Progress in optics, volume 60

EDITORIAL ADVISORY BOARD

G.S. Agarwal

Stillwater, USA

T. Asakura

Sapporo, Japan

M.V. Berry

Bristol, England

C. Brosseau

Brest, France

A.T. Friberg

Joensuu, Finland

F. Gori

Rome, Italy

D.F.V. James

Toronto, Canada

P. Knight

London, England

G. Leuchs

Erlangen, Germany

J.B. Pendry

London, England

J. Perˇina

Olomouc, Czech Republic

J. Pu

Quanzhou, PR China

W. Schleich

Ulm, Germany

T.D. Visser

Amsterdam, The Netherlands


Elsevier
Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands

The Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB, UK
225 Wyman Street, Waltham, MA 02451, USA
First edition 2015
Copyright © 2015, Elsevier B.V. All rights reserved.
No part of this publication may be reproduced or transmitted in any form or by any means,
electronic or mechanical, including photocopying, recording, or any information storage and
retrieval system, without permission in writing from the publisher. Details on how to seek
permission, further information about the Publisher’s permissions policies and our
arrangements with organizations such as the Copyright Clearance Center and the Copyright
Licensing Agency, can be found at our website: www.elsevier.com/permissions.
This book and the individual contributions contained in it are protected under copyright by
the Publisher (other than as may be noted herein).
Notices
Knowledge and best practice in this field are constantly changing. As new research and
experience broaden our understanding, changes in research methods, professional practices,
or medical treatment may become necessary.
Practitioners and researchers must always rely on their own experience and knowledge in
evaluating and using any information, methods, compounds, or experiments described
herein. In using such information or methods they should be mindful of their own safety and
the safety of others, including parties for whom they have a professional responsibility.
To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors,
assume any liability for any injury and/or damage to persons or property as a matter of
products liability, negligence or otherwise, or from any use or operation of any methods,
products, instructions, or ideas contained in the material herein.
ISBN: 978-0-12-802284-9
ISSN: 0079-6638
For information on all Elsevier publications
visit our website at store.elsevier.com

CONTRIBUTORS
Ayman Alfalou
Vision ISEN-BREST Laboratory L@BISEN, Institut Supe´rieur de l’Electronique et du
Nume´rique, Brest, France
Mario Bertolotti
Dipartimento SBAI, Universita` di Roma “La Sapienza”, Rome, Italy
Fabio Bovino
Quantum Technologies Laboratory, SELEX-ES, Genoa, Italy
Christian Brosseau
Laboratoire des Sciences et Techniques de l’Information, de la Communication et de la
Connaissance, Universite´ de Brest, Brest, France
Natalie A. Cartwright
Department of Mathematics, State University of New York, New Paltz, New York, USA
Rafal Demkowicz-Dobrza
nski
Faculty of Physics, University of Warsaw, ul. Pasteura 5, Warszawa, Poland
Marcin Jarzyna
Faculty of Physics, University of Warsaw, ul. Pasteura 5, Warszawa, Poland
Brandon A. Kemp
College of Engineering, Arkansas State University, Jonesboro, Arkansas, USA
Jan Kołody
nski
Faculty of Physics, University of Warsaw, ul. Pasteura 5, Warszawa, Poland
Yuri A. Obod
R&D Company “Fotron – Auto”, Moscow, Russia
Kurt E. Oughstun
College of Engineering and Mathematics, University of Vermont, Burlington, Vermont,
USA
Alexander B. Shvartsburg
Joint Institute for High Temperatures, Russian Academy of Sciences; Institute for Space

Researches Russian Academy of Sciences, Moscow, and Far Eastern Federal University,
Vladivostok, Russia
Concita Sibilia
Dipartimento SBAI, Universita` di Roma “La Sapienza”, Rome, Italy
Oleg D. Volpian
R&D Company “Fotron – Auto”, Moscow, Russia

ix


PREFACE
In this 60th volume of Progress in Optics, six review articles are presented.
Chapter 1, contributed by Bertolotti, Bovino, and Sibilia, takes a historical approach to single-photon sources. They discuss photon statistics and
entangled states. Recently developed plasmonic sources and possible applications in quantum information processing are also described.
In Chapter 2, Alfalou and Brosseau discuss a variety of image processing
techniques. They compare the more traditional digital approach with newly
emerging, all optical setups. Because of their parallel nature, these can produce real-time results.
Chapter 3, by Cartwright and Oughstun, is a continuation of their article
in Volume 59. In this second part, they discuss recent developments in the
theory of pulse propagation through dispersive media. One of these is the
extension of the classic Sommerfeld–Brillouin theory to both the weak
and singular dispersion limits.
In Chapter 4, written by Demkowicz-Dobrza
nski, Jarzyna, and
Kołody
nski, the authors describe how nonclassical light states can be used
to enhance the performance of optical interferometers. This is especially
important, for example, for the ongoing search for gravitational waves.
Chapter 5 is a contribution by Kemp. Starting with the Minkowski–
Abraham controversy, different expressions for the electromagnetic force

density are analyzed. This ongoing discussion has direct consequences for
the growing number of optical trapping applications.
The final chapter, Chapter 6, is written by Shvartsburg, Obod, and
Volpian. They discuss how the classical effect of tunneling through optical
barriers takes on new and unexpected forms when the barrier consists
of a dielectric gradient metamaterial. Because of the scalability of these
effects, these also apply to microwave tunneling phenomena in transmission lines.
It has been my honor to serve as editor of Progress in Optics for over
50 years. But, as I used to tell my children when they were young, “All good
things must come to an end.” I would like to thank the members of the
editorial advisory board for their help, advice, and friendship over these
many years. Beginning with Volume 61, Dr. Taco Visser will take over

xi


xii

Preface

as editor of this series. I have complete confidence that he will keep readers
of Progress in Optics well informed of the most important advances being
made in the field. I wish him well.
EMIL WOLF
Rochester, NY
April 2015


ACKNOWLEDGEMENT
The editorial staff would like to extend a special thanks for the support and

expert assistance provided by Dr. Taco Visser during the compilation of
this work.

xiii


CHAPTER ONE

Quantum State Engineering:
Generation of Single and Pairs
of Photons
Mario Bertolotti*, Fabio Bovino†, Concita Sibilia*
*Dipartimento SBAI, Universita` di Roma “La Sapienza”, Rome, Italy
†
Quantum Technologies Laboratory, SELEX-ES, Genoa, Italy

Contents
1.
2.
3.
4.
5.
6.
7.
8.

Introduction
Fock States
The Problem of Localizing Photons
Antibunching of Single-Photon States

Photon Statistics and Spectral Purity
The Purcell Effect and the Control of Emission of Electromagnetic Radiation
Preparation of Single-Photon States: Quantum Engineering
Realization of Single-Photon Sources
8.1 Strongly Attenuated Sources
8.2 Single-Photon Sources Emitting one Photon in a Random Way
8.3 Single-Photon Sources on Demand
9. Entangled States
10. Plasmonic Sources
11. Application to Quantum Information Processing
12. Conclusions
Addendum
References

1
3
6
7
9
12
13
20
21
23
69
70
82
86
95
95

96

1. INTRODUCTION
The introduction of quantization of energy to discuss the interaction
of electromagnetic radiation with matter was done in 1900 by Planck (1900a,
1900b). Einstein (1905) surmised that also free radiation had a granular structure. The name photon was later proposed by Lewis (1926). The quantization
of the electromagnetic field was made by Dirac (1926, 1927). A review of
some historical papers on the subject has been made by Keller (2007).
Progress in Optics, Volume 60
ISSN 0079-6638
/>
#

2015 Elsevier B.V.
All rights reserved.

1


2

Mario Bertolotti et al.

What a photon is, exactly nobody can say. It is the quantum representation of a mode of the electromagnetic field and is an exclusively quantum
concept. With this definition, photons have associated plane waves of definite wave vector k and definite polarization s. A monochromatic wave
implies delocalization in time and space; in practice, a single photon localized to some degree in time and space can be described as superposition of
monochromatic photon modes.
When Glauber (1963a, 1963b) completed the model of radiation detection, discussing from a quantum point of view the interaction of radiation
and matter, and constructed a quantum theory of coherence, a number of
interesting states of radiation received a full reconnaissance as useful and possible states: among them coherent and single-photon states are perhaps the most

interesting, together with squeezed and entangled ones. Full description of
these states may be found in many excellent textbooks, like, e.g., Mandel
and Wolf (1995).
The generation of quantum states of the radiation field started to
receive great attention from the 1980s. Single-photon states, in particular,
are studied because of possible applications in quantum communication,
quantum lithography, quantum metrology, information processing, and
quantum computing, such as quantum random-number generation,
quantum networks, secure quantum communications, and quantum cryptography (see, for example, Beveratos, Brouri, et al., 2002; Cerf &
Flurasek, 2006; Dusek, Lutkenhas, & Hendrych, 2006; Gisin, Ribordy,
Tittel, & Zbinden, 2002; Gisin & Thew, 2007; Grangier & Abram, 2003;
Kilin, 2001).
For example, the security in some schemes of quantum cryptography
is based on the fact that each bit of information is coded on a single photon.
The fundamental impossibility of duplicating the complete quantum state
of a single particle (no cloning theorem; Cerf & Flurasek, 2006; Diecks,
1982; Ghirardi & Weber, 1983; Wootters & Zurek, 1982) prevents any
potential eavesdropper from intercepting the message without the receiver’s
noticing.
An ideal single-photon source would produce exactly one photon in a
definite quantum state, in contrast with a “classical” source, such as attenuated laser pulses, for which the photon number follows a Poisson distribution. A more stringent request would be to have the single-photon
generation on demand, that is, at a determined time. Additional requests
could be room temperature operation, high repetition frequency, high
efficient extraction into free space or fiber, good coherence, and


Quantum State Engineering

3

Fourier-transformed linewidth. Much progress has been made in the years
toward such devices, especially in suppressing the probability of two photons
in the same pulse.
Historically, the first experiment with single photons was made using an
atomic cascading process in which an excited atomic level decayed with the
emission of two photons of different frequencies (Clauser, 1974). The detection of one of them established the presence of the other; we will describe
this experiment later.
In the following, we will give a brief introduction to Fock states, remember the problem of localization of single photons, focus on their antibunching property and photon statistics, and remember the Purcell effect,
which allows a control of emission probability. We then discuss the preparation of single-photon states, the so-called quantum engineering, the different
kinds of single-photon sources, entangled states, plasmonic sources, and
applications to quantum information processing. The problem of detection
is deliberately not treated.
A number of review papers already exist on the subject such as Moerner
(2004), Lounis and Orrit (2005), Oxborrow and Sinclair (2005), Scheel
(2009), and Eisaman, Fan, Mugdall, & Polyakov (2011). Quite recently, a
Single-Photon Workshop has been held at Oak Ridge National Laboratory,
October 15–18, 2013. The presented papers are available to attendees only.
In the following, we will follow an approximate historical presentation
enlightening the single contributions and the evolution of the methods to
obtain single-photon sources. The survey may not be complete; we apologize for any omission.

2. FOCK STATES
States with a prescribed number of photons are called number states or
Fock states.
They were first introduced and discussed by Fock (1932) (see also
Faddeev, Khalfin, & Komarov, 2004).
A Fock state is strictly quantum mechanical and contains a precisely definite number of quanta of field excitation; hence, its phase is completely
undefined.
As well known, the Hamiltonian for the free electromagnetic field can be
written as

À
Á
À
Á
H ~a{ , ~a ¼ ℏω ~a{ ~a + 1=2
(1)


4

Mario Bertolotti et al.

where ћ is h/2π with h Planck’s constant, ω is the frequency (pulsation) of
the field, and a˜ and a˜† are the annihilation and creation operators, respectively.
They obey the commutation relations
 {Ã
Â
Ã
~a, ~a ¼ 1; ½~a, ~aŠ ¼ ~a{ , ~a{ ¼ 0
(2)
The product a˜†a˜ is called the photon number operator n˜ which accounts for
the photons in the chosen spatial–temporal mode
n~ ¼ ~a{ ~a

(3)

The eigenstate of the photon number operator n˜
n~jni ¼ njni

(4)

has a perfectly fixed photon number n. Since n˜ is a Hermitian operator, the
number n is real.
If jni is an eigenstate of n˜, then a˜jni must be an eigenstate as well, with the
eigenvalue n À 1. In fact,
À
Á
n~~ajni ¼ ~a{ ~a2 jni ¼ ~a~a{ ~a À ~a jni ¼ ðn À 1Þ~ajni
(5)
In a similar way, a˜†jni is an eigenstate of n˜ with eigenvalue n + 1. So there
are the fundamental relations
~ajni ¼ √njn À 1i

(6)

~a{ jni ¼ √ ðn + 1Þjn + 1i

(7)

The prefactors have been obtained using the fact that
 { 
nj~a ~ajn
must be equal to the eigenvalue n.
The state ~aj0i ¼ 0 exists and it is the vacuum state.
States with a prescribed number of photons can be created by applying
the creation operator to the vacuum state
jni ¼ ~a{n =√n!j0i

(8)

The Fock states must be complete
1
X
n¼0

and orthonormal

jnihnj ¼ 1

(9)


5

Quantum State Engineering

hnjn0 i ¼ δnn
The wave function ψ(q) of the state j0i is
À
Á
ψ o ðqÞ ¼ π À1=4 exp Àq2 =2

(10)

(11)

In momentum representation, it is
À
Á
ψ o ðpÞ ¼ π À1=4 exp Àp2 =2

(12)

The expectation value of the electric field and its square are
hnjEjni ¼ 0

(13)

 2 
njE jn ¼ ðћω=εo V Þðn + 1=2Þ

(14)

and

where εo is the vacuum dielectric constant and V is the volume.
The electromagnetic wave amplitude can be represented by the quantity
Eo ¼ ð2ћω=εo V Þ1=2 ðn + 1=2Þ1=2

(15)

The phase of the wave is of course completely uncertain.
The use of number states could lead to great signal to noise improvement
(Yuen, 1986).
Coherent states are the eigenstates of the annihilation operator a˜
~ajαi ¼ αjαi

(16)

jαi is an eigenstate of the annihilation operator a˜ with eigenvalue α, which

is in general a complex number since a˜ is not a Hermitian operator.
A coherent state is the closest analogue to a classical light field and exhibits
a Poisson photon number distribution with an average photon number
jα2j. Coherent states have relatively well-defined amplitude and phase,
with minimal fluctuations permitted by the Heisenberg uncertainty
principle.
Coherent states are an overcomplete set.
Because the number states are a complete set, it is possible to express
coherent states as a superposition of n states
X
(17)
jαi ¼ expðÀ1=2Þjαj2 αn =ðn!Þ1=2 jni


6

Mario Bertolotti et al.

Vice versa n states may be expressed as a superposition of coherent states
Z
À
Á
jni ¼ ð1=π Þ exp Àjαj2 =2 α*n =ðn!Þ1=2 jα > d2 α
(18)
where d2 α ¼ dðReαÞdðImαÞ:
The other states of interest here are squeezed and entangled states. The real
and imaginary part of the complex amplitude of the electromagnetic field
fluctuate with equal dispersion in a coherent state. The phenomenon of vacuum fluctuations is a manifestation of this effect because the vacuum state is
an example of a particular coherent state. In a squeezed state, one part of the
field fluctuates less and the other part fluctuates more than in the vacuum

state. A number state can be considered a special case of a squeezed state.
Entangled states will be discussed in Section 9.

3. THE PROBLEM OF LOCALIZING PHOTONS
The number operator n˜ refers to the total photon number in all space.
It is therefore not expected to be accessible to direct measurement. From a
practical point of view, one could interpret the electronic signal registered
by a photodetector as due to a photon that has been localized in the detector
volume. More precisely, the counts registered by a detector whose surface is
normal to the incident field and exposed for some finite time Δt could be
interpreted as a measurement of the number of photons in a cylindrical volume whose base is the sensitive surface of the detector and whose height is
cΔt. The integral of the intensity I(r,t) over such volume can therefore be
interpreted as a configuration space number operator, at least in an approximate sense, provided the linear dimensions of the volume are all large compared with the wavelengths of contributing modes (see, for example,
Mandel & Wolf, 1995 for a detailed discussion).
This is of course a very simplified point of view. A photon wave function
concept in the coordinate representation was introduced originally by
Landau and Peierls (1930), connected with the Maxwell electromagnetic
field solutions. This wave function is a highly nonlocal object. Its modulus
squared has the right dimensionality to be interpreted as a probability density
to find a photon. However, as noted already by Pauli (1933), it has serious
drawbacks. A more refined theory was presented by Oppenheimer (1931),
but this theory too is not completely satisfying. A fundamental point is that a
position operator cannot exist in a relativistic quantum theory. Even for
massive particles, the localization is not perfect because it is not


Quantum State Engineering

7

relativistically invariant. Two observers in relative motion would not quite
agree as to the localization region of a relativistic particle.
We need not to enter in this matter and refer, among others, to review
papers by Acharya and Sudarshan (1960), Bialynicki-Birula (1996), Scully
and Zubairy (1997), and Keller (2007). We will assume the pragmatic position that a single-photon state is there if a detector, that can determine the
number of incident photons with 100% accuracy, clicks.

4. ANTIBUNCHING OF SINGLE-PHOTON STATES
One-photon state presents a peculiar anticorrelation effect, which
does not exist for a classical wave. If we send a one-photon state on a beam
splitter and place photon-counting detectors on the reflected and transmitted beams—a disposition first used by Hanbury Brown and Twiss (1956a,
1956b, 1958) to study intensity correlations in astronomy—we never
observe any coincidence between counts measured by the two detectors,
as this would violate energy conservation. The photon cannot split to be
present at the two output ports contemporarily. As a consequence of the
principles of quantum mechanics, the wave function of the photon has to
collapse onto either one or the other of the two detectors. This absence
of coincidences of detection events on the two detectors has been dubbed
photon antibunching and is utterly irreconcilable with a classical description
of light. This effect was predicted by Carmichael and Walls (1976) and
Kimble and Mandel (1976). Antibunching can be assessed measuring the
joint probability p(tjt + τ) of detecting two photoelectric pulses at time t
within Δt and at time t + τ within Δτ. The normalized quantity
pðtjt + τÞ=pðtÞ ¼ hI ðtÞI ðt + τÞi=hI ðtÞi2 ¼ gð2Þ ðτÞ

(19)

is called the second-order correlation function g2(τ). Antibunching means
g(2)(0) ¼ 0.
Naively the antibunching can be understood very simply. In the emission

process from a single atom or molecule, the correlation function measures
the joint probability for the arrival of a photon at time t ¼ 0 and the arrival
of a photon at t > 0. After the emission of a photon at t ¼ 0, the quantum
system is projected in its ground state since it just emitted a photon. Hence,
the simultaneous emission of a second photon is impossible because the molecule cannot emit from the ground state. It needs some time to have a finite
probability to be again in the excited state and emit a second photon. On the


8

Mario Bertolotti et al.

average a time a half a Rabi period1 has to elapse to have a finite probability
for the molecule to be in the excited state and emit a second photon.
Antibunching was first observed in pioneering experiments (Kimble,
Dagenais, & Mandel, 1977; Walls, 1979).
In the nanosecond time regime within a single bunch of photons, the
emitted photons from a single-quantum system are expected to show antibunching; that is, the probability for two photons to arrive at the same time
is zero.
To observe antibunching correlations, the second-order correlation
function g(2)(τ) is generally measured by determining the distribution of time
delays N(τ) between the arrival of successive photons in a dual beam detector
gð2Þ ðτÞ ¼ hnðtÞnðt + τÞi=hni2 ¼ pðt=t + τÞ=pðt Þ

(20)

In Equation (20), n(t) is the number of photons counted at time t.
For a number state with n photons, it is
gð2Þ ð0Þ ¼ 1 À 1=n

(21)

The Hanbury Brown and Twiss experimental arrangement is shown in
Figure 1.
Liquid filter

Half-silvered mirror

Interference filter

Photomultiplier tube

Mercury arc

Lens

Rectangular aperture

Slide

Photomultiplier tube
Correlator

Amplifier

Amplifier

Integrating motor

Figure 1 The Hanbury Brown and Twiss experimental arrangement.

1

The Rabi frequency Ω ¼ jμ12 Ej/ћ, where μ12 is the transition dipole moment and E is the laser field
with which the atom is excited.


9

Quantum State Engineering

120

2.0

80
c(t)

g(2)(t)

1.5
1.0

40

0.5
0.0

0
−20

0

20
t (ns)

40

Figure 2 An example of antibunching. From Beveratos, Brouri, Gacoin, Poizat, and
Grangier (2001).

An example of the result obtained when antibunching is present is shown
in Figure 2.
The correlation function never reaches exactly zero because of a number
of experimental drawbacks which are characteristics of the particular
experiment.
In any case, the time response of the detectors will determine a minimum
value of the measured g(2)(0) even with an ideal g(2)(0) ¼ 0 source.

5. PHOTON STATISTICS AND SPECTRAL PURITY
The probability distribution pm(T) of m photons in the quantum field
during an observation time T (photon number distribution) can be connected to the probability of counting n photons at a detector (Mandel &
Wolf, 1995).
The photon-count distribution p(n, T, t) of the detection of n counts in
the time interval (t, t + T) by a photodetector may be written as
Z
pðn, T, t Þ ¼ ½ðηW Þn =n!Š expðÀηW ÞPN ðW ÞdW
(22)
where PN(W) is the probability distribution of the integrated intensity W
Z

W¼

I ðt0 Þdt0

(23)

η is the photodetection efficiency and I(t) is the intensity of light incident
on a photocathode. If η ¼ 1, Equation (22) is also the probability of detecting


10

Mario Bertolotti et al.

n photons in a time interval t, t + T. Quantum mechanics arrives at an
equation formally similar to Equation (22) (Perina, 1984; Perina,
Hradil, & Jur, 1994)
n
pðn, T , tÞ ¼ Tr
Z fρ : ½ðηW Þ =n!Š expðÀηW Þ : PN ðW ÞdW g

¼

½ðηW Þn =n!Š expðÀηW ÞPN ðW ÞdW

(24)

where : : represent the normal order of the field operator, PN(W) is the
quasi-probability distribution of the normal integral intensity W, that may
have also negative values, and ρ is the density matrix.

We briefly recall the possibility of describing the most general statistical
quantum system by means of the so-called density matrix ρ. This is a
Hermitian operator, time independent in the Heisenberg picture, such that
the quantum and statistical expectation value of any operator A is expressed as
fhAigav ¼ TrðρAÞ
where the trace Tr of an operator O is defined as
X
TrðOÞ ¼
hΨ jOjΨ i

(25)

(26)

the jΨ is forming a set of states verifying the completeness relation
ΣjΨ ihΨ j ¼ 1

(27)

It is immediately apparent that for a system in a pure state jΨ i
corresponding to no statistical indetermination, the density matrix ρ is given
by the operator jΨ ihΨ j.
The photon-counting distribution relative to chaotic fields if the counting time T is much less than the inverse frequency bandwidth of the radiation is (Bose–Einstein distribution)
pm ðT Þ ¼ hmim =ð1 + hmiÞm + 1

(28)

For light from a laser, the distribution is a Poisson distribution
pm ðT Þ ¼ hmim expðÀhmiÞ=m!

(29)

The moments of the distribution are
hni ¼ ηhW iN
hn2 i ¼ ηhW iN + η2 hW 2 iN

(30)


11

Quantum State Engineering

Quantum mechanically
hn~i ¼ ~a{ ~a
 2

À Á2
  
n~ ¼< ~a{ ~a > ¼ ~a{ ~a + ~a{2 ~a2

(31)

The variance of the number of absorbed photons (η ¼ 1) is expressed by
À

À

(32)
Δ~

nÞ2 ¼ hn~i + ΔW Þ2 N
For a coherent field h(ΔW)2iN ¼ 0 and h(Δn˜)2i ¼ n˜2, i.e., such a field is
Poissonian. Classical fields have h(ΔW)2iN > 0 and they are super-Poissonian,
for example, for the chaotic field of a natural source h(ΔW)2i ¼ hIi2T2 and
À

(33)
ΔnÞ2 ¼ hnið1 + hniÞ
corresponding to the Bose–Einstein distribution (28).
For quantum fields having no classical analogues, it may be h(ΔW)2iN < 0
and h(Δn)2i < hni, and such light is sub-Poissonian.
The photon number distribution can be calculated using the density
matrix ρ
pðmÞ ¼ hmjρjmi

(34)

From an experimental point of view p(m) is related to p(n,T,t) by the
Bernoulli transformation
X
½m!=n!ðm À nÞ!Šηn ð1 À ηÞmÀn pðmÞ
(35)
pðn, T, t Þ ¼
À

Introducing the so-called Fano factor Fn ¼ ΔnÞ2 =hni, the fluctuation
formula (32) can be written in terms of the photocount number n and the
photon number m in the form of the quantum Burgess variance theorem
(Perina, 1984)
Fn À 1 ¼ ηðFm À 1Þ

(36)

For the Fock state Fm ¼ 0.
Although both effects are nonclassical, are exhibited only by a quantum
field, and are often associated, sub-Poissonian statistics and antibunching are
distinct effects that need not necessarily occur together; one may occur
without the other (Singh, 1983).
However for a stationary single-photon source, the nonclassical nature of
the emitted radiation would lead to sub-Poissonian photon statistics with
g(2)(0) < 1.


12

Mario Bertolotti et al.

Observation of sub-Poissonian photon statistics was first made by Short
and Mandel (1983) using fluorescence from a single two-level atom which
emitted single photons.
For some application, it is necessary that the single photons be indistinguishable. The produced packets should be fully coherent, characterized by
a Fourier-transform relationship between their temporal and spectral profiles, that is
ΔνΔτ ¼ 1=2π

(37)

where Δν ¼ 1/(2πT2) and Δτ ¼ 2T1 designate, respectively, the Lorentzian
half-width and the spontaneous decay time of the field amplitude. This
property is required for the implementation of any photon-based quantum
information processing system (Knill, Laflamme, & Milburn, 2001).

6. THE PURCELL EFFECT AND THE CONTROL OF
EMISSION OF ELECTROMAGNETIC RADIATION
Spontaneous emission of an atom is a result of the interaction between
the atom dipole and the vacuum electromagnetic fields. Therefore, it is not an
intrinsic property of an isolated emitter but rather a property of the coupled
system of the emitter and the electromagnetic modes in its environment.
Purcell (1946) first predicted that nontrivial boundary conditions of an
electromagnetic field in the vicinity of an excited atom could alter its
decay rate.
The rate Γ for spontaneous transitions from an initial state jii with no
photons to a final state jf i with one photon is given by the well-known Fermi
golden rule (Loudon, 2000)
Γ ¼ h2 ρðvc Þjh f jHjiij2

(38)

where H is the interaction Hamiltonian and ρ(νc) is the density of states at the
transition frequency νc, that for radiation in free space is
 2
4πvc
(39)
ρðvc Þ ¼ 2
c3
The spontaneous emission rate can therefore be changed if an atomic system finds itself in a space region where ρ(νc) is modified; for example, it is
placed close to a metal surface or a dielectric interface. First experimental
demonstrations were carried out by Drexhage, Kuhn, and Schafer (1968),


Quantum State Engineering

13

Kuhn (1970), and Chance, Prock, and Silbey (1974). Approximate solutions
of the electromagnetic boundary value problem were reviewed, for example, by Drexhage (1974). More detailed calculations are in Lukosz and Kunz
(1977a, 1977b) and references therein. In a more effective way, the radiation
emitted by a source can be altered by suitably modifying the surrounding
vacuum fields in a cavity.
The application to a small cavity for which the density of modes may be
modified was considered by Kleppner (1981). In particular, when the transition frequency νc is near resonance with a mode eigenfrequency, the spontaneous emission rate can be considerably increased.
The effect was experimentally observed (Goy, Raimond, Gross, &
Haroche, 1983) with a sodium Rydberg atom set through a resonant superconducting cavity. Also inhibited spontaneous emission was observed by
studying the cyclotron motion of a single electron (Gabrielse & Dehmelt,
1985). Therefore, a reduction of the two-photon probability relative to a
Poisson distribution may be achieved acting on the density of modes.
Periodic dielectric structures in the form of photonic crystals can alter
the emission properties ( John, 1987; Yablonovitch, 1987). The existence
of forbidden electromagnetic frequencies inside the structure may inhibit
the emission of radiation which is, on the contrary, enhanced at the frequencies where the density of modes is maximum.
Enhanced spontaneous emission by quantum semiconductor boxes in a
monolithic optical microcavity, that can be considered the ancestor of quantum dots’ experiments, was studied, for example, by Gerard et al. (1996,
1998 and; Gerard & Gayral, 1999) (see also Bulu, Caglayan, & Ozbay,
2003). A general review of 1D photonic structures is, for example,
Bertolotti (2006). In semiconductor systems, enhanced and inhibited spontaneous emission from GaAs quantum well excitons was demonstrated using
a planar microcavity (Yamamoto, Machida, Horikoshi, & Igeta, 1991;
Yokoyama et al., 1990). The properties of band gap structures have been
largely studied (see, for example, Yang, Fleischhauer, & Zhu, 2003;
Yang & Zhu, 2000). The effect of nanotubes on the spontaneous emission
of a single InAs quantum dot was considered by Bleuse et al. (2011).

7. PREPARATION OF SINGLE-PHOTON STATES:
QUANTUM ENGINEERING
How to obtain an arbitrary quantum state is a task that may be accomplished by so-called quantum-engineering methods that allow one to prepare a


14

Mario Bertolotti et al.

previously specified quantum state by a series of elementary operations on
the system that is to be prepared. Different approaches have been considered
to create nonclassical states. The general approach is to find an appropriate
Hamiltonian which transforms via unitary time evolution a given initial state
to the desired state. Another way is to operate on the system projecting it on
another state. In general, the creation of Fock states was considered via the
interaction of a suitably prepared two-level atom with the electromagnetic
field. A way to single out atoms or molecules and have efficient interaction
with an electromagnetic field is to use microcavities.
In the years 1980s and 1990s, a great number of proposals were done on
possible production of states with defined number of photons by using the
interaction of atoms with quantum microcavities (see, for example, Meystre,
1992 for a description of the interaction in microcavities).
One fundamental aspect in the strong coupling regime of the Jaynes–
Cummings ( Jaynes & Cummings, 1963) approximation is that, at resonance,
an atom initially in an excited state jei and an empty cavity, periodically
exchange a quantum. The atom–field state wave function jΨ (t)j oscillates
between an excited atom and no photon state je,0i and a state in which the
atom is on the ground level and a photon is in the cavity jg,1i, according to
Ψ ðtÞ ¼ cos ðΩo t=2Þje, 0i + sin ðΩo t=2Þjg, 1i

(40)

where Ωo is the Rabi frequency and t is the interaction time. So, instead of
simply emitting a photon and going on its way, an excited atom in a resonant
cavity oscillates back and forth between its excited and unexcited states. The
emitted photon remains in the cavity in the vicinity of the atom and is
promptly reabsorbed. The atom–cavity system oscillates between two states,
one consisting of an excited atom and no photon and the other a deexcited
atom and a photon trapped in the cavity.
The basic idea of many schemes is that measurement of the atom after its
interaction with the field in a cavity provides information about the field.
The collapse of the entangled state of the atom–field system makes the field
part of the wave function to jump into a different state from the one before
the measurement. Based on the interaction between the radiation field and
atoms, many theoretical schemes were therefore proposed for the generation
of Fock states. We consider here, without pretending to be exhaustive, a few
of them. Some of them, with suitable changes, were later experimentally
realized (see Section 8.2). Eventually, as will be shown later, the real sources
of single photons have been implemented on a much simpler and direct way
choosing with skill the emitter in solid state.


15

Quantum State Engineering

The group at Max-Planck-Institute, Garching, had great experience
with quantum cavities. Filipowicz, Javanainen, and Meystre (1986a,
1986b) were probably the first ones to make the proposal to use the interaction of atoms with the electromagnetic field in a microwave microcavity.
They considered a lossless, single-mode cavity in which two-level atoms

were injected at such a low rate that at most one atom at a time is present
inside the cavity. Assuming that only one atom at the time was coupled to
the field, the authors investigated the interaction classically and quantum
mechanically using the Jaynes–Cummings model ( Jaynes & Cummings,
1963). Controlling the initial conditions, if atoms in their excited state
are considered, a highly excited Fock state of the cavity mode was predicted.
The situation in which atoms in their excited state are injected in a highQ micromaser cavity was considered also by Krause, Scully, and Walther
(1987), but from a completely different point of view. In their proposed
experiment, Rydberg two-level atoms in their upper state were injected into
the micromaser cavity. After they leave the cavity, they are probed with a
static electric field which ionizes all atoms which are in their upper level.
All the atoms that are not ionized have emitted a photon in the cavity. When
these atoms are counted, the total number of photons in the maser can be
inferred. Therefore by the determination of the state of the outgoing atoms,
the photon number in the field is exactly known; i.e., the state of the field is
reduced to a pure number state. Since at start no radiation is in the cavity, the
field is always in a number state when an atom enters the cavity. By the interaction of the atom with the field, which is in a state │ni, the field state will be
changed to a superposition of states │ni and │n + 1i. Due to the measurement of the atomic state, afterward this superposition is reduced to one of the
states │ni and │n À 1i, depending on the result of the measurement. In an
experiment, thermal photons in the cavity have to be suppressed. It means
that the cavity should be cooled at temperatures close to absolute zero. It also
should be emphasized that the number of atoms leaving the cavity in the
lower state is equal to the number of photons in the cavity only for a lossless
cavity. Moreover, a feature central to the proposed scheme is the long lifetime of the photons in the cavity, which is demanding for very high
Q-factors. The decay time for photons of frequency f is linked to the quality
factor Q of the cavity by
τ ¼ Q=f

(41)

Quality factors Q of the order of 1010 at frequencies in the microwave region
f ¼ 1010 Hz give long lifetimes of the order of a second.


16

Mario Bertolotti et al.

In a second paper, the control, via an electric accelerating field, of the
interaction time of ionic Rydberg atoms with the maser field was proposed
(Krause, Scully, Walther, & Walther, 1989). If the velocity of the ions is
adjusted in such a way that every ion emits a photon, then the total number
of photons is exactly known via the total number of passing ions.
The important ingredients in this type of proposals are superconducting
cavities together with the laser preparation of highly excited atoms. These
atoms are Rydberg atoms that have quite remarkable properties: the probability of induced transitions between neighboring states of a Rydberg atom
scales as n4, where n denotes the principal quantum number. Consequently,
a single photon is enough to saturate the transition between adjacent levels.
Moreover, the spontaneous lifetime of a highly excited state is very large. In
the experiments, a transition between two neighboring Rydberg levels is
resonantly coupled to a single mode of a cavity field.
The limitation of the Krause method is that it does not predict the eventual Fock state, but only the probability of its occurrence. Moreover, this
technique seems to be useful to generate single-photon states, which however cannot be made free, that is to say, are not available for further
experiments.
Also at Ecole Normale Superieure in Paris, the production of Fock states
was considered. A method for their production was proposed by Raimond,
Brune, Lepape, and Haroche (1989). The scheme utilized the passage of a
two-level atom through a microwave cavity, the resonant frequency of
which is continuously tuned (during the atomic transition time) so as to produce an adiabatic transformation of the initial eigenstate describing the
atom–cavity system into a final eigenstate corresponding to the cavity mode

being in a Fock state. In this application, the adiabatic theorem (see Kramers,
1957) is used which applies to a system described by a time-varying
Hamiltonian H(t). Applied to this system for the interval of time from to
to t1, provided the evolution in this time interval is sufficiently slow, the
theorem says that if the system is initially in an eigenstate of H(to), it will pass
into the eigenstate of H(t1) that derives from it by continuity.
Brune, Haroche, Lefevre, Raimond, and Zagury (1990) at Ecole
Normale Superieure, Paris, suggested a method for the preparation of a Fock
state based on the quantum nondemolition measurement. The photon number is measured in a cavity by coupling the field to a beam of Rydberg atoms
and measuring the atomic phase shift. After a sequence of atomic measurements, the cavity field collapses into a Fock state with an unpredictable number of photons.


Quantum State Engineering

17

Holland, Walls, and Zoller (1991) proposed a quantum nondemolition
measurement of the photon number of a cavity field mode by measuring the
deflection of atoms passing through the field.
A variation of the scheme proposed by Brune et al. (1990), involving an
initial coherent state and single-atom interaction and detection, can generate
a Schr€
odinger cat state of the field (Brune, Haroche, Lefevre, Raimond, &
Zagury, 1992; Savage, Braunstein, & Walls, 1990). The approach, capable of
yielding both Schr€
odinger cats and Fock states, was based on the detection of
the dispersive field-induced phase shift acquired by the initially polarized
state of nonresonant atoms in the cavity (Brune et al., 1992). The random
variation of these phase shifts with the atomic transit time (velocity) in a
sequence of such measurements results in repeated splitting of the phasespace field distribution and interference between the split parts. The first step

in the sequence yields a phase difference Schr€
odinger cat, while subsequent
steps gradually lead to phase diffusion and decimation (through interference)
of the photon number distribution, converging to a Fock state. Also this
method cannot determine in advance the eventual Fock state.
A method for reconstructing the average evolution of the photon number distribution of a field decaying in a high-Q cavity was later presented
(Sayrin et al., 2012).
At low temperature, steady-state macroscopic superpositions can be generated in the field of a micromaser pumped by a stream of two-level atoms
injected in a coherent superposition of their upper and lower states (Slosser,
Meystre, & Wright, 1990). Under appropriate conditions, the field in a lossless micromaser can evolve to pure number states.
Generation of Schr€
odinger cats by many atoms initially prepared (by a
π/2 pulse) in a polarized state whose sequential transit through the cavity
gradually builds up an approximate trapping state of the field, described
by a superposition of the vacuum and a number state, was proposed by
Meystre, Slosser, and Wilkens (1990).
Vogel, Akulin, and Schleich (1993) discussed in general the way to construct any desired quantum state of the radiation field by using a simple
Hamiltonian in a single-mode resonator. A single-mode cavity field state
could be prepared by injection of a prepared and later probed two-level
atom (see also Vogel & Wellentowitz, 2001). They considered constructing
any superposition of the first N + 1 number states from the vacuum state by
injecting N appropriately prepared atoms into a cavity and detecting all of
them in the ground state. Since the successful preparation of the desired cavity field state depends on the outcomes of the electronic state measurements


18

Mario Bertolotti et al.

on the atoms after having passed through the cavity, the technique is probabilistic and attains a successful engineered quantum state only for a specific

series of measurement outcomes.
Injection of atoms with their Zeeman electronic levels initially prepared
in specific superpositions interacting adiabatically with a cavity was proposed
(Parkins, Marte, Zoller, Carnal, & Kimble, 1995; Parkins, Marte, Zoller, &
Kimble, 1993).
A scheme based upon the same mechanism described by Vogel et al.
(1993) but employing two-photon interactions in a micromaser has been
described by Garraway, Sherman, Moya-Cessa, Knight, and Kurizki
(1994) who proposed a scheme that allows the generation of a variety of
nonclassical field states with controllable (predetermined) photon number
and phase distribution. It was based on the two-photon-resonant interaction
of a single electromagnetic field mode in a high-Q cavity with initially
excited atoms crossing the cavity sequentially (one at the time). The
sequence duration should be much shorter than the cavity lifetime. The
states are generated by selecting only those sequences wherein each atom
is measured to be in the excited state after the interaction. The field distribution resulting from a sequence of N such measurements is peaked about
2N positions in the phase plane. When these peaks are chosen to overlap,
part of the field can be made to interfere giving rise to decimation of the
photon number distribution and Fock states may be generated.
At variance with the practice of measuring atoms once they came out of
the cavity, Law and Eberly (1996) discussed a method in which quantum
states of the atom (source) are manipulated during the atom–field interaction
process, i.e., inside the cavity. They presented a cavity model which allows
the vacuum state to evolve to an arbitrarily prescribed superposition of Fock
states.
Later, Law and Kimble (1997) discussed the possibility to produce a prescribed sequence of single-photon pulses controlling the emission of an atom
with a Λ-type three-level structure in an optical cavity. In 2003, the Caltech
group (Duan, Kuzmich, & Kimble, 2003) extended the study of the interaction of an atom trapped in a cavity with hot atoms with an inhomogeneous
distribution in position and/or a time-varying location, using an adiabatic
passage technique.

The previously considered schemes find at least two types of difficulties.
First, the quantum states of the sources of radiation (the atomic states) have
to be manipulated in an arbitrarily controllable manner. Second, the source
must “teach” the field to evolve toward the desirable quantum state.


19

Quantum State Engineering

A new method was proposed by Kuhn, Hennrich, Bondo, and Rempe
(1999) at Garching, for the generation of single photons into a single mode
of the radiation field. The idea was based on proposals by Law and Eberly
(1996) and Law and Kimble (1997). A single atom, strongly coupled to an
optical cavity, which defines the active mode and ensure photon emission
into a well-defined direction, was used as the active medium for generating
the photon. The excitation scheme is shown in Figure 3. A Λ-type threelevel atom with two long-lived states jui and jgi, typically two Zeeman or
hyperfine states of the atomic ground state, and an electronically excited state
jei is in a cavity with a resonant frequency close to the atomic transition jei to
jgi but far off resonance from jei to jui. Hence, only the states je,0i with zero
photon and jg,1i with one photon are coupled to the cavity mode. Initially,
the system is prepared in state ju,0i. To trigger a photon emission, the atom
is exposed to a light pulse at Rabi frequency coupling the states ju,0i and
je,0i. Provided the trigger pulse rises sufficiently slowly, an adiabatic evolution is assured. If the two-photon resonance condition is fulfilled, a STIRAP
(STImulated Raman scattering involving Adiabatic Passage)-type adiabatic
passage takes place and ju,0i goes to jg,1i generating the emission of one
photon. Recycling can then be provided.
The use of nonlinear interactions was also discussed.
Leonski and Tanas (1994) proposed a scheme in which a one-photon state
can be obtained in a cavity that is periodically kicked by a sequence of classical

light pulses and is filled with nonlinear Kerr medium. If the amplitude
between the kicks and the time between the kicks are appropriately chosen,

)
TIR

AP

Atom–cavity
coupling g

Emission of a
single photon

aba

tic

tran

sfe

r (S

Trigger
pulse
WT

Ig,1>

Ig,0>

Adi

Energy of the atomic bare states

Ie,0>

Iu,0>

Repumping pulse

Figure 3 Scheme of the atomic levels coupled by the trigger pulse, the cavity, and a
possible repumping pulse. From Kuhn et al. (1999).