dissertation two sided printingphoton wave mechanics and experimental quantum state determination

8/9/2019 Dissertation Two Sided PrintingPHOTON WAVE MECHANICS AND EXPERIMENTAL QUANTUM STATE DETERMINATION

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PHOTON WAVE MECHANICS AND EXPERIMENTAL QUANTUM STATE

DETERMINATION

by

BRIAN JOHN SMITH

A DISSERTATION

Presented to the Department of Physicsand the Graduate School of the University of Oregon

in partial fulfillment of the requirementsfor the degree of

Doctor of Philosophy

March 2007


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Photon Wave Mechanics and Experimental Quantum State Determination, a dissertation

prepared by Brian John Smith in partial fulfillment of the requirements for the

Doctor of Philosophy degree in the Department of Physics. This dissertation has

been approved and accepted by:

Dr. Hailin Wang, Chair of the Examining Committee

Date

Committee in charge: Dr. Hailin Wang, ChairDr. Michael G. Raymer, Research AdvisorDr. Jens NockelDr. Stephen HsuDr. Andrew H. Marcus

Accepted by:

Dean of the Graduate School


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cMarch 2007

Brian John Smith


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An Abstract of the Dissertation of

Brian John Smith for the degree of Doctor of Philosophy

in the Department of Physics to be taken March 2007

Title: PHOTON WAVE MECHANICS AND EXPERIMENTAL

QUANTUM STATE DETERMINATION

Approved:Dr. Michael G. Raymer

In this dissertation, a new method of quantum state tomography (QST) for light

is presented and demonstrated. This QST approach characterizes the transverse-

spatial state of an ensemble of single photons by measuring the transverse-spatial

Wigner function of the ensemble. The first experimental measurements of the full

transverse-spatial state at the single-photon level for light are presented. To perform

these measurements, we developed a novel photon-counting, parity-inverting Sagnac

interferometer.

We also show how this method may be generalized to determine the transverse-

spatial state of an ensemble of photon pairs, which may be entangled. This allows

characterization of the continuous-variable entanglement properties that can arise


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in photon-pair states. The method introduced measures the two-photon, transverse-

spatial Wigner function, which may be used to demonstrate a Bell-inequality violation.

In treating photons as particle-like entities, as we do in the interpretation of these

experiments, the question of the most appropriate theoretical description comes to

the fore. In order to describe these experiments, we extend a quantum theory of light

called photon wave mechanics, based on a single-particle viewpoint, and we show it

to be equivalent to the standard quantum field theory of light. We show that the

wave mechanics for multi-photon states is identical to the evolution of the coherence

matrices that appear in classical, vector coherence theory. The connection between

classical coherence theory (CCT) and photon wave mechanics allows us to utilize the

well-developed tools of CCT to describe the propagation of multi-photon states. We

present two example calculations to show the utility of the photon wave mechanics

treatment.


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John Borneman Prize (to an outstanding student in the fields of physicsand mathematics), Gustavus Adolphus College, St. Peter, Minnesota,

1999 - 2000

John Chindvall Scholarship in Physics (to an outstanding physicsstudent), Gustavus Adolphus College, St. Peter, Minnesota, 1998 -1999

PUBLICATIONS:

B. J. Smith, M. G. Raymer, Two-photon wave mechanics, Phys. Rev.A, 74, 062104, (2006).

B. J. Smith and M. G. Raymer, Photon Wave Mechanics, inCLEO/QELS and PhAST, Technical Digest (CD) (Optical Society ofAmerica, 2006), paper QThD3.

B. J. Smith, B. Killett, M. G. Raymer, I. A. Walmsley, and K. Banaszek,Measurement of the transverse spatial quantum state of light at thesingle-photon level, Opt. Letters 30, 3365-3367 (2005).

M. G. Raymer, B. J. Smith, The Maxwell wave function of the photon,Proc. SPIE 5866, 293 (2005).

B. J. Smith, B. Killett, A. Nahlik, M. G. Raymer, K. Banaszek, and I. A.Walmsley, The One- and Two-Photon Transverse Wave Functions:Theory and Experiment, in CLEO/QELS and PhAST, TechnicalDigest (CD) (Optical Society of America, 2005), paper QTuA3.

B. J. Smith, M. G. Raymer, B. Killett, K. Banaszek, and I. A. Walmsley,The photon transverse wave function and its measurement, in FiO,

OSA Technical Digest Series (Optical Society of America, 2004), paperFMO1.

B. J. Smith, M. G. Raymer, B. Killett, K. Banaszek, and I. A. Walmsley,The photon transverse wave function and its measurement, inCLEO/IQEC and PhAST, Technical Digest (CD) (Optical Society ofAmerica, 2004), paper ITuM4.


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ACKNOWLEDGEMENTS

I would like to thank my advisor Professor Michael Raymer, who provided me with

numerous opportunities to learn and grow as an individual and a scientist. Thank

you for your support, guidance, leadership and teaching I will always carry these

with me. I would also like to acknowledge and thank Professor Ian Walmsley at the

University of Oxford, United Kingdom, and Professor Peter Smith at the University

of Southampton, United Kingdom for helpful discussions, and their hospitality when

I visited for collaborative research. I thank Professor Jaewoo Noh at Inha University,

Korea, for helpful hints with down conversion.

I have greatly benefited from the many discussions with, helpful hints from, and

camaraderie of my peers in the lab. To all the members of the Raymer lab during

my tenure I say, Thank you. Dr. Ethan Blansett, Dr. Andy Funk, Guoqiang Cui,

Justin Hannigan, Wenhai Ji, PengFei Nie, Chunbai Wu, Cody Leary, and Hayden

McGuinness I wish you well in all that you do.

I am fortunate to have a wonderful, caring, supportive family. To my mom, Jackie

Gerard, dads Al Gerard and Tom Smith, brother and sister, Rick Smith and Dorothy

Gerard, in-laws John, Marcy and Charlie Colvin, and grandparents, aunts, uncles,

and cousins thank you for all the love, support, and encouragement you have given

me over the years. You have always believed in me, and I am forever grateful.


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And last but not in the least, the most important person in my life, my exquisite,

loving, darling, sweet wife, Kelly. You are my cheerleader, first line of support, partner

in life, best friend, and so much more. For your patience, understanding, and all that

you do for me I thank and love you from the bottom of my heart.


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TABLE OF CONTENTS

Chapter Page

1 . I N T R O D U C T I O N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Quantum State Determination/Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Photon as a Particle (Photon Wave Function) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

K e y I s s u e s A d d r e s s e d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0

2. QUANTUM OPTICS AND PHOTON WAVE MECHANICS . . . . . . . . . . . . . . 23

I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3

Quantum Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

From Einstein to Maxwell Deriving the Single-photon Wave Function . . . . . 31

Quantization of the Single-photon Wave Function .. . . . . . . . . . . . . . . . . . . . . . . . . 43

Photon Wave Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Connections to Classical Coherence and Photo-detection Theories . . . . . . . . . . 56

Modes Versus States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Measurement-induced Photon Interactions .................................64

3. MEASURING THE TRANSVERSE SPATIAL STATE OF LIGHT ATTHE SINGLE-PHOTON LEVEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 9

Electromagnetism in the Paraxial Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 71

The Photon Wave Function in the Paraxial Approximation .................77

The Wigner Distribution Function and Its Properties. . . . . . . . . . . . . . . . . . . . . . . 79The Transverse Spatial Wigner Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .83

Parity-inverting Sagnac Interferometer .....................................87

Sagnac Interferometer Diffraction Theory ..................................95

E x p e r i m e n t a l S e t u p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 3

Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4. TWO-PHOTON TRANSVERSE SPATIAL-STATECHARACTERIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119


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Chapter Page

I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 9Two-photon Transverse Wave Function and Wigner Function . . . . . . . . . . . . . . . 121Transverse Spatial Disentanglement of a Photon Pair.......................123Spontaneous Parametric Down Conversion .................................129Experimental Down-conversion Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135Ghost Imaging and Ghost Wigner Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155Two-photon Transverse Spatial Wigner Function Measurement and BellI n e q u a l i t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6 6

5 . C O N C L U S I O N S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 7 2

Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

A. SINGLE-PHOTON WAVE FUNCTION LORENTZ TRANSFORMATIONPROPERTIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

B. SPONTANEOUS PARAMETRIC DOWN CONVERSION . . . . . . . . . . . . . . . . 197

C. GHOST IMAGING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

D. FOURIER OPTICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

B I B L I O G R A P H Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 1


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LIST OF FIGURES

Figure Page

1. Schematic set up for measurement-induced interaction .. . . . . . . . . . . . . . . . . . . . 652. Longitudinal and transverse wave functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723. Converging and diverging Gaussian beams and associated transverse

s p a t i a l W i g n e r f u n c t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 64. Parity-inverting Sagnac interferometer with a Dove prism. . . . . . . . . . . . . . . . . . 925. Parity-inverting Sagnac interferometer with the top mirror. . . . . . . . . . . . . . . . . 94

6. Rotation of the transverse spatial state caused by the top-mirror. . . . . . . . . . . 957. Parity-inverting Sagnac interferometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .978. Linear-optical evolution of the transmitted and reflected fields that pass

through the parity-inverting Sagnac interferometer. ........................999. Experimental setup to measure the transverse spatial Wigner function. . . . . . 10410. Steering mirror motion control setup.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10511. Detection electronics for single-photon Wigner measurements. . . . . . . . . . . . . . . 10612. Raster scan of phase space for a one-dimensional field. . . . . . . . . . . . . . . . . . . . . . 10813. Experimental and theoretical plots of the transverse spatial Wigner

function of a diverging Gaussian beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .110

14. Intensity and field amplitude of the HG10 mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11215. Experimental and theoretical plots of the transverse spatial Wigner

function of a HG10 mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11216. Top-hat field amplitude and experimental arrangement for its

c o n s t r u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 417. Experimental and theoretical plots of the transverse spatial Wigner

function for a propagated top-hat field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11418. Displaced double-top-hat field amplitude and experimental arrangement

for its construction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11719. Experimental and theoretical transverse spatial Wigner functions for the

d i s p l a c e d t o p - h a t fi e l d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 720. Paraxial two-photon source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12221. Entangled photon pair transmission through a turbulent atmosphere. . . . . . . 12522. Concurrence and transmission fidelity of an OAM-entangled photon pair

after propagation through a turbulent atmosphere.. . . . . . . . . . . . . . . . . . . . . . . . . 12723. Measuring the OAM density matrix using Laguerre-Gauss holograms. . . . . . . 12824. Parametric amplification process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13025. Spontaneous parametric down-conversion process. . . . . . . . . . . . . . . . . . . . . . . . . . 13226. Four possible spontaneous parametric down-conversion configurations.. . . . . . 133


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CHAPTER 1

INTRODUCTION

Light

Light has played an instrumental role in many developments of our understanding

of nature, and continues to be at the forefront of modern-day, fundamental physical

research. From the time of Euclid (c. 300 B.C.E.), who thought that light traveled

in straight lines, but originated from the eye and strikes the objects seen by the

observer, light has been a continuing theme in scientific discovery. The microscope,

thought to be first developed by the Dutch lens maker, Zacharias Janssen (c. 1590),

opened the way for many discoveries in the medical and biological sciences. The

telescopes of Galileo Galilei (1609) led to the discovery of moons circling Jupiter,

which reinforced the heliocentric model of the universe, leading to his persecution by

the church. Fermats principle of least time, (1657) in which light travels from one

point to another along the path taking the least transit time, may be taken as a pre-

cursor to the principle of least action, which is at the foundation of Hamiltonian

and Lagrangian dynamics, as well as modern quantum field theory. Isaac Newton

(1672) and Christian Huygens (1678) put forth the competing corpuscle and wave

theories of light respectively. In 1801, Thomas Young presented his famous double-


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slit experiment results, and laid to rest (for a time) the particle-wave debate of light,

solidifying Huygens wave theory.

One of the pinnacles of 19th century science, Maxwells theory of electromagnetism

(1864) was the first unified field theory, uniting electricity, magnetism and optical

phenomena in a single theory. Unbeknownst to him, Maxwell had also discovered

the first relativistic quantum theory of photons (light corpuscles). In order for the

equations that now bear his name to take the same form in any non-accelerating

(inertial) frame of reference, the Galilean transformations assumed to transform

position and time coordinates from one inertial frame to another had to be modified.

These required modifications of Galilean relativity led Einstein (1905) to formulate

the theory of special relativity.

Quantum mechanics (QM), one of the most successful physical theories we have

produced, was developed at the beginning of the 20th century, and brought the

particle-wave debate back to center stage. The origins of quantum theory can be

traced to Max Plancks theory of the emission of light by heated material bodies

(1900). In order to correctly predict the observed spectrum, he needed to assume

that light was only emitted with certain discrete energies. Building on the idea

that light was emitted and absorbed with discrete energies, Einstein carried the idea

further, introducing the light quanta, to describe the photoelectric effect, work for

which he later won his only Nobel Prize. Arthur Comptons experiments on the

scattering of X-rays by free electrons followed the same law as the collision between


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two elastic spheres. Thus the novel idea of light quantization was established both

experimentally and theoretically in the early 1900s, and the wave-particle debate was

upon us once again.

The advent of the laser in 1960, which allowed the creation of light in highly

coherent states, ushered in the field of quantum optics and quantum coherence theory,

and brought with it many technological advances in society from fiber optics

communications to compact-disc data storage. It enabled many experiments to be

carried out such as the ultra-fast probing of molecular bonds with femtosecond laser

pulses and opened the door to completely new fields of research, such as nonlinear

optics. Additionally, the laser enabled the first conclusive experiments testing the

Bell-inequality [1], which highlights the strange (non-local, non-classical) behavior of

correlated (entangled) quantum systems spatially separated from one another, to be

performed using polarization-entangled photons from an atomic-ion cascade emission

[2, 3].

Even today, light is at the fore of modern research. The fast-growing field of

quantum information has relied on many of its proof-of-principle experiments to

be carried out with entangled photons produced with spontaneous parametric down

conversion. In addition, the now well-established field of quantum state determination,

also called quantum state tomography, which was first experimentally carried out in

1993 to characterize the quantum state of light [4], is central to the emerging fields

of experimental quantum information and quantum technologies. Moreover, light is


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typically the main tool used to investigate or change properties of matter. Therefore,

by gaining as much information as possible about light, we can better understand the

behavior and properties of materials.

In this dissertation, we focus on the characterization of single-photon and two-

photon quantum states of light, which play an important role in quantum information

science, quantum metrology, and communications. In the process, we find that it is

useful to discuss the state of the photon, or photons, in terms of coordinate-space

wave functions which obey certain wave equations. This formalism we call photon

wave mechanics. However, before proceeding we first present some background on

the subjects that we will discuss.

Quantum State Determination/Tomography

Physics is the study of nature at its most fundamental level. Owing to the fact

that mathematics seems to be the language in which physical concepts are the easiest

to express, physics is often mathematically very detailed. However, one should not

think that simply because a mathematical concept is beautiful, or intriguing,

that it necessarily corresponds to a physical theory. This is a common mistake made

by non-physicists associating mathematics with physics. Physics is a scientific

study of nature and, as such, is deeply rooted in its observational and experimental

details. Without experimental data to support and drive physical theories, they

become nothing more than mathematical objects, not reflecting any actual physical


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system. Today there seems to be a lag in the amount and development of experimental

data and discoveries that push our envelope of understanding. Experiments at the

edges of scientific knowledge are becoming increasingly difficult to perform (all of the

easy and obvious experiments have already been done). This lack of experimental data

leads to theoreticians developing pseudo-theories of everything with little real-world

support to back them up. The difficulty of experiments also drives the experimentalist

to devise more creative ways in which to probe the fundamental workings of nature.

The state of a physical system represents our knowledge of the system and provides

information about it in the past and future. In standard quantum mechanics the state

of a system is represented mathematically by a wave function in either the momentum-

space representation (p, t), or the coordinate-space representation (x, t) (or a

statistical ensemble of wave functions when the state is mixed). The representations

are typically related by a three-dimensional Fourier-transform relationship. The

modulus squared of the momentum-space (coordinate-space) wave function is equal

to the probability per unit momentum-space (coordinate-space) volume to find the

system with a particular value of momentum (position). Quantum state determination,

or quantum state tomography, refers to a method by which one may experimentally

gain all possible information about the state of a system. This enables one to

make the best possible prediction (in a probabilistic sense) about the results of

any measurement or experiment that may be performed on the system. In classical

mechanics, the state of a system is represented by a set of numbers that label the


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Risken [11], and the first quantum state determination experiments carried out by

Smithey, et. al. at the University of Oregon in 1993, in which the quantum state of

a single mode of the electromagnetic field was measured [4]. In these experiments,

the amplitude and phase structure of a single electromagnetic field mode, specified

by the polarization, and spatio-temporal mode, were determined by a procedure

called optical homodyne tomography (OHT). The technique of OHT was used to

tomographically reconstruct the Wigner distribution function of the quantum state

from several measured probability distributions. The Wigner function, introduced

in 1932 by Wigner to simplify quantum statistical mechanics problems [12], is a

quasi-probability phase-space distribution that is directly related to the wave function

(density matrix) for a pure- (mixed-) state quantum system. For pure (mixed) states,

this direct relationship between the Wigner function and the wave function (density

matrix) allows for us to obtain one from the other. Thus complete knowledge of the

Wigner function implies complete knowledge of the state of the system [13]. Since

these pioneering experiments were performed over a dozen years ago, the field of

quantum state determination has become widespread, and the various techniques

have become a standard tool in laboratories around the world [14].

A closely related subject, optical phase retrieval aims to characterize the amplitude

and phase structure of a fully coherent, time-stationary, quasi-monochromatic electro-

magnetic wave field as a function of position, E(x), assuming the polarization and

frequency are known. Here E(x) refers to the complex field amplitude of a scalar


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electro-magnetic wave, as a function of the spatial coordinate x = (x,y,z). In the case

of partially-coherent light, the field may be characterized by the Wigner distribution

function [15]

W(x, k) = 1

3

d3x

E(x x) E(x + x) ei2kx, (1.1)

where k = (kx, ky, kz) is the wave vector, and brackets , imply an ensemble average

over all statistical realizations of the field. The task of spatially mapping out the field

amplitude is quite easy, and can be accomplished by using photographic paper, or

charge-coupled-device (CCD) cameras for example. However the ability to determine

the phase as a function of spatial position for an optical field is quite challenging.

The ability to characterize the amplitude and phase information of an optical

field, or amplitude and coherence information in the case of partially-coherent light,

is critical to several areas of study [16]. Optical coherence tomography, which has

found many applications in the biological sciences, relies on the known coherence of

the incident radiation to determine the structure of objects from which it scatters.

There are several other areas of practical importance that the coherence of an optical

field plays a critical role, such as the testing of optical equipment, the study of fluid

dynamics, and photolithography, to name a few. Indeed, one can go as far as to say

that coherence properties of light are the most important aspect, due to their role in

determining interference and other optical correlation effects. Various light sources,

both man-made and naturally occurring, have varying degrees of coherence, and thus

a study of such sources requires the ability to characterize coherence. This makes


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techniques to measure the coherence of optical fields not only of practical use, but

also of great interest in studying the fundamental nature of light.

There have been several proposed techniques to measure the coherence, (we use the

term coherence for spatial coherence), of optical fields [1719]. The first experimental

methods suffered from an inability to measure fields of arbitrary coherence. Then

in 1995, at the University of Oregon, McAlister et. al. introduced a tomographic

method to measure fields with any state of coherence [20]. This method characterized

the transverse field, assuming the paraxial approximation, and required tomographic

reconstruction of the Wigner function, which led to the possibility of errors in the

inverse transform. After McAlisters development, there were several other transverse-

coherence measurement methods introduced [2123]. However, none of the proposed

techniques worked at the level of single-photon fields.

In this dissertation, we present and demonstrate the first experimental technique

to fully characterize the transverse spatial state of light (i.e. transverse spatial

coherence) at the single-photon level. The method measures the transverse spatial

Wigner function W(r, k), for a single photon, in the two-dimensional plane (x, y)

located atz= 0, of a wave field with arbitrary state of coherence. Herer = (x, y), and

k= (kx, ky) are the transverse position and wave vectors in the plane perpendicular

to the propagation axis (z). Our method utilizes a parity-inverting Sagnac (common-

path) interferometer to scan the phase space (r, k). This is done by taking advantage

of the fact that the Wigner function is proportional to the expectation value of the


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at the same time, one observes no coincidences between the detectors. This effect is

known as photon bunching, and stems from the boson nature of light. Indeed, if

the same experiment were performed with electrons, one would observe electron anti-

bunching, reflecting the fermion nature of electrons. Now, as one of the photons

arrival time to the beam splitter is varied with respect to the others, coincidence

counts between the two detectors begin to emerge. When the arrival time difference

is greater than the coherence time of the photons, the coincidence counts reach the

classical level when 50 percent of the time one will observe coincidences. Thus, a plot

of the probability of coincidence as a function of arrival time delay shows a drop, or

dip, at zero time delay, known as the Hong-Ou-Mandel dip.

The HOM interference experiment probes the temporal correlations of photon

states by assuming that the photon spatial states are identical. Non-identical photon

states lead to degradation of the visibility in the HOM dip. To address the spatial

state of the photon pair, which generally includes any entanglement between them,

one must devise a new characterization method. The ability to characterize the spatial

state of two-photon states would not only be of interest at a fundamental level, but

it would also find use to characterize photon sources used for quantum information,

metrology and communications schemes.

Another often cited set of experiments that highlight the non-classical nature of

spatially-entangled two-photon states are the quantum-imaging experiments of the

mid 1990s [2629]. These quantum-imaging experiments utilize entangled photon


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pairs derived from a spontaneous parametric down-conversion (SPDC) source. The

SPDC source is of particular interest and use in quantum-optical implementations of

quantum-information schemes, and tests of fundamental quantum theory [3032]. In

the process of SPDC, an intense laser beam, called the pump, is incident on a nonlinear

optical crystal. Through the nonlinear, optical interaction, a pump photon has a

small, but non-zero probability to split into a pair of daughter photons, traditionally

called the signal and idler photons. Energy and momentum are typically conserved

in the interaction, resulting in correlation, or entanglement, between the daughter

photons.

In the quantum-imaging experiments, the entangled photons from the SPDC

source are directed along two different, spatially-separated paths. In one path, say

the idler-photon path, an aperture (amplitude mask) followed by a large-area photon-

counting detector, is placed. In the other path, the signal-photon path, a lens, followed

by a small, point-like photon-counting detector, is placed. As the point-like detector is

scanned in the plane perpendicular to the beam axis, the coincidence rate between the

two detectors is recorded. The coincidence rate as a function of the signal detectors

position in the transverse plane maps out the aperture in the idler-photon path.

Each individual detector count rate remains relatively constant, but the coincidence

rate reflects the aperture transmission function. The distance between the SPDC

source and aperture, and the SPDC source, imaging lens, and point-like detector

are determined by a thin-lens-like equation [26]. These experiments depend only


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on amplitude correlations and not phase correlations, and thus do not give complete

information about the two-photon state. It has been shown that these experiments do

not require entangled photons to observe the coincidence image, classically-correlated

photons suffice. However, there are some benefits, such as increased spatial resolution

and visibility, when using the entangled-photon source.

In this dissertation, we present an experimental technique to fully characterize

the two-photon transverse spatial state of light. The method measures the two-

photon transverse spatial Wigner function W(r1, k1; r2, k2), a generalization of

the single-photon transverse spatial Wigner function, in a pair of planes perpendicular

to the propagation axis. This allows characterization of not only amplitude correlations,

but also phase correlations between photon pairs.

Our approach is based on the single-photon Wigner function method described

above. In the two-photon case, two parity-inverting Sagnac interferometers are used

to measure the two-photon Wigner function of spatially separated photons traveling

in different directions, such as the entangled photons encountered in a SPDC source.

The signals of the detectors placed at the outputs of the interferometers are sent to

a coincidence counter, whose count rate is proportional to the two-photon transverse

spatial Wigner function. This method enables one to violate a Bell inequality based

on the Wigner function [33].

The question of interpretation arises as to what is measured for single-photon and

two-photon states in these experiments. We advocate that it is not the electromagnetic


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field that is characterized; we go so far as to claim the electromagnetic field does not

exist for a single photon. Rather we explain our results in terms of Wigner functions

derived from single-photon wave functions, and that the electromagnetic field is an

emergent quantity when considering many photons.

Photon as a Particle (Photon Wave Function)

The concept of the light quantum was first introduced by Einstein in 1905 [34]

to describe the then-recently-observed photo-electric effect [35]. In his paper [34],

Einstein writes, According to the assumption considered here, when a light ray

starting from a point is propagated, the energy is not continuously distributed over

an ever increasing volume, but it consists of a finite number of energy quanta, localized

in space, which move without being divided and which can be absorbed or emitted

only as a whole. This statement captures the essence of the view of the photon

as a quantum particle that many physicists hold in one form or another. More

evidence of the particle-like nature of the light quantum was provided by the results of

Arthur Comptons electron-X-ray scattering experiments [36]. In these experiments,

an electron and X-ray scatter from one another. The resulting change in energy and

momentum of the two objects can be easily described from a billiard-like collision, in

which a point-like electron scatters from a similarly point-like light quantum.

In spite of these early developments, an acceptable quantum theory of electro-

magnetism based upon the standard particle-wave-function viewpoint did not develop.


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The first satisfactory quantum treatment of electromagnetism was not given until 1927

by Dirac [37], and later clarified by Fermi [38]. Diracs quantum treatment of light

was not given in terms of wave mechanics, in which particles are the fundamental

quantum objects described by wave functions. Rather, he presented a quantized field

theory (QFT), where the field is the fundamental physical entity. The term photon,

which we typically use instead of light quantum, was not used until 1926 by Gilbert

N. Lewis to describe the interaction of neutral-atom valence bonds [39], and not to

describe the light quanta of Planck and Einstein. Indeed, the term photon has been

met with opposition for its catch-all nature [40]. However, photon is a very convenient

word to describe what we now mean as a fundamental excitation of the quantized

electromagnetic field.

There is good reason that the particle view of the photon did not lead to a

quantum theory of light, as opposed to the quantum theory of the electron, where

particles abound. Much of the difficulty in developing a quantum theory of the

photon as a particle stems from its inherent relativistic nature, due to its zero rest

mass. The absence of rest mass, along with its internal, spin-1 degree of freedom,

led Newton and Wigner to the conclusion that the photon is, strictly speaking, non-

localizable [41, 42]. To what extent the photon may be localized has been carefully

examined [4348]. Bialynicki-Birula has found that photons can be localized in space

with an exponential falloff in the energy spatial density and photo-detection rate

[48]. Nevertheless, faster than exponential falloff cannot be achieved, as far as is


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known. The strict non-localizability of the photon implies that there is no position

operator, and thus no position eigenstates for the photon. This leads us to the

conclusion that there is, in the usual sense, no probability density for the position

of the photon, and thus a coordinate-space (position-representation) wave function

cannot be consistently introduced. This has led to the course grained, photo-detection

model of the photon probability amplitude [4953], that has been used to explain

the majority of experimental results. Non-canonical position operators have been

introduced [5457] to try avoiding these difficulties, but still leave something to be

desired.

Nevertheless, several candidates have been proposed for the photon wave function

in coordinate-space [52, 53, 5863]. The first attempt to introduce a coordinate-

space photon wave function, viewed as a description of a single particle, was given by

Landau and Peierls [58]. However, it was quickly noted that the Landau-Peierls (LP)

wave function is a highly non-local object [9]. This function was also independently

rediscovered in the 1980s [6467].

By extending what one means by wave function, to a complex vector-function

of space and time coordinates (x, t), that adequately describes the quantum state

of a single photon, it is possible to define a wave function for the photon. The

mathematical object that is now accepted by many as the photon wave function is

closely related to the Riemann-Silberstein (RS) vector

F (x, t) =D (x, t)

20+ i

B (x, t)20

, (1.2)


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where D (B) is a field analogous to the electric- (magnetic-) displacement field, and

0(0) is the permittivity (permeability) of the vacuum [68]. The RS vector obeys

the complex form of the Maxwell equations, which in free space may be written as

itF (x, t) =c F (x, t) ,

F (x, t) = 0.(1.3)

The use of the RS vector as the photon wave function has been advocated by many

over the past 75 years [59, 60, 6885]. There are several reasons for choosing the

RS vector, and other equivalent formulations, as the single-photon wave function in

coordinate space. For example, one can arrive at the RS vector from a particle view,

by starting with Einstein kinematics, and derive the photon wave function in much

the same way that Dirac did for the electron (see chapter II).

The subtlety of using the RS vector as the photon wave function lies in the fact that

|F (x, t)|2 is not the position probability density of standard non-relativistic quantum

mechanics, but rather is the local spatial energy density. Thus, the standard scalar

product between two RS vectors F and F

F | F =

F (x, t) F (x, t) d3x, (1.4)

cannot be interpreted as the probability amplitude for finding a photon in state F,

when it is known to be in state F. If one tried to push this interpretation, there

are several problems, the least of which is the fact that the integral in Eq. (1.4)

is not Lorentz invariant [68, 73]. Lorentz invariance is expected since a probability

amplitude is a number. The proper, Lorentz-invariant scalar product is found to take


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It is often useful, and sometimes easier to use the wave mechanics approach to

solving a problem than to use the full QFT. For example, one typically does not use

QFT to treat the helium atom, but rather one uses Schrodinger wave mechanics. The

ability to attack a problem from different approaches allows for deeper insight into the

issues at hand, and can lead to a better understanding of the problem. The quantum-

field theoretic approach gives the correct answer, however, it is useful to be able to

solve problems using several different methods. Indeed this opinion was emphasized

in Feynmans 1965 Nobel Lecture [86], in which he notes, I, therefore, think that a

good theoretical physicist today might find it useful to have a wide range of physical

viewpoints and mathematical expressions of the same theory (for example, of quantum

electrodynamics) available to him. Thus the photon-wave-mechanics approach can

lead to a more intuitive understanding of experiments, and give different insights into

the physics occurring. It is also satisfying to note that photons can be treated in the

same way as electrons, at the level of a quantum particle.

In this dissertation, we show, for the first time, how to treat multi-photon states

in a consistent photon-wave-mechanics approach. We develop the wave mechanics for

multi-photon states by expanding on the single-photon wave-mechanics formalism.

We define the multi-photon wave functions, and determine their equations of motion.

In treating mixed states, we point out how to obtain the reduced density matrix from

a given multi-photon state. In the process, we find useful connections between photon

wave mechanics (PWM) and several other well-known theories, such as classical


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coherence theory. As an example calculation using PWM, we show how to treat

the problem of multi-photon interference on a beam splitter.

We find that classical vector coherence theory [87] is closely related to photon

wave mechanics. We show that the two-photon wave function and its equation of

motion are equivalent in form to the second-order, classical coherence matrices [87].

This implies that the evolution of a two-photon state can be described using the well-

developed tools of classical coherence theory. As a demonstration of the utility of this

close relationship, we show how a pair of photons, entangled in their orbital-angular-

momentum degrees of freedom, disentangle as they propagate through a turbulent

atmosphere.

Key Issues Addressed

We begin the dissertation with the theoretical descriptions of light used to describe

our experiments. A review of standard quantum optics is presented to show the

connection between QFT and PWM. The monochromatic mode expansion of the

electric field operator, which is widely known, and the non-monochromatic mode

expansion [88], which is not as well known, are discussed. We present a derivation

of the single-photon wave function in coordinate space [59, 60]. When the canonical-

quantization procedure is carried out on the single-photon wave-function theory,

the non-orthogonality of the non-monochromatic wave-packet modes of Titulaer and

Glauber [88] naturally arise. We note that the scalar product for the wave function


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in the coordinate representation, which is a non-local integral Eq. (1.5), is tied to the

non-orthogonality of the non-monochromatic wave packet modes of quantum optics.

We extend the single-photon theory to multi-photon states by constructing multi-

photon wave functions and determining their proper wave equations. It is shown

that the PWM for two photons is closely related to second-order vector, classical

coherence theory (CCT) [87], and the two-photon detection amplitude of quantum

optics [52, 87], which is also called the biphoton amplitude [89]. These connections

are generalized ton-photon wave mechanics,n-th order CCT, andn-photon detection

amplitudes, highlighting the connections between the theories. We perform a calculation

of measurement-induced photon interaction as an example of the utility of the PWM

approach and how it can be implemented.

After developing the theory of photon wave mechanics, we turn to characterization

of a single-photon state. In particular, we focus on the transverse spatial state of the

photon in the paraxial approximation. Here the single-photon wave function may be

represented by the transverse spatial wave function, which we introduce in analogy

to the paraxial treatment of classical radiation. From this transverse spatial wave

function we construct the transverse spatial Wigner function for such a state.

We present an experimental technique to directly measure the transverse spatial

Wigner function of an ensemble of single photons. This is done through use of a

novel, parity-inverting Sagnac interferometer. The first complete measurements of


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the transverse spatial state of light at the single-photon level are given for a series of

different field distributions in order to demonstrate the technique.

We then move from characterization of single-photon states, to characterization

of two-photon states. Again, we examine the transverse spatial state of the photons.

The multi-photon generalizations of the single-photon transverse spatial wave and

Wigner functions are given. In particular, we discuss the two-photon transverse

spatial wave function and Wigner function. We show how one can utilize the close

relationship between PWM and CCT to calculate the disentanglement of a pair of

spatially-entangled photons traversing a realistic turbulent atmosphere.

An experimental technique to measure the two-photon transverse spatial Wigner

function is presented. We show how one can use this method to violate a Bell

inequality. Our experimental progress towards realization of this experiment is given

by presenting HOM interference and quantum-imaging results from our SPDC source.

The idea of quantum-imaging can be generalized to a non-local Wigner function,

which we introduce.

The dissertation concludes with comments on current and future work. There are

several appendices that cover the Lorentz-transformation properties of the photon

wave function, the Lorentz-invariance of the photon-wave-function scalar product,

the theory of spontaneous parametric down conversion, and quantum-imaging.


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CHAPTER 2

QUANTUM OPTICS AND PHOTON WAVE MECHANICS

Introduction

Quantum field theory (QFT) has its origin in Diracs exposition on the quantization

of the electromagnetic field [37]. In this treatment, Dirac noted that the Hamiltonian

of the electromagnetic field may be expressed as a sum of harmonic-oscillator modes

with different resonant frequencies. He then proceeded to quantize the electro-

magnetic field by imposing the now-famous commutation relations on the canonically-

conjugate variables (field amplitudes, or field quadratures) that arise from the Poisson

brackets in the classical discussion. The fields were raised to the status of operators,

and expanded in terms of creation and annihilation operators of the fictitious harmonic

oscillator. Photons were interpreted as excitations of the field that arose from the

application of the creation operator acting on the vacuum state of the electromagnetic

field. This is the basis of QFT and quantum optics (QO), in which the details have

since been developed and fleshed out. Yet there is still debate as to the nature of

the photon. There is a discrete click in a photo-detector, which signals the arrival

of a photon. The recent development of controlled single-, pair-, and few-photon


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states, which may be manipulated, and measured, begs the question, Is a photon

just a monochromatic field excitation, as in the canonical Dirac theory?

To contrast this development, consider the story of the electron. It was first

discovered experimentally in cathode-ray-tube experiments, and viewed as a click

in an electron detector. The model of atomic physics, in which negatively-charged

electrons orbit a positively-charged nucleus, relies on a particle view of the electron,

and is well seated in its predictive power. Indeed, the Dirac equation of the electron,

which was the first quasi-successful attempt to unite special relativity and the quantum

theory of electrons, resulted in a relativistic wave equation for the electron (viewed

as a particle), and predicted the existence of a new particle, the positron, or anti-

electron. The electron was still treated as a particle in this theory. It is only when one

quantizes the Dirac wave function of the electron, by elevating it to the status of

an operator, that the true QFT of light and matter arises. In this theory, localized

electrons can be described in terms of non-relativistic wave-packet modes.

In this chapter, we begin with a review standard quantum optics the Dirac,

monochromatic theory of electromagnetism in free space [37], and its wave-packet

(non-monochromatic) counterpart, developed by Titulaer and Glauber in 1967 [88].

This is followed by a derivation and review of the single-photon wave function in

coordinate space, which has developed over the past decade [59, 60, 68, 73]. We

then show that both the monochromatic and non-monochromatic quantized theories

can be derived directly from quantization of the single-photon wave function. In


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doing so, we also show that the scalar product for the single-photon wave function,

which is a non-local integral in coordinate space [68, 73], gives the appropriate

overlap of the Titulaer-Glauber (TG) wave-packet states. This is closely connected to

the well-known non-orthogonality of the TG wave-packet modes under the standard

scalar product. The single-photon wave function theory is then extended to multi-

photons states, culminating in a complete wave mechanics theory of photons. Close

connections between classical coherence theory, photo-detection theory, and photon

wave mechanics are given explicitly. We discuss how to treat entanglement in the state

of two photons, and how to correctly reduce a two-photon state to a single-photon

density matrix. The distinction between the modes of a quantum field and states

of a particle are then discussed. We end the chapter with an example calculation of

multi-photon interference.

Quantum Optics

The quantized electromagnetic theory first developed by Dirac starts from the

Maxwell theory of classical electromagnetism. The positive-frequency part of the

classical electric and magnetic-induction fields may be expanded in monochromatic

modes in free space as [25, 52, 87, 90]

E(+) (x, t) =i

d3k

(2)3

k20

k,uk,(x) eikt, (2.1)

B(+) (x, t) =i

d3k

(2)3

k20

k,k

c |k| uk,(x) eikt, (2.2)


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where is the Planck constant divided by 2. Here 0 is the permittivity of the

vacuum, the sum is taken over two orthogonal polarization indices , and k =c |k|is the frequency associated with a given wave vector k. Here E(+) (x, t) is implicitly

assumed to be the transverse part of the electric field. The monochromatic, plane-

wave modes are

uk,(x) = ek,eikx, (2.3)

where the ek, are unit polarization vectors. The Hamiltonian for the transverse-

electromagnetic field expressed in terms of the normal-mode expansion coefficients

k,, is

H=

d3k

(2)3k

2

k,k,+ k,

k,

. (2.4)

One may express these quantities in terms of the real-valued, canonically-conjugate

variablesqk, and pk,, known as the quadrature amplitudes, given by

qk, =

k2

k,+

k,

, pk, =i

1

2

k, k,

, (2.5)

which arise in the classical Hamiltonian and Lagrangian treatments of electrodynamics.

With the help of the inverse relations,

k, =

k2

qk,+ i

pk,k

, k, =

k2

qk, ipk,

k

, (2.6)

this leads to a Hamiltonian of the form

H=1

2

d3k

(2)3

2kq2k,+ p

2k, ik {qk,, pk,}

, (2.7)


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where {qk,, pk,} = 0 is the Poisson bracket of the two canonically-conjugate variables.

This is formally equivalent to a sum of Hamiltonians for classical harmonic oscillators

with resonant frequencies k. In the canonical-quantization scheme of fields, the

fields become operators, in which canonically-conjugate amplitudes, now operators,

obey the following commutation relations (note that for Fermion fields, the conjugate

operators obey anti-commutation relations)

[qk,,pk,] =i,(2)2 (3) (k k), (2.8)

where the caret notation emphasizes that these are now operators. Here , is a

Kronecker delta, which is non-zero only when the subscripts are equal, and (3) (k k)

is a three-dimensional Dirac delta function. Upon quantization, the normal-mode

amplitudes k, and k, become annihilation and creation operators ak, and a

k,,

respectively [25, 52, 87, 90]. These operators obey the inherited commutation relations

ak,, a

k,

= ,(2)

2 (3) (k k) . (2.9)

The state space of the free electromagnetic fieldHR, (R implies radiation field)

on which the field operators act, consists of a tensor product of the state spaces of

an infinite number of harmonic oscillator states,

HR=

j=1Hj , (2.10)

where Hj,j = (k, ), is the harmonic-oscillator state space associated with the wave-

vector and polarization pair j = (k, ). One possible orthonormal basis ofHj is


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This may be written in terms of the number operators nk, = ak,ak, as

H=

d

3

k(2)3

k

nk,+ 12

. (2.16)

This is the standard approach taken in the quantum-mechanical treatment of the

electromagnetic field. The interaction of the quantized electromagnetic field with

atomic systems is typically introduced through an interaction term (usually the dipole

interaction in non-relativistic treatments) in the full atom-field Hamiltonian. We will

not go into the details of this, as it has been treated elsewhere [25, 52, 87, 90], and it

is not imperative to the current discussion.

The free-space field operators in Eqs. (2.13) and (2.14) are expanded in terms of

infinite plane waves, which work fine for simple models. However, when interactions

with localized objects are considered, such as atoms or molecules, the plane-wave

description of the electromagnetic field operators becomes insufficient. In such a

case, one may expand the electric and magnetic-induction field operators for a given

polarization =1, in terms of non-orthogonal, polychromatic, spatio-temporal

wave-packet modes vl,(x, t) [88]. The non-orthogonal, polychromatic modes are

related to the monochromatic, orthonormal, plane-wave modes uk,(x), through the

unitary transformationUl,(k) by

vl,(x, t) =i

2

d3k

(2)3

kUl(k) uk,(x). (2.17)

This may be inverted using the unitary relation

l

Ul(k) Ul (k

) = (2)3 (3) (k k) , (2.18)


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to give

uk,(x) = il

2k e

ikt

U

l (k) vl,(x, t). (2.19)

The annihilation and creation operators ak,and ak, are also changed by this unitary

transformation, leading to new annihilation and creation operators bl, and bl, given

by

bl, =

d3k

(2)3Ul (k) ak,, (2.20)

which obey boson commutation relations

bl,, b

m,

= l,m,. (2.21)

Assuming circular polarization, as we do throughout this chapter, the positive-

frequency parts of the electric and magnetic-induction field operators may then be

expressed in terms of the non-monochromatic modes for each polarization , as

E(+) (x, t) = 1

0

l

vl,(x, t) bl,, (2.22)

and

B(+) (x, t) =ic

0

l

vl,(x, t) bl,. (2.23)

Here we have made use of the following relationship between the unit polarization

vectors (for circular polarization) to simplify the expression for the magnetic-induction

field [68, 73]

k

|k| ek, = iek,. (2.24)

The full positive-frequency parts of the electric and magnetic-induction field operators

are given by adding together the two polarization parts.


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One particular advantage to using the plane-wave expansion, or any other complete,

orthonormal expansion of monochromatic mode functions, denoted generically by

uk,(x), is that they are orthonormal under the standard definition of the scalar

product given by

uk,| uk, =

uk,(x) uk,(x) d3x= (2)3(3) (k k) ,, (2.25)

where the last equality holds only for the monochromatic modes. This is not the case

for the non-monochromatic modes, in which the non-orthogonality under the scalar

product in Eq. (2.25) can be seen to arise from different weightings given to different

frequency components due to the

k factor in Eq. (2.17). The wave-packet modes

vl,(x, t), are not orthogonal under a scalar product of the form in Eq. (2.25). This

is one major disadvantage to using such an expansion. However, as we will show, the

weighting of the different monochromatic modes by the

k factor may be canceled

out by defining a new scalar product for the wave-packet modes, which leads to a

different interpretation of these mode functions.

From Einstein to Maxwell Deriving the Single-photon Wave Function

In the previous section we reviewed the standard Dirac quantum theory of electro-

magnetism in vacuum [37]. The classical Maxwell fields were raised to the status

of operators acting on a Hilbert space, or state space, of the electromagnetic field.

These operators obey the Maxwell equations. This is in contrast to the historical


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and contemporary quantum-mechanical treatments of the electron, in which case

the single-particle wave function is first introduced to describe the evolution of one

electron (typically in the context of non-relativistic Schrodinger evolution). Then the

relativistic Dirac equation of the electron field is introduced and the electron field is

quantized.

Here we aim to show that this particle-like approach to the electron may also be

applied to the case of the photon. We begin by following closely Diracs approach

to finding the equation of motion for the single electron, a spin-1/2 particle, from

Einstein kinematics. After arriving at the equations of motion for a single photon,

taken as a particle-like object with zero rest mass and spin-1, we discuss the scalar

product and normalization of the wave function, showing that the scalar product

must be a non-local integral in coordinate space.

We begin by reviewing the approach taken by Dirac to arrive at the relativistic

equation of motion for the electron, now called the Dirac equation, which led to

the prediction of its anti-particle, the positron. Starting from the Einstein energy-

momentum-mass relationship

E2 =c2 |p|2 + mc22 , (2.26)

one can easily arrive at a relativistic theory for scalar fields, the well-known Klein-

Gordon equation. This is done by multiplying both sides of Eq. (2.26) by the

scalar wave function (x, t), and replacing the energy and momentum with their


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corresponding quantum operators

E it, (2.27)

p i, (2.28)

which leads to

2t (x, t) =c22 (x, t) +

mc2

2 (x, t) . (2.29)

Note that the Schrodinger equation may be derived from Eq. (2.26) by taking

the positive square root of both sides, and expanding the right-hand side (RHS) for

non-relativistic particles, i.e., c |p| mc2, to give

E=

c2 |p|2 + (mc2)2 mc2 +c

2 |p|22mc2

. (2.30)

Dropping the constant rest energy of the particle mc2, making the canonical operator

substitutions (2.27) and (2.28), and multiplying by the wave function (x, t), we

arrive at the free-space Schrodinger equation

it (x, t) = 222m

(x, t) . (2.31)

There are difficulties that arise when one tries to treat (x, t) in the Klein-Gordon

equation, Eq. (2.29), as a wave function for a particle [9193]. The modulus squared

of the function (x, t) is not positive definite, and therefore cannot be interpreted

as a probability. This negative probability arises from the square energy term

in Eq. (2.26), or the second derivative in time in Eq. (2.29), which do not enter

into the Schrodinger equation. To remedy this negative-probability issue, Dirac tried


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form

i (t+ c ) 0m

c2

I44 (x, t) = 0. (2.34)

The Dirac matrices are also related to the generators of rotation for spin-1/2 particles,

usually taken to be the Pauli matrices. Equation (2.34) is one form of the Dirac

equation, which may be recast into a more explicit Lorentz-covariant form, by changing

the representation of the Dirac matrices so that

(i m) (x, t) = 0. (2.35)

Here we have used the God given units [92] in which = 1, c = 1, and the 4 4

identity is implicitly assumed present with the mass term. From here one typically

identifies two components of the four-component wave function with the two-spinor

wave function for the electron, and the other two components are identified with the

two-spinor wave function of the positron. The wave function is then quantized in the

usual way [92, 93]. The term spinor, short for spin-tensor, arises from the treatment

of general internal degrees of freedom (the number of internal degrees of freedom is

related to the spin) of particles under rotations. In particular, a spin-1/2 particle,

such as the electron, has two internal degrees of freedom, leading to two-independent

geometric components.

This treatment of the electron may be replicated for the photon, treated as a

spin-1 particle with zero rest mass. Note that the number of components for a spin-j

particle, j = 0, 1/2, 1, 3/2, 2 . . ., is given byn = (2j+ 1), and comes from the general

treatment of rotations forn-component wave functions in three dimensions [94]. This


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leads us to a three-component wave function for the photon. Beginning with Eq.

(2.26) and setting m= 0, we take the square root of both sides, leading to

E=c

p p. (2.36)

In the energy-momentum representation, where a momentum-space wave function

can be well defined, we focus on the transverse, three-component momentum-space

wave function (p). Transversality implies that

p (p) = 0, (2.37)

so that we may make use of the vector identity

p p (p) = p p (p) + pp (p)

= p p (p) ,

(2.38)

to linearize Eq. (2.36). This leads us to the conclusion that the proper choice for the

Hermitian Hamiltonian operator on the RHS of Eq. (2.36) is

H=icp, (2.39)

where we have introduced the label =1, which we will see corresponds to the

helicity of the photon. When this Hamiltonian is substituted into Eq. (2.36) it gives

the following momentum-space wave equation

E(p) =icp (p) =c |p| (p) . (2.40)

This equation can be put into a form that more closely resembles the Dirac equation,

with explicit spin dependence, by noting the following feature of the spin-1 matrices


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and the vector cross product

a b= i (a s) b, (2.41)

where a and b are ordinary three-component vectors, and s = (sx, sy, sz) is a three-

component vector composed of the three spin-1 matrices (generators of rotations for

spin-1 particles)

sx=

0 0 0

0 0 i

0 i 0

, sy =

0 0 i

0 0 0

i 0 0

, sz =

0 i 0i 0 0

0 0 0

. (2.42)

This leads to the following form of the photon Hamiltonian

H=icp =c (s p) , (2.43)

and the corresponding momentum-space wave equation

E(p) =c (s p)(p) =c |p| (p) . (2.44)

The helicity dependence is now explicitly present, as can be seen by noting that the

helicity operator, i.e., the projection of the spin onto the direction of propagation, is

h= s p|p| . (2.45)

For completeness, one must treat both helicities on equal footing, which can be

done by creating a six-component, spinor wave function [68, 73]. However, since the

helicities do not mix in free space, we treat each helicity independently.


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Interpretation of the momentum-space photon wave function (p), must now be

addressed prior to transformation into coordinate space. The momentum-space wave

function (p), is typically interpreted as the probability amplitude in momentum

space [60]. This means that(p)2 d3p(2)3 gives the probability of finding a

photon with helicity, and momentum in a momentum-space volume d3paboutp. In

the standard non-relativistic quantum mechanics of massive particles, the momentum-

space wave function and the coordinate-space wave function, which is interpreted as

the probability amplitude in coordinate space, are related by a Fourier transform

relationship. However, it is well-known that photons, being inherently relativistic

particles, are non-localizable, and thus have no well-defined coordinate-space wave

function in this usual sense [41]. Thus one may not interpret the Fourier transform

of(p), given by

(LP) (x, t) =

d3p

(2)3ei(pxc|p|t)/(p) , (2.46)

as a coordinate-space wave function. Nonetheless, this has been done in the past, and

interpreted, albeit mistakenly, as the photon wave function [58, 64, 66]. Here we have

denoted this pseudo wave function with the superscript LP for Landau-Peierls, the

first to propose this form of the wave function in coordinate space [58]. There are

several reasons for not choosing this function as the true single-photon wave function,

including the fact that the wave function is non-locally connected to the classical

electromagnetic field [59, 60, 68, 73].


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To obtain the coordinate-space representation of the wave function and wave

equation (2.44), instead of the standard Fourier transformation of the momentum-

space wave function, we opt to weight the transformation with a function of the

magnitude of the momentum (equivalently, the energy) [59, 60, 68, 73], and make the

standard operator substitutions Eqs. (2.27) and (2.28). This leads to the following

form of the coordinate-space wave function

(x, t) =

d

3

p(2)3

ei(pxc|p|t)/f(|p|)(p) , (2.47)

where the weighting function f(|p|), is yet to be determined. One way of obtaining

the weighting function f(|p|), is to note that the only localizable, scalar quantity

that can be associated with a photon is its energy [60, 68, 73, 95]. Indeed, it has

been shown that for massless particles with spin greater than one, even the energy is

non-localizable [95]. This leads us to a weighting function of the form

f(|p|) =

c |p|, (2.48)

so that the coordinate-space wave function defined in Eq. (2.47) is equal to the

energy-density amplitude. For this reason we refer to the wave function defined in

Eq. (2.47), with f(|p|) = c |p|, as the energy-density amplitude or energy-density

wave function or simply the Bialynicki-Birula-Sipe (BB-S) wave function, after the

people who proposed this as the wave function [59, 60, 68, 73]. The wave equation

obeyed by any function of the form given in Eq. (2.47) is given by

it(x, t) = c (x, t) = ic (s ) (x, t) . (2.49)


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Note that Eq. (2.49), with the zero-divergence condition, Eq. (2.37), which in

coordinate space is given by

(x, t) = 0, (2.50)

are equivalent to the complex form of the Maxwell equations given in Chapter I, Eq.

(1.3). Also note, the non-local Landau-Periels wave function in Eq. (2.46) obeys the

same wave equation as the BB-S wave function. However, they have very different

interpretations, Lorentz-transformation properties, and normalizations [59, 60, 68,

73].

The probabilistic interpretation of the photon wave function, and scalar product

of two different single-photon wave functions are most clearly defined in momentum

space. The probabilistic interpretation of quantum mechanics requires a definition of

the scalar product between two different states and , that is used in calculating

transition probabilities. We denote the scalar product of photon wave functions with

the non-standard notation ( ), to emphasize that this is not the usual scalar

product. The Born rule states that the modulus squared of the scalar product of

two normalized wave functions |( )|2, is to be interpreted as probability of finding

a photon in state , when it is known to be in state . The probability is a real,

dimensionless number, and, being a true observable, must be invariant under all

Lorentz transformations. A choice as to the dimensions and interpretation of the

photon wave functions in momentum-space must be made prior to the determination

of the form of the Lorentz-invariant scalar product. As stated above, the single-photon


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Then upon substitution into the momentum-space scalar product in Eq. (2.51), this

gives the scalar product in coordinate space

( ) = 122c

d3x

d3x

(x, t) (x, t)

|x x|2 . (2.55)

For a more detailed discussion of the Lorentz-transformation properties of the photon

wave function see Appendix A, and references therein.

Note that this is a non-local integral, which is not surprising in light of the fact

that the photon number is not a local quantity in coordinate space [51]. It is also

interesting to note the following forms for the energy and momentum expectation

values in the coordinate-space representation [59, 68, 73]

H|

=

1

22c

d3x

d3x

1

|x x|2 (x, t)

H (x, t)

= 1

22

c d3x d3x

1

|x x

|2 (x

, t)

(

ics

) (x, t)

=

d3x (x, t) (x, t) ,

(2.56)

and P |

=

1

22c

d3x

d3x

1

|x x|2 (x, t)

P (x, t)

= 1

22c

d3x

d3x

1

|x x|2 (x, t)

(i) (x, t)

=

1

2ic

d3

x (x, t)

(x, t) .

(2.57)

Here the notation

O |

implies that the operator O act on the ket to its

right, | ), and the double lined bra ( , indicates the double integral, and non-

local weighting occur in front of the operator. The last line in Eq. (2.56) indicates

that it is the energy (equivalently, the momentum), and not photon number, that is


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a localizable scalar quantity in the coordinate-space representation. Thus we see that

the wave functions in coordinate space (x, t), are interpreted as energy amplitudes.

Quantization of the Single-photon Wave Function

We may now proceed to quantize the single-photon theory in the same manner

that one does for the Dirac equation [92, 93, 96]. We raise the photon wave function

(x, t), to the status of a field operator. We expand the wave function in modes

{j,(x, t)} that are orthonormal with respect to the non-local norm defined in Eq.

(2.55). The subscripts j, represent spatial and spin (helicity) degrees of freedom.

The expansion amplitudes become annihilation and creation operators, bj, and bj,

respectively. The photon field operator may then be expressed as

(x, t) =j,

j,(x, t) bj,+ H.c., (2.58)

where H.c. stands for Hermitian conjugate. The canonical boson commutation relations

for the bj, and bj, operators are

bj,, bl,= jl. (2.59)

The mode functions are orthonormal with respect to the non-local scalar product

defined for the single-photon wave function

(j, l,) = 122c

d3x

d3x

j,(x, t) l,(x

, t)

|x x|2 =jl. (2.60)


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(2.14),

F(+)(x, t) =

02

[E(+)(x, t) + icB(+)(x, t)]

=i

d3k

(2)3

k

4 ak,[uk,(x) + ic

k

c|k| uk,(x)]eikt,

(2.75)

where the mode functions uk,(x), are plane wave amplitudes, Eq. (2.1). When the

electromagnetic field operators are expanded in a discrete sum of plane waves instead

of the continuous spectrum given above, the equivalence of the two theories is clear

F(+)(x, t) =k,

k

4

uk,(x, t) + i

k

|k| uk,(x, t)

ak,+ H.c.

=k,

k4V

ek,+ i

k

|k| ek,

ei(kxkt)ak,+ H.c..

(2.76)

Comparing Eqs. (2.73) and (2.76), we see that they are equivalent (up to

2). We

also note that they both obey the complex form of the Maxwell equations, Eqs. (2.49)

and (2.50). This may be summarized by stating that in quantum field theory, the

photon wave functions are the mode functions of the quantized Reimann-Silberstein

vector. Conversely, the quantum field theory of light is constructed by canonically-

quantizing the single-photon wave function.

Indeed, if we examine the real and imaginary parts of the single-photon wave

function and their equations of motion derived from Eq. (2.49), we find

tR(x, t) =c I(x, t) , (2.77)

and

tI(x, t) = c R(x, t) . (2.78)


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Along with the zero divergence conditions

R(x, t) = I(x, t) = 0, (2.79)

these equations are identical to the Maxwell equations, with the electric field identified

with the real part of the photon wave function, and the magnetic induction field

identified with the imaginary part of the photon wave function (up to a couple

constants)

R= 0E, I= 0cB. (2.80)

One should be cautious interpreting this as proof that the electric and magnetic fields

for a single photon actually exist, however. We suggest that the photon wave function

be taken as the fundamental physical object, and that the macroscopic electric and

magnetic fields appear as emergent properties of a collection of many photons. This

is similar to the macroscopic spin associated with a collection of several atoms, which

is determined by weak measurements on the entire ensemble [9799].

In this section, we have shown that the monochromatic Dirac, and polychromatic

Titulaer-Glauber quantized field theories of electromagnetism can be derived from the

photon energy-density amplitude wave function and its equations of motion, in much

the same way that one arrives at the quantum field theory for electrons. The photon

wave function and its equations of motion are found by linearizing the Einstein energy-

momentum-mass relation for massless, spin-1 particles, and then the single-particle

theory is canonically quantized. We presented the Lorentz-invariant scalar product

of the photon wave function, which is non-local in the coordinate representation.


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Photon Wave Mechanics

In the previous sections we reviewed the theory of single-photon wave functions.

The main results are the form of the single-photon wave function, its interpretation

as the energy-density probability amplitude, whose modulus squared is related to the

probability of localizing the photon energy at one point in space-time, and the form

of the normalization integral and scalar product, which are non-local in coordinate

space. We also noted a useful connection of this non-local scalar product with the

wave-packet theory of quantum optics. In doing so, we showed how such wave-packet

modes can be made orthogonal with respect to a new inner product.

In this section, we will build upon the single-photon wave mechanics theory,

developing a two-photon wave mechanics, and show explicitly the relationship of this

theory to other well-known theories, such as photo-detection theory, classical and

quantum optical coherence theory [49, 50, 87], and the biphoton amplitude [89, 100

102] that is used in most discussions of spontaneous parametric down conversion

experiments. The theory is then extended to multi-photon states with known photon

number.

The two-photon wave function (2) (x1, x2, t), which is related to the probability

amplitude for localizing the energies of the two photons at two different spatial points

x1 and x2 at the same time t, can be expressed in free space as a sum over tensor


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in which the differential operator acts on the appropriate tensor component.

The spatially-varying refractive index of a linear medium may be treated in a

phenomenological manner, resulting in modified single-photon Hamiltonians [68, 73].

The two-photon wave function correspondingly changes to

it(2) = v1

(2)1 (1+ 1L1) (2) +v2 (2)2 (2+ 2L2) (2), (2.86)

where the material dependent quantities are evaluated at the local coordinatesL1(2)=

Lx1(2)

and v1(2) = v

x1(2)

. The divergence condition is also modified and becomes

(j+ jLj) (2) = 0, j = 1, 2. (2.87)

Here v1(2)=vx1(2)

is the local value of the speed of light in the medium given by

v (x) = 1/ (x) (x), (2.88)where (x) and (x) are the local values of permittivity and permeability of the

medium. The matrixLj, (j= 1, 2), is also given in terms of the local values of the

permittivity and permeability of the medium, and is defined as

L (x) =I ln

(x) (x) + 1ln

(x) / (x)

2 (2.89)

where

1=

0 1

1 0

. (2.90)

Tracing over the tensor product of the Hermitian conjugate of the two-photon wave

function wit

dissertation two sided printingphoton wave mechanics and experimental quantum state determination

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