described mathematically by P = |A〉〈A|−|D〉〈D|.

The post-selection probability is then 〈Pf |Pi〉 = δ and the weak value is Pw =

2/δ � 1. The small displacement caused by the calcite crystal has been ampli-

18

Figure 1.6: The electric fields at several points in the weak value beam deflec-tion experiment are shown. An initial pre-selected Gaussian beam in arbitraryunits with radius w is shown in (a). The beam that has been perturbed by thecalcite crystal along with the individual diagonal and antidiagonal polarizationcomponents of the beam are shown in (b). The vertical lines show the locationsof the centroids of the diagonal and antidiagonal polarized beam components.The calcite crystal perturbs the beam by 5% of the radius. The perturbed beamalong with the final post-selected beam is shown in (c). The vertical lines repre-senting the polarization component centroids is shown again. The post-selectedbeam is seen to have decreased intensity but its deflection is seen to lie outsidethe range allowed by the pure polarizations. The post-selection is set to give apost selection probability of Pps = 20% and a weak value amplification of A = 10.

fied by the weak value, however the remaining beam power is also attenuated.

The amplification and attenuation in this case are inversely related. The electric

field at various points in the experiment is shown in Fig. 1.6.

1.2.3 Controversy

Because of this strange behavior, weak values have been a controversial

subject in physics—beginning with their introduction in the provocatively titled

paper “How the result of a measurement of a component of the spin of a spin-1/2

particle can turn out to be 100” [32]. Various objections to weak values have

been made including that weak values violate the uncertainty principle [37], and

include an incorrect description of the measurement process [37, 38].

Despite the objections, there has been considerable research into the foun-

19

dations of quantum mechanics involving weak values, including an attempt to

resolve quantum mechanical paradoxes [39], the simultaneous observation of the

wave and the particle nature of a photon [40, 41], and research clarifying the

mathematical description of the measurement process [42].

There has also been more applied weak value research involving amplifying

optical nonlinearities [43, 44], measuring phase and frequency shifts [45–48], and

even incorporating the amplification effect into gravimeters [49].

1.2.4 What Is Classical and What Isn’t

An interesting aspect of weak values is that although they were introduced

in the context of an esoteric quantum phenomenon, in many quantum optics

implementations their description reduces to a purely classical one [35, 36]. This

is due to the fact that superposition and interference—features thought to be

purely quantum mechanical in many cases—are considered to be classical effects

in optics. Indeed, by using a combination of classical tilt and lead interferometry,

one can implement a weak value apparatus.

There are several cases when a weak value experiment is non-classical

and must use a quantum mechanical description. These cases include: when the

particles used in the experiment are massive particles for which wave-like behavior

is inherently quantum, when the entangling interaction leads to non-local effects,

and when the quantity being measured requires a quantum description. This final

case is considered in chapter 3 where light fluctuations, which require a quantum

mechanical description, are measured.

20

1.2.5 Weak Value Investigations

I study the amplification effect of weak values. For low dimensional imag-

ing applications that involve beam deflections, simple analysis suggests that am-

plifying a deflection by a factor of A could boost the noisy channel’s mutual

information from

I(X;Y ) =1

2log

(1 +

S

N

)(1.31)

to

I(X;Y ) =1

2log

(1 + A2 S

N

). (1.32)

This fails to include the possibility of a changed noise spectrum, however. Addi-

tionally, it neglects the fact that the measurement is made on a small percentage

of the photons, affecting a measure of information per detected photon. A more

detailed investigation of the effects of weak values on beam deflection measure-

ments is given in chapters 3 and 4. Chapter 3 investigates the smallest disturbance

that can be measured using weak values. Chapter 4 considers the amplification

as well as well as the effect on noise, determining the optimum signal to noise

ratio that weak values can be used to achieve.

1.3 High Dimensional Images and Entanglement

In the common usage of the term, an image is a continuous spatial distri-

bution of intensities. In this sense, even an image that is small in size is infinite

dimensional—practically however, the continuous distribution can be discretized

into pixels and the intensities can be digitized. This results in a very high but

finite dimensional representation of the image.

The applications of high dimensional imaging are incredibly diverse, rang-

ing from television and movie applications, to free space communication and

21

cryptography. In the experiments described in this thesis, I create quantum me-

chanical images for image based communication channels using entanglement and

then use classical techniques to quantify the capabilities of these channels, or to

improve their capabilities. This is the inverse of the low dimensional image sys-

tems where I use standard techniques for creating images and novel techniques

to improve their performance.

1.3.1 Entanglement

The concept of entanglement in quantum mechanics is older than weak val-

ues but is similarly striking. Two quantum sub-systems are said to be entangled

if one sub-system cannot be accurately described independent of the other one

(see for example ref. [50]). A typical example of entanglement involves the decay

of a spin-zero particle into two spin-1/2 particles, particles A and B. Because this

interaction must conserve angular momentum, if particle A has spin “up” in some

axis, the other particle must have spin “down”in that axis and vice versa. We can

write the quantum state of the composite system as |ψ〉 = |+〉A|−〉B + |−〉A|+〉B.

We see that the spin degree of freedom of particle A cannot be accurately de-

scribed without considering the spin of particle B as well—the composite state

is entangled.

A key aspect of entanglement is that this correlation is independent of

measurement axis, referred to as “rotational invariance” for spin based entan-

glement. The generalization of this aspect to other variables is the presence of

simultaneous quantum correlations in conjugate variables. Quantum systems en-

tangled in position are also entangled in momentum; systems entangled in energy

are also entangled in time.

22

1.3.2 Paradoxes

Although entangled systems do not necessarily involve two spatially sep-

arated particles, this case has led to famous paradoxical “thought experiments”

including the Einstein, Podolsky, and Rosen (EPR) paradox [51], Karl Popper’s

paradox [52], and Hardy’s Paradox [53]. All paradoxes invoke entangled states in

an attempt to show quantum theory to be either incomplete or in disagreement

with other aspects of the physical world.

The EPR paradox is particularly famous and involves a system of two

particles that have interacted such that they are position-momentum entangled—

meaning the two particles have correlated locations as well as momenta. The

paradox is as follows: the two particles travel in opposite directions, moving

far from each other and the momentum of particle A is measured. Because

of the correlations, once we know the momentum of particle A we also know

the momentum of particle B. Relativity prevents signals from particle A being

transmitted instantaneously to particle B, and since the distances between the

particles can in principle be as large as we like, they argued that the now known

momentum of B must have been that value even before the measurement on

particle A took place. At the same time that the momentum of particle A is

measured, the position of particle B is measured. Therefore it seems that we

have simultaneously measured both position and momentum of particle B—a

violation of Heisenberg’s quantum mechanical uncertainty principle.

An equivalent formulation of the EPR thought experiment can be made

with photon polarization [54], the setup is shown in Fig. 1.7. In this formulation

there is a source that creates two photons with correlated polarizations in the

state |ψ〉 = |+〉A|−〉B + |−〉A|+〉B where |+〉 and |−〉 indicate horizontal and

vertical polarizations respectively. The measurement on photon A is done in

23

Figure 1.7: An EPR experiment using photon polarization is shown. Photon Agoes to the left where it is measured in the horizontal/vertical polarization basis.Photon B goes to the right where is is measured in the diagonal/anti-diagonalpolarization basis.

the horizontal/vertical polarization basis, while the measurement on photon B

is done in the diagonal/anti-diagonal polarization basis, and by the same argu-

ment, simultaneous values of these non-commuting variables have been made—a

seeming violation of the uncertainty principle.

EPR were arguing that all variables have definite values at all times, a

concept known as “realism,” and that these values are described locally, a concept

known as “locality.” EPR did not believe that quantum theory was necessarily

wrong, just that it was incomplete—that a locally realistic variable could be

added to the theory to make it more complete. The idea was that this added

“hidden variable” would not modify predictions of standard quantum theory, but

simply allow the modified theory to have a greater range of applicability than the

standard quantum theory [55].

1.3.3 Bell Inequalities

Paradoxical thought experiments and conflicting interpretations of quan-

tum mechanics were thought to have little potential for experimental investiga-

tion. This was shown to be incorrect in 1964 when John Bell derived an inequality

obeyed by all locally realistic “hidden variable” theories and violated by stan-

dard quantum theory [56]. Bell proposed a straightforward experiment to test

24

Figure 1.8: A Bell experiment using photon polarization is shown. The mea-surement bases for photons A and B are changed by rotating the correspondingdetection apparatus.

the inequality using a pair of spatially separated, entangled particles. Like the

EPR experiment, Bell’s experiment can be cast in terms of photon polarization.

This formulation is identical to the polarization EPR experiment with one key

difference—each measurement apparatus is rotated through several measurement

bases. The experimental setup is shown in Fig. 1.8

Following the presentation of Sakurai [50], I label the measurement bases

by their rotation angle θ1, θ2, and θ3. According to local realistic hidden variable

theories, photons have definite polarizations in each of these three measurement

bases. For any ensemble then, the mutually exclusive possibilities are:

Number Left Photon Right PhotonN1 θ1+, θ2+, θ3+ θ1−, θ2−, θ3−N2 θ1+, θ2+, θ3− θ1−, θ2−, θ3+N3 θ1+, θ2−, θ3+ θ1−, θ2+, θ3−N4 θ1+, θ2−, θ3− θ1−, θ2+, θ3+N5 θ1−, θ2+, θ3+ θ1+, θ2−, θ3−N6 θ1−, θ2+, θ3− θ1+, θ2−, θ3+N7 θ1−, θ2−, θ3+ θ1+, θ2+, θ3−N8 θ1−, θ2−, θ3− θ1+, θ2+, θ3+

The probability of jointly measuring the left traveling photon as horizontally

polarized in the θ1 rotated measurement basis and the right traveling photon as

25

horizontally polarized in the θ2 rotated measurement basis is

P (θ1+, θ2+) =N3 +N4∑

iNi

, (1.33)

similarly

P (θ1+, θ3+) =N2 +N4∑

iNi

, (1.34)

and

P (θ3+, θ2+) =N3 +N7∑

iNi

. (1.35)

Since N3 +N4 ≤ (N2 +N4) + (N3 +N7) we can write

P (θ1+, θ2+) ≤ P (θ1+, θ3+) + P (θ3+, θ2+). (1.36)

This equation holds for any locally realistic hidden variable description of the

experiment.

We now compare this to the quantum mechanical prediction. Using the

state (in the unrotated basis) |ψ〉 = |+〉A|−〉B + |−〉A|+〉B, the probability that

photon A is measured as horizontal in the θ1 basis is 1/2. Because the polariza-

tions are perfectly correlated, photon B is known to be vertically polarized in this

same basis. Given that this measurement is made on photon A, the probability

that photon B is measured as horizontal in the θ2 basis is given by Malus’s law

as sin2(θ12), where θ12 = θ1 − θ2. This results in

P (θ1+, θ2+) =1

2sin2 (θ12) . (1.37)

Similarly

P (θ1+, θ3+) =1

2sin2 (θ13) , (1.38)

and

P (θ3+, θ2+) =1

2sin2 (θ32) . (1.39)

26

The local hidden variable inequality then becomes

sin2 (θ12) ≤ sin2 (θ13) + sin2 (θ32) . (1.40)

This inequality is violated for a range of values. For example if we let θ1 = 0,

θ2 = π/4, and θ3 = π/8 the inequality becomes 0.50 ≤ 0.29.

One can use similar reasoning to come up with many types of Bell like

inequalities, involving different types of particles besides photons and different

degrees of freedom besides polarization [57, 58].

Experiments measuring Bell type inequalities have been performed. The

results support the quantum mechanical prediction [59, 60]. This indicates that it

very unlikely that the the observed correlations can be accounted for by a locally

realistic hidden variable theory.

1.3.4 Nonlocality

Bell inequalities show that quantum theory violates local realism. How-

ever, the concepts of locality and realism are distinct and in fact, when considering

light polarization, realism is violated by classical optics. It is therefore of inter-

est to be able to determine when a system violates locality as an independent

concept.

If we assume a system is described by quantum mechanics we allow for

realism to be violated, we also accept the Heisenberg uncertainty principle for

conjugate variables position and momentum, denoted by x and p, respectively:

∆x∆p ≥ ~2

(1.41)

where, for example ∆x =√〈(x− 〈x〉)2〉. We consider a joint system in spatially

separated regions—described by position variables x1 and x2, and corresponding

27

momentum variables p1 and p2. If the system obeys locality, then measure-

ments relating to x1 are independent of conditioning measurements made on

x2—indicating that the variance of x1 conditioned upon a measurement of x2,

written as ∆x1|x2 , is equal to the unconditioned variance of x1. Using the same

reasoning for momentum we can see that, by assuming locality, a system obeys

the uncertainty principle

∆x1|x2∆p1|p2 ≥~2. (1.42)

Violating this inequality shows that the system is nonlocal [61, 62]. Moreover,

since a violation indicates the system cannot be factored into spatially separated

x1 and x2 subsystems, the system is entangled.

1.3.4.1 Entropic Uncertainty

Heisenberg’s uncertainty principle is the most well known uncertainty prin-

ciple, however it is possible to derive other uncertainty principles using quantum

theory. Because of the connections to information theory and channel capacity

it is useful, for our purposes, to make use of an entropic uncertainty principle.

I follow the presentation of Bialynicki-Birula and Mycielski [63] and derive

an entropic uncertainty principle by considering the position and momentum

wavefunctions of a particle, 〈~x|ψ〉 = ψ(~x) and 〈~k|ψ〉 = φ(~k) respectively. These

wavefunctions are related through the Fourier transform

φ(~k) =1

(2π)n/2

∫ψ(~x)ei

~k·~xd~x (1.43)

where n is the spatial dimension (images are two dimensional). The (p, q)-norm

of a Fourier transform is defined as the smallest number k(p, q) that satisfies the

relation

‖ψ‖q ≤ k(p, q)‖φ‖p, (1.44)

28

where, for example,

‖ψ‖q =

(∫|ψ(~x)|qd~x

)1/q

. (1.45)

The form of the (p, q)-norm is as follows:

k(p, q) =

(2π

q

)n/2q (2π

p

)−n/2p. (1.46)

Of course we are interested in the case when p = q = 2, such that the norms

relate to probabilities, but for the purposes of the derivation we allow p and q to

vary. Similarly, we impose the restriction that

1

p+

1

q= 1, (1.47)

such that p is a function of q, leaving only one free parameter and enforcing p = 2

when we set q = 2. We rewrite the definition of the (p, q)-norm as follows

W (q; p) ≡ k(q; p)‖φ‖p − ‖ψ‖q ≥ 0. (1.48)

For q = 2 it is clear that W (q; p) is simply subtracting unit probabilities, indi-

cating that W (2, 2) = 0. This, along with the fact that W (q; p) ≥ 0, requires the

derivative at q = 2 to be non-negative as well:

d

dqW (q; p)|q=2 ≥ 0. (1.49)

The derivative is:

d

dqW (q; p) =−

(n‖ψ‖p2kq2

)log

(2πe

q

)−(

n‖ψ‖p2kp2(q − 1)2

)log

(2πe

p

)−(

k‖ψ‖pp(q − 1)2

)∫|ψ(~x)|p log

(|ψ(~x)|p

)d~x+

(k‖ψ‖qp(q − 1)2

)log(‖ψ‖p

)−(‖φ‖qq

)∫|φ(~k)|q log

(|φ(~k)|q

)d~k +

(‖φ‖qq

)log(‖φ‖q

).

(1.50)

29

At q = 2 we have p = 2, k = 1, and ‖ψ‖q = ‖φ‖p = 1, as well as W (2, 2) = 1, so

the derivative becomes:

d

dqW (q; p)|q=2 =

−1

2

(n log

(πe)

+

∫|φ(~k)|2 log

(|φ(~k)|2

)d~k

+

∫|ψ(~x)|2 log

(|ψ(~x)|2

)d~x

),

(1.51)

which is non-negative so,

−(∫|φ(~k)|2 log

(|φ(~k)|2

)d~k +

∫|ψ(~x)|2 log

(|ψ(~x)|2

)d~x

)≥ n log

(πe).

(1.52)

The terms |ψ(~x)|2 and |φ(~k)|2 are recognized to be probability densities, indicat-

ing the integrals are continuous variable entropies, resulting in

Hc(X) +Hc(K) ≥ n log(πe), (1.53)

where X and K represent the position and momentum variables in this case, but

more generally can represent any two conjugate variables related through Eq.

1.43. Changing the base of the logarithms introduces only a multiplicative factor

to both sides, so the equation is valid for any logarithm base as long as it is the

same for all terms in the inequality.

This formula uses continuous probability distributions, whereas measured

data will be discrete probabilities. As shown previously, in the limit of small

discretization widths, entropies for continuous and discrete probabilites are equal

up to an additive offset. The offset serves to increase the entropic bound H(X)+

H(K) ≥ n log(πe) + log(b1b2

), where b1 and b2 are the number of position and

momentum discretization bins, respectively. As a result, neglecting this offset

does not affect the validity of the inequality, and the inequality for discrete prob-

abilities holds, independent of discretization scheme,

H(X) +H(K) ≥ n log(πe). (1.54)

30

As noted earlier, n = 2 for images, so the entropic uncertainty relation

for images is H(X) + H(K) ≥ 2 log(πe)≈ 6.18. By the same logic used for

Heisenberg’s uncertainty relation, violating this entropic inequality shows that

the system is nonlocal (and therefore entangled).

1.3.5 Related Concepts

It should be stressed that entanglement, nonlocality, and non-realism are

all distinct concepts. Non-realism is more general than entanglement and nonlo-

cality. It is possible to violate a Bell like inequality relying solely on non-realistic

properties of quantum systems. The violation results from the modified prob-

ability laws required for non-realistic descriptions of reality. Entanglement and

nonlocality are very similar and both require non-realistic systems. A system

is entangled if it is non-factorable in any two degrees of freedom; a system is

only nonlocal if it is non-factorable in spatially separate degrees of freedom. All

nonlocal systems are therefore entangled however not all entangled systems are

nonlocal. The relationship between these concepts is shown conceptually in Fig.

1.9

1.3.6 Spontaneous Parametric Down-Conversion

A common method of generating entangled particles is the process of spon-

taneous parametric down-conversion (SPDC) [64–67]. This process, which in-

volves a nonlinear optical interaction to create pairs of entangled photons, is well

understood and reliably creates entangled photons that are easily manipulated

in the lab.

Nonlinear optical materials are those where the polarization P does not

vary linearly with the electric field E—it can often be represented as a power

31

Figure 1.9: A “map” of related concepts in quantum mechanics is shown. Classi-cal particle systems and quantum mechanical systems exist on separate islands.All quantum mechanical systems can exhibit non-realism, a subset of these sys-tems exhibit entanglement. A subset of the entangled systems exhibit nonlocality.

series in field strength. In Gaussian-cgs units this can be writen as [68]:

P ∝ χ(1)E + χ(2)E2 + χ(3)E3 + ... (1.55)

The process of SPDC is a three wave mixing process and so requires a medium

with a χ(2) nonlinearity. It involves a field at frequency ωi producing two fields at

ω2 and ω3, such that ωi = ω2 + ω3. Quantum mechanically this is a photon of a

specific energy being converted into two photons with their energy sum equal to

the original photon’s energy. This conservation of energy results in entanglement.

Other conservation principles apply as well and indeed photons from SPDC can be

entangled in polarization, time and energy, and position and momentum, which

is the entanglement I will focus on in this thesis.

Momentum conservation restrains the wave-vector output of the SPDC

interaction. This so-called “phase matching” requirement is common in nonlinear

optical interactions [68]. In SPDC the result is that there are two relevant k-

32

Figure 1.10: A conceptual SPDC interaction is shown where a beam of frequencyωi is incident upon a crystal of length L with a χ(2) nonlinearity. The SPDC in-teraction in the crystal produces a beam at frequency ωo. Quantum mechanicallythis is a single ωi photon being transformed into two ωo photons.

vector widths: the spread in SPDC output k-vectors (governed by phase-matching

considerations) and the initial spread of k-vectors from the input beam (governed

by laser specifications). In many cases, including the experiments I perform, we

can approximate the resulting two photon state as a double Gaussian with two

widths [69, 70]. In the momentum basis this is:

|φo〉 ∝∫ ∫

exp

(−(k1 − k2)2

2b2

)exp

(−(k1 + k2)2

2a2

)a

(†)1 a

(†)2 |0〉dk1dk2, (1.56)

where subscripts label the two SPDC output photons, a is the k-vector spread

in the k1 + k2 direction, and b is the k-vector width in the k1 − k2 direction. By

Fourier transforming this state we can represent it in the position basis:

|ψo〉 ∝∫ ∫

exp

(−(x1 − x2)2

2/b2

)exp

(−(x1 + x2)2

2/a2

)a

(†)1 a

(†)2 |0〉dx1dx2. (1.57)

This double Gaussian state is shown in both the position and momentum basis in

Fig. 1.11. A separable state with the same single photon widths as the entangled

state is shown in Fig. 1.12. It is seen that the conditional widths of the separable

state are different than those of the entangled state.

33

Figure 1.11: A double Gaussian entangled state from Eq. 1.57 and Eq. 1.56 isshown in (a) and (b) respectively. The state has width in the k1 − k2 directionof a = 2, and width in the k1 + k2 direction of b = 1/2. The Fourier transformedstate has the inverse of this: the width in the x1 + x2 direction is 1/a = 2, andthe width in the x1−x2 direction is 1/b = 2. The experimentally accessible singlephoton width and conditional width are shown in (a).

34

Figure 1.12: A separable state with similarities to the state shown in Fig. 1.11is shown. The state in the position basis is shown in (a) and the state in themomentum basis is shown in (b). The single photon widths for this state are thesame as those of the state shown in Fig. 1.11, however the conditional widths aredifferent.

35

1.3.7 Entanglement in High Dimensional Imaging Applications

In an entangled state the conditioned width is only accessible in correlation

measurements, and it is this feature that we take advantage of for creating secure

communication channels. Considering the mutual information of this channel

with the probability density in position given by p(x1, x2) = |〈x1, x2|ψo〉|2 sug-

gests experimentally realizable parameters give mutual information ranging up

to 10 or more bits of information per joint photon detection event. I investi-

gate this in chapter 6, where the experimental realization of over 7 bits/photon

using position-momentum entangled photons is described. The use of position-

momentum entangled states however requires free space propagation. In chapter

5 I use position-momentum entangled states and investigate possibly negative

effects from free space propagation and how to minimize them.

36

Chapter 2

Weak Values and Deflection

Aside from the fundamental physics interest in weak values, they also are

useful. If we consider the spin of the system as a small signal, the fact that the use

of weak values maps this small signal onto a large shift of a measuring device’s

pointer may be seen as an amplification effect. Like any amplifier, something

must be sacrificed in order to achieve the enhancement of the signal. For weak

values the sacrifice comes in the form of throwing away most of the data in the

post-selection process. If the detector is shot noise limited, then the advantage

gained in the amplification via the weak value is negated by the loss of data in

the post-selection. However, most experiments are limited by technical noise, and

the weak value technique can improve existing experimental set-ups by orders of

magnitude. While this feature was briefly noted in the original weak value paper

[32], its utility has been dramatically demonstrated by Hosten and Kwiat [34]

who were able to detect a polarization-dependent beam deflection of 1 A.

2.1 Introduction

This chapter describes the development of a weak value amplification tech-

nique for any optical deflection. In particular, our weak value measurement uses

the which-path information of a Sagnac interferometer, and can obtain dramat-

ically enhanced resolution of the deflection of an optical beam. This technique

has several advantages for amplification: the post-selection consists of a photon

37

emerging from the interferometer, and the post-selection attenuation originates

from the destructive interference between the two paths, it is therefore completely

independent of the source of the optical deflection. In the experiment reported

here, the weak measurement consists of monitoring the transverse position of

the photon, which gives partial information about the system. The deflection

is caused by a slight mirror motion, which, for this geometry, causes opposite

deflections for the two interferometer paths. For other geometries the beams can

be deflected in the same way. However, this is not a problem for this proposal,

because the source of the deflection may be placed asymmetrically in the inter-

ferometer, causing one path to be longer (corresponding to a larger spatial shift),

and the other path to be shorter (corresponding to a shorter spatial shift).

2.2 Theoretical Description

Consider the schematic of the weak value amplification scheme shown in

Fig. 2.1. A light beam enters an optical Sagnac interferometer composed of a

50/50 beam splitter and mirrors to cause the beam to take one of two paths and

eventually exit the 50/50 beam splitter. For an ideal, perfectly aligned Sagnac

interferometer, all of the light exits the input port of the interferometer. The

port that all of the light exits is referred to as the bright port, the other port

as the dark port. The constructive interference at the entrance port occurs due

to the sum of two π/2 phase shifts which occur on reflection in the beam split-

ter. This symmetry is broken with the presence of a Soleil-Babinet compensator

(SBC), which introduces a relative phase φ between the paths, allowing one to

continuously change the dark port to a bright port. In presenting the theory, we

assume a single photon undergoes a weak measurement.

The beam travels through the interferometer, and the spatial shift of the

38

beam exiting the dark-port is monitored. We refer to the which-path information

of the interferometer as the system, described with the states {|�〉, |〉}. The

transverse position degree of freedom, labeled by the states |x〉, is referred to

as the meter. A slight periodic tilt is given to the mirror at the symmetric

point in the interferometer. This tilt corresponds to a shift of the transverse

momentum of the beam. Importantly, the tilt also breaks the symmetry of the

Sagnac interferometer, with one propagation direction being deflected to the left,

and the other being deflected to the right.

This effect entangles the system with the meter via the impulsive interac-

tion Hamiltonian Hi = xAk, where x is the transverse position of the meter, k is

the transverse momentum shift given to the beam by the mirror, and the system

operator A = |�〉〈�| − |〉〈| describes the fact that the momentum-shift is

opposite, depending on the propagation direction.

The splitting of the beam at the 50/50 beam splitter, plus the Soleil-

Babinet compensator (causing the phase-shift φ) results in an initial system state

of |ψi〉 = (ieiφ/2|〉 + e−iφ/2|�〉)/√

2. The entangling of the position degree of

freedom with the which-path degree of freedom results in the state

|Ψ〉 =

∫dxψ(x)|x〉 exp(−ikAx)|ψi〉. (2.1)

Where ψ(x) is the wavefunction of the meter in the position basis. This evolution

is part of a standard analysis on quantum measurement, where the above trans-

formation would result in a momentum-space shift of the meter, Φ(p)→ Φ(p±k),

if the initial state is an eigenstate of A.

The weak value analysis then consists of expanding exp(−iAkx) to first

order (assuming ka < 1, where a =√〈x2〉 is the beam initial size) and post-

selecting with a final state |ψf〉 = (|〉 + i|�〉)/√

2 (describing the dark-port of

39

the interferometer). This leaves the state as

〈ψf |Ψ〉 =

∫dxψ(x)|x〉[〈ψf |ψi〉 − ikx〈ψf |A|ψi〉]. (2.2)

We now assume that ka|〈ψf |A|ψi〉| < |〈ψf |ψi〉| < 1, and can therefore factor out

the dominant state overlap term to find

〈ψf |Ψ〉 = 〈ψf |ψi〉∫dxψ(x)|x〉 exp(−ixkAw), (2.3)

where we have re-exponentiated to find an amplification of the momentum shift

by the weak value

Aw =〈ψf |A|ψi〉〈ψf |ψi〉

(2.4)

with a post-selection probability of Pps = |〈ψf |ψi〉|2 = sin2 φ/2. The new momen-

tum shift kAw will be smaller than the width of the momentum-space wavefunc-

tion, 1/a, but the weak value can greatly exceed the [−1, 1] eigenvalue range of A.

In the case at hand, the weak value is purely imaginary, Aw = −i cotφ/2 ≈ −2i/φ

for small φ. This has the effect of causing a shift in the position expectation,

〈x〉 = 2ka2|Aw| = 4ka2/φ, (2.5)

assuming a symmetric spatial wavefunction.

In these investigations, further enhancement is possible by extending be-

yond the collimated beam analysis described above, and putting a lens before

the interferometer, with a negative image distance si, corresponding to a diverg-

ing beam (we neglect diffractive effects). Taking paraxial beam propagation into

account, the result analogous to Eq. (2.5) is found to have an additional factor

of F = `im(`im + `md)/s2i , where `im is the distance from the lens image to the

moving mirror, and `md is the distance from the moving mirror to the detector

[71].

40

From an experimental point of view, it is convenient to express the deflec-

tion in terms of easily measurable quantities. This can be done through using

the beam size at the detector, σ = a(`im + `md)/si, and the initial beam size at

the lens, a, to eliminate si from the equation, and express it in terms of `lm, the

distance from lens to the moving mirror. This gives

〈x〉 = 2k|Aw|σ2`lm + σa`md`lm + `md

. (2.6)

Finally, we compare this result to the unamplified deflection (without the

interferometer) of δ = k`md/k0, where k0 is the wavenumber of the light so that

θ = k/k0 is the small angle the mirror imparts to the light beam. This gives an

amplification factor of A = 〈x〉/δ which is independent of k. Under the conditions

described below, this gives a magnification about about M = 70|Aw|. However,

we expect that it is possible to further amplify the signal with modifications to

the optics.

2.3 Experiment and Results

A fiber coupled 780 nm laser beam is collimated using a 10× microscope

objective. Just after the objective, the beam has a Gaussian radius (denoted by

a in the theory) of 640 µm. The beam can be made to be converging or diverging

by moving the fiber end relative to the microscope objective. After collimation,

the beam passes through a polarizing beam splitter giving a pure horizontal

polarization. Half and quarter wave plates are used to adjust the intensity of

the beam passing through the polarizing beam splitter. The beam then enters

a Sagnac interferometer input port (the pre-selection process). Passing through

the interferometer in the clockwise direction, the beam first passes through a

half wave plate which rotates the polarization to vertical, the beam then passes

41

Figure 2.1: Experimental setup for interferometric weak values beam deflectionmeasurement. The objective lens collimates a 780 nm beam. After passingthrough polarization optics, the beam enters a Sagnac interferometer consist-ing of three mirrors and a 50/50 beamsplitter arranged in a square. The outputport is monitored by both a quadrant detector and a CCD camera. The SBCand halfwave plate in the interferometer allow the output intensity of the inter-ferometer to be tuned. The piezo mirror gives a small beam deflection.

through a Soleil-Babinet compensator which adds a tunable phase to the beam

(the compensator is set to add this phase to vertically polarized beams relative

to those polarized horizontally). Passing counterclockwise, the beam first passes

through the Soleil-Babinet compensator which now has no relative effect, then

through the half wave plate, changing the polarization to vertical. A piezo electric

actuator scans the tilt of one of the interferometer mirrors back and forth. A

gimbal mount is used so the center of the mirror is the fulcrum. The tilt of the

mirror gives the two propagation directions opposite deflections. The small beam

deflection is the weak interaction between transverse beam deflection (meter) and

which path degree of freedom (system).

Post-selection is achieved simply by monitoring the light that exits the

dark port of the interferometer. Tuning the Soleil-Babinet compensator to add a

42

small but nonzero relative phase allows a small amount of light out of the dark

port. This light is split by a 50/50 beamsplitter and sent to a CCD camera (New-

port model LBP-2-USB) which monitors the beam structure, and to a quadrant

detector (New Focus model 2921) which monitors beam deflection as well as total

power.

The interferometer is roughly square with sides of approximately 15 cm.

The distance from the microscope objective to the piezo driven mirror is `lm =

48cm. The distance from the piezo driven mirror to the detectors is `md = 114cm

(the same distance to both the CCD camera and the quadrant detector). The

piezo driven mirror has a lever arm of 3.5 cm.

Piezo deflection was calibrated by removing the 50/50 beam splitter from

the interferometer and observing beam centroid position on the CCD camera. In

this configuration the beam experiences no interference and ray optics describes

the beam deflection. Driving the mirror, the piezo response was found to be 91

pm/mV. The piezo response was verified from 500 Hz down to D.C.

To characterize the system the interferometer was first aligned well, min-

imizing the light exiting the dark port. The Soleil-Babinet compensator relative

phase was then tuned away from zero, allowing light to exit the interferometer.

The piezo driven mirror was given a 500 mV peak to peak amplitude, 100 Hz,

sinusoidal driving voltage and the beam deflection was observed using the quad-

rant detector connected to an oscilloscope. This was done over a range of beam

sizes, σ, for three values of Soleil-Babinet compensator phase difference. These

measurements, as well as the corresponding theoretical prediction curves given by

Eq. (2.6) are shown in Fig. 2.2. The measured data is, in general, well described

by the theory.

At the smallest Soleil-Babinet compensator angle (7.2◦) the small overlap

43

Figure 2.2: Effect of beam radius on interferometric weak values beam deflectionmeasurement. Measured beam deflection is plotted as a function of beam radius,σ. The value of SBC angle (φ) for each data set is labeled. The scale on theleft is the measured beam deflection, 〈x〉. The scale on the right is the amplifi-cation factor, A. The unamplified deflection is δ = 2.95 µm. The solid lines aretheoretical predictions based on Eq. 2.6.

between pre and post-selected states allows only a small amount of light to exit

the dark port. With this light at low intensities it begins to be of roughly equal

intensity to stray light incident on the quadrant detector. This leads to less than

ideal amplification, as shown in Fig. 2.2.

For fixed interferometer output intensity, the range of detectable deflec-

tions was also explored. The interferometer was again aligned such that the

beams only had a small phase offset from each other. The alignment was ad-

justed to give the maximum quadrant detector output while still having a large

weak value amplification factor. For these measurements the beam size at the

detector was σ = 1235 µm and the weak value amplification factor was 86. The

amplification factor was found by driving the piezo with a 500 mV peak to peak

44

Figure 2.3: Angular mirror displacement in interferometric weak values beam de-flection measurement. The angular displacement of the mirror is plotted versusthe piezo driving voltage. Weak value signal amplification allows small deflec-tions to be measured. The solid line shows the expected deflection based onan interpolation of calibrated measurements of the piezo actuator’s linear travelat higher voltages. These data were taken using a weak value amplification ofapproximately 86.

45

signal and comparing the measured beam deflection with the aligned interfer-

ometer to the measured beam deflection with the interferometer beam splitter

removed. The piezo driving voltage was varied over five orders of magnitude

while the output of the quadrant detector was sent to a lock-in amplifier and

the signal was observed. The smallest driving voltage that yielded measurable

beam deflection was 220 nV corresponding to an angular deflection of the mirror

of 560 femtoradians (the mirror angle is half the beam deflection angle). These

measurements are shown in Fig. 2.3.

There is another, perhaps, more interesting point; the deflection indirectly

measured the linear travel resolution of the piezo electric actuator. The piezo

actuator moved approximately 20 fm in making this measurement. This distance

is on the order of large atomic nucleus diameters (Uranium is 15 fm) and is almost

six orders of magnitude more resolution than the manufacturer’s specifications

of 10 nm. Steps to achieve this resolution increase include: using a quadrant

detector with a larger active area which allows a larger beam size to be used,

decreasing stray light on the detector by carefully minimizing any back reflections

from optics, and aligning the interferometer to have an improved dark port,

possibly by using a deformable mirror.

2.4 Channel Analysis

By considering the deflecting mirror as the source of the message and the

measured signal as the received message, through a noisy communication chan-

nel, we can analyze these experimental investigations in terms of communication

theory. It is reasonable to model the mirror deflection as a Gaussian random

distribution with variance S, and the total noise as an additive Gaussian random

distribution with variance N . The maximum channel capacity I is then described

46

by the equation:

I(X;Y ) = log

(1 +

S

N

). (2.7)

In this case, since we are not detecting single photons, but rather classical pho-

tocurrents over an extended period of time, it makes sense to use mutual infor-

mation units of bits per optical power. There is an additionally ambiguity as one

can either use the detected optical power, or the sent optical power. The advan-

tage of the bits per detected power measure is that it is invariant to changes in

detector efficiencies or other sources of loss that could in principle be overcome by

sufficient engineering. The downside is that it neglects any sources of loss that are

fundamental—possibly skewing the utility of the measure as a meaningful system

characterization. In this experiment the power that is not post-selected on is not

lost and can still be utilized, indicating that a measure of bits per detected power

may be more natural. It should be noted that these different measures will have

different noise powers associated with them.

If we naively neglect the effect of the weak value amplification on the

noise power, the improvement of the mutual information can be characterized in

the following way: By amplifying the signal variable by a factor A we amplify

its variance by a factor A2, similarly through post-selection the detected optical

power is decreased by the factor Pps. The amplified mutual information capacity

Ia in bits per detected optical power is then:

Ia(X;Y ) =1

Ppslog

(1 + A2 S

N

). (2.8)

In the limit of small signal to noise ratio S � N—which was the case for the

small deflections in this experiment—the mutual information is amplified by the

factor

AI = Ia/I =A2

Pps. (2.9)

47

In the measurement sensitivity experiment with A = 86 and Pps = sin2(2◦) ≈

0.02, resulting in an amplification of the mutual information of AI ≈ 400, 000.

This extends to the single photon level as well. The weak value experiment ampli-

fied the mutual information per detected photon by a factor of 400, 000, however

the initial mutual information per photon was so low that the amplified value

is still very small and requires many photon measurements to get appreciable

information.

It should be noted that this analysis is incomplete in that it ignores the

changing noise spectrum between I and Ia. The changed noise will affect the

value of A, and it is this topic that we explore in depth in chapter 3. This

channel analysis, despite its lack of noise analysis, does point to the utility of the

interferometric weak value amplification technique.

2.5 Concluding Remarks

In this chapter we have described and demonstrated a method of amplify-

ing small beam deflections using weak values. The amplification is independent

of the source of the deflection. In this experiment a small mirror deflection in

a Sagnac interferometer provides the beam deflection. By tuning the interfer-

ometer misalignment and monitoring the resulting small amount of light exiting

the interferometer dark port, weak value amplification factors of over 100 are

achieved. The weak-value experimental setup, in conjunction with a lock-in am-

plifier, allows the measurement of 560 femtoradians of beam deflection which is

caused by 20 femtometers of piezo actuator travel. Analysis of the experiment

as a communication channel suggests the weak value technique can amplify the

channel capacity by several orders of magnitude.

48

Chapter 3

Weak Values SNR for Deflections

In this chapter the amplification obtained using weak values is quantified

through a detailed investigation of the signal to noise ratio for an optical beam

deflection measurement. We show that for a given deflection, input power and

beam radius, the use of interferometric weak values allows one to obtain the

optimum signal to noise ratio using a coherent beam. This method has the

advantage of reduced technical noise and allows for the use of detectors with a

low saturation intensity. We report on an experiment which improves the signal

to noise ratio for a beam deflection measurement by a factor of 54 when compared

to a measurement using the same beam size and a quantum limited detector.

The ultimate limit of the sensitivity of a beam deflection measurement is of

great interest in physics. The signal to noise ratio (SNR) of such measurements

is limited by the power fluctuations of coherent light sources such as a laser,

providing a theoretical bound known as the standard quantum limit [30]. It was

found that interferometric measurements of longitudinal displacements and split-

detection of transverse deflections have essentially the same ultimate sensitivity

[25].

Standard techniques to optimize the SNR of a beam deflection measure-

ment include focusing the beam onto a split detector or focusing the beam onto

the source of the deflection. The improvement of the SNR is of great inter-

est in not only deflection and interferometric phase measurements but also in

49

spectroscopy and metrology [72, 73], anemometry [74], positioning [24], micro-

cantilever cooling [28], and atomic force microscopy [29, 75]. In particular, atomic

force microscopes are capable of reaching atomic scale resolution using either a

direct beam deflection measurement [29] or a fiber interferometric method [75].

We show that for any given beam radius, interferometric weak value amplifica-

tion (WVA) can improve (or, at least match) the SNR of such beam deflection

measurements. It has also been pointed out by Hosten and Kwiat that WVA

reduces technical noise [34], which combined with our result provides a powerful

technique.

3.1 Theoretical Description

In chapter 2 I described an interferometric weak value setup measuring

beam deflection (caused by a piezo-actuated (PA) mirror) that used the which-

path degree of freedom (the system observable) of a Sagnac interferometer coupled

with the transverse degree of freedom (the meter variable) of a laser beam (see

Fig. 3.1). This chapter is concerned with the same type of experimental setup.

The analogy between interferometry and beam deflection described in a

paper by Barnett et al. [31] allows one to predict the SNR for a deflection of an

arbitrary optical beam (a coherent beam or a squeezed beam for example). For

a coherent beam with a horizontal Gaussian intensity profile at the detector of

I(x) =1√2πσ

e−x2/2σ2

, (3.1)

Barnett et al. show that the SNR is given by

R =

√2

π

√Nd

σ, (3.2)

where N is the total number of photons incident on the detector, d is the trans-

verse deflection, and σ is the beam radius defined in Eq. (3.1). Equation 3.2

50

Figure 3.1: Experimental setup for interferometric weak values signal to noisemeasurement. A fiber coupled laser beam is launched into free space before pass-ing through a polarizer, producing a horizontally polarized single mode Gaussianbeam. The laser enters the input port of a Sagnac interferometer via a 50/50beamsplitter (BS). The light is divided equally and travels through the inter-ferometer clockwise and counterclockwise, encountering three mirrors before re-turning to the BS. The piezo-actuated mirror (PA), positioned symmetrically inthe interferometer, causes a slight opposite deflection for the two different paths,altering the interference at the BS. The dark port is monitored with both a CCDcamera and a quadrant cell detector (QCD) positioned at equal lengths from thesecond BS. The CCD is used only to verify the mode quality of the dark port.

51

represents the ultimate limit of the SNR for position detection with a coherent

Gaussian beam.

We now incorporate weak values by describing the amplification of a de-

flection at a split detector as a multiplicative factor A. Thus, da = Ad is the

amplified deflection caused by the weak value. Also, the post-selection prob-

ability Pps modifies the number of photons incident on the detector such that

Na = PpsN. The beam radius is not altered. In chapter 2 I showed that for a

collimated Gaussian beam passing through a Sagnac interferometer (see Fig. 3.1)

the WVA factor and the post-selection probability are given by

A =2k0σ

2

lmdcot(φ/2), Pps = sin2(φ/2), (3.3)

where lmd is the distance from the piezo-actuated mirror to the detector, k0 is

the wave number of the light and φ is the relative phase of the two paths in the

interferometer.

Using Eqs. (3.3) and making the substitutions d → Ad and N → PpsN

into Eq. (3.2), we find the weak value amplified SNR,

RA = αR, (3.4)

where α = 2k0σ2 cos(φ/2)/lmd. For a typical value of φ we note that cos(φ/2) ≈ 1.

The experiment can be modified by inserting a negative focal length lens

before the interferometer, creating a diverging beam. This changes the WVA

such that the new SNR is given by

R′A = αRllm + almd/σ

llm + lmd= C

(σ + a

lmdllm

), (3.5)

where C =√

(8N)/π(k0llmd cos(φ/2))/(lmd(llm + lmd)) and a is the radius of the

beam at the lens which is a distance llm from the piezo-actuated mirror. It is

52

interesting to note that the dependence of the SNR is proportional to the beam

radius at the detector in the amplified case [Eq. (3.5)] but inversely proportional

when there is no amplification (Eq. 3.2).

Equations 3.4 and 3.5 are the main theoretical results of this chapter. We

see that it is possible to greatly improve the SNR in a deflection measurement

with experimentally realizable parameters. Typical values for the experiment to

follow are φ/2 = 25◦, σ = 1.7 mm, lmd = 14 cm and k0 = 8× 106 m−1 such that

the expected SNR amplification is α ≈ 300.

We notice that for small φ, the value of α is the ratio of the SNR for a

beam deflection measurement in the far-field and the near-field. The far-field

measurement can be obtained at the focal plane of a lens. This is recognized

as a typical method to reach the ultimate precision for a beam deflection mea-

surement [25]. Consider a collimated Gaussian beam with a large beam radius σ

which acquires a transverse momentum shift k given by a movable mirror. The

beam then passes through a lens with focal length f followed by a split detector.

The total distance from the source of the deflection to the detector is lmd, and

the detector is at the focal plane of the lens. This results in a new deflection

d′ = fk/k0 and a new beam radius σ′ = f/2k0σ at the detector. Making the

substitutions d → d′ and σ → σ′ into Eq. 3.2, we see that when the beam is

focused onto a split detector the SNR is amplified:

Rf = αf R, (3.6)

where αf = 2k0σ2/lmd is the improvement in the SNR relative to the case with

no lens [i.e. Eq. (3.2)]. Yet this is identical to the improvement obtained using

interferometric weak values, up to a factor of cos(φ/2) ≈ 1 for small φ. Thus we

see that the improvement factors are equal using either WVA or a lens focusing

53

the beam onto a split detector, resulting in the same ultimate limit of precision.

However, WVA has three important advantages: the reduction in technical noise,

the ability to use a large beam radius and lower intensity at the detector due to

the post selection probability Pps = sin2(φ/2).

3.2 Technical Noise

We now consider the contribution of technical noise to the SNR of a beam

deflection measurement. Suppose that there are N photons contributing to the

measurement of a deflection of distance d. In addition to the Poisson shot noise

ηi, there is technical noise ξ(t) that we model as a white noise process with

zero mean and correlation function 〈ξ(t)ξ(0)〉 = S2ξ δ(t). The measured sig-

nal x = d + ηi + ξ(t) then has contributions from the signal, the shot noise,

and the technical noise. The variance of the time-averaged signal x is given by

∆x2 = (1/N2)∑N

i,j=1〈ηiηj〉+ (1/t2)∫ t

0dt′dt′′〈ξ(t′)ξ(t′′)〉, where the shot noise and

technical noise are assumed to be uncorrelated with each other. For a coherent

beam described in Eq. (3.1), the shot noise variance is 〈ηiηj〉 = σ2δij. Therefore,

given a photon rate Γ (so N = Γt), the measured distance (after integrating for

a time t) is given by

〈x〉 = d± σ√Γt± Sξ√

t. (3.7)

We now compare this with the weak value case. Given the same number

of original photons N, we will only have PpsN post-selected photons, while the

technical noise stays the same. Taking d→ Ad this gives

〈x〉 =1√Pps

(αd± σ√

Γt±Sξ√Pps√t

). (3.8)

In other words, once we re-scale, we have the same enhancement of the SNR

by α as discussed in Eq. 3.4, but additionally the technical noise contribution is

54

reduced by√Pps from using the weak value post-selection. Therein lies the power

of weak value amplification for reducing the technical noise of a measurement.

3.3 Experimental Setup

The experimental setup is shown in Fig. 3.1. A 780 nm fiber-coupled laser

is launched and collimated using a 20× objective lens followed by a spherical

lens with f = 500 mm (not shown) to produce a collimated beam radius of

σ = 1.7 mm. For smaller beam radiuses, the lens is removed and the 20×

objective is replaced with a 10× objective. A polarizer is used to produce a

pure horizontal linear polarization. The beam enters the interferometer (this

is the pre-selection) and is divided, traveling clockwise and counterclockwise,

before returning to the beamsplitter (BS). A piezo-actuated mirror on a gimbal

mount at a symmetric point in the interferometer is driven (horizontally) with

a 10 kHz sine wave with a flat peak of duration 10 µs. The piezo actuator

moves 127 pm/mV at this frequency with a lever arm of 3.5 cm. Due to a

slight vertical misalignment of one of the interferometer mirrors, the output port

does not experience total destructive interference (this is the post-selection on

a nearly orthogonal state) and contains approximately 20% of the total input

power, corresponding to φ/2 = 25◦. A second beamsplitter sends this light to

a quadrant cell detector (QCD) (New Focus model 2921) and a charge coupled

device (CCD) camera (Newport model LBP-2-USB). The output from the CCD

camera is monitored and the output from the quadrant cell detector is fed into

two low-noise preamplifiers with frequency filters (Stanford Research Systems

model SR560) in series. The first preamplifier is ac coupled with the filter set

to 6 dB/oct band-pass between 3 and 30 kHz with no amplification. The second

preamplifier is dc coupled with the filter set to 12 dB/oct low-pass at 30 kHz and

55

Figure 3.2: A comparison of signal to noise ratio of interferometric weak valuesmetrology to standard metrology techniques is shown. The blue curve shows thequantum limited SNR for SD setup calculated using Eq. (3.2). The red circlesshow the measured SNR was measured for SD setup. The black diamonds showthe measured SNR for the WVA setup. As predicted by Eq. (3.4) the data showslinear dependence on driving voltage (and hence deflection d). The data for theSD setup (blue and red) use the right axis whereas the data for WVA setup(black) use the left axis. The lines are linear fits to the measured data withy-intercepts forced to zero. The statistical variations are smaller than the datapoints.

an amplification ranging from 100 to 2000. The low-pass filter limits the laser

noise to the 10–90% risetime of a 30 kHz sine wave τ = 10.5 µs) and so we take

this limit as our integration time such that the number of photons incident on

the detector is N = Pτ/Eλ where P is the power of the laser and Eλ is the energy

of a single photon at λ = 780 nm.

In what follows, we compare measurements using two separate configura-

tions: WVA setup is shown in Fig. 3.1 and produces the weak value amplification

SNR found in Eq. 3.4; SD setup (for standard detection) is the same as WVA

setup except with the first 50/50 beamsplitter removed, resulting in the SNR

given by Eq. 3.2. The theoretical SNR curves in figures 3.2 and 3.3 assume a

56

Figure 3.3: A comparison of signal to noise ratio of interferometric weak valuesmetrology to standard metrology techniques is shown. The blue curve shows thequantum limited SNR for SD setup calculated using Eq. (3.2). The red circlesshow the measured SNR was measured for SD setup. The black diamonds showthe measured SNR for the WVA setup. The data show the linear vs. inversedependence on beam radius as predicted by Eqs. (3.2) and (3.5). The lines arelinear or inverse fits to the measured data. The statistical variations are smallerthan the data points.

57

noiseless detector with perfect detecton efficiency; this is what we refer to as an

“ideal measurement.” We see reasonable agreement of the data with theory by

noting the trends in figures 3.2 and 3.3 follow the predictions of Eqs. 3.4 and 3.5.

The quoted error comes from the measured data’s standard deviation from the

linear fits.

Data were taken for various piezo voltage amplitudes with a beam radius

σ = 1.7 mm, a detector distance lmd = 14 cm, and an input power of 1.32 mW

(Fig. 3.2). Using SD setup, we measured a SNR that was 1.77± 0.07 times worse

than an ideal measurement; with WVA, i.e. WVA setup, we measured a SNR

that was 21.8± 0.5 times better than an ideal measurement, corresponding to an

amplification of the SNR by α = 39± 3.

Next, the beam radius at the detector σ was varied from 0.38 mm to 1.1

mm while the beam radius at the lens was roughly constant at a = 850 µm. For

these measurements, the input power was 1.32 mW, the distances were llm = 0.51

m and lmd = 0.63 m, and the driving voltage amplitude was 12.8 mV. The results

are shown in Fig. 3.3. Using SD setup, we find that the SNR varies inversely with

beam radius as predicted by Eq. 3.2. However, using WVA setup, we see a linear

increase in the SNR as the beam radius is increased as predicted by Eq. 3.5.

It is interesting to note that equation 3.4 is independent of the detector

location lmd. The deflection in the numerator d = lmd ∆θ depends on lmd which

cancels with the lmd in the denominator, leaving only the angular deflection ∆θ.

this was verified experimentally and suggests a much smaller interferometer can

be advantageous by combining amplification effects with increased stability. To

demonstrate this we constructed an interferometer with a lmd = 42 mm and a

smaller beam radius σ = 850 µm. For this geometry with 2.9 mW of input light

and 390 µW of output light, the predicted amplification is α = 260, however the

58

measured value was only α = 150.

An important note is that although the small interferometer did not

achieve the expected value of α = 260, the smaller size allowed it to approach

this expected value much better than the larger interferometer. Indeed, the small

interferometer’s measured α was only 260/150 = 1.7 times below the expected

value, while the large interferometer’s measured α was 300/39 = 7.7 times below

the expected value.

3.4 Channel Analysis

Just as I did in chapter 2, by considering the deflecting mirror as the

source of the message and the measured signal as the received message, through

a noisy communication channel, I can analyze this set of experiments in terms

of communication theory. Again, I model the mirror deflection as a Gaussian

random variable X with variance S. The noise can in general come from several

sources including the beam itself (in the form of shot noise) and the detection

system (technical noise), which I model as independent additive Gaussian noises

with variances Ns and Nt, respectively. The mutual information for such a system

without weak value amplification is

I(X;Y ) = log

(1 +

S

(Ns +Nt

). (3.9)

By incorporating weak value amplification we can increase this mutual informa-

tion. The effect is slightly different based on which noise terms dominates.

For the case of technical noise being the dominant noise term Nt � Ns,

weak values amplifies the deflection by the factor A, leaves the noise unaffected,

and post-selects on a small portion of the beam power given by Pps. Because the

noise is unaffected, this results in the mutual information in bits per detected

59

power being amplified by the factor A2/Pps, as described in by Eq. 2.9 (assuming

a small initial SNR).

For the case of shot noise being the dominant noise term Ns � Nt, weak

values amplifies the SNR by the factor α ≈ A/φ meaning S/Ns is amplified by

α2, again the post selection probability is Pps, resulting in

Ia(X;Y ) =1

Ppslog

(1 + α2 S

Ns

). (3.10)

In the limit of small initial SNR the mutual information is amplified by the factor

AI = Ia/I =α2

Pps. (3.11)

The small interferometer had an expected SNR amplification of α = 260,

and the post-selection probability of Pps = 0.14. Assuming this system was shot

noise limited, these values give an expected mutual information amplification

of AI = 480, 000. The measured SNR amplification was smaller than expected

however, with α = 150, resulting in a mutual information amplification of AI =

160, 000.

It should be noted that, for these interferometers, having a shot noise

limited detection system will only gain a factor of 10 in the mutual information

amplification factor AI . This is a result of the fact that φ ≈ 1/10 in these

experiments and α ≈ A/φ. This limitation would not apply to more sophisticated

interferometers or other weak value experiments where Pps could be much smaller.

A more important effect related to technical noise is that the detector itself

has different requirements. As a result of the post-selection, the optical power

incident on the detector is reduced. This allows for a different, possibly lower

noise detector to be used. In this way, the technique allows for the detection

system to more easily (and cheaply) approach being shot noise limited.

60

3.5 Concluding Remarks

While the interferometric weak value technique does not beat the ultimate

limit for a beam deflection measurement, it does have a number of improvements

over other schemes: it reduces technical noise, it allows for the use of high power

lasers with low power detectors while maintaining the optimal SNR, and it al-

lows one to obtain the ultimate limit in deflection measurement with a large

beam radius. Additionally, we point out that, while weak values can be under-

stood semi-classically, the SNR in a deflection measurement requires a quantum

mechanical understanding of the laser and its fluctuations.

It is interesting to note that interferometry and split detection have been

competing technologies in measuring a beam deflection [25]. Here we show that

the combination of the two technologies leads to an improvement that can not be

observed using only one, i.e. that measurements of the position of a large radius

laser beam with WVA allows for better precision than with a quantum limited

system using split detection for the same beam radius. In terms of communication

channels, the signal amplification as well as the altered noise is analyzed. The

technique is found to amplify the channel capacity in bits/photon by 5 orders of

magnitude for standard laboratory optical setups.

61

Chapter 4

Ghost Imaging Through Turbulence

This chapter begins my investigation of position-momentum entangled

states. I use these states not in metrology settings (as with weak values) but in

direct two party communication settings. Before characterizing the capabilities

of such states, it is important to know whether they have utility in realistic

communication settings. In this chapter I investigate the effect of turbulence on

quantum ghost imaging using position-momentum entangled photons.

4.1 Introduction

The phenomenon of ghost imaging, first observed by Pittman et al. in 1995

[15], is a method of generating the image of an object from correlation measure-

ments. Pittman’s experiment made use of pairs of entangled photons. One of the

photons passed through a transmission object and then to a photon counter with

no spatial resolution. The other photon passed directly to a spatially resolving

photon counter. When looking at coincident photon detections, the detectors

were able to see the object despite the fact that the object and the spatially

resolving detector were in different arms of the experiment. While it was initially

thought to be a quantum mechanical effect reliant upon the entanglement be-

tween the two photons, similar results were later obtained using classical sources

[76].

In addition to clarifying the boundary between quantum and classical

62

Figure 4.1: Experimental setup for ghost imaging through turbulence measure-ment. A pump beam undergoes SPDC at a nonlinear crystal (NLC), the outputpasses a beamsplitter (BS). One beam is sent through a lens and onto a trans-mission object. The other beam is sent through a lens and onto a scanning slit.The ghost image of the object is profiled by the slit. Photons are detected withsingle-photon avalanche diodes (SPAD). Detection events are then correlated.

effects [77–79], ghost imaging has been used for lensless imaging [80], super-

resolution [22, 81], and entanglement detection [82]. More recently, research has

recognized connections between ghost imaging and compressive sensing [20, 21].

For many optical applications, imaging through turbulence is unavoidable

[83, 84], and research on the effect of turbulence on ghost imaging has recently

witnessed a surge of interest [85–88]. In this chapter, I experimentally investigate

the effect of turbulence on ghost imaging using position-momentum entangled

photons. I present the first experimental demonstration that entangled-photon

ghost imaging is affected by turbulence and how the effect can be reduced.

63

Figure 4.2: Conceptual setup for ghost imaging through turbulence measurement.The experiment is shown using the Klyshko picture [64], the object (on the right)is ghost imaged onto the scanning slit (on the left). The nonlinear crystal isoffset from the central image plane by a distance ∆. The top picture shows theturbulence—represented by wavy lines—between the crystal and the lens. Thebottom picture shows the turbulence located between the lens and the object.Experimentally relevant distances are labelled.

Figure 4.3: Conceptual setup for ghost imaging through turbulence measurement.The experiment is shown using the Klyshko picture [64], the object (on the right)is ghost imaged onto the scanning slit (on the left). The nonlinear crystal isoffset from the central image plane by a distance ∆. The top picture shows theturbulence—represented by wavy lines—between the crystal and the lens. Thebottom picture shows the turbulence located between the lens and the object.Experimentally relevant distances are labelled.

64

4.2 Theoretical Description

The experimental apparatus is depicted in Fig. 4.1. A biphoton state |ψ〉

is created at a nonlinear crystal [65] and then split by a 50/50 beamsplitter,

sending the biphoton into two arms of the apparatus.

In the object arm, the biphoton travels a distance 2f + ∆, to a lens

which has focal length f . The biphoton then travels a distance 2f to a photon

detector with no spatial resolution (a “bucket” detector). A transmission object—

consisting of alternating opaque and clear vertical bars—is placed just in front

of the detector. In the image arm, the biphoton travels a distance 2f − ∆ to a

lens which again has focal length f . The biphoton then travels a distance 2f to

a spatially-resolving detector.

For ∆ = 0 the detectors and crystal are all located at image planes of each

other. As one arm’s lens/detector is moved towards the crystal by a distance ∆,

the other arm’s lens/detector is moved away by the same distance, keeping the

sum of the arm’s length constant, see figures 4.2 and 4.3.

Turbulent air flow is introduced into the beam path of the object arm. For

turbulence between the crystal and the lens, it is a distance l1 from the crystal—

or a distance l1 − ∆ from the central image plane. For turbulence between the

lens and the object, it is a distance ∆− l1 from the object.

The relevant function for GI is the second order degree of coherence

G(2)(x1, x2), where x1 is a transverse position variable in the plane of the spatially-

resolving detector and x2 is a transverse position variable in the plane of the

bucket detector. We begin with the standard quantum mechanical form and in-

clude an additional ensemble averaging—represented by large outer brackets—to

65

account for the statistical effect of turbulence:

G(2)(x1, x2) =

⟨〈ψ|E†i (x1)E†s(x2)Es(x2)Ei(x1)|ψ〉

⟩. (4.1)

Neglecting overall normalization, this can be represented in the following way:

G(2)(x1, x2) =

⟨∫ (4)

ψ?(xs, xi)H?(xi, x1)H?(xs, x2; xt)

×H(xs, x2;xt)H(xi, x1)ψ(xs, xi)dxidxsdxsdxi

⟩.

(4.2)

Subscript s and i indicate variables in the crystal plane and subscript t indicates

variables in the plane of the turbulence. The function ψ(xs, xi) is the transverse

biphoton wavefunction which we approximate as a plane-wave with delta function

correlations ψ(xs, x1) = δ(xs − xi). The function H(xs, x2;xt) is a propagation

operator going from the crystal plane to the object arm detection plane, passing

through the plane of turbulence; H(xi, x1) is a propagation operator going from

the crystal plane to the image arm detection plane. These operators can be

represented in the following way:

H(xs, x2;xt) =

∫exp

[−ik (x2 − xt)2

2(l1 −∆)

]T(xt)

× exp

[ik(xt − xs)2

2l1

]dxt,

(4.3)

H(xi, x1) = exp

[−ik2∆

(xi − x1)2

]. (4.4)

In our theoretical treatment, we assume a narrow sheet of turbulent air, whose

effect on propagation can be characterized by a multiplicative operator T(xt).

We also assume that the lenses are sufficiently large that they capture all of the

light from the SPDC source. As a result, both turbulence locations (shown in

Fig. 4.2 and Fig. 4.3) are governed by the same operators.

We model the turbulence as a 6/3 scaling law effect:⟨T?(xt)T(xt)

⟩=

exp [−α (xt − xt)2/2], where α parameterizes the strength of the turbulence and

66

has units 1/m2 [83, 84]. The resulting expression for G(2)(x1, x2) is:

G(2)(x1, x2) = exp

[−k2

(x1 − x2

)2

2α (l1 −∆)2

]. (4.5)

The ghost image I(x1) is then the product of the object and G(2)(x1, x2),

integrated over x2. We represent the object as: O(x2) = exp [−x 22 /2w

2](1 +

cos(ko x2)). Here w is the spatial width of the illuminating beam and ko is

wavenumber for the object’s pattern spacing. Assuming (l1 −∆)√α � k w, the

ghost image is found to be

I(x1) = exp

[−1

2

(x1

w

)2](

1 + V cos(ko x1)). (4.6)

I(x1) has the same form as O(x1) with the object’s unity visibility replaced by

the GI visibility V:

V = g × exp

[−α(l1 −∆

)2

2 (k/k0)2

]. (4.7)

Where g is the the optimum GI visibility with no turbulence. As either the

turbulence increases in strength (increasing α) or the turbulence is moved away

from the central image plane or the detector (increasing l1 − ∆), the detected

visibility V decreases—thus obscuring the detected pattern.

4.3 Experiment

Collimated light from a 3 mW, 325 nm HeCd laser with a 1/e2 full width

of approximately 1600 µm pumped a 10 mm thick BBO nonlinear crystal. The

crystal was oriented for degenerate type-I collinear spontaneous parametric down-

conversion (SPDC). After the crystal, the pump beam was blocked by colored

glass filters and the SPDC bandwidth was limited by a 3nm wide spectral filter

centered at 650 nm. The remaining SPDC beam was split into two arms by a

50:50 beamsplitter.

67

In the image arm, a lens was located 1000 mm − ∆ from the crystal; in

the object arm, a lens was located 1000 mm + ∆ from the crystal. Both lenses

had focal length f = 500 mm. Detectors were located 1000 mm from the lenses.

The transmission object was a test pattern located 1000 mm from the lens.

The bucket detector consisted of a 10× microscope objective which collected the

transmitted light into a multimode optical fiber. The pattern had unit visibility

and 3.6 cycles per mm, which resulted in an object pattern wavenumber of ko =

7.2×π mm−2. The spatially-resolving detector consisted of a computer controlled

scanning slit located 1000 mm from the lens, which was again followed by a 10×

microscope objective which collected light into a multimode optical fiber. The

slit was approximately 40 µm wide and was scanned in 5 µm increments, giving

spatial resolution.

The optical fibers were connected to Perkin Elmer single-photon detectors.

The outputs of these detectors were time correlated using a PicoHarp 300 from

PicoQuant. Photon counts were integrated at each slit location for between 1

and 4 seconds. The spatially resolved coincident detections made up the ghost

image profiles.

A heat gun was mounted above the setup, providing turbulent air flow

across the beam path. The effect of the turbulence was fitted to the model’s wave

structure function αx2 [84]. From the fit we determined α = 2.5± 1.5 mm−2. It

should be noted that although our theoretical model makes use of a thin sheet of

turbulence, experimentally the turbulent region was approximately 10 cm wide.

The turbulence was therefore present for a significant portion of the apparatus

arm.

Data was taken for an unshifted configuration with ∆ = 0, and for a

shifted configuration with ∆ = 330 mm. In each configuration, ghost images were

68

recorded with turbulence present in the object arm: both between the crystal and

lens, and between the lens and the object. Ghost images were also recorded with

no turbulence. The recorded ghost image profiles were fitted to I(x1) from Eq.

4.6. The fit included a visibility term which constituted our measurement of the

visibility V.

While allowing access to the central image plane of the apparatus, the

shifted configuration introduced two experimental limitations: the detected flux

decreased significantly as a result of the detectors being away from the beam

focus, and fewer spatial frequencies contributed to the ghost image as a result of

the nonlinear crystal having a stronger aperturing effect. Representative ghost

images are shown in Fig. 4.4. With no turbulence, the unshifted configuration

produced GI visibilities of 1.00 ± 0.05. The shifted configuration produced GI

visibilities of only 0.65 ± 0.05, which can be explained by the true correlation

area having finite extent (this is explained in the detailed theoretical analysis of

ref. [89]). The scans also show the decreased flux and the broader beam profile

associated with the shifted configuration.

Visibilities for turbulence between the lens and the object are shown in

Fig. 4.5. When turbulence was close to the object, the observed visibility was near

its no turbulence levels. As the turbulence was moved away from the object, the

GI visibility decreased. The visibility for the unshifted configuration remained

above the visibility for the shifted configuration for all turbulence locations.

Visibilities for turbulence between the crystal and the lens are shown in

Fig. 4.6. This is the main result of the experiment. Visibilities decreased as the

turbulence was moved away from the crystal, however, the unshifted configuration

had lower fringe visibility than the shifted configuration. Indeed, for turbulence

located 432 mm from the crystal, the visibility was V = 0.15 ± 0.04 for the

69

Figure 4.4: Representative ghost images for the unshifted configuration (left),and shifted configuration (right). The top row shows images with no turbulence.The middle row shows images for turbulence between the lens and the object, 203mm (right) and 229 mm (left) from the object. The bottom row shows imagesfor turbulence between the crystal and the lens, 432 mm from the crystal. Pointsare experimental data while curves are fits to the data. Counts are measured incoincident photon detections per second.

70

Figure 4.5: Ghost image visibilities turbulence near the object. GI visibilities areshown for turbulence between the lens and the object. Visibilities are plotted asa function of distance from the object to the turbulence (l1−∆ in Eq. 4.7). Datafor the unshifted configuration are shown as blue circles. Data for the shiftedconfiguration are shown as purple squares. Curves are plots from Eq. 4.7. Thesolid curve is for the unshifted configuration, with g = 1.00. The dashed curve isfor the shifted configuration, with g = 0.65. For both curves α = 2.0 mm−2.

71

Figure 4.6: Ghost image visibilities for turbulence near the illumination source.GI visibilities are shown for turbulence between the crystal and the lens. Visibil-ities are plotted as a function of distance from the crystal to the turbulence (l1in Eq. 4.7). Data for the unshifted configuration are shown as blue circles whiledata for the shifted configuration are shown as purple squares. Curves are plotsfrom Eq. 4.7. The solid curve is for the unshifted configuration, with g = 1.00.The dashed curve is for the shifted configuration, with g = 0.65. For both curvesα = 2.0 mm−2. The vertical line marks the location of the central image plane.

72

unshifted configuration, while for the shifted configuration it was V = 0.42±0.04.

Moving to the shifted configuration tripled the visibility. This effect can be

understood physically by recognizing that the image of an object is unaffected

by perturbing the phase of the illumination source—images consist of intensities

only. By placing the turbulence near one of the image planes, it is as if we are

perturbing the phase of the illumination source only, not the propagation.

4.4 Channel Analysis

In this experiment I can make direct use of the probability distributions

that give rise to the coincident photon detections to calculate the mutual infor-

mation. When normalized, the G(2)(x1, x2) function, along with the overall beam

profile exp [−x 22 /2w

2] make up the joint measurement probability density:

p(x1, x2) =1

CWπexp

(−(x1 − x2)2

C2

)exp

(−x2

2

W 2

), (4.8)

where C is the width from the G(2)(x1, x2) function: C = k/(√

2α(l1 −∆)).

Using this probability distribution to directly calculate the mutual infor-

mation (as explained in chapter one), where the x1 position corresponds to the

sent message and the x2 position corresponds to the received message, results in:

I =1

2log

(1 +

W 2

C2

). (4.9)

This measures bits of information per joint photon detection event.

We can see that by removing the turbulence entirely (α→ 0), or offsetting

the crystal such that l1 = ∆, the mutual information tends to infinity. This is

a consequence of assuming delta function correlations in the quantum state as

well as assuming infinitely large lenses. In reality, the finite correlation area and

the optical system’s point spread function will limit the mutual information to a

finite but still large number.

73

Figure 4.7: The additional mutual information is shown as a function of l1 −∆for the case of turbulence near the crystal (shown in figure 4.3), with the crystalshifted by ∆ = 330 mm. This behavior is described by Eq. 4.10

For the situation with turbulence near the crystal, one can analyze the

increased mutual information inherent in the shifted configuration. Unlike in the

weak value experiments, the experimental parameters here are such that W � C,

resulting in additional mutual information per biphoton given by

Is − I =1

2log

(1 +

W 2

C2s

)− 1

2log

(1 +

W 2

C2s

)=

1

2log

(l1

l1 −∆

),

(4.10)

where Cs is the width of the G(2)(x1, x2) function for the shifted configuration

and C is the width for the unshifted configuration. The effect is shown in Fig. 4.7

using the experimentally implemented shift of ∆ = 330 mm. At the l1 = 432 mm

point that resulted in a tripling of the fringe visibility, there is 1 additional bit

per biphoton resulting from the shifted configuration benefit. This counteracts

much of the loss of bits due to the turbulence and a sufficiently engineered system

should be able to completely cancel turbulence induced loss of information.

74

4.5 Concluding remarks

By moving the crystal from the central image plane we were able to place

turbulence in this plane. This decreased the observed effect of turbulence, in

fact it more than made up for the inherent loss of visibility associated with

the shifted configuration. This technique has use in free space GI applications

where turbulence is involved. By arranging detectors to place an image plane

at the location of the turbulence, image degradation from the turbulence can

be diminished. Although we used optical fields and turbulent airflow, our result

applies to any type of propagating wave and a broad class of random or complex

media including, for example, biological tissue or metamaterials.

Although we have used entangled photons, similar results are expected

for thermal light ghost imaging. It should also be noted that the theoretical

description assumes delta function correlations for the biphoton state and a thin

region, non-Kolmogorov turbulence model [83, 84, 90]. The limitations of the

theoretical do not extend to the experimental results, indeed the biphoton state

had a correlation size of approximately 50 µm. The turbulence was in reality

volume turbulence approximately 10 cm in length, and it did not truly have the

Gaussian structure function of our approximation. Indeed, a more sophisticated

theoretical description of the experiment has been developed [89]. This model

takes into account volume turbulence and more realistic quantum states and

agrees better with the experimental data than the theoretical description provided

here.

In this chapter I have demonstrated a method of ameliorating the effects

of turbulence on GI systems, and have provided a theoretical model which accu-

rately describes the experimental data. I shift the source of entangled photons

away from a quantum ghost imaging system’s central image plane, and place

75

turbulence near this plane. This dramatically increases the ghost image con-

trast. For turbulence located 432 mm from the crystal, this technique took the

observed pattern visibility from V = 0.15± 0.04 to V = 0.42± 0.04, tripling the

system’s imaging visibility. Analyzed in terms of communication channel and

mutual information—the novel configuration allows for additional bits of infor-

mation per biphoton to be transmitted. For turbulence located 432 mm from the

crystal, each biphoton transmitted 1 additional bit of information, thus counter-

acting much of the loss of information due to turbulence.

76

Chapter 5

Mutual Information

In this chapter I investigate the capabilities of position-momentum entan-

gled states as a communication channels. High dimensional Hilbert spaces used

for quantum communication channels offer the possibility of large data transmis-

sion capabilities. I propose a method of characterizing the channel capacity of an

entangled photonic state in high dimensional position and momentum bases. I

use this method to measure the channel capacity of a parametric downconversion

state, achieving a channel capacity over 7 bits/photon in either the position or

momentum basis, by measuring in up to 576 dimensions per detector. The chan-

nel violated an entropic separability bound, suggesting the performance cannot

be replicated classically.

5.1 Introduction

Quantum systems can be entangled in various degrees of freedom. Typi-

cal examples include photonic polarization states [91] and atomic or ionic energy

levels [92]. These systems enable various technologies such as quantum com-

munication, quantum cryptography, and quantum computation [93–95], however

the systems are not in principle limited to two states [18, 19, 96–99]. Indeed,

for communication purposes—such as quantum key distribution [100, 101] or

dense coding communication [102]—higher dimensional states increase quantum

communication channel capacity and offer additional benefits such as increased

77

security [103, 104]. The photonic position degree of freedom is a good candidate

for practical high-dimensional entangled systems due to the wide availability of

off the shelf technology for manipulating this degree of freedom [105]. In this

chapter I describe a quantum channel capacity characterization that considers

both the quantum state and the measurement apparatus. I use this method

to measure the channel capacity of a high dimensional position and momentum

entangled photonic state. The measurements include up to 576 dimensions per

detector and demonstrate channel capacities of over 7 bits/photon.

The capacity of a quantum channel using entangled photons characterizes

information transfer in joint detection events. The locations where joint photon

detections can take place can be thought of as characters in an alphabet; the

size of this alphabet is the number of distinguishable joint detection locations

available within the beam envelopes (see for example [26]). Theoretically, the

channel capacity calculation the best possible measurement for a given state used

as the channel. Practically however, the measurement technique is not necessarily

optimal and the channel capacity depends on the details of this measurement.

Experimental characterization of channel capacities has consisted of per-

forming quantum state tomography and then using the result in channel capacity

calculations. Recently Pors et al. [106] proposed a more direct way of character-

izing the channel capacity without recreateding the full quantum state. By con-

sidering the quantum state in conjunction with the measurement apparatus, they

define an effective channel dimensionality called the Shannon dimensionality. We

propose a similar measure: an effective entropic channel capacity—rather than

dimensionality—that considers both the state and the measurement apparatus,

and measures bits of information per detection event. This quantity character-

izes the information capacity that can be effectively probed for a given system.

78

Figure 5.1: Experimental setup for high dimensional quantum mutual informa-tion characterization. A collimated laser beam undergoes spontaneous paramet-ric down-conversion at a nonlinear crystal. The output passes a focusing lensfollowed by a beam-splitter. The outputs from the beam-splitter are sent to dig-ital micro-mirror devices at either image planes or Fourier planes of the crystal.The micro-mirror devices are set to retro-reflect the beams, a quarter wave plateand a polarizing beam-splitter send the retro-reflected beam to a single-photondetector. A coincidence circuit correlates these measurements.

It depends solely on each party’s measurements and is independent of character

coding scheme.

5.2 Theoretical description

I use mutual information to quantify the channel capacity. Mutual in-

formation describes how much information can be determined about a random

variable A, by knowing the value of a correlated random variable B [26, 107].

Discrete variables A and B are characterized by the values they take a and b,

respectively, and the probability of these discrete values P (a) and P (b), respec-

tively. The mutual information can be written as

I(A;B) = H(A) +H(B)−H(A,B), (5.1)

79

where, for example

H(A) = −∑a∈A

P (a) logP (a) (5.2)

is the marginal entropy of A, and

H(A,B) = −∑a∈Ab∈B

P (a, b) logP (a, b) (5.3)

is the joint entropy of A and B. The function P (a, b) is the joint probability

distribution which characterizes the correlation between A and B.

We created a position-momentum entangled state using spontaneous parametric-

downconversion (SPDC). The state, represented both in position and in momen-

tum, is approximated as [12]

|ψ〉 =

∫d~xad~xb f(~xa, ~xb) a

†aa†b |0〉

=

∫d~kad~kb f(~ka, ~kb) a

†aa†b |0〉

(5.4)

where a† is the photon creation operator. Subscript a or b indicates the photon is

created in the signal or idler mode, which are sent to Alice or Bob, respectively.

The function

f(~xa, ~xb) = N exp

(−(~xa − ~xb)2

4w21

)exp

(−(~xa + ~xb)

2

16w22

)(5.5)

is the entangled biphoton wavefunction in the position basis, and

f(~ka, ~kb) = (4w1w1)2N exp(−w21(~ka − ~kb)2)

× exp(−4w22(~ka + ~kb)

2)(5.6)

is the biphoton wavefunction in the momentum basis. In these equations N =

(2πw1w2)−1 is a normalization constant, w2 is Gaussian width in the x1 + x2

direction, and 2w1 is the Gaussian width in the x1 − x2 direction. These rep-

resentations are related through a Fourier transform, which inverts the relative

80

Gaussian function widths, such that position correlations become momentum

anti-correlations.

Experimentally measured widths however are the single photon width

sigmap and the conditional width σc. These widths are related to w1 and w2

in the following way:

σ2p = w2

2 +(w1

2

)2

(5.7)

σ2c =

4w21w

22

4w22 + w2

1

(5.8)

For w1 � w2 we can approximate the single photon width as σp = w2 and

the conditional photon width as σc = w1.

To measure position correlations we put spatially-resolving single-photon

detectors at image planes of the SPDC source: to measure momentum correla-

tions we put the detectors at Fourier transform planes of the source. For our

purposes then, random variable A corresponds either to the position or momen-

tum of Alice’s photon and B corresponds either to the position or momentum of

Bob’s photon.

The theoretical maximum mutual information for the wavefunction in Eq.

5.5 (measuring in the position basis), is:

I(A;B) = −∫p(~xa, ~xb) log

(p(~xa, ~xb)

p(~xa)p(~xb)

)d~xad~xb (5.9)

where p(~xa, ~xi) = |f(~xa, ~xb)|2 and p(~xa) =∫|f(~xa, ~xb)|2d ~xb. are discrete probabil-

ity densities For the momentum basis, the same relations hold, but the position

variables are replaced by the momentum variables and the position wavefunction

is replaced by the momentum wavefunction of Eq. 5.6.

For either basis, this theoretical maximum simplifies to

I(A;B) = log

(σpσc

)2

= log

(4w2

2 + w21

4w1w2

)2

, (5.10)

81

which is independent of detector characteristics. The ratio σp/σc is the familiar

Fedorov ratio for quantifying entanglement [69]. Our physical SPDC state had a

beam envelope width of σp = 1500 µm and a correlation width of approximately

σc = 40 µm, resulting in an optimum mutual information from Eq. 5.10 of Ic ∼= 10

bits/photon.

This mutual information is slightly different than what we might ex-

pect from Shannon’s noisy channel calculations: if we think of σp as the sig-

nal channel width and σc as the noise width, then we would expect I(A;B) =

log(1 + (σp/σc)2). The reason for the discrepancy is that the σc noise is not ad-

ditive as Shannon’s formula requires. Additive noise can be reduced by signal

averaging over time, and indeed for σc = σp this is not the case. We can cast the

formula in terms of an effective additive noise N as follows:

I(A;B) = log

(1 +

(σpN

)2)

(5.11)

N =σ2p

1− σ2c/σ

2p

. (5.12)

We can see that N ≈ σc for σc � σp but diverges as σc → σp.

The measurement apparatus consisted of a digital micromirror device

(DMD) chip reflecting a portion of the signal or idler beam onto a single photon

counting module. The DMD chip allowed us to raster scan over the face of the

beam in a controllable number of detection pixels, giving varying detector res-

olution. To incorporate the effects of the measurement apparatus, we integrate

the probability density over the pixel area giving a true porbability (rather than

a probability density) corresponding to the value of a discrete variable. For posi-

tion correlations between the mth pixel on Alice’s detector, and the nth pixel on

Bob’s detector, the joint detection probability is:

P (m,n) =

∫m

d~xa

∫n

d~xb |f(~xa, ~xb)|2. (5.13)

82

Similarly, for momentum correlations between the mth pixel on Alice’s

detector, and the nth pixel on Bob’s detector, the joint detection probability is:

P (m,n) =

∫m

d~ka

∫n

d~kb |f(~ka, ~kb)|2. (5.14)

The detected mutual information in either position or momentum is then

I(A;B) =∑m

P (m) logP (m) +∑n

P (n) logP (n)

−∑m,n

P (m,n) logP (m,n),(5.15)

where, for example,

P (m) =∑n

P (m,n) (5.16)

is the marginal probability for a pixel on Alice’s detector.

It should be noted that the mutual information calculation only takes into

account coincident photon detections. Thus a non-ideal detection efficiency does

not change the form of the mutual information equations, rather it reduces the

effective SPDC beam intensity. This effect reduces the system’s information per

unit time, but not the information per detected photon pair.

5.3 Experiment

The experimental setup is shown in Fig. 5.1. A 325 nm wavelength laser

beam with diameter a diameter of σp = 1500 µm pumped a 10 mm long BBO

nonlinear crystal aligned for Type-I degenerate collinear SPDC. The SPDC out-

put had a correlation width of approximately σc = 40 µm. For these parameters

the optimum mutual information from Eq. 5.9 is I ∼= 10 bits/photon. For mea-

suring position correlations the SPDC output passed through a 125 mm focal

length lens followed by a beamsplitter and a spatially resolving detectors were

located at the resulting image planes of the crystal. For measuring momentum

83

Figure 5.2: Mutual information for position correlation measurements are shownas a function of detector resolution. Data for detector resolutions of 8× 8 pixels,16 × 16 pixels, and 24 × 24 pixels are shown. The dark blue points with errorbars are experimental data. The light blue bars are numerical simulations basedon Eq. 5.15 both for the case of perfectly relative transversely aligned detectorsand the case of a relative transverse misalignment of half a pixel. The red curveis the maximum mutual information that can be detected for the correspondingnumber pixels per detector.

84

Figure 5.3: Mutual information for momentum correlation measurements areshown as a function of detector resolution. Data for detector resolutions of 8× 8pixels, 16×16 pixels, and 24×24 pixels are shown. The dark blue points with errorbars are experimental data. The light blue bars are numerical simulations basedon Eq. 5.15 both for the case of perfectly relative transversely aligned detectorsand the case of a relative transverse misalignment of half a pixel. The red curveis the maximum mutual information that can be detected for the correspondingnumber pixels per detector.

85

correlations the SPDC output passed through a 150 mm focal length lens followed

by a beamsplitter and a spatially resolving detector was placed at the resulting

Fourier transform planes of the crystal.

The spatially resolving single-photon detectors consisted of a computer

controlled digital micro-mirror device from Texas Instruments in conjunction with

a Perkin Elmer single photon avalanche diode (SPAD) running in geiger mode.

The micro-mirror displays had 1064 × 768 resolution which selectively reflected

portions of the SPDC signal and idler beams to SPADs. Photon detection events

were correlated with a PicoHarp 300 from PicoQuant with a 3 ns coincident

window.

The micro-mirror displays were each raster scanned and counts for each

pixel pair were recorded for between 1 and 5 seconds. These double raster scans

were set such that they were centered on the signal and idler beams, divided into

8× 8 pixels, 16× 16 pixels, and 24× 24 pixels, encompassing 80% of each beam.

For ideal alignment, a given pixel on Alice’s detector will correlate very

well to only one pixel on Bob’s detector. In practice however, a relative lat-

eral shift of pixels between Alice’s and Bob’s detectors—both vertically and

horizontally—spreads correlations to four pixels at best. However, pixels far

from the correlated pixel will still have no correlation. This was verified experi-

mentally and it allowed us, for a given pixel on Alice’s detector, to scan only in

a region of interest around the correlated pixel on Bob’s detector, thus reducing

the time required to complete a double raster scan.

Both the predicted and experimentally measured values for mutual infor-

mation are shown in figures 5.2 and 5.3. Values for position correlation measure-

ments are shown in 5.2 and values for momentum correlation measurements are

shown in 5.2. Uncertainties in detected photon number N for each point in the

86

double raster scan were assumed to be√N. This uncertainty was then propa-

gated through the entropy calculations, giving the uncertainties for the measured

mutual information values. These values were found to be in agreement with

the statistics found by taking multiple data scans for a given detector resolution.

It should be noted that this uncertainty calculation method does not take into

account detector dark counts. Since the dark counts from each detector are un-

correlated, the dark coincident rate is much less than the coincident rate from

the highly correlated SPDC state.

Light blue bars represent predicted mutual information values from nu-

merical calculations of Eq. 5.15. The tops of the bars correspond to perfect lateral

pixel alignment between Alice and Bob, and the bottoms correspond to relative

lateral shifts, both horizontally and vertically, of half a pixel. These cases repre-

sent the maximum and minimum mutual information possible for a given number

of detector pixels. The dark blue circles represent experimentally measured chan-

nel capacities. The red curve gives the maximum mutual information that can

be detected I = log(n× n) for n× n pixel resolution per detector. Data for the

detectors scanning in 8× 8 pixels, 16× 16 pixels, and 24× 24 pixels are shown.

The experimental data agrees well with the theoretically predicted values.

For momentum correlations, a maximum mutual information of 7.2± 0.3

bits/photon was achieved; for position correlations only 7.1 ± 0.7 bits/photon

were achieved. In principle, the two measurement bases should give the same

mutual information. However, the alignment for position correlations was more

sensitive—the reduction of mutual information for this basis most likely resulted

from slight system misalignments.

The 576 dimensional measurement space is 16 times larger than the pre-

vious maximum for position-momentum entangled photons, and had an increase

87

of bit capacity of more than 50% [99]. It should be noted that channel capac-

ity characterization is different from using the channel for communication. When

used for key distribution or communication, the characterized channel will indeed

transmit 7 bits of information for a single joint detection event, despite the fact

that our characterization method requires many photon detection events. The

use of this channel for key distribution or communication does however require

some additional structure (such as in ref. [100] or ref. [102]). We are further

investigating the ultimate experimental realization these structures. Although

our entropic channel characterization measure is similar to the Shannon dimen-

sionality of Pors et al. it has several important differences: the units of the

measures are not the same and the the weighting of the different pixel proba-

bilities differs between the measures—however for a flat probability distribution

the measures have identical magnitudes. By measuring bits of information rather

than dimensionality, our measure is more directly linked to channel information

capacity.

By taking data in two mutually unbiased bases (position and momentum) I

am able to test the separability of the state used as the communication channel. A

separable state satisfies the inequality H(A|B)P +H(A|B)M ≥ 2 log2(πe) ≈ 6.18

where subscripts P or M indicate measurements in the position or momentum

bases respectively, and H(A|B) = H(A,B)−H(B) is the conditional entropy of

A given B [63, 108, 109]. From the 8× 8 pixel scan data, we calculate

H(A|B)P +H(A|B)M = 3.7± 0.3 (5.17)

H(B|A)P +H(B|A)M = 3.9± 0.2, (5.18)

88

from the 16× 16 pixel scan data, we calculate

H(A|B)P +H(A|B)M = 3.2± 0.8 (5.19)

H(B|A)P +H(B|A)M = 3.1± 0.8, (5.20)

and from the 24× 24 pixel scan data, we calculate

H(A|B)P +H(A|B)M = 2.2± 0.7 (5.21)

H(B|A)P +H(B|A)M = 2.2± 0.6. (5.22)

As the scan resolution increases, we probe stronger correlations and the separa-

bility sums decrease. For 24 × 24 resolution the separability bound is violated

by more than 5 standard deviations, indicating it is unlikely that the channel

performance can be replicated classically.

Although our characterization method is independent of character coding

scheme, it suggests a simple one: we assign an alphabet character to each of the

pixels. This scheme achieves the measured channel capacities, and errors can be

minimized by reducing the size (but not location) of pixels on Alice’s detector,

such that the system is unaffected by small relative lateral pixel misalignments

between Alice and Bob.

The experiment was limited only by pump laser flux. Higher resolution

scans are in principle possible, however they are not practical for our setup due

to the scan time exceeding the relaxation time of the optical setup. A more

powerful pump laser would enable higher resolution scans while maintaining fea-

sible times. Such scans would come closer to experimentally demonstrating the

optimum capacity for a position momentum entangled state.

89

5.4 Concluding Remarks

In this chapter I have proposed and demonstrated a simple method of char-

acterizing the quantum mutual information based channel capacity of a quantum

communication channel using position and momentum entangled photons and

a controllable pixel mirror. I measured up to 576 dimensions per detector, in

both the position and the momentum basis, which resulted in a measured chan-

nel capacity of more than 7 bits/photon for either basis. The channel violated

an entropic separability bound, indicating the performance cannot be replicated

classically.

90

Chapter 6

Conclusion

In this dissertation I have demonstrated the utility of several nonclassi-

cal quantum imaging techniques—weak values and position-momentum entan-

glement. I carefully investigated their uses and limitations and found that both

have very promising practical applications.

The broad applicability of classical information theory allows for the anal-

ysis of seemingly disparate ideas under a common theoretical framework. Weak

values in the field of metrology, and entangled states in the field of data trans-

mission, are two such disparate ideas linked by their relationship to information.

When analyzed in these information terms, natural similarities emerge.

The two experiments described in chapters 2 and 3 show that interferomet-

ric weak values is a method for dramatically amplifying information transmission

rates. Chapter 2 shows this method can be used to amplify incredibly small

signals and both chapter 2 and 3 show that the subsequent information trans-

mission rate can be amplified by over 5 orders of magnitude. The applicability of

interferometric weak values is limited however to cases where the initial informa-

tion transmission rate is very low—useful information is gained only over a large

ensemble of detected photons.

Although weak values are thought of as a quantum phenomenon, the signal

amplification obtained in chapter 2 can be described classically. This is because

the quantum behavior is related to interference, which is thought of as being

91

a classical phenomenon for light. Noise calculations in optics however are an

inherently quantum process and subsequently the amplification of signal to noise

ratio described in chapter 3 cannot be described classically.

The main benefit of interferometric weak values in practical realizations

however may not be this amplification itself, but rather the way in which the

amplification is realized. Indeed, as described in chapter 3, the amplification

limit does not exceed standard amplification techniques, however it does allow

for this amplification to be realized while only making a destructive measurement

on a small subset of the photons and with fewer requirements on the measuring

detector.

On the other hand, the two experiments described in chapters 4 and 5

show that position-momentum entanglement can be utilized to create commu-

nication channels with incredibly large information transmission rates with no

amplification needed, where many bits of information are transmitted for a single

photon pair. Chapter 4 shows the negative effects of the communication envi-

ronment can be overcome in some cases by a novel experimental configuration.

Chapter 5 shows that over 7 bits of information transmitted per photon pair is

experimentally achievable. The use of a more sophisticated optical system or the

implementation of a multimode quantum dense coding apparatus [102] should

bring this number to over 10 bits per photon pair. An additional benefit of a

quantum dense coding apparatus is that the communicated message could be

controlled rather than being random as in the apparatus in chapter 5.

Although the novel configuration in chapter 4 was demonstrated using

non-classical, entangled states, it is expected that the benefits would apply to

classical states as in thermal ghost imaging experiments. In this way, chapter

4 demonstrates the utility of a classical technique on a quantum system. The

92

channel analysis in chapter 5 however is characterizing this quantum system. The

channel performance explicitly violated a separability bound, indicating that it

cannot be described classically.

As these experiments demonstrate, both interferometric weak values and

position-momentum entangled states offer unique advantages for metrology and

communication applications. I have investigated the applications and limits of

these effects in terms of quantum imaging and it is expected that continued

research will further their use in practical applications.

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