HOA IN INTERMEDIATE STATES AND OPTICAL...

CHAPTER 2

HOA IN INTERMEDIATE STATES AND

OPTICAL PROCESSES

In the last chapter we have brie�y described di�erent intermediate states andhave mentioned that di�erent lower order nonclassical e�ects have already beenobserved in these states (see Table 1.1). But before the present study higherorder antibunching was predicted in only one type of intermediate state, whichis known as shadowed negative binomial state. In recent past Pandey, Guptaand Pathak have shown that the higher order antibunching is not a rare phe-nomenon [28]. To establish their claim further, in this chapter we have shownthat the higher order antibunching can be seen in di�erent intermediate states,such as binomial state, reciprocal binomial state, hypergeometric state, gener-alized binomial state, negative binomial state and photon added coherent state.We have also studied the possibility of observing the higher order antibunchingin di�erent limits of intermediate states. The e�ects of di�erent control param-eters on the depth of nonclassicality have also been studied in this connectionand it has been shown that the depth of nonclassicality can be tuned by control-ling various physical parameters. Potential single photon sources are expectedto satisfy the criterion of HOA. As the intermediate states mentioned abovesatisfy the condition of HOA, they form a set of potential SPSs. A quantitativemeasure of quality of single photon source is also introduced in this chapter anddi�erent intermediate states are compared using that criterion. Further, since

39

CHAPTER 2. HOA IN INTERMEDIATE STATES AND OPTICAL PROCESSES

the experimental generation of intermediate states are di�cult we have alsoshown existence of HOA in some simple optical systems (many wave mixingprocesses).

2.1 Introduction

Since the introduction of BS as an intermediate state, it was always been of in-terest to quantum optics, nonlinear optics, atomic physics and molecular physicscommunity. Consequently, di�erent properties of binomial states have beenstudied [15-59]. In these studies, it has been observed that the nonclassicalphenomena (such as, antibunching, squeezing and higher order squeezing) canbe seen in BS. Nonclassicalities have been reported in di�erent intermediatestates1 (such as, excited binomial state [58], odd excited binomial state [59],hypergeometric state [60], negative hypergeometric state [61], reciprocal bi-nomial state [62], shadowed state [63], shadowed negative binomial state [64]and photon added coherent state [66] etc.). The studies in the nineties weremainly limited to theoretical predictions but the recent developments in the ex-perimental techniques made it possible to experimentally verify some of thosetheoretical predictions. For example, we can note that, as early as in 1991 Agar-wal and Tara [66] had introduced photon added coherent state (1.55) but theexperimental generation of the state has happened only in recent past when Za-vatta, Viciani and Bellini [67] succeed to produce it in 2004. Before the presentstudy it was known that most of these intermediate states show antibunchingand squeezing, (see Table 1.1) etc. but higher order antibunching was reportedin only one type of intermediate state which is known as shadowed negativebinomial state [64].

Most of the interesting recent developments in quantum optics have arisenthrough the nonclassical properties of the radiation �eld only. For example,antibunching and squeezing, which do not have any classical analogue [10], have

1The de�nitions of most of these states have been described in Chapter 1.

40


extensively been studied in last thirty years. But the majority of these studiesare focused on lowest order nonclassical e�ects. Higher order extensions of thesenonclassical states have only been introduced in recent past [11-24]. Amongthese higher order nonclassical e�ects, higher order squeezing (Hillery type) hasalready been studied in detail [10, 11, 16, 86] but the higher order antibunching(HOA) is not yet studied rigorously. The idea of HOA was introduced by Lee ina pioneering paper [23] in 1990, since then it has been predicted in two photoncoherent state [23], shadowed negative binomial state [64], trio coherent state[25] and in the interaction of intense laser beam with an inversion symmetricthird order nonlinear medium [27]. From the fact that in �rst 15 years after itsintroduction, HOA was reported only in some particular cases, HOA appearedto be a very rare phenomenon. But recently Gupta, Pandey and Pathak haveshown that the HOA is not a rare phenomenon [28] and it can be seen in simpleoptical processes like six wave mixing process, four wave mixing process andsecond harmonic generation. To establish that claim further, here we haveshown the existence of HOA in di�erent intermediate states, namely, binomialstate, reciprocal binomial state, photon added coherent state, hypergeometricstate, Roy-Roy generalized binomial state and negative binomial state. Wehave also shown the existence of HOA in some simple optical processes.

Here we would like to note that Vyas and Singh [26] have shown that theregime of nonclassicality represented by the criterion of HOA is not the same asone represented by the usual antibunching criterion. Further, recently Prakashand Mishra [29] have shown that the higher order subpoissonian photon statis-tics can be used to detect Hong Mandel squeezing and amplitude squaredsqueezing. These two types of squeezing can be seen in intermediate states.These facts have primarily motivated us to study intermediate state in thisnovel regime of nonclassicality [26]. The present work is also motivated by therecent success in experimental observation of intermediate state [67], theoreticalobservation of possibility of observing HOA in some simple optical systems [28]

41


and the fact that the intermediate states, which frequently show di�erent kind ofnonclassicality, form a big family of quantum state. But till now HOA has beenpredicted in only one member (Shadowed negative binomial state) of such a bigfamily of quantum states [64]. Motivated by these facts the present work aimsto study the possibility of HOA in all the popularly known intermediate states.The theoretical predictions of the present study can be experimentally veri�edwith the help of various intermediate state generation schemes and homodyneexperiment, since the criteria for HOA appears in terms of factorial moments,which can be measured by using homodyne photon counting experiments [87,88].

2.2 Criteria of HOA

The criterion of HOA is expressed in terms of higher order factorial momentsof number operator. There exist several criteria [18, 27, 28, 34, 35] for the samewhich are essentially equivalent. Initially, using the negativity of P function[10], Lee introduced the criterion for HOA as

R(l,m) =

⟨N

(l+1)x

⟩⟨N

(m−1)x

⟩⟨N

(l)x

⟩⟨N

(m)x

⟩ − 1 < 0, (2.1)

where N is the usual number operator,⟨N (i)

⟩= 〈N(N − 1)...(N − i+ 1)〉 is

the ith factorial moment of number operator, 〈〉 denotes the quantum average,l and m are integers satisfying the conditions 1 ≤ m ≤ l and the subscript xdenotes a particular mode. Ba An [25] choose m = 1 and reduced the criterionof lth order antibunching to

Ax,l =

⟨N

(l+1)x

⟩⟨N

(l)x

⟩〈Nx〉

− 1 < 0, (2.2)

42


or, ⟨N (l+1)x

⟩<⟨N (l)x

⟩〈Nx〉 . (2.3)

We can further simplify (2.3) as⟨N (l+1)x

⟩<⟨N (l)x

⟩〈Nx〉 <

⟨N (l−1)x

⟩〈Nx〉2 <

⟨N (l−2)x

⟩〈Nx〉3 < ... < 〈Nx〉l+1 ,

(2.4)and obtain the condition for l − th order antibunching as

d(l) =⟨N (l+1)x

⟩− 〈Nx〉l+1 < 0. (2.5)

This simpli�ed criterion (2.5) coincides exactly with the physical criterion ofHOA introduced by Pathak and Garcia [27] and the criterion of Erenso, Vyasand Singh [38]. All these criteria essentially lead to same kind of nonclassicalitywhich belong to the class of strong nonclassicality according to the classi�cationscheme of Arvind et al [95]. Here we can note that d(l) = 0 and d(l) > 0

corresponds to higher order coherence and higher order bunching (many photonbunching) respectively.Actually,

⟨a†lal

⟩=⟨N (l)

⟩is a measure of the probabil-

ity of observing l photons of the same mode at a particular point in space timecoordinate. Therefore, the physical meaning of inequalities (2.4) is that theprobability of detection of single photon is greater than that of two photon in abunch and that is greater than the probability of detection of three photon in abunch and so on. This is exactly the characteristic that is required in a proba-bilistic single photon source used in quantum cryptography. HOA is essentiallymultiphoton version of antibunching. But the region of nonclassicality repre-sented by the HOA is di�erent from that of usual antibunching [26]. Now sincethe maximum number of photon that can come out of a probabilistic singlephoton source is not limited to 2, this is a better criterion than the lower orderone. In other words all the probabilistic single photon sources used in quan-tum cryptography should satisfy the criteria (2.5) of HOA [41]. Keeping thesefacts in mind we have studied the possibilities of observing HOA in di�erent

43


intermediate states. Corresponding observations are reported in the followingsections.

2.3 HOA in Binomial State

Binomial state is de�ned in (1.46), from which it is straight forward to see that

a|p,M〉 =∑M

n=0

{M !

(n−1)!(M−n)!pn(1− p)M−n

} 12 |n− 1 >

=∑M−1

l=0

{M(M−1)!l!(M−1−l)!p

l+1(1− p)M−1−l} 1

2 |l > (assuming n− 1 = l)

= [Mp]12

∑M−1l=0

{(M−1)!

l!(M−1−l)!pl(1− p)M−1−l

} 12 |l >

= [Mp]12 |p,M − 1〉.

(2.6)Similarly, we can write,

a2|p,M〉 = [M(M − 1)p2]12 |p,M − 2〉

a3|p,M〉 = [M(M − 1)(M − 2)p3]12 |p,M − 3〉

... ... ...al|p,M〉 = [M(M − 1)....(M − l + 1)pl]

12 |p,M − l〉

=[

M !(M−l)!p

l] 1

2 |M − l, p〉.

(2.7)

Therefore,

〈M, p|a†l = 〈M − l, p|[

M !

(M − l)!pl] 1

2

, (2.8)

and consequently,

〈M, p|n(l)|p,M〉 = 〈M, p|a†lal|p,M〉 =

[M !

(M − l)!pl]. (2.9)

44


Now substituting (2.9) in equation (2.5) we obtain the condition for lth orderantibunching as

d(l) =

[M !

(M − l − 1)!pl+1

]− [Mp]l+1 < 0, (2.10)

or,(M − 1)(M − 2)....(M − l) < M l, (2.11)

which is always satis�ed for any M > l and both M and l are positive (sinceevery term in left is < M). As M is the number of photons present in the �eldand d(l) is a measure of correlation among (l+1) photons, thereforeM ≥ (l+1)

or M > l. Consequently, a binomial state always shows HOA and the highestpossible order of antibunching in a binomial state is M − 1, where M is thenumber of photon present in the �eld. From (2.10) it is straight forward tosee that the number state is always higher order antibunched and in the otherextreme limit (when p → 1, M → ∞ and the BS reduces to coherent state)d(l) = 0, which is consistent with the physical expectation.

2.4 HOA in Generalized Binomial State

Roy and Roy have introduced the generalized binomial state (GBS) as de�nedin (1.47)-(1.49). This intermediate state reduces to vacuum state, number state,coherent state, binomial state and negative binomial state in di�erent limits ofparameters α, β and N . The Pochhammer symbol used to de�ne GBS in (1.48)and (1.49) satis�es following identity:

a(a+ 1)n = (a)n+1. (2.12)

Now it is easy to see that the above identity (2.12) yields the following usefulrelations:

(α+ 1)l+1 = (α+ 1)(α+ 2)l , (2.13)

45


and

(α+β+2)N = (α+β+2)(α+β+3)N−1 = (α+β+2)(α+2+β+1)N−1. (2.14)

Using (1.47) and (1.48) we can obtain

a|N,α, β〉 =∑N

n=0

{N !

(α+β+2)N

(α+1)n

(n−1)!(β+1)N−n

(N−n)!

} 12 |n− 1〉

=∑N−1

l=0

{N(N−1)!

(α+β+2)N

(α+1)l+1

l!(β+1)N−1−l

(N−1−l)!

} 12 |l〉

, (2.15)

where n = l + 1 has been used. Now we can apply (2.13) and (2.14) on (2.15)to obtain

a|N,α, β〉 ={

N(α+1)(α+β+2)

} 12 ∑N−1

l=0

{(N−1)!(α+2)l(β+1)N−1−l

(α+2+β+1)N−1l!(N−1−l)!

} 12 |l〉

={

N(α+1)(α+β+2)

}12 ∑N−1

n=0

√ω(n,N − 1, α+ 1, β)|n〉,

(2.16)

where dummy variable l is replaced by n. Therefore,

〈N,α, β|a†a|N,α, β〉 = N(α+1)(α+β+2)

〈N,α, β|a†2a2|N,α, β〉 = N(N−1)(α+1)(α+2)(α+β+2)(α+β+3)... ... ...

〈N,α, β|a†lal|N,α, β〉 = [N(N−1).....(N−l+1)][(α+1)(α+2).....(α+l)](α+β+2)(α+β+3).....(α+β+l+1)

= N !(α+l)!(α+β+1)!(N−l)!α!(α+β+l+1)! ,

and

dGBS(l) = N !(α+l+1)!(α+β+1)!(N−l−1)!α!(α+β+l+2)! −

{N(α+1)(α+β+2)

}l+1

= [N(N−1).....(N−l)][(α+1)(α+2).....(α+l+1)](α+β+2)(α+β+3).....(α+β+l+2) −

{N(α+1)(α+β+2)

}l+1,

(2.17)

The physical condition N ≥ l+1 ensures that all the terms in d(l) are positive.The expression of d(l) is quite complex and it depends on various parameters(e.g. α, β and N). Fig. 2.1 shows that for particular values of these parametersHOA is possible. The depth of nonclassicality is more in case of dGBS(9) than

46


Figure 2.1: Higher order antibunching can be seen in Generalized binomial state.Existence of 8th and 9th order antibunching (for α = 2 and β = 1)and variationof depth of nonclassicality with N has been shown.

in dGBS(8). This is consistent with earlier observation [27]. A systematic studyreveals that the probability of observing HOA increases with the increase of αbut it decreases (i.e. the probability of higher order bunching increases) withthe increase of β. This can be seen clearly in Fig. 2.2 and Fig. 2.3. Further it isobserved (from Fig. 2.1 and Fig. 2.2) that for lower values of α the probabilityof bunching increases with the increase ofN but for a comparatively large valuesof α (larger compare to β) the probability of HOA increases with the increasein N (see Fig. 2.2) but the situation is just opposite in the case of β (as it isseen from Fig. 2.3). While studying di�erent limiting cases of Roy and Roygeneralized binomial state, we have observed that binomial state and numberstate always show HOA and d(l) = 0 for coherent state. This is consistent withthe physical expectation and the conclusion of the last section.

2.5 HOA in other intermediate states

As it is mentioned in the earlier sections, there exist several di�erent inter-mediate states. For the systematic study of possibility of observing HOA inintermediate states, we have studied all the well known intermediate states.

47


Figure 2.2: Variation of dGBS(2) with α and N for β=1.

Figure 2.3: Variation of dGBS(2) with β and N for α=10.

48


Since the procedure adapted for the study of di�erent states is similar, math-ematical detail has not been shown in the subsections below. But from theexpression of d(l) and the corresponding plots it would be easy to see that theHOA can be observed in all the intermediate states studied below.

2.5.1 Reciprocal binomial state

Reciprocal binomial state (RBS) [62] is de�ned in (1.58), antinormal ordering2

and procedure adapted in previous sections, we can obtain;

dRBS(l) =

l+1∑i=0

(−1)i (l+1)!2

(l+1−i)!2i!ntr(N+l+1−i )!

N ! −N l+1

= π csc(π(l+N))(l+1)!2Gamma(l+1−N)M !(Gamma(2+l)Gamma(−N))2

−N l+1,

(2.18)

where Gamma denotes the Gamma function. The possibility of observingHOA in reciprocal binomial state can be clearly seen from the Fig. 2.4. Butit is interesting to note that the nature of singularity and zeroes present in thesimpli�ed expression of dRBS(l) as expressed in the last line of (2.18) can provideus some important information. For example, the underlying mathematicalstructure of the criterion of HOA and that of reciprocal binomial state demandsthat l and N are integers but if both of them are integer then dRBS(l) hasa singularity as the csc[π(l + N)] term present in the numerator blows up.But this local singularity can be circumvented by assuming l → integer andN → integer. In this situation (i.e. when l and N tends to integer value)〈N (l)〉 is �nite and consequently d(l) is also �nite. This is the reason that thesingular nature of the simpli�ed expression of dRBS(l) is not re�ected in theFig. 4. In the analysis of the dRBS(l) it is also interesting to observe thatGamma(−N) = ∞ for N = integer, and in an approximated situation whenl → integer and M = integer, the csc[π(l +N)] term in the numerator is no

2antinormal ordering already de�ned in Chapter 1

49


Figure 2.4: Variation of dRBS(8) and dRBS(9) with photon number N .

Figure 2.5: Variation of dNBS(8) with η and M .

more singular and as a result 〈N (l)〉 = 0 and dRBS = −N (l+1). In this situationone can observe HOA for arbitrarily large values of l and N . Thus physically, itis expected that in reciprocal binomial state, higher order antibunching of anyarbitrary order will be seen and HOA will not be destroyed with the increaseof N , as it happens (for some particular values of α and β) in the case ofgeneralized binomial state (see Fig. 2.1).

2.5.2 Negative Binomial StateBarnett's de�nition [72] of Negative Binomial state (NBS) is stated in (1.50).This intermediate state interpolates between number state and geometric state.Following the mathematical techniques adapted in the sections 2.3 and 2.4,weobtain

dNBS(l) =1

ηl+1

((l +M + 1)! 2F1 (−l − 1,−l − 1;−l −M − 1; η)

M !− (M + 1)l+1

), (2.19)

50


Figure 2.6: Variation of dNBS(8) with η for M = 10.

where 2F1(a, b; c; z) is a conventional hypergeometric function. Variation ofd(l) with various parameters such as η, l and M have been studied and areshown in Fig. 2.5- Fig. 2.6. From these �gures one can observe that the stateis not always antibunched. From Fig. 2.6, we can see that dNBS(8) shows theexistence of HOA for a smaller value of η (compared to dNBS(9)) but the depthof nonclassicality is more in dNBS(9). The broad features remain same for theother orders (other values of l) of antibunching.

In the limit M → 0 the negative binomial state reduces to geometric state(GS). In this limit dNBS(l) reduces to

dGS(l) =1

ηl+1

((1− η)l+1(l + 1)!− 1

). (2.20)

It is interesting to observe that the above expression has a singularity at η = 0

and dGS(l) → −∞ in the limit η → 0. Consequently one negative valuesof dGS can be seen at very small values of η but this is not the signature ofHOA, rather this is the signature of the existence of a strong singularity in theneighborhood. This can further justi�ed by the fact that for any �nite valueof l there does not exist any real root (whose value is close to zero or whichis negligibly small compared to 1) of dGS(l) = 0. Thus there is no oscillationbetween bunching and antibunching. We further observe that dGS(l) → −1

in the limit η → 1 and for l ≥ 3 there exists only one physically acceptablereal root of dGS(l) = 0. By physically acceptable real root we mean that it

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Figure 2.7: Variation of dGS(8), dGS(9) and dGS(10) with respect to η.

Figure 2.8: Variation of dPACS(4) with α and m

appears in [0, 1]. Before this value of η (or before the physically acceptable realroot) the state shows higher order bunching but immediately after the root itbecomes negative and thus shows HOA. As we increase l the real root shifts inright side of the real axis (i.e. towards η = 1). It can be clearly seen in theFig. 2.7. From this �gure it can be easily seen that it satis�es all the physicalproperties of HOA derived in [27].

2.5.3 Photon added coherent state

Photon added coherent state (1.55) or PACS, which was introduced by Agarwaland Tara [66] is de�ned as (1.56). Rigorous operator algebra yields

52


dPACS(l) =exp(−α2)α2l+2((l+m+1)!)2 PFQ({1,2+l+m,2+l+m};{2+l,2+l,m+1};α2)

(m!(l+1)!)2 1F1(−m;1;−α2)

−(

exp(−α2)(−m+m 1F1(1+m;1;α2)+(1+m)α21F1(2+m;2;α2))

1F1(−m;1;−α2)

)l+1

,

(2.21)where, PFQ is the generalized hypergeometric function which is de�ned as

pFq(a; b; z) =∞∑k=0

(a1)k.....(ap)k(b1)k.....(bq)k

k!

zk. (2.22)

The analytic expression for dPACS(l) is quite complicated and it is di�cultto conclude anything regarding its photon statistics directly from (2.21) but wehave investigated the variation of dPACS with α, l andm and could not �nd anyregion which does not show HOA. Therefore, HOA can be seen in this particularintermediate state. This fact is manifested in Fig. 2.8 and Fig. 2.9. From these�gures it is easy to observe that depth of nonclassicality increases monotonicallywith the increase of m and l. The variation of depth of nonclassicality withα has a deep for a small value of α (see Fig. 2.8 and Fig. 2.9). AlthoughdPACS is always negative, initially its magnitude (dPACS(l) without the negativesign) increases, then decreases and then become a monotonically increasingfunction. Actually for the smaller values of α, an e�ective contribution fromthe combination of all the hypergeometric functions appears and dominatesbut as soon as α increases a bit, the exp(−α2) term starts dominating andas a consequence depth of nonclassicality increases monotonically. Here wewould also like to note that in contrast to the nonclassical properties of thephoton added coherent state, d(l) is always positive for the analogous state|α,−m〉, introduced by Shivakumar3 [89]. Thus |α,−m〉 always shows higher

3|α,−m〉 is photon subtracted coherent state which is intermediate betweenvacuum state (as α→ 0) and coherent state (asm→ 0) is de�ned as |α,−m〉 =∑

m!√n!αn

(n+m)! c0 |n〉, where c0 is a constant can be determined by normalization.

53


order superpoissonian photon statistics. Further, we would like to note thatphoton added coherent state which, is intermediate between Coherent and Fockstate has already been experimentally generated in 2004 [67]. Therefore, it istechnically feasible to observe higher order antibunching for an intermediatestate.

Figure 2.9: Variation of 10dPACS(3) and dPACS(4) with α for m = 15, the solid line denotesdPACS(3) and the dashed line denotes dPACS(4) to show the variation in the same scaledPACS(3) is multiplied by 10. The plot shows that depth of nonclassicality of dPACS(4) isalways greater than that of dPACS(3) which is consistent with the properties of HOA

Figure 2.10: Variation of dHS(8) with η and M , when lowest allowed values of L have beenchosen at every point.

54


2.5.4 Hypergeometric state

Following [60] hypergeometric state (HS) is de�ned in (1.51)-(1.54). Using thetechniques adopted in the sections 2.3 and 2.4, and a bit of operator algebrawe obtain a closed form analytic expression for d(l) as

dHS(l) = − (Mη)l+1 +(L− l − 1)!M !(Lη)!

L!(M − l − 1)!(Lη − 1− l)!(2.23)

From Fig. 2.10, it is clear that HOA can be observed in Hypergeometric state.It is also observed that the depth of nonclassicality increases with the increasein η and M . Hypergeometric state reduces to binomial state, coherent state,number state and vacuum state in di�erent limits of M, L and η. It has beenveri�ed that if we impose those limits on dHS then we obtain correspondingphoton statistics.

2.6 HOA in optical processes

The main goal of the present thesis is to study the possibilities of observinghigher order nonclassicalities in intermediate states. In this chapter we havealready shown that HOA may be seen in the intermediate states. We have alsomentioned that PACS has been experimentally generated and several schemesfor generation of other intermediate states have been proposed but so far otherintermediate states have not been produced experimentally. Keeping this inmind we went beyond the domain of intermediate states and have studied thepossibilities of observing HOA in some simple optical processes which are com-paratively easier to achieve experimentally. A single photon source (SPS) is veryimportant for quantum computation. In particular, it is essential for securedquantum cryptography. But there is no perfect SPS in reality. Therefore, prob-abilistic SPS, where probability of simultaneous emission of two, three, four andmore photon is less than the emission of a single photon are used. In the well-known antibunched state the rate of simultaneous emission of two photon is less

55


than that of single photon. But the requirement of quantum cryptography is amany photon version of the antibunched state or the higher order antibunchedstate. Recently Pathak and Garcia have reported a simpli�ed mathematicalcriterion for observing higher order antibunching (HOA) [27] and with the helpof that criterion Gupta, Pandey and Pathak shown that the HOA is not arare phenomenon [28] and it can be seen in simple optical processes. But stillwe don't know what kind of interaction produces HOA? How the signature ofHOA is a�ected by the power of modes in the interaction term of a Hamiltonianfor an optical process? In the following subsections we have tried to provideanswers to these questions. To do so we have chosen a new six wave mixingprocess; explore the possibility of HOA in all modes. We have also studied var-ious other multi-wave mixing processes. Further in the preceding subsectionswe have shown the existence of HOA in intermediate states.

2.6.1 Six wave mixing process

Six wave mixing may happen in di�erent ways. To generalize the possibilityof HOA in di�erent ways we have chosen such a way that three photon offrequency ω1 are absorbed (as pump photon) and two photon of frequency ω2

and another of frequency ω3 are emitted. The Hamiltonian representing thisparticular six wave mixing process is

H = a†aω1 + b†b ω2 + c†c ω3 + g(a†3b2c+ a3b†2c†), (2.24)

where a† and a are creation and annihilation operators in pump mode whichsatisfy [a†, a] = 1, similarly b†, b and c†, c are creation and annihilation opera-tors in stokes mode and signal mode respectively and g is the coupling constant.Substituting A = a eiω1t, B = b eiω2t and C = c eiω3twe can write the Hamilto-nian (2.24 ) as

56


H = A†Aω1 +B†B ω2 + C†C ω3 + g(A†3B2C + A3B†2C†). (2.25)

2.6.1.1 Time evolution of A

Since the Hamiltonian is known, we can use Heisenberg's equation of motion(with ~ = 1):

A = ∂A∂t + i[H,A], (2.26)

and short time approximation method [28] to �nd out the time evolution of theessential operators. From equation (2.25) we have

[H,A] = −Aω1 − 3gA†2B2C . (2.27)

From (2.26) and (2.27) we have

A = iAω1 − iAω1 − i3gA†2B2C = −i3gA†2B2C . (2.28)

We can �nd the second order di�erential of A using (2.26) and (2.28) as

A = ∂A∂t + i[H, A]

= 3g2[6A†A2B†2B2C†C + 6AB†2B2C†C − 4A†2A3B†BC†C

−2A†2A3C†C − A†2A3B†2B2 − 4A†2A3B†B − 2A†2A3]

(2.29)

Now by using Taylor's series expansion,

f(t) = f(0) + t(∂f(t)∂t

)t=0

+ t2

2!

(∂2f(t)∂t2

)t=0

+ .............. (2.30)

and substituting (2.28) and (2.29) in (2.30) we get

57


A(t) = A− 3igtA†2B2C + 32g2t2[6A†A2B†2B2C†C + 6AB†2B2C†C

−4A†2A3B†BC†C − 2A†2A3C†C − A†2A3B†2B2 − 4A†2A3B†B − 2A†2A3]

(2.31)The Taylor series is valid when t is small, so this solution is valid for a shorttime and that is why it is called short time approximation. This is a very strongtechnique since this straight forward prescription is valid for any optical processwhere interaction time is short. After obtaining the analytic expression for timeevolution of annihilation operator, now we can use it to check whether it satis�escondition (2.5) or not. Let us start with the possibility of observing �rst orderantibunching in pump mode. From equation (2.31), Hermitian conjugate ofA(t) can be written as

A†(t) = A† + 3igtA2B†2C† + 32g2t2[6A†A2B†2B2C†C + 6A†B†2B2C†C

−4A†3A2B†BC†C − 2A†3A2C†C − A†3A2B†2B2 − 4A†3A2B†B − 2A†3A2].

(2.32)Hence using (2.31) and (2.32), number operator of A mode NA(t) can bederived as

NA(t) = A†(t)A(t)

= A†A− 3igt(A†3B2C − A3B†2C†)+ 3g2t2[9A†2A2B†2B2C†C + 18A†AB†2B2C†C

+6B†2B2C†C − 4A†3A3B†BC†C − 4A†3A3B†2B2 − 2A†3A3C†C

−A†3A3B†2B2 − 2A†3A3].

(2.33)Taking expectation value of NA(t) with respect to |α > |0 > |0 >, we can get

〈NA(t)〉α = |α|2 − 6g2t2 |α|6 , (2.34)

where A|α〉 = α|α〉. By using this straight forward description we can get

58


⟨N

(2)A (t)

⟩α

=⟨A†2(t)A2(t)

⟩= |α|4 − 12g2t2

[|α|6 + |α|8

], (2.35)

and

⟨N

(3)A (t)

⟩α

=⟨A†3(t)A3(t)

⟩= |α|6 − 6g2t2

[2 |α|6 + 6 |α|8 + 3 |α|10

].

(2.36)Now by using (2.34)-(2.36) and (2.5) we can show that the pump mode for sixwave mixing process satis�es the criterion of normal antibunching and secondorder antibunching,

dA(1) =⟨N

(2)A (t)

⟩α− 〈NA(t)〉2α

= |α|4 − 12g2t2[|α|6 + |α|8

]− |α|4 + 12g2t2 |α|8

= −12g2t2 |α|6 ,

(2.37)

and

dA(2) =⟨N

(3)A (t)

⟩α− 〈NA(t)〉3α

= |α|6 − 6g2t2[2 |α|6 + 6 |α|8 + 3 |α|10

]− |α|6 + 18g2t2 |α|10

= −12g2t2[|α|6 + 3 |α|8

],

(2.38)Since dA(1) and dA(2) are always negative. Hence normal antibunching andhigher order antibunching (second order) exist in pump mode. Expectationvalues are taken over |α〉|0〉|0〉 physically means that initially a coherent state(Laser) is used as pump and before the interaction of the pump with atom,there was no photon in signal mode B or stokes mode C. Thus the pumpinteracts with the atom and causes excitation followed by emission. Now our

59


aim for studying this particular problem is whether stokes mode (expectationvalue over |0〉|β〉|0〉) and signal mode (expectation value over |0〉|0〉|γ〉) alsoshow antibunching or not for time evolution of B and C in six wave mixingprocess.

2.6.1.2 Time evolution of B

Following same procedure as mentioned in last subsection, we can get

B = −2igA3B†C† , (2.39)

and

B = 2g2[A†3A3B†B2 + 2A†3A3BC†C + 2A†3A3B − 9A†2A2B†B2C†C

−18A†AB†B2C†C − 6B†B2C†C].

(2.40)Hence Time evolution of B can be written as, using Taylor's expansion,

B(t) = B − 2igtA3B†C† + 2g2t2[A†3A3B†B2 + 2A†3A3BC†C + 2A†3A3B

−9A†2A2B†B2C†C − 18A†AB†B2C†C − 6B†B2C†C],

(2.41)and number operator correspond to stokes mode is,

NB(t) = B†(t)B(t)

= B†B − 2igt[A3B†2C† − A†3B2C

]+2g2t2

[A†3A3B†2B2 + 4A†3A3B†BC†C + 4A†3A3B†B + 2A†3A3C†C

−9A†2A2B†2B2C†C − 18A†AB†2B2C†C − 6B†2B2C†C + 2A†3A3].

(2.42)

Expectation value of NB(t) with respect to |α〉|0〉|0〉 is〈NB(t)〉α = 4g2t2|α|6,with respect to |0〉|β〉|0〉 is 〈NB(t)〉β = |β|2and for |0〉|0〉|γ〉 is 〈NB(t)〉γ = 0.Similarly we can obtain Expectation values of second factorial moment N (2)

B (t)

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and third factorial moment N (3)B (t) of number operator NB(t) with respect to

|α〉|0〉|0〉 are⟨N

(2)B (t)

⟩α

= 4g2t2|α|6 and⟨N

(3)B (t)

⟩α

= 0, with respect to

|α〉|β〉|0〉 are⟨N

(2)B (t)

⟩β

= |β|4 and⟨N

(3)B (t)

⟩β

= |β|6, �nally for |0〉|0〉|γ〉 are0. Therefore normal and second order antibunching for B over pump mode aredBα(1) = 4g2t2|α|6 and dBα(2) = 0, over stokes mode and signal mode areabsent.

2.6.1.3 Time evolution for C

Time evolution of C can be derived by using Taylor's expansion,

C(t) = C − igtA3B†2 + 12g2t2

[A†3A3B†2B2C + 4A†3A3B†BC + 2A†3A3C

−A†3A3B†2B2 − 9A†2A2B†2B2 − 18A†AB†2B2 − 6B†2B2],

(2.43)

and Number operator NC(t) is,

NC(t) = C†C − igt[A3B†2C† − A†3B2C

]+1

2g2t2

[2A†3A3B†2B2C†C + 8A†3A3B†BC†C + 4A†3A3C†C

−A†3A3B†2B2C† − 9A†2A2B†2B2C† − 18A†AB†2B2C† − 6B†2B2C†

−A†3A3B†2B2C − 9A†2A2B†2B2C − 18A†AB†2B2C − 6B†2B2C].

(2.44)

Expectation values of NC(t) with respect to |α〉|0〉|0〉 is〈NC(t)〉α = 0, withrespect to |0〉|β〉|0〉 is 〈NC(t)〉β = 0 and with respect to |0〉|β〉|0〉 is 〈NC(t)〉γ =

|γ|2. Similarly we can obtain Expectation values of second factorial momentN

(2)C (t) and third factorial moment N (3)

C (t) of number operator N(t) with re-spect to |α〉|0〉|0〉, |0〉|β〉|0〉 are 0 and �nally with respect to |0〉|0〉|γ〉 are⟨N

(2)C (t)

⟩γ

= |γ|4 and⟨N

(3)C (t)

⟩γ

= |γ|6. Hence dC(1) and dC(2) are alwayszero for all modes. Therefore neither the normal antibunching nor the secondorder antibunching is present in mode C. Earlier Pandey, Gupta and Pathakhave reported normal and higher order antibunching for pump mode in an-other kind of six wave mixing [28] whose interaction term in the Hamiltonianwas a†2b3c. In the same manner we have obtained the results for all other modes

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Table 2.1: Possibility of observing HOA in di�erent modes of six wave mixing process.Expectation values with respect to

Sr.No.

Opticalpro-cesses

Interactionterm

d pa-rame-ter

mode A mode B mode C

|α〉 |β〉 |γ〉 |α〉 |β〉 |γ〉 |α〉 |β〉 |γ〉

1

Six wavemixing(new) A†3B2C d(1) −12g2t2|α|6 0 0 4gt2|α|6 0 0 0 0 0

d(2) −12g2t2[|α|6+3|α|8]

0 0 0 0 0 0 0 0

2

Six wavemixing(earlier) A†2B3C d(1) −12g2t2|α|4 0 0 36gt2|α|4 0 0 0 0 0

d(2) −36g2t2|α|6 0 0 0 0 0 0 0 0

in six wave mixing (a†2b3c). Results4 are provided in Table 2.1.

2.6.2 Other optical processes

Earlier Gupta, Pandey and Pathak [28] and our group [75] have reported normaland higher order antibunching for pump mode only in other optical processes.For example, four wave mixing, second harmonic generation [28] and �ve wavemixing [75]. In the present section we have studied normal and higher orderantibunching in two types of six wave mixing processes for all modes. Nowwe want to generalize our results on the basis of earlier and new observationsrelated to HOA in optical processes. All the results5 are mentioned in Table2.2.

From Table 2.2, it is clear that except tri-linear parametric process all thephysical systems studied show antibunching and HOA in pump mode only. Forstokes mode and signal mode antibunching and HOA is absent. When the

4Expectation values are taken with respect to |α〉|0〉|0〉, |0〉|β〉|0〉| and |0〉|0〉|γ〉respectively foreach mode. These states physically correspond to the initial states. Vacuum state is not written inthe top of the Table (i.e. |α〉|0〉|0〉 is written as |α〉).

5Expectation values are taken with respect to |α〉|0〉|0〉, |0〉|β〉|0〉| and |0〉|0〉|γ〉respectively foreach mode. These states physically correspond to the initial states. Vacuum state is not written inthe top of the Table (i.e. |α〉|0〉|0〉 is written as |α〉).

62


Table 2.2: Possibility of observing HOA in di�erent modes of many wave mixing processes.Expectation values with respect to

Sr.No.

Opticalprocesses

Interactionterm

d pa-ram-eter

mode A mode B mode C

|α〉 |β〉 |γ〉 |α〉 |β〉 |γ〉 |α〉 |β〉 |γ〉

1Six wavemixing(new)

A†3B2C d(1) −12g2t2|α|6 0 0 4gt2|α|6 0 0 0 0 0

d(2) −12g2t2[|α|6+3|α|8] 0 0 0 0 0 0 0 0

2Six wavemixing(earlier)

A†2B3C d(1) −12g2t2|α|4 0 0 36gt2|α|4 0 0 0 0 0

d(2) −36g2t2|α|6 0 0 0 0 0 0 0 0

3Four wavemixing A†2BC d(1) −2g2t2|α|4 0 0 0 0 0 0 0 0

d(2) −6g2t2|α|6 0 0 0 0 0 0 0 0

4Secondharmonicgeneration

A†2B d(1) −2g2t2|α|4 0 0 0 0 0 0 0 0

d(2) −6g2t2|α|6 0 0 0 0 0 0 0 0

5Five wavemixing A†3B2 d(1) −12g2t2|α|6 0 0 0 0 0 0 0 0

d(2) −12g2t2[|α|6+3|α|8] 0 0 0 0 0 0 0 0

6Third

harmonicgeneration

A†3B d(1) −6g2t2|α|6 0 0 0 0 0 0 0 0

d(2) −6g2t2[|α|6+3|α|8] 0 0 0 0 0 0 0 0

7Tri-linearparametricprocess

A†BC d(1) 0 0 0 0 0 0 0 0 0

d(2) 0 0 0 0 0 0 0 0 0

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expectation values are taken over |α〉|0〉|0〉, physically that means a choice ofa coherent state (Laser) as pump and before the interaction of the pump withatom, there was no photon in signal mode b or stokes mode c. Thus the pumpinteracts with the atom and causes excitation followed by emission. This state ismuch more physical compared to the other two states used in the present work.Interestingly all the interesting results appeared for this particular choice ofinitial state. It is also observed that depth of nonclassicality can be tuned withthe help of number of photons present in pump mode. It is clear that the HOAwould not have been observed if we would have considered �rst order operatorsolutions (�rst order in g), on the other hand if we use second order operatorsolutions then the depth of nonclassicality is found to increase monotonicallywith the increase of input photon number (|α|2). This monotonic growth maybe seized in presence of higher order terms. Following the work of Vyas andSingh [26] one can easily make an experimental setup to observe HOA (see [39])and verify the prediction of the present work. In such an experiment it wouldnot be wise to choose a large value of (|α|2) as in that case the detector willessentially record a superposition of antibunched and coherent pulse and withincrease of (|α|2) the probability of detecting original (non-absorbed) coherent(Poissonian) photon will also increase. Most of the physical systems shown inthe present work are simple and easily achievable in laboratories and thus itopens up the possibility of experimental observation of HOA and will decidesuitable probabilistic single photon source.

2.7 Quality of single photon sources: How to compare

two potential candidates of single photon source?

In this chapter we have studied possibilities of observing HOA in di�erent phys-ical systems. Following Pathak and Garcia's physical interpretation of HOA,we have also mentioned that the possible single photon sources are expected to

64


satisfy the criterion of HOA. Here we would like to note that negativity of d(l)is just an indication of possible use of the physical system to construct a SPSbut its not a good quantitative measure of quality of SPS. Keeping this in mindwe have introduced a parameter ηk in this section as a quantitative measureof quality SPS and use it to compare the potential sources of SPS proposed inthe present thesis. We have considered the intermediate states showing HOAas potential SPS and have compared them with the help of ηk. In the construc-tion of the criterion we assume that the quantum state of light is representedin Fock space (i.e. the state is expanded in number state, |n〉 as

|ψ〉k =∑n

cn,k(α1, α2, ....αj)|n〉, (2.45)

where the coe�cient of the number state (cn,k) is a function of j parameters(namely (αj), the index k is used to distinguish between the quantum states.Now to compare the single photon generation capacity of two quantum states,one logical way is to compare them for the situation when average photonnumber is same. Thus we put a constrain condition

Nk =k 〈ψ|a†a|ψ〉k = γ, (2.46)

where γ is a positive constant and normally 0 < γ < 1 as the number of photonin a probabilistic single photon source is expected to be less than 1. Now underthis constrain condition one can easily �nd out the probability that the pulsecontain no photon (P0,k) and the probability that the pulse contain 1 photon(P1,k) are respectively P0,k = |c0,k|2 and P1,k = |c1,k|2. Therefore the probabilitythat the pulse contain two or more photon is

P (n ≥ 2) = 1− (P0,k + P1,k) = 1− (|c0,k|2 + |c1,k|2). (2.47)

To provide a quantitative measure of the single photon generation probabilitywe de�ne a parameter ηk as the ratio of the number of pulses containing just

65


one photon to the number of pulses containing more than one. Therefore,

ηk =P1,k

1− (P0,k + P1,k)=

|c1,k|2

1− (|c0,k|2 + |c1,k|2). (2.48)

The greater is ηk the better is the single photon source (SPS). So the next task isto maximize ηk with respect to photon number γ and all other parameters (αj)on which it depends to obtain ηk,Max, now if we obtain η1,Max > η2,Max, thenwe will be able to conclude that the quantum state |ψ〉1 will be more useful as aSPS than the quantum state |ψ〉2. In earlier sections we have already discussedthe possibility of construction of SPS using intermediate states. Now we wishto compare the single photon generation probability of di�erent intermediatestates.

2.7.1 Example: ηk for BS, GBS and HS

Binomial state (BS) is de�ned in (1.46). Using the de�nition of BS (1.46) andparameter ηk (2.48) we obtain

ηBS =MC1

(γM

) (1− γ

M

)M−1

1−[(

1− γM

)M−1 (MC0(1− γ

M

)+ MC1

(γM

))] . (2.49)

In binomial state, parameter ηBS is observed to decrease with increasing valuesof γ or M . In a similar way we can obtain analytic expression of ηk for otherstates too. For example, here we report ηk for generalized binomial state (GBS)de�ned in (1.47-1.49) and hypergeometric state de�ned (HS) in (1.51-1.54).Therefore using their de�nitions and (2.48) we obtain

ηGBS =

N(β+1)γ(β+1)N−1

(N−γ)(N(β+1)N−γ

)N1− (β+1)N(

N(β+1)N−γ

)N

−N(β+1)γ(β+1)N−1

(N−γ)(N(β+1)N−γ

)N

, (2.50)

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Figure 2.11: Comparison of single photon generation probability of Binomial State withGeneralized Binomial state (GBS) and Hypergeometric State (HS).

and

ηHS =

(LγM )

L(1− γ

M

)M − 1

L

M

−1

1−

L(1− γ

M

)M

L

M

−1

−(LγM )

L(1− γ

M

)M − 1

L

M

−1. (2.51)

Comparison of single photon generation probability of Binomial state ηBS withηGBS and ηHS is shown in Fig 2.11 keeping same value of γ = 0.1 and othercommon parameters6. In case of GBS β = 10 is chosen. The comparison showsthat HS is a better candidate of SPS. Similarly one can get expression of η inany other intermediate state.

2.8 Conclusions

In essence, all the intermediate states studied in the present work show HOA.But all the intermediate states are not higher order antibunched (for example,

6Among BS, GBS and HS, value of M is equivalent to N. Mean value γ isde�ned in (2.46) can be obtained in BS as γ = Mp, in GBS as γ = N(α+1)

(α+β+2) andin HS as γ = Mp.

67


|α,−m〉 is always higher order bunched). We have also observed that an in-termediate state which shows HOA may also show higher order bunching orhigher order coherence for particular values of the control parameters (for ex-ample, negative binomial state and generalized binomial state show both higherorder bunching and higher order antibunching for di�erent parametric values).Thus we can conclude that, as far as HOA is concerned there does not existany common characteristics among the di�erent intermediate states but mostof them show HOA. Further, we have seen from (Fig. 2.1- Fig. 2.10) that thedepth of nonclassicality of a higher order antibunched state varies with di�erentcontrol parameters (e.g. α, N , m etc.). These parameters represent some phys-ical quantity and their value may be controlled and consequently by controllingthese parameters we can control the depth of nonclassicality.

Figure 2.12: Schematic of the experimental setup for the homodyne detection of the lightproduced from the intermediate state (IS); φ is the phase of the local oscillator (LO) relativeto the IS; BS is loss less beam splitter; D is detector; α, αl and βi correspond to theannihilation operators for the IS, LO and the output of the BS respectively.

A potential scheme for the experimental observation of HOA in intermedi-ate state can be introduced in analogy with the work of Vyas and Singh (seeFig. 1 in [26]). The schematic experimental setup for homodyne detection ofthe higher order photon statistics of the light in intermediate state is providedin Fig. 2.12. In this experiment the frequency of the local oscillator is the

68


same as that of the signal. The homodyne light incidents on the detector andgenerates random photoelectric pulses. One can easily obtain the statistics ofthese pulses. Now factorial moments of photo-electron counts can be used todescribe the statistics of these pulses. This method essentially converts the pho-ton statistics of the homodyne light into the photo electron statistics (recordedby the detector). Dodson and Vyas [90] have shown that by using the prop-erty of the �eld variables a generating function for the photo electron statisticscan be obtained and that can be used to �nd analytic expression for factorialmoments of the photo electron counting distribution. The nonclassicality ob-served in homodyne light will actually represent the inherent nonclassicality ofthe intermediate state since the local oscillator does not introduce nonclassical-ity [90]. Dodson-Vyas method [90] has already been successfully used by Vyasand Singh [88] and Erneso, Vyas and Singh [38] to study HOA in degenerateoptical parametric oscillator and in intracavity second harmonic generation re-spectively. Thus we can conclude that the Photon statistics (factorial moment)of an intermediate state can be obtained experimentally by using homodynedetection (photon counting) technique. This fact along with the recent suc-cess in experimental production of intermediate state open up the possibilityof experimental observation of HOA in intermediate state.

The prescription followed in the present chapter is easy and straight forwardand it can be used to study the possibilities of observing higher order anti-bunching in other intermediate states (such as negative hyper geometric state,excited binomial state and odd excited binomial state) and other physical sys-tems. Thus it opens up the possibility of studying higher order nonclassicale�ects from a new perspective. This is also important from the applicationpoint of view because any probabilistic single photon source used for quantumcryptography has to satisfy the condition for higher order antibunching.

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