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De Broglie Optics References 1139

reduce coherence, as has been shown with fullerene

molecules [77.84]. Finally, coherence is lost when atoms

interact with random electromagnetic ﬁelds. This has

become relevant for atom reﬂection from evanescent

light because of the roughness of the dielectric surface

used [77.85] (see also [77.86]). The coherent operation

of integrated atom optics near metallic surfaces is lim-

ited by thermally excited electromagnetic near ﬁelds

as shown in experiments by Harber et al. [77.87](see

also [77.88]).

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Part F 77

1141

Quantized Fiel

78. Quantized Field Effects

The electromagnetic ﬁeld appears almost every-

where in physics. Following the introduction of

Maxwell’s equations in 1864, Max Planck initi-

ated quantum theory when he discovered h =2π

in the laws of black-body radiation. In 1905 Al-

bert Einstein explained the photoelectric effect

on the hypothesis of a corpuscular nature of

radiation and in 1917 this paradigm led to a de-

scription of the interaction between atoms and

electromagnetic radiation.

The study of quantized ﬁeld effects requires

an understanding of the quantization of the

ﬁeld which leads to the concept of a quantum

of radiation, the photon. Speciﬁc nonclassical

features arise when the ﬁeld is prepared in

particular quantum states, such as squeezed

states. When the radiation ﬁeld interacts with an

atom, there is an important difference between

a classical ﬁeld and a quantized ﬁeld. A classical

ﬁeld can have zero amplitude, in which case it

does not interact with the atom. On the other hand

a quantized ﬁeld always interacts with the atom,

even if all the ﬁeld modes are in their ground

states, due to vacuum ﬂuctuations. These lead to

variouseffectssuchasspontaneousemissionand

the Lamb shift.

The interaction of an atom with the many

modes of the radiation ﬁeld can conveniently be

described in an approximate manner by a master

equation where the radiation ﬁeld is treated as

a reservoir. Such a treatment gives a microscopic

and quantum mechanically consistent description

of damping.

78.1 Field Quantization ............................... 1142

78.2 Field States .........................................1142

78.2.1 Number States ..........................1143

78.2.2 Coherent States.........................1143

78.2.3 Squeezed States ........................1144

78.2.4 Phase States .............................1145

78.3 Quantum Coherence Theory ..................1146

78.3.1 Correlation Functions.................1146

78.3.2 Photon Correlations...................1146

78.3.3 Photon Bunching

and Antibunching ..................... 1147

78.4 Photodetection Theory ......................... 1147

78.4.1 Homodyne

and Heterodyne Detection..........1147

78.5 Quasi-Probability Distributions .............1148

78.5.1 s-Ordered Operators .................. 1148

78.5.2 The P Function..........................1149

78.5.3 The Wigner Function..................1149

78.5.4 The Q Function..........................1151

78.5.5 Relations

Between Quasi-Probabilities ......1151

78.6 Reservoir Theory .................................. 1151

78.6.1 Thermal Reservoir .....................1152

78.6.2 Squeezed Reservoir ...................1152

78.7 Master Equation ..................................1152

78.7.1 Damped Harmonic Oscillator....... 1153

78.7.2 Damped Two-Level Atom ...........1153

78.8 Solution of the Master Equation ...........1154

78.8.1 Damped Harmonic Oscillator.......1154

78.8.2 Damped Two-Level Atom ........... 1155

78.9 Quantum Regression Hypothesis ...........1156

78.9.1 Two-Time Correlation Functions

and Master Equation ................. 1156

78.9.2 Two-Time Correlation Functions

and Expectation Values..............1156

78.10 Quantum Noise Operators.....................1157

78.10.1 Quantum Langevin Equations .....1157

78.10.2 Stochastic Differential Equations . 1158

78.11 Quantum Monte Carlo Formalism ..........1159

78.12 Spontaneous Emission in Free Space .....1159

78.13 Resonance Fluorescence .......................1160

78.13.1 Equations of Motion ..................1160

78.13.2 Intensity of Emitted Light...........1160

78.13.3 Spectrum

of the Fluorescence Light ...........1161

78.13.4 Photon Correlations...................1161

78.14 Recent Developments...........................1162

78.14.1 Literature .................................1162

78.14.2 Field States ..............................1162

78.14.3 Reservoir Theory .......................1162

References .................................................. 1163

Part F 78

1142 Part F Quantum Optics

78.1 Field Quantization

This section provides the basis for the quantized ﬁeld

effects discussed in this Chapter [78.1]. We expand the

ﬁeld in a complete set of normal modes which reduces

the problem of ﬁeld quantization to the quantization of

a one dimensional harmonic oscillator corresponding to

each normal mode.

The classical free electromagnetic ﬁeld, i. e., the ﬁeld

in a region without charge and current densities, obeys

the Maxwell equations

∇· B = 0 ,

(78.1)

∇· D =0 , (78.2)

∇× E+

∂ B

∂t

= 0 ,

(78.3)

∇× H−

∂ D

∂t

= 0 ,

(78.4)

where B = µ

H, D = 

E. The magnetic permeabil-

ity µ

connects the magnetic induction B with the

magnetic ﬁeld H and the electric permittivity 

free space connects the displacement D with the elec-

tric ﬁeld E. In the case of a free ﬁeld, E and B may be

obtained from

B =∇× A,

(78.5)

E =−

∂ A

∂t

(78.6)

where the vector potential A obeys the Coulomb gauge

condition ∇ · A = 0 and satisﬁes a wave equation. In

order to solve this wave equation we expand the vector

potential

A(x, t) =



σ=1



2ω





1/2



kσ



kσ

i(k·x−ω

+c.c.



(78.7)

in a set of normal modes V

−1/2

exp(ik· x)

kσ

which are

orthonormal in the volume V. Due to the gauge condition

∇· A = 0, we obtain two orthogonal polarization vec-

tors 

and 

with 

kσ

· k= 0 for each wave vector k.

The dispersion relation is ω

= c|k|. The Fourier ampli-

tudes α

kσ

are complex numbers in the classical theory.

The ﬁeld is quantized by replacing the classical am-

plitude α

kσ

by the mode annihilation operator a

kσ

.The

complex conjugate α

∗

kσ

is replaced by the mode creation

operator a

†

kσ

. They obey the commutation relation



kσ

, a

†





= δ

kk‘

σσ



. (78.8)

The representation of the electric ﬁeld operator

E(x, t) = i



k,σ



2



1/2



kσ



kσ

i(k·x−ω

−h.c.



≡ E

(x, t) + E

−

(x, t) (78.9)

in terms of these operators follows from (78.6)andthe

operator for the vector potential. Note also the often

used decomposition of the electric ﬁeld operator into

the positive and negative frequency parts E

and E

−

respectively. A similar relation holds for the operator

describing the magnetic induction B.

Using the operators for the electric and magnetic

ﬁeld, one can transform the ﬁeld energy

H =







+ B

/µ



(78.10)

into the form

H =



k,σ



†

kσ

+1/2



, (78.11)

which is a sum of independent harmonic oscillator

Hamiltonians corresponding to each mode (k,σ).The

number operator N

kσ

= a

†

kσ

represents the number

of photons in the mode (k,σ), while

/2 is the energy

of the vacuum ﬂuctuations.

Hence each mode of the electromagnetic ﬁeld is

equivalent to a harmonic oscillator. In the next section

we discuss speciﬁc states of a single mode. The general

quantum state of the electromagnetic ﬁeld consisting of

many modes is given by a superposition of product states

that are composed out of these single mode states.

78.2 Field States

This section summarizes the properties of several im-

portant states of the electromagnetic ﬁeld. From the

independence of the normal modes, the discussion may

be restricted to a single normal mode. With the mode

index (k,σ)suppressed, a single mode Hamiltonian is

H =



†

a +1/2



. (78.12)

Part F 78.2

Quantized Field Effects 78.2 Field States 1143

In the second step, the quadrature operators

x =

√



a +a

†



(78.13)

p =

√



a −a

†



(78.14)

are introduced, which are equivalent to scaled position

and momentum operators of a massive particle in a har-

monic potential. The quadratures of a quantized ﬁeld

are measurable with the help of homodyne detection as

discussed in Sect. 78.4.1

We shall now describe several states of this quantized

ﬁeld mode: number states, coherent states, squeezed

states, Schrödinger cats, and phase states. A quantized

ﬁeld in a coherent state shows the most classical be-

havior. A superposition of two coherent states, which is

a Schrödinger cat, already shows nonclassical features.

Number states and squeezed states are further typical

examples of nonclassical states.

78.2.1 Number States

The eigenstates of the Hamiltonian (78.12) are the eigen-

states of the number operator N = a

†

N|n=n|n ,

(78.15)

where n = 0, 1, 2,... denotes the excitations or the

number of photons in the mode. The vacuum state of

the mode |0,isdeﬁnedby

a|0=0 .

(78.16)

The ladder of excitations can be climbed up and down

via the application of creation and annihilation operators

†

|n=

√

n +1|n +1 , (78.17)

a|n=

√

n|n −1 , (78.18)

on a Fock state |n. These number or Fock states form

a complete and orthonormal set of states so that

∞



n=0

|nn|=1 , n|k=δ

. (78.19)

Their quadrature representations are

x|n=



√

π2



−1/2

(x)e

−x

, (78.20)

p|n=



√

π2



−1/2

(−i)

( p) e

−p

(78.21)

The states |x and |p are eigenstates of the quadrature

operators x and p,(78.13)and(78.14).

Number states provide a frequently used representa-

tion of a pure quantum state

|ψ=

∞



n=0

|n , (78.22)

or a mixed quantum state given by the density operator

ρ =



n,k

|nk| (78.23)

(Chapt. 7).

78.2.2 Coherent States

The coherent state is a speciﬁc superposition of num-

ber states. In contrast to a number state, a coherent

state does not possess a deﬁnite number of photons:

the photon distribution is Poissonian. For a large aver-

age photon number, the electric and magnetic ﬁelds have

rather well deﬁned amplitudes and phases with vanish-

ing relative quantum ﬂuctuations. Hence the Poissonian

photon distribution frequently serves as a borderline

between classical and nonclassical ﬁeld states. Non-

classical states show a sub-Poissonian behavior. An

extreme example is a ﬁeld prepared in a number state.

A parameter which quantiﬁes the deviations from Pois-

sonian behavior is the Q parameter introduced by

Mandel [78.2].

We deﬁne the coherent state |α as an eigenstate of

the annihilation operator

a|α=α|α

(78.24)

with the complex amplitude α =|α|e

iθ

. The coherent

state can be represented by

|α=e

αa

†

−α

∗

|0=D(α)|0 , (78.25)

that is, by the action of the displacement operator D(α)

on the vacuum.

The number state representation of |α reads

|α=e

−|α|

∞



n=0

√

|n .

(78.26)

A coherent state |α

 that evolves in time according to

the free ﬁeld Hamiltonian (78.12) stays coherent, i. e.,

|ψ(t)=exp

(

−iHt/

)

|α

=e

−iωt/2

|α(t) ,

(78.27)

with amplitude α(t) =α

exp[−iωt].

Part F 78.2

1144 Part F Quantum Optics

Another important representation of a coherent state

is the x representation

x|α=π

−1/4

exp



−[Re(α)]



×exp



−x

/2 +

√

2αx



, (78.28)

where |x denotes again the x quadrature eigenstate.

The photon distribution in a coherent state

|n|α|

|α|

−|α|

(78.29)

is a Poisson distribution with average photon number

N=|α|

and variance



∆N







−N

=|α|

Hence the relative ﬂuctuations (∆N )/N=N

−1/2

vanish for a large average photon number.

The Mandel Q parameter

Q ≡

(∆N )

−N

N

(78.30)

vanishes for a ﬁeld in a coherent state. A nonclassical

ﬁeld may show sub-Poissonian behavior with Q < 0. As

an example, the Schrödinger cat state is a macroscopic

superposition of two coherent states

|cat=



2 +2cos



sin φ



−2α

sin

(φ/2)



−1/2





α e

iφ/2





α e

−iφ/2





(78.31)

where α is assumed to be real. The Q parameter for this

superposition state, shown in Fig. 78.1, takes on negative

values for speciﬁc angles φ. The nonclassical behavior

of such a |cat-state can be explained [78.3]asare-

sult of quantum interference between the two coherent

states present in (78.31). The incoherent superposition

described by the density operator

ρ =





α e

iφ/2



α e

iφ/2



α e

−iφ/2



α e

−iφ/2





(78.32)

does not have this nonclassical character: its Q param-

eter vanishes. Coherent states have a direct physical

signiﬁcance: the quantum state of a stabilized laser op-

erating well above threshold can be approximated by

a coherent state.

78.2.3 Squeezed States

Squeezed states [78.4–6] minimize the uncertainty prod-

uct of the quadrature components of the electromagnetic

ﬁeld. The quadrature components x and p of the single

mode ﬁeld are deﬁned in (78.13)and(78.14). They obey

the commutation relation [x, p]=i. Their uncertainties

(∆x)

≡





−x

and (∆ p)

≡





−p

fulﬁll the

Heisenberg inequality

∆ x∆ p ≥ 1/2 .

(78.33)

The coherent state is a special minimum uncertainty state

with equal uncertainties ∆x = ∆ p = 1/

√

2. Squeezed

states comprise a more general class of minimum uncer-

tainty states with reduced uncertainty in one quadrature

at the expense of increased uncertainty in the other.

These states |α, are obtained by applying the displace-

ment operator D(α) and the unitary squeeze operator

S() = e



∗

−

a

†2

(78.34)

to the vacuum

|α, =D(α)S()|0.

(78.35)

The squeeze operator S() transforms a and a

†

according

†

()aS() = a cosh r −a

†

−2iφ

sinh r , (78.36)

†

()a

†

S() = a

†

cosh r −a e

2iφ

sinh r , (78.37)

where  = r e

−2iφ

. The rotated quadratures

= x cos φ − p sin φ, (78.38)

= p cos φ +x sin φ, (78.39)

transform according to

†

()(X

+iX

)S() = X

−r

+iX

, (78.40)

which yields the uncertainties

∆X

= e

−r

√

2 , (78.41)

∆X

= e

√

2 . (78.42)

1.5

0.5

–0.5

–1

0.2 0.3 0.4

0.1

Fig. 78.1 The Q parameter for a Schrödinger cat state

(78.31) with amplitude α = 4

Part F 78.2

Quantized Field Effects 78.2 Field States 1145

In particular, for φ =0 the squeezed state |α, ris a min-

imum uncertainty state for the quadratures x and p with

∆ x = e

−r

√

2and∆ p = e

√

2. The degree of squeez-

ing in the quadrature x is determined by the squeeze

factor r.

The average photon number

N=|α|

+sinh

r (78.43)

of a squeezed state and its photon number variance

(∆N )



α cosh r −α

∗

−2iφ

sinh r



+2cosh

r sinh

r (78.44)

contain the coherent contribution α as well as squeez-

ing contributions expressed by r and φ. In particular,

for φ = 0, the Q parameter becomes negative for a large

enough amplitude α and r > 0. The photon number dis-

tribution W

=|α, r|n|

becomes narrower than the

one for the corresponding coherent state with the same α.

This sub-Poissonian behavior is one of the nonclassical

features of a squeezed state. Furthermore, W

shows os-

cillations [78.7] for larger squeezing. The two regimes

with sub-Poissonian and oscillating photon statistics W

are shown in Fig. 78.2.

A second representation of squeezed states has been

introduced by Yuen [78.8]. In his notation, a squeezed

state is an eigenstate of the operator

b = µa +νa

†

, (78.45)

with |µ|

−|ν|

= 1 and eigenvalue β. This eigenstate

can be written in the form

|, β=S()D(β)|0 ,

(78.46)

which connects the squeezing operator S(r e

−2iφ

) with

the parameters µ = cosh r and ν = e

−2iφ

sinh r. In con-

trast to the deﬁnition (78.35), the displacement operator

D(β) and the squeezing operator S() are applied

now in reversed order. Nevertheless, the two equations

(78.35)and(78.46) deﬁne the same state if the relation

α = βµ +β

∗

ν is fulﬁlled.

Several experiments have demonstrated the gener-

ation of squeezed light. Slusher et al. [78.9] obtained

squeezing in the sidemodes of a four-wave mix-

ing process. An optical parametric oscillator below

threshold has been used by Wu et al. [78.10]inor-

der to generate squeezed light. Nonclassical features

can also be found in a down conversion process.

This second-order process creates so-called signal and

idler photons from one pump photon. Signal and

idler beam are distinguished by frequency or po-

larization. Heidmann et al. [78.11]haveshownthat

the difference intensity of these twin beams may ex-

hibit reduced quantum ﬂuctuations. Pulsed twin beams

also contain reduced noise in the difference of their

intensities.

78.2.4 Phase States

The problem of a correct quantum mechanical de-

scription of phase has a long history in quantum

mechanics [78.12]. First attempts to deﬁne a quantum

phase are due to London and Dirac. The London phase

state is

|φ=

√

2π

∞



n=0

inφ

|n , (78.47)

which is an eigenstate of the exponential phase operator



iφ

∞



n=0

|nn +1| . (78.48)

Since this operator is not unitary, it does not deﬁne

a Hermitian operator

φ for the phase. Nevertheless

many treatments of the phase of a quantum state

|ψ=



|n are based on the London phase distri-

bution

Pr (φ) =



φ|ψ|

2π





−inφ



. (78.49)

0.1

0.05

Fig. 78.2 The photon number distribution W

of a squeezed

state |α, rwith the coherent amplitude α =7. For a squeez-

ing parameter r =0, the Poisson distribution of a coherent

state |α =7 is just visible. When r increases the pho-

ton distribution ﬁrst becomes sub-Poissonian and then

oscillatory

Part F 78.2

1146 Part F Quantum Optics

Later treatments [78.13] rely on the Hermitian oper-

ators



sin φ =





iφ

−



iφ

†



(78.50)

cos φ =





iφ



iφ

†



(78.51)

for the sine and cosine function of the phase.

Recently [78.14] a Hermitian phase operator was

constructed starting from the phase state (78.47),

restricted to a ﬁnite Hilbert space. An operational

phase description has been proposed [78.15]inwhich

a classical phase measurement is translated to the

quantum realm by using an eight-port homodyne

detector.

78.3 Quantum Coherence Theory

This section introduces the correlation functions of the

electromagnetic ﬁeld. Ideal photon correlation measure-

ments can bring out the phenomenon of photon bunching

and antibunching.

78.3.1 Correlation Functions

Correlation functions were originally introduced to de-

scribe an ideal photodetection process. Glauber [78.16]

has presented a treatment based on an absorption mech-

anism in the detector which is sensitive to the positive

frequency part E

of the electric ﬁeld evaluated at the

detector’s space-time position x ≡ (x, t). This leads to

an average ﬁeld intensity

I(x) = Tr



ρE

−

(x)E

(x)



(78.52)

at point x. Here the density operator ρ describes the state

of the ﬁeld. The ordering of the operators, i. e., E

−

∼

†

a, is known as normal ordering with all annihilation

operators to the right of all creation operators.

The expression (78.52) now immediately general-

izes to the correlation function of ﬁrst order

(1)

, x

) = Tr



ρE

−

)



, (78.53)

with x

= (x

, t

) and x

= (x

, t

). The classical in-

terference experiments, such as Young’s double slit

experiment, can be described in terms of G

(1)

.Further-

more, the correlation function of ﬁrst order is connected

to the power spectrum S(ω) of a quantized ﬁeld via

the Wiener–Khintchine theorem [78.17]. Under the

assumption of a stationary process, i. e., when the au-

tocorrelation function



−

(t)E



)



depends only on

the time difference τ =t −t



,then

S(ω) =

2π

∞



dτ



−

(τ)E

(0)



−iωτ

+c.c .

(78.54)

This relation between the spectrum and the ﬁrst-order

correlation function is known as Wiener–Khintchine

theorem.

In order to analyze the Hanbury–Brown and Twiss

experiment [78.18] it is necessary to deﬁne higher order

correlation functions. The general nth order correlation

function is deﬁned by

(n)

,... ,x

, x

n+1

,... ,x

)

= Tr



ρE

−

) ···E

−

)

× E

n+1

) ···E

)



, (78.55)

where the ﬁeld operators are again normal ordered.

These correlation functions fulﬁll a generalized

Schwartz inequality

(1)

, x

(1)

, x

) ≥



(1)

, x

)



(78.56)

which becomes, for the nth order functions,

(n)

, .., x

, x

, .., x

)

× G

(n)

n+1

, .., x

, x

, .., x

n+1

)

≥



(n)

, .., x

, x

n+1

, .., x

)



. (78.57)

A ﬁeld is said to be ﬁrst-order coherent when its

normalized correlation function

(1)

, x

) =

(1)

, x

)



(1)

, x

(1)

, x

)



1/2

(78.58)

satisﬁes



(1)

, x

)



= 1. In a Young type experiment,

this case gives maximum fringe visibility. A more gen-

eral deﬁnition of ﬁrst order coherence is the condition

that G

(1)

, x

) factorizes

(1)

, x

) = G

∗

)G(x

), (78.59)

where G denotes some complex function. This deﬁnition

can be readily generalized to the nth order case. The nth

order coherence applies when the relation

(n)

,... ,x

)

= G

∗

) ···G

∗

)G(x

n+1

) ···G(x

) (78.60)

Part F 78.3

Quantized Field Effects 78.4 Photodetection Theory 1147

holds. A ﬁeld in a coherent state possesses nth order

coherence.

78.3.2 Photon Correlations

The Young experiment demonstrates the appearance

of ﬁrst-order correlations. However, experiments that

can distinguish between the classical and quantum

domains have to be based on measurements of

second-order correlations. These experiments are of

the Hanbury–Brown and Twiss type, and determine

the arrival of a photon at detector position x and

time t and another photon at time t +τ. Following

the theory of Glauber, the second-order correlation

function

(2)

(τ) =



−

(t)E

−

(t +τ)E

(t)



(78.61)

is measured. In this formula we have omitted the variable

for the position x. Usually the normalized correlation

function

(2)

(τ) =

(2)

(τ)



(1)

(0)



(78.62)

is introduced. The function g

(2)

is always positive, which

is true for classical as well as for quantum ﬁelds; but

there exists a purely quantum domain given by

0 ≤ g

(2)

(0)<1 . (78.63)

For example, for a number state |n,

(2)

|n

(0) = 1 −1/n , (78.64)

with n ≥ 1. In contrast, a coherent state |α yields

(2)

|α

(0) = 1.

78.3.3 Photon Bunching and Antibunching

In a realistic theory (but not in an oversimpliﬁed

one-mode model), the correlation function G

(2)

(τ) al-

ways factorizes on a sufﬁciently long time scale, and

(2)

(τ) →1. The photons are then no longer correlated,

and they arrive randomly as in the case of coherent light;

see for example (78.195).

If g

(2)

(0)>1, the photons show a tendency to ar-

rive in bunches, an effect known as photon bunching.

This effect has been observed for chaotic light. The op-

posite situation with 0 ≤ g

(2)

(0)<1 demonstrates the

reverse effect, namely photon antibunching. As seen

from (78.63), this is a regime only accessible to non-

classical light. An example is given by the resonance

ﬂuorescence of a two-level atom, treated in Sect. 78.13.

Note that we can rewrite g

(2)

(0) with the help of Man-

del’s Q parameter

(2)

(0) =



†



a

†

a

= 1 +Q . (78.65)

Hence, a ﬁeld state with Q < 0 shows the effect of

photon antibunching.

78.4 Photodetection Theory

So far we have used a very simple theory of photode-

tection: any absorbed photon leads to a photoelectric

emission which can be observed. But in any real ex-

periment, these photons are counted over some time

interval T and the observed photoelectric emissions

are dominated by two statistics: (i) the statistics of

photoelectric emission which is also present for a clas-

sical ﬁeld and (ii) the speciﬁc quantum statistics of

a quantized ﬁeld. A detailed discussion of the quan-

tum theory of photoelectric detection has been given

by Kelley and Kleiner [78.19]. A central result is the

formula

p(n, t, T) =



exp(−I) :



(78.66)

for the probability of counting n photoelectrons in the

time interval from t to t +T . This photocounting distri-

bution contains the integrated intensity operator

I = η

t+T





−



) (78.67)

containing the quantum efﬁciency η of the detector. The

notation :···:indicates a quantum average where the

operators have to be normally ordered and time or-

dered. This operator ordering reﬂects the process on

which a photodetector is based. It annihilates or ab-

sorbs photons, one after the other. A good treatment of

photoelectric detection can be found in [78.20].

78.4.1 Homodyne

and Heterodyne Detection

These detection methods allow the extraction of spe-

ciﬁc quantum features of a single mode quantum ﬁeld,

Part F 78.4

1148 Part F Quantum Optics

the signal ﬁeld. Figure 78.3 summarizes the principle of

optical homodyning. Two quantum ﬁelds described by

the annihilation operators a and b are mixed at a 50/50

beam splitter BS. Both ﬁelds have the same frequency.

The mode a represents the signal mode whose quantum

state is given by the density operator ρ. Mode b serves

as a reference ﬁeld, the local oscillator. The coherent

state |α=||α|e

iθ

 determines the quantum state of the

local oscillator. Two ideal photodetectors 1 and 2 meas-

ure the number of photons in the output modes of the

beam splitter. For a highly excited coherent state, i. e.,

a classical local oscillator, the statistics of the photocur-

rent difference ∆I can be described by the moments of

the signal mode operator

√



a e

−iθ

†

iθ



(78.68)

⌬I

–

ⱍα⬎

Fig. 78.3 The principle of optical homodyning

For example, the photocurrent difference

∆I∼X

=Tr(ρX

) (78.69)

is proportional to the expectation value of X

.Inpar-

ticular, for θ =0andθ =

one is able to measure all

the moments of the two quadratures x and p (78.13)and

(78.14) of the signal mode. In general, the statistics of

the photocurrent ∆I reveal the probability distribution

Pr (X

) =X

|ρ|X

 (78.70)

of the observable X

when the signal mode is in the

state ρ. The states

=π

−1/4

exp



− X



∞



n=0

√

(

)

inθ

|n (78.71)

are eigenstates of the operator X

, and are known as

rotated quadrature states.

The heterodyne technique [78.21,22] relies on a sim-

ilar mixing of a signal ﬁeld with a local oscillator at

a beam splitter, but this time the local oscillator fre-

quency is offset by the intermediate frequency ∆ω with

respect to the frequency ω

of the signal mode. Filters

select the beat frequency components in the photocur-

rent of the detectors. This photocurrent contains the

quantum statistics of the two quadratures of the signal

ﬁeld [78.21, 22].

78.5 Quasi-Probability Distributions

Quasi-probability distributions play an important role

in quantum optics for three reasons. First, they

are a complete representation of the density oper-

ator of a quantum ﬁeld. Second, they allow one

to calculate expectation values in the spirit of

classical statistical physics. Third, they offer the

possibility of converting a master equation for the

density operator into an equivalent c-number partial

differential equation. In this section, we relate a spe-

ciﬁc quasi-probability function to a speciﬁc operator

ordering.

78.5.1 s-Ordered Operators

A normally ordered product of a and a

†

is a product of

the form



†



: the annihilation operators a stand to

the right of the creation operators a

†

. In an antinormally

ordered product like a



†



, the order of a and a

†

has

changed. A generalized s-ordered product can be deﬁned





†





≡



∂

∂ξ





−

∂

∂ξ

∗



× D(ξ, ξ

∗

, s)



ξ=ξ

∗

(78.72)

with the generalized displacement operator

D(ξ, ξ

∗

, s) ≡ exp



ξa

†

−ξ

∗

a +

sξξ

∗



(78.73)

For s =1 we ﬁnd again normal ordering. The values

s = 0ands =−1 produce symmetric and antinormal or-

dered products. As an example we note {a

†

= a

†

a −

(s −1)/2.

Part F 78.5