THEORETICAL QUANTUM OPTICS - uni-freiburg.de

x

z

y

L

L

L

zR

√2ω0ω0

ω(z)

phase fronts

x

z

y

−Lx2

Lx2

Lz2

−Lz2

−Ly2

Ly2

E(r) =∫d 3k∑

λ=1,2

i

√~ωk

16π 3ε0e ik·r

ak,λek,λ + h.c.

B(r) =∫d 3k∑

λ=1,2

i

√~

16π 3ε0ωke ik·r

ak,λk × ek,λ + h.c.

HF =

∫d

3r

1

2ε0

Π2 (r) +

1

2µ0

[∇⊗ A(r)

]:[∇⊗ A(r)

]

=

∫d

3r

1

2ε0

Π2 (r) +

1

2µ0

A(r) ·∆A(r)

=

∫d

3r∑j,k

1

2ε0

ε20ωjωk

[−(((

(((((

Aj(r) ·Akajak+Aj(r) ·A∗kaja

†k

+ A∗j(r) ·Aka

†jak−

A∗j(r) ·A∗ka

†ja†k

]

+1

2µ0

ω2k

c2

[((((((((

Aj(r) ·Akajak+Aj(r) ·A∗kaja

†k

+ A∗j(r) ·Aka

†jak

+

A∗j(r) ·A∗ka

†ja†k

]

=∑j

ε0

2ω

2j

~2ωjε0

2(aja†j+ a†ja†j

)

=∑j

~ωj2

(aja†j+ a†jarj

)

=∑j

~ωj(nj +

1

2

)

E0 = 〈0|H|0〉 =∑j

~ωj2

(〈0|aj a†j |0〉︸︷︷︸

=1

+ 〈0|a†j aj |0〉︸︷︷︸=0

)=∑j

~ωj2

THEORETICALQUANTUM OPTICS

Stefan Y. BuhmannAlbert-Ludwigs University Freiburg

2017

LATEX: Joshua [email protected]

Contents

Introduction: What is quantum optics? 1

Prelude: Quantum mechanics and canonical quantization 5

1 Basic postulates of quantum mechanics 71.1 Postulates of quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.1 Postulate 1: States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.1.2 Postulate 2: Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.1.3 Postulate 3: Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 91.1.4 Postulate 4: Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . 101.1.5 Postulate 5: Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Quantum mechanics of composite systems . . . . . . . . . . . . . . . . . . . . 11

2 Canonical quantisation 152.1 Langrangian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 Hamiltonian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Poisson bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4 The correspondence principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Part I: Photons 23

3 Canonical quantisation of electrodynamics in free space 253.1 Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Lagrangian electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 Hamiltonian electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4 Poisson bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.5 The correspondence principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Normal modes, creation and annihilation operators 374.1 Normal modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1.1 Plane propagating wave in a finite free space region . . . . . . . . . . . 384.1.2 Plane propagating waves in free space . . . . . . . . . . . . . . . . . . . 414.1.3 Standing waves in a cuboid cavity . . . . . . . . . . . . . . . . . . . . . 424.1.4 Other sets of mode functions . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2 Creation and annihilation operators . . . . . . . . . . . . . . . . . . . . . . . . 45

5 The quantum vacuum and its consequences 515.1 Quantum Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2 The Casimir force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.3 Other consequences of the quantum vacuum . . . . . . . . . . . . . . . . . . . 56

6 Quantum states of the electromagnetic field 616.1 Fock states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.2 Thermal states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686.3 Coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.4 Squeezed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.5 Futher quantum states of the electromagnetic field . . . . . . . . . . . . . . . . 88

6.5.1 Field strength states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.5.2 Phase states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7 Quasiprobability distributions in phase space 917.1 P -, Q- and Wigner functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

8 Spatiotemporal coherence of the electromagnetic field 1038.1 Classical coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1038.2 Relevance of classical coherence in experiments . . . . . . . . . . . . . . . . . 1088.3 Quantum coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Part II: Atoms & Photons 116

9 Quantum coherence and its applications 1199.1 Quantum coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1199.2 No-cloning theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1209.3 Quantum cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1219.4 Quantum computing and Shor’s algorithm . . . . . . . . . . . . . . . . . . . . 125

10 Canonical quantization of electrodynamics in the presence of charged particles 12710.1 The Maxwell–Lorentz equations . . . . . . . . . . . . . . . . . . . . . . . . . . 12710.2 Lagrangian electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12910.3 Hamiltonian electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13210.4 Canonical quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13310.5 Atom-field interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

10.5.1 Minimal coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13710.5.2 Multipolar coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

11 Atom–field dynamics 14311.1 Spontaneous decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14311.2 Optical Bloch equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14811.3 Resonance fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Excursus Index

Excursus 1: Longitudinal and transverse vector fields . . . . . . . . . . . . . . . . . 27Excursus 2: Functional derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Excursus 3: Probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Excursus 4: Lie algebra of the harmonic oscillator . . . . . . . . . . . . . . . . . . . 71Excursus 5: SU(1,1) Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Excursus 6: Operator ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Introduction: What is quantum optics?

Quantum optics is a very active, dynamically growing and evolving field. Before delvinginto the details of this intriguing subject, we will in this chapter give several alternativedefinitions before giving an overview over key aspects and achievements of the field.

Alternative definitions

Let us begin by considering four different possible definitions of the term “quantum optics”which each emphasises different aspects of the field.

• Definition 1: The quantum version of classical optics

According to the Oxford Dictionary, optics is "the scientific study of sight and thebehaviour of light, or the properties of transmission and deflection of other forms ofradiation." Within physics, the emphasis in classical optics is less on sight, but more onthe behaviour of light and its interaction with matter as well as devices to manipulatelight such as mirrors, lenses etc. Quantum optics as its quantum version then dealswith aspects of light in which its quantum or particle nature becomes manifest. Theclassical devices to manipulate light become complemented by those relevant to theparticle nature such as the photodetector. In addition, the emphasis in the light-matterinteraction has a strong focus on microscopic objects such as atoms or molecules.

• Definition 2: Table–top light-matter phenomena

As an experimentally-inspired definition, one could say that quantum optics covers allthose light–matter interactions which can be studied and tested in table-top experiments.This definition again invokes associations with the common devices used in suchexperiments such as lasers, mirrors, lenses, photodetectors and atomic traps. It alsostresses one of the major appeals of the field: that they involve fundamental quantumeffects that can be studied with relatively modest experimental means—in stark contrastto strong-field quantum electrodynamics or particle physics set-ups.

• Definition 3: Non-relativistic quantum electrodynamics

This is the theoretical analogue of the previous definition. Quantum optics is regardedas the applied or non-relativistic branch of quantum electrodynamics in a limit wherethe creation of virtual electron–positron pairs can be safely neglected and the internalas well as the centre-of-mass motion of all involved particles and macroscopic objectsare much slower than the speed of light. This is a very precise theoretical definition thatplaces an emphasis on the fundamental description of light–matter interactions ratherthan on the many applications and effects emerging from these.

• Definition 4: A collection of closely related sub-fields and techniques

2 Introduction

This definition gives the most inclusive notion of quantum optics by simply subsumingan ever-growing number of sub-fields. It is useful when structuring the content of alecture such as this one. It does not, however, give a strong idea of the common essenceof all these sub-fields.

Sub-fields of quantum optics

We shall list the fields that may be included as parts of quantum optics:

• Photonics

Photonics, the science of photons, is concerned with the generation, manipulation anddetection of quantum light with a strong emphasis on applications to information andcomputer science. Photonics may be regarded as a sub-field of quantum optics, analternative term for the field of quantum optics with a slightly different focus or a fieldin its own right alongside quantum optics.

• Quantum tomography

Quantum tomography is concerned with the reconstruction of a system’s state vectoror density matrix or of the operator for a physical observable from measurementdata. When applied to photons, it may be viewed as a sub-field of quantum optics orphotonics.

Tomography is the main tool for observing the various quantum states of light thatfeature in this lecture.

• Laser physics

The laser is both main tool and key application of quantum optics. Laser physics isconcerned with the basic principles, design and applications of the laser.

Lasers will be a recurrent theme throughout this lecture.

• Cavity quantum electrodynamics

Cavity quantum electrodynamics is the study of strong atom–light interactions in aresonator or cavity. Typically, the present atoms interact reversibly with one or a fewcavity modes.

The field has successful implemented many of the model interactions that we are goingto introduce in this lecture.

• Quantum information and communication

Quantum information and communication uses photons or other quantum systems tocoherently store, manipulate and transmit quantum information.

We will examine some paradigmatic schemes of these fields towards the end of thelecture.

3

• Cold atom and ion physics

This field is concerned with techniques to cool and trap ensembles of atoms and ionsfor probing fundamental quantum phenomena. One of its greatest achievements is thephysical realisation of a Bose–Einstein condensate as a new collective quantum state ofmatter. Methods from cold atom and ion physics have been used to implement some ofthe schemes of quantum information and communication.

• Optomechanics

Optomechanics is a fast-emerging field that is concerned with the coherent interactionof light with macroscopic objects such as mirrors or cantilevers.

• Matter wave optics

As the matter complement to quantum optics, this field is concerned with the wave na-ture of particles such as electrons, atoms or molecules. Analogues of many paradigmaticsetups from light optics have been realised.

Achievements of quantum optics

The genesis of quantum optics is the result of a number of important achievements. Thesecan be exemplified by three Nobel prizes awarded for discoveries that directly laid thefoundations for the field:

• Planck (1918): Blackbody radiation

The emission spectrum of an idealised black body at finite temperature was poorlyunderstood in terms of classical physics. Planck was able to explain this spectrumby postulating that the energy of electromagnetic waves of frequency ν is quantisedin discrete packets of hν where the constant h now bears his name. Although Planckhimself considered the discretisation hypothesis as a primarily mathematical devicethat should not be taken too literally, it was the seed for our modern notion of a photon.

We will revisit the blackbody spectrum when discussing the statistic properties ofthermal light.

• Einstein (1921): Photoelectric effect

Einstein took the idea of quantised light more seriously in his explanation of thephotelectric effect: light impinging on a metal surface can liberate electrons fromthe metal. In contrast to intuition from classical physics, their kinetic energy does notdepend on the amplitude of the responsible radiation, but only on its frequency. Einsteinconcluded that the electron can absorb light of given frequency ν only in discrete quantaof hν, the photons.

The photoelectric effect is the operating principle behind the photodetector, a key devicein quantum optics that we shall encounter later on.

• Tomonaga, Schwinger, Feynman (1965): Quantum electrodynamics

The three Nobel laureates of 1965 made important contributions to quantum elec-trodynamics, from which quantum optics is derived. In particular, they introduced

4 Introduction

renormalization as a methods to obtain finite results for matter–field interactions andinvented Feynman diagrams as a powerful means to study these.

Further pioneering achievements in experiment and theory have been made since theformation of quantum optics as a field in its own right. Some of these have again beenhonoured with a Nobel prize:

• Chu, Cohen-Tannoudji, Philipps (1997): Laser cooling of atoms

Chu, Cohen-Tannoudji and Philipps invented experimental methods to cool the randomthermal motion of atoms by means of lasers, so that they can subsequently be trapped.

These techniques from the basis for detailed investigations of the atom–field interactionsthat feature prominently in many models of quantum optics.

• Cornell, Ketterle, Wieman (2001): Bose–Einstein condensation

Cornell, Ketterle and Wieman made use of laser cooling techniques to prepare anensemble of trapped atoms in a Bose–Einstein condensate, where a macroscopic fractionof the atoms condenses to the ground state of the centre-of-mass motion.

Bose–Einstein condensation has since developed into a vast field of its own right.

• Glauber, Hall, Hänsch (2005): Coherent states of light, precision spectroscopy

Glauber’s development of theoretical tools for describing quantum states of light (interalia the coherent states now carrying his name) in 1963 may be considered as thebirth of quantum optics. Hall and Hänsch used lasers to develop the frequency-combtechniques allowing for spectroscopic measurements of unprecedented precision.

We shall intensely discuss Glauber’s coherent states when introducing and comparingdifferent quantum states of light.

• Haroche, Wineland (2012): Microscopic atom-field interactions

Haroche used techniques of experimental quantum optics to study the interaction ofindividual atoms in a cavity with individual standing-wave photons. Wineland studiedsimilar interactions of trapped ions with individual quanta of light.

The paradigmatic experimental setups of Haroche and Wineland are practical realisa-tions of some of the atom–field interaction models that we shall study in the secondpart of the course.

PRELUDE:Quantum mechanics and canonical

quantization

1 What is quantum?

Basic postulates of quantum mechanics

Intended learning outcomes

The students should be able to:

• recall the basic properties of state vectors in a Hilbert space and linear operators,

• relate the Hilbert space formalism to the concept of a quantum measurement,

• be aware of the equations of motion in the Schrödinger and Heisenberg picturesas well as their formal solution and

• know how to describe a composite quantum system.

As a prerequisite to formulating a quantum theory of light, we will in the following brieflyrecapitulate the basic postulates the much simpler quantum mechanics.

1.1 Postulates of quantum mechanics

In classical mechanics, a system is described by a set of coordinates and its dynamics is givenby the Newton equations, which are second-order differential equations. When specifyingthe values of the coordinates and their time derivatives at some initial time, one can solvethese equations uniquely to find the complete trajectory of the system as described bythe coordinates at arbitrary time. Other observables can be expressed as functions of thecoordinates, so that their values at arbitrary times can also be predicted with certainty.

In quantum mechanics, the system coordinates and other observables no longer assumedeterministic values. Instead, only averages of observables as obtained from measuring alarge number of identically prepared systems are predicted by the theory. In particular, thefollowing postulates are at the core of quantum mechanics:

Postulate 1: StatesThe state of an isolated quantum system is described by a vector in a Hilbert space.Postulate 2: ObservablesAn observable is represented by a hermitian operator.Postulate 3: MeasurementsA measurement yields as a result an eigenvalue of the measured observable with the probability beingdetermined by the overlap of the respective eigenstate and the state of the system.

8 Basic postulates of quantum mechanics

Postulate 4: CommutatorsThe commutator of two hermitian operators is determined by the Poisson bracket of their classicalcounterparts.Postulate 5: DynamicsThe time evolution of a state is governed by the Schrödinger equation.

In the following, we will briefly review the concepts involved in these postulates andintroduce some basic notation.

1.1.1 Postulate 1: StatesConcepts:

• Hilbert spaceH: complex linear vector space with a scalar product.

• Complex numbers:

c1, c2, ... ∈ C . (1.1)

• Linear combinations:

c1|ψ1〉+ c2|ψ2〉 ∈ H . (1.2)

• Scalar product:

〈ψ1|ψ2〉 ∈ C . (1.3)

• Orthonormal basis:

〈ei|ej〉 = δij . (1.4)

• Completeness:

|ψ〉 =∑i

ci|ei〉 with ci = 〈ei|ψ〉 (1.5)

⇒∑i

|ei〉〈ei|= I . (1.6)

Remarks:

• Two vectors |ψ2〉 = c|ψ1〉with c ∈ C represent the same physical state.

• We usually work with normalized stated vectors: ‖|ψ〉‖= 1.

1.1.2 Postulate 2: Operators

Concepts:

• Operator A:

A :H → H|ψ〉 → A|ψ〉 .

(1.7)

• Linear operator:

A(c1|ψ1〉+ c2|ψ2〉

)= c1A|ψ1〉+ c2A|ψ2〉 . (1.8)

1.1 Postulates of quantum mechanics 9

• Adjoint operator A†:

〈φ|A|ψ〉 = 〈ψ|A†|φ〉∗ . (1.9)

• Inverse operator A−1:

A−1A = AA−1 = I . (1.10)

• Hermitian operator H :

H = H† . (1.11)

• Unitary operator U :

U †U = U U † = I . (1.12)

• Trace:

Tr(A) =∑i

〈ei|A|ei〉 . (1.13)

• Eigenvalues and eigenvectors:

A|ai〉 = ai|ai〉 ⇒ A =∑i

ai|ai〉〈ai| . (1.14)

Remarks:

• Hermitian operators have real eigenvalues. The corresponding eigenvectors can bechosen to be an orthonormal basis.

1.1.3 Postulate 3: Measurements

Concepts:

• Probabilities pi for outcome ai (normalized state):

pi = |〈ai|ψ〉|2∈ [0, 1] . (1.15)

• Average :

〈A〉 =∑i

piai =∑i

〈ψ|ai|ai〉〈ai|ψ〉 =∑i

〈ψ|A|ai〉〈ai|ψ〉 = 〈ψ|A|ψ〉 . (1.16)

• Variance:(∆A)2 = 〈ψ|

(A− 〈A〉

)2|ψ〉 = 〈ψ|A2|ψ〉 − 2〈A〉〈ψ|A|ψ〉+ 〈A〉2〈ψ|ψ〉= 〈A2〉 − 〈A〉2 .

(1.17)

• Basis representation:

A =∑i,j

cij|ei〉〈ej| with cij = 〈ei|A|ej〉 . (1.18)

Remarks:

• The probabilities for all possible outcomes add up to unity:∑i

pi =∑i

〈ψ|ai〉〈ai|ψ〉 = 〈ψ|ψ〉 = 1 (1.19)


1.1.4 Postulate 4: Commutators

Concepts:

• Commutator:

[A, B] = AB − BA . (1.20)

• Commuting operators:

[A, B] = 0 . (1.21)

• Properties:

[A, B] = −[B, A] , (anti-symmetry) (1.22)

[A+ B, C] = [A, C] + [B, C] , (additivity) (1.23)

[AB, C] = A[B, C] + [A,C] B . (product rule) (1.24)

Remarks:

• Commuting operators possess common eigenvectors:

A|a, b〉 = a|a, b〉, B|a, b〉 = b|a, b〉⇒ [A, B] = 0

(1.25)

The measurement of observable A does then not affect a subsequent measurement ofobservable B.

• The Heisenberg uncertainty relation

(∆A)2(∆B)2 ≥ 1

4|〈[A, B]〉|2 (1.26)

dictates that the variances of non-commuting observables cannot simultaneously bemade arbitrarily small.

• A state for which the equality holds in the uncertainty relation (1.26) is called a minimumuncertainty state.

1.1.5 Postulate 5: Dynamics

Concepts:

• Schrödinger equation:

i~d

dt|ψ(t)〉 = H(t)|ψ(t)〉 (1.27)

• Hamiltonian H(t): hermitian operator

1.2 Quantum mechanics of composite systems 11

• Formal solution for the Schrödinger equation: |ψ(t)〉 = U(t, t0)|ψ(t0)〉

U(t, t0) = T exp

− i

~

∫ t

t0

dτ H(τ)

≡ 1− i

~

∫ t

t0

dτ H(τ) +

(i

~

)2 ∫ t

t0

dτ1

∫ τ1

t0

dτ2 H(τ1)H(τ2)

+ ...+

(i

~

)n ∫ t

t0

dτ1

∫ τ1

t0

dτ2 · · ·∫ τn−1

t0

dτn H(τ1)H(τ2) · · · H(τn) (1.28)

= exp

− i

~

∫ t

t0

dτ H(τ)

(if H(t) commutes at different times) (1.29)

= exp

− i

~(t− t0)H

(if H is time-independent) (1.30)

• Solution in terms of energy eigenstates: H|Ei〉 = Ei|Ei〉|ψ(t)〉 =

∑i

e−i~ (t−t0)Ei〈Ei|ψ(t0)〉|Ei〉 . (1.31)

• Heisenberg picture:

〈A(t)〉 = 〈ψ(t)|A|ψ(t)〉 = 〈ψ(t0)|U †(t, t0)AU(t, t0)|ψ(t0)〉= 〈ψ(t0)|A(t)|ψ(t0)〉 (1.32)

with A(t) = U †(t, t0)AU(t, t0).

• Heisenberg picture dynamics:d

dtA(t) =

d

dt

[U †(t, t0)AU(t, t0)

]=

1

i~U †(t, t0)AHU(t, t0)− 1

i~U †(t, t0)HAU(t, t0)

=1

i~[A(t), H(t)] . (1.33)

1.2 Quantum mechanics of composite systems

In many cases, our system of interest is not an isolated system, but a part of a larger, compositesystem. In this case, the state of the subsystem can be characterised by a density matrixrather than a state vector. Some of the above concepts have to be generalised as follows toaccommodate for such a situation.Concepts:

• Composite Hilbert space:

H = HA ⊗HB . (1.34)

• Basis states:

|aibj〉 = |ai〉 ⊗ |bj〉 (1.35)

1.2 Quantum mechanics of composite systems 13

Remarks:

• Note that the dynamics of the density matrix is different from that of the observables,as the density matrix represents the state of a system rather than an observable.

2 How do we get there?

Canonical quantisation



• verify a given Langrangian by deriving the respective equations of motion,

• identify canonically conjugate coordinates and momenta and use these to con-struct a Hamiltonian,

• understand the relation between Poisson brackets and commutators and

• construct the quantum formulation of a given classical theory by means of thecorrespondence principle.

In the last chapter, we have reminded ourselves of the basic principles of quantummechanics. Whereas a classical theory is deterministic, i.e., it makes definite predictions ofall future values of a set of observables for given initial conditions, a quantum theory onlyrenders probabilities for finding certain values when measuring an observable. These can beused to predict the averags of the observables when repeating the measurement many timesfor identically prepared systems.

The theory underlying quantum optics is quantum electrodynamics. It is the quantumversion of classical electrodynamics, just as quantum mechanics is the quantum versionof classical mechanics. As a prerequisite for constructing and understanding quantumelectrodynamics, we will in the following review how quantum mechanics is obtained fromclassical mechanics by means of a heuristic procedure called canonical quantisation: theequations of motion of the classical theory are cast in a specific form in a series of steps,starting from a Lagrangian theory, introducing canonically conjugate momenta and endingup with a Hamiltonian formulation. This formulation is particularly suitable for constructingthe quantum analogue of the theory by means of the correspondence principle.

In the next chapter, we will then review classical electrodynamics and apply canonicalquantisation to obtain quantum electrodynamics as the foundation of quantum optics.

16 Canonical quantisation

Classical Mechanics

canonicalquantization

Quantum Mechanics

ClassicalElectrodynamics

canonicalquantization Quantum

Electrodynamics

X X

X ?

Focussing on mechanics for the moment, we ask ourselves the following two questions:

1) How are classical and quantum mechanics related to each other?

2) By which mechanism can this relation be guaranteed?

Answering the first question, we have seen that operators have taken the role of observ-ables in quantum mechanics, replacing the c-number variables of the classical theory. Inorder for the classical theory to emerge on the quantum average, we have to require that theequations of motion for these operators, the Heisenberg equations of motion, have the samefunctional form as the classical equations of motion. In answer to the second question, thiscan be ensured by following the procedure of canonical quantisation.

2.1 Langrangian mechanics

According to Newtonian mechanics, the dynamics of a system of point-like particles ofpositions rα = rα(t) and masses mα is described by the Newton equations of motion:

mαrα = Fα (2.1)

These second-order differential equations have a unique solution once the initial conditionsrα(t0) and rα(t0) are specified. In the Newtonian description, the forces Fα carry the maininformation determining the equations of motion.

Example: Point particle in earth’s gravitational potential

Equations of motion and initial conditions:

F = −mgez ⇔ mr = −mgez , r(t0) = r0 , r(t0) = v0 . (2.2)

Solution:r(t) = r0 + v0t−

1

2gt2ez . (2.3)

Langrangian mechanics provides an alternative description of the dynamics which isbased on an extremum principle. Here, a system is characterized by a set of generalizedcoordinates qk = qk(t) which can, but do not have to be, position variables. One defines a

2.1 Langrangian mechanics 17

Lagrangian L = L(qk, qk), typically the difference between a system’s kinetic and potentialenergies. The trajectory from an initial state qk(t0) to a final state qk(t1) then has to be suchthat the action

S =

∫ t1

t0

dt L(qk, qk) (2.4)

is an extremum, i.e., it must not change if we change the trajectory by an infinitesimal amountqk(t)→ qk(t) + δqk(t). The variation of the action is given by

0 = δS (2.5)

=

∫ t1

t0

dt∑k

(∂L

∂qkδqk +

∂L

∂qkδqk

)(2.6)

=

∫ t1

t0

dt∑k

(∂L

∂qk− d

dt

∂L

∂qk

)δqk +

∑k

∂L

∂qkδqk

∣∣∣∣∣t1

t0

(2.7)

where we have used integration by parts and assumed that the variation of the trajectoryleaves the end points fixed: δqk(t0) = δqk(t1) = 0. The requirements that the variation of theaction shall vanish for every possible variation δqk(t) leads to the Euler–Lagrange equations

d

dt

∂L

∂qk=∂L

∂qk. (2.8)

In Lagrangian mechanics, all the relevant information about the equations of motion is con-tained in the Lagrangian L(qk, qk). One advantage with respect to the Newtonian formulationis the fact that symmetries and other general properties of a system can often be identifiedmore easily. What is more important for our purpose is the fact that Lagrangian mechanics isa crucial step towards canonical quantization.


Obviously, the three generalized coordinates can be chose as q1 = x, q2 = y, q3 = z withr = (x, y, z).Expressing the particle’s kinetic energy

T =1

2mv2 (2.9)

and its potential energyV = mgz (2.10)

in terms of these coordinates, we postulate a Lagrangian

L = T − V =1

2m

3∑k=1

q2k −mgq3 . (2.11)

With this we find

d

dt

∂L

∂qk=

d

dtm

3∑l=1

ql∂ql∂qk

=d

dtm

3∑l=1

qlδlk = mqk (2.12)


and∂L

∂qk= −mgδ3k (2.13)

so that the Euler-Lagrange equations read

mqk = −mgδ3k . (2.14)

They are equivalent to the Newtonian equations of motion above.

2.2 Hamiltonian mechanics

The next step toward canonical quantization is the Hamiltonian formulation. Central ideais the description of the system not in terms of qk and qk, but instead via the pairs qk and pkwhere

pk =∂L

∂qk(2.15)

are the canonically conjugate momenta. Using a Legendre transformation, we define aHamiltonian

H = H(qk, pk) =∑k

qkpk − L (2.16)

which is typically the sum of the systems kinetic and potential energies. Given that theEuler-Lagrange equations hold, we can equivalently describe the dynamics of the system viathe Hamilton equations of motion. To derive them, we calculate the total differential of theHamiltonian in two ways. On the one hand, we have

dH =∑k

(∂H

∂qkdqk +

∂H

∂pkdpk

)(2.17)

while on the other hand, one finds

dH = d

(∑k

qkpk − L)

=∑k

(pkdqk + qkdpk −

∂L

∂qkdqk −

∂L

∂qkdqk

)(2.18)

here we have used the definition of the canonical momenta pk from Eq. (2.15). By comparingcoefficients, we find the Hamilton equations

qk =∂H

∂pk, pk = −∂H

∂qk, (2.19)

where we have used the Euler–Lagrange equations.In Hamiltonian mechanics, the state of a system is characterized by two sets of indepen-

dent variables qk and pk instead of just one set qk (with qk being determined by qk). Thisenlargement of the coordinate space is compensated for by the fact that the equations ofmotion are only first-order equations. The trajectory (qk, pk) of the system is often illustratedin a phase-space diagram.

2.2 Hamiltonian mechanics 19


Using the Lagrangian (2.11), we find the canonically conjugate momenta

pk =∂

∂qk

1

2m

3∑l=1

q2l −mgq3 , = mqk (2.20)

so that the Hamiltonian reads

H =3∑

k=1

qkpk − L =3∑

k=1

pkmpk −

[1

2m

3∑k=1

(pkm

)2

−mpq3

]=

3∑k=1

p2k

2m+mgq3 . (2.21)

The resulting Hamilton equations are

qk =∂H

∂pk=pkm

, (2.22)

pk = −∂H∂qk

= −mgδ3k . (2.23)

As seen by taking the time derivative of the first equation and substituting it into thesecond, this set of equations is equivalent to the Euler–Lagrange equations

qk =pkm

= −gδ3k . (2.24)

The phase diagram of the system looks like this:

q3

p3


2.3 Poisson bracket

As a last and crucial preparation for the canonical quantization of classical mechanics, weintroduce the Poisson bracket: for two given functions f(qk, pl) and g(qk, pk), it is defined as

f, g =∑k

(∂f

∂qk

∂g

∂pk− ∂f

∂pk

∂g

∂qk

). (2.25)

Examples for f and g include the Hamiltonian, the canonical variables qk and pk themselvesor any other observable of the system. The Poisson bracket has the following propertieswhich are reminiscent of the properties of the commutator in quantum mechanics:

• Anti-symmetry:

f, g = −g, f . (2.26)

• Additivity:

f + g, h = f, h+ g, h . (2.27)

• Product rule:

fg, h = f g, h+ f, h g . (2.28)

The canonically conjugate variables play a prominent role and their Poisson bracket is unity:

qk, pl =∑n

(∂qk∂qn

∂pl∂pn−∂qk

∂pn

∂pl∂qn

)=∑n

δknδln = δkl . (2.29)

Using the Poisson bracket, the Hamilton equations assume the unified form

qk, H =∑l

(∂qk∂ql︸︷︷︸=δkl

∂H

∂pl−∂qk

∂pl

∂H

∂ql

)=∂H

∂pk= qk , (2.30)

pk, H = −∂H∂qk

= pk . (2.31)

There are special cases of the more general dynamical equation

f =∑l

(∂f

∂qlql +

∂f

∂plpl

)=∑k

(∂f

∂ql

∂H

∂pl− ∂f

∂pl

∂H

∂ql

)= f,H (2.32)

2.4 The correspondence principle

By introducing the canonically conjugate coordinates and momenta, the Hamiltonian and thePoisson bracket, we have cast the equations of motion of classical mechanics into a form thatis strikingly similar to the Heisenberg equations of quantum mechanics.

2.4 The correspondence principle 21

Indeed, by using the correspondence principle

f −→ f , (2.33)

f, g −→ 1

i~[f , g] , (2.34)

the generalized Hamilton equation is analogous to the Heisenberg equation of motion:

˙f =

1

i~[f , H] . (2.35)

As the mathematical properties of Poisson bracket and commutators are identical and becausethe canonically conjugate variables have a particularly simple commutator,

[qk, pl] = i~δkl , (2.36)

the Heisenberg equations indeed have the same functional form as the classical equations ofmotion. It is instructive to again illustrate this with our example.


Hamiltonian of the system:

H =3∑

k=1

p2k

2m+mgq3 . (2.37)

Using the Heisenberg equation of motion (2.35) for the canonical conjugate variablesyields:

˙qk =1

i~

[qk, H

]=

1

i~

3∑l=1

1

2m

[qk, p

2l

]=

1

i~1

2m

3∑l=1

(pl[qk, pl]︸︷︷︸

=i~δkl

+ [qk, pl]︸︷︷︸=i~δkl

pl)

=pkm

, (2.38)

˙pk =1

i~

[pk, H

]=

1

i~[pk,mgq3] = −mgδ3k , (2.39)

which have the same form as Eqs. (2.22) and (2.23).

The mechanism underlying the correspondence principle is the fact that taking the com-mutators of a function with one of the canonically conjugate variables is equivalent to takingpartial derivatives with respect to the other. This is ensured by the product rule:

• Quadratic functions:

f(pk) = p2k (2.40)

⇒ [qk, f(pk)] =[qk, p

2k

]= 2i~pk = i~

∂f(pk)

∂pk. (2.41)

• Power functions:

f(pk) = pnk (2.42)

⇒ [qk, f(pk)] = i~(n− 1)pn−1k = i~

∂f(pk)

∂pk. (2.43)


• Analytic functions:

f(pk) =∞∑n=0

anpnk (2.44)

⇒ [qk, f(pk)] = i~∞∑n=0

an(n− 1)pn−1k = i~

∂f(pk)

∂pk. (2.45)

PART I:Photons

3 What is quantum electrodynamics?

Canonical quantisation ofelectrodynamics in free space



• recall the Maxwell equations and recast them into their Lagrangian and Hamilto-nian forms,

• understand the analogy between discrete and continuous degrees of freedom,

• use functional derivatives and

• apply the correspondence principle to the electromagnetic field.

Having reminded ourselves of canonical quantisation as a general procedure for turninga classical theory into a quantum theory, we next want to apply this knowledge to classicalelectrodynamics. To that end, we start by reviewing the basic equations of electrodynamics,which we will then cast into their Lagrangian and Hamiltonian forms in order to performcanonical quantisation.

3.1 Maxwell’s equations

As in the case of mechanics, we start from the equations of motion, i.e. the Maxwell equationsfor the electromagnetic field in free space in the absence of charges or currents:

∇ ·E = 0 , (3.1)∇ ·B = 0 , (3.2)

∇×E + B = 0 , (3.3)

∇×B − 1

c2E = 0 . (3.4)

26 Canonical quantisation of electrodynamics in free space

Using these equations, one can show that the following quantities are conserved:

H =

∫d3r

(ε0

2E2 +

1

2µ0

B2

), (energy) (3.5)

P =

∫d3r ε0E ×B , (momentum) (3.6)

J =

∫d3r r × (ε0E ×B) . (angular momentum) (3.7)

To demonstrate the conservation of the field energy, one calculates

H =

∫d3r

(ε0E · E +

1

µ0

B · B)

=

∫d3r

(((((

(((((

ε0c2E · (∇×B) −

1

µ0

B · (∇×E)

)= ε0c

2

∫d3r∇ · (B ×E) = 0 (3.8)

(ε0µ0 = 1/c2) where we have assumed that all fields vanish at infinity. The conservation ofthe other two quantities may be shown analogously.

3.2 Lagrangian electrodynamics

As a first step towards reformulating electrodynamics in its Lagrangian form, we need toidentify the independent variables of the system. To that end, we introduce scalar and vectorpotentials Φ andA according to

E = −∇Φ− A , (3.9)B =∇×A . (3.10)

The Maxwell equations (3.1) and (3.3) are then fulfilled by construction while the remainingtwo lead to:

−∆Φ−∇ · A = 0 , (3.11)1

c2∇Φ +

1

c2A+∇ (∇ ·A)−∆A = 0 (3.12)

where we have used∇× (∇×A) =∇(∇ ·A)−∆A.The potentials are not uniquely defined because the above electromagnetic fields are

invariant under a gauge transformation:

Φ′ = Φ− Λ , (3.13)A′ = A−∇Λ . (3.14)

In quantum optics, one almost exclusively uses the Coulomb gauge

∇ ·A = 0 . (3.15)

3.2 Lagrangian electrodynamics 27

(Note that in relativistic quantum electrodynamics, the Lorentz gauge is more conventional,because the resulting vector potential is manifestly covariant.) In Coulomb gauge, the scalarpotential satisfies the Laplace equation

∆Φ = 0 (3.16)

while the vector potential obeys a Helmholtz equation

1

c2A−∆A = 0 . (3.17)

The Laplace equation is trivially solved by Φ = 0, so the dynamical variable of our system isthe vector potential.

Noting that the vector potential in Coulomb gauge as well as the electromagnetic fieldin the absence of sources are purely transverse, let us briefly review the properties of suchfields.

Excursus 1: Longitudinal and transverse vector fields

According to Helmholtz’s theorem of vector calculus, every vector field v(r) can beuniquely decomposed into longitudinal and transverse components

v(r) = v‖(r) + v⊥(r) . (3.18)

Vector field longitudinal transverse

Illustration

Defintion ∇× v‖ = 0 ∇ · v⊥ = 0

Construction v‖/⊥(r) =

∫d3r′ δ‖/⊥(r − r′) · v(r′)

Delta functions δ‖(r) = −∇⊗∇ 1

4πr

δ⊥(r) =∇× (∇× 1)1

4πr

= δ(r) +∇⊗∇ 1

4πr


Explicit deltafunctions

δ‖(r) =1

3δ(r)

+1− 3er ⊗ er

4πr3

δ⊥(r) =2

3δ(r)

− 1− 3er ⊗ er4πr3

Definitions:

δ(r) =1

(2π)3

∫d3k eik·r , (scalar delta function) (3.19)

1ij = δij , (unit tensor) (3.20)δ(r) = 1δ(r) , (tensor delta function) (3.21)

(a⊗ b)ij = aibj , (dyadic product) (3.22)(T · a) = Tijaj . (tensor–vector product) (3.23)

We note that the longitudinal and transverse components of a vector field are non-local,i.e. v⊥(r) depends on values v(r′) with r 6= r′. The separation into longitudinal andtransverse components becomes much simpler in Fourier space

v(k) =1

(2π)3

∫d3r e−ik·r v(r) , (3.24)

where the operation is local.

Vector field Longitudinal Transverse

Illustration

k

v‖

v⊥ v

Defintion k × v‖ = 0 k · v⊥ = 0

Construction v‖ =k(k · v)

k2

v⊥ = −k × (k × v)

k2

= v − k · (k · v)

k2


Delta functionsδ‖(r) =

1

(2π)3

∫d3k

× eik·r k ⊗ kk2

δ⊥(r) =1

(2π)3

∫d3k

× eik·r(1− k ⊗ k

k2

)

The independent degrees of freedom of classical electrodynamics are the valuesA(r) ofthe vector potential at all possible points in space r with the constraint thatA(r) is transverse.To construct a Lagrangian formulation, we proceed in analogy to the classical-mechanics case,bearing in mind that a discrete and finite set of degrees of freedom has been replaced with acontinuous and infinite set.

Analogy Classical mechanics Classical electrodynamics

Degrees offreedom qk A(r)

Label k r

Illustration

k

qk

×

1×2

×

3

×

· · · · · ·

×

Nr

A(r)

Sums∑k

∫d3r

Lagrangian L(qk, qk) =∑k

L(qk, qk)L[A, A

]=∫

d3r′ L(A(r′),∇⊗A(r′), A(r′)

)where L is the Lagrangian density

Partial deriva-tives

∂L

∂qk=∑l

∂L∂ql

∂ql∂qk

=∂L∂qk

δL

δA(r)=

∫d3r′

δL

δA(r′)· δA(r′)

δA(r)

=∂L

∂A(r)−∇ · ∂L

∂(∇⊗A(r))


As a consequence of having a continuous set of degrees of freedom, partial derivativeshave to be replaced with functional derivatives, which we will introduce in the following.

Excursus 2: Functional derivative

A functional is a mapf : A 7−→ f [A] ∈ C (3.25)

which assigns c-numbers to entire vector fields

A : R3 −→ R3

r 7−→ A(r) . (3.26)

To indicate that the functional depends on the entire vector field and not just itsvalueA(r) at a given position r, we use square brackets and either drop the positionargument altogether, f [A], or we use a dummy argument r′, f [A(r′)].The functional derivative is the variation of the functional f with respect to a changein the input vector fieldA at position r. It can be defined via a difference quotient:

δf

δAk(r)= lim

h→0

f[Aj(r

′) + hδjkδ(r′ − r)

]− f [Aj(r

′)]

h. (3.27)

Here δf/δAk(r) describes a sensitivity of f with respect to a change of the kth compo-nent of the input vector fieldA(r′). We use the vector notation δf/δA(r) where[

δf

δA(r)

]k

=δf

δAk(r). (3.28)

Alternatively, we can define the functional derivative by relating an infinitesimalchange δA of the vector fieldA to the resulting change

δf = f [A+ δA]− f [A] (3.29)

via an integral equation:

δf =

∫d3r

δf

δA(r)· δA(r) . (3.30)

In this definition, δf/δA(r) is the quantity which ensures that the above equationholds for all δA.Unfortunately, these elementary definitions are not quite suitable for our purpose, be-cause we want to work exclusively with transverse functions. To ensure the functionalderivative remains within the space of transverse functions, we have to modify ourfirst definition to

δf

δAk(r)= lim

h→0

f[Aj(r

′) + hδ⊥jk(r′ − r)

]− f [Aj(r

′)]

h. (3.31)

where the transverse delta function appears in place of the ordinary delta function.Our integral definition remains valid where now δA has to be a transverse function.


To illustrate the use of these definitions, let us consider a simple example. The func-tionals

fl : A 7−→ Al(r0) (3.32)

map a vector field to the value of its lth component at a given position r0. Theirfunctional derivatives read

δflδAk(r)

= limh→0

f[Aj(r

′) + hδ⊥jkδ(r′ − r)

]− f [Aj(r

′)]

h

= limh→0

Al(r0) + hδ⊥lk(r0 − r)− Al(r0)

h= δ⊥lk(r0 − r) . (3.33)

Renaming variables, this implies that

δAi(r)

δAj(r′)= δ⊥ij(r − r′) . (3.34)

Using the short-hand notation [δA(r)

δA(r′)

]ij

=δAi(r)

δAj(r′), (3.35)

we can writeδA(r)

δA(r′)= δ⊥(r − r′) . (3.36)

Using the alternative integral definition, we find

δfl = fl[A+ δA]− fl[A] = Al(r0) + δAl(r0)− Al(r0) = δAl(r0)

!=

∫d3r

δflδAk(r)

δAk(r) (3.37)

which again impliesδfl

δAk(r)= δ⊥lk(r0 − r) . (3.38)

As in the discrete case, we make use of the Lagrangian

L[A, A

]=

∫d3r L

(A(r′),∇′ ⊗A(r′), A(r′)

)(3.39)

with L(A(r),∇⊗A(r), A(r)

)being the Lagrange density to define an action

S =

∫ t1

t0

dt L[A,∇⊗A, A

](3.40)


and we require that this action is an extremum

0 = δS =

∫ t1

t0

dt

∫d3r

[δL

δA(r)· δA(r) +

δL

δA(r)· δA(r)

]=

∫ t1

t0

dt

∫d3r

[δL

δA(r)− d

dt

δL

δA(r)

]· δA(r) +

∫d3r

δL

δA(r)δA(r)

∣∣∣∣t1t0

(3.41)

where we have assumed the variations to vanish at initial and final times δA(r, t0) ≡δA(r, t1) ≡ 0. This leads to an Euler-Lagrange equation

d

dt

δL

δA(r)− δL

δA(r)= 0 . (3.42)

To be more explicit, we need to evaluate the functional derivatives. We calculate

δL = L[A+ δA]− L[A]

=

∫d3r′ L

(A(r′) + δA(r′),∇′ ⊗ (A(r′) + δA(r′)), A(r′)

)−∫

d3r′ L(A(r′),∇′ ⊗A(r′), A(r′)

)=((((

((((((((

((((((∫

d3r′ L(A(r′),∇′ ⊗A(r′), A(r′)

)+

∫d3r′

[∂L

∂A(r′)· δA(r′) +

∂L∂(∇′ ⊗A(r′))

: (∇′ ⊗ δA(r′))

]−(((

(((((((

((((((((

∫d3r′ L

(A(r′),∇′ ⊗A(r′), A(r′)

)=

∫d3r′

[∂L

∂A(r′)· δA(r′)−∇′ · ∂L

∂(∇′ ⊗A(r′))· δA(r′)

]=

∫d3r′

[∂L

∂A(r′)−∇′ · ∂L

∂(∇′ ⊗A(r′))

]· δA(r′) (3.43)

with S : T = SijTij (Frobenius product). We have used integration by parts and the fact thatthe vector potential either vanishes at the boundary of our region of interest or obeys periodicboundary conditions. We have introduced the shorthand tensor notation[

∂L∂(∇⊗A(r))

]ij

=∂L

∂(∂iAj(r)). (3.44)

By virtue of the above integral definition (3.55), the functional derivative hence reads

δL

δA(r)=

∂L∂A(r)

−∇ · ∂L∂(∇⊗A(r))

(3.45)

The Euler-Lagrange equations in their explicit form hence read:

d

dt

∂L∂A

=∂L∂A−∇ · ∂L

∂(∇⊗A). (3.46)

3.3 Hamiltonian electrodynamics 33

We next need to find a specific Lagrangian such that the Euler-Lagrange equation is equivalentto the Helmholtz equation fot the vector potential. A possible Lagrangian density is

L =ε0

2E2 − 1

2µ0

B2 =ε0

2A2 − 1

2µ0

(∇×A

)2 . (3.47)

Using the identity (a× b)2 = a2b2 − (a · b)2, we calculate∫d3r

(∇×A

)2=

∫d3r

[(∇⊗A

):(∇⊗A

)−(∂iAj

)(∂jAi

)](3.48)

=

∫d3r

[(∇⊗A

):(∇⊗A

)−((((((

(((∇ ·[(A ·∇)A

]+((((

((((((

A ·∇)(∇ ·A

)](3.49)

=

∫d3r

(∇⊗A

):(∇⊗A

)(3.50)

where the second term can be transformed to a (vanishing) surface integral and the thirdterm vanishes due to the Coulomb gauge. We can hence use the simpler Lagrangian density

L =ε0

2A2 − 1

2µ0

(∇⊗A

):(∇⊗A

). (3.51)

To verify that this is a correct Lagrangian density, we calculate

d

dt

∂L∂A

= ε0A , (3.52)

∂L∂A

= 0 , (3.53)

∇ · ∂L∂(∇⊗A)

= − 1

µ0

∆A , (3.54)

which, when substituted into the Euler-Lagrange equation, leads to

ε0A =1

µ0

∆A . (3.55)

This is indeed the Helmholtz equation for the vector potential.

3.3 Hamiltonian electrodynamics

Having established the Lagrangian formulation as the basis for canonical quantization, ournext step is the Hamiltonian reformulation. Again, we establish an analogy between thediscrete and continuous cases:


Canonical momenta pk =∂L

∂qkΠ(r) =

δL

δA(r)

Legendre transformation H =∑k

qkpk − L H =

∫d3r′ A(r′) ·Π(r′)− L


With our choice of Lagrangian, we find canonically conjugate momenta

Π(r) =δL

δA(r)= ε0A(r) . (3.56)

Physically, the canonically conjugate momenta for the free electromagnetic field are related tothe electric field:

Π(r) = −ε0E(r) . (3.57)The Legendre transformation then results in a Hamiltonian

H = H[A,Π

]=

∫d3r′

[A(r′) ·Π(r′)− ε0

2A2(r′) +

1

2µ0

[∇′ ⊗A(r′)

]:[∇′ ⊗A(r′)

]]=

∫d3r′ H

(A(r′),∇′ ⊗A(r′),Π(r′)

)(3.58)

with a Hamilton density

H =1

2ε0

Π2 +1

2µ0

[∇⊗A

]:[∇⊗A

]. (3.59)

Recall that we can equivalently use

H =1

2ε0

Π2 +1

2µ0

[∇×A

]2 . (3.60)

Physically, the Hamiltonian is energy of the (transverse) electromagnetic field:

H =

∫d3r

(ε0

2E2 +

1

2µ0

B2

). (3.61)

The Hamilton equations

A(r) =δH

δΠ(r), Π(r) = − δH

δA(r)(3.62)

can be written in the more explicit form

A(r) =∂H

∂Π(r), Π(r) = − ∂H

∂A(r)+∇ · ∂H

∂(∇⊗A(r)). (3.63)

With our choice of Hamiltonian, they read

A(r) =1

ε0

Π(r) , (3.64)

Π(r) =1

µ0

∆A(r) . (3.65)

Taking the time derivative of the first Hamilton equation and combining the two, we againrecover the Helmholtz equation (3.17):

A(r) =1

ε0µ0

∆A(r) . (3.66)

3.4 Poisson bracket

The final concept that we need to transfer to the continuous case is the Poisson bracket:

3.5 The correspondence principle 35


Poisson bracket f, g =∑k

(∂f

∂qk

∂g

∂pk− ∂f

∂pk

∂g

∂qk

) f, g =

∫d3r′

(δf

δA(r′)· δg

δΠ(r′)

− δf

δΠ(r′)· δg

δA(r′)

)

Fundamental Pois-son bracket

qk, pl = δkl A(r),Π(r′) = δ⊥(r − r′)

Let us first evaluate the fundamental Poisson brackets

A(r),Π(r′) =

∫d3r′′

[δA(r)

δA(r′′)· δΠ(r′)

δΠ(r′′)−δA(r)

δΠ(r′′)·δΠ(r′)

δA(r′′)

](3.67)

=

∫d3r′′ δ⊥(r − r′′) · δ⊥(r′′ − r′) (3.68)

= δ⊥(r − r′) . (3.69)

Using the Poisson brackets, the Hamilton equations can be written in the compact form

A(r) = A(r), H , (3.70)

Π(r) = Π(r), H (3.71)

which are special cases of the general dynamical equation

f = f,H . (3.72)

3.5 The correspondence principle

As in the case of quantum mechanics, we can now use the correspondence principle

f −→ f , (3.73)

f, g −→ 1

i~[f , g] , (3.74)

as our last step of canonical quantization. All observables hence become operators. In thecase of electrodynamics, one such field A(r) is a whole continuum of operators (as opposedto the finite and discrete set of operators qk). The canonically conjugate variables obey thefundamental commutation relations[

A(r), Π(r′)]

= i~δ⊥(r − r′) , (3.75)[A(r), A(r′)

]=[Π(r), Π(r′)

]= 0 . (3.76)


The Hamiltonian

H =

∫d3r

1

2ε0

Π2(r) +1

2µ0

[∇⊗ A(r)

]:[∇⊗ A(r)

](3.77)

generates the following Heisenberg equations of motion

˙A(r) =

1

i~

[A(r), H

]=

1

i~

∫d3r′

1

2ε0

[A(r), Π2(r′)

]=

1

i~

∫d3r′

1

ε0

[A(r), Π(r′)

]· Π(r′) =

1

i~

∫d3r′

1

ε0

i~δ⊥(r − r′) · Π(r′)

=1

ε0

Π(r) , (3.78)

˙Π(r) =

1

i~

[Π(r), H

]=

1

i~

∫d3r′

1

2µ0

[Π(r),

[∇′ ⊗A(r′)

]:[∇′ ⊗A(r′)

]]=

1

i~

∫d3r′

1

µ0

[Π(r), A(r′)

]·(−∆′A(r′)

)=

1

µ0

∫d3r′ δ⊥(r − r′) ·∆′A(r′)

=1

µ0

∆A(r) . (3.79)

Together, they yield the operator-valued Helmholtz equation

¨A(r) =

1

ε0µ0

∆A(r) . (3.80)

As in the case of quantum mechanics, our Hamiltonian reformulation together with thealgebraic properties of the Poisson bracket have ensured that the Heisenberg equations ofmotion are the operator-valued analogue of the classical equations of motion.

4 What are the solutions of quantumelectrodynamics?

Normal modes, creation andannihilation operators



• construct some basic solutions of the electromagnetic field,

• know a range of different mode functions for the electromagnetic field,

• express the quantised electromagnetic field terms of mode creation and annihila-tion operators and

• understand the analogy between the electromagnetic field and a set of harmonicoscillators.

In the previous chapter, we have applied the procedure of canonical quantisation toelectrodynamics. The result is a quantum field theory described in terms of operator-valuedcanonically conjugate variables. Their dynamics and that of all other observables can beobtained from the Hamiltonian of the electromagnetic field by means of the Heisenbergequation of motion, where the arising commutators can be evaluated by means of thecanonical commutation relations.

However, two central ingredients are still missing from a fully practical quantum theory.Firstly, we require explicit solutions of the Heisenberg equations of motion, or the operator-valued Maxwell equations, respectively. Secondly, we need to construct the concrete Hilbertspace of quantum states on which the operators for the electromagnetic field can act. In thischapter, we will mainly address the first question while getting a first glimpse at the answerto the second one.

As seen in the last chapter, the operator-valued vector potential satisfies the Helmholtzequation

∆A(r, t)− 1

c2

¨A(t) = 0 (4.1)

with the constraint∇ · A(r, t) = 0 (4.2)

as imposed by the Coulomb gauge. As in the classical case, we may solve the Helmholtz

38 Normal modes, creation and annihilation operators

equation via separation of variables:

A(r, t) =∑j

[Aj(r)aj(t) +A∗j(r)a†j(t)

](4.3)

where the purely classical mode functionsAj(r) must satisfy

∆Aj(r) +ω2j

c2Aj(r) = 0 (4.4)

and the operator-valued coefficients aj(t) evolve in time according to

¨aj(t) + ω2j aj(t) = 0 . (4.5)

4.1 Normal modes

For a given region of interest V with well-defined boundary conditions on ∂V , the solutionto the Helmholtz equation in frequency space form a complete set of functions which can bechosen to be orthogonal. We hence have∫

V

d3rAj(r) ·A∗k(r) = |cj|2δjk . (4.6)

The normalization constants are usually chosen to be

cj =

√~

2ωjε0

. (4.7)

In this way, one can ensure that the expansion for the vector potential takes the simple formabove where the operators aj and a†j obey Bosonic commutation relations. We will confirmthis in the next section.

The completeness of the modes means that any classical transverse vector potentialA(r)that obeys the boundary conditions at the surface ∂V of our region of interest V can bewritten as

A(r) =∑j

[Aj(r)aj +A∗j(r)a∗j

](4.8)

with expansion coefficients

aj =1

2|cj|2∫V

d3rA∗j(r) ·A(r) . (4.9)

In the following, let us consider a few common examples for mode functions.

4.1.1 Plane propagating wave in a finite free space region

An infinite free-space region can be problematic due to the absence of a well-defined boundarysurface. To avoid this problem, we use a trick invented by Rayleigh and consider instead a

4.1 Normal modes 39

cubic box of length L and volume V = L3 for which we impose periodic boundary conditions:r ∈ [−L/2, L/2]3,

A(x = −L/2, y, z) = A(x = L/2, y, z) , (4.10)A(x, y = −L/2, z) = A(x, y = L/2, z) , (4.11)A(x, y, z = −L/2) = A(x, y, z = L/2) (4.12)

and similarly for the derivatives∇⊗A.

x

z

y

-L2

L2

L2

-L2

-L2

L2

The simplest solutions to the Helmholtz equation that obey these boundary conditions areplane propagating waves:

Aj(r) = Aj eik·r . (4.13)

Substituting into the Helmholtz equation yields the dispersion relation

k2 =ω2j

c2. (4.14)

In addition, the periodic boundary conditions restrict the allowed wave vectors to discretevalues

k =2π

Ln , n ∈ Z3 (4.15)

so thatωn = c

2π

L

√n2x + n2

y + n2z (4.16)

while the Coulomb gauge implies that

k ·Aj = 0 . (4.17)

The wave is hence transverse: it oscillates in the plane perpendicular to the propagationdirection. For a given propagation direction ek = k/k, we can choose two transversepolarization vectors ek,λ (λ = 1, 2) of unit length such that the three vectors form a completeorthonormal basis:

ek ⊗ ek +∑λ=1,2

ek,λ ⊗ e∗k,λ = 1 (4.18)


With these polarization vectors, we can write the solutions to the wave equation as

An,λ =

√~

2ωnε0Ven,λ e2πin·r/L (4.19)

so that their generalized mode index is j = (n, λ). They are normalized such that∫V

d3rAn,λ(r) ·A∗n′,λ′(r) =~

2ε0V√ωnωn′

en,λ · e∗n′,λ′3∏i=1

∫ L2

−L2

dxi e2πi(ni−n′i)xi/L

︸︷︷︸=V δnn′

=~

2ε0ωnen,λ · e∗n,λ′︸︷︷︸

=δλλ′

δnn′ = |cn,λ|2δnn′δλλ′ (4.20)

as required.Let us briefly comment on the possible polarizations of the modes. Without loss of

generality, we assume that the wave vector k points along the negative z-direction. Therespective classical plane wave reads (eλ ≡ ek,λ)

Ak,λ(r, t) ∝

Re[eλ,x ei(kzz−ωt)

]Re[eλ,y ei(kzz−ωt)

]0

. (4.21)

Writing eλ in its most general form as a Jones vector

eλ =

cos θ eiϕx

sin θ eiϕy

0

, (4.22)

we find that

Ak,λ(r, t) ∝

cos θ cos(kzz + ϕx − ωt)sin θ cos(kzz + ϕy − ωt)

0

. (4.23)

From this result, we can read off the behaviour of the field vector as the wave propagates todistinguish different types of polarization:

• Linear polarization: ϕx = ϕy.

The field vector follows a straight line. We distinguish horizontal (θ = 0) and vertical(θ = π/2) as well as two diagonal polarizations (θ = ±π/4).

x

Horizontal

y

x

Vertical

y

x

Diagonal

y

4.1 Normal modes 41

• Circular polarization: ϕx = ϕy ± π/2 , θ = π/4.

The field vector maps out a circle. We distinguish left- vs right-handed circular polar-ization or clockwise vs anti-clockwise. In our convention, this refers to the motion ofthe field vector when moving along the propagation direction of the wave. (Note thatthe opposite convention also exists).

x

Circular (anti-clockwise)

y

x

Circular (clockwise)

y

x

Elliptical

y

• Elliptical polarization: all other cases. The field vector maps out an ellipse.

4.1.2 Plane propagating waves in free space

A truly infinite region can be modelled by starting from a finite volume with periodicboundary conditions and then letting the dimension of the volume tend to infinity. In thislimit, the wave vector takes arbitrary continuous values and mode sums have to be replacedby integrals. From n = L/(2π)k, one sees that∑

n

=V

(2π)3

∫d3k . (4.24)

It is then useful to renormalize the modes to

Ak,λ(r) =

√~

16π3ε0ωkek,λ eik·r (4.25)

with generalized mode index j = (k, λ) such that∫d3rAk,λ(r) ·A∗k′,λ′(r) =

~2ε0√ωkωk′

ek,λ · e∗k′,λ′1

(2π)3

∫d3r ei(k−k′)·r

=~

2ε0ωkδ(k − k′)δλλ′ . (4.26)

The completeness relation reads∫d3k

∑λ

Ak,λ(r)⊗A∗k,λ(r′) =~

2ε0ωk

1

(2π)3

∫d3k

∑λ=1,2

ek,λ ⊗ e∗k′,λ′︸︷︷︸=1−ek⊗ek

eik·(r−r′)

=~

2ε0ωk

1

(2π)3

∫d3k

(1− k ⊗ k

k2

)eik·(r−r′)

=~

2ε0ωkδ⊥(r − r′) (4.27)


In other words, the complete sum of modes is proportional to the transverse delta function,which is the unit operator in the space of transverse functions.

4.1.3 Standing waves in a cuboid cavity

As a less trivial example, let us consider a cuboid cavity of dimensions Lx, Ly, Lz which isbound by perfectly conducting walls. The perfect-conductor boundary conditions requirethe parallel components of the electric field to vanish. By virtue of E(r) = −A(r), the samemust hold for the vector potential:

Ay(r) = Az(r) = 0 for x = 0, Lx , (4.28)Ax(r) = Az(r) = 0 for y = 0, Ly , (4.29)Ax(r) = Ay(r) = 0 for z = 0, Lz . (4.30)

x

z

y

L

L

L

As a result, the plane-wave solutions become standing waves. The wave vector takes discretevalues

k =

(πnxLx

,πnyLy

,πnzLz

)Twith n ∈ N3 (4.31)

with associated eigenfrequencies

ωn = c

√(πnxLx

)2

+

(πnyLy

)2

+

(πnzLz

)2

. (4.32)

As before, we can chose two orthogonal polarization unit vectors en,λ for each wave vector.The modes can then be written as

An,λ(r) =

√4~

ε0ωnV

en,λ,x cos

(πnxLxx)

sin(πnyLyy)

sin(πnzLzz)

en,λ,y sin(πnxLxx)

cos(πnyLyy)

sin(πnzLzz)

en,λ,z sin(πnxLxx)

sin(πnyLyy)

cos(πnzLzz)

. (4.33)

4.1 Normal modes 43

Note that no modes exist for the case of two or more ni = 0, as the respective modes simplyvanish. If only one ni = 0, then exactly one mode exists, take for instance nx = 0. Then

An,λ(r) =

√4~

ε0ωnV

en,λ,x sin

(πnyLyy)

sin(πnzLzz)

0

0

. (4.34)

For the mode not to vanish, the polarization vector hence needs to point in the x-direction.

4.1.4 Other sets of mode functions

Depending on the geometry of the problem at hand, a vast range of possible sets of modefunctions exist.

• Spherical vector wave functions:In the case of spherical symmetry (e.g. when the electromagnetic waves have their originin a point-like emitter), it is useful to employ spherical coordinates r, θ, ϕ. The solutionto the Helmholtz equations are then the spherical vector wave functions Alm(r, θ, ϕ).They carry an orbital angular momentum that is parametrised by l and m, and canbe expressed in terms of spherical Bessel functions jl(kr) with k = ω/c and sphericalharmonics Ylm(θ, ϕ).

• The fundamental Gaussian mode:In contrast to plane waves, realistic electromagnetic waves typically have a finite extentin the transverse direction. A prototypical solution of this kind is the fundamentalGaussian mode. Assuming that it propagates in the z-direction and is polarized in thex-direction, it can explicitly by given as (ρ =

√x2 + y2)

A0(x, y, z) =

√2

π

1

w(z)e− ρ2

w2(z) eik

[z+ ρ2

2R(z)

]−iΨ0(z)

ex (4.35)

with

w(z) = w0

√1 +

(z

zR

)2

, (width along beam)

w0 , (beam waist)

zR =kw2

0

2, (Rayleigh length)

R(z) = z +z2R

z, (curvaturve radius of wave fronts)

Ψ0 = arctan

(z

zR

). (Gouy’s phase)

The relevant parameters of the fundamental Gaussian mode care illustrated in thefollowing sketch:


w(z)

zzR

√2w0

w0

phase fronts

As suggested by its name, the Gaussian mode has a Gaussian intensity profile:

|A0(r)|2 =2

πw2(z)e− 2ρ2

w2(z) . (4.36)

Deviating from our usual convention, the mode is normalized according to∫ ∞−∞

dx

∫ ∞−∞

dy |A0(r)|2 = 1 . (4.37)

The Rayleigh length is the characteristic length scale over which the functions ω(z), R(z)and Ψ0(z) vary. As we have kw0 1 for typical realistic Gaussian beams, this lengthis much greater than the wavelength, zR 1/k, so that all three functions are slowlyvarying.

• Hermite–Gauss modes:Hermite–Gauss modes are a generalization of the fundamental Gaussian mode inCartesian coordinates, where the x- and y-dependences of the mode factorize. They aregiven by

Am,n(x, y, z) =√2

π

1

ω(z)Hm

(√2x

w(z)

)e− x2

w2(z) eik x2

2R(z) Hn

(√2y

w(z)

)e− y2

w2(z) eik y2

2R(z) eikz−iΨmn(z) ex

(4.38)

with

Hn(x) = (−1)n ex2 dn

dxne−x

2

, (Hermite polynomials)

Ψmn(z) = (1 +m+ n) arctan

(z

zR

). (generalized Guoy’s phase)

The transverse intensity profile of some Hermite-Gauss modes is illustrated by thefollowing pictures:

4.2 Creation and annihilation operators 45

m/n 0 1 2

0

1

2

• Laguerre–Gauss modes: Similarly, Laguerre–Gauss modes generalize the fundamentalGaussian modes in polar coordinates ρ, φ in the xy-plane:

An,m(ρ, φ, z) =

√2

π

1

w(z)

[ρ

w(z)

]mLmn

(2ρ2

w2(z)

)e−imφ e

ik

[z+ ρ2

2R(z)

]−iΨmn(z)

ex (4.39)

with Lmn (z) being the generalized Laguerre polynomials. In contrast to the Hermite–Gauss modes, Laguerre–Gauss modes carry orbital angular momentum as parametrizedby m.

We have hence seen that the spatial structure of the quantized electromagnetic field isgoverned by mode functions which are solutions to the classical Helmholtz equation andparametrized by a set of continuous (k, k, ...) and or discrete (λ,m, n, ...) parameters.

Accordingly, the sum over the generalized collective mode index in

A(r, t) =∑j

[Aj(r)aj(t) +A∗j(r)a†j(t)

](4.40)

represents sum and/or integrals. Similarly, the delta in the orthonormality relation∫V

d3rAj(r) ·A∗k(r) = |cj|2δjk (4.41)

represents a product of Kronecker deltas and/or delta functions.

4.2 Creation and annihilation operators

Let us next turn our attention to the objects aj and a†j which carry both the operator natureand the time dependence of the electromagnetic field. Solving the equation of motion

¨aj(t) + ω2j aj(t) = 0 , (4.42)


the latter is given by

aj(t) = e−iωjt aj (4.43)

a†j(t) = eiωjt a†j (4.44)

where aj = aj(t0), a†j = a†j(t0) denotes the respective operators in the Schrödinger picture.In order to find the commutation relations of aj and a†j , we have to use our knowledge of

the commutation relations for the canonically conjugate variables A and Π. We hence haveto invert

A(r, t) =∑j

[Aj(r)aj e−iωj(t−t0) +A∗j(r)a†j eiωj(t−t0)

], (4.45)

Π(r, t) = −ε0

∑j

[iωjAj(r)aj e−iωj(t−t0)−iωjA

∗j(r)a†j eiωj(t−t0)

], (4.46)

where we have recalled that Π = −ε0E = ε0˙A. This is achieved by multiplying the above

equations withAj(r) orA∗j(r), integrating over space and using the orthonormality of themodes. One finds

aj =1

~

∫V

d3rA∗j(r) ·[ε0ωjA(r) + iΠ(r)

], (4.47)

a†j =1

~

∫V

d3rAj(r) ·[ε0ωjA(r)− iΠ(r)

]. (4.48)

Recalling the canonical commutation relations[A(r), Π(r′)

]= i~δ⊥(r − r′) , (4.49)[

A(r), A(r′)]

=[Π(r), Π(r′)

]= 0 , (4.50)

we hence derive[aj, a

†j′

]=

1

~2

∫V

d3r

∫V

d3r′ A∗j(r) ·−iε0ωj

[A(r), Π(r′)

]+ iε0ωj′

[Π(r), A(r′)

]·Aj′(r)

=ε0(ωj + ωj′)

~

∫V

d3r

∫V

d3r′ A∗j(r) · δ⊥(r − r′) ·Aj′(r′)

=ε0(ωj + ωj′)

~

∫V

d3rA∗j(r) ·Aj′(r)︸︷︷︸= ~

2ωjε0δjj′

= δjj′ (4.51)

and similarly [aj, aj′

]=[a†j, a

†j′

]= 0 . (4.52)

The operators aj and a†j for a given mode hence fulfil the bosonic commutation relations thatwe know from the harmonic oscillator in quantum mechanics.


The analogy goes even further, as we can see by expressing the Hamiltonian in terms ofaj and a†j :

HF =

∫d3r

1

2ε0

Π2 +1

2µ0

[∇⊗ A

]:[∇⊗ A

]=

∫d3r

1

2ε0

Π2 − 1

2µ0

A ·∆A

=

∫d3r

∑j,k

1

2ε0

ε20ωjωk

[−(((((((Aj ·Akaj ak +Aj ·A∗kaj a†k +A∗j ·Aka

†j ak −

A∗j ·A∗ka†j a†k]

+1

2µ0

ω2k

c2

[((((

(((Aj ·Akaj ak +Aj ·A∗kaj a†k +A∗j ·Aka†j ak +

A∗j ·A∗ka†j a†k]

=∑j

ε0

2ω2j

~2ωjε0

2(aj a†j + a†j aj

)=∑j

~ωj2

(aj a†j + a†j aj

)=∑j

~ωj(nj +

1

2

)(4.53)

with nj = a†j aj . Here we have used partial integration and recalled the orthonormality of thefield modes: ∫

d3r Aj(r) · A∗k(r) = |cj|2δjk =~

2ωjε0

δjk , (4.54)∫d3r Aj(r) · Ak(r) = 0 if ωj 6= ωk . (4.55)

When decomposing the quantized electromagnetic field into normal modes and expressingit in terms of the bosonic operators aj and a†j , the total field energy is that of a collectionof harmonic oscillators. Our choice of mode normalization factors cj has ensured that thebosonic commutation relations for a and a† as well as the Hamiltonian assume their simplestforms without additional factors.

With the above Hamiltonian, the equations of motion for a and a† can be derived in aparticularly straightforward manner:

˙aj =1

i~

[aj, H

]=

1

i~

[aj,∑k

~ωk(a†kak +

1

2

)]=

1

i~∑k

~ωk[aj, a

†k

]︸︷︷︸=δjk

ak = −iωj aj , (4.56)

˙a†j =1

i~[a†j, H

]= iωj a

†j , (4.57)

confirming our earlier solutions:

aj(t) = e−iωj(t−t0) aj , (4.58)

a†j(t) = eiωj(t−t0) aj . (4.59)

The particular simple form of the Hamiltonian relies on the fact that the chosen modesare energy eigenfunctions. Other observables take a similarly simple form when using theappropriate eigenfunctions: for instance, we can express the total field momentum in termsof plane waves (which are momentum eigenfunctions):

P =

∫d3r ε0E × B = −ε0

∫d3r

˙A× (∇× A) = −ε0

∫d3r (∇⊗ A) · ˙

A . (4.60)


Substituting

A(r) =

∫d3k

∑λ=1,2

√~

16π3ε0ωk

[eik·r ak,λek,λ + e−ik·r a†k,λe

∗k,λ

](4.61)

and using1

(2π)3

∫d3r ei(k−k′)·r = δ(k − k′) (4.62)

as well asek,λ · e∗k,λ′ = δλλ′ , (4.63)

one finds

P =

∫d3k

∑λ=1,2

~k(nk,λ +

1

2

). (4.64)

Each plane-wave mode hence contributes a momentum which is an integer multiple of ~k.Finally, let us analyse the total angular momentum of the field:

J =

∫d3r r × (ε0E × B) = −ε0

∫d3r r ×

[˙A× (∇× A)

]= −ε0

∫d3r

[r × (∇⊗ A) · ˙

A− r × (˙A ·∇)A

]= −ε0

∫d3r

r × (∇⊗ A) · ˙A− (

˙A ·∇)(r × A) +

˙A · (∇⊗ r︸︷︷︸

=1

)× A

= −ε0

∫d3r

˙A ·

(r ×∇

)A+ ε0

∫d3r A× ˙

A

≡ L+ S (4.65)

where we have used partial integration together with the fact that ∇ · ˙A = 0. Using the

plane-wave mode expansion above, one finds

J =

∫d3r

~2

−i [(k ×∇k)ak] · a†k + iak × a†k

+ h.c. (4.66)

where h.c. denotes the hermitian conjugate and we have defined

ak =∑λ=1,2

akek,λ , (4.67)

a†k =∑λ=1,2

a†ke∗k,λ . (4.68)

The first term in Eq. (4.65) acts on the wave vector k and represents orbital angular momen-tum, while the second term involves the polarization unit vectors and hence represents spin.Let us investigate the spin of the electromagnetic field in more detail. The contribution of asingle plane wave of given wave vector k and polarization ek,λ to the spin is given by

Sk,λ =i~2ek,λ × e∗k,λ −

i~2e∗k,λ × ek,λ . (4.69)


It vanishes for linearly polarized light with ek,λ = e∗k,λ. For circularly polarized light withk = kez and ek,λ = 1/

√2(1, i, 0)T , we find

Sk,λ =i~4

1i0

× 1−i0

− i~4

1−i0

×1

i0

= ~ez (4.70)

The spin hence correctly represents the rotation of the polarization unit vector along thepropagation direction of the wave.

As an alternative to the mode creation and annihilation operators, it is common to use thedimensionless quadrature operators

Xj =1

2(aj + a†j) , Yj =

i

2(aj − a†j) . (4.71)

They are hermitian dimentionless position and momentum operators. By inverting the aboverelations,

aj = Xj − iYj , a†j = Xj + iYj , (4.72)

one can reexpress the field Hamiltonian in term of these operators as

H =∑j

~ωj(a†j aj +

1

2

)=∑j

~ωj(X2j + Y 2

j − i[Xj, Yj

]+

1

2

)=∑j

~ωj(X2j + Y 2

j

)(4.73)

where we have used the commutator[Xj, Yj

]=

i

4

[aj + a†j, aj − a†j

]= − i

2. (4.74)

The Heisenberg uncertainty relation

(∆A)2(∆B)2 ≥ 1

4|〈[A, B]〉|2 (4.75)

hence takes the form(∆Xj)

2(∆Yj)2 ≥ 1

16. (4.76)

5 What is the simplest state of theelectromagnetic field?

The quantum vacuum and itsconsequences



• know the properties of the quantum electrodynamic vacuum,

• calculate the Casimir force between two perfectly conducting plates and

• be aware of a number of quantum vacuum phenomena.

In the last chapter, we formulated explicit solutions for the operator-valued electromag-netic field in the absence of free charges or currents. We have seen that these can be givenin terms of mode functions and bosonic creation and annihilation operators. Having thuselaborated on the properties of the operators describing the field, we must next ask ourselves:What is the Hilbert space of quantum states on which these operators act? We begin in thischapter with the simplest possible state, the ground state of the electromagnetic field.

5.1 Quantum Vacuum

This so called quantum vacuum is defined by

aj|0〉 = 0 ∀j . (5.1)

In other words, each of the modes constituting the electromagnetic field is in it ground state.The energy of the ground state is

E0 = 〈0|H|0〉 =∑j

~ωj2

(〈0|aj a†j|0〉︸︷︷︸

=1

+ 〈0|a†j aj|0〉︸︷︷︸=0

)=∑j

~ωj2

, (5.2)

which is infinite. To understand the origin of this infinite ground-state energy, we recall thatthe electric and magnetic fields are non-commuting observables: the canonical commutationrelation [

A(r), Π(r′)]

= i~δ⊥(r − r′) (5.3)

52 The quantum vacuum and its consequences

together with E = −Π/ε0 and B =∇× A implies[E(r), B(r′)

]=

i~ε0

∇× δ(r − r′) (5.4)

where we have used∇× δ⊥ =∇× δ because of∇× δ‖ = 0. According to the Heisenberguncertainty relation, it is hence impossible to simultaneously determine the electric andmagnetic fields to arbitrary precision. In particular, they cannot simultaneously be completelyzero. Instead, both fields are random quantities whose average may vanish on the ground-state average ⟨

E(r)⟩

=⟨B(r)

⟩= 0 (5.5)

but which exhibit quantum fluctuations around this average. As a result, the energy⟨H⟩

=

∫d3r

⟨ε0

2E2(r) +

1

2µB2(r)

⟩≡∫

d3r 〈H(r)〉 (5.6)

can also never vanish completely and assumes the above infinite ground-state value in-stead. One can however identify a finite spectral energy density ρ0(ω) of the ground-stateelectromagnetic field. Using our mode expressions

E(r) =

∫d3k

∑λ=1,2

i

√~ωk

16π3ε0

eik·r ak,λek,λ + h.c. , (5.7)

B(r) =

∫d3k

∑λ=1,2

i

√~

16π3ε0ωkeik·r ak,λk × ek,λ + h.c. , (5.8)

one finds⟨H(r)

⟩=

∫d3k

∑λ=1,2

[ε0

2

~ωk16π3ε0

ek,λ · e∗k,λ +1

2µ0

~16π3ε0ωk

(k × ek,λ) · (k × e∗k,λ)︸︷︷︸k2ek,λ·e∗k,λ

]=

~8π3

∫d3k ωk =

∫ ∞0

dω ρ0(ω) (5.9)

with a spectral density

ρ0(ω) =~ω3

2π2c3. (5.10)

The respective vacuum fluctuations of the electric field alone in a given frequency interval[ω, ω + dω] read ⟨

E2(r)⟩

[ω,ω+dω]=

~ω3

2π2ε0c3dω . (5.11)

At first glance, one may be tempted to dismiss the infinite ground-state energy of the electro-magnetic field, because it represents a constant contribution to the energy. Indeed, redefiningthe Hamiltonian according to

H =∑j

~ωj(nj +

1

2

)−→

∑j

~ωjnj (5.12)

leaves the equations of motion unchanged. However, there is a number of observablephenomena which can be attributed to the quantum vacuum fluctuations.

5.2 The Casimir force 53

5.2 The Casimir force

The most famous consequence of the quantum vacuum is an attractive force that arisesbetween two neutral metal plates. It was first predicted by H. B. G. Casimir in 1948 for thecase of two infinitely extended, parallel and perfectly conducting plates [1]. Following hisargument, let us consider a cuboid cavity as introduced in the previous chapter. With themodes having frequencies

ωn = c

√(πnxLx

)2

+

(πnyLy

)2

+

(πnzLz

)2

, (5.13)

the vacuum energy of the electromagnetic field inside the cavity reads

E0 =∑j

~ωj2

=~c2

∑n∈N3

∑λ=1,2

√(πnxLx

)2

+

(πnyLy

)2

+

(πnzLz

)2

. (5.14)

Let us say we are interested in the Casimir force between the plates at z = 0 and z = Lz. Tothat end, we assume that Lz Lx, Ly.

Lx

LyLz

In this case, kx = πnx/Lx and ky = πny/Ly become quasi-continuous, so that we may replace

∞∑nx,ny

−→ Lxπ

Lyπ

∫ ∞0

dkx

∫ ∞0

dky . (5.15)

The vacuum energy hence reads

E0 =~cLxLy

2π2

∫ ∞0

dkx

∫ ∞0

dky

∞∑nz=0

∑λ=1,2

√k2x + k2

y +

(πnzLz

)2

=~cA4π

∫dk‖ k‖

∞∑nz=0

∑λ=1,2

√k2‖ +

(πnzLz

)2

(5.16)

where we have introduced polar coordinates in the kx-ky-plane (kx = k‖ cosϕ, ky = k‖ sinϕ,ϕ ∈ [0, π/2]) and noted that A = LxLy is the area of each of the two opposing plates. Recallingthat two polarizations exist for each nz 6= 0, but only one for nz = 0, the sum over λ gives

E0(L) =~cA2π

∫ ∞0

dk‖ k‖∞∑′

n=0

√k2‖ +

(πnL

)2

(5.17)

[1] H. B. G. Casimir, On the attraction of two perfectly conducting plates, Proc. K. Ned. Wet. 51, 793 (1947)[Casimir48].


with n ≡ nz, L ≡ Lz and where the primed sum indicates that the n = 0 term carries a weightof 1/2.

We have hence obtained an expression for the vacuum energy between two perfectlyconducting plates which depends on their area and separation. In principle, the negativegradient of this energy should render the Casimir force. However, our vacuum energy is stillinfinite. In order to obtain a finite result, we employ a two-step procedure commonly used inquantum field theories:

1 Cut-off regularization:

The vacuum-energy divergence stems from the unbounded contributions from largewave vectors (ultraviolet divergence). This divergence is due to our unrealistic as-sumption of plates which are perfect conductors, reflecting the electromagnetic field atall frequencies. Any realistic material ultimately becomes transparent in the limit oflarge frequencies, so that the respective photons are not influenced by the plates andhence cannot contribute to the Casimir force. To account for this fact, we introduce anexponential cut-off function exp(−λpk) which suppresses contributions from photonswhose wave vector exceeds the plasma wavelength λp = 2π/ωp. The resulting Casimirenergy

E0(L) =~cA2π

∫ ∞0

dk‖ k‖∞∑′

n=0

e−λp

√k2‖+(πn/L)2

√k2‖ +

(πnL

)2

(5.18)

is finite for any given positive λp, where the perfect-conductor limit is recovered viaλp → 0+.

2 Renormalization:

The regularized vacuum energy contains a large, physically unobservable self-energythat arises as each plane individually interacts with the quantum vacuum. The self-energy of the individual plates can be found in the limit L→∞. Here, the mode sumin the z-direction also becomes quasi continuous and we find

E0(L→∞) =~cA2π

∫ ∞0

dk‖ k‖L

π

∫ ∞0

dkz e−λp

√k2‖+k

2z

√k2‖ + k2

z . (5.19)

We renormalize the vacuum energy by subtracting this self-energy from the aboveresult:

E0(L) −→ ~cA2π

∫ ∞0

dk‖ k‖

[ ∞∑′

n=0

e−λp

√k2‖+(πn/L)2

√k2‖ +

(πnL

)2

−Lπ

∫ ∞0

dkz e−λp

√k2‖+k

2z

√k2‖ + k2

z

]. (5.20)

Finally, we perform the perfect-conductor limit L λp by means of the Euler-Maclaurinformula

∞∑′

n=0

f(n)−∫ ∞

0

dx f(x) = − 1

12f ′(0) +

1

720f ′′′(0) + ... (5.21)

5.2 The Casimir force 55

With

f(n) =~cA2π

∫ ∞0

dk‖ k‖ e−λp

√k2‖+(πn/L)2

√k2‖ +

(πnL

)2

= −π2~cA4L3

∫ ∞n2

dx e−λpπ√x/L√x (5.22)

(where x = (k‖L/π)2 + n2, dk‖k‖ = π/(2L2) dx) one finds

f ′(n) = −π2~cA2L3

n2 e−λpπn/L , (5.23)

f ′′(n) = −π2~cA2L3

(2n− λpπ

Ln2

)e−λpπn/L , (5.24)

f ′′′(n) = −π2~cA2L3

(2− 4

λpπ

Ln+

λ2pπ

2

L2n2

)e−λpπn/L , (5.25)

so that

f ′(0) = f ′′(0) = 0 , f ′′′(0) = −π2~cAL3

. (5.26)

The Casimir energy of two perfectly conducting plates is hence

E0(L) = −π2~cA

720L3, (5.27)

corresponding to an attractive Casimir force per unit area

F (L)

A= − π2~c

240L4ez . (5.28)

Remarks

• Dependence on natural constants:

The Casimir energy between two perfectly conducting plates depends on the naturalsconstants ~ and c, signifying its quantum origin and its dependence on electromagneticwaves. It does not seem to depend on the electronic coupling strength, i.e. the electroncharge e. However, we note that the perfect-conductor limit implicitly assumes a limitof infinite charge density: ω2

p = e2η/(ε0me), so λp = 2πc/ωp → 0+ implies η → ∞. Weare hence in a certain limit where the dependence on e has reached a plateau.

• Distance-dependence:

The Casimir force for perfect conductors is governed by a pure 1/L4 power law. It fallsextremely rapidly with increasing distance. To observe it, electrically neutral metal ordielectric objects with very smooth surfaces have to be brought into close proximity [2].It can become a dominant effect at short ranges of 100nm – 10µm.

[2] S. K. Lamoreaux, Demonstration of the Casimir force in the 0.6 to 6µm range, Phys. Rev. Lett. 78, 5 (1997)[Lamoreaux97].


• Cut-off independence:

The leading contribution in λp/L is cut-off independent, i.e. it does not depend on theparticular shape of the ultraviolet cut-off used. This is not true for the higher-ordercorrections in λp/L that are relevant for finite λp and small distances.

• Realistic materials:

For real materials, the Casimir force is determined by the specific frequency-dependentreflectivities of the plates. The resulting force is typically below the perfect-conductorresult which is recovered in the large distance limit for metals. For short distances, a1/L3 power law is observed instead of the 1/L4 behaviour.

• Sign of the force:

For non-magnetic objects in free space at zero temperature, the Casimir force is alwaysattractive and decreasing with distance. The fact that the Casimir force is unavoidabledue to its origin the ground-state quantum fluctuations causes the notorious problemof stiction in nano technology. Repulsive forces would hence be desirable. These arepredicted when a purely magnetic object interact with a purely electric one. However,due to the additional presence of attractive electric-electric force components, this effecthas not been observed to date.

• Physical sign of the force:

Our calculation of the Casimir force suggests that it is due to vacuum fluctuations ofthe electromagnetic field. An alternative approach developed by Lifshitz derives theCasimir force between real dielectric or metallic materials from dipole fluctuationsinside the plates [3]. These fluctuating dipoles inside one of the plates give rise toelectromagnetic fields which then exert a force on the polarizable matter of the otherplate. It seems to be generally true that for each physical phenomenon derived fromvacuum fluctuations of the electromagnetic field, there is an alternative description orderivation of the effect in terms of ground-state fluctuations of polarizable matter.

5.3 Other consequences of the quantum vacuum

A range of vacuum-related phenomena exist, which we shall only briefly review in thefollowing. One class of phenomena stems from the interaction of the vacuum field with

[3] E. M. Lifshitz, The theory of molecular attractive forces between solids, Sov. Phyis. JETP 2, 73 (1956)[Lifshitz56].

5.3 Other consequences of the quantum vacuum 57

microscopic matter in free space; these belong to the most precisely tested predictions ofquantum electrodynamics:

• Anomalous magnetic moment of the electron:

The spin S of a single electron gives rise to a magnetic moment

m =gµB~S (5.29)

where µB = e~/(2m) is the Bohr magneton and g is the electron g-factor. It has the valueg = 2 in classical electrodynamics. As first predicted by Schwinger [4], the presence ofvacuum fluctuations changes the electron’s magnetic moment which can be interpretedas a change in the electron g-factor.

+ =

• Lamb shift of an atom:

The interaction of the electric dipole moment of an atom with the vacuum electricfield leads to a shift of all its internal energy eigenstates, the Lamb shift [5]. It can beinterpreted as a quadratic AC Stark shift induced by the quantum vacuum.

+ =

• Spontaneous decay of an excited atom:

As postulated by Einstein, an atom in an excited energy eigenstate will spontaneouslydecay to lower-energy states with a rate governed by the Einstein A-coefficient [6].Roughly speaking, spontaneous decay can be viewed as decay induced by the fluctuat-ing vacuum field.

+ =

[4] J. Schwinger, On quantum-electrodynamics and the magnetic moment of the electron, Phys. Rev. Lett. 73, 416 (1948)[Schwinger48].

[5] W. E. Lamb and R. C. Retherford, Fine structure of the hydrogen atom by a microwave method, Phys. Rev. 72, 241(1947) [Lamb47].

[6] A. Einstein, Zur Quantentheorie der Strahlung, Phys. Z. 18, 121 (1917) [Einstein1917.pdf].


• Van der Waals force between two atoms:

As calculated by Casimir and Polder, the interaction of two atoms via the vacuum fieldleads to an effective electromagnetic force between them, the van der Waals force [7].

+ =

Another class of vacuum-induced phenomena arises in the presence of macroscopicobjects and is hence derived from the quantum vacuum in the presence of boundaries. In thiscase, the emerging phenomena typically become position–dependent. The prime exampleis obviously the Casimir force. It is worth mentioning two other environment-dependenteffects:

• Casimir-Polder force between an atom and a body:

When an atom is placed near a macroscopic object such as a perfectly conducting plate,then its Lamb shift for a given internal states becomes position dependent. As recog-nized by Casimir and Polder [7], such a position-dependent energy can be interpretedas a potential, giving rise to the Casimir-Polder force between an atom an a metalor dielectric body. It is typically attractive for ground-state atoms interacting withnon-magnetic bodies

z

U(z)

• Purcell effect:

As predicted by Purcell, the rate of spontaneous decay also becomes position-dependentin the presence of macroscopic bodies [8]. While always remaining positive, the rate canbe significantly enhanced or suppressed in this way, leading to very long-lived excitedstates or a rapid emission of photons.

[7] H. B. G. Casimir and D. Polder, The influence of retardation on the London-van der Waals force, Phys. Rev. 73, 360(1948) [CasimirPolder48].

[8] E. M. Purcell, Spontaneous emission probabilities at radio frequencies, Phys. Rev. 69, 681 (1946) [Purcell46].

5.3 Other consequences of the quantum vacuum 59

z

Γ(z)

6 How can we describe the state of theelectromagnetic field?

Quantum states of the electromagneticfield



• construct Fock states as eigenstates of the number operator,

• derive the Bose–Einstein statistics of the thermal state,

• identify coherent states as eigenstates of the annihilation operator, and

• know the properties of squeezed states.

In the previous chapter, we have intensively discussed the quantum vacuum as the groundstate of the electromagnetic field. Although it is responsible for a number of phenomena,quantum optics is of course concerned with a number of different states in which the electro-magnetic field can be prepared. We will learn the mathematical description of a few of themand their properties in this chapter.

6.1 Fock states

To describe arbitrary quantum states of the electromagnetic field, we require a set of (ideallyorthonormal) basis states. The most obvious choice of such basis states are the eigenstates ofthe Hamilton operator

H =∑j

~ωj(nj +

1

2

). (6.1)

We will start with the eigenstates for a given mode j of the electromagnetic field: nj ≡ n,ωj ≡ ω. The eigenstates of the Hamiltonian can obviously constructed from eigenstates of therespective single-mode number operator

n = a†a . (6.2)

To solve the eigenvalue relationn|n〉 = n|n〉 (6.3)

62 Quantum states of the electromagnetic field

we first make use of the bosonic commutation relation[a, a†

]= 1 (6.4)

to show that

nak|n〉 = akn|n〉+[a†a, ak

]|n〉 = akn|n〉+

[a†, ak

]︸︷︷︸=−kak−1

a|n〉 = (n− k)ak|n〉 . (6.5)

This implies that application of ak to an eigenstate |n〉 again yields an eigenstate

ak|n〉 ∝ |n− k〉 . (6.6)

Similarly one can show that (a†)k |n〉 ∝ |n+ k〉 . (6.7)

For a given eigenstate |n〉, we can hence construct new eigenstates |n− k〉 and |n+ k〉 whoseeigenvalues differ by positive and negative integers from that of |n〉. The operators a and a†

that accomplish this are hence referred to as lowering and raising operators or annihilationand creation operators, respectively. This procedure cannot be applied indefinitely. Therelation

n = n〈n|n〉 = 〈n|a†a|n〉 = (〈n|a†) (a|n〉) ≥ 0 (6.8)

shows that n is non-negative. So n can only assume non-negative integer values n = 0, 1, 2, ...Furthermore, the above relation shows that

a|n〉 =√n|n− 1〉 (6.9)

if both states |n〉 and |n− 1〉 are to normalized. Repeatedly applying a hence terminates withthe ground state of eigenvalue n = 0:

a|0〉 = 0 . (6.10)

This is the quantum vacuum discussed in the previous chapter. Similarly, one can show that

a†|n〉 =√n+ 1|n+ 1〉 . (6.11)

All eigenstates of the number operator can be obtained by repeated operation of the creationoperator:

|n〉 =

(a†)n

√n!|0〉 . (6.12)

The Fock states [1] from a complete orthonormal basis set for the Hilbert space of a singlemode of the quantum electromagnetic field:

〈m|n〉 = δmn ,∑n

|n〉〈n|= 1 . (6.13)

[1] V. A. Fock, Kofigurationsraum und zweite Quantelung, Z. Phys. 75, 622 (1932) [Fock32].

6.1 Fock states 63

An arbitrary given state |Ψ〉 can hence be expanded in terms of Fock states as

|Ψ〉 =∑n

〈n|Ψ〉|n〉 (6.14)

wherepn = |〈n|Ψ〉|2 (6.15)

is the probability to find n photons of frequency ω when the field is in state |Ψ〉. Beforediscussing the statistics of the electromagnetic field when it is prepared in a Fock state, let usreview some common probability distributions.

Excursus 3: Probability distributions

3.1 Binomial distributionThe binomial distribution

pk(p,N) =

(N

k

)pk(1− p)N−k (6.16)

describes the probability of finding exactly k positive outcomes in a series of N inde-pendent random events, were the (constant) probability for a positive outcome in asingle event is p. The average number of positive outcomes is

n =N∑k=0

kpk =N∑k=0

k

(N

k

)pk(1− p)N−k = p

d

dp

N∑k=0

(N

k

)pkqN−k

∣∣∣∣∣q=1−p

= pd

dp(p+ q)N

∣∣∣∣q=1−p

= Np(p+ q)N∣∣q=1−p = Np . (6.17)

Similarly, one can show that

n2 = pd

dpp

d

dp(p+ q)N−1

∣∣∣∣q=1−p

= N2p , (6.18)

so that the variance is given by

∆n2 = (n− n)2 = n2 − n2 = N2p−N2p2 = N2p(1− p) . (6.19)

3.2 Poisson distribution

For a given number of events N , the number of positive outcomes k cannot exceedN . The binomial distribution is hence not very suitable for describing photon numberstatistics, where the observed photon number for a measurement can take arbitrarynon-negative values. To account for this, we perform the limit N →∞with n = Npfixed. In this limit we may approximate(

N

k

)=N(N − 1) · · · (N − k + 1)

k!' Nk

k!(6.20)


and

(1− p)N−k ' (1− p)N =N∑l=0

(N

l

)(−p)l '

N∑l=0

1

l!(−Np)l ' e−Np (6.21)

so that the binomial distribution approaches a Poisson distribution:

pk(n) =nke−n

k!. (6.22)

The following figure illustrates this limit where we have fixed n = 3 for the Poisson dis-tribution and the binomial distributions have been plotted for N = 7, 8, 10, 20, 50, 500with p = n/N .

0 5 10 15 20k

pkBinomial distributions

Poisson distribution

Again, we determine the average

n =∞∑k=0

kpk =∞∑k=0

knk e−n

k!= e−n n

d

dn

∞∑k=0

nk

n!= e−n n

d

dnen = n . (6.23)

Calculating

n2 =∞∑k=0

k2nk e−n

k!= e−n n

d

dnn

d

dnen = n(1 + n) = n2 + n , (6.24)

we find the variance∆n2 = n2 − n2 = n2 + n− n2 = n . (6.25)

It is one of the defining features of the Poisson distribution that its average and varianceare equal. Other probability distributions are classified as super- or sub-Poissonian iftheir variance is greater or smaller that the average.

6.1 Fock states 65

As originally conceived by Poisson, the Poisson distribution applies to situationswhere independent events occur with a constant probability rate such that the averagenumber of events in a given time interval is n.

3.3 Normal distribution

In the limit of large n, the discrete character of the Poisson distribution becomesless and less important. In the vicinity of n, the Poisson distributions is hence welldescribed by a continuous distribution, the normal or Gaussian distribution. To seethis, we introduce a continuous variable

x = n(1 + δ) (6.26)

with n 1, δ 1. Using Stirling’s formula

x! =√

2πx e−x xx , (6.27)

we then find

p(x) =nx e−n

x!=

nn(1+δ) e−n√2π en(1+δ)[n(1 + δ)]n(1+δ)+1/2

=enδ(1 + δ)−n(1+δ)−1/2

√2πn

=e−nδ

2/2

√2πn

=1√2πn

exp

[−(x− n)2

2n

](6.28)

where we have used

ln[(1 + δ)−n(1+δ)−1/2

]= [−n(1 + δ)− 1/2] ln(1 + δ)

' −n(1 + δ)

(δ − δ2

2

)for n 1 , δ 1

' −nδ − nδ2

2. (6.29)

The Poisson distribution hence approaches a normal or Gaussian distribution

p(x;µ, σ) =1√

2πσ2exp

[−(x− µ)2

2σ2

](6.30)

with average µ = n and variance σ2 = n.The following figure illustrates how the Poisson distribution approaches a normaldistribution for large n:


0 10 20 30 40 50k/x

pk/p(x)

Poisson distributionGaussian distributions

Poisson and Gaussian distributions for n = 5, 20, 40

After this little digression, let us determine the average and variance of key observablesfor a field mode prepared in a Fock state |n〉. The average photon number n is given by

〈n〉 = 〈n|n|n〉 = n〈n|n〉 = n . (6.31)

Calculating

〈n2〉 = 〈n|n2|n〉 = n2〈n|n〉 = n2 , (6.32)

one finds a variance

(∆n2) = 〈n2〉 − 〈n〉2 = n2 − n2 = 0 . (6.33)

The photon number statistics for a Fock state is hence an extreme example of a sub-Poissoniandistribution. This is rather obvious, because the probability distribution reads

pk = |〈k|n〉|2= δkn . (6.34)

There is hence only one possible outcome for the photon number, so that the distribution hasa vanishing variance. The following figure illustrates this by comparing the photon numberstatistics for a Fock state |n = 3〉 (black) with the corresponding Poissonian distribution(white)

6.1 Fock states 67

0 2 4 6 8 10

1

k

pk

Other observables of interest are the electromagnetic fields, such as the electric or magneticfields or the vector potential. The contribution of a single mode to these field operators canbe given as

F = Ca+ C∗a† . (6.35)

The average of a field operator vanishes for a Fock state:

〈F 〉 = 〈n|(Ca+ C∗a†)|n〉 = C√n 〈n|n− 1〉︸︷︷︸

=0

+C∗√n+ 1 〈n|n+ 1〉︸︷︷︸

=0

= 0 . (6.36)

This shows that a classical electromagnetic field is not simply a field prepared in a Fock statewhith large photon number n.

The field variance on the other hand does not vanish for a Fock state and it reads

(∆F )2 = 〈F 2〉 − 〈F 〉2︸︷︷︸=0

= 〈n|(Ca+ C∗a†)(Ca+ C∗a†)|n〉

= C2 〈n|a2|n〉︸︷︷︸=0

+|C|2〈n|(aa† + a†a)|n〉+ C∗2 〈n|(a†)2|n〉︸︷︷︸=0

= |C|2〈n|(2n+ 1)|n〉 = (2n+ 1)|C|2 . (6.37)

So the field fluctuations grow with increasing photon number and become proportional to thelatter in the large-n limit. At the opposite extreme, for n = 0, we find the vacuum fluctuations

(∆F )2 = |C|2 (6.38)

which we have extensively discussed in Chapter 5.One particular set of field operators are the quadrature operators

X =1

2(a+ a†), Y =

i

2(a− a†) . (6.39)


The above general findings imply that

〈X〉 = 〈Y 〉 = 0 , (6.40)

(∆X)2 = (∆Y )2 =1

4+n

2. (6.41)

The Heisenberg uncertainty relation holds with

(∆X)2(∆Y )2 =1

16(1 + 2n)2 ≥ 1

16, (6.42)

showing that only the state |n = 0〉 is a minimum uncertainty state for the quadratureoperators.

Fock states can easily be generalized to the multimode case: we introduce product states

|n1, n2, ...〉 =⊗j

|nj〉 (6.43)

withnj|nj〉 = nj|nj〉 . (6.44)

These states are eigenstates of the total photon number operator

N =∑j

nj, N |n1, n2, ...〉 = N |n1, n2, ...〉 (6.45)

with eigenvaluesN =

∑j

nj . (6.46)

They are also eigenstates of the Hamiltonian of the free electromagnetic field:

H|n1, n2, ...〉 =∑j

~ωj(nj +

1

2

)|n1, n2, ...〉 =

∑j

~ωj(nj +

1

2

)|n1, n2, ...〉 . (6.47)

The multimode Fock states form an orthonormal basis for the Hilbert space of the electro-magnetic field:

〈n1, n2, ...|m1,m2, ...〉 =∏i

δnjmj ,∏j

∞∑nj=0

|n1, n2, ...〉〈n1, n2, ...|= 1 (6.48)

6.2 Thermal states

A state which arises quite naturally for equilibrium systems as a consequence of statisticalphysics is the thermal state of the electromagnetic field at temperature T . Concentratingon the statistics of a single field mode, the thermal state is described by a density matrixaccording to the Boltzmann distribution

ρT =e−HF /(kBT )

Tr[

e−HF /(kBT )] =

e−~ω(n+1/2)/(kBT )

Tr[

e−~ω(n+1/2)/(kBT )] =

e−~ωn/(kBT )

Tr[

e−~ωn/(kBT )] (6.49)

6.2 Thermal states 69

where kB is the Boltzmann constant. Note that the vacuum energy does not affect the thermalstate, as the respective factors exp(−~ω/(2kBT )) cancel. The thermal density matrix can beexpanded in terms of Fock states as

ρT =∞∑

m,n=0

|m〉〈m|ρT |n〉〈n|=∞∑n=0

pn|n〉〈n| . (6.50)

As |n〉 is an eigenstate of the field Hamiltonian and thus also of ρT , the thermal state is anincoherent (i.e., purely diagonal) superposition of energy eigenstates. The probability offinding n photons is given by

pn = 〈n|ρT |n〉 =〈n|e−~ωn/(kBT )|n〉∑∞k=0〈k|e−~ωn/(kBT )|k〉 =

e−~ωn/(kBT )∑∞k=0 e−~ωk/(kBT )

= e−~ωn/(kBT )[1− e−~ω/(kBT )

](6.51)

where we have used the geometric sum∞∑k=0

qk =1

1− q (6.52)

with q = exp(−~ω/(kBT )).We can now easily evaluate the average photon number for a thermal state:

〈n〉 = Tr[ρT n] =∞∑n=0

npn =∞∑n=0

nqn(1− q)∣∣∣q=e−~ω/(kBT )

= (1− q)q∞∑n=0

d

dqqn∣∣∣q=e−~ω/(kBT )

= (1− q)q d

dq

(1

1− q

) ∣∣∣q=e−~ω/(kBT )

=q

1− q∣∣∣q=e−~ω/(kBT )

=e−~ω/(kBT )

1− e−~ω/(kBT )

=1

e~ω/(kBT )−1≡ n(ω) . (6.53)

This is the famous Bose-Einstein statistics [2], [3]. Similarly, one can calculate

〈n2〉 =∞∑n=0

n2pn =q(1 + q)

(1− q)2

∣∣∣q=e−~ω/(kBT )

, (6.54)

so that the thermal photon number-variance reads

(∆n)2 = 〈n2〉 − 〈n〉2 =

[q(1 + q)

(1− q)2+

q2

(1− q)2

]q=e−~ω/(kBT )

=q

(1− q)2

∣∣∣q=e−~ω/(kBT )

=e−~ω/(kBT )

[1− e−~ω/(kBT )]2 . (6.55)

One easily rewrites this as

(∆n)2 =

[q

1− q +q2

(1− q)2

]q=e−~ω/(kBT )

= 〈n〉+ 〈n〉2 ≥ 〈n〉 , (6.56)

[2] S. N. Bose, Plancks Gesetz und Lichquantentheorie, Z. Phys. 26, 178 (1924) [Bose24].[3] A. Einstein, Quantentheorie des einatomigen idealen Gases, Sitzungsber. Preuss. Akad. Wiss. 245 (1924)

[Einstein24].


showing that a thermal state exhibits super Poissonian photon statistics. The thermal photonnumber statistics and the respective Poissonian statistics for 〈n〉 = 6 are better shown in thefigure below:

n

pn

Boltzmann distribution

Poisson distribution

Averages of the mode creation and annihilation operators vanish for a thermal state

〈a〉 = Tr[ρT a] =∞∑n=0

pn 〈n|a|n〉︸︷︷︸=0

= 0 , (6.57)

〈a†〉 = 0 , (6.58)

while their products have non trivial averages:

〈a†a〉 = 〈n〉 ≡ n(ω) (6.59)

〈aa†〉 = 〈a†a+ 1〉 = 〈n〉+ 1 = n(ω) + 1 , (6.60)

〈a2〉 = 〈(a†)2〉 = 0 . (6.61)

These relations can easily be generalized to the multimode case. The thermal state densitymatrix in this case reads

ρT =e−HF /(kBT )

Tr[

e−HF /(kBT )] =

e−∑j ~ωj nj/(kBT )

Tr[

e−∑j ~ωj nj/(kBT )

] (6.62)

leading to

〈aj〉 = 〈a†j〉 = 0 , (6.63)

〈a†j aj′〉 = n(ωj)δjj′ , (6.64)

〈aj a†j′〉 = [n(ωj) + 1]δjj′ , (6.65)

〈aj aj′〉 = 〈a†j a†j′〉 = 0 . (6.66)

With these relations, we can determine the spectral energy density of the free-space electro-magnetic field for a thermal state. Following similar steps as in Chapter 5, the energy densitycan be written as

〈H(r)〉 =

⟨ε0

2E2(r) +

1

2µ0

B2(r)

⟩=

∫ ∞0

dω ρ(ω) (6.67)

6.3 Coherent states 71

with a spectral density at finite temperature

ρ(ω) =~ω3

2π2c3[1 + 2n(ω)] ≡ ρ0(ω) + ρT (ω) . (6.68)

Here ρ0 is the spectral energy density of the ground-state fluctuations of the electromagneticfield as described in Chapter 5, whereas

ρT (ω) =~ω3

π2c3n(ω) (6.69)

is the spectral energy density due to thermal fluctuations, which is the famous Planckspectrum for thermal radiation [4].

6.3 Coherent states

The number states introduced in the first section from a discrete basis for the Hilbert spaceof the electromagnetic filed which is relatively easy to use. We have seen that they mark anextreme case of a sub-Poissonian photon statistics and that a number state with large photonnumber is by no means a classical state.

Next, we are going to discuss coherent states, which are also known as Glauber states aftertheir inventor Roy Glauber [5]. These states form a continuous, overcomplete set of stateswhich satisfy Poissonian statistics and which will later in Chapt. 7 enable us to visualizequantum states. Coherent states are also much easier to prepare experimentally than Fockstates: they describe the state of the electromagnetic field produced by a laser far abovethreshold.

Coherent states can be constructed from the vacuum state by means of a specific unitarytransformation

|α〉 = D(α)|0〉 (6.70)

whereD(α) = eαa

†−α∗a (6.71)

is the coherent displacement operator.To proceed, we will need to evaluate similarity transformations of the type D(α)aD†(α)

and to disentangle the exponential D(α) into product forms. Both of these tasks can beaccomplished very elegantly by means of Lie algebra methods.

Excursus 4: Lie algebra of the harmonic oscillator

4.1 Lie algebraA lie algebra is a vector space G of linear operators with an inner product

[4] M. Planck, Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum, Verh. Dt. Phys. Ges. 2, 202 (1900)[Planck1900].

[5] R. J. Glauber, Coherent and incoherent sates of the radiation field, Phys. Rev. 131, 2766 (1963) [Glauber63].


[ , ] : G × G → G(A, B) 7→ [A, B] (6.72)

called Lie bracket, which has the following properties:

[aA+ bB, C] = a[A, C] + b[B, C] ,[A, bB + cC] = b[A, B] + c[A, C] ,

(Bilinearity) (6.73)

[A, A] = 0 , (Alternativity) (6.74)[A, [B, C]

]+[B, [C, A]

]+[C, [A, B]

]= 0 , (Jacobi-Identity) (6.75)

In our case, the commutator obviously plays the role of the Lie bracket.In particular, a Lie algebra must be closed under the operation of commutation, i.e.the commutator of any two operators of the space must again be an operator withinthis space. For a finite-dimensional space spanned by a basis X1, X2, ..., Xn , this isobviously fulfilled if

[Xi, Xj] =n∑k=1

cijkXk . (6.76)

The basis operators are called generators of the Lie algebra and the constants cijk areknown as structure constants.The set of operators a, a†, n and 1 generates the harmonic oscillator algebra. Thestructure constants can be read off from the commutation relations

[a, a†] = 1 (6.77)

[n, a] = [a†a, a] = [a†, a]︸︷︷︸=−1

a = −a , (6.78)

[n, a†] = [a†a, a†] = a† [a, a†]︸︷︷︸=1

= a† , (6.79)

i.e.

caa†1 = 1 , ca†a1 = −1 ,cnaa = −1 , cana = 1 , (6.80)cna†a† = 1 , ca†na† = −1 ,

and all other structure constants vanish.

4.2 Similarity transformation

For a invertible operator S, the transformation

A→ SAS−1 (6.81)


is called similarity transformation. One can show that for any analytic function

f(z) =∞∑k=0

fkzk , (6.82)

the similarity transformation can be pulled inside the function:

Sf(A)S−1 =∞∑k=0

fkSAkS−1 =

∞∑k=0

fk(SAS−1

)k= f(SAS−1) (6.83)

where we have inserted multiple identities 1 = S−1S.In particular, we are interested in the similarity transformation

Xi(θ) = eθZ Xi e−θZ (6.84)

where Xi is a generator and

Z =n∑i=1

αiXi (6.85)

is an arbitrary member of the Lie algebra. This similarity transformation can beevaluated as follows: we differentiate the transformation with respect to θ to find

X ′i(θ) = eθZ ZXi e−θZ − eθZ XiZ e−θZ = eθZ[Z, Xi] e−θZ . (6.86)

Next, we insert the explicit expression of Z and evaluate the commutators by means ofthe structure constants to find

X ′i(θ) = eθZ

[n∑j=1

αjXj, Xi

]e−θZ =

n∑j=1

n∑k=1

αjcjik eθZ Xk e−θZ

=n∑j=1

n∑k=1

αjcjikXk(θ) . (6.87)

The similarity transformation can thus be found by solving a set of coupled first-orderlinear differential equations with the initial conditions Xi(0) = Xi.Let us apply this to an example from the harmonic-oscillator algebra. With

Z = αa† − α∗a , (6.88)

the respective similarity-transformed annihilation operator

a(θ) = eθZ a e−θZ (6.89)

satisfies the differential equation

a′(θ) = α ca†a1︸︷︷︸=−1

1(θ) = −α . (6.90)


It is solved bya(θ) = −αθ + c (6.91)

where the operator-valued constant c is determined by the initial condition

a(0) = c = a , (6.92)

so thata(θ) = a− αθ . (6.93)

In particular for θ = 1, we find that the similarity transformation for the coherentdisplacement operator

D(α) = eZ = eαa†−α∗a (6.94)

is given byD(α)aD†(α) = a− α . (6.95)

4.3 Disentangling an exponential

Lie algebra methods can also be used to disentangle an exponential of operators intoproducts:

eθZ = ef1(θ)X1 ef2(θ)X2 · · · efn(θ)Xn (6.96)

where fi(θ) are unknown functions. To find then, we differentiate the equation withrespect to θ and multiply it from the right with its inverse:

n∑i=1

αiXi =

(d

dθeθZ)

e−θZ

=d

dθ

(ef1(θ)X1 ef2(θ)X2 · · · efn(θ)Xn

)e−fn(θ)Xn e−f2(θ)X2 · · · e−f1(θ)X1

= f ′1(θ)X1 + f ′2(θ) ef1(θ)X1 X2 e−f1(θ)X1 + · · ·+ f ′n(θ) ef1(θ)X1 · · · efn−1(θ)Xn−1 Xn e−fn−1(θ)Xn−1 · · · e−f1(θ)X1 . (6.97)

After carrying out the similarity transformations SXiS−1 by using the techniques

of the previous section, the right hand side becomes a linear combination of the Xi.Comparing coefficients, one hence obtains coupled first-order differential equationsfor the functions fi(θ) which have to be solved with the initial conditions fi(0) = 0.We again apply this to an example from the harmonic-oscillator algebra:

eθ(αa†−α∗a) = ef1(θ)a† ef2(θ)a ef3(θ) . (6.98)

The above procedure yields

αa† − α∗a = f ′1(θ)a† + f ′2(θ) ef1(θ)a† a e−f1(θ)a†

+ f ′3(θ) ef1(θ)a† ef2(θ)a1 e−f2(θ)a e−f1(θ)a†

= f ′1(θ)a† + f ′2(θ)[a− f1(θ)] + f ′3(θ) (6.99)


where we have used the similarity transformation

ef1(θ)a† a e−f1(θ)a† = a− f1(θ) . (6.100)

Equating coefficients leads to

f ′1(θ) = α ,f ′2(θ) = −α∗ , (6.101)f ′3(θ) = f ′1(θ)f ′2(θ) .

Recalling the initial conditions fi(0) = 0, these equations can easily be solved:

f1(θ) = αθ ,f2(θ) = −α∗θ , (6.102)

f ′3(θ) = −|α|2θ ⇒ f3(θ) = −|α|2

2θ2 .

The disentangled exponential is then given by

eθ(αa†−α∗a) = eθαa

†eθα

∗a e−|α|2θ2/2 (6.103)

so that the displacement operator in normal ordering reads

D(α) = eαa†−α∗a = e−

12|α|2 eαa

†e−α

∗a . (6.104)

Here, normal ordering means that all arising operator products have been arrangedsuch that annihilation operators are positioned to the right and creation operatorsappear to the left. The opposite, antinormally ordered product form can be found asfollows:

D(α) = e−12|α|2 eαa

†e−α

∗a = e−12|α|2 eαa

†e−α

∗a e−αa†eαa

†︸︷︷︸=1

= e−12|α|2 exp

[eαa

†(−α∗a) e−αa

†]

eαa†

= e−12|α|2 exp[−α∗(a− α)] eαa

†

= e−12|α|2 e−α

∗a e|α|2

eαa†

= e12|α|2 e−α

∗a eαa†

(6.105)

where we have inserted a unity factor, pulled the similarity transformation inside theexponential function and subsequently evaluated it.

After this digression, let us return our attention to the coherent states

|α〉 = D(α)|0〉 . (6.106)

We begin by determining the action of a and a† on |α〉. Starting from

a|0〉 = 0 , (6.107)


so that〈α|β〉 = e−

12|α|2− 1

2|β|2+α∗β = e−

12|α−β|2+ 1

2(α∗β−αβ∗) . (6.119)

The coherent states for α 6= β are hence only approximately orthogonal for sufficientlyseparate α, β:

|〈α|β〉|2 = e−|α−β|2 ' 0 for |α− β| 1 . (6.120)

Nevertheless, one can show that the coherent states are over complete. To show this, we needto first expand the coherent states in terms of the number states, which do form a completebasis:

|α〉 =∑n

|n〉〈n|α〉 . (6.121)

For the coefficients, we find

〈n|α〉 = 〈n|D(α)|0〉 = e−12|α|2〈n|eαa† e−α

∗a|0〉 (disentangling exponential)

= e−12|α|2〈n|eαa†|0〉 = e−

12|α|2

∞∑k=0

αk

k!〈n|(a†)k|0〉 (Taylor series of exponential)

= e−12|α|2

∞∑k=0

αk√k!〈n|k〉︸︷︷︸=δnk

=αn√n!

e−12|α|2 . (6.122)

Using this expansion in terms of Fock states, we can now calculate∫d2α |α〉〈α|=

∑m,n

∫d2α |m〉〈m|α〉〈α|n〉〈n|=

∑m,n

|m〉〈n|√m!n!

∫d2α αmα∗n e−|α|

2

. (6.123)

Introducing polar coordinates α = |α|eiϕ, the integral can be performed to give∫d2α αmα∗n e−|α|

2

=

∫ 2π

0

dϕ

∫ ∞0

d|α| |α| |α|meimϕ|α|ne−inϕ e−|α|2

=

∫ 2π

0

dϕ ei(m−n)ϕ︸︷︷︸=2πδmn

∫ ∞0

d|α| |α|m+n+1e−|α|2

x=|α|2= π

∫ ∞0

dx xme−x︸︷︷︸=m!

δmn = πm!δmn , (6.124)

so that ∫d2α |α〉〈α|=

∑m,n

|m〉〈n|√m!n!

πm!δmn = π∑m

|m〉〈m|= π1 (6.125)

and the completeness relation reads

1

π

∫d2α |α〉〈α|= 1 . (6.126)

The coherent states are hence an overcomplete set of non-orthogonal states. In particular, acoherent state |β〉 has a non trivial expansion

|β〉 =1

π

∫d2α |α〉〈α|β〉 . (6.127)


This expansion is illustrated in the following figure for the state |β〉with β = 2 + 3i :

β

2

4

2

4

Re(α)

Im(α)

1π |〈α|β〉|2

As a consequence of the overcompleteness, a given state can be expanded in terms of a subsetof all coherent states. For instance, a number state |n〉, can be spanned by coherent states |α〉on a circle α = |α|eiϕ

|n〉 =

∫ 2π

0

dϕ cn(α)|α〉 =

√n!

2π

e12|α|2

|α|n∫ 2π

0

dϕ e−inϕ|α〉 . (6.128)

To show this, we calculate√n!

2π

e12|α|2

|α|n∫ 2π

0

dϕ e−inϕ∑m

|m〉〈m|α〉 (Completeness of number states)

=

√n!

2π

e12|α|2

|α|n∑m

|m〉 |α|m

√m!

e−12|α|2∫ 2π

0

dϕ ei(n−m)ϕ︸︷︷︸=2πδmn

=

√n!

2π

e12|α|2

|α|n|α|n√n!

e−12|α|2 2π|n〉 = |n〉 . (6.129)

The photon number statistics of the coherent states follows immediately from their overlapwith the number states:

pn = |〈n|α〉|2 =|α|2nn!

e−|α|2

. (6.130)

This is a Poissonian statistics with average photon number

〈n〉 = 〈α|n|α〉 = 〈α|a†a|α〉 = 〈α|α∗α|α〉 = |α|2〈α|α〉 = |α|2 . (6.131)

Indeed, the variance of the photon number reads

〈(∆n)2〉 = 〈α|n2|α〉 − 〈α|n|α〉2 = 〈α|a†aa†a|α〉 − |α|4

= 〈α|(a†a†aa+ a†a)|α〉 − |α|2= |α|4+|α|2−|α|4= |α|2= 〈n〉 . (6.132)


For a field strength operator F = Ca+ C∗a†, we find an average

〈F 〉 = 〈α|(Ca+ C∗a†)|α〉 = Cα + C∗α∗ (6.133)

and a variance

(∆F )2 = 〈α|F 2|α〉 − 〈α|F |α〉2 = 〈α|(Ca+ C∗a†)(Ca+ C∗a†)|α〉 − (Cα + C∗α∗)2

= 〈α|(C2a2 + 2|C|2a†a+ C∗2(a†)2 + |C|2)|α〉 − (Cα + C∗α∗)2

= (Cα + C∗α∗)2 + |C|2−(Cα + C∗α∗)2 = |C|2 . (6.134)

In particular, for the quadrature operators

X =1

2(a+ a†) , Y =

i

2(a− a†) (6.135)

we have〈X〉 =

1

2(α + α∗) , 〈Y 〉 =

i

2(α− α∗) (6.136)

and(∆X)2 = (∆Y )2 =

1

4(6.137)

Just like the vacuum state, every coherent state is hence an equal-variance minimal uncertaintystate of the quadrature operators.

The Glauber coherent states can be generalized to coherently displaced number states

|n, α〉 = D(α)|n〉 . (6.138)

From the eigenvalue relationn|n〉 = n|n〉 , (6.139)

it follows that

D(α)nD†(α)|n, α〉 = D(α)nD†(α)D(α)|n〉 = D(α)n|n〉 = nD(α)|n〉 = n|n, α〉 . (6.140)

So the coherently displaced number states are eigenstates

n(α)|n, α〉 = n|n, α〉 (6.141)

of the coherently displaced number operator

n(α) = D(α)nD†(α) = D(α)a†aD†(α) = D(α)a†D†(α)D(α)aD†(α)

= a†(α)a(α) = (a† − α∗)(a− α) . (6.142)

The overlap of the coherently displaced number states with the original number states reads

〈n|m,α〉 = e−12|α|2√n!m!

min(n,m)∑l=0

(−1)(m−l)α∗(m−l)α(n−l)

l!(n− l)!(m− l)! (6.143)


and these states are again overcomplete:

1

π

∫d2α |n, α〉〈n, α|= 1 . (6.144)

The original Glauber coherent states can easily be generalized to the multimode case:

|α1, α2, ...〉 =⊗j

|αj〉 (6.145)

withaj|αj〉 = αj|αj〉 . (6.146)

These multimode coherent states are eigenstates of the positive-frequency field strengthoperators

F (+) =∑j

Cj aj (6.147)

withF (+)|α1, α2, ...〉 = F (+)|α1, α2, ...〉 (6.148)

and eigenvaluesF (+) =

∑j

Cjαj . (6.149)

6.4 Squeezed states

As seen in the last section, Glauber coherent states are minimum uncertainty states where thevariances of the quadrature operators X and Y are equal. While the product of the variancescan obviously not be further reduced, it is possible to to reduce the variance in X uponenlarging the variance in Y or vice versa. This is the case for the so-called squeezed states.

Just like coherent states, squeezed states can be obtained by applying a unitary transfor-mation:

|ξ〉 = S(ξ)|0〉 (6.150)

whereS(ξ) = e(ξ∗a2−ξa†2)/2 (6.151)

is the squeezing operator. To determine its action on the creation and annihilation operators,we again require Lie algebra methods.

Excursus 5: SU(1,1) Lie algebra

5.1 Lie algebra

The operators a†2/2, a2/2 and (n+ 1/2)/2 generate the SU(1,1) algebra. The respectivestructure constants can be read off from the commutation relations

6.4 Squeezed states 81

[a†2

2,a2

2

]=

1

4

(a†[a†, a2

]+[a†, a2

]a†)

=1

4

(a†a[a†, a

]+ a†

[a†, a

]a+ a

[a†, a

]a† +

[a†, a

]aa†)

= −1

2

(a†a+ aa†

)= −

(n+

1

2

)= −2

1

2

(n+

1

2

), (6.152)

[1

2

(n+

1

2

),a2

2

]= − a

2

2, (6.153)[

1

2

(n+

1

2

),a†2

2

]=a†2

2. (6.154)

5.2 Similarity transformation

To apply the squeezing operator to the annihilation operator, we consider the similaritytransformation

a(θ) = eθZ a e−θZ (6.155)

with

Z = ξ∗a2

2− ξ a

†2

2. (6.156)

Taking the derivative with respect to θ, we find

a′(θ) = eθZ[Z, a

]e−θZ = eθZ

[ξ∗a2

2− ξ a

†2

2, a

]e−θZ = ξ eθZ a† e−θZ = ξa†(θ) (6.157)

wherea†(θ) = eθZ a† e−θZ . (6.158)

To form a closed system of equations, we next require a differential equation for a†(θ).Noting that

a†(θ) = eθZ a† e−θZ =(

eθZ a e−θZ)†

= [a(θ)]† , (6.159)

this can be obtained by simply taking the Hermitian conjugate:

a†′(θ) = [a′(θ)]†

= ξ∗a(θ) . (6.160)

Taking the derivative of the equation for a′(θ) and substituting the equation for a′†(θ),we have

a′′(θ) = |ξ|2a(θ) (6.161)

which is solved bya(θ) = C1 e|ξ|θ +C2 e−|ξ|θ . (6.162)


The initial conditions

a(0) = C1 + C2 = a , (6.163)

a′(0) = |ξ|C1 − |ξ|C2 = ξa† = |ξ|eiϕξ a† ⇒ C1 − C2 = eiϕξ a† (6.164)

can be easily solved to give

C1 =1

2

(a+ eiϕξ a†

), (6.165)

C2 =1

2

(a− eiϕξ a†

). (6.166)

⇒ a(θ) = cosh (|ξ|θ) a+ eiϕξ sinh (|ξ|θ) a† . (6.167)

In particular, setting θ = 1, we find that

S(ξ)aS†(ξ) = cosh (|ξ|) a+ eiϕξ sinh (|ξ|) a† . (6.168)

5.3 Disentangling an exponential

To obtain the squeezing operator in normal ordering, we write

eθZ = ef1(θ)a†2/2 ef2(θ)(n+1/2)/2 ef3(θ)a2/2 (6.169)

with

Z = ξ∗a2

2− ξ a

†2

2. (6.170)

Following our procedure for disentangling an exponential from the last section, wetake the derivative of the above equation with respect to θ, multiply the result withexp(−θZ), carry out all arising similarity transformations on the right hand side andcompare coefficients on both sides of the equation. A lengthy calculation eventuallyresults in

f ′1 − f1f′2 + f 2

1 f′3 e−f2 = −ξ, (6.171)

f ′2 − 2f1f′3 e−f2 = 0, (6.172)

f ′3 e−f2 = ξ∗ (6.173)

with initial conditions f1(0) = f2(0) = f3(0) = 0. This non linear system of ordinarydifferential equations leads to a Ricatti-type differential equation. Its solution yields

f1(θ) = − eiϕξ tanh (|ξ|θ) , (6.174)f2(θ) = −2 ln [cosh (|ξ|θ)] , (6.175)

f3(θ) = e−iϕξ tanh (|ξ|θ) . (6.176)

Substituting these results back into Eq. (6.169) above, we find that the disentangledexponential reads

eθ(ξ∗a2/2−ξa†2/2) = e− e

iϕξ tanh(|ξ|θ)a†2/2(

1

cosh(|ξ|θ)

)n+1/2

ee−iϕξ tanh(|ξ|θ)a2/2 . (6.177)


The squeezing operator in its normal-ordered product form is hence given by (θ = 1):

S(ξ) = e(ξ∗a2−ξa†2)/2 = e− eiϕξ tanh|ξ|a†2/2

(1

cosh|ξ|

)n+1/2

ee−iϕξ tanh|ξ|a2/2 . (6.178)

With these preparations at hands, we return our attention to the squeezed vacuum

|ξ〉 = S(ξ)|0〉 . (6.179)

To determine its properties, we require the transformed creation and annihilation operators.As shown in the excursus, they can be given as

a(ξ) ≡ S(ξ)aS†(ξ) = cosh|ξ|a+ eiϕξ sinh|ξ|a† = µa+ νa† , (6.180)

a†(ξ) ≡ S(ξ)a†S†(ξ) = ν∗a+ µa† . (6.181)

Here, we have introduced the two parameters

µ = cosh|ξ| , (6.182)

ν = eiϕξ sinh|ξ| (6.183)

for a more compact notation. They fulfill the constraint

µ2 − |ν|2= 1 (6.184)

which is a consequence of the unitary of the squeezing operator:

1 =[a, a†

]=[a(ξ), a†(ξ)

]=[µa+ νa†, ν∗a+ µa†

]= µν∗ [a, a]︸︷︷︸

=0

+µ2[a, a†

]︸︷︷︸=1

+|ν|2[a†, a

]︸︷︷︸=−1

+νµ[a†, a†

]︸︷︷︸=0

= µ2 − |ν|2 . (6.185)

We further note that the squeezing operator has the property

S†(ξ) = S(−ξ) (6.186)

with

µ(−ξ) = cosh|−ξ|= µ(ξ) , (6.187)

ν(−ξ) = eiϕ′−ξ sinh|−ξ|= −ν(ξ) . (6.188)

To expand the squeezed vacuum in terms of Fock states, we make use of the normal-orderedsqueezing operator as derived in the previous Excursus:

S(ξ) = e(ξ∗a2−ξa†2)/2

= e− eiϕξ tanh(|ξ|θ)a†2/2

(1

cosh(|ξ|θ)

)n+1/2

eeiϕξ tanh(|ξ|θ)a2/2

= e−ν/(2µ)a†2(

1

µ

)n+1/2

eν∗/(2µ)a2 (6.189)


which has also been conveniently re-expressed in terms of the parameters µ and ν. Thisshows that

|ξ〉 = S(ξ)|0〉 = e−ν/(2µ)a†2(

1

µ

)n+1/2

eν∗/(2µ)a2|0〉︸︷︷︸

=|0〉

= e−ν/(2µ)a†2(

1

µ

)n+1/2

|0〉︸︷︷︸(1/µ)1/2|0〉

=1√µ

e−ν/(2µ)a†2|0〉 . (6.190)

Using a Taylor expansion of the exponential, we find

|ξ〉 =∑n

|n〉〈n|ξ〉 (6.191)

with

〈n|ξ〉 =

1√µ

√n!

(n/2)!

(ν

2µ

)n2

for n even,

0 for n odd.(6.192)

By combining the squeezing operator with the coherent displacement operator, we maydefine squeezed coherent states:

|ξ, α〉 = S(ξ)D(α)|0〉 = S(ξ)|α〉 (6.193)

According to this definition, |ξ, α〉 is obtained by just coherently displacing the vacuum andthen applying the squeezing operator. What happens, on the other hand , if we first squeezethe vacuum and then coherently displace to construct a coherently displaced squeezed state?To answer this question, we recall that the squeezing operator is a similarity transformation:

S(ξ)D(α)S†(ξ) = S(ξ) eαa†−α∗a S†(ξ) = eαa

†(ξ)−α∗a(ξ)

= eα(ν∗a+µa†)−α∗(µa+νa†) = e(αµ−α∗ν)a†−(α∗µ−αν∗)a

= D(α) (6.194)

withα = µα− να∗ . (6.195)

A squeezed coherent state is hence also a coherently displaced squeezed state, but with amodified displacement parameter:

|ξ, α〉 = S(ξ)D(α)|0〉 = S(ξ)D(α)S†(ξ)S(ξ)|0〉 = D(α)S(ξ)|0〉 = D(α)|ξ〉 ≡ |α, ξ〉 . (6.196)

The squeezed coherent states are eigenstates of the transformed annihilation operator:

a(ξ)|ξ, α〉 = S(ξ)aS†(ξ)S(ξ)|α〉 = S(ξ)a|α〉 = αS(ξ)|α〉 = α|ξ, α〉 . (6.197)

For a given fixed squeezing parameter, they form an overcomplete set of non-orthogonalstates, just like the original coherent states:

〈ξ, α|ξ, β〉 = 〈α|S†(ξ)S(ξ)|β〉 = 〈α|β〉 = e−12|α−β|2+ 1

2(α∗β−αβ∗) , (6.198)


we have

S†(ξ)aS(ξ) = S(−ξ)aS(ξ) = µa− νa† , (6.208)

S†(ξ)a†S(ξ) = (µa− νa†)† = µa† − ν∗a . (6.209)

Substitution of these results yields the average photon number

〈n〉 = 〈ξ, α|n|ξ, α〉 = 〈α|(µa† − ν∗a)(µa− νa†)|α〉= 〈α|(µα∗ − ν∗α)(µα− να∗)|α〉+ νν∗〈α|

[a, a†

]|α〉

= |µα− να∗|2+|ν|2= |α|2+|ν|2= 〈α|n|α〉+ 〈ξ|n|ξ〉 (6.210)

where we have recalled α = µα − να∗. We see that the average photon number has twocomponents. The coherent photon number |α|2 is that of the coherent displacement neces-sary to generate the corresponding coherently displaced squeezed vacuum (as opposed tothe squeezed coherent state). The incoherent photon number is the contribution from thesqueezed vacuum (which is not the ground state and hence has photons present, on average).

Lets us next turn our attention to the field strength operator F = Ca+ C∗a†. Noting that

〈a〉 = 〈ξ, α|a|ξ, α〉 = 〈α|S†aS(ξ)|α〉 = 〈α|(µa− νa†)|α〉 = µα− να∗ = α , (6.211)

we find that

〈F 〉 = 〈ξ, α|(Ca+ C∗a†)|ξ, α〉 = 〈ξ, α|(Cα + C∗α∗)|ξ, α〉 = Cα + C∗α∗ . (6.212)

So the field strength for the squeezed coherent state |ξ, α〉 is the same that one would find forthe coherent state |α〉 of the corresponding coherently displaced squeezed vacuum.

The variance of the field strength operator will reveal why the squeezed state carries itsname. To calculate it, we require

〈a2〉 = 〈0|S†(ξ)D†(α)a2D(α)S(ξ)|0〉 = 〈0|S†(ξ)(a− α)2S(ξ)|0〉= 〈0|(µa− νa† − α)2|0〉 = α2 − µν . (6.213)

This leads to

(∆F )2 = 〈F 2〉 − 〈F 〉2 = 〈C2a2 + |C|2(aa† + a†a) + C∗2a†2〉 − 〈F 〉2= C2(α2 − µν) + |C|2+|C|2(|α|2+|ν|2) + C∗(α2 − µν)∗2 − (Cα + C∗α∗)2

= |Cµ− C∗ν∗|2 . (6.214)

After using C = |C|eiϕ, µ =√

1 + |ν|2, ν = |ν|eiϕν , the field strength variance can be re-expressed as

(∆F )2 = |C|2∣∣∣eiϕ

√1 + |ν|2 − e−i(ϕ−ϕν)|ν|

∣∣∣= |C|2

1 + 2|ν|2

[1−

√1 +

1

|ν|2 cos(2ϕ− ϕν)]

. (6.215)


The field strength variance can hence be larger or smaller (i.e., squeezed) than the value |C|2found for a non-squeezed state. It takes its minimum when the phases of the field strengthoperator and the squeezing are such that

2ϕ− ϕν = 2nπ, n ∈ Z : (∆F )2 = |C|2e−2|ξ| . (6.216)

On the other hand, maximum field strength fluctuations are found for

2ϕ− ϕν = (2n+ 1)π, n ∈ Z : 〈(∆F )2〉 = |C|2e2|ξ| . (6.217)

Let us apply this to the quadrature operators

X =1

2(a+ a†) , Y =

i

2(a− a†) , (6.218)

which are maximally out of phase (|C|= 1/2 with ϕ = 0 for X and ϕ = π/2 for Y ). Thevariance of X is minimized for ϕν = 0 with

(∆X)2 =1

4e−2|ξ| (6.219)

whereas the variance of Y for the same squeezing parameter takes its maximum value

(∆Y )2 =1

4e2|ξ| . (6.220)

The product of the variances reads

(∆X)2(∆Y )2 =1

16, (6.221)

so the squeezed state with maximal squeezing of (∆X)2 is a minimum uncertainty state forX and Y . Note that states with different values of ϕξ (such that the squeezing is not maximal)are not minimum uncertainty states.

Just like the Fock states and the coherent states, squeezed states can be generalized to themultimode case. A multimode squeezed state can be obtained from the vacuum by applyingthe multimode squeezing operator

S = exp

[∑jj′

(ξ∗jj′ aj aj′ − ξjj′ a†j a†j′

)]. (6.222)

As an example, let us consider the two-mode squeezing operator (ξ12 = ξ21 = ξ/2, ξ11 = ξ22 =0)

S(ξ) = e(ξ∗a1a2−ξa†1a†2) . (6.223)

It implies a similarity transformation

a1(ξ) = µa1 + νa†2 , a2(ξ) = µa2 + νa†1 . (6.224)


Using the normal-ordered form

S(ξ) = e−ν/µa†1a†2

(1

µ

)n1+n2+1

eν∗/µa1a2 , (6.225)

we find that the two-mode squeezed vacuum reads

S(ξ)|0, 0〉 =1

µ

∞∑n

(−νµ

)n|n, n〉 . (6.226)

This is a highly entangled state, because finding n photons in mode 1 immediately impliesthat there are also n photons in mode 2.

6.5 Futher quantum states of the electromagnetic field

So far, we have studied Fock states |n〉 as eigenstates of the number operator n and coherentstates |α〉 as eigenstates of the annihilation operator a. Further non-trivial quantum stateswhich are being considered in quantum optics are eigenstates of other relevant operators.

6.5.1 Field strength states

An operator whose statistics we have investigated in all past examples is the field strength:

F = Ca+ C∗a† . (6.227)

Writing C = |C|eiϕ, the field strength can be viewed as a one-parameter family of operators

F (ϕ) = |C|(a eiϕ +a† e−iϕ) . (6.228)

For a given value of the phase-parameter ϕ, the field strength states are defined as theeigenstates of F (ϕ):

F (ϕ)|F, ϕ〉 = F |F, ϕ〉 . (6.229)

These states form an orthonormal basis:

〈F, ϕ|F ′, ϕ〉 = δ(F − F ′) , (6.230)∫ ∞−∞

dF |F, ϕ〉〈F, ϕ|= 1 . (6.231)

They can be expanded in terms of Fock states according to

〈n|F, ϕ〉 =1

(2|C|2π)1/4e−F

2/(4|C|2) einϕ

√n!2n

Hn

(F√2C

)(6.232)

where Hn(x) are again the Hermite polynomials. Particular examples of the field strengthstates are the quadrature eigenstates

X|X〉 = X|X〉 , (6.233)

Y |Y 〉 = Y |Y 〉 (6.234)

6.5 Futher quantum states of the electromagnetic field 89

which are eigenstates of the quadrature operators

X =1

2(a+ a†) (6.235)

Y =i

2(a− a†) . (6.236)

Noting that |C|= 1/2 in both cases and ϕ = 0 (for X) and ϕ = π/2 (for Y ), the quadratureeigenstates have the Fock-state components

〈n|X〉 =

(2

π

)1/4

e−X2 1√

n!2nHn(√

2X) , (6.237)

〈n|Y 〉 =

(2

π

)1/4

e−Y2 eiπn/2

√n!2n

Hn(√

2Y ) . (6.238)

6.5.2 Phase statesThe classical amplitudes α and α∗ can be decomposed into modulus and phase according to

α = |α|eiϕ =√αα∗ eiϕ , (6.239)

α∗ = |α|e−iϕ =√αα∗ e−iϕ . (6.240)

In analogy, we write

a = V√n , (6.241)

a† =√nV (6.242)

and define phase states as eigenstates of V :

V |ϕ〉 = eiϕ|ϕ〉 . (6.243)

Special care needs to be taken, because V is not a unitary operator. To see this, we expand

a =∞∑n=0

√n+ 1|n〉〈n+ 1|=

∞∑n=0

|n〉〈n+ 1|√n

!= V√n (6.244)

to deduce

V =∞∑n=0

|n〉〈n+ 1| , (6.245)

V † =∞∑n=0

|n+ 1〉〈n| . (6.246)

We then have

V V † =∞∑

m,n=0

|n〉〈n+ 1|m+ 1〉︸︷︷︸=δnm

〈m|=∞∑n=0

|n〉〈n|= 1 , (6.247)

7 What do quantum states of light look like?

Quasiprobability distributions inphase space



• express quantum averages as averages over quasiprobability distributions inphase space,

• understand the connection between different operator orderings and the respec-tive quasiprobability distributions and

• know the different quasiprobability distributions of some common quantumstates of light.

In the previous chapter, we have encountered a variety of different quantum states oflight and we have learned how to evaluate averages and fluctuations of different observablesof the electromagnetic field for these quantum states. However, our treatment of both statesand averages was very abstract. In order to develop a more visual understanding, we will inthe following present an alternative representation that is analogous to classical averages.

Recall that the classical electromagnetic field can be given in terms of normal modes as

A(r, t) =∑j

[Aj(r)αj e−iωj(t−t0) +A∗j(r)α∗j eiωj(t−t0)

](7.1)

where αj are the classical amplitudes which completely determine the modulus and the phaseof the electromagnetic field, αj = |αj|eiϕj . A classical random electromagnetic field is onewhere the amplitudes are not known precisely, but only probabilities for finding differentvalues αj can be given. For a single mode, the classical random state is described by aprobability distribution

p : C→ [0,∞)

α 7→ p(α)(7.2)

which must be normalized: ∫d2α p(α) = 1 .. (7.3)

92 Quasiprobability distributions in phase space

Classical averages of an observable f(α) (such as the above vector potential, the amplitude αitself, the electric and magnetic fields or the field energy) are given by

f(α) =

∫d2α p(α)f(α) .. (7.4)

The classical random state of the electromagnetic field can be easily be visualized by display-ing the respective probability distribution for all possible values of α.

7.1 P -, Q- and Wigner functions

In the quantum case, we have operators a and a† in place of the classical amplitudes α and α∗

and an observable is represented by a function f(a, a†). To obtain a description of quantumaverages that resembles the above classical average, we are aiming to find a function f(α)and a probability distribution p(α) such that

〈f(a, a†)〉 =

∫d2α p(α)f(α) .. (7.5)

It would be tempting to obtain the function f(α) by simply replacing operators with classicalamplitudes in the operator function: f(a, a†) 7−→ f(α, α∗) = f(α). However, when doingso we are faced with an ambiguity, because equivalent forms of f(a, a†) lead to differentfunctions f(α). For instance,

f(a, a†) = aa† (7.6)

leads to

f(α) = αα∗ = |α|2 . (7.7)

One the other hand, the same operator function can be given as

f(a, a†) = aa† = a†a+ 1 (7.8)

leading to the different result

f(α) = α∗α + 1 = |α|2+1 . (7.9)

In order to avoid such ambiguities, we need to specify the operator ordering.

Excursus 6: Operator ordering

Operator ordering is procedure whereby an operator function f(a, a†) is mapped to anew, and in general, different function where operator products have been re-arrangedto obey a desired ordering. The most common type of ordering are normal ordering,antinormal ordering and symmetric ordering. We begin by defining these for the

7.1 P -, Q- and Wigner functions 93

product aa† and a†a, respectively:

N (aa†) ≡ N (a†a) ≡ a†a , (normal ordering) (7.10)

A(aa†) ≡ A(a†a) ≡ aa† , (antinormal ordering) (7.11)

S(aa†) ≡ S(a†a) ≡ 1

2(a†a+ aa†) . (symmetric ordering) (7.12)

Note that the alternative notation :a†a: is commonly used instead of N (a†a). So insimple operator products, annihilation operators are shuffled to the right of creationoperators in normal ordering and being placed to the left in antinormal orderingwhereas symmetric ordering is an average over the two possible choices. Note that allthree of the above orderings are special cases of s-ordering where operator productsare mapped to a weighted average

1 + s

2a†a+

1− s2

aa† with s ∈ [−1, 1] . (7.13)

Here normal ordering corresponds to s = 1, antinormal ordering to s = −1 andsymmetric ordering to s = 0.To extend our definition to higher operator products, we denote by π(a†man) an arbi-trary permutation of the (m+ n) operators a†m and an. We then define

N (π(a†man)) = a†man for all π (7.14)

A(π(a†man)) = ana†m for all π (7.15)

S(π(a†man)) =1

(m+ n)!

∑σ

σ(a†man) for all π (7.16)

where the sum runs over all possible permutations σ. For instance we have

N (aa†2) = N (a†aa†) = N (a†2a) = a†2a , (7.17)

A(aa†2) = A(a†aa†) = A(a†2a) = aa†2 , (7.18)

S(aa†2) = S(a†aa†) = S(a†2a) =1

3(aa†2 + a†aa† + a†2a) . (7.19)

Operator ordering for analytical functions can then be introduced by applying thisprocedure to their Taylor expansion:

N(

∞∑m,n=0

fmnπmn(a†man)

)=

∞∑m,n=0

fmna†man for all πmn , (7.20)

A(

∞∑m,n=0

fmnπmn(a†man)

)=

∞∑m,n=0

fmnana†m for all πmn , (7.21)

S(

∞∑m,n=0

fmnπmn(a†man)

)=

∞∑m,n=0

fmn(m+ n)!

∑σmn

σmn(a†man) for all πmn . (7.22)


Note that none of these ordering procedures is a linear operation. To demonstrate this,we calculate

a†a = N (a†a) = N (aa†) = N (1 + a†a) 6= N (1) +N (a†a) = 1 + a†a . (7.23)

This means that the respective ordering procedure has to be applied immediatelyafter expressing the operator function as a Taylor series without prior use of anycommutation relations.As a nontrivial example, consider the coherent displacement operator

D(α) = eαa†−α∗a . (7.24)

A Taylor expansion of ex followed by an expansion of terms (αa† − α∗a)n reveals thatthe displacement operator in this for is automatically symmetrically ordered, so that

S(D(α)) = eαa†−α∗a (7.25)

On the other hand using the Taylor expansion in the identity ex+y = ex ey and reshuf-fling operators a and a† (without applying commutation relations) such that terms witha and a† can be separated into two factors, we find the normally- and antinormallyordered displacement operators

N (D(α)) = N(

eαa†−α∗a

)= eαa

†e−α

∗a , (7.26)

A(D(α)) = A(

eαa†−α∗a

)= e−α

∗a eαa†

. (7.27)

Note that the normally-ordered and antinormally ordered displacement operators aredifferent from the original displacement operator.This is to be distinguished from a rearrangement of the displacement operator bymeans of the commutation relations, which lead to equivalent representations of thesame displacement operator in normally and antinormally-ordered forms (equivalentre-ordering):

D(α) = eαa†−α∗a = e−|α|

2/2 eαa†e−α

∗a = e|α|2/2 e−α

∗a eαa†

. (7.28)


With these preparations at hand, we can construct a representation of quantum observ-ables which resembles classical random observables. To avoid the ambiguities arising whenreplacing operators with c-numbers, we have to fix our operator ordering. Let us begin byusing normal ordering. Starting from a given observable f(a, a†), we make use of the com-mutation relation [a, a†] = 1 to rearrange the observable in its equivalent normally-orderedform:

f(a, a†) = f (N )(a, a†) . (7.29)

Here f (N )(a, a†) is the equivalent expression of f(a, a†) which agrees with its own normal-ordered version:

N (f (N )) = f (N )(a, a†) . (7.30)

As an example, for f(a, a†) = aa†, we have

f(a, a†) = a†a+ 1 = f (N )(a, a†) , (7.31)

so that

N (f (N )(a, a†)) = N (a†a+ 1) = a†a+ 1 = f (N )(a, a†) . (7.32)

We then read off the corresponding classical function by making the replacements a → α,a† → α∗:

f (N )(α) ≡ f (N )(α, α∗) . (7.33)

To write averages as an integral over this function, we expand

f (N )(α) =

∫d2β δ(α− β)f (N )(β) (7.34)

where

δ(α) = δ(Reα)δ(Imα) =1

(2π)2

∫ ∞−∞

dx

∫ ∞−∞

dy ei(xReα+y Imα) (7.35)

with α = αR + iαI is the two-dimensional delta function. By substituting γ = (−y + ix)/2, itcan be given in the more compact form

δ(α) =1

π2

∫d2γ eγα

∗−γ∗α . (7.36)

Using the explicit delta function, we have

f (N )(α) =1

π2

∫d2β

∫d2γ eγ(α∗−β∗)−γ∗(α−β) f (N )(β) . (7.37)

Finally, in order to find an expression for the operator-valued observable f (N )(a, a†), we makethe reverse replacement α → a, α∗ → a†. Special care has to be taken at this point. Afterthe replacement, the operators on the right hand side of the above equation will appear in


a certain ordering. After carrying out the integrals, the resulting expression on the left willhave the same ordering. In order to return to our original operator expression in normalordering, we hence have to use the normal-ordered version of the right hand side:

f (N )(a, a†) =1

π2

∫d2β

∫d2γ eγ(a†−β∗) e−γ

∗(a−β) f (N )(β)

=1

π2

∫d2β

∫d2γ N

(eγ(a†−β∗)−γ∗(a−β)

)f (N )(β) . (7.38)

Introducing the operator-valued delta function

δ(a− α) =1

π2

∫d2β eβ(a†−α∗)−β∗(a−α) , (7.39)

we may hence write

f (N )(a, a†) =

∫d2αN (δ(a− α))f (N )(α) . (7.40)

Finally, by taking the expectation values, we have

〈f (N )(a, a†)〉 =

∫d2α P (α)f (N )(α) . (7.41)

Indeed, the quantum average can be written as a phase-space integral over the respectivefunction f (N )(α) representing the observable and a quasiprobability distribution

P (α) = 〈N (δ(a− α))〉 (7.42)

which is known as the Glauber–Sudarshan P -function [1], [2]. The P -function is normalized,as can easily be verified in the special case f (N )(a, a†) = 1:∫

d2α P (α) = 〈1〉 = Tr[ρ1] = Trρ = 1 . (7.43)

However, it can take negative values and is unbounded, as we will see from the examplesbelow. The P -function is hence not a genuine probability distribution in the classical sense.Instead, we refer to it as a quasi-probability distribution.

The fact that the P -function is not a classical probability distribution is ultimately due tothe Heisenberg uncertainty relation. It prohibits a simultaneous knowledge of the observablesa and a† (or equivalently, the the phase-space observables X and Y ) with arbitrary precision.We hence cannot expect to be able to construct a classical probabilistic representation whichis based on sharply determined variables α and α∗.

The P -function for a given state can be obtained by evaluating the expectation value ofthe normal-ordered delta function. Alternatively, we note that in normal ordering,

f (N )(α) = 〈α|f (N )(a, a†)|α〉 = Tr[|α〉〈α|f (N )(a, a†)

](7.44)

[1] R. J. Glauber, Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams, Phys. Rev.131, 2766 (1963) [Glauber63].

[2] E. C. G. Sudarshan, Coherent and incoherent states of the radiation field, Phys. Rev. Lett. 10, 277 (1963)[Sudarshan63].


so that

Tr[ρf (N )(a, a†)] = 〈f (N )(a, a†)〉 =

∫d2α P (α)f (N )(α) = Tr

[∫d2α P (α)|α〉〈α|f (N )(a, a†)

](7.45)

which implies

ρ =

∫d2α P (α)|α〉〈α| . (7.46)

By writing the density operator of a given state in this form, the P -function can then be readoff.

The above considerations can be preformed in complete analogy using different operatororderings. For instance, starting from an observable which has been recast into its equivalentantinormally ordered form

f(a, a†) = f (A)(a, a†) (7.47)

with

A(f (A)(a, a†)) = f (A)(a, a†) , (7.48)

we arrive at the representation

〈f (A)(a, a†)〉 =

∫d2α Q(α)f (A)(α) (7.49)

where

Q(α) = 〈A(δ(a− α))〉 (7.50)

is the so-called Husimi Q-function [3]. To determine it for a given state, we calculate

A(δ(a− α)) =1

π2

∫d2β A

(eβ(α∗−a†)−β∗(α−a)

)=

1

π2

∫d2β eβα

∗−β∗α eβ∗a e−βa

†

=1

π3

∫d2β

∫d2γ eβα

∗−β∗α eβ∗a|γ〉〈γ|e−βa† (completeness)

=1

π3

∫d2β

∫d2γ eβ(α∗−γ∗)−β∗(α−γ)|γ〉〈γ|

=1

π|α〉〈α| (7.51)

which implies

Q(α) = 〈A(δ(α− a))〉 =1

πTr[ρ|α〉〈α|] =

1

π〈α|ρ|α〉 . (7.52)

[3] K. Husimi, Some formal properties of the density matrix, Proc. Phys. Math. Soc. Jpn. 22, 264 (1940) [Husimi40]


Due to the properties of a density matrix, 〈ψ|ρ|ψ〉 ∈ [0, 1], the Q-function is non-negative andbounded:

0 ≤ Q(α) ≤ 1

π. (7.53)

It hence resembles a classical probability distribution more closely than the P -function, butin contrast to the former, it cannot become arbitrarily sharply peaked.

Finally, by using symmetric operator ordering

f(a, a†) = f (S)(a, a†) (7.54)

with

S(f (S)(a, a†)) = f (S)(α) (7.55)

we can represent quantum averages

〈f (S)(a, a†)〉 =

∫d2αW (α)f (S)(a, a†) (7.56)

via the Wigner function

W (α) = 〈δ(α− a)〉 . (7.57)

Note that our definition of the delta function is already symmetric, so no additional orderingis required. Symmetric ordering is maybe the most natural representation, as it involveshermitian operators corresponding to quantum-mechanical observables.

The properties of the Wigner function lie between those of the P - and Q-functions. Toreveal its general range of possible values, we perform the following steps: as seen fromEq. (11.60), we have

D(α) = e−|α|2/2N

(D(α)

)(7.58)

which implies

δ(a− α) =1

π2

∫d2β e−

12|β|2 N

(eβ(a†−α∗)−β∗(a−α)

)=

1

π2

∫d2β e−

12|β|2 eβ

∗α−βα∗ N(

eβa†−β∗a

)=

1

π2

∫d2β e−

12|β|2 eβ

∗α−βα∗∫

d2γ eβγ∗−β∗γ N (δ(a− γ))

(normal-ordereddelta function)

=1

π2

∫d2γ

∫d2β e−

12|β|2 eβ

∗(α−γ)−β(α∗−γ∗)N (δ(a− γ)) . (7.59)

The integral over β can then be performed using∫d2β e−

12|β|2+β∗α−βα∗ =

∫d2β e−

12|β−2α|2 e−2|α|2 = e−2|α|2

∫d2β e−

12|β|2 = 2π e−2|α|2 (7.60)

leading to

δ(a− α) =2

π

∫d2γ e−2|α−γ|2 N (δ(a− γ)) , (7.61)

7.2 Examples 99

which expresses the symmetric operator-valued delta function in terms of the normal-ordereddelta function. It implies the useful relation

W (α) =2

π

∫d2β e−2|α−β|2 P (β) (7.62)

for obtaining the Wigner function from the P -function. The integral in Eq. (7.61) can becarried out to give

δ(a− α) =2

πN(

e−2(a−α)(a†−α∗))

=2

πD(α)N

(e−2n

)D†(α) . (7.63)

Finally, we use the relation

N(nk

k!e−n)

= |k〉〈k| (7.64)

which can be proven by taking its matrix elements for arbitrary coherent states 〈α| and |β〉 tofind

δ(a− α) =2

πD(α)N

(e−n e−n

)D†(α) =

2

πD(α)

∑k

(−1)kN(nk

k!e−n)D†(α)

=2

πD(α)

∑k

(−1)k|k〉〈k|D†(α) =2

πD(α)(−1)n

∑k

|k〉〈k|D†(α)

=2

πD(α)(−1)−nD†(α) (completeness)

=2

π(−1)n(α) . (similarity tranformation) (7.65)

The (symmetric) delta function is hence related to the displacement parity operator and hencethe Wigner function has a range

W (α) =2

π〈(−1)−n(α)〉 ∈

[− 2

π,

2

π

]. (7.66)

It is hence bounded but can assume negative values, in stark contrast to classical probabilitydistributions.

7.2 Examples

Let us next study examples of quasi-probability distributions for some simple states. Webegin with a coherent state |α0〉. The relation

ρ =

∫d2α P (α)|α〉〈α| (7.67)

in this particular case reads

|α0〉〈α0|=∫

d2α δ(α− α0)|α〉〈α| (7.68)


from which the P -function can be read off as

P (α) = δ(α− α0) . (7.69)

The state thus appears to have no fluctuations (its P -function being infinitely sharp). Thisis of course not the case, as the fluctuations reside in the c-number representation of thequantum observable in this case: we find

〈n〉 =

∫d2α |α|2δ(α− α0) = |α0|2 , (7.70)

but(∆n)2 = n2 − 〈n〉2 = a†aa†a− 〈n〉2 = a†2a2 + a†a− 〈n〉2 (7.71)

in normal order, so that

(∆n)2 =

∫d2α

(|α|4+|α|2

)δ(α− α0)− |α0|4= |α0|4+|α0|2−|α0|4= |α0|2 . (7.72)

The Wigner function can be obtained from the P -function by means of Eq. (7.62),

W (α) =2

π

∫d2β e−2|α−β|2 δ(β − α0) =

2

πe−2|α−α0|2 . (7.73)

We see that this quasi-probability distribution does exhibit fluctuations in the form of a finitewidth. The largest amount of fluctuations resides in the Q-function:

Q(α) =1

π〈α|α0〉〈α0|α〉 =

1

πe−|α−α0|2 . (7.74)

The three quasiprobability distributions are illustrated in the following figure for the state|α0 = 2 + 3i〉:

α0

2

4

2

4

Re(α)

Im(α)

P (α)

α0

2

4

2

4

Re(α)

Im(α)

W (α)

α0

2

4

2

4

Re(α)

Im(α)

Q(α)

For a Fock state |n〉, the P -function becomes even more singular,

P (α) =n∑k=0

(n

k

)1

k!

∂k

∂αk∂k

∂α∗kδ(α) (7.75)

while the Wigner function is oscillatory,

W (α) =2

π(−1)n e−2|α|2 Ln(4|α|2) (7.76)

with Ln(x): Laguerre polynomial, and the Q-function displays a Poisson-like profile

Q(α) =1

π〈α|n〉〈n|α〉 =

1

π

|α|2nn!

e−|α|2

. (7.77)

These results may be shown as an exercise. The Wigner and Q-functions of the Fock staten = 4 are displayed in the following figure

7.2 Examples 101

−2

2−2

2

Re(α)

Im(α)

Q(α)

−2

2−2

2

Re(α)

Im(α)

W (α)

One can further show that the P -function of a thermal state reads

P (α) =1

πn(ω)e−|α−λ|

2/n(ω) (7.78)

withn(ω) =

1

e~ω/(kBT )−1. (7.79)

Its width is hence determined solely by the (classical) thermal fluctuations.Finally, the Q-function of a squeezed state |ξ〉 reads

Q(α) =1

π〈α|ξ〉〈ξ|α〉 =

1

πµe−|α|

2− 12µ

(να∗2+ν∗α2) . (7.80)

This is a squeezed Gaussian whose equiprobability lines are ellipses, as seen from the follow-ing figure for µ = cosh(2), ν = sinh(2)

−2

2−2

2

Re(α)

Im(α)

Q(α)


We

summ

arizethe

resultsofthis

chapterin

atable:

Ordering

Norm

alSym

metric

Antinorm

al

c-number

functionN

(f(N

)(a,a†))

=f

(N)(a,a†)

S(f

(S)(a,a†))

=f

(S)(a,a†)

A(f

(A)(a,a†))

=f

(A)(a,a†)

Quasiprobability

P(α

)= ⟨N

(δ(a−α

)) ⟩W

(α)

= ⟨δ(a−α

) ⟩Q

(α)

= ⟨A(δ(a−

α)) ⟩

Range

unbounded−

2π≤W

(α)≤

2π0≤

Q(α

)≤1π

Fluctuationsin

c-number

functionm

ixedin

quasiprobability

Coherentstate|α

0 〉P

(α)

=δ(α−α

0 )W

(α)

=2π

e −2|α−

α0 | 2

Q(α

)=

1πe −|α−

α0 | 2

Fockstate|n〉

P(α

)=

n∑k

=0 (

nk )1k!

∂k

∂αk

∂k

∂α∗kδ(α

)W

(α)

=2π

(−1)n

e −2|α| 2

Ln (4|α| 2 )

Q(α

)=

1π

|α| 2n

n!

e −|α| 2

8 How can we show that light isnon-classical?

Spatiotemporal coherence of theelectromagnetic field



• understand the concepts of first- and second-order coherence,

• their implications for interference experiments,

• model classical deterministic and chaotic light

• use coherence properties to distinguish quantum from classical light

In the previous chapters, we have introduced different quantum states of light, determinedby the associated statistics of the field operators and visualized these states graphically viaquasiprobability functions. Along the way, we have encountered a range of signatures ofthe quantum nature of light such as sub-Poissonian number statistics or a negative Wignerfunction.

In this chapter, we intend to elucidate the practical implications of different states oflight for quantum optics interference experiments. The key concept is the (spatiotemporal)coherence of the electromagnetic field. We will first introduce this concept classically andthen generalize it to the quantum case. Comparing the coherence properties of light in thetwo cases, we will be able to find criteria for identifying non-classical light.

8.1 Classical coherence

Classically, electromagnetic waves are generated by moving charges. Let us consider a collec-tion of classical oscillating electric dipoles located in a finite source region and investigatethe properties of the resulting electromagnetic wave created far from the source region. Inparticular, we are interested in whether the radiation generated by the individual dipolescombines to a coherent wave with stable phase fronts.

104 Spatiotemporal coherence of the electromagnetic field

?

Source regionObservation region

The answer to this question obviously depends on whether the frequencies and phases of theemitters are stable. A single emitter α whose phase ϕα and frequency ωα are constant in timegenerates an electric field

Eα(r, t) = Eα(r) e−iωαt+iϕα (8.1)

at a given observation point. Note that in the following, we will for simplicity neglect thevector nature of the electric field, so that the scalar field E may represent field componentsin a chosen, fixed direction. This implies that we do not take into account polarization fornow. The coherence of the electromagnetic wave can be demonstrated by comparing the fieldamplitude Eα(r, t) with its value Eα(r, t + τ) at the same observation point, but later timet+ τ : the product

E∗α(r, t)Eα(r, t+ τ) = |Eα(r)|2e−ωατ (8.2)

shows stable oscillations as a function of the time delay for all times t.Next, we adress the question as to what degree the coherence persists under less perfect

conditions. Firstly, we consider a large number N of independent emitters α whose phasesϕα are uncorrelated. Assuming the source region to be small with respect to its distancefrom the observation point, the fields emitted by the individual atoms only differ by theirfrequencies and phases,

Eα(r, t) = E0 e−iωαt+iϕα (8.3)

so that the total emitted electric field can be given as

E(r, t) = E0

N∑α=1

e−iωαt+iϕα . (8.4)

The resulting correlation then reads

E∗(r, t)E(r, t+ τ) = |E0|2N∑

α,β=1

ei(ωα−ωβ)t e−iωβτ e−i(ϕα−ϕβ) (8.5)

In the limit of large N , the uncorrelated phases lead to a vanishing of the off-diagonalcontributions α 6= β of the sum

N∑α=1

eiϕα = 0 =⇒N∑

α,β=1

e−i(ϕα−ϕβ) · · · =N∑

α,β=1

δαβ · · · , (8.6)

so that only the diagonal terms remain

E∗(r, t)E(r, t+ τ) = |E0|2N∑α=1

e−iωατ . (8.7)

8.1 Classical coherence 105

Secondly, we take into account that the frequencies of the emitters are not exactly equal,but are distributed around an average ω0. This can be accounted for by the normalized powerspectral density F (ω) which represents the probability of a given dipole to emit radiation offrequency ω: ∫ ∞

−∞dω F (ω) = 1 . (8.8)

In the large-N limit, the sum over emitters α becomes a continuous average over the normal-ized power spectrum according to

N∑α=1

−→ N

∫ ∞−∞

dω F (ω) , (8.9)

so that

E∗(r, t)E(r, t+ τ) = N |E0|2∫ ∞−∞

dω F (ω) e−iωτ . (8.10)

Here, the bar indicates the classical average over the emitters. To remove the prefactors Nand |E0|2, we note that

E∗(r, t)E(r, t) = N |E0|2∫ ∞−∞

dω F (ω) = N |E0|2 (8.11)

and measure the field-field correlation function via the normalized first-order coherence

g(1)(τ) =E∗(r, t)E(r, t+ τ)

|E(r, t)|2. (8.12)

The above Eq. (8.10) then shows that

g(1)(τ) =

∫ ∞−∞

dω F (ω) e−iωτ . (8.13)

This is the Wiener–Khinchine theorem [1], [2] relating the field–field correlation function inthe time domain to the spectrum of the emitters.

To evaluate the coherence of the light generated by our ensemble of dipoles more explicitly,we need to specify the normalized power spectral density F (ω). Typical mechanisms leadingto a finite emission line width for atoms are radiative, collisional and Doppler broadening.Radiative broadening is associated with the spontaneous decay of an exited atomic state withrate Γ, accompanied by the emission of a photon. This process, which we will be studied inmore detail in Part II of this lecture series, leads to a Lorentzian profile

F (ω) =1

π

Γ

(ω0 − ω)2 + Γ2. (8.14)

[1] N. Wiener, Generalized harmonic analysis, Acta Mathematica 55, 117 (1930) [Wiener30].[2] A. Khinchine, Korrelationstheorie der stationären stochastischen Prozesse, Math. Ann. 109, 604 (1934)

[Khinchine34].


The associated first-order coherence can easily be evaluated by contour integration

g(1)(τ) =Γ

π

∫ ∞−∞

dωe−iωτ

(ω0 − ω)2 + Γ2=

Γ

π

∫ ∞−∞

dωe−iωτ

[ω − (ω0 + iΓ)] [ω − (ω0 − iΓ)]

= e−iω0τ−Γ|τ | (8.15)

where we have closed the integration contour by infinite semicircles in the lower (for τ > 0)or upper (for τ < 0) halves of the complex frequency plane, leading to residues from the polesat ω = ω0 − iΓ and ω = ω0 + iΓ, respectively. The result shows that the first-order coherenceof light from a source with radiative broadening decays exponentially on a time scale set bythe coherence time τc = 1/Γ:

g(1)(τ) = e−iω0τ−|τ |/τc . (8.16)

Collisional broadening results from interruptions of emission from atoms due to their colli-sions. Typically, the collision time is much shorter than the average emission time and theemission is resumed after the collision, albeit with a new, random initial phase. The spectrumof emitters with collisional broadening is given by a Lorentzian spectrum (8.14) where thewidth Γ is now the inverse of the average time between collisions.

Finally, Doppler broadening is due to the frequency shift arising when the emitters aremoving. For an ensemble of randomly moving emitters, the resulting spectrum reads

F (ω) =1√

2π∆e−(ω0−ω)2/2∆2

. (8.17)

and the first-order coherence is hence given by

g(1)(τ) = e−iω0τ− 12

∆2τ2 = e−iω0τ−π2 (τ/τc)2 (8.18)

where the coherence time is now τc =√π/∆.

We compare these results for classical chaotic light emitted by radiatively, collisionally orDoppler broadened sources with classical deterministic (monochromatic) light with

E(r, t) = E(r) e−iω0t+iϕ , (8.19)

which has a first-order coherence

g(1)(τ) =E∗(r, t)E(r, t+ τ)

|E(r, t)|2 = eiω0τ (8.20)

and is hence coherent for all delays τ .While first-order coherence refers to field–field correlations as a function of time delay,

second-order coherence is a measure of intensity-intensity correlations:

g(2)(τ) =I(r, t)I(r, t+ τ)

I(r, t)2 (8.21)

whereI(r, t) = E∗(r, t)E(r, t) (8.22)

8.1 Classical coherence 107

is a normalized intensity. For deterministic light where no averaging is required, we obviouslyhave

g(2)(τ) = 1 . (8.23)

To determine the second-order coherence of chaotic light as generated from our ensemble ofdipole emitters, we calculate

I(r, t)I(r, t+ τ) =N∑

α,β,γ,δ=1

E∗α(r, t)Eβ(r, t)E∗γ(r, t+ τ)Eδ(r, t+ τ)

=N∑α=1

E∗α(r, t)Eα(r, t)E∗α(r, t+ τ)Eα(r, t+ τ)

+N∑

α,β=1α 6=β

E∗α(r, t)Eα(r, t)E∗β(r, t+ τ)Eβ(r, t+ τ)

+N∑

α,β=1α6=β

E∗α(r, t)Eα(r, t+ τ)E∗β(r, t+ τ)Eβ(r, t) (8.24)

where we have recalled Eq. (8.3) and noted that all other terms vanish in the large-N limitdue to the uncorrelated phases. Finally, we note that the first term above scales as N and ishence dominated by the other two terms in the large-N limit, which each scale as N(N − 1).We hence have

I(r, t)I(r, t+ τ) ' |E(r, t)|22+ E∗(r, t)E(r, t+ τ) E∗(r, t+ τ)E(r, t) (8.25)

since |E(r, t+ τ)|2 = |E(r, t)|2 for our stationary light source. Using the definitions of first-and second-order coherence, this implies

g(2)(τ) = 1 + |g(1)(τ)|2 (8.26)

for chaotic light, leading tog(2)(τ) = 1 + e−2|τ |/τc (8.27)

for a radiatively/collisional broadened source with Lorentzian spectrum and

g(2)(τ) = 1 + e−π(τ/τc)2 (8.28)

for a Doppler broadened source with a Gaussian spectrum.To finish this section, we discuss some general properties of first- and second-order

coherence in the classical case. By definition, the first-order coherence at zero time delay isalways unity,

g(1)(0) = 1 . (8.29)

By virtue of the Cauchy-Schwartz inequality for classical averages, the second-order coher-ence at zero time delay is subject to the constraint

g(2)(0) =I2(r, t)

I(r, t)2 ≥ 1 . (8.30)


For a finite time delay, we haveg(2)(τ) ≥ 0 , (8.31)

because the averaged quantity I(r, t)I(r, t+ τ) is non-negative. Again, by using a Cauchy–Schwartz type inequality, one can show that the second-order coherence is decreasingmonotonously:

g(2)(0) ≥ g(2)(τ) . (8.32)

We summarize the coherence properties of deterministic versus chaotic classical light in thefollowing table.

Classical light DeterministicChaotic

Lorentzian Gaussian

First-order coherence: g(1)(τ) e−iω0τ e−iω0τ−|τ |/τc e−iω0τ−π2 (τ/τc)2

• zero delay |g(1)(0)| 1 1 1

• long delay |g(1)(∞)| 1 0 0

• general properties g(1)(0) = 1

Second-order coherence: g(2)(τ) 1 1 + e−2|τ |/τc 1 + e−π(τ/τc)2

• zero delay |g(2)(0)| 1 2 2

• long delay |g(2)(∞)| 1 1 1

• general properties g(2)(0) ≥ 1 , g(2)(τ) ≥ 0 , g(2)(0) ≥ g(2)(τ)

8.2 Relevance of classical coherence in experiments

Physically, the first-and second-order coherences characterize the ability of light to forminterference patterns. An important element in optical interference is the beam splitter

8.2 Relevance of classical coherence in experiments 109

E1 E ′1

E2

E ′2

It takes two input fields E1, E2 and creates two output fields E ′1, E ′2 via transmission andreflection (

E ′1E ′2

)= R

(E1

E2

)(8.33)

where

R =

(T RR T

)(8.34)

contains the reflection and transmission coefficients of the beam splitter. Its symmetric natureis reflected by the fact that one and the same reflection coefficient R appears twice in thereflection/transmission matrixR (and the same holds for the transmission coefficient T). Inthe absence of losses, we must have

|E ′1|2+|E ′2|2=

(E1

E2

)†· R†R ·

(E1

E2

)!

=

(E1

E2

)†·(E1

E2

)(8.35)

showing that the reflection/transmission matrix must be unitary:

R†R = 1 . (8.36)

Two identical symmetric beam splitters can be combined with two perfect mirrors to form aMach–Zehnder interferometer [3], [4]:

E

z2

z1

E ′

The output field is given by

E ′(t) = TRE(t− z1/c) +RTE(t− z2/c) , (8.37)

[3] L. Zehnder, Ein neuer Interferenzrefraktor, Z. Instrum. 11, 275 (1891) [Zehnder1891].[4] L. Mach, Über eine Interferenzrefraktor, Z. Instrum. 12, 89 (1892) [Mach1892].


where we have taken into account that the optical paths z1 and z2 are traversed with thespeed of light c. The resulting output intensity thus reads

I ′(t) = |R|2|T |2[|E(t− z1/c)|2 + |E(t− z2/c)|2

+ E∗(t− z1/c)E(t− z2/c) + E∗(t− z2/c)E(t− z1/c)]

. (8.38)

Noting that for the stationary light sources considered so far, we may shift time argumentsunder averages by arbitrary amounts without changing the result and recalling our definitionof the first-order coherence, we find

I ′(t) = 2|R|2|T |2I(t)

[1 + Re g(1)

(z1 − z2

c

)]. (8.39)

For instance, for chaotic light with a Lorentzian spectrum, this leads to

I ′(t) = 2|R|2|T |2I(t)

[1 + e−|z1−z2|/(cτc) cos

(z1 − z2

cω0

)]. (8.40)

As expected, the Mach-Zehnder interferometer shows an oscillatory signal as a function ofthe optical path difference z1 − z2. The visibility V of this interference pattern is given by thefirst-order coherence

V =Imax − Imin

Imax + Imin

=

∣∣∣∣g(1)

(z1 − z2

c

)∣∣∣∣ ; (8.41)

it exponentially decreases for chaotic light with increasing optical path difference.We can generalize our notion of first-order coherence to characterize the ability of light to

form spatiotemporal interference patterns:

g(1)(r1, t1, r2, t2) =E∗(r1, t1)E(r2, t2)√|E(r1, t1)|2

√|E(r2, t2)|2

. (8.42)

It can be used to determine the visibility of interference, e.g. in the double-slit experiment.The second-order coherence, on the other hand, governs intensity–intensity correlations,

which can for instance be observed in the Hanbury Brown–Twiss interferometer [5].

I

I2

I1

D2

D1

C

[5] R. Hanbury Brown, R. Q. Twiss, A test of a new type of stellar interferometer on Sirius, Nature 177, 27 (1956)[Hanbury56]

8.3 Quantum coherence 111

Here, a light beam of intensity I is incident on a 50 : 50 beam splitter

R =1√2

(1 ii 1

). (8.43)

The output intensities I1 and I2 are detected by detectors D1 and D2 and the signal is trans-fered to a correlator which is designed to measure

C(τ) =

[I1(t)− I1(t)

] [I2(t+ τ)− I2(t+ τ)

]I1(t) I2(t+ τ)

=I(t)I(t+ τ)− I(t)

2

I(t)2 = g(2)(τ)− 1 (8.44)

for a stationary light source. For a chaotic light source with a Lorentzian spectrum, we expect

C(τ) = e−2τ/τc . (8.45)

The stationary Hanbury Brown–Twiss interferometer hence provides a direct observation ofsecond-order coherence. For more general spatiotemporal interference experiments, one candefine a second-order coherence

g(2)(r1, t1, r2, t2) =I(r1, t1)I(r2, t2)

I(r1, t1) I(r2, t2). (8.46)

We say that a light source is first- and second-order coherent if∣∣g(1)(r1, t1, r2, t2)∣∣ = 1 , (8.47)

g(2)(r1, t1, r2, t2) = 1 (8.48)

hold, respectively.

8.3 Quantum coherence

We have defined the first- and second-order coherence of the classical electromagnetic fieldvia correlations of field amplitudes and intensities, averaged over an ensemble of dipoleemitters in the source. To transfer these notions to the quantum case, we need to replaceclassical averages over the source by quantum averages over the prepared state of the field.

Again neglecting the vector nature of the electromagnetic field, the electric-field operatorcan be given as

E(r, t) =∑j

[iωjAj(r)aj e−iωjt−iωjA

∗j(r)a†j eiωjt

]≡ E(+)(r, t) + E(−)(r, t) (8.49)

where the positive-frequency part of the field E(+) contains only annihilation operatorsand the negative-frequency part E(−) contains only creation operators. When replacingclassical averages with quantum averages of field operators, one has to bear in mind thatphotodetectors probe the quantum electromagnetic field via the absorption of real photonsaccording to the photoelectric effect. Rather than measuring 〈E(r, t)E(r, t)〉, they are only


sensitive to those terms in field products which are in normal order, e.g. 〈E(−)(r, t)E(+)(r, t)〉,where only real photons present in the electromagnetic field can lead to a signal, whereas thevacuum fluctuations cannot be observed directly:

〈0|E(−)(r, t)E(+)(r, t)|0〉 = 0 . (8.50)

We hence define the observable first- and second-order coherence of the electric field as

g(1)(τ) =〈E(−)(r, t)E(+)(r, t+ τ)〉〈E(−)(r, t)E(+)(r, t)〉

, (8.51)

g(2)(τ) =〈E(−)(r, t)E(−)(r, t+ τ)E(+)(r, t+ τ)E(+)(r, t)〉

〈E(−)(r, t)E(+)(r, t)〉2(8.52)

with obvious generalizations to the spatiotemporal case:

g(1)(r1, t1, r2, t2) =〈E(−)(r1, t1)E(+)(r2, t2)〉√

〈E(−)(r1, t1)E(+)(r1, t1)〉√〈E(−)(r2, t2)E(+)(r2, t2)〉

, (8.53)

g(2)(r1, t1, r2, t2) =〈E(−)(r1, t1)E(−)(r2, t2)E(+)(r2, t2)E(+)(r1, t1)〉〈E(−)(r1, t1)E(+)(r1, t1)〉〈E(−)(r2, t2)E(+)(r2, t2)〉

. (8.54)

Let us apply these notions to the single-mode case. With

E(+)(r, t) ∝ a e−iωt , E(−)(r, t) ∝ a† eiωt , (8.55)

we find

g(1)(τ) =〈a†a〉 e−iωτ

〈a†a〉 = e−iωτ . (8.56)

Irrespective of the prepared state, single-mode quantum light is hence always first-ordercoherent with |g(1)(τ)|= 1, just like deterministic classical light. This is due to the fact thatsingle-mode quantum light is perfectly monochromatic. In particular, the identical first-ordercoherences of deterministic classical and single-mode quantum light implies that field–fieldinterference experiments such as Mach–Zehnder or double-slit interference cannot be used todistinguish quantum from classical light. When reducing the intensity incident on a doubleslit such that single photons enter the interference region one after another, then the signalaccumulated from many such photons will form the same interference pattern expected fromclassical wave optics.

So in order to demonstrate quantum effects, we must turn our attention to the second-order coherence

g(2)(τ) =〈a†2a2〉〈a†a〉2 = g(2)(0) . (8.57)

It turns out to be independent of the delay τ , but unlike the first-order coherence, it is sensitiveto the specific quantum state of the electromagnetic field.

Let us consider a few examples. For a Fock state with n 6= 0, we have

g(2)(0) =〈n|a†2a2|n〉〈n|a†a|n〉2 =

〈n|n2 − n|n〉〈n|n|n〉2 =

n2 − nn2

= 1− 1

n. (8.58)


This violates the classical constraint

g(2)(0) ≥ 1 . (8.59)

The fact that g(2)(0) < 1 is equivalent to the respective quantum state having sub-Poissoniannumber statistics, as is easily seen from

g(2)(0) =〈n2〉 − 〈n〉〈n〉2 = 1 ⇔ (∆n)2

〈n〉 =〈n2〉 − 〈n〉2〈n〉 = 1 . (8.60)

For a thermal state, the results of Sect. 6.2 can be used to show that

g(2)(0) = 2 , (8.61)

which agrees with our findings for the classical chaotic field, showing that thermal quantumlight is equivalent to classical chaotic light in terms of its coherence properties.

For a coherent state with α 6= 0, we find

g(2)(0) =〈α|a†2a2|α〉〈α|a†a|α〉2 =

|α|4|α|4 = 1 . (8.62)

This explains the name coherent state: light in this state is first- and second-order coherent,just like classical deterministic light.

Finally, for the squeezed vacuum with |ν|6= 0, we find

g(2)(0) = 3 +1

|ν|2 = 3 +1

〈n〉 > 1 . (8.63)

This is an example of light with a super-Poissonian photon number statistics.In general, the second-order coherence does not necessarily obey the classical constraint

g(2)(0) ≥ 1 , (8.64)

but only the weaker relationg(2)(0) ≥ 0 (8.65)

which follows from Eq. (8.54) together with the positivity of the operator a†2a2.Our results found for the coherence of single-mode quantum light are summarized in the

following table:

Quantum light Fock Coherent Thermal Squeezed

First-order coherence: g(1)(τ) e−iωτ

• general properties |g(1)(τ)|= 1

Second-order coherence: g(2)(τ) 1− 1

n1 2 3− 1

〈n〉

• general properties g(2)(τ) = g(2)(0) ≥ 0


Our result that the second-order coherence is independent of the delay is again due to ourimplicit assumption of single-mode, i.e. perfectly monochromatic light. This assumption isparticularly unrealistic for thermal light, which is intrinsically a broadband phenomenon.

When studying the dynamics of second-order coherence allowing for multimode states,one finds that the classical constraint

g(2)(0) ≥ g(2)(τ) (8.66)

is not necessarily always obeyed by quantum light. This phenomenon is known as photonantibunching. It describes a tendency of photons to appear one after another — in contrast tothe classical bunching where the intensity–intensity correlation is largest for small delays.We will encounter antibunching later when investigating the light created by a single atom.

The intensity–intensity correlations can again be measured in a straightforward way usingthe Hanbury Brown–Twiss interferometer. In the quantum case, the beamsplitter relatescreation and annihilation operators of incoming and outgoing modes via(

a′1a′2

)=

(T RR T

)(a1

a2

)= R

(a1

a2

)(8.67)

whereR is again a unitary matrix withR†R = 1, implying

|R|2+|T |2= 1 , RT ∗ +R∗T = 0 . (8.68)

Suppose that we have a single-photon input at arm 1 of the beamsplitter:

|Ψin〉 = |11, 02〉 = a†1|0〉 . (8.69)

The output state is obtained from the inverse transformation(a1

a2

)= R†

(a′1a′2

)=

(T ∗ R∗

R∗ T ∗

)(a′1a′2

)(8.70)

to be

a†1 = T a′†1 +Ra′†2 =⇒ |Ψout〉 =(T a′†1 +Ra′†2

)|0〉 = T |1′1, 0′2〉+R|0′1, 1′2〉 . (8.71)

The Hanbury Brown–Twiss experiment now measures the correlation

〈n′1n′2〉 = [T ∗〈1′1, 0′2|+R∗〈0′1, 1′2|] n′1n′2 [T |1′1, 0′2〉+R|0′1, 1′2〉] = T ∗R +R∗T = 0 . (8.72)

As expected, the second-order correlation at zero delay vanishes in the quantum case for ann = 1 Fock state, recall that g(2)(0) = 1− 1/n. This is seen to be consequence of the fact thatonly one photon passes the interferometer which cannot make both detectors click at thesame time.

A single 50 : 50 beamsplitter can be used to observe another quantum effect of light, theHong–Ou–Mandel effect [6]. We start from a two-photon input state

|Ψin〉 = |11, 12〉 = a†1a†2|0〉 (8.73)

[6] C. K. Hong, Z. Y. Ou and L. Mandel, Measurement of subpicosecond time intervals between two photons byinterference, Phys. Rev. Lett. 59 2044 (1987) [Hong87].


and calculate the output state via(a1

a2

)= R†

(a′1a′2

)=

1√2

(1 −i−i 1

)(a′1a′2

)=

1√2

(a′1 − ia′2−ia′1 + a′2

)(8.74)

to find|Ψout〉 =

1

2(a†′1 + ia†′2 )(ia†′1 + a†′2 )|0〉 =

i√2|21, 02〉+

i√2|01, 22〉 . (8.75)

In particular, the amplitude of the state |11, 12〉 vanishes for the 50 : 50 beamsplitter, so thatthe photons always leave the experiment in a common arm. The effect sensitively depends onthe input photons being identical (in terms of frequency, temporal profile and polarization)and can hence be used to test the distinguishability of photons.

PART IIAtoms & Photons

9 How can we exploit the non-classicalproperties of light?

Quantum coherence and itsapplications



• distinguish coherent and incoherent quantum states,

• be aware of the consequences of the no-cloning theorem,

• understand the basic idea of quantum cryptography and

• be aware of the essence of quantum computing and the Shor algorithm.

In the previous chapter, we have discussed the spatiotemporal coherence of the electro-magnetic fields which governs its ability to form interference patterns. In this chapter, weshall introduce quantum coherence as a different, more fundamental property that can befound not only in the electromagnetic field, but for any quantum system.

We will further see how quantum coherence could be exploited in applications to crypto-graphy and computing.

9.1 Quantum coherence

The quantum state of a physical system is described by a normalized vector in a Hilbert space.The space being linear, two given vectors |ψ1〉 and |ψ2〉 can form coherent superposition states

|ψ〉 = cos θ|ψ1〉+ sin θ eiφ|ψ2〉 (9.1)

with a well-defined relative phase φ. The density matrix of such a coherent superpositionmatrix reads

ρ = |ψ〉〈ψ|= cos2 θ|ψ1〉〈ψ1|+ sin θ cos θ e−iφ|ψ1〉〈ψ2|+ sin θ cos θ eiφ|ψ2〉〈ψ1|+ sin2 θ|ψ2〉〈ψ2| .(9.2)

Its diagonal elements encode the probability of the system to be in either of the two super-posed quantum states,

p1 = 〈ψ1|ρ|ψ1〉 = cos2 θ , p2 = 〈ψ2|ρ|ψ2〉 = sin2 θ , (9.3)

120 Quantum coherence and its applications

while the off-diagonal elements carry information about the relative phase of the superposi-tion. This is a genuine quantum concept; it has no analogue in a classical description, wherewe only talk about probability as opposed to states.

If the phase of a superposition state is completely undetermined, one ends up with anincoherent superposition state

ρincoherent =1

2π

∫ 2π

0

dφ ρ = cos2 θ|ψ1〉〈ψ1|+ sin2 θ|ψ2〉〈ψ2| (9.4)

which is described by a diagonal density matrix and encodes classical randomness. Note thatcoherent superposition states are pure states while incoherent superposition states are mixedstates.

The coherent and squeezed states encountered in Chapter 6 are coherent superpositionstates of Fock states in this sense, while the thermal state is an incoherent superposition ofFock states. To see that coherent and incoherent superposition states can lead to very differentstatistics, recall that a coherent state

|α〉 =∑n

αn√n!

e−12|α|2|n〉 (9.5)

has an average field strength

〈F 〉 = 〈α|(Ca+ C∗a†)|α〉 = Cα + C∗α∗ (9.6)

while the respective incoherent superposition state

ρ =∑n

|α|2nn!

e−|α|2|n〉〈n| (9.7)

has a vanishing field strength

〈F 〉 = Tr[ρF]

=∑n

|α|2nn!

e−|α|2 (C〈n|a|n〉︸︷︷︸

=0

+ C∗〈n|a†|n〉︸︷︷︸=0

)= 0 (9.8)

9.2 No-cloning theorem

The quantum state of a system encodes its complete statistics, i.e. it can be used to predictaverages and fluctuations for repeated measurements of any observable. A measurementtypically destroys or changes the quantum state. The full statistical information encodedin the state can hence only be extracted by performing a large number of measurements ona number of systems prepared in identical states. This is the central task of quantum statetomography.

Now suppose that we only have one system prepared in a unknown quantum state |ψ〉 andthat we want to determine this state via measurements. A single measurement is obviouslynot sufficient to gain all information encoded in the state, so it would be advantageous toprepare several copies of the state and them perform measurements on each copy.

9.3 Quantum cryptography 121

The no-cloning theorem [1] states that this is not possible: suppose that we have a singlesystem prepared in an arbitrary unknown quantum state |ψ〉. A hypothetical cloning device,described by a unitary evolution operator U , transforms the pair of states |ψ〉 ⊗ |φ〉 (where |φ〉is the initial state of a second system) to |ψ〉 ⊗ |ψ〉, so that we end up with two systems in thestate |ψ〉:

U |ψ〉 ⊗ |φ〉 = |ψ〉 ⊗ |ψ〉 . (9.9)

If this device is universal, it should work for arbitrary states

U |ψ1〉 ⊗ |φ〉 = |ψ2〉 ⊗ |ψ1〉 , U |ψ2〉 ⊗ |φ〉 = |ψ2〉 ⊗ |ψ2〉 . (9.10)

We can now construct a contradiction by considering the superpostion state

|ψ〉 =1√2

(|ψ1〉+ |ψ2〉

). (9.11)

Due to U being unitary, it leads to a result

U |ψ〉 ⊗ |φ〉 =1√2

(U |ψ1〉 ⊗ |φ〉+ U |ψ2〉 ⊗ |φ〉

)=

1√2|ψ1〉 ⊗ |ψ1〉+

1√2|ψ2〉 ⊗ |ψ2〉 (9.12)

which is not the required result

|ψ〉 ⊗ |ψ〉 =1

2

(|ψ1〉 ⊗ |ψ1〉+ |ψ1〉 ⊗ |ψ2〉+ |ψ2〉 ⊗ |ψ1〉+ |ψ2〉 ⊗ |ψ2〉

). (9.13)

We have to conclude that no universal cloning device exists. The no-cloning theorem showsthat a quantum state is a precious resource that is invariably lost when making a measure-ment.

On the contrary, it is possible to transfer an unknown state |ψ〉 from one physical systemto a second one without destroying it

|ψ〉 ⊗ |φ〉 −→ |φ′〉 ⊗ |ψ〉 . (9.14)

This procedure is called quantum teleportation.

9.3 Quantum cryptography

The fact that a quantum state can only be measured once (and in particular, only by a singleperson) can be used to securely encrypt messages. To understand the principle of quantumcryptography, let us first consider a simple classical scheme to encrypt a message. One of thesimplest encryption procedures is the Cesar cipher, whereby every letter on the alphabet isreplaced by a shifted letter, as illustrated in the following for shift parameter s = 3:

Plain:

Cypher:

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

D E F G H I J K L M N O P Q R S T U V W X Y Z A B C

[1] W. Wootters and W. Zurek, A single quantum cannot be cloned, Nature 299, 802 (1982) [Wootters82].


In this way the words "QUANTUM OPTICS" become:

Q U A N T U M O P T I C S

T X D Q W X P R S W L F V

Once the shift parameter (which plays the role of the encryption key) is known, one can easilydecipher the encrypted message by reversing the shift. However, it turns out that sufficientlylong messages which are encrypted via the Cesar cipher with constant shift can easily bedecrypted by a third party without knowing the shift. For instance, one can compare thefrequency of letters in the encrypted message to the known average frequencies of lettersin a given language to deduce the shift (frequency analysis). For instance, the letter whichappears most frequently in the German language is "E". For a long encrypted message withconstant shift, one can hence simply check which letter is most frequent and deduce that thisletter must correspond to "E" in the original message. This identity can be used to infer theshift.

A completely secure alternative would be Cesar cipher where each letter is being en-crypted by an independent random shift. The resulting encrypted message is a completelyrandom sequence of letters which cannot be decrypted by a third party with statisticalmethods. Decryption is only possible by knowing the sequence of shifts as a key, which iscommonly known as a one-time-pad. The one-time-pad is as long as the message itself and itcan only be used once in order to maintain security.

One-time-pad encryption is a secure way to transmit messages from a sender Alice (A)to a receiver Bob (B), provided that Alice and Bob share a sufficiently long one-time-pad,i.e. they possess two identical sequences of random numbers which are not known to anyeavesdropper Eve (E). This can be achieved by means of quantum cryptography, where wewill discuss the BB84 protocol as an example [2].

We assume a scheme where the key is encoded in the polarization of a photon. Consider aplane electromagnetic wave travelling in the negative z-direction whose electric-field operatoraccording to the results of Chapter 4 is given by (k = −kez)

E(r) = i

√~ω

16π3ε0

e−ikz∑λ=1,2

ek,λak,λ + h.c. (9.15)

If we choose horizontal and vertical polarization unit vectors

ek,1 ≡ e =

100

, ek,2 ≡ e =

010

, (9.16)

so that the electric-field operator can be written as

E(r) = i

√~ω

16π3ε0

e−ikz∑

λ= ,

eλaλ + h.c. (9.17)

[2] C. H. Bennett and G. Brassard, Quantum cryptography: Public key distribution and coin tossing in Proceedingsof IEEE international Conference of Computers, Systems and Signal Processing, p. 175 (Bangalore 1985)[Bennett84].

9.3 Quantum cryptography 123

Alternatively, we can use diagonal polarization unit vectors as our basis:

ek,1 ≡ e =1√2

110

, ek,2 ≡ e =1√2

1−10

. (9.18)

The same electric-field operator then reads

E(r) = i

√~ω

16π3ε0

e−ikz∑

λ= ,

eλaλ + h.c. (9.19)

This implies that ∑λ= ,

eλaλ =∑

λ= ,

eλaλ . (9.20)

Using the fact that (ee

)=

1√2

(1 11 −1

)(ee

)(9.21)

we hence find (aa

)=

1√2

(1 11 −1

)(aa

). (9.22)

Single-photon Fock states |λ〉 ≡ a†λ|0〉 in the two bases can hence be related to one another via

| 〉 =1√2

(| 〉+ | 〉

), | 〉 =

1√2

(| 〉 − | 〉

), (9.23)

and

| 〉 =1√2

(| 〉+ | 〉

), | 〉 =

1√2

(| 〉 − | 〉

). (9.24)

The BB84 protocol for establishing a shared one-time-pad now proceeds as follows: a sequenceof polarization measurements corresponds to a sequence of bits 0, 1 according to

= 0 , = 1 , = 0 , = 1 . (9.25)

• Alice prepares a random sequence of photons which are each prepared in one of thefour states | 〉, | 〉, | 〉 or | 〉. She transmits these states to Bob and records both thecorresponding bits and polarization bases or .

• Bob generates a random sequence of measurement bases or . For each incomingphoton, he measures the polarization in the randomly chosen basis and records thebasis and the resulting bit.


• After the measurement, Alice and Bob openly declare to each other which basis theyhave used for which photon. Whenever these agree (i.e. for 50% of the events) they canbe sure that their recovered bits (which they keep secret) agree. E.g., if Alice has sentthe photon |ψ〉 = | 〉 from the basis (recorded bit: 0) and Bob has also used basis ,then he will have found the result with certainty:

p = |〈 |ψ〉|2 = |〈 | 〉|2 = 1 . (9.26)

If on the other hand, their measurement bases disagree, then Bob’s bits only agreewith Alice’s in 50% of the cases. E.g. if Bob had measured in the basis in the aboveexample, then he would have found (recorded bit: 0) with probability

p = |〈 |ψ〉|2 =1

2|(〈 |+〈 |)| 〉|2=

1

2(9.27)

or he could have found (recorded bit: 1) with the same probability

p = |〈 |ψ〉|2 =1

2|(〈 |−〈 |)| 〉|2=

1

2. (9.28)

In the latter case, his bit 1 would differ from that of Alice 0. By retaining only thosebit where they had used the same basis, Alice and Bob can hence establish a sharedrandom sequence of bits, to be used as a one-time-pad:

AlicePhotonBasis → tell Bob afterwardsBit 0 1 1 0 1 1 0 0 1 1 0 0

BobBasis → tell Alice afterwardsBits 0 1 1 1 0 1 1 0 1 1 0 0Shared bits 0 1 1 0 1 1 0

The protocol is secure against eavesdropping by virtue of the no-cloning theorem. Ifthe no-cloning theorem did not exist, then a potential eavesdropper Eve could intercepteach transmitted photon, make a faithful copy of its quantum state and pass the originalphoton to Bob. After learning about the openly declared basis sets used on each photon, shecould perform the identical measurements on her copied photons and reproduce the sharedsequence of bits.

However, due to the no-cloning theorem, Eve can only gain information on the transmittedphoton by making a measurement on that photon. Moreover, she has to do so before knowingwhich basis Bob is going to use (he only shares his basis after he has measured the photon,i.e. when it is too late for Eve to intercept it). Eve thus has to guess the measurement basis,make the measurement and then prepare a new photon according to her result in order topass it to Bob. This can then be noticed by Alice and Bob when comparing notes. Assume an

9.4 Quantum computing and Shor’s algorithm 125

event where Alice and Bob have used the same basis, say . If, for instance Alice has senta photon (bit: 0), then without interception Bob would observe the correct polarization(bit: 0) with certainty. If Eve intercepts and measured in the wrong basis , then she wouldrandomly find or (bits: 0 or 1) with 50% probability and have obtained no informationabout the shared bits. Even worse, when passing on the observed photon | 〉 or | 〉, thenBob would in each case have a 50% chance of measuring (bit: 1) in disagreement fromAlice’s transmitted bits. By comparing the bits where they had used the same basis, Aliceand Bob can thus detect if someone has been eavesdropping.

The BB84 protocol is hence secure in the sense that Alice and Bob can notice whether theirmessage has been intercepted. In a practical implementation, they would use some of theshared bits to encrypt a message and some to check whether eavesdropping has taken place.

9.4 Quantum computing and Shor’s algorithm

Quantum cryptography relies on the fact that unknown quantum states cannot be copied byan eavesdropper due to the no-cloning theorem and that a measurement of a state potentiallydestroys or alters this state.

Quantum computing relies on the fact that quantum systems can be prepared in coherentsuperposition states, allowing to explore a large parameter space in one operation. Forinstance, lets us assume that we have a function

f : 0, 1, ..., N − 1 → 0, 1x 7→ f(x)

(9.29)

and that we want to determine its period r:

f(x+ r) = f(x) . (9.30)

In principle, this would involve computing f(x) for each possible value of xwhich can be verytime-consuming. By means of quantum coherence, one can perform all these computationssimultaneously via a simple step, which is the essence of Shor’s algorithm [3].

To this end, one prepares a quantum system in a coherent superposition state

|ψx〉 =1√N

N−1∑x=0

|x〉 . (9.31)

One can show that for any given function f there is a unitary operator Uf such that

|ψyx〉 = Uf |ψx〉 ⊗ |0〉 =1√N

N−1∑x=0

|x〉 ⊗ |f(x)〉 . (9.32)

This state vector hence contains all values of f(x) at once and it was obtained in a singleoperation. However, these values can not all be read out simultaneously as each measurementdestroys the state.

[3] P. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, inProceedings of the 75th Annual Symposium on Foundations of Computer Science, p. 124 (Santa Fe, 1994)[Shor94].


Nevertheless, for determining the period of f , this is not necessary. Instead, one measuresthe system state to find f(x) = y. The state |ψxy〉 then collapses to the state

|ψ′x〉 =1√M

M−1∑j=0

|xj〉 (9.33)

where f(xj) = y. From this state, one can deduce the period of f via a second unitarytransformation (a discrete Fourier transform) and subsequent measurement.

In general, quantum computing exploits the fact that coherent superpositions allow forperforming many computations at once, i.e. in parallel. Since a coherent superposition statecan only be measured once, this principle can only be used for specific situations where sucha single measurement yields the answer by implicitly taking all intermediate results intoaccount. Apart from Shor’s algorithm, which can be used to factorize integers into productsof two prime numbers, this is the case, for instance, in Grover’s algorithm [4] which findsa given object among a set of N objects in a number of O(

√N) of steps (as opposed to the

O(N) steps required classically).The main challenge in implementing quantum computation schemes is the requirement

to prepare and coherently manipulate large coherent superposition states.

[4] L. K. Grover, A fast quantum mechanical algorithm for database search, in Proceeding of the 28th Annual ACMSymposium on the Theory of Computing, p. 122 (Philadelphia, 1996) Grover96.

10 What are the sources of theelectromagnetic field?

Canonical quantization ofelectrodynamics in the presence ofcharged particles



• recall the Maxwell-Lorentz equations and recast them into their Lagrangian andHamiltonian forms,

• use the correspondence principle to construct the Hamiltonian of nonrelativisticquantum electrodynamics in the presence of charged particles and

• be aware of a range of equivalent approximate schemes for describing atom–fieldinteractions.

In the first part of this lecture, we have been concerned with describing the electromagneticfield and its quantum-statistical properties without ever asking about the origin of this field.This question brings us to the second part of the lecture, which is concerned with light–matterinteractions. In line with our definition of quantum optics from the introduction, we willfocus on the interaction of the electromagnetic field with microscopic objects such as atoms ormolecules. In this chapter, we will lay the ground-work for studying atom-field interactionsby developing nonrelativistic quantum electrodynamics in the presence of charged particles.

10.1 The Maxwell–Lorentz equations

Let us assume a collection of charged particles with masses mα and charges qα which aresituated at positions rα. The particles will later be specified as electrons and nuclei formingone or several atoms or molecules. The charges give rise to a charge density

ρ(r) =∑α

qαδ(r − rα) (10.1)

and a current densityj(r) =

∑α

qαrαδ(r − rα) (10.2)

128 Canonical quantization of electrodynamics in the presence of charged particles

which naturally obey the continuity equation

ρ+∇ · j = 0 . (10.3)

The charge and current densities act as source for the electromagnetic field according to theMaxwell equations

ε0∇ ·E = ρ , (10.4)∇ ·B = 0 , (10.5)

∇×E + B = 0 , (10.6)1

µ0

∇×B − ε0E = j . (10.7)

Conversely, the electromagnetic field acts on the charged particles via Lorentz force

mrα = Fα = qαE(rα) + qαrα ×B(rα) . (10.8)

Together, the Maxwell–Lorentz equations completely specify the coupled particle–field dy-namics: they can be used to predict the positions and velocities of the particles and the valuesof the electromagnetic fields at all times provided that these values are known at some initialtime.

As in the case of electrodynamics without sources, one show that certain quantities areconserved with respect to the dynamics of the Maxwell–Lorentz equations:

H =∑α

1

2mαr

2α +

∫d3r

(ε0

2E2 +

1

2µ0

B2

), (energy) (10.9)

P =∑α

mαrα +

∫d3r ε0E ×B , (momentum) (10.10)

J =∑α

rα ×mαrα +

∫d3r r × (ε0E ×B) , (angular momentum) (10.11)

where now these quantities contain contributions from both particles and fields. For instance,using the Maxwell–Lorentz equations, one finds

H =∑α

mαrα · rα +

∫d3r

(ε0E · E +

1

µ0

B · B)

=∑α

[qαrαE(rα) + qα rα · (rα ×B(rα))︸︷︷︸

=0

]+

∫d3r

[E ·

(1

µ0

∇×B − j)− 1

µ0

B · (∇×E)

](Mawell–Lorentzequations)

=

∫d3r j ·E −

∫d3r j ·E − 1

µ0

∫d3r∇ · (E ×B)

(definition ofcurrent density)

= 0 , (10.12)

as the last term can be transformed into a vanishing surface integral.


10.2 Lagrangian electrodynamics

As in the case of the free electromagnetic field, we need to identify the independent degreesof freedom as a prerequisite for constructing a Lagrangian formulation of the theory. Oncemore, we introduce vector and scalar potentials for the electromagnetic fields according to

E = −∇Φ− A , (10.13)B =∇×A (10.14)

and work in Coulomb gauge:∇ ·A = 0 (10.15)

where E‖ = −∇Φ and E⊥ = −A. With these definitions, the inhomogeneous Maxwellequations lead to

∆Φ = − ρ

ε0

, (10.16)

∆A− 1

c2A− 1

c2∇Φ = −µ0j . (10.17)

Recalling that ∆(1/r) = −4πδ(r), the first equation is solved by the Coulomb potential

Φ(r) =∑α

qα4πε0|r − rα|

, (10.18)

hence the name Coulomb gauge. That scalar potential is hence not a dynamical variableof the system, as it can be given explicitly in terms of the position of the charged particles.In order to substitute this solution into the dynamical equation for the vector potential, wedecompose the current density into its longitudinal and transverse parts

j(r) = j‖(r) + j⊥(r)

=∑α

qαrα · δ‖(r − rα) +∑α

qαrα · δ⊥(r − rα) (10.19)

whereδ(r) = −∇⊗∇ 1

4πr, δ⊥(r) = δ(r) +∇⊗∇ 1

4πr(10.20)

are the longitudinal and transverse delta functions. One can then calculate

∇Φ(r) =1

ε0

∑α

qαd

dt∇ 1

4π|r − rα|= − 1

ε0

∑α

qαrα ·∇⊗∇1

4π|r − rα|=

1

ε0

j‖(r) , (10.21)

so that the Helmholtz equation for the (transverse) vector potential reads

∆A− 1

c2A = −µ0j

⊥ . (10.22)

Finally, we express equations of motion of the particles in terms of the vector potential andthe known Coulomb potential:

mrα = Fα =∑β 6=α

qαqβ(rα − rβ)

4πε0|rα − rβ|3− qαA(rα) + qαrα × [∇×A(rα)] (10.23)


where we have discarded the divergent Coulomb self-interaction arising for α = β. Theequations of motion (10.22) and (10.23) determine the complete dynamics of the system inthe Coulomb gauge as described by the independent degrees of freedom rα andA(r).

To find a Lagrangian formulation, we need to find a Lagrangian which reproduces theabove equations of motion via the Euler-Lagrange equations. In generalization of the free-field Lagrangian, we postulate

L =∑α

1

2mαr

2α +

∫d3r

(ε0

2E2 − 1

2µ0

B2

)+

∫d3r (j ·A− ρΦ) . (10.24)

This approach is known as minimal coupling, as it only involves an interaction of thevector potential with the current and charge densities of the particles themselves in theLorentz-invariant form jµA

µ, where no coupling via higher multipole moments of the chargedistribution arises. The minimal coupling term can be justified in quantum field theory byrequiring local gauge invariance. Calculating∫

d3rε0

2E2 =

∫d3r

ε0

2(∇Φ + A)2

= −ε0

2

∫d3r Φ∆Φ +

∫d3r ε0A ·∇Φ︸︷︷︸

=0

+

∫d3r

ε0

2A2 (partial integration)

=1

2

∫d3r Φρ+

∫d3r

ε0

2A2 , (10.25)

we can explicitly express the Lagrangian in terms of the dynamical variables:

L = Lp + LF + LPF . (10.26)

Here,

Lp =∑α

1

2mαr

2α −

1

2

∫d3r ρΦ =

∑α

1

2mαr

2 −∑α 6=β

qαqβ8πε0|rα − rβ|

(10.27)

is the particle Lagrangian, from which we have again removed divergent the self-interactionterms α = β;

LF =

∫d3r LF (10.28)

with the two equivalent Lagrange densities

LF =ε0

2A2 − 1

2µ0

(∇×A)2 (10.29)

orLF =

ε0

2A2 − 1

2µ0

(∇⊗A) : (∇⊗A) (10.30)

is the field Lagrangian; and

LPF =∑α

qαrα ·A(rα) =

∫d3r LPF (10.31)


withLPF = j⊥ ·A (10.32)

is the particle–field interaction. To verify that this is a legitimate Lagrangian, we have toderive the respective Euler–Lagrange equations. Starting from the particle coordinates, wehave

d

dt

∂L

∂rα=

d

dt

(∂LP∂rα

+∂LPF∂rα

)=

d

dt[mαrα + qαA(rα)]

= mαrα + qα(rα ·∇)A(rα) + qαA(rα) (chain rule) (10.33)

and∂L

∂rα=∂LP∂rα

+∂LPF∂rα

=∑β 6=α

qαqβ(rα − rβ)

4πε0|rα − rβ|3+ qα∇ [rα ·A(rα)] (10.34)

Combining these results, the Euler–Lagrange equations

d

dt

∂L

∂rα=

∂L

∂rα(10.35)

take the form

mαrα =∑β 6=α

qαqβ(rα − rβ)

4πε0|rα − rβ|3− qαA(rα) + qαrα × [∇×A(rα)] (10.36)

where we have used the identity rα × (∇ ×A) = ∇(rα ·A) − (rα ·∇)A. The Lagrangianhence generates the correct Lorentz equations (10.23) for the particle motion.

Next, we consider the Euler-Lagrange equations

d

dt

∂L∂A

=∂L∂A−∇ · ∂L

∂(∇⊗A)(10.37)

for the field. Recalling results from Chapter 3, we find

d

dt

∂L∂A

=d

dt

∂LF∂A

= ε0A , (10.38)

∂L∂A

=∂LPF∂A

= j⊥ , (10.39)

∇ · ∂L∂(∇⊗A)

= − 1

µ0

∆A , (10.40)

so that the Euler-Lagrange equation reads

ε0A = j⊥ +1

µ0

∆A , (10.41)

in agreement with the inhomogeneous Helmholtz equation (10.22).


10.3 Hamiltonian electrodynamics

As in the free-field case, we proceed by introducing canonical momenta. With our choice ofLagrangian, the canonical particle momenta read

pα =∂L

∂rα=∂LP∂rα

+∂LPF∂rα

= mαrα + qαA(rα) (10.42)

while the field momentum reads

Π =∂L∂A

=∂LF∂A

= ε0A = −ε0E⊥ . (10.43)

In contrast to the free-field case, the field momentum only provides the transverse part ofthe electric field, while an additional longitudinal part resides in the Coulomb potential. ALegendre transformation

H =∑α

rα · pα +

∫d3r A(r) ·Π(r)− L , (10.44)

followed by an elimination of the particle and field velocities via the above two relationsresults in

H = HP +HF +HPF (10.45)

with

HP =∑α

p2α

2mα

+∑α6=β


(10.46)

being the particle Hamiltonian;

HF =

∫d3rHF (10.47)

being the field Hamiltonian with Hamiltonian densities

HF =1

2ε0

Π2 +1

2µ0

(∇×A)2 (10.48)

or

HF =1

2ε0

Π2 +1

2µ0

(∇⊗A) : (∇⊗A) ; (10.49)

and

HPF = −∑α

qαmα

pα ·A(rα) +∑α

q2α

2mα

A2(rα) =

∫d3rHPF (10.50)

10.4 Canonical quantization 133

being the particle–field coupling with

HPF (r) = −∑α

qαmα

pα · δ⊥(r − rα) ·A(r) +∑α

q2α

2mα

A(r) · δ⊥(r − rα) ·A(r) . (10.51)

We leave it to the reader to show that the Hamiltonian equations for the particles

rα =∂H

∂pα, pα = − ∂H

∂rα(10.52)

and the fields

A =δH

δΠ, Π = −δH

δA(10.53)

indeed lead to the correct Maxwell–Lorentz equations.Finally, we generalize the free-field Poisson-bracket to include the particle degrees of

freedom

f, g =∑α

(∂f

∂rα

∂g

∂pα− ∂f

∂pα

∂g

∂rα

)+

∫d3r

(δf

δA· δgδΠ− δf

δΠ· δgδA

)(10.54)

so that the fundamental Poisson-bracket for the fields

A(r),Π(r′) = δ⊥(r − r′) (10.55)

becomes complemented by the particle Poisson-bracket

rα,pβ = δαβ1 . (10.56)

10.4 Canonical quantization

As in the free-field case, we may now use the correspondence principle

f −→ f

f, g −→ 1

i~[f , g]

(10.57)

to construct quantum electrodynamics in the presence of charged particles. The operator-valued field variables again obey the fundamental commutation relations

[A(r), Π(r′)] = i~δ⊥(r − r′) , (10.58)

[A(r), A(r′)] = [Π(r), Π(r′)] = 0 , (10.59)

while the particle coordinates an momenta obey the well-known commutation relations fromquantum mechanics

[rα, pβ] = i~1δαβ , (10.60)[rα, rβ] = [pα, pβ] = 0 . (10.61)


The Hamiltonian of the particle–field system reads

H = HP + HF + HPF (10.62)

with

HP =∑α

p2α

2mα

+∑α6=β


, (10.63)

HF =

∫d3r

1

2ε0

Π2(r) +1

2µ0

[∇× A(r)]2

=

∫d3r

1

2ε0

Π2(r) +1

2µ0

[∇⊗ A(r)] : [∇⊗ A(r)]

, (10.64)

HPF = −∑α

qαmα

pα · A(rα) +∑α

q2α

2mα

A2(rα) . (10.65)

Note that the replacements of pα → pα, rα → rα may be ambiguous in terms containingproducts of pα and rα (much in the same way as the replacements α→ a, α∗ → a† ambiguousin Chapter 7). However, the only potentially problematic term here is pα ·A(rα) where theCoulomb gauge ensures that

pα · A(rα) =~i∇α · A(rα)︸︷︷︸

=0

+A(rα) · pα = A(rα) · pα . (10.66)

The Heisenberg equations of motion generated by this Hamiltonian are the operator-valuedequivalents of the Maxwell–Lorentz equations. To demonstrate this, let us start with theparticle coordinates. This particle velocities are given by

˙rα =1

i~[rα, H

]=

1

i~[rα, HP

]+

1

i~[rα, HPF

]=

1

i~∑β

1

2mβ

[rα, p

2β

]− 1

i~∑β

qβmβ

[rα, pβ · A(rβ)

]=pα − qαA(rα)

mα

, (10.67)

in agreement with the classical relation (10.42) between velocities and canonical momenta.Considering the Heisenberg equation for ˙rα, we find

mα¨rα =

1

i~[mα

˙rα, H]

=1

i~[pα, HP

]+

1

i~[pα, HPF

]− 1

i~[qαA(rα), HP

]− 1

i~[qαA(rα), HF

]− 1

i~[qαA(rα), HPF

], (10.68)

We evaluate the different contributions1

i~

[pα, HP

]=

1

i~~i∇α

∑β 6=γ

qβqγ8πε0|rβ − rγ|

=∑β 6=α

qαqβ(rα − rβ)

4πε0|rα − rβ|3(10.69)

where we have used [pα, f(rα)] = ~i∇αf(rα) and∇f(r) = f ′(r)er;

1

i~[rα, HPF

]=

1

i~

[pα,−

∑β

qαmβ

pβ · A(rβ) +∑β

q2β

2mβ

A2(rβ)

]

=qαmα

∇α

[pα · A(rα)

]− q2

α

mα

[∇αA(rα)

]· A(rα) = qα∇

[˙rα · A(r)

]∣∣∣r=rα

(10.70)

10.4 Canonical quantization 135

where the above relation (10.67) for ˙rα has been used;

− 1

i~[qαA(rα), HP

]= − qα

mα

[pα ·∇

]A(r)

∣∣∣r=rα

, (10.71)

− 1

i~[qαA(rα), HF

]= − 1

i~

∫d3r

qα2ε0

[A(rα), Π2(r)

]︸︷︷︸=2i~Π(r)δ⊥(r−rα)

= −qαε0

Π(rα) = −qα ˙A(rα) (10.72)

(the last equality will be derived below);

− 1

i~[qαA(rα), HPF

]=

q2α

mα

[A(rα) ·∇

]A(r)

∣∣∣r=rα

. (10.73)

Combining the results, invoking Eq. (10.67) and using the rule ∇( ˙r · A) − ( ˙r · ∇)A =˙r × (∇× A), we recover the Lorentz equations

mα¨rα =

∑β 6=α

qαqβ(rα − rβ)

4πε0|rα − rβ|3− qα ˙

A(rα) + qα ˙rα ×[∇× A(rα)

]. (10.74)

For the field, we find

˙A(r) =

1

i~[A(r), H

]=

1

i~[A(r), HF

]=

1

i~

∫d3r′

1

2ε0

[A(r), Π2(r′)

]=

1

i~ε0

∫d3r′ δ⊥(r − r′) · Π(r′) =

1

ε0

Π(r) (10.75)

as in the free-field case and

¨A(r) =

1

i~

[1

ε0

Π(r), H

]=

1

i~

[1

ε0

Π(r), HF

]+

1

i~

[1

ε0

Π(r), HPF

](10.76)

with1

i~

[1

ε0

Π(r), HF

]=

1

ε0µ0

∆A(r) (10.77)

as in the free-field case and

1

i~

[1

ε0

Π(r), HPF

]=

1

i~

[1

ε0

Π(r),−∑α

qαmα

pα · A(rα) +∑α

q2α

2mα

A2(rα)

]

=1

ε0

∑α

qαmα

pα · δ⊥(r − rα) +1

ε0

∑α

q2α

mα

A(rα) · δ⊥(r − rα)

=1

ε0

∑α

qα ˙rα · δ⊥(r − rα) =1

ε0

j⊥(r) . (10.78)

Combining the results, we recover the Helmholtz equation for the electromagnetic vectorpotential:

∆A− 1

c2

¨A = −µ0j

⊥ . (10.79)

As in the free-field case discussed in Chapter 3, canonical quantization has led from theclassical equations of motion to the corresponding Heisenberg equations of motion. Inderiving them, we have had to carefully account for non-vanishing commutators of bothparticle and field variables.


10.5 Atom-field interactions

In most situations of practical interest to quantum optics, the charged particles α form oneor several electrically neutral bound aggregates, i.e. atoms or molecules (labeled here byA,B,C, ...). We restrict our attention to the single-atom case where the particle Hamilto-nian becomes the Hamiltonian of an atom A as well known from nonrelativistic quantummechanics:

HA =∑α∈A

p2α

2mα

+∑

α 6=β∈A


. (10.80)

To account for the fact that the atom is a bound system, we introduce centre-of-mass andrelatives coordinates:

rA =∑α

mαrαmA

, rα = rα − rA (10.81)

with mA =∑

αmα. The associated momenta are given by

pA =∑α∈A

pα , pα = pα −mα

mA

pA . (10.82)

The atomic Hamiltonian then takes the form

HA =p2A

2mA

+ H intA (10.83)

where

H intA =

∑α

p2

α

2mα

+∑

α 6=β∈A


=∑n

EAn |nA〉〈nA| (10.84)

is the internal atomic Hamiltonian governing the dynamics of the relative coordinates witheigenenergies EA

n and eigenstates |nA〉 as found in nonrelativistic quantum mechanics. Thespecific form of this Hamiltonian, together with the canonical commutation relations for theparticle coordinates and momenta, implies the useful rule

∑α

qαmα

〈mA|pα|nA〉 = iEAm − EA

n

~〈mA|

∑α∈A

qαrα|nA〉 = iωmndmn (10.85)

where ωmn = (EAm − EA

n )/~ are the atomic transition frequencies and dmn = 〈mA|d|nA〉 arematrix elements of the atomic dipole operator

d =∑α

qαrα . (10.86)

10.5 Atom-field interactions 137

10.5.1 Minimal couplingExpressed in terms of the relative atomic coordinates, the atom–field coupling Hamiltonianreads (HPF → HAF )

HAF = −∑α∈A

qαmα

(mα

mA

pA + pα

)· A(rA + rα) +

∑α∈A

q2α

2mα

A2(rA + rα) . (10.87)

In many situations of practical interest, the size of the atom as encoded in the relativecoordinates is much smaller than the wavelength of the electromagnetic field:

+ −

−

−

−

Field

Atom

In this case, we may perform the long-wavelength approximation by writing rA + rα ' rA,leading to a simplified interaction Hamiltonian

HAF = −∑α∈A

qαmα

pα · A(rA) +∑α∈A

q2α

2mα

A2(rA) (10.88)

where we have used the fact that the atom is neutral,∑

α∈A qα = 0.Note that the second atom–field coupling term does no longer depend on the relative

atomic coordinates and is hence independent of the internal atomic state. It gives rise toconstant atom–field energies in close analogy to the vacuum energy of the free electromagneticfield encountered in Chapter 5. Only position–dependent changes of this energy such as theCasimir–Polder potential can be observed.

The major disadvantage of the minimal coupling scheme is the fact that the canonicalparticle momenta do not coincide with the physical momenta and that the interaction is givenin terms of the vector potential rather than the physical electromagnetic fields.

Finally, a number of interaction terms has to be considered. Not only does the atom–fieldcoupling consist of two terms, but the Coulomb potential has to be carefully take into account:its value changes from the free-space solution in the presence of boundaries and it directlycouples atoms to one another. Many of this disadvantages may be overcome by using adifferent, equivalent coupling scheme.

10.5.2 Multipolar coupling

In the minimal coupling scheme, the total system Hamiltonian

H = HA + HF + HAF (10.89)


is split into a part HA that only acts on the atomic Hilbert space spanned by the basis set |nA〉;a part HF that acts only of the field Hilbert space as spanned by the multimode Fock states|n1, n2, ...〉 and an atom–field coupling term HAF that connects the two Hilbert spaces. Whileeigenstates of HA + HF are simple products of atomic and field eigenstates, eigenstates ofthe total Hamiltonian H are complicated superpositions of such product states that may befound, e.g. by using perturbation theory.

Ultimately, we are interested in predicting quantum averages in this composite Hilbertspace

〈O〉 = 〈Ψ|O|Ψ〉 . (10.90)

Such averages are unaffected by a unitary transformation

|Ψ〉 −→ |Ψ′〉 = U |Ψ〉 , (10.91)

O −→ O′ = UOU † . (10.92)

As this unitary transformation couples atomic and field variables, it may redistribute atomicand field degrees of freedom and in this way simplify the atom–field coupling. One suchtransformation is the Göppert-Mayer transformation [1], which is the long-wavelengthapproximation to the more general Power–Zienau–Woolley transformation [2]. We restrictour attention to an atom at a fixed centre-of-mass position, which may be treated as a classicalparameter rA → rA. The Göppert-Mayer transformation reads

U = ei~ d·A(rA) . (10.93)

To determine the transformed particle and field variables, we use the Baker–Campbell–Hausdorff formula

eA B e−A = B + [A, B] +1

2!

[A, [A, B]

]+ . . . (10.94)

As U depends only on rα and not on pα, the transformation leaves the relative particlecoordinates unchanged:

r′α = U rαU

† = rα . (10.95)

The relative particle momenta, on the other hand, are transformed to

p′α = U pαU

† = pα +

[i

~d · A(rA), pα

]= pα +

i

~∑β

qβ[rβ · A(rA), pα

]= pα − qαA(rA) .

(10.96)

[1] M. Göppert-Mayer, Über Elementarakte mit zwei Quantensprüngen, Ann. Phys. (Leipzig) 401, 273 (1931)[Goeppert31].

[2] E. A. Power, S. Zienau, Coulomb gauge in non-relativistic quantum electrodynamics and the shape of spectral lines,Philos. Trans. R. Soc. Lond. Ser. A 251, 427 (1959) [Power59].R.G. Woolley, Molecular quantum electrodynamics, Proc. R. Soc. Lond. Ser. 321, 557 (1971) [Woolley71].


Recalling that

mα˙rα = pα − qαA(rα) ' pα − qαA(rA) (10.97)

in the long-wave length approximation, and that ˙rα = ˙r′α (see above), the transformedcanonical momenta coincide with the physical momenta.

The price paid for this more intuitive picture on the particle side can be seen whentransforming the field variables. In particular, while the vector potential is unaffected by thetransformation, the electric field transforms according to

E⊥′ = E⊥(r) +i

~[d · A(rA), E⊥(r)

]= E⊥(r)− i

~ε0

[d · A(rA), Π(r)

]= E⊥(r) +

1

ε0

d · δ⊥(r − rA) (10.98)

where we have used Π = −ε0E⊥. Noting that

P (r) = d · δ⊥(r − rA) (10.99)

is the atomic polarization in long-wavelength approximation, the transformed electric fieldhence has the physical meaning of an electric displacement field with respect to the atomiccharge distribution.

We now express the total system Hamiltonian in terms of the transformed variables.Neglecting the atomic centre-of-mass motion, the Hamiltonian reads

H =∑α∈A

[pα − qαA(rA)

]22mα

+∑

α 6=β∈A

qαqβ

8πε0|rα − rβ|+

∫d3r

[1

2ε0

Π2 +1

2µ0

(∇⊗ A) : (∇⊗ A)

].

(10.100)

Using the transformation rules (10.95) and (10.96) above together with

Π′(r) = Π(r)− d · δ⊥(r − rA) (10.101)

which follows from Eq. (10.98) above, we find that

H = H ′A + H ′F + H ′AF (10.102)

where

H ′A =∑α∈A

p′2α

2mα

+∑

α 6=β∈A

qαqβ

8πε0|r′α − r

′β|

+1

2ε0

d′ · δ⊥(0) · d′ (10.103)

depends only on the new atomic variables;

H ′F =

∫d3r

[1

2ε0

Π′2 +1

2µ0

(∇⊗ A′) : (∇⊗ A′)]

(10.104)

depends only on the field variables; and

H ′AF =1

ε0

d′ · Π′(rA) (10.105)

is the atom–field interaction.


Remarks:• Atomic Hamiltonian:

The atomic Hamiltonian has new eigenstates |n′A〉 and energies EA′n such that

H ′A =∑n

EA′n |n′A〉〈n′A| . (10.106)

The additional term containing the transverse delta function is a self energy. It has to beregularized in order to be consistent with the long-wavelength approximation. It maybe regarded as a contribution to the Lamb shift in the multipolar coupling scheme.

• Field Hamiltonian:

Expressing the transformed electromagnetic field in terms of normal modes

A′(r) = U∑j

[Aj(r)aj +A∗j(r)a†j

]U † =

∑j

[Aj(r)a′j +A∗j(r)a†′j

](10.107)

we see that it takes a similar form to the original mode expansion, but with transformedcreation and annihilation operators:

a′j = aj +

[i

~d · A(rA), aj

]= aj +

i

~d ·∑k

A∗k(rA)[a†k, aj] = aj +i

~d ·A∗j(rA) ,

(10.108)

a†′j = a†j −i

~d ·Aj(rA) . (10.109)

Similarly, the new field Hamiltonian can be expressed in terms of a′j and a†′j in exactlythe same way as the old field Hamiltonian:

H ′F =∑j

~ωj(n′j +

1

2

). (10.110)

The transformation being unitary, the new creation and annihilation operators obey thesame bosonic commutation relations as the old ones:

[a′j, a†′k ] = δjk . (10.111)

We can work with the new field operators in exactly the same way as we did with theold ones by simply putting primes on all the well-known expressions from Chapter 6.

• Atom–field coupling:

In the multipolar coupling scheme, the coupling consists of a single electric-dipole term

H ′AF = −d′ · E⊥′(rA) (10.112)

which is the main advantage of this approach. One has to bear in mind, however, thatE⊥′ is not the physical transverse electric field, but rather a displacement field withrespect to the atomic polarization.


While the total Hamiltonians in the minimal and multipolar coupling schemes are iden-tical, the separations into atom, field and coupling parts are not. When approximatingthe coupling by means of perturbation theory, two different approximations may result,where none of the two schemes is in any way more legitimate that the other.

• Multipolar coupling

The Power–Zienau–Woolley transformation without long-wavelength approximation isgiven by

U = exp

[i

~

∫d3r P · A

](10.113)

where

P (r) =∑α∈A

qαrα

∫ 1

0

ds δ(r − rA − srα) (10.114)

is the atomic polarization according to ρ = −∇ · P . With this more exact transformation,one obtains an atom–field interaction

H ′AF = −∫

d3r P ′ · E′ −∫

d3r M ′ · B′ (10.115)

where M is the atomic magnetization such that j =˙P +∇ × M . This interaction

allows for a systematic expansion in terms of electric and magnetic multipoles, hencethe name multipolar coupling.

• Alternative descriptions:

Rather than using a unitary transformation in quantum electrodynamics, the multipolarcoupling scheme can be derived by using alternative Lagrangians and Hamiltoniansprior to canonical quantization. In this way, the Power–Zienau–Woolley transformationis equivalent to a change in Lagrangian to

L′ = L+dF

dt(10.116)

with

F = −∫

d3r P ·A , (10.117)

where the addition of a total time derivative dF/dt leaves the Euler–Lagrange equationsinvariant. In the Hamiltonian formulation, this leads to a canonical transformation

p′α = pα +∂F

∂rα, (10.118)

Π′α = Πα +∂F

∂A. (10.119)

11 How does the electromagnetic fieldinteract with a single atom?

Atom–field dynamics



• derive spontaneous decay from the interaction of an atom with the vacuum field,

• describe the internal dynamics of an atom driven by a monochromatic electricfield via the optical Bloch equations, and

• understand that the resonance fluorescence emitted by such an atom is anti-bunched.

In the previous chapter, we have extended quantum electrodynamics by including thecoupling of the electromagnetic field to charged particles and, in particular, atoms. Theextended formalism with its two equivalent minimal and multipolar coupling schemes formsthe basis for studying a range of different atom–field dynamics in this chapter. For simplicity,we will use the multipolar coupling scheme throughout, discarding the primes that indicatetransformed atom and field variables.

11.1 Spontaneous decay

Spontaneous decay is the process where by an initially excited atom makes a transition to theground state upon emitting a real photon. To follow this process in the Heisenberg picture, weintroduce the time-dependent atomic flip operators Amn = |mA〉〈nA|. Note that we suppressthe time argument in the following for notational convenience. They are related to the atomicdensity matrix ρA

〈Amn〉 = Tr(ρA|mA〉〈nA|

)= 〈nA|ρ|mA〉 = ρAnm (11.1)

i.e. their expectation value renders the associated atomic density-matrix elements. We areinterested in particular in the diagonal density matrix elements

pAn = ρAnn = 〈Ann〉 (11.2)

which give the probability of the atom to be in a given state |nA〉. As a preparation forstudying the atom-field dynamics, we recast the atomic and coupling Hamiltonians in terms

144 Atom–field dynamics

of the flip operators:

HA =∑n

EAn |nA〉〈nA|=

∑n

EAn Ann , (11.3)

HAF = −d · E⊥(rA) = −∑m,n

|mA〉〈mA|d|nA〉〈nA|·E⊥(rA) = −∑m,n

Amndmn · E⊥(rA) (11.4)

with dmn = 〈mA|d|nA〉. We now study the coupled atom–field dynamics by means of theHeisenberg equations, starting from the atomic equation of motion:

˙Amn =

1

i~[Amn, H] =

1

i~[Amn, HA] +

1

i~[Amn, HAF ] . (11.5)

With

[Amn, Akl] = |mA〉〈nA|kA〉︸︷︷︸=δnk

〈lA|−|kA〉〈lA|mA〉︸︷︷︸=δlm

|nA〉 = Amlδnk − Aknδlm (11.6)

we find1

i~[Amn, HA] =

1

i~∑k

EAk [Amn, Akk] =

1

i~∑k

EAk

(Amkδnk − Aknδkm

)= i

EAm − EA

n

~Amn = iωmnAmn (11.7)

where ωmn = (EAm − EA

n )/~ are the atomic transition frequencies, and

1

i~[Amn, HAF ] =

1

i~∑k,l

[Amn, Akl]dkl · E⊥(rA) = − 1

i~∑k,l

(Amlδnk − Aknδlm

)dkl · E⊥(rA)

=i

~∑k

(Amkdnk − Akndkm

)· E⊥(rA) . (11.8)

The atomic equation of motion hence reads

˙Amn = iωmnAmn +

i

~∑k

(Amkdnk − Akndkm

)· E⊥(rA) . (11.9)

To proceed, we next need to consider the field dynamics. Recalling that the electric-fieldoperator can be written as

E⊥(rA) =∑j

[iωjAj(rA)aj − iωjA

∗j(rA)a†j

]= E⊥(+)(rA) + E⊥(−)(rA) , (11.10)

we need to first derive the dynamics of aj and a†j . To this end, we write the field and couplingHamiltonians in the forms

HF =∑j

~ωj(a†j aj +

1

2

)(11.11)

HAF = −∑m,n

∑j

Amndmn ·[iωjAj(rA)aj − iωjA

∗j(rA)a†j

]. (11.12)

11.1 Spontaneous decay 145

This results in an equation of motion

˙aj =1

i~[aj, HF

]+

1

i~[aj, HAF

]=

1

i~∑k

~ωk[aj, a

†kak]

+1

i~∑m,n

∑k

iωkAmndmn ·A∗k(rA)[aj, a

†k

]︸︷︷︸=δjk

= −iωj aj +1

~∑m,n

ωjdmn ·A∗j(rA)Amn . (11.13)

This inhomogeneous linear differential equation can be formally integrated according to

aj(t) = e−iωj(t−t0) aj(t0) +1

~∑m,n

ωjdmn ·A∗j(rA)

∫ t

t0

dt′ e−iωj(t−t′) Amn(t) (11.14)

where aj(t0) = aj is the time-independent operator in the Schrödinger picture. We substitutethis solution back into the atomic equation of motion and aim toward deriving a closedsystem of equations for the atomic state populations pn(t) = 〈Ann(t)〉. This can be achievedvia the following steps:

1 Normal ordering:

Noting that atomic and field operators commute at equal times, we can rearrangeatom–field products on the right hand side of the atomic equation of motion in normalorder:

˙Amn = iωmnAmn(t) +

i

~∑k

[Amk(t)dnk − Akn(t)dkm

]· E⊥(+)(rA, t)

+i

~∑k

E⊥(−)(rA, t) ·[Amk(t)dnk − Akn(t)dkm

]. (11.15)

2 Expectation values:

We substitute our solution for aj into the above equation and take expectation values.With the field initially being in its vacuum state |0〉, we have

aj(t0)|0〉 = 0 , (11.16)

so that the terms aj(t0) and a†j(t0) do not contribute to the expectation value in ourchosen normal order.

3 Weak atom–field coupling:

The resulting equation related 〈 ˙Amn〉 to products 〈Akl(t)Amn(t)〉 on the right hand side.

We assume weak atom–field coupling which is valid unless an atomic transition isresonant with an isolated, narrow mode of the electromagnetic field. To leading orderin the atom–field coupling, we may hence assume the terms on the right hand side tofollow their free dynamics:

˙Akl = iωklAkl ⇒ Akl(t) = eiωkl(t−t′) Akl(t

′) . (11.17)


We then have terms∫ t

t0

dt′ e−iωj(t−t′)〈Akl(t)Amn(t′)〉 = 〈Akl(t)Amn(t)〉∫ t

t0

dt′ e−i(ωj−ωlk)(t−t′)

= δlm〈Akm(t)〉∫ t

t0

dt′ e−i(ωj−ωlk)(t−t′) (11.18)

on the right hand side. Again for weak coupling, the integral is well approximated byletting the lower time limit to −∞:∫ t

t0

dt′ e−i(ωj−ωlj)(t−t′) '∫ t

−∞dt′ e−i(ωj−ωlj)(t−t′) =

∫ ∞0

dτ e−i(ωj−ωlk)τ (τ = t− t′)

= πδ(ωj − ωlk)− iP

ωj − ωlk(11.19)

where P denotes the principal value. This is known as the Markov approximation; itimplies that the system has no memory.

4 Neglect of oscillating terms:

We finally concentrate on the equations for pn(t) = 〈Ann(t)〉 and neglect all the termson right hand side which are proportional to 〈Amn(t)〉 with m 6= n. Such terms areoscillating with eiωmnt, so that their contribution averages to zero.

We also note that the atom has no permanent dipole moment, so that dnn = 0. For theremaining terms on the right hand side, contributions from the principal part mutuallycancel.

As a result of these steps, we arrive at the rate equations for spontaneous decay

pn(t) = −∑m

pn(t)Γnm +∑m

pm(t)Γmn (11.20)

where

Γnm =2π

~2

∑j

ω2jdnm ·Aj(rA)⊗A∗j(rA) · dmnδ(ωj − ωnm) (11.21)

are the rates of spontaneous transitions from an upper atomic energy eigenstate |nA〉 to alower state |mA〉.

Let us apply this general result to an atom situated in free space where the modes aregiven by

Ak,λ =

√~

16π3ε0ωkek,λ eik·r . (11.22)

We evaluate the mode sum∑j

ω2jAj(rA)⊗A∗j(rA)δ(ωj − ωnm) =

∫d3k

∑λ=1,2

~16π3ε0ωk

ω2kek,λ ⊗ e∗k,λδ(ωk − ωnm)

=~

16π3ε0

∫ ∞0

dk k2ωkδ(ωk − ωnm)

∫dΩ

∑λ=1,2

ek,λ ⊗ e∗k,λ

(11.23)

11.1 Spontaneous decay 147

in spherical coordinates and solve the angular integral upon recalling the completenessrelation for the polarization unit vectors∫

dΩ∑λ=1,2

ek,λ ⊗ e∗k,λ =

∫dΩ

(1− ek ⊗ ek

)= 4π

(1− 1

31

)=

8π

31 . (11.24)

Using ωk = ck, we then find

∑j

ω2jAj(rA)⊗A∗j(rA)δ(ωj − ωnm) =

~ω3nm

6π2ε0c31 for ωnm > 0 . (11.25)

so that

Γnm =

ω3nm|dnm|23πε0~c3

for ωnm > 0 ,

0 for ωmn > 0 .

(11.26)

This is the famous Einstein A-coefficient [1] as postulated in 1917. We have now derived itsvalue from the interaction of an initially excited atom with the quantum electromagnetic field,initially in its vacuum state.

When assuming the field to be in a thermal state ρT , the above calculation yields the moregeneral rates

Γnm =

ω3nm|dnm|23πε0~c3

[n(ωnm) + 1

]for ωnm > 0 ,

ω3mn|dnm|23πε0~c3

[n(ωmn)

]for ωmn > 0 ,

(11.27)

where

n(ω) =1

e~ω/(kBT )−1(11.28)

is the thermal photon number. The terms proportional to n(ω) obviously represent theabsorption and stimulated emission of the thermal photons; they are related to the EinsteinB-coefficients.

|mA〉

|nA〉

Γmn ∝ n(ω) Γnm ∝ n(ω) + 1

[1] A. Einstein, Phys. Z. 18, 121 (1917) [Einstein1917].


11.2 Optical Bloch equations

In the last section, we have considered the dynamics of an initially excited atom in theabsence of external electromagnetic fields. We now ask what happens if we subject the atomto a monochromatic electromagnetic field of frequency ωj which is near-resonant with asingle transition of the atom between an upper state |1A〉 and a lower state |0A〉. The ensuingdynamics can be well described by restricting our attention to the two levels in question.

The flip operators for the resulting two-level atom are commonly represented in thermsof Pauli operators

σ = A01 , σ† = A10 , (11.29)

σx = σ + σ† = A01 + A10 , (11.30)

σy = i(σ − σ†) = i(A01 − A10) , (11.31)

σz = A11 − A00 , (11.32)

which obey the angular-momentum commutation rules[σi, σj

]= 2iεijkσk ,

[σ, σz

]= 2σ ,

[σ†, σz

]= −2σ† ,

[σ, σ†

]= −σz . (11.33)

With these definitions, the atomic Hamiltonian of the two-level atom

HA = EA1 A11 + EA

0 A00 =1

2(E1 − E0)(A11 − A00) +

1

2(E1 + E0)(A11 + A00)

=1

2~ωAσz +

1

2(E1 + E0)1 (11.34)

can be redefined as

HA =1

2~ωAσz (11.35)

where ωA = ω10 is the atomic transition frequency and we have discarded the constant term12(E1 + E0)1 which does not contribute to the dynamics. The atom–field coupling reads

HAF = −(d01A01 + d10A10

)· E⊥(rA) = −

(dσ + d∗σ†

)· E⊥(rA) (11.36)

where d01 = d∗10 ≡ d and d11 = d00 = 0 for an atom without permanent electric dipolemoment. As we will see below, the free Pauli operators (i.e., in the absence of atom–fieldcoupling) oscillate as σ(t) e−iωAt whereas the free field operators carry a time dependence

E⊥(+)(rA, t) ∝ e−iωjt and E⊥(−)(rA, t) ∝ eiωjt . (11.37)

Products of the type

σ(t)E⊥(+)(rA, t) ∝ e−i(ωA+ωj)t and σ†(t)E⊥(−)(rA, t) ∝ ei(ωA+ωj)t (11.38)

are hence rapidly oscillating compared to

σ(t)E⊥(−)(rA, t) ∝ e−i(ωA−ωj)t and σ†(t)E⊥(+)(rA, t) ∝ ei(ωA−ωj)t (11.39)

11.2 Optical Bloch equations 149

and can be neglected in the near-resonant case ωj ' ωA. This leads to the electric-dipoleHamiltonian in rotating-wave approximation

HAF = −σd · E⊥(−)(rA) + σ†d∗ · E⊥(+)(rA) . (11.40)

With these preparations, we may easily derive the Heisenberg equations of motion for thePauli operators via their above commutation relations:

˙σz =1

i~[σz, HAF

]=

1

i~[σz, σ

]︸︷︷︸=−2σ

d · E⊥(−)(rA) +1

i~[σz, σ

†]︸︷︷︸=2σ†

d∗ · E⊥(+)(rA)

= −2i

~σd · E⊥(−)(rA) +

2i

~σ†d∗ · E⊥(+)(rA) , (11.41)

˙σ =1

i~[σ, HA

]+

1

i~[σ, HAF

]= −iωAσ −

i

~σzd

∗ · E⊥(+)(rA) . (11.42)

The equations are special cases of the multilevel equations (11.15) from the previous sectionin the rotating-wave approximation. Similarly, the equation of motion for the electromagneticfield simplifies to

˙aj = iωj aj +ωj~A∗j(rA) · dσ . (11.43)

It can again be formally integrated to give

aj(t) = e−iωj(t−t0) aj(t0) +ωj~A∗j(rA)

∫ t

t0

dt′ e−iωj(t−t′) σ(t′) . (11.44)

As in the previous section, we assume weak atom–field coupling (i.e. σ(t′) = σ(t) eiωA(t−t′) onthe right hand side) and apply the Markov approximation to the second term:∫ t

t0

dt′ e−iωj(t−t′) σ(t′) = σ(t)

∫ t

t0

dt′ ei(ωA−ωj)(t−t′) ' πδ(ωj − ωA)− iP

ωj − ωA. (11.45)

We substitute this solution into the atomic equations of motion and define

E⊥(+)free (rA, t) =

∑j

iωjAj(rA)aj(t0) e−iωjt (11.46)

as the free part of the evolving electric field; its dynamics is that of the field in absence of theatom. One eventually finds

˙σz = −Γ(1 + σz)−2i

~σd · E⊥(−)

free (rA) +2i

~σ†d∗ · E⊥(+)

free (rA) (11.47)

˙σ =

[−i(ωA + δω)− Γ

2

]σ − i

~σzd

∗ · E⊥(+)free (rA) (11.48)

where

Γ =2π

~∑j

ω2jd∗ ·Aj(rA)⊗A∗j(rA) · dδ(ωj − ωA) (11.49)


is the two–level decay rate as a special case of Eq. (11.21) and

δω = − 1

~2P∑j

ω2j

ωj − ωAd∗Aj(rA)⊗Aj(rA) · d (11.50)

is the Lamb shift, i.e. a modification of the atomic transition frequency due to the interactionof the atom with the vacuum electromagnetic field.

We note that in the absence of a free electromagnetic field, i.e. when the field is in itsvacuum state, taking expectation values of Eq. (11.47) leads to

p1 − p0 = −Γ[1 + (p1 − p0)

](11.51)

and using the constraint

p1 + p0 = 1 , p1 + p0 = 0 (11.52)

we recover the equation of spontaneous decay:

p1 = −Γp1 . (11.53)

However, we are interested in the case where the atom is subject to a monochromatic laserfield which can be characterized by a single-mode coherent state |αj〉. Taking expectationvalues with respect with respect to the field state then leads to

〈αj|E⊥(+)free (rA, t)|αj〉 = iωjAj(rA)αj e−iωjt , (11.54)

so that the atomic equations of motion read

˙σz(t) = −Γ[1 + σz(t)

]− Ω∗σ(t) eiωjt−Ωσ†(t) e−iωjt (11.55)

˙σ(t) =

[−iωA − δω −

Γ

2

]σ(t) + Ωσz(t) e−iωjt (11.56)

with

Ω =2ωjαj~

Aj(rA) · d∗ (11.57)

being the Rabi frequency. It is a complex parameter describing the atom–field couplingstrength. To remove the time-dependent factors e±iωjt, we rewrite these equations in aco-rotating frame by defining

ˆσ(t) = σ(t) eiωjt (11.58)˙σ(t) = ˙σ eiωjt +iωjσ(t) eiωjt (11.59)

leading to

˙σz = −Γ(1 + σz)− Ωˆσ − Ω∗ ˆσ† , (11.60)

˙σ =

(−i∆− Γ

2

)˙σ +

Ω

2σz . (11.61)

11.2 Optical Bloch equations 151

where

∆ = ωA + δω − ωj (11.62)

is the atom–field detuning.Finally, taking expectation values with respect to the atomic state, we obtain the optical

Bloch equation for the internal dynamics of an atom driven by a coherent single–mode field.This dynamics is equivalent to that of a spin under the influence of an external magnetic field.It may be visualized by parametrising the atomic density matrix according to

ρ =1

2

(1 + u · σ

)(11.63)

by a real vector u = (ux, uy, uz)T . Using

Tr(σi) = 1 , Tr(σiσj) = 2δij (11.64)

its elements are found to be

ui = Tr(ρσi) = 〈σi〉 . (11.65)

In order for ρ to be a density matrix, the vector u has to fulfil the constraint |u|≤ 1. It hencelies inside a sphere of radius 1, the Bloch sphere:

ux

uy

uz

|1A〉

|0A〉

ϕ

1√2

(|1A〉+ eiϕ|0A〉

)1√2

(|1A〉+ eiϕ|0A〉

)

The density matrix of a two-level system is hence described by a vector in the Bloch spherewhere vectors on the surface represent pure states. In particular u = (0, 0, 1) correspondsto the excited state |1A〉〈1A|, u = (0, 0,−1) to the ground state |0A〉〈0A|, states on the equator(uz = 0) to coherent superposition states (|1A〉 + eiϕ|0A〉)/

√2 and u = 0 to the maximally

mixed state 12(|1A〉〈1A|+|0A〉〈0A|).


In terms of the Bloch vector, the optical Block equations following from Eqs. (11.60) and(11.61) read

u =

−Γ2

−∆ Re Ω

∆ −Γ2− Im Ω

−Re Ω Im Ω Γ

u− Γez . (11.66)

In the case of negligible decay (Γ = 0), exact resonance between atom and field (∆ = 0) and areal Rabi frequency, an atom initially prepared in the ground state u = (0, 0,−1) undergoesRabi oscillations

u(t) =

− sin(Ωt)0

− cos(Ωt)

. (11.67)

11.3 Resonance fluorescence

So far, he have concentrated on the dynamics of the atom. Resonance fluorescence is theradiation emitted by the atom while being driven by the laser field. Recalling our resultsfrom earlier, the field emitted by the atom is governed by σ(t). The coherence properties ofthis field are hence given by

g(1)(τ) =〈σ†(t)σ(t+ τ)〉〈σ†(t)σ(t)〉 , (11.68)

g(2)(τ) =〈σ†(t)σ†(t+ τ)σ(t+ τ)σ(t)〉

〈σ†(t) ˆσ(t)〉2. (11.69)

This implies that

g(1)(0) =〈σ†σ〉〈σ†σ〉 = 1 (11.70)

is first-order coherent at zero time delay, while

g(2)(0) =〈σ†σ†σσ〉〈σ†σ〉2 = 0 (11.71)

due to σσ = |0A〉〈1A|0A〉〈1A|= 0. The field emitted by the driven two-level atom hencedisplays sub-Poissonian photon number statistics.

In the limit of an infinite time delay τ →∞, we may assume the atomic density matrix att+ τ to be completely independent of its earlier value at t, so that

limτ→∞

g(2)(τ) = limτ→∞

〈σ†(t)σ†(t+ τ)σ(t+ τ)σ(t)〉〈σ†(t) ˆσ(t)〉2

=〈σ†(t)σ(t)〉〈σ†(t+ τ)σ(t+ τ)〉

〈σ†(t)σ(t)〉2 = 1 (11.72)

in the stationary limit t 1/Γ. We hence have

g(2)(τ →∞) > g(2)(τ = 0) . (11.73)

The field emitted by a single driven two-level atom shows antibunching where the probabilityof finding a second photon increases with the time having passed after the observation of afirst photon. This is a purely nonclassical effect due to the particle nature of light.

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