[Kok P., Lovett B.] Introduction to Optical Quantu(BookFi.org)
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Introduction to Optical Quantum
Information Processing
Quantum information processing offers fundamental improvements over classical infor- mation processing, such as computing power, secure communication, and high-precision measurements. However, the best way to create practical devices is not yet known. This textbook describes the techniques that are likely to be used in implementing optical quantum information processors.
After developing the fundamental concepts in quantum optics and quantum information theory, this book shows how optical systems can be used to build quantum computers according to the most recent ideas. It discusses implementations based on single photons and linear optics, optically controlled atoms and solid-state systems, atomic ensembles, and optical continuous variables.
This book is ideal for graduate students beginning research in optical quantum information processing. It presents the most important techniques of the field using worked examples and over 120 exercises.
Pieter Kok is a Lecturer in Theoretical Physics in the Department of Physics andAstronomy,
the University of Sheffield. He is a member of the Institute of Physics and the American Physical Society, and his Ph.D. thesis won the Institute of Physics Quantum Electronics and Photonics thesis award in 2001.
Brendon W. Lovett is a Royal Society University Research Fellow in the Department of
Materials and a Fellow of St. Anne’s College at the University of Oxford. He is a member of the Institute of Physics and the Materials Research Society. He has been a visiting Fellow at the University of Queensland, Australia, and is an Academic Visitor at the National University of Singapore.
Introduction to
Optical Quantum
Information Processing
Pieter Kok
University of Sheffield
Brendon W. Lovett
University of Oxford
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
© P. Kok and B. W. Lovett 2010 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2010
ISBN-13 978-0-511-77618-2 eBook (NetLibrary)
ISBN-13 978-0-521-51914-4 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
To Rose, Xander and Janet
Contents
Preface page3.6 References and further reading
3 Figures of merit
3.1 Density operators and superoperators
3.2 The fidelity
3.3 Entropy, information, and entanglement measures
3.4 Correlation functions and interference of light
3.5 Photon correlation measurements
2.5 References and further reading
Part II Quantum information in photons and atoms
4 Photon sources and detectors
4.1 A mathematical model of photodetectors
4.2 Physical implementations of photodetectors
4.3 Single-photon sources
4.4 Entangled photon sources
Part I Quantum optics and quantum information
1.5 Quantum states of the electromagnetic field
1 The quantum theory of light
1.1 The classical electromagnetic field
1.2 Quantization of the electromagnetic field
1.3 Mode functions and polarization
1.4 Evolution of the field operators
2.4 Quantum computation with continuous variables
1.6 References and further reading
2 Quantum information processing
2.1 Quantum information
2.2 Quantum communication
2.3 Quantum computation with qubits
4.5 Quantum non-demolition photon detectors
5 Quantum communication with single photons
8.6 References and further reading
8 Quantum communication with continuous variables
8.1 Phase space in quantum optics
8.2 Continuous-variable entanglement
8.3 Teleportation and entanglement swapping
8.4 Entanglement distillation
8.5 Quantum cryptography
9 Quantum computation with continuous variables
7.6 References and further reading
9.1 Single-mode optical qunat gates
9.2 Two-mode Gaussian qunat operations
9.3 The Gottesman–Knill theorem for qunats
9.4 Nonlinear optical qunat gates
9.5 The one-way model for qunats
9.6 Quantum error correction for qunats
9.7 References and further reading
10 Atomic ensembles in quantum information processing
Part III Quantum information in many-body systems
5.1 Photons as information carriers
6.3 Building quantum computers with probabilistic gates
5.2 Quantum teleportation and entanglement swapping
5.3 Decoherence-free subspaces for communication
5.4 Quantum cryptography
5.5 References and further reading
6 Quantum computation with single photons
6.1 Optical N -port interferometers and scalability
6.2 Post-selection and feed-forward gates
6.4 Photon counting and quantum memories
7.5 Other entangling gates
6.5 Threshold theorem for linear-optical quantum computing
6.6 References and further reading
7 Atomic quantum information carriers
7.1 Atomic systems as qubits
7.2 The Jaynes–Cummings Hamiltonian
7.3 The optical master equation and quantum jumps
7.4 Entangling operations via path erasure
10.1 An ensemble of identical two-level atoms
10.3 Quantum memories and quantum repeaters
References
13.6 References and further reading Appendix A Baker–Campbell–Haussdorff relations Appendix B The Knill–Laflamme–Milburn protocol Appendix C Cross–Kerr nonlinearities for single photons
13.5 Optical quantum metrology
13.4 Entanglement-assisted parameter estimation
13.3 The dynamical evolution of states
13.2 The statistical distance
13.1 Parameter estimation and Fisher information
13 Quantum metrology
12.6 References and further reading
12.5 Strong coupling effects
12.4 Overcoming decoherence
10.4 The atomic ensemble as a single qubit
12.3 The master equation for electrons and phonons
12.2 Electron–phonon coupling
12.1 Phonons
12 Decoherence of solid-state qubits
11.6 References and further reading
11.5 Scalability of solid-state devices
11.4 Entangling two-qubit operations
11.3 Interactions in solid-state qubit systems
11.2 Definition and optical manipulation of solid-state qubits
11.1 Basic concepts of solid-state systems
11 Solid-state quantum information carriers
10.6 References and further reading
10.5 Photon–photon interactions via atomic ensembles
Index
Preface
The field of quantum information processing has reached a level of maturity, and spans such a wide variety of topics, that it merits further specialization. In this book, we consider quantum information processing with optical systems, including quantum communication, quantum computation, and quantum metrology. Optical systems are the obvious choice for quantum communication, since photons are excellent carriers of quantum information due to their relatively slow decoherence. Indeed, many aspects of quantum communication have been demonstrated to the extent that commercial products are now available. The importance of optical systems for quantum communication leads us to ask whether we can construct integrated systems for communication and computation in which all processing takes place in optical systems. Recent developments indicate that while full-scale quantum computing is still extremely challenging, optical systems are one of the most promising approaches to a fully functional quantum computer.
This book is aimed at beginning graduate students who are starting their research career in optical quantum information processing, and it can be used as a textbook for an advanced master’s course. The reader is assumed to have a background knowledge in classical elec- trodynamics and quantum mechanics at the level of an undergraduate physics course. The nature of the topic requires familiarity with quantized fields, and since this is not always a core topic in undergraduate physics, we derive the quantum mechanical formulation of the free electromagnetic field from first principles. Similarly, we aim to present the topics in quantum information theory in a self-contained manner.
The book is organized as follows: in Part I, we develop the quantum theory of light, give an introduction to quantum communication and computation, and we present a num- ber of advanced quantum mechanical techniques that are essential for the understanding of optical quantum information processing. In Part II, we consider quantum information processing using single photons and atoms. We first develop the theory of photodetec- tion and explore what we mean by photon sources, followed by an exposition of quantum communication with single photons, quantum computation with single photons and linear optics, and quantum computing where the information carriers, the qubits, are encoded in atoms. In Part III, we explore quantum information processing in many-body systems. We revisit linear optical quantum communication and computation, but now in the context of quantum continuous variables, rather than qubits. We discuss how atomic ensembles can be used as quantum memories and repeaters, and we study in detail how to define robust qubits in solid-state systems such as quantum dots and crystal defects. The last chapter of the book deals with quantum metrology, where we explore how quantum states of light can be exploited to attain a measurement precision that outperforms classical metrology. As is to length restrictions. We have not included quantum information processing in ion traps, photonic band-gap materials, optical lattices, and Bose–Einstein condensates. We have also omitted the topic of quantum imaging.
We wish to thank a number of colleagues who have made valuable comments, and suggested many improvements: Charles Adams, Simon Benjamin, Samuel Braunstein, Earl Campbell, Jim Franson, Erik Gauger, Dominic Hosler, Nick Lambert, Peter van Loock, Janet Lovett, Ahsan Nazir, Todd Pittman, Nusrat Rafiq, Andrew Ramsay, Marshall Stoneham, Joachim Wabnig, David Whittaker, and Marcin Zwierz. We thank Joost Kok for suggesting the artist Victor Vasarely for the cover image. BWL thanks the Royal Society for financial support. Finally, we would like to thank Andrew Briggs and the Quantum Information Processing Interdisciplinary Research Collaboration (QIP IRC) for continued support.
PART I
QUANTUM OPTICS ANDQUANTUM INFORMATION
1 The quantum theory of light Classically, light is an electromagnetic phenomenon, described by Maxwell’s equations.
However, under certain conditions, such as low intensity or in the presence of certain nonlinear optical materials, light starts to behave differently, and we have to construct a ‘quantum theory of light’. We can exploit this quantum behaviour of light for quantum information processing, which is the subject of this book. In this chapter, we develop the quantum theory of the free electromagnetic quantum field. This means that we do not yet consider the interaction between light and matter; we postpone that to Chapter 7. We start from first principles, using the canonical quantization procedure in the Coulomb gauge: we derive the field equations of motion from the classical Lagrangian density for the vector potential, and promote the field and its canonical momentum to operators and impose the canonical commutation relations. This will lead to the well-known creation and annihilation operators, and ultimately to the concept of the photon. After quantization of the free electromagnetic field we consider transformations of the mode functions of the field. We will demonstrate the intimate relation between these linear mode transformations and the effect of beam splitters, phase shifters, and polarization rotations, and show how they naturally give rise to the concept of squeezing. Finally, we introduce coherent and squeezed states.
The first two sections of this chapter are quite formal, and a number of subtleties arise when we quantize the electromagnetic field, such as the continuum of modes, the gauge freedom, and the definition of the creation and annihilation operators with respect to the classical modes. Readers who have not encountered field quantization procedures before may find these sections somewhat daunting, but most of the subtleties encountered here have very little bearing on the later chapters. We mainly include the full derivation from first principles to give the field of optical quantum information processing a proper physical foundation, and derive the annihilation and creation operators of the discrete optical modes from the continuum of modes that is the electromagnetic field.
1.1 The classical electromagnetic field
Classical electrodynamics is the theory of the behaviour of electric and magnetic fields in the presence of charge and current distributions. It was shown by James Clerk Maxwell (1831–1879) that the equations of motion for electric and magnetic fields, the Maxwell equations, allow for electromagnetic waves. In vacuum, these waves propagate with a
−1 frequency range with light. In this section, we define the electric and magnetic fields in terms of the scalar and vector potentials, and construct the field Lagrangian density in the presence of charge and current distributions. Variation of the Lagrangian density with respect to the potentials then leads to Maxwell’s equations. Subsequently, we consider the Maxwell equations for the vacuum, and derive the wave equation and its plane-wave solutions. The source-free Lagrangian density is then used to define the canonical momenta to the potentials, which in turn allow us to give the Hamiltonian density for the free field. These are the ingredients we need for the canonical quantization procedure in Section 1.2.
The electric and magnetic fields E(r, t) and B(r, t) are related to a scalar and a vector potential (r, t) and A(r, t): ∂A(r, t) and (1.1)
E(r, t) B(r, t) = −∇ (r, t) − = ∇×A(r, t) .
∂ t The most elegant way to construct a classical field theory is via the Lagrangian density. We can use the potentials as the dynamical variables of our classical field theory, which means that we can write the Lagrangian density L as a function of the potentials and their time derivatives
L (1.2) = L ( , ˙ ; A, ˙A) .
The equations of motion for the potentials and A are then given by the Euler–Lagrange equations
d δL δL
(1.3) − = 0
dt δ
δ ˙ and
d δL δL
(1.4) − = 0 .
dt δ A
δ ˙ A j
j
Here δ denotes the functional derivative, since the potentials are themselves functions of space and time, and each component of A, denoted by A , obeys a separate Euler–Lagrange
j equation.
In the presence of a charge density ρ(r, t) and a current density J(r, t) the general Lagrangian density of classical electrodynamics can be written as
1 ε
2
2 L
(1.5)
E (r, t) B (r, t) ,
= J(r, t) · A(r, t) − ρ(r, t) (r, t) + − 2 2μ
2
2
2
2
where E and B depend on and A according to Eq. (1.1). When the ≡ |E| ≡ |B|
Lagrangian density is varied with respect to we obtain the Euler–Lagrange equation in Eq. (1.3), which can be written as Gauss’ law
(1.6) −ε ∇ · E(r, t) + ρ(r, t) = 0 . When we vary the Lagrangian density with respect to the components of A, we find the Euler–Lagrange equations in Eq. (1.4). These can be reformulated as the Maxwell–Ampère law
∂E
1 J(r, t) (r, t) (1.7) + ε − ∇×B(r, t) = 0 .
t
∂ μ The relations in Eq. (1.1) and Eqs. (1.6) and (1.7) are equivalent to Maxwell’s equations, as can be seen by taking the curl of E in Eq. (1.1): ∂B
. (1.8) ∇×E = −
t
∂ The last Maxwell equation, ∇ · B = 0, is implicit in B = ∇×A since the divergence of any curl vanishes.
It is well known that we have a gauge freedom in defining the potentials and A that constitute the fields E and B. Since we are interested in radiation, it is convenient to adopt the Coulomb, or radiation, gauge
(1.9) ∇ · A = 0 and = 0 . In addition to the gauge choice, in this chapter we consider only the vacuum solutions of the electromagnetic fields:
ρ (1.10) = 0 and J = 0 . When we now write Eq. (1.7) in terms of the potentials, we obtain the homogeneous wave equation for A
2
∂ A
2
(1.11)
A μ ∇ − ε = 0 .
2 t
∂ The classical solutions to this equation can be written as
ik k t ·r−iω
dk A (k)ǫ (k)e
λ λ
(1.12)
A(r, t)
= √ + c.c.,
3
ε 2π) 2ω
(
k λ
where A (k) denotes the amplitude of the mode with wave vector k and polarization λ, and
λ
c.c. denotes the complex conjugate. The vector ǫ gives the direction of the polarization,
λ
which we will discuss in Section 1.3. The dispersion relation for the free field is given by
2
ω
2
2 2 k
(1.13) μ ω
|k| − ε k ≡ k − = 0 ,
2 c
where c is the phase velocity of the wave with frequency ω . Any well-behaved potential
k
A(r, t) that can be expressed as a superposition of Fourier components is a solution to the
wave equation. This is exemplified by the fact that we can see different shapes, colours, etc., rather than just uniform plane waves.
Finally, the Lagrangian density can be used to find the Hamiltonian density of the field. To this end, we define the canonical momenta of and A as
δL δL and . (1.14)
A
≡ ≡ We can now take the Legendre transform of the Lagrangian density with respect to the dynamical variables ˙ and ˙ A to obtain the Hamiltonian density of the free electromagnetic field
H( , ; A, ) (1.15)
A A = ˙ + · ˙A − L .
In the Coulomb gauge, the canonical momenta are ˙A . (1.16)
= 0 and A = ε This leads to the Hamiltonian density for the free field
ε
1
2
2 H
B A
= · ˙A − L | ρ =
- E . (1.17)
=J=0
2 2μ We now have all the necessary ingredients to proceed with the quantization of the electromagnetic field.
Derive the homogeneous wave equation in Eq. (1.11) and show that the
Exercise 1.1: solutions are given by Eq. (1.12).
1.2 Quantization of the electromagnetic field
We are now ready to quantize the classical electromagnetic field. First, we have to decide which of the fields A, E (or B) we wish to quantize. In later chapters, we discuss the coupling between light and matter, and it is most convenient to express that coupling in terms of the vector potential A. We therefore apply the quantization procedure to A, rather than to E. In the quantization procedure we have to ensure that the quantum fields obey Maxwell’s equations in the classical limit, and this leads to the introduction of a modified Dirac delta function. After the formal quantization, we explore the properties of the mode functions and the mode operators that result from the quantization procedure, and establish a fundamental relationship between them. We then construct eigenstates of the Hamiltonian, and define the discrete, physical modes. This leads to the concept of the photon. The final part of this section is devoted to the construction of the quantum mechanical field observables associated with single modes.
1.2.1 Field quantization
We denote the difference between classical and quantum mechanical observables by writing the latter with a hat. In the quantum theory of light, A and then become operators
A
satisfying the equal-time commutation relations. In index notation these are written as
j
j k k ′ ′, t) , t) (1.18) ˆA (r, t), ˆ A (r ˆ (r, t), ˆ (r = A = 0 .
A The field consists of four variables: three from A and one from . We again work in the Coulomb gauge, where = 0 and ∇ · A = 0 ensures that we end up with only two dynamical variables.
Standard canonical quantization prescribes that, in addition to Eq. (1.18), we impose the following commutation relation:
j
3 ′ ′
, t) , (1.19) ˆA (r, t), ˆ (r δ (r )
i ij
= i δ − r
A j
where we must remember the difference between upper and lower indices, A ,
j
= −A because electrodynamics is, at heart, a relativistic theory. Unfortunately, given that in the
k k
Coulomb gauge , this commutation relation is not compatible with Gauss’ law in ∝ E
A ′
on both sides vacuum: ∇ · E = 0. If we take the divergence with respect to the variable r of Eq. (1.19), the left-hand side will be zero, but the divergence of the delta function does not vanish. We therefore have to modify the delta function such that its divergence does vanish. For the ordinary Dirac delta function we use the following definition:
dk ′
3 ik )
′ ·(r−rδ δ (r ) δ e . (1.20)
ij ij
− r ≡
3
( 2π) , because after the redefinition of the delta function
We have included the Kronecker delta δ
ij
the internal degree of freedom j and the external degree of freedom r may no longer be independent (in fact, they will not be). Taking the divergence of Eq. (1.20) with respect to
r yields dk ′
3 ik ) ′ ·(r−r
. (1.21) ∂ δ δ (r ) k e
i ij j
− r = i
3
2π) (
i
Therefore, we have to subtract something like this from the redefined delta function. We write
dk ′ dk ′ ik ) ik )
′ ·(r−r ·(r−r
, (1.22) (r ) δ e α k e
ij ij i j
− r = − i
3
3
2π) 2π) ( (
′
and we want to find α such that ∂ (r )
i i ij
− r = 0:
dk ′ ik )
′ ·(r−r
∂ (r ) ∂ e δ α
i ij i ij j i
− r = − ik
3
( 2π)
i i dk ′ ik )
·(r−r e ik δ )( ik )α i ij j i i
= − (ik
3
( 2π)
i
(1.23) = 0 . We therefore have that
k i ik δ k α , (1.24) i ij i j i i
- k = 0 , or α = −i
2
|k|
i
′
and the ‘transverse’ delta function becomes (r )
ij − r dk k k
′ i j ik )
′ ·(r−r
(r ) e δ . (1.25)
ij ij
− r = −
3
2
( 2π) |k|
Using this modified delta function, we can complete the quantization procedure by imposing the equal-time canonical commutation relation
j ′ ′
, t) . (1.26) ˆA (r, t), ˆ (r (r )
i ij
= i − r
A
That this leads to the correct covariant Hamiltonian and momentum is shown, for example, in Bjorken and Drell (1965). We can now write the three space components of the quantum field as
2 † ∗ ∗
, (1.27) ˆA (r, t) dk ǫ (k) (k)u(k; r, t) (k) (k)u (k; r, t)
j λ j λ
= ˆa + ǫ λ j ˆa
λ
ε
λ =1
where the u(k; r, t) are mode functions that are themselves solutions to the wave equation in Eq. (1.11), and λ again indicates the polarization of the electromagnetic field. The classical amplitudes A (k) are replaced by the operators (k), and ˆ A is now a quantum field. Note
λ ˆa λ
that ˆ A is now an operator, and unlike its classical counterpart does not directly represent a particular vector potential. Specific quantum mechanical vector potentials are represented by quantum states.
The equal-time commutation relation in Eq. (1.26) determines the commutation relation
† ∗
(k) and (k), given the mode functions u(k; r, t) and u (k, r, t). From Eq. (1.12) we
λ
for ˆa ˆa
λ
can read off the plane-wave solutions with continuum normalization:
ik t ·r−iω k e u(k; r, t) , (1.28)
=
3
( 2π) 2ω
k
where k is the wave vector of a wave with frequency ω . Plane waves are of constant
k
intensity throughout space and time, and are therefore unphysical. However, they are mathematically very convenient. When the mode functions are the plane waves defined in Eq. (1.28), we find explicitly that
†
3 ′ ′ ′ , (1.29)
(k), (k ) δ (k )
λ ′ λλ
ˆa ˆa = δ − k
λ
and
† † ′ ′
(k), ′ (k ) (k), (k ) (1.30) ˆa λ ˆa λ = ˆa ˆa ′ = 0 .
λ λ
(k) and its Hermitian conjugate are the ‘mode operators’ of the quantized The operator ˆa λ electromagnetic field. In the next section we will see that any operators that obey these commutation relations are good mode operators.
We are now done with the quantization of the classical electromagnetic field, and the remainder of this section is devoted to the exploration of the direct consequences of this procedure.
Exercise 1.2: Derive the commutation relations in Eqs. (1.29) and (1.30).
1.2.2 Mode functions and mode operators
We will now discuss some of the fundamental properties of the mode functions u(k; r, t) and
† ∗
u (k; r, t), and the mode operators (k) and (k). In order to study the mode functions of
λ
ˆa ˆa
λ
the field (its ‘shape’, if you like), we must first define a scalar product that allows us to talk about orthogonal mode functions. This is given by the ‘time-independent scalar product’
↔ ∗ ∗ ∗
(φ dr φ ∂ ψ dr φ (∂ ψ ) φ )ψ . (1.31)
t t t
, ψ) ≡ i = i − (∂ From the general structure of the scalar product in Eq. (1.31) we see that
∗ ∗ ∗
, ψ) , ψ (1.32) (φ ) = (ψ, φ) and (φ = −(ψ, φ) .
This scalar product finds its origin in the continuity equation of the field, which determines the conserved currents (see Bjorken and Drell, 1965). It is therefore time-independent. The completeness relation of the mode functions u(k; r, t) is then derived as follows: consider a function f (r, t) that is a superposition of different mode functions
∗ f dk α(k)u(k) (k) , (1.33)
= + β(k)u where we have suppressed the dependence on r and t in f (r, t) and u(k; r, t) for notational brevity. Using the orthogonality of the mode functions defined by the scalar product, we can write the coefficients α(k) and β(k) as
∗
(1.34) α(k) (k), f ) . = (u(k), f ) and β(k) = −(u
This leads to an expression for f
∗ ∗ f dk ( u(k), f ) u(k) (k), f ) u (k) . (1.35)
= − (u For a second superposition of mode functions g the scalar product (g, f ) can be written as
∗ ∗
(
g, f ) dk (
g, u(k))(u(k), f ) (k))(u (k), f ) . (1.36)
= − (g, u This constitutes the ‘completeness relation’ for the mode functions u(k; r, t), and it holds only if the mode functions are, in fact, complete.
Using the definition of the time-independent scalar product, we can show that plane-wave solutions are orthonormal: k t ik t ′
−ik·r+iω ·r−iω k′ ↔
e e
( u , u ′ ) dr ∂
k t k ≡ i
3
3
( 2π) 2ω ( 2π) 2ω ′
k k ′
) ) t −i(k−k ·r+i(ω k −ω k′ dr (ω ′ ) e k
- ω k =
3
( 2π) 2√ω ω ′
k k
3 ′
. (1.37) (k )
= δ − k
∗
We also find by direct evaluation that (u , u )
k ′ = 0. This can be understood physically as
k
the orthogonality of waves moving forward in time, and waves moving backwards in time and in opposite directions. The plane waves therefore form a complete orthonormal set of mode functions.
We can define a new set of mode functions v(κ; r, t), which are a linear combination of plane waves
ik k t ·r−iω v e
. (1.38) (κ dk V (κ, k)u(k; r, t) dk V (κ, k)
; r, t) = =
3
2π) 2ω (
k
Two symbols are needed for wave vectors here, namely k and κ. We emphasize that we will normally reserve k for describing wave vectors. When V (κ, k) is unitary, the new mode functions v(κ; r, t) are also orthonormal. When we express ˆ
A in terms of the new mode ′ , since the mode operators
(k) to ˆb (κ) functions, we should also change the operators ˆa λ λ depend on k and will generally change due to the transformation V (κ, k). The field operator then becomes
2 † ∗ ∗
′ ′ ; r, t) . (1.39)
ˆA (r, t) dκ ǫ (κ) ˆ b (κ)v(κ (κ) ˆ b (κ)v (κ
j λ j λ ; r, t) + ǫ ′ ′
= λ j
λ
ε
′ λ =1
Note that here we have also included a possible change in the polarizarion degree of freedom
′ λ , which can be incorporated straightforwardly in the time-independent scalar product.
Prove the orthonormality of v(κ; r, t) if V is unitary.
Exercise 1.3:
We next explore the precise relationship between mode functions and mode operators. The
† ∗
λ
(k) and (k) are related to the mode functions u(k; r, t) and u (k; r, t) mode operators ˆa λ ˆa
via the time-independent scalar product ε
, ˆ
′ (k) u(k)ǫ ′ A λ λ
ˆa ≡
↔
ε
∗ ∗ dr u (k; r, t) ∂ ǫ (k) (1.40) t ′ = i · ˆA(r, t) .
λ
We can then extract the operator ˆb ′ , associated with the mode function v(κ; r, t), using (κ)
λ
the procedure
↔
ε ε v ∗ ∗ ˆb ′ (κ) (κ)ǫ ′ , ˆ A dr v (κ ; r, t) ∂ ǫ (κ) (1.41)
t ′ λ ≡ λ = i · ˆA(r, t) .
λ
This is a definition of the operator ˆb ′ , and is completely determined by the mode function (κ) v λ and polarization vector ǫ ′ . Now suppose that we have an expression for ˆ
(κ) (κ) A(r, t) in
λ
(k), and polarization vectors ǫ (k) given terms of mode functions u(k), mode operators ˆa λ λ in Eq. (1.27). The mode operator ˆb ′ then becomes
(κ)
λ † ∗ ∗ ∗ ∗ v
ˆb ′ (κ) dk ǫ (κ) (k) (v, u) (k) (κ) (k) , u (k) , (1.42)
λ ′ λ λ ′
= · ǫ ˆa + ǫ · ǫ λ ˆa
λ λ λ λ
†
. The spatial where we express the mode operator ˆb in terms of mode operators ˆa and ˆa
∗
must be evaluated before the integration integration in the scalar products (v, u) and (v, u ) over k in order to make the scalar product of the two polarization vectors ǫ
′ (κ) (k) λ λ
· ǫ definite. This demonstrates that the mode operators have a notion of orthogonality that is directly inherited from the orthogonality of the mode functions. Up to addition of a complex constant, Eq. (1.42) is the most general linear transformation of the mode operators, and is called the ‘Bogoliubov transformation’. In principle it can mix the mode operators with
∗ their adjoints when the scalar product (v, u is non-zero.
) Using Eq. (1.42), show that
Exercise 1.4: †
3 ′ ′
ˆb , ˆb ′ , (1.43) (κ) (κ ) δ (κ )
λ ′ λλ
= δ − κ
λ
and
† † ′ ′
ˆb (κ) , ˆb ′ (κ ) ˆb (κ) , ˆb (κ ) (1.44)
λ λ = ′ = 0
λ λ These are the expected commutation relations for the mode operators.1.2.3 Photons as excitations of the electromagnetic field
The revolutionary aspect of the quantum mechanical description of the electromagnetic field is the notion that the field can deliver its energy only in discrete amounts. This leads to the concept of the ‘photon’. In order to derive this from the quantum theory, we first consider the Hamiltonian and momentum operators for the quantum field. We then con- struct energy eigenstates, and regularize them to obtain well-behaved physical states of the electromagnetic field.
From the quantum mechanical version of Eq. (1.17) we can formally derive the Hamiltonian operator H of the free field as
ω
k † †
H
dk (k) (k) (k) (k) λ λ
= ˆa ˆa + ˆa ˆa
λ λ
2
λ
(1.45)
dk H (k) , λ
≡
λ
where H (k) will be called the ‘single-mode Hamiltonian operator’. Similarly, the ‘field
λ
momentum operator’ is k
† † dk (k) (k) (k) (k) . (1.46)
λ λ
ˆP = ˆa ˆa + ˆa ˆa
λ λ
2
λ
This operator is similar to H, but ω is replaced by k. Therefore, the properties we derive
k
for the Hamiltonian can easily be translated into properties for the momentum. The field momentum can be formally derived from Maxwell’s stress tensor, but this is beyond the
†
(k) and (k) obey the following commutation The Hamiltonian and the operators ˆa λ ˆa
λ
relations:
† †
H and H (1.47) (k) (k) (k) (k) .
λ k λ , ˆa k
, ˆa = − ω ˆa = ω ˆa
λ λ n
We can define |ψ as an eigenstate of the Hamiltonian H such that H (1.48)
n n n
|ψ = E |ψ , with E the corresponding energy eigenvalue (which depends on ω ). From the commutation
n k
relations in Eq. (1.47) we find
† † †
H (1.49)
(k) (k) (H ) ) (k)