Cambridge University Press
978-1-107-00421-4 - The Quantum Theory of Nonlinear Optics
Peter D. Drummond and Mark Hillery
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The Quantum Theory of Nonlinear Optics
Playing a prominent role in communications, quantum science, and laser physics, quantum
nonlinear optics is an increasingly important field. This book presents a self-contained
treatment of field quantization, and covers topics such as the canonical formalism for fields,
phase-space representations, and the encompassing problem of quantization of electrodynamics in linear and nonlinear media. Starting with a summary of classical nonlinear
optics, it then explains in detail the calculation techniques for quantum nonlinear optical
systems and their applications, quantum and classical noise sources in optical fibers, and
applications of nonlinear optics to quantum information science. Supplemented by end-ofchapter exercises and detailed examples of applications to different systems, this book is
a valuable resource for graduate students and researchers in nonlinear optics, condensed
matter physics, quantum information, and atomic physics. A solid foundation in quantum
mechanics and classical electrodynamics is assumed, but no prior knowledge of nonlinear
optics is required.
Peter D. Drummond is a Distinguished Professor in the Faculty of Science, Engineering and
Technology, Swinburne University of Technology, Melbourne. His current research focuses
on ultra-cold atomic physics, quantum information, and bio-informatics.
Mark Hillery is a Professor at the Department of Physics and Astronomy, Hunter College,
and in the Physics Graduate Program, Graduate Center, City University of New York. His
research focuses on the field of quantum information.
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Cambridge University Press
978-1-107-00421-4 - The Quantum Theory of Nonlinear Optics
Peter D. Drummond and Mark Hillery
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This book forms part of an informal series of books, all of which originated as review articles published in Acta Physica Slovaca. The journal can be accessed for free
at www.physics.sk/aps.
Vladimir Buzek, editor of the journal
“This book is a valuable contribution to the scientific literature by addressing issues that fall
at the boundary between quantum optics and nonlinear optics. It is exactly at this boundary
where one might expect exciting advances to develop in the coming years. The authors have
done a good job in selecting the topics for inclusion in their very fine text.”
Robert W. Boyd
Canada Excellence Research Chair in Quantum Nonlinear Optics, University of Ottawa
Professor of Optics and of Physics, University of Rochester
“Two of the pioneers of quantum optics have produced a clear introduction to the quantum
theory of nonlinear optical processes with applications ranging from laser physics to quantum information. The powerful method of P representations to treat quantum stochastic
processes is introduced with exemplary clarity and many examples. This is an essential
introduction for graduate students, who will appreciate the carefully chosen problem sets,
and a valuable reference for experienced researchers in the field.”
Gerard J. Milburn
Centre Director, Centre for Engineered Quantum Systems, University of Queensland
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Cambridge University Press
978-1-107-00421-4 - The Quantum Theory of Nonlinear Optics
Peter D. Drummond and Mark Hillery
Frontmatter
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The Quantum Theory of
Nonlinear Optics
PETER D. DRUMMOND
Swinburne University of Technology, Melbourne
MARK HILLERY
Hunter College, City University of New York
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978-1-107-00421-4 - The Quantum Theory of Nonlinear Optics
Peter D. Drummond and Mark Hillery
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iv
University Printing House, Cambridge CB2 8BS, United Kingdom
Published in the United States of America by Cambridge University Press, New York
Cambridge University Press is a part of the University of Cambridge.
It furthers the University’s mission by disseminating knowledge in the pursuit of
education, learning and research at the highest international levels of excellence.
www.cambridge.org
Information on this title: www.cambridge.org/9781107004214
C
P. D. Drummond & M. Hillery 2014
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
Cover artwork by Igor Minarik.
First published 2014
Printed in the United Kingdom by Clays, St Ives plc
A catalogue record for this publication is available from the British Library
Library of Congress Cataloguing in Publication data
Drummond, P. D. (Peter D.), 1950–
The quantum theory of nonlinear optics / Peter D. Drummond, Swinburne
University of Technology, Melbourne, Mark Hillery, Hunter College, City University
of New York.
pages cm
Includes bibliographical references and index.
ISBN 978-1-107-00421-4 (hardback)
1. Nonlinear optics. 2. Quantum optics. 3. Quantum theory. I. Hillery,
Mark Stephen. II. Title.
QC446.2.D78 2014
535 .2 – dc23
2013030430
ISBN 978-1-107-00421-4 Hardback
Cambridge University Press has no responsibility for the persistence or accuracy of
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and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
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978-1-107-00421-4 - The Quantum Theory of Nonlinear Optics
Peter D. Drummond and Mark Hillery
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Contents
Preface
page ix
Introduction
1 Classical nonlinear optics
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Linear polarizability
Nonlinear polarizability
Frequency dependence and dispersion
Power and energy
Order-of-magnitude estimates
The two-level atom
Local-field corrections
Propagation in a nonlinear medium
Raman processes
Additional reading
Problems
2 Field quantization
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
Quantum theory
Fock space for bosons
Many-body operators
Fock space for fermions
Canonical quantization
One-dimensional string
Scattering matrix
Quantized free electromagnetic field
Constrained quantization
Exponential complexity
Additional reading
Problems
3 Quantized fields in dielectric media
3.1
3.2
3.3
3.4
Dispersionless linear quantization
Scattering in linear media
Quantizing a nonlinear dielectric
Homogeneous nonlinear dielectric
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978-1-107-00421-4 - The Quantum Theory of Nonlinear Optics
Peter D. Drummond and Mark Hillery
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Contents
vi
3.5 Inhomogeneous nonlinear dielectric
3.6 Dispersion
3.7 One-dimensional waveguide
Additional reading
Problems
4 Microscopic description of media
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
The Coulomb gauge
The multipolar gauge
Hamiltonian for a polarizable medium
Dipole-coupling approximation
Linear medium
Quantization of the linear model
Two-level atomic medium
Polaritonic limit
Additional reading
Problems
5 Coherence and quantum dynamics in simple systems
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
Photon counting and quantum coherence
Quadratures and beam-splitters
Coherent states and P-representations
Nonclassical states
Two-mode states
Mode entanglement
Parametric interactions
Anharmonic oscillator and Schrödinger’s cat
Jaynes–Cummings dynamics
Parametric approximation
Additional reading
Problems
6 Decoherence and reservoirs
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
Reservoir Hamiltonians
Absorption
Gain
Phase decoherence
Input–output relations
Photon flux and density
Two-time correlation functions
Master equations
Gain and damping rates
Driven linear cavity example
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978-1-107-00421-4 - The Quantum Theory of Nonlinear Optics
Peter D. Drummond and Mark Hillery
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Contents
vii
Additional reading
Problems
7 Phase-space distributions
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
Diffusion processes
Fokker–Planck equations
Stochastic differential equations
Phase-space representations
Wigner and Q-representations
Nonclassical representations
Operator identities and quantum dynamics
Quasi-probability Fokker–Planck equation
Linearized fluctuation theory
Functional phase-space representations
Additional reading
Problems
8 Single-mode devices
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
Linear cavity
Phase-space representation methods
Driven nonlinear absorber
Squeezing and photon anti-bunching
High-Q laser
Laser linewidth
Laser quantum state: number or coherent?
Open nonlinear interferometer
Additional reading
Problems
9 Degenerate parametric oscillator
9.1
9.2
9.3
9.4
9.5
Hamiltonian and stochastic equations
Classical results
Fokker–Planck and stochastic equations
Adiabatic approximation
Multi-mode treatment of parametric down-conversion in a cavity
Additional reading
Problems
10 Quantum field dynamics
10.1
10.2
10.3
10.4
10.5
Kerr medium
Quantum solitons
Time-dependent Hartree approximation
Quantum solitons in phase space
Parametric down-conversion
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Peter D. Drummond and Mark Hillery
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Contents
viii
10.6
Maxwell–Bloch equations
Additional reading
Problems
312
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318
11 Quantum propagation in fibers and waveguides
319
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11.1
11.2
11.3
11.4
11.5
11.6
11.7
11.8
11.9
11.10
Order-of-magnitude estimates
Waveguide modes
Dispersive energy
Nonlinear Hamiltonian
Fiber optic Hamiltonian
Raman Hamiltonian
Gain and absorption
Combined Heisenberg equations
Phase-space methods
Polarization squeezing
Additional reading
Problems
12 Quantum information
12.1
12.2
12.3
12.4
12.5
12.6
The Einstein–Podolsky–Rosen paradox
Bell inequality
Schrödinger cat paradoxes
Probabilistic simulations of Bell violations
Quantum cloning
Teleportation
Additional reading
Problems
List of symbols
Index
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978-1-107-00421-4 - The Quantum Theory of Nonlinear Optics
Peter D. Drummond and Mark Hillery
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Preface
This book grew out of our work in the field of the quantum theory of nonlinear optics.
Some of this work we have done together and some following our own paths. One major
emphasis of this work has been the quantization of electrodynamics in the presence of
dielectric media. This is a subject that is often given short shrift in many treatments, and
we felt that a book in which it receives a more extensive discussion was warranted.
M.H. would like to thank his thesis advisor, Eyvind Wichmann, for an excellent education
in quantum mechanics and quantum field theory, and M. Suhail Zubairy for introducing
him to the field of nonlinear optics with quantized fields. Others who played a major role
are Leonard Mlodinow, with whom the initial work on quantization in nonlinear media was
done, and Janos Bergou and Vladimir Buzek, long-time collaborators with whom it has
been a pleasure to work. He also thanks Carol Hutchins for many things.
P.D.D. wishes to acknowledge his parents and family for their invaluable support. The
many colleagues who helped form his approach include Crispin Gardiner and the late
Dan Walls, who pioneered quantum optics in New Zealand. Subhash Chaturvedi, Howard
Carmichael, Steve Carter, Paul Kinsler, Joel Corney, Piotr Deuar and Kaled Dechoum have
contributed greatly to this field. He also thanks Margaret Reid, who has played a leading
role in some of the developments outlined in the quantum information section, and Qiongyi
He, who provided illustrations.
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