VOLUME 10, NUMBER 3 PHYSICAL RKVIKW LKYTKRS 1 FEBRU~RV 1963

domain (evidently bordering the region) appearsto keep individual experimental observation un-likely.

The foregoing treatment remains independentof fundamental constants by replacing Hall's quan-tum mechanical parameter v =(8/2m) 1n(b/a), interms of Planck's constant 8, the mass m of thehelium atom, and the inner and outer radii of quan-tized vortices a and b, respectively. Implicationsare not intended, however, that rotating heliumremains devoid of macroscopic quantum effects.In imagining such features, the domain structureindicating ability to preserve the irrotational state"(quantum index, n= 0) over prescribed regions mayin fact, constitute a form of macroscopic uncer-tainty principle. Such considerations mill bediscussed more fully in a later detailed treat-ment.

Dork supported in part by the National Science Foun-dation.

'J. R. Pellam, Phys. Rev. Letters 9, 289 (1962).H. Hall and%. Vinen, Proc. Roy. Soc. (London)

A238, 204 (1956).SL Onsager, Suppl. Nuovo Cimento 2, 249 (1949).4R. Feynman, Progress in Low-Temperature Physics,

edited by J. C. Gorter (North-Holland Publishing Com-pany, Amsterdam, 1955), Vol. 1, Chap. II, p. 36.

H. Hall and W. Vinen, Proc. Roy. Soc. {London)A238, 215 (1956).

H. Hall, Phil. Trans. Roy. Soc. London A250, 359

(1957); Proc. Roy. Soc. (London) A245, 546 (1958);Adv. Phys. 9, 89 (1960).

Such magnus force 2ps(Q xv) exerted per unit vol-ume against vortex lines constituted the product of totalcirculation (2Q) residing per unit area in vortex linestimes the relative momentum density (psv).

SProcedures employed by Hall in reference 6 for eval-uating curvature in terms of the circulation vector 7'

x v retain validity here. Application of the k V' opera-stion to V' x vs results in the same differential functionaldependence of domain axis velocity v on system coordi-nate position as appear in Eqs. (5)-(7) of that referencefor vortices.

Space forbids equation-by-equation detail, and thereader is therefore referred to the following equationsof the Hall 1958 treatise, reference (6): (a) Eq. (1),(b) Eqs. {5)-(8), (c) Eqs. {9)-{10),and (d) Eqs. {16)-(17).

~~In this connection, note the interesting observationsthat magnitude of the angular momentum and rotationalenergy content for counter-rotating domains become,respectively,

angular momentum=2mp a Q/p ra2s s

=3.4 & 10 3 cm /sec

per gram of superfluid, and

rotational energy =4' a Qs

=3.5 && 10 ~p g-cm/sec2s

per unit length of domain, each evidently independent ofangular rotation rate Q.

PHOTON CORRELATIONS*

Roy J. GlauberLyman Laboratory, Harvard University, Cambridge, Massachusetts

(Received 27 December 1962)

In 1956 Hanbury Bromn and Tmiss' reportedthat the photons of a light beam of narrow spec-tral width have a tendency to arrive in correlatedpa. irs. We have developed general quantum me-chanica1 methods for the investigation of suchcorrelation effects and shall present here re-sults for the distribution of the number of pho-tons counted in an incoherent beam. The factthat photon correlations are enhanced by narrow-ing the spectral bandwidth has led to a prediction'of large-scale correlations to be observed in thebeam of an optical maser. We shall indicatethat this prediction is misleading and followsfrom an inappropriate model of the maser beam.In considering these problems we shall outline

a method of describing the photon field mhich ap-pears particularly mell suited to the discussionof experiments performed with light beams, wheth-er coherent or incoherent.

The correlations observed in the photoioniza-tion processes induced by a light beam were giv-en a simple semiclassical explanation by Purcell, 'who made use of the methods of microwave noisetheory. More recently, a number of papers havebeen written examining the correlations in con-siderably greater detail. These papers'&' ' re-tain the assumption that the electric field in alight beam can be described as a classical Gaus-sian stochastic process. In actuality, the be-havior of the photon field is considerably more

VOLUME 10, NUMBER 3 PHYSICAL RKVIKW LKTTKRS 1 FEBRUARY 1963

ln ) =exp(--,' ln l')Q [n /(& l)'"]l&)k= k' nkwhere Q.k is an arbitrary complex amplitude.We shall call the l nit) coherent states; their useis well known in discussions of the harmonic os-cillator in the classical limit. The expectationvalue in the state l n~) of the contribution of thekth mode to the total field is a monochromaticwave with complex amplitude proportional to ak.The coherent states l n&), for all complex n&,form a complete set in a sense best expressedby the relation

(1/~)fin )(n ld'"nk k k

(2)

where d"'zk is a real element of area of the com-plex nk plane. It follows that any state may beexpanded linearly in terms of coherent states.The most general light beam can thus be de-scribed by a density operator of the form

p= f+((n, n '})ll ln )(n 'ld'~n d'"n ', (3)k k k'

which deals with all the modes of the field at once.An incoherent light beam must be described as

a statistical mixture of all the excitation statesavailable for each mode excited. For the kthmode, the probability to be associated with thestate ln&) is proportional to ((N&)/(1+(&t, ))} ",

varied than such an assumption would indicate.Whereas a stationary Gaussian stochastic processis described completely by its frequency-depend-ent power spectrum, a great deal more informa-tion in the form of amplitude and phase relationsbetween differing quantum states may be requiredto describe a steady light beam. Beams of identi-ca,l spectral distributions may exhibit altogetherdifferent photon correlations or, alternatively,none at all. There is ultimately no substitutefor the quantum theory in describing quanta.

We assume, for convenience, that the field hasdiscrete propagation modes labeled by an indexk (which in free space, for example, specifiespropagation vector and polarization). To describethe quantum state of the kth mode we must specifyan infinite set of complex amplitudes, one foreach quantum occupation state [nt,), nt = 0, 1, 2, ~ ~ ~ .Since the states me wish to describe include onesin which the phase of the kth mode is fairly melldefined, and a, large number of states ln~) mustthen be superposed, it is preferable to use an al-together different set of basis states. We takethese to be of the form

where (Nt) is the mean number of photons occu-pying the mode. A simple theorem expressesthis mixture in terms of the coherent states de-fined earlier: The density operator (3) reducesto a product of operators of the form

where the probability p(n&) is a Gaussian func-tion,

p(n ) =(~(N )}-'exp(-ln l'/(X )).

In particular, blackbody radiation may be de-scribed as a mixture of coherent waves by sub-stituting for (N&) the familiar value for thermalexcitation of a field oscillator.

To discuss photon correlations me examine thephotoionization probability of a pair of atoms,labeled 1 and 2, which lie at r, and r, within thelight beam. We assume that the incident beamis of narrow enough spectral bandwidth that anyvariation of frequency-dependent parameters en-tering the ohotoionization probabilities may beneglected. Then, if we sum the transition prob-abilities over final electron energies, there isno difficulty in defining a time at mhich eachelectron emission takes place. The probabilitydensity for ionization of atom 1 at time t, andfor atom 2 at t2 may be mritten as

w(t, t, ) = w, w2C(r, t,r, t,), (6)

where M, and x, are the constant transition prob-abilities for each atom placed individually in thebeam, and C is the function whose departure fromunity expresses a tendency for the tmo events tobe correlated.

We assume, to simplify notation, that photonsof only one polarization are present. The appro-priate vector component of the electric field op-erator has a positive-frequency part 8 '+'(r, t)and a negative-frequency part 8 ' '(r, t). Thecorrelation function may be expressed in termsof these operators as

C (r,t,r2t2)

trg h ' '(r, t,) 8' '(r2t2) 8+'(r~t, ) 8'+'(r2t2)}trQ g' '(r, t,) g'+'(r, t,)}trQ 8' '(r, t,) 8'+'(r, t,)}'

(t)where tr stands for trace, and p is a density op-erator of the general form (3).

It is easily shown that coherent states of thefield lead to no photoionization correlations at

85

VOLUME 10, NUMBER 3 P 8YSI CAL R E VI E%' LETTERS 1 FEBRUARY 1963

C =1+!ft(t,- t, -c-'(x, -x,))!', (8)

all. If the state of the field is specified by anydensity operator of the form IIy I o't, )(of, !, thecorrelation function C reduces to unity. A cor-relation between photons only appears when inco-herent mixtures or superpositions of the coherentstates are present. For collimated, completelyincoherent beams of the type described earlier(e.g. , filtered thermal radiation), we find

(ywt)a(n, t) =—,— S [t(y'+2yW)"']1

n! (y +2yW)'

(N~) - j(~t, - &u)'+ y') ', with central frequency cu,

may be stated as follows: Let W be the averagerate at which photons are recorded; then theprobability that n photons are recorded in a timet, long compared to 1/y, is

where the coordinates x& are the components of

rj ln the propagation dlrectlon* and the f nctlonR is given by

x exp[-t[(y'+2yW) '-yD. (10)

(9)

This result, for incoherent beams, correspondsin the classical limit, when the modes are treatedas forming a continuum, to that derived usingstochastic models. ' ' It may be associated, in

this limit, with a tendency of the complex totalfield strength of the beam to fluctuate in modu-lus with time.

The density operator which represents an actualmaser beam is not yet known. It is clear thatsuch a beam cannot be represented by a productof individual coherent states, II&!a&)(a&!, un-less the phase and amplitude stability of the de-vice is perfect. On the other hand, a maserbeam is not at all likely to be described by theideally incoherent classical model which under-lies the calculation of Mandel and Wolf, andleads them to results corresponding to Eqs. (8)and (9). More plausible models for a steady ma-ser beam are much closer in behavior to theideal coherent states. They may be shown tolead to photon correlations only to the extent thatrandom amplitude modulation is present in thestatistically averaged beam.

If photoionization processes tend to be corre-lated in time, the distribution of the number ofphotons recorded by a counter in a fixed intervalof time should differ from the Poisson distribu-tion. We have developed a general technique forfinding this distribution for incoherent light beams. 'The result for the important case in which thespectral distribution has the Lorentz line shape,

The functions S„(x) are nth order polynomials in

x ', familiar in the theory of modified Besselfunctions of half-integral order. They may befound from the relations Sp Sy 1 and Spy+ $ ~pg'

+(1+nx ')S„. The full set of moments of the dis-tribution is given by the averages of products ofthe form n!/(n —j)!. These are

(n! /(n -j)!) = (Wt) S.(yt). (11)av

When the number of photons recorded during arelaxation time is small, S'«y, e. g. , when thelinewidth becomes appreciable, the distribution(10) approaches the Poisson distribution as alimit.

The author is grateful to Dr. Saul Bergmannfor bringing these problems to his attention, andto Dr. S. M. MacNeille and the American OpticalCompany for partial support of their solution.

'%'ork supported in part by the Air Force Office ofScientific Research.

'R. Hanbury Brown and R. Q. Twiss, Nature 177, 27(1956); G. A. Rebka and R. V. Pound, Nature 180, 1035(1957).

2L. Mandel and E. Wolf, Phys. Rev. 124, 1696 (1961).3E. M. Purcell, Nature 178, 1449 (1956).4F. D. Kahn, Optica Acta 5, 93 (1958).5L. Mandel, Proc. Phys. Soc. (London) 71, 1037

(1958).8L. Mandel, Proc. Phys. Soc. (London) 74, 233 (1959).~Incoherent light beams of exceedingly narrow band-

width may, in principle, be formed by superposing theoutputs of many identical but independent masers.

86