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

1 —

±’·±— ’|·|2 /2

d2 ·e· e χN (·)

W (±) =

π2

1 —

’±— ) ’· — (β’±) ’|·|2 /2

d2 βP (β) d2 · e·(β

=2 e e . (5.194)

π

The ·-integral is readily done by converting to real variables, and the relation between

the Wigner distribution and the P -distribution is

2

d2 βe’2|β’±| P (β) .

2

W (±) = (5.195)

π

An interesting consequence of this relation is that a classical state automatically

yields a positive Wigner distribution, i.e.

P (±) 0 implies W (±) 0, (5.196)

but the opposite statement is not true:

W (±) 0 does not imply P (±) 0. (5.197)

This is demonstrated by exhibiting a single example”see Exercise 5.7”of a state with

a positive Wigner function that is not classical.

It is natural to wonder why P (±) 0 should be chosen as the de¬nition of a

classical state instead of W (±) 0. The relations (5.196) and (5.197) give one reason,

since they show that P (±) 0 is a stronger condition. A more physical reason is that

counting rates are described by expectation values of normal-ordered products, rather

than Weyl products. This means that P (±) is more directly related to the relevant

experiments than is W (±).

Multimode phase space—

5.6.4

In Section 5.5 we de¬ned multimode coherent states |± by aκ |± = ±κ |± , where aκ

is the annihilation operator for the mode κ and

± = (±1 , ±2 , . . . , ±κ , . . .) . (5.198)

For states in which only a ¬nite number of modes are occupied, i.e. aκ |± = 0 for

κ > κ , the characteristic functions de¬ned previously have the generalizations

†

’· — a

χW · = Tr ρe··a , (5.199)

† —

χN · = Tr ρe··a e’· a

, (5.200)

½ Coherent states

where · ≡ (·1 , ·2 , . . .), and

· · a† = ·κ a† . (5.201)

κ

κκ

The corresponding distributions are de¬ned by multiple Fourier transforms. For ex-

ample the P -distribution is

⎡ ¤

2

d ·κ ¦ ’··±— +·— ·±

⎣

P (±) = e χN · , (5.202)

π2

κκ

and the density operator is given by

⎡ ¤

⎣ d2 ±κ ¦ |± P (±) ±| .

ρ= (5.203)

κκ

All this is plain sailing as long as κ remains ¬nite, but some care is required to

get the mathematics right when κ ’ ∞. This has been done in the work of Klauder

and Sudarshan (1968), but the κ ’ ∞ limit is not strictly necessary in practice. The

reason is to be found in the alternative characterization of coherent states given by

|± ’ |w , where

A(+) [v] |w = (v, w) |w , (5.204)

and the wave packets w, v, etc. are expressed as expansions in the chosen modes,

w (r) = ±κ wκ (r) . (5.205)

κ

The vector ¬elds v and w belong to the classical phase space “em de¬ned in Section

3.5.1, so the expansion coe¬cients ±κ must go to zero as κ ’ ∞. Thus any real ex-

perimental situation can be adequately approximated by a ¬nite number of modes.

With this comforting thought in mind, we can express the characteristic and distribu-

tion functions as functionals of the wave packets. In this language, the normal-ordered

characteristic function and the P -distribution are respectively given by

χN (v) = Tr ρ exp A(+) [v] exp ’A(’) [v] (5.206)

and

D [v] exp {(w, v) ’ (v, w)} χN (v) .

P (w) = (5.207)

The symbol D [v] stands for a (functional) integral over the in¬nite-dimensional

space “em of classical wave packets; but, as we have just remarked, it can always

be approximated by a ¬nite-dimensional integral over the collection of modes with

non-negligible amplitudes.

Gaussian states— ½

Gaussian states—

5.7

In classical statistics, the Gaussian (normal) distribution has the useful property that

the ¬rst two moments determine the values of all other moments (Gardiner, 1985, Sec.

2.8.1). For a Gaussian distribution over N real variables”with the averages of single

variables arranged to vanish”all odd moments vanish and the even moments satisfy

(2q)!

x1 · · · x2q = xk xl · · · xm xn ]sym ,

[ xi xj (5.208)

q!2q

where i, j, k, l, m, n range over 1, . . . , 2q and the subscript sym indicates the average

over all ways of partitioning the variables into pairs. Two fourth-order examples are

4! 1

x1 · · · x4 = { x1 x2 x3 x4 + x1 x3 x2 x4 + x1 x4 x2 x3 }

2!22 3

= x1 x2 x3 x4 + x1 x3 x2 x4 + x1 x4 x2 x3 (5.209)

and

x4 = 3 x2 x2 . (5.210)

1 1 1

This classical property is shared by the coherent states, as can be seen from the

general identity

m

± a†m an ± = ±—m ±n = ± a† ±

n

( ± |a| ± ) . (5.211)

A natural generalization of the classical notion of a Gaussian distribution is to de¬ne

Gaussian states (Gardiner, 1991, Sec. 4.4.5) as those that are described by density

operators of the form

ρG = N exp ’G a, a† , (5.212)

where

1 1

G a, a† = La† a + M a†2 + M — a2 , (5.213)

2 2

L and M are free parameters, and the constant N is ¬xed by the normalization

condition Tr ρ = 1.

For the special value M = 0, the Gaussian state ρG has the form of a thermal state,

and we already know (see eqn (5.148)) how to calculate the Wigner characteristic

function for this case. We would therefore like to transform the general Gaussian state

into this form. If the operators a and a† were replaced by complex variables ± and ±— ,

this would be easy. The c-number quadratic form G (±, ±— ) can always be expressed

as a sum of squares by a linear transformation to new variables

± = µ± + ν±— ,

(5.214)

±— = µ— ±— + ν — ± .

What is needed now is the quantum analogue of this transformation, i.e. the new and

old operators are related by