In order to justify this approach, we next demonstrate that the average, Tr ρX, of

any operator X can be expressed in terms of the moments of the Wigner distribution.

The representation (5.127) of the delta function and the identities

m n

‚ ‚

m —n · — ±’·±— —

±’·±—

’ e·

±±e = (5.129)

‚· — ‚·

allow the moments of W (±) to be evaluated in terms of derivatives of the characteristic

function, with the result

m n

‚ ‚

m —n

χW (·, · — )

’—

2

d ± ± ± W (±) = . (5.130)

‚· ‚· ·=0

The characteristic function can be cast into a useful form by expanding the exponential

in eqn (5.125) and using the operator binomial theorem (C.44) to ¬nd

∞

1 k

—

Tr ρ ·a† ’ · — a

χW (·, · ) =

k!

k=0

∞ k

1 k! k’j

· k’j (’· — ) a†

j

aj

= Tr ρS ,

j! (k ’ j)!

k! j=0

k=0

(5.131)

k’j

where the Weyl”or symmetrical”product S a† aj is the average of all distinct

orderings of the operators a and a† . Using this result in eqn (5.130) yields

n

d2 ±±m ±—n W (±) = Tr ρS a† am . (5.132)

By means of the commutation relations, any operator X that has a power series

expansion in a and a† can be expressed as the sum of Weyl products:

∞ ∞

n

a†

Xnm S

W

am ,

X= (5.133)

n=0 m=0

where the Xnm s are c-number coe¬cients. The expectation value of X is then

W

∞ ∞

n

a†

Xnm S

W

am

X=

n=0 m=0

d2 ±X W (±) W (±) ,

= (5.134)

where

∞ ∞

Xnm ±m ±—n .

X W W

(±) = (5.135)

n=0 m=0

Thus the Wigner distribution carries the same physical information as the density

operator.

½ Coherent states

ω0 /2 0 a ’ a† is the electric

As an example, consider X = E 2 , where E = i

¬eld amplitude for a single cavity mode. In terms of Weyl products, E 2 is given by

ω0

S a2 ’ 2S a† a + S a†2

E2 = ’ , (5.136)

20

and substituting this expression into eqn (5.134) yields

ω0

d2 ± 2 |±| ’ ±2 ’ ±—2 W (±) .

2

E2 = (5.137)

20

Existence of the Wigner distribution—

C

The general properties of Hilbert space operators, reviewed in Appendix A.3.3, guar-

antee that the unitary operator exp ·a† ’ · — a has a complete orthonormal set of

(improper) eigenstates |Λ , i.e.

exp ·a† ’ · — a |Λ = eiθΛ (·) |Λ , (5.138)

where θΛ (·) is real, ’∞ < Λ < ∞, and Λ |Λ = δ (Λ ’ Λ ). Evaluating the trace in

the |Λ -basis yields

dΛ Λ |ρ| Λ eiθΛ (·) .

χW (·) = (5.139)

This in turn implies that χW (·) is a bounded function, since

|χW (·)| < dΛ | Λ |ρ| Λ | = d· Λ |ρ| Λ = Tr ρ = 1 , (5.140)

where we have used the fact that all diagonal matrix elements of ρ are positive. The

Fourier transform of a constant function is a delta function, so the Fourier transform

of a bounded function cannot be more singular than a delta function. This establishes

the existence of W (±)”at least in the delta function sense”but there is no guarantee

that W (±) is everywhere positive.

D Examples of the Wigner distribution

In some simple cases the Wigner function can be evaluated analytically by means of

the characteristic function.

Coherent state. Our ¬rst example is the characteristic function for a coherent state,

ρ = |β β|. The calculation of χW (·) in this case can be done more conveniently by

applying the identities (C.69) and (C.70) to ¬nd

†

’· — a /2 ·a† ’· — a /2 ’· — a ·a†

= e’|·| = e|·|

2 2

e·a e e e e . (5.141)

The ¬rst of these gives

†

’· — a † —

/2 ·β — ’· — β

= e’|·| β e·a e’· β = e’|·|

2 2

χW (·) = Tr ρe·a /2 a

e . (5.142)

½

Phase space description of quantum optics

This must be inserted into eqn (5.126) to get W (±). These calculations are best done

by rewriting the integrals in terms of the real and imaginary parts of the complex

integration variables. For the coherent state this yields

2 ’2|±’β|2

W (±) = e . (5.143)

π

The fact that the Wigner function for this case is everywhere positive is not very

surprising, since the coherent state is quasiclassical.

Thermal state. The second example is a thermal or chaotic state. In this case, we

use the second identity in eqn (5.141) and the cyclic invariance of the trace to write

— † † — †

χW (·) = e|·| Tr ρe’· a e·a = e|·| Tr e·a ρe’· a e·a

2 2

/2 /2

. (5.144)

Evaluating the trace with the aid of eqn (5.72) leads to the general result

d2 ± ·±— ’·— ±

± |ρ| ± .

χW (·) = e (5.145)

π

According to eqn (2.178) the density operator for a thermal state is

∞

nn

|n n| ,

ρth = (5.146)

n+1

n=0 (n + 1)

where n = Nop is the average number of photons. The expansion (5.23) of the

coherent state yields

2

|±|

1

± |ρth | ± = exp ’ , (5.147)

n+1 n+1

so that

2

|±|

d2 ±

1

exp (·±— ’ · — ±) exp ’

χth (·) =

W

n+1 π n+1

1

|·|2 .

= exp ’ n + (5.148)

2

The general relation (5.126) de¬ning the Wigner distribution can be evaluated in the

same way, with the result

2

|±|

1 1

exp ’

Wth (±) = , (5.149)

π n + 1/2 n + 1/2

which is also everywhere positive.

½ Coherent states

Number state. For the third example, we choose a pure number state, e.g. ρ =

|1 1|, which yields

†

’· — a † —

= e’|·| 1 e·a e’·

2

χW (·) = Tr ρe·a /2 a

1. (5.150)

Expanding the exponential gives

—

e’· |1 = |1 ’ · — |0 ,

a

(5.151)

so the characteristic function and the Wigner function are respectively

χW (·) = 1 ’ |·|2 e’|·|

2

/2

(5.152)

and

1 —

±’·±—

e’|·|

2

2

1 ’ |·|

d2 ·e· /2

W (±) =

π2

2 ’2|±|2

2

= ’ 1 ’ 4 |±| e . (5.153)

π

In this case W (±) is negative for |±| < 1/2, so the Wigner distribution for a number

state |1 1| is a quasiprobability density. A similar calculation for a general number

state |n yields an expression in terms of Laguerre polynomials (Gardiner, 1991, eqn

(4.4.91)) which is also a quasiprobability density.

The Q-function

5.6.2

A Antinormal ordering

According to eqn (5.76) ρ is uniquely determined by its diagonal matrix elements in

the coherent state basis; therefore, complete knowledge of the Q-function,

1

± |ρ| ± ,

Q (±) = (5.154)

π

is equivalent to complete knowledge of ρ. The real function Q (±) satis¬es the inequality

1

0 Q (±) , (5.155)

π

and the normalization condition

d2 ±Q (±) = 1 .

Tr ρ = (5.156)

The argument just given shows that Q (±) contains all the information needed to

calculate averages of any operator, but it does not tell us how to extract these results.

½

Phase space description of quantum optics

The necessary clue is given by eqn (5.78) which expresses any operator X as a sum of

antinormally-ordered terms. With this representation for X, the expectation value is

∞ ∞

Xmn Tr ρam a†n

A

X = Tr (ρX) =

m=0 n=0

∞ ∞

d2 ±

± a†n ρam ±

Xmn

A

=

π

m=0 n=0

d2 ±Q (±) X A (±) ,

= (5.157)

where X A (±) is de¬ned by eqn (5.80). In other words the expectation value of any

physical quantity X can be calculated by writing it in antinormally-ordered form, then

replacing the operators a and a† by the complex numbers ± and ±— respectively, and

¬nally evaluating the integral in eqn (5.157).

The Q-function, like the Wigner distribution, is di¬cult to calculate in realistic

experimental situations; but there are idealized cases for which a simple expression

can be obtained. The easiest is that of a pure coherent state, i.e. ρ = |±0 ±0 |, which

leads to

2

exp ’ |± ’ ±0 |

2

| ± |±0 |

Q (±) = = . (5.158)

π π

Despite the fact that this state corresponds to a sharp value of ±, the probability

distribution has a nonzero spread around the peak at ± = ±0 . This unexpected feature

is another consequence of the overcompleteness of the coherent states.

At the other extreme of a pure number state, ρ = |n n|, the expansion of the

coherent state in number states yields

e’|±| |±|2n

2

| ± |n |2

Q (±) = = , (5.159)

π π n!

√

which is peaked on the circle of radius |±| = n.

Di¬culties in computing the Q-function—

B

For any state of the ¬eld, the Q-function is everywhere positive and normalized to

unity, so Q (±) is a genuine probability density on the electromagnetic phase space “em .

The integral in eqn (5.157) is then an average over this distribution. These properties

make the Q-function useful for the display and interpretation of experimental data

or the results of approximate simulations, but they do not mean that we have found

the best of all possible worlds. One di¬culty is that there are functions satisfying the

inequality (5.155) and the normalization condition (5.156) that do not correspond to

any physically realizable density operator, i.e. they are not given by eqn (5.154) for

any acceptable ρ. The irreducible quantum ¬‚uctuations described by the commutation

relation a, a† = 1 are the source of this problem. For any density operator ρ,

aa† = a† a + 1 1. (5.160)

½¼ Coherent states

Evaluating the same quantity by means of eqn (5.157) produces the condition

2

d2 ±Q (±) |±| 1 (5.161)

on the Q-function. As an example of a spurious Q-function, consider

4

|±|

2

Q (±) = √ 2 exp ’ 4 . (5.162)

π πσ σ

√

This function satis¬es eqns (5.155) and (5.156) for σ 2 > 2/ π, but the integral in eqn

(5.161) is

σ2

2

d2 ±Q (±) |±| = √ . (5.163)

π

√ √

Thus for 2/ π < σ 2 < π, the inequality (5.161) is violated. Finding a Q-function

that satis¬es this inequality as well is still not good enough, since there are similar

inequalities for all higher-order moments a2 a†2 , a3 a†3 , etc. This poses a serious problem

in practice, because of the inevitable approximations involved in the calculation of the

Q-function for a nontrivial situation. Any approximation could lead to a violation of

one of the in¬nite set of inequalities and, consequently, to an unphysical prediction for

some observable.

The dangers involved in extracting the density operator from an approximate Q-

function do not occur in the other direction. Substituting any physically acceptable

approximation for the density operator into eqn (5.154) will yield a physically accept-

able Q (±). For this reason the results of approximate calculations are often presented

in terms of the Q-function. For example, plots of the level lines of Q (±) can provide

useful physical insights, since the Q-function is a genuine probability distribution.

The Glauber“Sudarshan P (±)-representation

5.6.3

A Normal ordering

We have just seen that the evaluation of the expectation value, X , using the Q-

function requires writing out the operator in antinormal-ordered form. This is contrary

to our previous practice of writing all observables, e.g. the Hamiltonian, the linear

momentum, etc. in normal-ordered form. A more important point is that photon-

counting rates are naturally expressed in terms of normally-ordered products, as we

will see in Section 9.1.

The commutation relations can be used to express any operator X a, a† in normal-

ordered form,

∞ ∞

Xnm a†n am ,

N

X= (5.164)

m=0 n=0

so we want a representation of the density operator which is adapted to calculating the

averages of normal-ordered products. For this purpose, we apply the coherent state

½½

Phase space description of quantum optics

diagonal representation (5.79) to the density operator. This leads to the P -function

representation introduced by Glauber (1963) and Sudarshan (1963):

d2 ± |± P (±) ±| .

ρ= (5.165)

If the coherent states were mutually orthogonal, then Q (±) would be proportional to