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Zbigniew Ficek Stuart Swain
Quantum Interference
and Coherence
Theory and Experiments
With 179 Figures
ZbigniewFicek StuartSwain
DepartmentofPhysics SchoolofMathematicsandPhysics
TheUniversityofQueensland Queen’sUniversityBelfast
Brisbane,QLD4072 Belfast,BT71NN
Australia UK
[email protected] [email protected]
LibraryofCongressCataloging-in-PublicationData
Ficek,Zbigniew.
Quantuminterferenceandcoherence:theoryandexperiments/ZbigniewFicekand
StuartSwain.
p. cm.—(Springerseriesinopticalsciences,ISSN0342-4111)
Includesbibliographicalreferencesandindex.
ISBN0-387-22965-5(alk.paper)
1.Quantuminterference. 2.Coherentstates. 3.Interference(Light) 4.Coherence
(Nuclearphysics) 5.Quantumtheory. I.Swain,Stuart. II.Title. III.Series.
QC174.17.Q33F53 2004
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Preface
The field that encompasses the term “quantum interference” combines a
number of separate concepts, and has a variety of manifestations in dif-
ferent areas of physics. In the sense considered here, quantum interference
is concerned with coherence and correlation phenomena in radiation fields
and between their sources. It is intimately connected with the phenomenon
of non-separability (or entanglement) in quantum mechanics. On account of
this, it is obvious that quantum interference may be regarded as a compo-
nent of quantum information theory, which investigates the ability of the
electromagnetic field to transfer information between correlated (entangled)
systems. Since it is important to transfer information with the minimum of
corruption, the theory of quantum interference is naturally related to the
theory of quantum fluctuations and decoherence.
Since the early days of quantum mechanics, interference has been de-
scribed as the real quantum mystery. Feynman, in his famous introduction
to the lectures on the single particle superposition principle, referred in the
following way to the phenomenon of interference: “it has in it the heart of
quantummechanics”,anditisreally‘theonlymystery’ofquantummechan-
ics.Withthedevelopmentofexperimentaltechniques,ithasbeenpossibleto
carryoutmanyoftheearlyGedankenexperimentsthatplayedanimportant
role in developing our understanding of the fundamentals of quantum inter-
ferenceandentanglement.Despiteitslonghistory,quantuminterferencestill
challenges our understanding, and continues to excite our imagination.
Quantum interference arises in some form or other in almost all the phe-
nomena of quantum mechanics and its applications. Obviously, we have to
be very selective in the topics we discuss here, and many important aspects
are dealt with only briefly, or not at all. In writing the book our intention
has been to concentrate on a systematic and consistent exposition of co-
herence and quantum interference phenomena in optical fields and atomic
systems and to discuss the details of the most recent theoretical and ex-
perimental work in the field. We begin in Chap. 1 by discussing the basic
principlesofclassicalandquantuminterferenceandsummarizingsomequite
elementary concepts and definitions that are frequently used in the analysis
of interference phenomena. The most important first- and second-order co-
herence effects are discussed including the welcher-weg problem, two-photon
VIII Preface
nonclassical interference, interferometric interaction-free measurements, and
quantum lithography. We also discuss important experiments that confirm
these basic interference predictions.
The mathematical formalism of quantum interference in atomic systems
is developed in Chap. 2 for multi-level and multi-atom systems in free space
andcavityenvironments.Forourpurposes,themasterequationofanatomic
system is derived in the Born−Markov and rotating-wave approximations.
The relation of the source field operators to the atomic dipole operators and
retardation effects are then discussed. In this way the correlation functions
of the electric field and their relationship to the atomic dipole operators
are developed as a basic formulation. The concept of superposition states is
then introduced in Chap. 3 and applied to three-level systems in Vee and
Lambda configurations. The concept of multi-atom entangled states is also
introduced so that one can see the relation between quantum interference
effects in multi-level and multi-atom systems. A full description of the quan-
tum beats phenomenon and its relation to quantum interference phenomena
is also included.
Chapter 4 discusses quantum interference effects induced by spontaneous
emission and the experimental evidence of spontaneously induced quantum
interference effects in a molecular multi-level system. This chapter includes
a discussion of decoherence free subspaces and the role of decoherence in the
formation of entanglement. A section on the effect of cavity and photonic
bandgap materials on spontaneous emission from an atomic system is in-
cluded here because these are examples of other practical systems to control
and suppress spontaneous emission.
The subject of coherence effects in multi-level systems is treated in
Chap. 5. The theory of two major quantum interference effects − coherent
population trapping and electromagnetically induced transparency in simple
three-level systems − are explored and described in terms of the density ma-
trixelementsofthesesystems.Theseprocessesdependonthecreationofco-
herentsuperpositionsofatomicstateswithaccompanyinglossofabsorption.
The chapter includes a general treatment of the spatial propagation of elec-
tromagnetic fields in optically dense media, and the absorption properties of
coherentlypreparedatomicsystems.Thischapteralsodiscussesapplications
of coherently prepared systems in the enhancement of optical nonlinearities
in electromagnetically induced transparency.
Material on the implementation of quantum interference is included in
Chap.6.Thischapteralsodiscussesthephasecontrolofquantuminterference
and extremely large values (superbunching) of the second-order correlation
functions. Methods for producing quantum interference effects in three-level
systems with perpendicular transition dipole moments is considered to show
howonecangetaroundthewell-knowndifficultyoffindingatomicormolec-
ularsystemswithparalleltransitiondipolemoments.Thischapterconcludes
Preface IX
with a fairly detailed description of Fano profiles, laser-induced continuum
structures and population trapping in photonic bandgap materials.
In Chap. 7 the theory of subluminal and superluminal propagation of a
weak electromagnetic field in coherently prepared media is formulated and
accompanied with many examples of the experimental observation of slow
and fast light, and the storage of photons. The concept of polaritons is then
introduced in terms of atomic and field operators.
The subject of quantum interference in a superposition of field states is
considered in Chap. 8. The phase space formalism is described and quan-
tum interference effects in phase space for several field states are discussed.
Examples of the experimental reconstruction of Wigner functions and of the
production of single-photon states are also included.
The final chapter discusses quantum interference effects with cold atoms.
This includes the subjects of diffraction of cold atoms, interference of two
Bose−Einstein condensates, collapses and revivals of an atomic interference
pattern and interference experiments in coherent atom optics.
Since this book is based to a large extent on the combined work of many
earlier contributors to the field of quantum interference, it is impossible to
acknowledge our debts on an individual basis. We should, however, like to
express our thanks to Peng Zhou who, during his stay at The Queen’s Uni-
versity of Belfast, carried out some of the work on control of decoherence
and field induced quantum interference presented in Chaps. 4 and 6. We
are greatly appreciative of the help and suggestions received from many col-
leagues, including Ryszard Tana´s, Helen Freedhoff, Peter Drummond, Bryan
Dalton, Shi-Yao Zhu, Christoph Keitel, Josip Seke, Gerhard Adam, Andrey
Soldatov,JoergEvers,TerryRudolphandUzmaAkram.Wearealsograteful
to Alexander Akulshin, Immanuel Bloch, Dmitry Budker, Milena D’Angelo,
Juergen Eschner, Edward Fry, Christian Hettich, Alexander Lvovsky, Steven
Rolston, and Lorenz Windholz for sending us originals of the reproduced
figures of their experimental results.
Brisbane, Belfast, Zbigniew Ficek
March 2004 Stuart Swain
Contents
1 Classical and Quantum Interference
and Coherence............................................ 1
1.1 Classical Interference and Optical Interferometers .......... 2
1.1.1 Young’s Double Slit Interferometer ................. 2
1.1.2 First-Order Coherence ............................ 4
1.1.3 Welcher Weg Problem ............................ 7
1.1.4 Experimental Tests of the Welcher Weg Problem ..... 11
1.1.5 Second-Order Coherence .......................... 15
1.1.6 Hanbury-Brown and Twiss Interferometer .......... 17
1.1.7 Mach−Zehnder Interferometer ..................... 19
1.2 Principles of Quantum Interference ....................... 20
1.2.1 Two-Photon Nonclassical Interference............... 21
1.2.2 The Hong−Ou−Mandel Interferometer ............. 25
1.3 Quantum Erasure ...................................... 28
1.4 Quantum Nonlocality ................................... 30
1.5 Interferometric Interaction-Free Measurements ............. 32
1.5.1 Negative-Result Measurements ..................... 33
1.5.2 Schemes of Interaction-Free Measurements........... 34
1.6 Quantum Interferometric Lithography .................... 38
1.7 Three-Photon Interference .............................. 42
1.7.1 Three-Photon Classical Interference................. 43
1.7.2 Three-Photon Nonclassical Interference ............. 44
2 Quantum Interference in Atomic Systems: Mathematical
Formalism ................................................ 47
2.1 Master Equation of a Multi-Dipole System................. 48
2.1.1 Master Equation of a Single Multi-Level Atom ....... 48
2.1.2 Master Equation of a Multi-Atom System ........... 67
2.2 Correlation Functions of Atomic Operators ................ 74
2.2.1 Correlation Functions for a Multi-Level Atom........ 74
2.2.2 Correlation Functions for a Multi-Atom System ...... 80
2.2.3 Spectral Expressions.............................. 82
XII Contents
3 Superposition States and Modification of Spontaneous
Emission Rates ........................................... 85
3.1 Superposition States in a Multi-Level System .............. 85
3.1.1 Superpositions Induced by Spontaneous Emission..... 87
3.2 Multi-Atom Superposition (Entangled) States.............. 91
3.2.1 Entanglement.................................... 91
3.2.2 Two Interacting Atoms ........................... 93
3.2.3 Entangled States of Two Identical Atoms............ 94
3.2.4 Entangled States of Two Nonidentical Atoms ........ 96
3.3 Experimental Evidence of the Collective Damping and
Frequency Shift ........................................ 104
3.4 General Criteria for Interference in Two-Atom Systems...... 110
3.4.1 Interference Pattern with Two Atoms ............... 111
3.4.2 Experimental Observation of the Interference Pattern
in a Two-Atom System............................ 113
3.5 Quantum Beats ........................................ 115
3.5.1 Theory of Quantum Beats in Multi-Level Systems .... 116
3.5.2 Quantum Beats in the Radiation Intensity from a
Multi-Level Atom ................................ 120
3.5.3 Quantum Beats in the Radiation Intensity from Two
Nonidentical Atoms............................... 126
3.5.4 Experimental Observation of Quantum Beats in a
Type I System ................................... 129
3.5.5 Quantum Beats in the Intensity−Intensity
Correlations ..................................... 131
3.6 Interference Pattern with a Dark Center................... 135
4 Quantum Interference as a Control of Decoherence ....... 139
4.1 Modified Spontaneous Emission .......................... 139
4.1.1 Effect of Environment on Spontaneous Emission...... 140
4.1.2 Modification by a Moderate Q Cavity............... 142
4.1.3 Modification by Photonic Crystals.................. 145
4.2 Quantum Interference in Vee Systems ..................... 146
4.2.1 Population Trapping and Dark States............... 148
4.2.2 Probing Quantum Interference in a Vee System ...... 150
4.3 Spectral Control of Spontaneous Emission ................. 156
4.4 Experimental Evidence of Quantum Interference............ 162
4.4.1 Energy Levels of the Molecular System.............. 162
4.4.2 Master Equation of the System..................... 163
4.4.3 Two-Photon Excitation ........................... 164
4.4.4 One- and Two-Photon Excitations.................. 166
4.5 Decoherence Free Subspaces ............................. 169
4.5.1 A Simple Example of a Decoherence Free Subspace ... 169
4.5.2 Experimental Verification of Decoherence Free
Subspaces ....................................... 171
