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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
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

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