Representations of the Electromagnetic Field

Abstract A full description of the electromagnetic field requires a quantum statistical treatment. The electromagnetic field has an infinite number of modes and each mode requires a statistical description in terms of its allowed quantum states. However, as the modes are described by independent Hilbert spaces, we may form the statistical description of the entire field as the product of the distribution function for each mode. This enables us to confine our description to a single mode without loss of generality.
In this chapter we introduce a number of possible representations for the density
operator of the electromagnetic field. One representation is to expand the density operator in terms of the number states. Alternatively the coherent states allow a number of possible representations via the P function, the Wigner function and the Q function.



4.1 Expansion in Number States


The number or Fock states form a complete set, hence a general expansion of ? is
? = ?Cnm|n)(m| . (4.1)

The expansion coefficients Cnm are complex and there is an infinite number of them. This makes the general expansion rather less useful, particularly for problems where the phase-dependent properties of the electromagnetic field are important and hence the full expansion is necessary. However, in certain cases where only the photon number distribution is of interest the reduced expansion
? = ?Pn|n)(n| , (4.2)

may be used. Here Pn is a probability distribution giving the probability of having n photons in the mode. This is not a general representation for all fields but may prove useful for certain fields. For example, a chaotic field, which has no phase information, has the distribution


57


1
Pn = (1 + n¯)

. n¯ .n
1 + n¯



, (4.3)


where n¯ is the mean number of photons. This is derived by maximising the entropy
S = ?TR{? ln ? } , (4.4)
subject to the constraint Tr{? a†a} = n¯, and is just the usual Planck distribution for black-body radiation with


n¯ =

1
eh¯ ?/kT ? 1


. (4.5)

The second-order correlation function g2(0) may be written according to (3.66)
g2(0)= 1 + V (n) ? n¯
n¯

where V (n) is the variance of the distribution function Pn. Hence, for the power-law distribution V (n)= n¯2 + n¯ we find g(2)(0)= 2. For a field with a Poisson distribution of photons


Pn =

e?n¯
n¯n n!


(4.7)

the variance V (n)= n¯, hence g(2)(0)= 1.
A coherent state has a Poisson distribution of photons. However, a measurement of g(2)(0) would not distinguish between a coherent state and a field prepared from an incoherent mixture with a Poisson distribution. In order to distinguish between these two fields a phase-dependent measurement such as a measurement of ?X1, ?X2 would need to be made.