634 J. Opt. Soc. Am. B/Vol. 17, No. 4 /April 2000 R. Vyas and S. Singh

Antibunching and photoemission waiting times

Reeta Vyas and Surendra Singh

Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701

Received July 16, 1999

We discuss antibunching of photons in terms of the distributions of waiting between successive photoemissionsand compare it with definitions of antibunching based on the two-time intensity correlation function. We il-lustrate our results for photon sequences emitted by parametric oscillators. Curves are presented to illustratethe behavior. © 2000 Optical Society of America [S0740-3224(00)02304-3]

OCIS codes: 270.0270; 270.5290; 270.2500; 190.4970.

Nonclassical properties of the electromagnetic field havecontinued to attract great attention, as they provide atesting ground for the predictions of quantum electrody-namics. Nonclassical properties of the electromagneticfield are reflected in squeezing,1 sub-Poissonianstatistics,2,3 antibunching,4–7 and violation of variousclassical inequalities.6–10 These nonclassical effects referto different aspects of the field. For example, squeezingrefers to the wavelike character of the field. It is mea-sured in an interference experiment. Antibunching andsub-Poissonian statistics, on the other hand, refer to theparticlelike character of the field and are measured inphotoelectric counting experiments.

Antibunching was one of the first nonclassical featuresof the electromagnetic field to be observed experi-mentally.4 It refers to the tendency of photons to beseparated from one another in time. Bunching refers tothe opposite tendency of photons to be bunched togetherin time. Whether a photon sequences exhibits bunchingor antibunching is intimately connected to the source dy-namics. Figure 1 shows three different photon-emissionsequences. Each vertical line represents a photoemis-sion event. We can also think of these sequences as pho-todetection events at the output of an ideal (unit detectionefficiency) photoelectric detector. For an ideal detectorthe distinction between a photodetection and a photo-emission sequence is not important. In what follows weshall assume this to be the case and speak of photoemis-sions and photoelectrons interchangeably. Furthermore,for simplicity, we shall restrict ourselves to stationary se-quences. All sequences in Fig. 1 have the same averagerate of occurrence of photons. Sequence (a) is a random(uncorrelated) photoemission sequence such as thatwhich might be generated by a laser operating high abovethreshold. Such a sequence is also called a Poisson se-quence. A comparison of sequences (a) and (b) showsthat photons in sequence (b) tend to bunch together.This is an example of a bunched photon sequence thatmight be generated by a thermal source or a laser operat-ing far below threshold. In sequence (c), photons tend tobe separated from one another. This is an example of anantibunched photon sequence that might be generated,for example, by a fluorescing single two-level atom.

The physical picture of photon bunching and anti-

0740-3224/2000/040634-04$15.00 ©

bunching developed in Fig. 1 can be quantified in terms ofthe distribution of waiting times between successive pho-toemissions or photodetections. This disruption is givenby10–14

w~T ! 5^T: I~t !$exp@2* t

t1T I~t8!dt8#% I~t 1 T !:&

^ I&, (1)

where I(t) is the intensity (photon flux from the source inunits of number of photons per second) operator at time tand ^T: :& stands for time ordering and normal orderingof the operator product between the colons. Note thatw(T) involves the detection of two successive photons attimes t and t 1 T and of no photons in the interval (t, t1 T). The probability of observing an interval betweenT and T 1 dT between successive photoemissions isw(T)dT. The waiting-time distribution w(T) refers tothe separation between photons and provides a clearphysical picture of photon bunching and antibunching inthe time domain.11,13,14

Waiting times for coherent light (sequence of randomphotons) are exponentially distributed according towc(T) 5 ^ I&exp(2^ I&T), where ^ I& is the average inten-sity (photon flux from the source). It is clear thatwc(0)/^ I& 5 1. The average separation between succes-sive photons is 1/^ I&, and the most probable waiting timeis zero. In an antibunched photon sequence, photonstend to be less bunched in time than photons in a randomphoton sequence. This means that, for an antibunchedphoton sequence, zero waiting time is less probable thanfor a random photon sequence. This leads to the crite-rion

w~0 ! , wc~0 ! (2)

for photon antibunching in terms of the waiting-time dis-tribution. Similar considerations hold for a bunchedphoton sequence, in which photons tend to be morebunched than in a random photon sequence; zero waitingtime will be more probable than for a Poisson sequence.Thus for a bunched photon sequence w(0) . wc(0).

The criterion for photon antibunching in terms of w(T)can be related to the traditional criteria in terms of thenormalized second-order intensity correlation functiong (2)(T). This correlation function is the joint probability

2000 Optical Society of America

R. Vyas and S. Singh Vol. 17, No. 4 /April 2000 /J. Opt. Soc. Am. B 635

of detecting a photon in the interval @t, t 1 D t) and an-other one in @t 1 T, t 1 T 1 dT), normalized by theprobability of recording two statistically independent pho-tons:

g ~2 !~T ! 5^T: I~t ! I~t 1 T !:&

^ I&2. (3)

Note that g (2)(T) involves the detection of any two pho-tons at times t and t 1 T, irrespective of what happens inthe interval (t, t 1 T). In contrast, w(T) involves thedetection of two successive photons. For a random pho-ton sequence, gc

(2)(T) 5 1, independently of T. Also notethat, in any photon sequence, photons separated by an in-terval T large compared with the intensity correlationtime are uncorrelated (statistically independent or ran-dom). In this limit g (2)(T) → ^ I(t)&^ I(t 1 T)&/^ I&2 5 1,which gives the probability of recording two uncorrelatedphotocounts. For a light beam whose intensity canbe treated as a classical stochastic variable, g (2)(T)must satisfy the following inequalities (Schwartzinequality)6–8,10:

inequality (I) g ~2 !~0 ! > 1, (4)

inequality (II) g ~2 !~0 ! > g ~2 !~T !. (5)

Inequality (I) means that for a classical photon sequencethe probability of detecting two photons in coincidence,g (2)(0), is always greater than the probability of detectingtwo random photons in chance coincidence. Inequality(II) means that the probability of detecting two photons incoincidence is always greater than the probability of de-tecting two photons separated by a finite time interval T.Quantum electrodynamics permits violations of both in-equalities. Note that a violation of inequality (I) alwaysimplies a violation of inequality (II). A violation of in-equality (II), however, does not imply a violation of in-equality (I). Light beams for which one or both of theseinequalities are violated are said to be nonclassical.Classical inequalities (I) and (II) are used to define bunch-ing and antibunching of photons. Antibunching is de-fined as violation of classical inequality (I) (Refs. 6, 10,and 13–15) or inequality (II).7 A violation of inequality(I) @ g (2)(0) , 1# implies that the probability of detectingtwo photons in coincidence is less than the probability ofdetecting two random photons in chance coincidence. Aviolation of inequality (II) @ g (2)(0) , g (2)(T)# implies

Fig. 1. (a) Random photon sequence, (b) bunched photon se-quence, and (c) antibunched photon sequence.

that the probability of detecting two photons in coinci-dence is less than the probability of detecting two photonsthat have nonzero separation in time.

Let us compare these definitions of antibunching basedon g (2)(T) with that based on waiting-time distributionw(T). The criterion for antibunching in terms of w(T),stated in inequality (II), leads to g (2)(0) , 1 as the condi-tion for antibunching. This corresponds to a violation ofinequality (I). We also see, from the definition of w(T) inEq. (1), that w(0) 5 ^ I&g (2)(0). Hence the conditiong (2)(0) , 1 is always reflected in the distribution of wait-ing times. Thus we have the equivalence

g ~2 !~0 ! < 1⇔w~0 ! < wc~0 !. (6)

On the other hand, antibunching defined as a violation ofinequality (II) need not be reflected in the distribution ofwaiting times. A violation of inequality (II) does not nec-essarily imply that the photons in the given sequencetend to be separated more from one another than in a ran-dom photon sequence.

We illustrate these conclusions by considering the pho-ton sequences generated when the light from a degener-ate parametric oscillator (DPO) is superimposed upon acoherent local oscillator (LO) beam. This is essentiallythe homodyne detection setup for the light from the DPO.We refer to the superimposed field as a homodyne detec-tion degenerate parametric oscillator (HMDPO). In thesteady state both g (2)(T) and w(T) can be computed bythe method outlined by Dodson and Vyas.10 Figure 2(a)shows g (2)(T) as a function of 2gT for the HMDPO field,where g is the DPO cavity decay rate. This figure showsa violation of inequality (I). Note that inequality (II) isalso violated. Figure 2(b) shows the corresponding dis-tribution of waiting times between photoemissions. Thewaiting-time distribution is peaked at a nonzero value,indicating that the photons tend to be separated from oneanother in time. This is what one would expect for anantibunched photon sequence. Figure 3(a) shows an ex-ample of g (2)(T) that violates inequality (II) but not in-equality (I). Figure 3(b) shows the correspondingwaiting-time distribution. Waiting-time distributionw(T) starts above unity and decreases monotonically tozero. In this case w(0) . wc(0), implying that succes-sive photons will tend to appear closer together thanthose in a random photon sequence. Consequently, ac-cording to this criterion of antibunching, based on a vio-lation of classical inequality (II), a photon sequence morebunched than a sequence of random photons will betermed antibunched. It is clear that, when the distribu-tion of waiting times between photons is considered,g (2)(0) , 1 is more appropriate for defining antibunch-ing, although the violations of both inequalities (I) and(II) are signatures of a nonclassical light beam.

In conclusion, waiting-time distribution w(T) providesa clearer physical picture of photon-emission sequences,corresponding photoelectron sequences, and their non-classical properties than does g (2)(T). The waiting-timedistribution is measurable in photoelectric detection ex-periments by use of a time-to-amplitude converter.16 Itis interesting to note that if the efficiency of photoelectricdetection is low, a measurement of w(T) with a time-to-amplitude converter effectively yields g (2)(T).14 Most ex-

636 J. Opt. Soc. Am. B/Vol. 17, No. 4 /April 2000 R. Vyas and S. Singh

periments on antibunching have been conducted in thislow-detection-efficiency regime. Because relatively fewcalculations of w(T) have been available in the literatureuntil recently,10–14 experiments have focused almost ex-clusively on the measurements g (2)(T). With the avail-ability of high-efficiency photoelectric detectors and colli-mated nonclassical light sources and because of recenttheoretical advances, measurements of w(T) should beforthcoming in the not-too-distant future.

Fig. 2. (a) Normalized second-order intensity correlationg (2)(T) as a function of 2gT, showing violations of classical in-equality (I) [inequality (4)]. (b) Waiting-time distributionw(T)/^ I& as a function of 2gT for the HMDPO field [see the textfollowing relation (6)]. The parameters for both of the curvesare the following: mean photon number for the DPO, nd5 0.28; mean photon number for the LO beam, n l 5 2; beam-splitter power transmissivity, T 5 0.75; DPO/LO phase angle,f 5 90°.

ACKNOWLEDGMENTThis research was supported in part by the National Sci-ence Foundation.

S. Singh’s e-mail address is ssingh@comp.uark.edu.

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Fig. 3. (a) Normalized second-order intensity correlation func-tion g (2)(T) as a function of 2gT, showing violations of classicalinequality (II) [inequality (5)]. (b) Waiting-time distributionw(T)/^ I& decreasing monotonically as a function of 2g T for theHMDPO field. The parameters for both of the curves are as fol-lows: mean photon number for the DPO, nd 5 0.28; mean pho-ton number for the LO, nl 5 2; beam-splitter power transmissiv-ity, T 5 0.75; DPO/LO phase angle, f 5 90°.

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