Channel

number

0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

Fig. 5.4 Data from Arecchi™s experiment measuring photoelectron statistics of a cw, heli-

um“neon laser. The number of counts of output pulses from a photomultiplier tube, binned

within a narrow window of pulse heights, is plotted against the voltage pulse height for two

kinds of light ¬elds: ˜L™ for ˜laser light™, which closely ¬ts a Poissonian, and ˜G™ for ˜Gaussian

light™, which closely ¬ts a Bose“Einstein distribution function. (Reproduced from Arecchi

(1965).)

|± (t)| = |±| is ¬xed by gain clamping, but the phase φ(t) is not locked to any

external source. Consequently, the phase wanders (or di¬uses) on a very long

coherence time scale „coh 0.1 s (the inverse of the laser line width). The phase-

wander time scale is much longer than the integration time, RC 1 ns, of the

very fast photon detection system. Furthermore, the Poissonian distribution p(n)

√

only depends on the ¬xed amplitude |±| = n, so the phase wander of the laser

output beam does not appreciably a¬ect the Poissonian photocount distribution

function.

(2) For the G-case, the coherence time „coh is determined by the time required for

a speckle feature to cross the pinhole. For a rapidly rotating disc this is shorter

than the integration time of the photon detection system. As explained above,

this results in incoherent light described by a Bose“Einstein distribution peaked

at n = 0.

(3) The measurement process occurs at the photocathode surface of the photomulti-

plier tube, which, for unit quantum e¬ciency, emits n photoelectrons if n photons

impinge on it. However, unity quantum e¬ciency is not an essential requirement for

this experiment, since an analysis for arbitrary quantum e¬ciencies, when folded

½½

Properties of coherent states

in with a Bernoulli distribution function, shows that the Poissonian photoelec-

tron distribution still always results from an initial Poissonian photon distribution

(Loudon, 2000, Sec. 6.10). Similarly, a Bose“Einstein photoelectron distribution

function always results from an initial Bose“Einstein photon distribution.

(4) The condition that the laser be far above threshold is often not satis¬ed by real

continuous-wave lasers. The Scully“Lamb quantum theory of the laser predicts

that there can be appreciable deviations from the exact Poissonian distribution

when the small-signal gain of the laser is comparable to the loss of output mirrors.

Nevertheless, a skewed bell-shape curve that roughly resembles the Poissonian

distribution function is still predicted by the Scully“Lamb theory.

In sum, Arecchi™s experiment gave the ¬rst partial evidence that lasers emit a

coherent state, in that the observed photon count distribution is nearly Poissonian.

However, this photon-counting experiment only gives information concerning the di-

agonal elements n |ρ| n = p(n) of the density matrix. It gives no information about

the o¬-diagonal elements n |ρ| m when n = m. For example, this experiment cannot

distinguish between a pure coherent state |± , with |±| = n, and a mixed state for

which n |ρ| n happens to be a Poissonian distribution and n |ρ| m = 0 for n = m.

We shall see later that quantum state tomography experiments using optical homo-

dyne detection are sensitive to the o¬-diagonal elements of the density operator. These

experiments provide evidence that the state of a laser operating far above threshold

is closely approximated by an ideal coherent state.

In an extension of Arecchi™s experiment, Meltzer and Mandel (1971) measured the

photocount distribution function as a laser passes from below its threshold, through

its threshold, and ends up far above threshold. The change from a monotonically

decreasing photocount distribution below threshold”associated with the thermal state

of light”to a peaked one above threshold”associated with the coherent state”was

observed to agree with the Scully“Lamb theory.

5.4 Properties of coherent states

One of the objectives in studying coherent states is to use them as an alternate set

of basis functions for Fock space, but we must ¬rst learn to deal with the peculiar

mathematical features arising from the fact that the coherent states are eigenfunctions

of the non-hermitian annihilation operator a.

5.4.1 The displacement operator

The relation (3.83) linking the Heisenberg and Schr¨dinger pictures combines with the

o

explicit solution (5.40) of the Heisenberg equation to yield U † (t) aU (t) = ae’iωt +

± (t). For N = a† a, the identity exp (iθN ) a exp (’iθN ) = exp (’iθ) a (see Appendix

C.3, eqn (C.65)) allows this to be rewritten as

U † (t) aU (t) = eiωN t ae’iωN t + ± (t) , (5.43)

which in turn implies

†

U (t) eiωN t a U (t) eiωN t = a + ± (t) . (5.44)

½¾ Coherent states

Thus the physical model for generation of a coherent state in Section 5.2 implies that

there is a unitary operator which acts to displace the annihilation operator by ± (t).

The form of this operator can be derived from the explicit solution of the model

problem, but it is more useful to seek a unitary displacement operator D (±) that

satis¬es

D† (±) aD (±) = a + ± (5.45)

for all complex ±. Since D (±) is unitary, it can be written as D (±) = exp [’iK (±)],

where the hermitian operator K (±) is the generator of displacements. A similar situ-

ation arises in elementary quantum mechanics, where the representation p = ’i d/dq

for the momentum operator implies that the transformation

T∆q ψ (q) = ψ (q ’ ∆q) (5.46)

of spatial translation is represented by the unitary operator exp (’i∆qp/ ) (Brans-

den and Joachain, 1989, Sec. 5.9). This transformation rule for the wave function is

equivalent to the operator relation

e’i∆qp/ qei∆qp/ = q + ∆q . (5.47)

The similarity between eqns (5.45) and (5.47) and the associated fact that a, a† (like

[q, p]) is a c-number together suggest assuming that K (±) is a linear combination of

a and a† :

K (±) = g (±) a† + g — (±) a , (5.48)

where g (±) is a c-number yet to be determined.

One way to work out the consequences of this assumption is to de¬ne the inter-

polating operator a („ ) by

a („ ) = ei„ K(±) ae’i„ K(±) . (5.49)

This new operator is constructed so that it has the initial value a (0) = a and the ¬nal

value a (1) = D† (±) aD (±). In the „ -interval (0, 1), a („ ) satis¬es the Heisenberg-like

equation of motion

da („ )

= [a („ ) , K (±)] = ei„ K(±) [a, K (±)] e’i„ K(±) .

i (5.50)

d„

In the present case, the ansatz (5.48) shows that [a, K (±)] = g (±), so the equation of

motion simpli¬es to

da („ )

= g (±) , (5.51)

i

d„

with the solution a („ ) = a ’ ig (±) „ . Thus eqn (5.45) is satis¬ed by the choice g (±) =

i±, and the displacement operator is

†

’±— a)

D (±) = e(±a . (5.52)

The displacement operator generates the coherent state from the vacuum by

½¿

Properties of coherent states

†

’±— a)

|± = D (±) |0 = e(±a |0 . (5.53)

The simplest way to prove that D (±) |0 is a coherent state is to rewrite eqn (5.45) as

aD (±) = D (±) [a + ±] , (5.54)

and apply both sides to the vacuum state.

The displacement operators represent the translation group in the ±-plane, so they

must satisfy certain group properties. For example, a direct application of the de¬nition

(5.45) yields the inverse transformation as

D ’1 (±) = D† (±) = D (’±) . (5.55)

From eqn (5.45) one can see that applying D (β) followed by D (±) has the same e¬ect

as applying D (± + β); therefore, the product D (±) D (β) must be proportional to

D (± + β):

D (±) D (β) = D (± + β) ei¦(±,β) , (5.56)

where ¦ (±, β) is a real function of ± and β. The phase ¦ (±, β) can be determined by

using the Campbell“Baker“Hausdor¬ formula, eqn (C.66), or”as in Exercise 5.6”by

another application of the interpolating operator method. By either method, the result

is

—

D (±) D (β) = D (± + β) ei Im(±β ) . (5.57)

5.4.2 Overcompleteness

Distinct eigenstates of hermitian operators, e.g. number states, are exactly orthogonal;

therefore, distinct outcomes of measurements of the number operator”or any other

observable”are mutually exclusive events. This is the basis for interpreting |cn |2 =

2

| n| ψ | as the probability that the value n will be found in a measurement of the

number operator. By contrast, no two coherent states are ever orthogonal. This is

shown by using eqn (5.23) to calculate the value

1

± |β = exp ’ |± ’ β| exp (i Im [±— β])

2

(5.58)

2

of the inner product. On the other hand, states with large values of |± ’ β| are ap-

proximately orthogonal, i.e. | ± |β | 1, for quite moderate values of |± ’ β|. The

2

lack of orthogonality between distinct coherent states means that | ±| ψ | cannot be

interpreted as the probability for ¬nding the ¬eld in the state |± , given that it is

prepared in the state |ψ .

Although they are not mutually orthogonal, the coherent states are complete. A

necessary and su¬cient condition for completeness of the family {|± } is that a vector

|ψ satisfying

ψ |± = 0 for all ± (5.59)

is necessarily the null vector, i.e. |ψ = 0. A second use of eqn (5.23) allows this

equation to be expressed as

½ Coherent states

∞

±n

√ c— = 0 ,

F (±) = (5.60)

n

n!

n=0

where c— = ψ |n . This relation is an identity in ±, so all derivatives of F (±) must

n

also vanish. In particular,

√—

n

‚

F (±) = n!cn = 0 , (5.61)

‚± ±=0

so that cn = 0 for all n 0. The completeness of the number states then requires

|ψ = 0, and this establishes the completeness of the coherent states.

The coherent states form a complete set, but they are not linearly independent

vectors. This peculiar state of a¬airs is called overcompleteness. It is straightforward

to show that any ¬nite collection of distinct coherent states is linearly independent,

so to prove overcompleteness we must show that the null vector can be expressed as a

continuous superposition of coherent states. Let u1 = Re ± and u2 = Im ±, then any

linear combination of the coherent states can be written as

∞ ∞

du2 z (u1 , u2 ) |u1 + iu2 ,

du1 (5.62)

’∞ ’∞

where z (u1 , u2 ) is a complex function of the two real variables u1 and u2 . It is custom-

ary to regard z (u1 , u2 ) as a function of ±— and ±, which are treated as independent

variables, and in the same spirit to write

du1 du2 = d2 ± . (5.63)

For brevity we will sometimes write z (±) instead of z (±— , ±) or z (u1 , u2 ), and the

same convention will be used for other functions as they arise. Any confusion caused

by these various usages can always be resolved by returning to the real variables u1

and u2 .

In this new notation the condition that a continuous superposition of coherent

states gives the null vector is

d2 ±z (±— , ±) |± = 0 ,

|z = (5.64)

where the integral is over the entire complex ±-plane and z (±— , ±) is nonzero on

some open subset of the ±-plane. The number states are both complete and linearly

independent, so this condition can be expressed in a more concrete way as

1 2

d2 ±e’|±| z (±— , ±) ±n = 0 for all n

n |z = √ /2

0. (5.65)

n!

By using polar coordinates (± = ρ exp iφ) for the integration these conditions become

∞ 2π

1 n+1 ’ρ2 /2

√ dφz (ρ, φ) einφ = 0 for all n

dρρ e 0. (5.66)

n! 0 0

½

Properties of coherent states

In this form, one can see that the desired outcome is guaranteed if the φ-dependence

of z (ρ, φ) causes the φ-integral to vanish for all n 0. This is easily done by choosing

z (ρ, φ) = g (ρ) ρm exp (imφ) for some m > 0; that is,

z (±— , ±) = g (|±|) ±m , with m > 0 . (5.67)

The linear dependence of the coherent states means that the coe¬cients in the generic

expansion

d2 ±F (±— , ±) |±

|ψ = (5.68)

are not unique, since replacing F (±— , ±) by F (±— , ±) + z (±— , ±) yields the same vector

|ψ .

In spite of these unfamiliar properties, the coherent states satisfy a completeness

relation, or resolution of the identity,

d2 ±

|± ±| = I , (5.69)

π

analogous to eqn (2.84) for the number states. To prove this, we denote the left side

of eqn (5.69) by I and evaluate the matrix elements

d2 ±

n |I| m = n |± ± |m

π

∞ 2π

ρn+m ’ρ2 dφ i(n’m)φ

√

= dρ ρ e e

π

n!m!

0 0

= δnm . (5.70)

Thus I has the same matrix elements as the identity operator, and eqn (5.69) is

established.

Applying this representation of the identity to a state |ψ gives the natural”but

not unique”expansion

d2 ±

|ψ = |± ± |ψ . (5.71)

π

The completeness relation also gives a useful formula for the trace of any operator:

∞

d2 ± d2 ±

|± ±| X n |± ± |X| n

Tr X = Tr =

π π n=0

d2 ±

± |X| ± .

= (5.72)

π

5.4.3 Coherent state representations of operators

The completeness relation (5.69) is the basis for deriving useful representations of

operators in terms of coherent states. For any Fock space operator X, we easily ¬nd

the general result

½ Coherent states

d2 ± d2 β

|± ±| X |β β|

X=

π π

d2 ± d2 β

|± ± |X| β β| .

= (5.73)

π π

Since the coherent states are complete, this result guarantees that X is uniquely de¬ned

by the matrix elements ± |X| β . On the other hand, the overcompleteness of the

coherent states suggests that the same information may be carried by a smaller set of

matrix elements.

A An operator X is uniquely determined by ± |X| ±

The diagonal matrix elements n |X| n in the number-state basis”or in any other

orthonormal basis”do not uniquely specify the operator X, but the overcompleteness

of the coherent states guarantees that the diagonal elements ± |X| ± do determine

X uniquely. The ¬rst step in the proof is to use eqn (5.23) one more time to write

± |X| ± in terms of the matrix elements in the number-state basis,

∞ ∞

m |X| n —m n

’|±|2

√

± |X| ± = e ± ±. (5.74)

m!n!

m=0 n=0

Now suppose that two operators Y and Z have the same diagonal elements, i.e.

± |Y | ± = ± |Z| ± ; then X = Y ’ Z must satisfy

∞ ∞

m |X| n —m n

√ ± ± = 0. (5.75)

m!n!

m=0 n=0

This is an identity in the independent variables ± and ±— , so the argument leading to

eqn (5.61) can be applied again to conclude that m |X| n = 0 for all m and n. The

completeness of the number states then implies that X = 0, and we have proved that

if ± |Y | ± = ± |Z| ± for all ± , then Y = Z . (5.76)

B Coherent state diagonal representation

The result (5.76) will turn out to be very useful, but it does not immediately supply us

with a representation for the operator. On the other hand, the general representation

(5.73) involves the o¬-diagonal matrix elements ± |X| β which we now see are appar-

ently super¬‚uous. This suggests that it may be possible to get a representation that

only involves the projection operators |± ±|, rather than the o¬-diagonal operators

|± β| appearing in eqn (5.73). The key to this construction is the identity

an |± ±| a†m = ±n ±—m |± ±| , (5.77)

which holds for any non-negative integers n and m. Let us now suppose that X has

a power series expansion in the operators a and a† , then by using the commutation

½

Multimode coherent states

relation a, a† = 1 each term in the series can be rearranged into a sum of terms in

which the creation operators stand to the right of the annihilation operators, i.e.

∞ ∞

Xnm an a†m ,

A

X= (5.78)

m=0 n=0

where Xnm is a c-number coe¬cient. Since this exactly reverses the rule for normal

A

ordering, it is called antinormal ordering, and the superscript A serves as a reminder

of this ordering rule. By combining the identities (5.69) and (5.77) one ¬nds

∞ ∞

d2 ±

|± ±| a†m

Xnm an

A

X=

π

m=0 n=0

d2 ±X A (±) |± ±| ,

= (5.79)

where

∞ ∞

1

Xnm ±n ±—m

X (±) =

A A

(5.80)

π m=0 n=0

is a c-number function of the two real variables Re ± and Im ±. This construction gives

us the promised representation in terms of the projection operators |± ±|.

5.5 Multimode coherent states

Up to this point we have only considered coherent states of a single radiation oscil-

lator. In the following sections we will consider several generalizations that allow the

description of multimode squeezed states.

5.5.1 An elementary approach to multimode coherent states

A straightforward generalization is to replace the de¬nition (5.18) of the one-mode

coherent state by the family of eigenvalue problems

aκ |± = ±κ |± for all κ , (5.81)

where ± = (±1 , ±2 , . . . , ±κ , . . .) is the set of eigenvalues for the annihilation operators

aκ . The single-mode case is recovered by setting ±κ = 0 for κ = κ. The multimode

coherent state |± ”de¬ned as the solution of the family of equations (5.81)”can

be constructed from the vacuum state by using eqn (5.53) for each mode to get

|± = D (±κ ) |0 , (5.82)

κ

where

D (±κ ) = exp ±κ a† ’ ±— aκ (5.83)

κ κ

is the displacement operator for the κth mode. Since there are an in¬nite number of

modes, the de¬nition (5.82) raises various mathematical issues, such as the convergence

½ Coherent states

of the in¬nite product. In the following sections, we show how these issues can be dealt

with, but for most applications it is safe to proceed by using the formal in¬nite product.

For later use, it is convenient to specialize the general de¬nition (5.82) of the

multimode state to the case of box-quantized plane waves, i.e.

|± = D (±) |0 , (5.84)

exp ±ks a† ’ ±— aks .

D (±) = D (±ks ) = (5.85)

ks

ks

ks ks

By combining the eigenvalue condition aks |± = ±ks |± with the expression (3.69) for

E(+) , one can see that

E(+) (r) |± = E (r) |± , (5.86)

where

ωk

E (r) = i ±ks eks eik·r (5.87)

2 0V

ks

is the classical electric ¬eld de¬ned by |± .

Coherent states for wave packets—

5.5.2

The incident ¬eld in a typical experiment is a traveling-wave packet, i.e. a superposition

of plane-wave modes. A coherent state describing this situation is therefore an example

of a multimode coherent state. From this point of view, the multimode coherent state

|± is actually no more complicated than a single-mode coherent state (Deutsch, 1991).

This is a linguistic paradox caused by the various meanings assigned to the word

˜mode™. This term normally describes a solution of Maxwell™s equations with some

additional properties associated with the boundary conditions imposed by the problem

at hand. Examples are the modes of a rectangular cavity or a single plane wave.

General classical ¬elds are linear combinations of the mode functions, and they are

called wave packets rather than modes. Let us now return to eqn (5.82) which gives

a constructive de¬nition of the multimode state |± . Since the operators ±κ a† ’ ±— aκ

κ κ

and ±κ a† ’±— aκ commute for κ = κ , the product of unitary operators in eqn (5.82)

κ

κ

can be rewritten as a single unitary operator,

±κ a† ’ ±— aκ

|± = exp |0

κ κ

κ

†

= exp a [±] ’ a [±] |0 , (5.88)

where

±— aκ

a [±] = (5.89)

κ

κ

is an example of the general de¬nition (3.191). In other words the multimode coherent

state |± is a coherent state for the wave packet

w (r) = ±κ wκ (r) , (5.90)

κ

½

Multimode coherent states

where the wκ (r)s are mode functions. The wave packet w(r) de¬nes a point in the

classical phase space, so it represents one degree of freedom of the ¬eld. This suggests

changing the notation by

|± ’ |w = D [w] |0 , (5.91)

where

D [w] = exp a† [w] ’ a [w] (5.92)

is the wave packet displacement operator, and a [w] is simply another notation for

a [±].

The displacement rule,

D† [w] a [v] D [w] = a [v] + (v, w) , (5.93)

and the product rule,

D [v] D [w] = D [v + w] exp {i Im (w, v)} , (5.94)

are readily established by using the commutation relations (3.192), the interpolating

operator method outlined in Section 5.4.1, and the Campbell“Baker“Hausdor¬ formula

(C.66). The displacement rule (5.93) immediately yields the eigenvalue equation

a [v] |w = (v, w) |w . (5.95)

This says that the coherent state for the wave packet w is also an eigenstate”with

the eigenvalue (v, w)”of the annihilation operator for any other wave packet v. To

recover the familiar single-mode form, a |± = ± |± , simply set w = ±w0 , where w0

is normalized to unity, and v = w0 ; then eqn (5.95) becomes a [w0 ] |±w0 = ± |±w0 .

The inner product of two multimode (wave packet) coherent states is obtained from

(5.91) by calculating

v |w = 0 D† [v] D [w] 0

= exp {i Im (v, w)} 0 |D [w ’ v]| 0

1 2

= exp {i Im (v, w)} exp ’ w ’ v , (5.96)

2

where u = (u, u) is the norm of the wave packet u.

Sources of multimode coherent states—

5.5.3

In Section 5.2 we saw that a monochromatic classical current serves as the source for a

single-mode coherent state. This demonstration is readily generalized as follows. The

total Hamiltonian in the hemiclassical approximation is the sum of eqns (3.40) and

(5.36),

d3 rA(’) (r, t) · ’∇2 A(+) (r, t) ’ d3 r J (r, t) · A (r, t) .

H = 2 0 c2 (5.97)

The corresponding Heisenberg equation for A(+) ,

½¼ Coherent states

‚A(+) (r, t) 1 ’1/2

1/2

= c ’∇2 A(+) (r, t) ’ ’∇2 J (r, t) ,

i (5.98)

‚t 2 0c

has the formal solution

1/2

A(+) (r, t) = exp ’i (t ’ t0 ) c ’∇2 A(+) (r, t0 ) + w (r, t) , (5.99)

where

t

i ’1/2

1/2

dt exp ’i (t ’ t ) c ’∇2 ’∇2 J (r, t ) ,

w (r, t) = (5.100)

2 0c t0

and the Schr¨dinger and Heisenberg pictures coincide at the time, t0 , when the current

o

is turned on. The classical ¬eld w (r, t) satis¬es the c-number version of eqn (5.98),

‚w (r, t) 1 ’1/2

1/2

= c ’∇2 w (r, t) ’ ’∇2 J (r, t) .

i (5.101)

‚t 2 0c

Applying this solution to the vacuum gives A(+) (r, t) |0 = w (r, t) |0 in the Heisen-

berg picture, and A(+) (r) |w, t = w (r, t) |w, t in the Schr¨dinger picture. The time-

o

dependent coherent state |w, t evolves from the vacuum state (|w, t0 = |0 ) under

the action of the Hamiltonian given by eqn (5.97).

Completeness and representation of operators—

5.5.4

The issue of completeness for the multimode coherent states is (in¬nitely) more com-

plicated than in the single-mode case. Since we are considering all modes on an equal

footing, the identity (5.69) for a single mode must be replaced by

d2 ±κ

|±κ ±κ | = Iκ , (5.102)

π

where Iκ is the identity operator for the single-mode subspace Hκ . The resolution of

the identity on the entire space HF is given by

d2 ±κ

|± ±| = IF . (5.103)

π

κ

The mathematical respectability of this in¬nite-dimensional integral has been estab-

lished for basis sets labeled by a discrete index (Klauder and Sudarshan, 1968, Sec.

7-4). Fortunately, the Hilbert spaces of interest for quantum theory are separable, i.e.

they can always be represented by discrete basis sets. In most applications only a few

modes are relevant, so the necessary integrals are approximately ¬nite dimensional.

Combining the multimode completeness relation (5.103) with the fact that op-

erators for orthogonal modes commute justi¬es the application of the arguments in

Sections 5.4.3 and 5.6.3 to obtain the multimode version of the diagonal expansion for

the density operator:

d2 ± |± P (±) ±| ,

ρ= (5.104)

where

d2 ±κ

d2 ± = . (5.105)

π

κ

½½

Multimode coherent states

Applications of multimode states—

5.5.5

Substituting the relation

2 0c 2 0c 1/2

d3 rw— (r) · ’∇2

A(+) [w] = A(+) (r)

a [w] = (5.106)

into eqn (5.95) provides the r-space version of the eigenvalue equation:

A(+) (r) |w = w (r) |w . (5.107)

2 0c

For many applications it is more useful to use eqn (3.15) to express this in terms of

the electric ¬eld,

E(+) (r) |w = E (r) |w , (5.108)

where

c 1/2

E (r) = i ’∇2 w (r) (5.109)

2 0

is the positive-frequency part of the classical electric ¬eld corresponding to the wave

packet w. The result (5.108) can be usefully applied to the calculation of the ¬eld

correlation functions for the coherent state described by the density operator ρ =

|w w|. For example, the equal-time version of G(2) , de¬ned by setting all times to

zero in eqn (4.77), factorizes into

— —

G(2) (x1 , x2 ; x3 , x4 ) = E1 (r1 ) E2 (r2 ) E3 (r3 ) E4 (r4 ) , (5.110)

where Ep (r) =s— · E (r). In fact, correlation functions of all orders factorize in the same

p

way.

Now let us consider an experimental situation in which the classical current is

turned on at some time t0 < 0 and turned o¬ at t = 0, leaving the ¬eld prepared in a

coherent state |w . The time at which the Schr¨dinger and Heisenberg pictures agree

o

is now shifted to t = 0, and we assume that the ¬elds propagate freely for t > 0. The

Schr¨dinger-picture state vector |w, t evolves from its initial value |w, 0 according to

o

the free-¬eld Hamiltonian, while the operators remain unchanged.

In the Heisenberg picture the state vector is always |w and the operators evolve

freely according to eqn (3.94). This guarantees that

E(+) (r, t) |w = E (r, t) |w , (5.111)

where E (r, t) is the freely propagating positive-frequency part that evolves from the

initial (t = 0) function given by eqn (5.109). According to eqn (5.110) the correlation

function factorizes at t = 0, and by the last equation each factor evolves independently;

therefore, the multi-time correlation function for the wave packet coherent state |w

factorizes according to

— —

G(2) (x1 , x2 ; x3 , x4 ) = E1 (r1 , t1 ) E2 (r2 , t2 ) E3 (r3 , t3 ) E4 (r4 , t4 ) . (5.112)

½¾ Coherent states

5.6 Phase space description of quantum optics

The set of all classical ¬elds obtained by exciting a single mode is described by a two-

dimensional phase space, as shown in eqn (5.1). The set of all quasiclassical states for

the same mode is described by the coherent states {|± }, that are also labeled by a

two-dimensional space. This correspondence is the basis for a phase-space-like descrip-

tion of quantum optics. This representation of states and operators has several useful

applications. The ¬rst is a precise description of the correspondence-principle limit.

The relation between coherent states and classical ¬elds also provides a quantitative

description of the departure from classical behavior. Finally, as we will see in Section

18.5, the phase space representation of the density operator ρ gives a way to convert

the quantum Liouville equation for the operator ρ into a c-number equation that can

be used in numerical simulations.

In Section 9.1 we will see that the results of photon detection experiments are ex-

pressed in terms of expectation values of normal-ordered products of ¬eld operators. In

this way, counting experiments yield information about the state of the electromagnetic

¬eld. In order to extract this information, we need a general scheme for representing

the density operators describing the ¬eld states. The original construction of the elec-

tromagnetic Fock space in Chapter 3 emphasized the role of the number states. Every

density operator can indeed be represented in the basis of number states, but there are

many situations for which the coherent states provide a more useful representation.

For the sake of simplicity, we will continue to emphasize a single classical ¬eld mode

for which the phase space “em can be identi¬ed with the complex plane.

5.6.1 The Wigner distribution

The earliest”and still one of the most useful”representations of the density operator

was introduced by Wigner (1932) in the context of elementary quantum mechanics. In

classical mechanics the most general state of a single particle moving in one dimension

is described by a normalized probability density f (Q, P ) de¬ned on the classical phase

space “mech = {(Q, P )}, i.e. f (Q, P ) dQdP is the probability that the particle has

position and momentum in the in¬nitesimal rectangle with area dQdP centered at the

point (Q, P ) and

dQ dP f (Q, P ) = 1 . (5.113)

In classical probability theory it is often useful to represent a distribution in terms of

its Fourier transform,

dP f (Q, P ) e’i(uP +vQ) ,

χ (u, v) = dQ (5.114)

which is called the characteristic function (Feller, 1957b, Chap. XV). In some ap-

plications it is easier to evaluate the characteristic function, and then construct the

probability distribution itself from the inverse transformation:

du dv

χ (u, v) ei(uP +vQ) .

f (Q, P ) = (5.115)

2π 2π

½¿

Phase space description of quantum optics

An example of the utility of the characteristic function is the calculation of the mo-

ments of the distribution, e.g.

‚nχ

2

2

Q = (i) ,

‚v 2 (u,v)=(0,0)

‚nχ

2 (5.116)

QP = (i) ,

‚v‚u (u,v)=(0,0)

.

.

.

A The Wigner distribution in quantum mechanics

In quantum mechanics, a phase space description like f (Q, P ) is forbidden by the

uncertainty principle. Wigner™s insight can be interpreted as an attempt to ¬nd a

quantum replacement for the phase space integral in eqn (5.114). Since the integral is

a sum over all classical states, it is natural to replace it by the sum over all quantum

states, i.e. by the quantum mechanical trace operation. The role of the classical dis-

tribution is naturally played by the density operator ρ, and the classical exponential

exp [’i (uP + vQ)] can be replaced by the unitary operator exp [’i (up + vq)]. In this

way one is led to the de¬nition of the Wigner characteristic function

χW (u, v) = Tr ρe’i(up+vq) , (5.117)

which is a c-number function of the real variables u and v. The classical de¬nition

(5.114) of the characteristic function by a phase space integral is meaningless for

quantum theory, but the inverse transformation (5.115) still makes sense when applied

to χW . This suggests the de¬nition of the Wigner distribution,

dv

du

χW (u, v) ei(uP +vQ) ,

W (Q, P ) = (5.118)

2π 2π

where the normalization has been chosen to make W (Q, P ) dimensionless. The Wigner

distribution is real and normalized by

dQdP

W (Q, P ) = 1 , (5.119)

but”as we will see later on”there are physical states for which W (Q, P ) assumes

negative values in some regions of the (Q, P )-plane. For these cases W (Q, P ) cannot be

interpreted as a probability density like f (Q, P ); consequently, the Wigner distribution

is called a quasiprobability density.

Substituting eqn (2.116) for the density operator into eqn (5.117) leads to the

alternative form

Pe Ψe e’i(up+vq) Ψe

χW (u, v) =

e

Ψe e’ivq e’iup Ψe ,

Pe e i uv/2

= (5.120)

e

½ Coherent states

where the last line follows from the identity (C.67). Since exp (’iup) is the spatial

translation operator, the expectation value can be expressed as

dQ Ψ— (Q ) e’iv(Q + u) Ψe (Q + u) .

Ψe e’ivq e’iup Ψe = (5.121)

e

Substituting these results into eqn (5.118) ¬nally leads to

dX 2iXP/

Ψe (Q + X) Ψ— (Q ’ X) ,

Pe

W (Q, P ) = e (5.122)

e

π

e

which is the de¬nition used in Wigner™s original paper. Thus the ˜momentum™ depen-

dence of the Wigner distribution comes from the Fourier transform with respect to the

relative coordinate X. Integrating out the momentum dependence yields the marginal

distribution in Q:

dP 2

Pe |Ψe (Q)| .

W (Q, P ) = (5.123)

e

Despite the fact that W (Q, P ) can have negative values, the marginal distribution in

Q is evidently a genuine probability density.

B The Wigner distribution for quantum optics

In the transition to quantum optics the mechanical operators q and p are replaced

by the operators q and p for the radiation oscillator. In agreement with our earlier

experience, it turns out to be more useful to use the relations (2.66) to rewrite the

unitary operator exp [’i (up + vq)] as exp ·a† ’ · — a , where

ω

u’i v, (5.124)

·=

2 2ω

so that eqn (5.117) is replaced by

†

’· — a

χW (·) = Tr ρe·a . (5.125)

The characteristic function χW (·) has the useful properties χW (0) = 1 and χ— (·) =

W

χW (’·). The Wigner distribution is then de¬ned (Walls and Milburn, 1994, Sec.

4.2.2) as the Fourier transform of χW (·):

1 —

±’·±—

d2 ·e·

W (±) = χW (·) . (5.126)

π2

After verifying the identity

d2 ± ·— ±’·±—

e = δ2 (·) , (5.127)

π2

where δ2 (·) ≡ δ (Re ·) δ (Im ·), one ¬nds that the Wigner function W (±) is normalized

by

½

Phase space description of quantum optics

d2 ±W (±) = 1 . (5.128)

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

P (±), but eqn (5.58) for the inner product shows instead that

d2 β 2

| ± |β | P (β)

Q (±) =

π

d2 β ’|β|2

= e P (± + β) . (5.166)

π

Thus the Q (±) is a Gaussian average of the P -function around the point ±.

The average of the generic normal-ordered product a†m an is

a†m an = Tr ρa†m an = Tr an ρa†m = d2 ±±n ±—m P (±) , (5.167)

which combines with eqn (5.164) to yield

X a, a† d2 ±X N (±) P (±) ,

= (5.168)

where

∞ ∞

Xnm ±—n ±m .

X N N

(±) = (5.169)

m=0 n=0

The normalization condition Tr ρ = 1 becomes

d2 ±P (±) = 1 , (5.170)

so P (±) is beginning to look like another probability distribution. Indeed, for a pure

coherent state, ρcoh = |±0 ±0 |, the P -function is

Pcoh (±) = δ2 (± ’ ±0 ) , (5.171)

where

δ2 (± ’ ±0 ) = δ (Re ± ’ Re ±0 ) δ (Im ± ’ Im ±0 ) . (5.172)

This is a positive distribution that exactly picks out the coherent state |±0 ±0 |, so

it is more intuitively appealing than the Q-function description of the same state by

a Gaussian distribution. Another hopeful result is provided by the P -function for a

½¾ Coherent states

thermal state. From eqn (2.178) we know that the density operator for a thermal or

chaotic state with average number n has the diagonal matrix elements

nn

n |ρth | n = ; (5.173)

(1 + n)n+1

therefore, the P -function has to satisfy

nn

d2 ±Pth (±) | n |± |2

= (5.174)

n+1

(1 + n)

|±|2n

’|±|2

2

= d ±Pth (±) e . (5.175)

n!

Expressing the remaining integral in polar coordinates suggests that P (±) might be

proportional to a Gaussian function of |±|, and a little trial and error leads to the

result

2

|±|

1

exp ’

Pth (±) = . (5.176)

πn n

Thus the P -function acts like a probability distribution for two very di¬erent states

of light. On the other hand, this is a quantum system, so we should be prepared for

surprises.

The interpretation of P (±) as a probability distribution requires P (±) 0 for all ±,

and the normalization condition (5.170) implies that P (±) cannot vanish everywhere.

The states with nowhere negative P (±) are called classical states, and any states

for which P (±) < 0 in some region of the ±-plane are called nonclassical states.

Multimode states are said to be classical if the function P (±) in eqn (5.104) satis¬es

P (±) 0 for all ±.

The meaning of ˜classical™ intended here is that these are quantum states with

the special property that all expectation values can be simulated by averaging over

random classical ¬elds with the probability distribution P (±). By virtue of eqn (5.171),

all coherent states”including the vacuum state”are classical, and eqn (5.176) shows

that thermal states are also classical. The last example shows that classical states need

not be quasiclassical,2 i.e. minimum-uncertainty, states.

Our next objective is to ¬nd out what kinds of states are nonclassical. A convenient

way to investigate this question is to use eqn (5.165) to calculate the probability that

exactly n photons will be detected; this is given by

2n

|±|

’|±|2

2

n |ρ| n = d ± | n |± | P (±) =

2 2

d ±e P (±) . (5.177)

n!

If ρ is any classical state”other than the vacuum state”the integrand is non-negative,

so the integral must be positive. For the vacuum state, ρvac = |0 0|, eqn (5.171) gives

P (±) = πδ2 (±), so the integral vanishes for n = 0 and gives 0 |ρvac | 0 = 1 for n = 0.

2 It is too late to do anything about this egregious abuse of language.

½¿

Phase space description of quantum optics

Thus for any classical state”other than the vacuum state”the probability for ¬nding

n photons cannot vanish for any value of n:

n |ρ| n = 0 for all n . (5.178)

Thus a state, ρ = ρvac , such that n |ρ| n = 0 for some n > 0 is nonclassical. The

simplest example is the pure number state ρ = |m m|, since n |ρ| n = 0 for n = m.

This can be seen more explicitly by applying eqn (5.177) to the case ρ = |m m|, with

the result

2n

’|±|2 |±| 1 for n = m ,

2

d ±e P (±) = (5.179)

n! 0 for n = m .

The conditions for n = m cannot be satis¬ed if P (±) is non-negative; therefore, P (±)

for a pure number state must be negative in some region of the ±-plane. A closer

examination of this in¬nite family of equations shows further that P (±) cannot even

be a smooth function; instead it is proportional to the nth derivative of the delta

function δ2 (±).

The normal-ordered characteristic function—

B

An alternative construction of the P (±)-function can be carried out by using the

† —

normally-ordered operator, e·a e’· a , to de¬ne the normally-ordered character-

istic function † —

χN (·) = Tr ρe·a e’· a . (5.180)

The corresponding distribution function, P (±), is de¬ned by replacing χW with χN

in eqn (5.126) to get

1 — —

d2 ·e· ±’·± χN (·) .

P (±) = 2 (5.181)

π

The identity (5.141) relates χN (·) and χW (·) by

χN (·) = e|·|

2

/2

χW (·) , (5.182)

so the argument leading to eqn (5.140) yields the much weaker bound |χN (·)| < e|·| /2

2

for the normal-ordered characteristic function χN (·). This follows from the fact that

† —

e·a e’· a is self-adjoint rather than unitary. The eigenvalues are therefore real and

need not have unit modulus. This has the important consequence that P (±) is not

guaranteed to exist, even in the delta function sense. In the literature it is often said

that P (±) can be more singular than a delta function.

We already know from eqn (5.171) that P (±) exists for a pure coherent state, but

what about number states? The P -distribution for the number state ρ = |1 1| can

be evaluated by combining the general relation (5.182) with the result (5.152) for the

Wigner characteristic function of a number state to get

1 —

±’·±— 2

1 ’ |·|

d2 ·e·

P (±) = . (5.183)

π2

This can be evaluated by using the identities

½ Coherent states

‚ ·— ±’·±— ‚ ·— ±’·±—

—

±’·±— — —

, · — e· ±’·± =

=’

·e· e e , (5.184)

‚±— ‚±

to ¬nd

‚‚

P (±) = δ2 (±) + δ2 (±) . (5.185)

‚± ‚±—

This shows that P (±) is not everywhere positive for a number state. Since P (±) is

a generalized function, the meaning of this statement is that there is a real, positive

test function f (±) for which

d2 ±P (±) f (±) < 0 , (5.186)

2

e.g. f (±) = exp ’2 |±| .

Let ρ be a density operator for which P (±) exists, then in parallel with eqn (5.130)

we have

m n

‚ ‚

—n m

χN (·, · — )

’—

2

d ± ± ± P (±) =

‚· ‚· ·=0

† —

= Tr ρa†n e·a am e’· a

·=0

= Tr ρa†n am . (5.187)

The case m = n = 0 gives the normalization

d2 ± P (±) = 1 , (5.188)

and the identity of the averages calculated with P (±) and the averages calculated with

ρ shows that the density operator is represented by

d2 ± |± P (±) ±| .

ρ= (5.189)

Thus the de¬nition of P (±) given by eqn (5.181) agrees with the original de¬nition

(5.165).

For an operator expressed in normal-ordered form by

∞ ∞

†

Xnm a†n am ,

N

X a ,a = (5.190)

m=0 n=0

eqn (5.187) yields

d2 ± P (±) X N (±— , ±) ,

Tr (ρX) = (5.191)

where

∞ ∞

—

Xnm ±—n ±m .

X N N

(± , ±) = (5.192)

m=0 n=0

½

Phase space description of quantum optics

The P -distribution and the Wigner distribution are related by the following argu-

ment. First invert eqn (5.181) to get

—

’· — ±

d2 ± e·±

χN (·) = P (±) . (5.193)

Combining this with eqn (5.126) and the relation (5.182) produces