Erenso et al. Vol. 19, No. 6 /June 2002/J. Opt. Soc. Am. B 1471

Higher-order sub-Poissonian photon statisticsin terms of factorial moments

Daniel Erenso, Reeta Vyas, and Surendra Singh

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

Received September 10, 2001

We introduce the concept of higher-order super-Poissonian and sub-Poissonian statistics and show that higher-order sub-Poissonian statistics is a signature of a nonclassical field. Fields generated in intracavity second-harmonic generation and single-atom resonance fluorescence are shown to exhibit higher-order sub-Poissonianstatistics. © 2002 Optical Society of America

OCIS codes: 270.0270, 270.5290, 190.4970, 270.2500.

1. INTRODUCTIONNonclassical properties of electromagnetic fields receive agreat deal of attention, as these properties provide a test-ing ground for the predictions of quantum electro-dynamics.1,2 Sub-Poissonian Photon statistics based onsecond-order intensity correlations provide one way tocharacterize the nonclassical nature of a light beam.3,4

Nonclassical effects as they relate to higher-order mo-ments have also been discussed in the literature.5–9

Agarwal and Tara discussed higher-order nonclassical ef-fects for single-mode fields in terms of normally orderedmoments.5 Perina and co-workers studied nonclassicalbehavior in optical parametric processes, Raman andBrillouin scattering,6 and nonlinear optical couplers7 interms of moments of integrated intensity. Lee consideredsingle-mode fields and defined higher-order nonclassicaleffects in terms of factorial moments of the photon distri-bution by using majorization theory.8 Vyas and Singhconsidered higher-order nonclassical effects in terms offactorial moments of the photocount distribution.9 Inthis paper we introduce criteria for evaluating higher-order sub-Poissonian statistics in terms of factorial mo-ments and show that higher-order sub-Poissonian statis-tics are indicative of nonclassical fields. The criteria thatwe introduce are independent of the efficiency of detec-tion. We then show that the light from intracavitysecond-harmonic generation (ISHG) and light fromsingle-atom resonance fluorescence exhibit higher-ordersub-Poissonian statistics.

2. HIGHER-ORDER SUB-POISSONIANSTATISTICSFor second-order sub-Poissonian statistics variance^(Dm)2& 5 ^m2& 2 ^m&2 of the photon-counting distribu-tion is less than the mean of the distribution, ^m&. Bynoting that second-order factorial moment ^m (2)&[ ^m(m 2 1)& 5 ^m2& 2 ^m&, we can write the criterionfor the second order sub-Poisson statistics as

^m ~2 !& 2 ^m&2 , 0. (1)

0740-3224/2002/061471-05$15.00 ©

Note that for a Poissonian distribution ^m (2)& 5 ^m&2, soinequality (1) holds as an equality. The departures fromPoisson statistics are then characterized in terms of theFano factor (F 5 ^(Dm)2&/^m&) or the Q parameter $Q5 @^m (2)& 2 ^m&2#/^m&%.3,4 For a sub-Poissonian distri-bution the Q parameter is negative. We now extend thiscriterion to higher-order factorial moments. The lth (l isa positive integer) order factorial moment of the photo-count distribution is defined by

^m ~l !& 5 (m5l

`

m~m 2 1 !...~m 2 l 1 1 !p~m, T !, (2)

where p(m, T) is the probability of detecting m photonsin the counting interval @0 –T#. Here we have sup-pressed the time argument in the factorial moments. Toextend the criteria for sub-Poissonian statistics to higher-order moments we introduce a parameter Sl :

Sl 5^m ~l !&

^m& l 2 1. (3)

It is easily proved that, for a Poisson distribution, Sl5 0 for all l. Parameter Sl for l > 2 provides a measureof the deviation of the lth factorial moment from that fora Poisson distribution with the same mean. Sl . 0 de-fines a super-Poissonian distribution, and Sl , 0 definesa sub-Poissonian distribution. Note that parameter S2 isnot equal to the Q parameter but is related to it byS2 5 Q/^m&. As the values of higher-order factorial mo-ments can be large, it is convenient to use normalized fac-torial moments rather than the analogs of Q to extend theconcept of sub- and super-Poissonian statistics to higher-order moments. Another advantage of using the Sl pa-rameters is that they are independent of the efficiency ofdetection.

We now show that negative values of parameter Sl forl . 2 (higher-order sub-Poissonian statistics) indicate thenonclassical nature of light. To establish this we notethat for a classical field the factorial moments must sat-isfy the inequality8,9

^m ~l1k !&^m ~ j2k !& > ^m ~l !&^m ~ j !&, (4)

2002 Optical Society of America

1472 J. Opt. Soc. Am. B/Vol. 19, No. 6 /June 2002 Erenso et al.

where l, j, and k are positive integers such that l > j> k. Parameter Sl can be expressed in the form

Sl 1 1 5^m ~l !&

^m& l 5 F ^m ~l !&

^m ~l21 !&^m&GF ^m ~l21 !&

^m ~l22 !&^m&G

3 F ^m ~l22 !&

^m ~l23 !&^m&G ...F ^m ~2 !&

^m&^m&G . (5)

On using inequality (4) we find that each of the factors onthe right-hand side of Eq. (5) is greater than unity.Hence it it follows that for a classical field Sl must bepositive, satisfying the inequality

Sl > 0. (6)

The equality holds for a Poisson distribution. It followsthat the negative values of Sl that imply sub-Poissonianstatistics are in violation of this classical inequality.Hence Sl , 0 provides a criterion for identifying the non-classical nature of light. Parameters Sl are measurablein photon-counting experiments and have the added ad-vantage that they are independent of the efficiency of de-tection. We now consider two light sources that both pro-duce light fields that exhibit higher order sub-Poissonianstatistics.

3. HOMODYNE STATISTICS OFINTRACAVITY SECOND-HARMONICGENERATIONFigure 1 shows a schematic of homodyne detection set upto detect the light from ISHG. The ISHG light is super-posed with a coherent field from a local oscillator at abeam splitter of reflectivity R and transmittivity T. Werefer to this superposed field as HISHG. A detector Dplaced at one of the output ports detects the HISHG field.By using the method of Dodson and Vyas10 we obtain ana-lytic results for the factorial moments of the photoelectriccounting distribution for the HISHG field by using thepositive-P representation to describe the nonclassicalfields generated in the ISHG. In the positive-Prepresentation11 the complex field amplitudes b i and b i* ,which correspond to the annihilation and creation opera-tors, respectively, at the output ports of the beam splitter,can be written as

b1 5 aAT 1 ua luexp~if !AR,

b2 5 ua luexp~if !AT 2 aAR, (7)

Fig. 1. Schematic of the setup for homodyne detection of thelight produced in IHSG: LO, coherent local oscillator; BS, loss-less beam splitter; D, detector.

b1* 5 a*AT 1 ua luexp~2if !AR,

b2* 5 ua luexp~2if !AT 2 a*AR. (8)

Here ua lu @a l 5 ua luexp(if )# is the amplitude and f is thelocal oscillator’s phase relative to the ISHG and a and a*are the complex field amplitudes that correspond to theannihilation and creation operators for the ISHG field.Note that in the positive-P representation a and a* arenot complex conjugates of each other. Here we focus ourattention on the b1 port of the beam splitter. One can ob-tain results for the b2 port by replacing AR with AT andAT with 2AR. The factorial moment generating functionG(s, T) for the photoelectron counting distribution isgiven by 10

G~s, T ! 5 K expF2sh E0

T

I~t !dtG L , (9)

where 0 < h < 1 is the quantum efficiency of detection, sis an auxiliary parameter, and @0 –T# is the counting in-terval. The averaging is done with respect to thepositive-P function for the HISHG field. The photonnumber flux variable I(t) is given by

I~t ! 5 2g ~b1b1* !

5 2g $Ta* a 1 Rua lu2

1 ARAT ua lu@a exp~2if ! 1 a* exp~if !#%. (10)

In terms of G(s, T) the factorial moments are given by

^m ~l !& 5 (m51

`

m~m 2 1 !...~m 2 l 1 1 !p~m, T !

5 ~21 !lF dl

dsl G~s, T !Gs50

. (11)

The equations of motion for complex-field amplitudes aand a* for the fundamental mode obtained by adiabaticelimination of the second-harmonic mode are 12–16

a 5 2g ~a 2 E ! 2ga* a2

no1 iA g

noaj, (12)

a* 5 2g ~a* 2 E ! 2gaa*

2

no1 iA g

noa* j* . (13)

Here g is the cavity linewidth at the fundamental fre-quency, E is the classical pump amplitude at the funda-mental, and no is the threshold photon number. Noisesources j and j* are real Gaussian white-noise processeswith zero mean and unit strength. By linearizing Eqs.(12) and (13) about steady-state amplitudes ass 5 a* ss5 Anon we find that the complex field amplitudes can bewritten as

a 5 Ano@An 1 i~u1 1 u2!#, (14)

a* 5 Ano@An 1 i~u1 2 u2!#. (15)

Here u1 and u2 are real Gaussian random variables withzero mean and correlations given by

Erenso et al. Vol. 19, No. 6 /June 2002/J. Opt. Soc. Am. B 1473

^ui~t !uj~t8!& 5 d ij

gn

4nol iexp~2l iut 2 t8u!,

i 5 1, 2, (16)

where l1 5 g (1 1 3n) and l2 5 g (1 1 n). Substitut-ing these expressions for a and a* in Eqs. (7) and (8) andusing the results in Eqs. (9) and (10), we can write thegenerating function G(s, T) in terms of u1 and u2 . Bymaking a Karhunen–Loeve expansion of u1 and u2 andusing their correlation properties we can evaluate thegenerating function in closed form by following a proce-dure similar to that outlined in Refs. 10 and 17–19. Thefinal result is

G~s, T ! 5 Q1~s, T !exp@2f1~s, T !#Q2~s, T !

3 exp@2f2~s, T !#, (17)

where

Qi~s, T !

5exp~l iT/2!

@cosh~ziT ! 1 1/2~l i /zi 1 zi /l i!sinh~ziT !#1/2, (18)

fi~s, T !

5 KiTFl i2

zi2 S 4 1 l iT

2 1 l iTD 1

2

l iBiT2 1 ~2Bi 2 l i!T 2 2

2 S 2Bil i

zi2 D Ci

BiT 2 1G , (19)

Ci 5cosh~ziT/2! 1 ~l i /zi!~4 1 l iT/l iT !sinh~ziT/2!

cosh~ziT/2! 1 ~zi /l i!sinh~ziT/2!,

(20)

zi2 5 l i

2 1 ~21 !i2shng2T, (21)

Bi 5 ~21 !ig2shnT

l i(22)

for i 5 1, 2 and

K1 5 2gshn0~AnT 1 AnlR cos f !2, (23)

K2 5 2gshn0nlR sin2 f. (24)

Differentiating the generating function with respect to sby using the Leibnitz theorem and substituting the resultinto Eq. (11), we obtain the factorial moments. Using theexpressions for the factorial moments in Eq. (3), we fi-nally obtain Sl . The results, although they are analytic,are rather cumbersome and are not be reproduced here.

In Fig. 2 we have plotted Sl for values of l that rangefrom 2 to 7 as a function of counting time for relativephase f 5 180°. This choice of phase permits partial re-moval of coherent background in the ISHG by destructiveinterference at the beam splitter and therefore enhancesthe nonclassical effects. For these parameters S2(5Q/^m&) is negative for all times that reflect the sub-Poissonian behavior of the light beam. ParametersS3 –S7 , however, are positive for short counting times, re-flecting higher-order super-Poissonian behavior. Withincreasing counting intervals T, all Sl for l 5 3 –7 be-come negative, indicating that the homodyne field

changes from super-Poissonian to sub-Poissonian, evenfor higher order. It is worth noting that for large count-ing intervals the higher-order sub-Poissonian character iseven more pronounced than the second-order sub-Poissonian behavior based on the Q parameter.

The sub-Poissonian behavior for the HISHG is due tointerference between quantum noise and a coherent field.The field from the ISHG has a large coherent componentthat suppresses nonclassical effects such as antibunchingand sub-Poissonian statistics. Homodyning the ISHGfield with the local oscillator with f 5 180° allows us toremove the coherent background of the ISHG by destruc-tive interference at the beam splitter. If the coherentcomponent is completely removed, the HISHG field exhib-its bunching and shows higher-order super-Poissonianstatistics. Partial removal of the coherent component,such that the noise terms and the coherent component arecomparable, enhances nonclassical effects.

4. SINGLE-ATOM RESONANCEFLUORESCENCEAnother example of higher-order sub-Poissonian behavioris provided by the fluorescence produced by a coherentlydriven single two-level atom. Photon statistics of thefluorescent light from a single atom have been treated ingreat detail by a number of authors.4,20–24 Here we usethe relevant results to illustrate the higher-order sub-Poissonian nature of the field produced in single-atomresonance fluorescence. Normalized factorial momentsfor the fluorescent photons in an interval T can be writtenas4

^m ~r !&

^m&r 5r!

Tr E0

T

dtr ...E0

t2

dt1@1 1 l~tr 2 tr21!#...@1

1 l~t2 2 t1!#, (25)

where

l~t! 5 2exp~23bt/2!F cos~V8bt! 13

2V8sin~V8bt!G ,

(26)

V8 5 @V/b!2 2 1/4]1/2, (27)

Fig. 2. Parameter Sl for the HISHG light as a function of 2gTfor l 5 2 –7. Parameters are no 5 106, n 5 0.2, n l 5 0.1998 (inunits of no), f 5 p, and transmittivity T 5 0.5. Sl , 0 indicatessecond- and higher-order sub-Poissonian statistics.

1474 J. Opt. Soc. Am. B/Vol. 19, No. 6 /June 2002 Erenso et al.

where b is half the Einstein-A coefficient for the resonanttransition and V is the Rabi frequency. Here it is as-sumed that the atom is resonant with the coherent driv-ing field.

Factorial moments given in Eq. (25) can be evaluatedby use of the Laplace transform. In the short countinglimit we can also evaluate Eq. (25) directly as

Sl 5^m ~r !&

^m&r 2 1

5 21 1 F2b2T2S 1 1V2

2b2D G r21 r!

~3r 2 2 !!. (28)

The ratio in Eq. (28) is less than unity for all l > 2 forshort counting intervals bT ! 1. It follows that Sl isnegative for all l > 2. Hence photon statistics are sub-Poissonian to all orders l > 2 in single-atom resonancefluorescence for short counting times. In the long-timecounting limit bT @ 1 we obtain the following expressionfor Sl :

Sl 5^m ~r !&

^m&r 2 1

5 21 1 r!(n51

r F r 2 1n 2 1 G 1

n! S 23/2bT

1 1 V2/2b2D r2n

(29)

5 23r~r 2 1 !

2bT

1

1 1 V2/2b2 1 O~1/b2T2!. (30)

Equation (30) clearly shows that Sl , 0, so the photo-count distribution in resonance fluorescence is sub-Poissonian to all orders in the long-time counting limit aswell.

Figure 3 shows Sl for single-atom resonance fluores-cence as a function of bT for Rabi frequency V 5 0.5 andl ranging from 1 to 4. For short counting times Sl is al-most 21, reflecting perfect second- and higher-order sub-Poissonian behavior. This is so because as soon as theatom emits a photon it is in the ground state and there-

Fig. 3. Parameter Sl for the light from a coherently drivensingle two-level atom as a function of bT for l 5 2 –5 and Rabifrequency V 5 0.5. Sl , 0 indicates second- and higher-ordersub-Poissonian statistics.

fore cannot emit a second photon until it is driven by theexternal field to the excited state. The inability of anatom to emit more than one photon simultaneously thusmakes the photon-counting distribution for the shortcounting interval extremely narrow, resulting in perfectsub-Poissonian behavior. In a short counting intervalthe probability of more than one emission is negligible, soall factorial moments higher than the first vanish or arenegligible. It is this behavior that is responsible forstrong sub-Poissonian behavior for short counting inter-vals. With an increase in counting time the probabilityof emission of many photons in the counting interval in-creases, and we find that Sl increases. However, Sl al-ways remains negative, indicating sub-Poissonian behav-ior to all orders. For a given counting time with increasein l the value of Sl decreases, indicating that the degree ofhigher-order sub-Poissonian behavior is larger than thesecond-order sub-Poissonian behavior, as can also be un-derstood because in a finite time the atom can emit a fi-nite number of photons. Therefore factorial momentsgreater than a certain number will be negligible, onceagain leading to a stronger sub-Poissonian behavior forhigher-order factorial moments. For large countingtimes Sl approaches zero, indicating Poissonian statisticsof the fluorescent light. For a large value of Rabi fre-quency V we find that Sl shows oscillations for a shortcounting time. However, Sl always remains less thanzero, indicating sub-Poissonian behavior.

In conclusion, we have defined higher-order sub- andsuper-Poissonian statistics in terms of factorial momentsand introduced a parameter that is measurable in photon-counting experiments. This parameter is independent ofthe efficiency of detection. We have also shown thathigher-order sub-Poissonian behavior is a nonclassical ef-fect and that, depending on the parameters, homodynedintracavity second-harmonic generation and single-atomresonance fluorescence can show these nonclassical ef-fects.

ACKNOWLEDGMENTSThis study was supported in part by the Office of NavalResearch and by the Arkansas Science and TechnologyAuthority. D. Erenso acknowledges support from the In-ternational Center for Scientific Culture—World Labora-tory.

R. Vyas’s e-mail address is rvyas@uark.edu.

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