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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