quantum mechanics

With the cross-correlation function rewritten as
G(1)(r1t1, r2t2) = |G(1)(r1t1, r2t2)|ei?(r1 t1 ,r2 t2 ) , (1.84) (1.81) becomes
(|E+(r, t)|2) = G(1)(r1t1, r1t1) + G(1)(r2t2, r2t2)
2|G(1)(r1t1, r2t2)| cos ?. (1.85)

The third term in (1.83) is responsible for the appearance of interferences.
We say that the highest degree of coherence corresponds to a light ?eld that produces the maximum contrast on the screen, where contrast is de?ned as
V = Imax ? Imin . (1.86)
Imax + Imin
Substituting (1.83) with cos ? = 1, we readily obtain
2|G(1)(r1t1, r2t2)|

V =
G(1)(r1t1, r1t1) + G(1)(r2t2, r2t2)

. (1.87)


The denominator in (1.85) doesn’t play an important role; G(1)(riti, riti) is just the intensity on the detector due to the ith slit and the denominator acts as a normalization constant. To maximize the contrast for a given source and
geometry, we need to maximize the numerator 2|G(1)(r1t1, r2t2)|. To achieve
this goal we note that according to the Schwarz inequality
G(1)(r1t1, r1t1)G(1)(r2t2, r2t2) ? |G(1)(r1t1, r2t2)|2 . (1.88)

The coherence function is maximized when equality holds, that is when
|G(1)(r1t1, r2t2)| = [G(1)(r1t1, r1t1)G(1)(r2t2, r2t2)]1/2 , (1.89)

which is the coherence condition used by Born and Wolf. As pointed out by Glauber, it is convenient to replace this condition by the equivalent expression
G(1)(r1t1, r2t2) = E ?(r1t1)E (r2t2) , (1.90)
where the complex function E (r1t1) is some function, not necessarily the electric ?eld. If G(1)(r1t1, r2t2) may be expressed in the form (1.88), we say that G(1) factorizes. This factorization property de?nes ?rst-order coherence: when (1.88) holds, the fringe contrast V is maximum.
This de?nition of ?rst-order coherence can be readily generalized to higher orders. A ?eld is said to have nth-order coherence if its mth-order correlation functions
G(m)(x1 ... xm, ym ... y1) = (E?(x1) ··· E?(xm)E+(ym) ··· E+(y1)) (1.91) factorize as


G(m)(x1 ... xm, ym ... y1) = E ?(x1) ···E ?(xm)E (ym) ···E (y1) (1.92)
for all m ? n. Here we use the compact notation xj = (rj , tj ), yj = (rm+j , tm+j ), and G(m) is a direct generalization of (1.80).
Before giving an example where second-order correlation functions play
a crucial role, we point out that although a monochromatic ?eld is coherent to all orders, a ?rst-order coherent ?eld is not necessarily monochromatic. One might be led to think otherwise because we often deal with stationary light, such as that from stars and cw light sources. By de?nition, the two- time properties of a stationary ?eld depend only on the time di?erence. The corresponding ?rst-order correlation function thus has the form
G(1)(t1, t2) = G(1)(t1 ? t2) . (1.93)

If such a ?eld is ?rst-order coherent, then with (1.88), we ?nd
G(1)(t1 ? t2) = E ?(t1)E (t2) , (1.94)


which is true only if


E (t1) ? e?i?t1 , (1.95)

that is, stationary ?rst-order coherent ?elds are monochromatic!
Let us now turn to the famous Hanbury Brown-Twiss experiment Fig. 1.5, which probes higher-order coherence properties of a ?eld. In this experiment, a beam of light (from a star in the original experiment) is split into two beams, which are detected by detectors D1 and D2. The signals are multiplied and averaged in a correlator. This procedure di?ers from the Young two-slit experiment in that light intensities, rather than amplitudes, are compared. Two absorption measurements are performed on the same ?eld, one at time t and the other at t + ? . It can be shown [Cohen-Tannoudji et al. 1989] that
this measures |E+(r,t + ?, )E+(r, t)|2 . Dropping the useless variable r and
averaging, we see that this is precisely the second-order correlation function
G(2)(t, t + ?, t + ?, t) = (E?(t)E?(t + ? )E+(t + ? )E+(t)) , (1.96)













Fig. 1.5. Diagram of Hanbury Brown-Twiss experiment


or for a stationary process,
G(2)(? ) = (E?(0)E?(? )E+(? )E+(0)) . (1.97)

According to (1.89), the ?eld is second-order coherent if (1.92) holds and
G(2)(? ) = E ?(0)E ?(? )E (? )E (0) . (1.98)

It is convenient to normalize this second-order correlation function as


(2)
g(2)(? ) =


(? )


. (1.99)

|G(1)(0)|2

Since a stationary ?rst-order coherent ?eld is monochromatic and satis?es (1.93), second-order coherence implies that

g(2)(? ) = 1 . (1.100)

that is, g(2)(? ) is independent to the delay ? .
The original experiment on Hanbury Brown-Twiss was used to measure the apparent diameter of stars by passing their light through a pinhole. A second-order correlation function like that in Fig. 1.6 was measured. Although the light was ?rst-order coherent, we see that it was not second-order coher- ent. The energy tended to arrive on the detector in bunches, with strong statistical correlations.
In contrast to the well-stabilized laser with a unity g(2) and the star-light with bunching, recent experiments in resonance ?uorescence show antibunch- ing, with the g(2) shown in Fig. 1.7. Chapter 16 discusses this phenomenon



g(2)(?)







1




0 ?

Fig. 1.6. Second-order correlation function (1.97) for starlight in original Hanbury Brown-Twiss experiment


g(2)(?)

1





0 ?

Fig. 1.7. Second-order correlation function showing antibunching found in reso- nance ?uorescence


in detail; here we point out that such behavior cannot be explained with classical ?elds. To see this, note that
(2)(0) ? |G(1)(0)|2

g(2)(0) ? 1 = G

In terms of intensities, this gives


|G(1)(0)|2

. (1.101)

g(2)(0) ? 1 = (I

2)? (I)2
(I)2


((I ? (I))2) (I)2


, (1.102)


where we do not label the times, since we consider a stationary system with ? = 0. Introducing the probability distribution P (I) to describe the average over ?uctuations, we ?nd for (1.100)


g(2)(0) ? 1 =

1 ¸
(I)2


dIP (I)(I ? (I))2 . (1.103)

Classically this must be positive, since (I ? (I))2 ? 0 and the probability
distribution P (I) must be positive. Hence g(2) cannot be less than unity, in
contradiction to the experimental result shown in Fig. 1.7. At the beginning of this chapter we say that the ?elds we use can usually be treated classically. Well we didn’t say always! To use a formula like (1.101) for the antibunched case, we need to use the concept of a quasi -probability function P (I) that permits negative values. Quantum mechanics allows just that (see Sect. 13.6).