Coherence Properties of the Electromagnetic Field

Abstract In this chapter correlation functions for the electromagnetic field are intro- duced from which a definition of optical coherence may be formulated. It is shown that the coherent states possess nth-order optical coherence. Photon-correlation measurements and the phenomena of photon bunching and antibunching are de- scribed. Phase-dependent correlation functions which are accessible via homo- dyne measurements are introduced. The theory of photon counting measurements is given.



3.1 Field-Correlation Functions


We shall now consider the detection of an electromagnetic field. A large-scale macroscopic device is complicated, hence, we shall study a simple device, an ideal photon counter. The most common devices in practice involve a transition where a photon is absorbed. This has important consequences since this type of counter is insensitive to spontaneous emission. A complete theory of detection of light re- quires a knowledge of the interaction of light with atoms. We shall postpone this until a study of the interaction of light with atoms is made in Chap. 10. At this stage we shall assume we have an ideal detector working on an absorption mechanism which is sensitive to the field E(+) (r,t) at the space-time point (r,t). We follow the treatment of Glauber [1].
The transition probability of the detector for absorbing a photon at position r and time t is proportional to
Tif = |( f |E(+)(r, t)|i)|2 (3.1)
if |i) and | f ) are the initial and final states of the field. We do not, in fact, measure the final state of the field but only the total counting rate. To obtain the total count
rate we must sum over all states of the field which may be reached from the initial state by an absorption process. We can extend the sum over a complete set of final
states since the states which cannot be reached (e.g., states | f ) which differ from |i)



29



by two or more photons) will not contribute to the result since they are orthogonal to E(+) (r,t)|i). The total counting rate or average field intensity is
I(r, t)= ?Tfi = ?(i|E(?)(r, t)| f )( f |E(+)(r, t)|i)
f f
= (i|E(?)(r, t)E(+)(r, t)|i) , (3.2)

where we have used the completeness relation
?| f )( f | = 1 . (3.3)
f
The above result assumes that the field is in a pure state |i). The result may be easily generalized to a statistical mixture state by averaging over initial states with the probability Pi, i.e.,
I(r, t)= ?Pi(i|E(?)(r, t)E(+)(r, t)|i) . (3.4)
i


This may be written as



I(r, t)= Tr {? E(?)(r, t)E(+)(r, t)} , (3.5)

where ? is the density operator defined by
? = ?Pi|i)(i| . (3.6)
i

If the field is initially in the vacuum state
? = |0)(0| , (3.7)


then the intensity is


I(r, t)= (0|E(?)E(+)|0) = 0 . (3.8)

The normal ordering of the operators (that is, all annihilation operators are to the right of all creation operators) yields zero intensity for the vacuum. This is a conse- quence of our choice of an absorption mechanism for the detector. Had we chosen a detector working on a stimulated emission principle, problems would arise with vacuum fluctuations. More generally the correlation between the field at the space-
time point x = (r,t) and the field at the space-time point x? = (r,t?) may be written
as the correlation function
G(1)(x, x?)= Tr{? E(?)(x)E(+)(x?)} . (3.9)

The first-order correlation function of the radiation field is sufficient to account for classical interference experiments. To describe experiments involving intensity cor- relations such as the Hanbury-Brown and Twiss experiment, it is necessary to define higher-order correlation functions. We define the nth-order correlation function of the electromagnetic field as

3.2 Properties of the Correlation Functions 31

G(n)(x1 ... xn, xn+1 ... x2n)= Tr{? E(?)(x1) ... E(?)(xn)
× E(+)(xn+1) ... E(+)(x2n)} . (3.10)

Such an expression follows from a consideration of an n-atom photon detector [1]. The n-fold delayed coincidence rate is

W (n)(t1 ... tn)= snG(n)(r1t1 ... rntn, rntn ... r1t1) , (3.11) where s is the sensitivity of the detector.