Optics Communications 284 (2011) 289–293

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Optics Communications

j ourna l homepage: www.e lsev ie r.com/ locate /optcom

Higher-order Hong–Mandel's squeezing in superposed coherent states

Hari Prakash a, Rakesh Kumar b, Pankaj Kumar c,⁎a Department of Physics, University of Allahabad, Allahabad, 211002 (U.P.), Indiab Department of Physics, Udai Pratap Autonomous College, Varanasi, 221002 (U.P.), Indiac Department of Physics, Bhavan's Mehta Mahavidyalaya (V. S. Mehta College of Science), Bharwari, Kaushambi-212201 (U.P.), India

⁎ Corresponding author.E-mail addresses: prakash_hari123@rediffmail.com (

r_rkumar123@rediffmail.com (R. Kumar), pankaj_k25@

0030-4018/$ – see front matter © 2010 Elsevier B.V. Aldoi:10.1016/j.optcom.2010.09.019

a b s t r a c t

a r t i c l e i n f o

Article history:Received 10 July 2010Received in revised form 5 September 2010Accepted 5 September 2010

Keywords:Non-classical features of lightCoherent stateSqueezingHigher-order squeezingSub-Poissonian photon statisticsDisplacement operatorPhase shifting operator

We study 2nth order Hong and Mandel's squeezing of the Hermitian operator, Xθ≡X1cosθ+X2sinθ in the mostgeneral superposition state, |ψ⟩=Z1|α⟩+Z2|β⟩ of two coherent states |α⟩ and |β⟩. Here operators X1,2 are definedby X1+i X2=a, the annihilation operator, angle θ and complex numbers Z1,2 , α, β are arbitrary and the onlyrestriction on these is the normalization condition of |ψ⟩.Wefindmaximum2nth-orderHong–Mandel squeezing

of Xθ in the superposed coherent state |ψ⟩ for an infinite combinations with α−β = A2nexp � i12π + iθ

h i,

Z2 = Z1 = exp 12βα*−β*αð Þ

h iandwith arbitrary values of (α+β) and θ. Here A2n is a constantwhich depends on

the order of squeezing. The maximum percentage of 2nth order Hong–Mandel squeezing and their respectiveconstants A2n have been reported for some values of n. We conclude that any large percentage of squeezing canbe obtained by suitably choosing of the order 2n. For the order greater than 128 we obtained more than 99%higher-order squeezing at very low intensity of the optical field. Variations of higher-order squeezing withdifferent parameters near its maxima have also been discussed.

H. Prakash),rediffmail.com (P. Kumar).

l rights reserved.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

States of light, whose properties cannot be explained on the basisof classical theory, are called non-classical states [1]. For these states,Glauber–Sudarshan P function [2,3] is less well behaved than aprobability density, i.e., it takes negative values and become moresingular than a delta function. The non-classical nature of a quantumstate can be manifested in different ways like antibunching, sub-Poissonian photon statistics and various kinds of squeezing, etc.Earlier study of such non-classical effects was largely in academicinterest [4], but now their applications in quantum information theorysuch as communication [5,6], quantum teleportation [7], dense coding[8] and quantum cryptography [9] are well realized. It has beendemonstrated that non-classicality is the necessary input forentangled state [10].

Squeezing, a well-known non-classical effect, is a phenomenon inwhich variance in one of the quadrature components become lessthan that in vacuum state or coherent state [11] at the cost ofincreased fluctuations in the other quadrature component. Thisdefinition of squeezing has been generalized to case of severalvariables [12–16]. Hong and Mandel [12] introduced the concept ofhigher-order squeezing by considering the 2nth order moments of thequadrature component and defined a state to be 2nth order squeezedif the expectation value of the 2nth power of the difference between a

field quadrature and its average value is less than what it would be ina coherent state. According to Hong and Mandel's definition [12], astate |ψ⟩ is said to be 2nth order squeezed for the operator,

Xθ = X1 cos θ + X2 sin θ; ð1Þ

if the 2nth order moment of Xθ,

⟨ψj ΔXθð Þ2njψ⟩b2−2n 2n−1ð Þ!!; ð2Þ

where Hermitian operators X1, 2 are defined by X1+iX2=a, a is theannihilation operator, θ is an arbitrary angle, ΔXθ=Xθ− ⟨ψ|Xθ|ψ⟩ and(2n−1)!! is product of factors, starting with (2n−1) and decreasingin steps of 2 and ending at 1. Note that the right hand side ininequality (Eq. (2)) is the value of the left hand side for the coherentstate. Hong and Mandel demonstrated [12], using examples, that thehigher-order squeezing invariably accompanies ordinary squeezingand, most remarkably, exhibits an even larger percentage noisereduction than ordinary squeezing. This feature makes higher-ordersqueezing especially interesting from the point of view of applica-tions. As emphasized by Hong and Mandel higher-order squeezingcan be exploited in all those contexts where second-order (ordinary)squeezing is useful. Such type of squeezing is quite distinct fromordinary squeezing because such squeezing does not require that theuncertainty product be a minimum and therefore both quadratures ofthe field can have higher-order squeezing. In other words, states existfor which product ⟨ψ|(ΔX1)2n|ψ⟩⟨ψ|(ΔX2)2n|ψ⟩ takes a value less thanthat for a coherent state [13].

290 H. Prakash et al. / Optics Communications 284 (2011) 289–293

Another form of higher-order squeezing in terms of real andimaginary parts of square of the amplitude, the so-called ‘amplitude-squared squeezing’, has been proposed by Hillery [14] by consideringthe operators Y1 and Y2, such that Y1+iY2=a2, a is annihilationoperator. Hillery [15] introduced another type of higher-ordersqueezing, called sum squeezing and difference squeezing byconsidering two mode systems and using sum and differences ofvarious bilinear combinations constructed from the creation andannihilation operators. Zhang et al. [16] defined kth-order squeezingbased on the operators A=(ak+a+ k) /2 and B=(ak−a+ k) /2ifollowing Hillery [14].

The higher-order squeezing defined [14–16] by different types ofdefinitions is different from Hong and Mandel's higher-ordersqueezing [12]. However all of the squeezed states produced bydifferent kinds of higher-order squeezing definitions are non-classicalin nature. Although these higher-order squeezing have received wideattention in the theory [17–21], there is no report of experimentaldetection of any higher-order squeezing. Recently, Prakash et al. [22]have however shown that Hong andMandel's higher-order squeezingand amplitude-squared squeezing can be detected experimentally byhomodyne method using higher-order sub-Poissonian photon statis-tics [22,23].

Many different schemes of generating such non-classical stateshave been proposed [17–21], such as four-wave mixing, resonancefluorescence, the use of free electron laser, cavities, harmonicgeneration, parametric amplification and J C model. Other possibilitiesfor generating such effect have been proposed by superposition of twoor more coherent states. It has been realized that a coherent state [11]does not exhibit non-classical effects but the superposition of two ormore coherent states exhibits [24–32] various non-classical effectslike squeezing, antibunching, higher-order squeezing and higher-order sub-Poissonian statistics etc. Buzek et al. [24] and Xia et al. [27]studied such effects in the superposition of two coherent states |α⟩and|α⟩ and reported that the even coherent state ~(|α⟩+|−α⟩)exhibits squeezing but not sub-Poissonian statistics while the oddcoherent state ~(|α⟩− |−α⟩) exhibits sub-Poissonian statistics butnot squeezing. Xia et al. also studied [27] such effects in the displacedeven and odd coherent states. Schleich et al. [25] studied such effectsin the superposition of two coherent states, |α ⟩and|α*⟩, and reportedthat such superposition can exhibit both squeezing and sub-Poissonian statistics when |α|2NN1. Recently, we reported [29,30]maximum squeezing and antibunching in the most general super-posed coherent state and in interaction of two two-level atoms with asingle-mode superposed coherent states. We also studied [31,32] themaximum fourth-order Hong–Mandel's squeezing and amplitude-squared squeezing in superposition of two arbitrary coherent states.In practice, the superposition of coherent states can be generated ininteraction of coherent state with nonlinear media [33] and inquantum nondemolition techniques [34].

In this paper we study 2nth-order Hong–Mandel squeezing of theHermitian quadrature operator Xθ in the most general superpositionstate,

jψi = Z1 jαi + Z2 jβi; ð3Þ

of two coherent states |α⟩ and |β⟩. Here, complex numbers Z1, Z2,α andβ are all-arbitrary and the only restriction on these is normalizationcondition of the superposed state|ψ⟩. We use the properties ofdisplacement operator and investigate conditions on the parametersZ1, Z2, α, β and θ for maximum nth-order squeezing of Xθ in thesuperposed coherent state |ψ⟩.Wegetmaximumnth-order squeezingofXθ in the superposed coherent state |ψ⟩ for an infinite combinationswith

α−β = A2n exp � i12π + iθ

h i, Z2 = Z1 = exp 1

2βα*−β*αð Þ

h iand with

arbitrary values of (α+β) and θ. Here A2n is a constant which dependson the order 2n of squeezing. The minimum values of 2nth-order

moments ⟨ψ|(ΔXθ)2n|ψ⟩ ofΔXθZXθ−⟨ψ|Xθ|ψ⟩ for some values of n havebeen found using computer programming. We conclude that any largepercentage of squeezing can be obtained by suitably choosing of theorder 2n. For the order greater than 128we findmore than 99% higher-order squeezing at very low intensity of the optical field. Variations ofhigher-order squeezingwithdifferent parameters at itsmaximumvaluehave also been discussed.

2. 2nth order moment of Xθ in the superposed coherent state |ψ⟩

A single-mode coherent state |α⟩ defined by a|α⟩=α|α⟩ can bewritten as

jαi = exp −12jαj2

� �∑∞

n=0

αnffiffiffiffiffin!

p jni = D αð Þ j0i; ð4Þ

where |n⟩ is the occupation number and D(α)=exp(α a+−α⁎a) isthe displacement operator. Using the relation D+(α)a D(α)=a+α,we have

⟨ψ′j ΔXθ; jψ′i� �2njψ′⟩= ⟨ψj ΔXθ; jψi

� �2njψ⟩; ð5Þ

where |ψ′⟩=D(α)|ψ⟩, ΔXθ, |ψ′⟩=Xθ− ⟨ψ′|Xθ|ψ′⟩ and ΔXθ, |ψ⟩=Xθ− ⟨ψ|Xθ|ψ⟩≡ΔXθ. From Eq. (5) we conclude that the 2nth order Hong andMandel's squeezing in any state |ψ⟩ is not affected by operation of thedisplacement operator. This observation and the relation [11],

D αð ÞD βð Þ = exp 12α β⁎−β α⁎� �h i

D α + βð Þ, suggests that we should

simplify the problem by writing the superposed coherent state |ψ⟩ as

jψi = D12

α+ β½ �� �

jψ1i; jψ1i = Z′1 jξi + Z′

2 j−ξi; ð6Þ

where ξ = 12α−βð Þ; Z′

1;2 = Z1;2 exp �14αβ⁎−βα⁎� �h i

. We can rewritethe state |ψ1⟩A in another form as a superposition of even and oddcoherent states in the form,

jψ1i = Z″1 jξi + j−ξi½ � + Z″

2 jξi− j−ξi½ �; ð7Þ

where, Z′1, 2=Z ″1±Z ″

2. Without any loss of generality, we can furtherwrite the state |ψ1⟩ considering the coefficient Z ″

1 to be real andpositive, as

jψ1i = cosχ2jξ; + i + sin

χ2ei ϕ jξ;−i; ð8Þ

where |ξ,± ⟩=K±[|ξ⟩±|−ξ⟩]; K� = 2 1� e−2 jξj2� �h i−12, are the

normalized even and odd coherent states respectively and anglesχ=2 tan −1(|Z ″

2| /Z ″1) and ϕ=arg(Z ″

2), lie in the domains 0≤χ≤πand −πbϕ≤π. Since ⟨ψ|(ΔXθ)2n|ψ⟩ in any state |ψ⟩ does not changeon operation of the displacement operator, and the state |ψ1⟩ containsonly four parameters while the state |ψ⟩ contains six parameters, it iseasier to calculate ⟨ψ1|(ΔXθ, |ψ1⟩

)2n|ψ1⟩ than ⟨ψ|(ΔXθ, |ψ⟩)2n|ψ⟩ andminimize it for study maximum 2nth-order squeezing of Xθ in thestate |ψ⟩.

Ifξ=Aexp(iθξ),we canwrite the state |ξ⟩=ei θξN|A⟩,whereN=a+a .We also have eiθNae− iθN=ae− iθ, and we can write

⟨ψ1j ΔXθ; jψ1i� �2njψ1⟩= ⟨ψ2j ΔXδ; jψ2i

� �2njψ2⟩ ð9Þ

where jψ2⟩= cos χ2 jA; +⟩+ sin χ

2 ei ϕ jA;−⟩, |A,±⟩=K′±[|A⟩±|−A⟩]and δ=θξ−θ. The 2nth order moment of Xδ can be written as

⟨ψ2j ΔXδ; jψ2i� �2njψ2⟩= ∑

n−1

i=0

2n! 12

� �3i

2n−2ið Þ!i!⟨ψ2j : ΔXδð Þ2n−2i : jψ2⟩ +2n−1ð Þ!!

22n ;

ð10Þ

(S2n

) min

χ,φ,

δ0.4

0.2

-0.2

-0.6

-0.8

-0.4

0

-00 1

A20.5 1.5 2.5

10

86

4

2

Fig. 1. Variation of squeezing factor (S2n)min χ,φ, δ with A for orders 2n=2, 4, 6, 8 and10. Orders 2n are shown in the figure on the curves.

0 20 40 60 80 100 120 140 1600.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

2n

A2n

Fig. 3. Variation of A2n with order 2n of squeezing.

291H. Prakash et al. / Optics Communications 284 (2011) 289–293

and the involved even and odd coherent states |A,±⟩ have theproperties

a jA;�i = KA� jA;�i; a2 jA;�i = A2 jA;�i; hA;+ jaþa jA;+i= tanh A2;

ð11Þ

where A� = A 1� e−2 A2� �−12 and K = 1−e−4 A2h i−1

2.

To find ⟨ψ2| : (ΔXδ, |ψ2⟩)2n : |ψ2⟩, we make binomial expansion in

powers of ⟨ψ2| :Xδm : |ψ2⟩. For even m, tedious but straight forward

calculations lead to

hA; + j : Xmδ : jA; + i =

Am cos δð Þm + −1ð Þm2 sin δð Þm e−2A2h i

2 cosh A2 e−A2 ; ð12Þ

hA;− j : Xmδ : jA;−i =

Am cos δð Þm− −1ð Þm2 sin δð Þm e−2A2h i

2 sinh A2 e−A2 ; ð13Þ

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-1

-0.95

-0.9

-0.85

-0.8

-0.75

-0.7

-0.65

-0.6

-0.55

Sm

in

A2n

Fig. 2. Variation of minimum squeezing factor Smin with A2n for 2n=2, 4, 6,…, 30, 40,50,..160. Points refer to these in increasing order for 2n from right to left starting withpoint for 2n=2 at A2n=0.8.

and finally to

hψ2 j : Xmδ : jψ2i =

Am cos2χ2

cos δð Þm + −1ð Þn sin δð Þm e−2A2

2 cosh A2 e−A2

+Am sin2 χ

2cos δð Þm− −1ð Þn sin δð Þm e−2A2

2 sinh A2 e−A2:

ð14Þ

Similarly, for odd m, we get

hA;� j : Xmδ : jA;�i = 0; ð15Þ

hA; + j : Xmδ : jA;−i = Am cos δð Þm + ið Þm sin δð Þm� �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−e−4A2

p ; ð16Þ

and finally to

hψ2 j : Xmδ : jψ2i =

Am sinχ cos δð Þmcosϕ + ið Þm + 1 sin δð Þmsinϕh i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−e−4A2

p :

ð17Þ

Using Eqs. (10), (14) and (17) we get 2nth-order moment of Xθ inthe state |ψ2⟩.

0 20 40 60 80 100 120 140 160-1

-0.95

-0.9

-0.85

-0.8

-0.75

-0.7

-0.65

-0.6

-0.55

2n

Sm

in

Fig. 4. Variation of Smin with order 2n of squeezing.

292 H. Prakash et al. / Optics Communications 284 (2011) 289–293

3. 2nth order squeezing of Xθ in the superposed coherent state |ψ⟩

Since ⟨ψ|(ΔXθ)2n|ψ⟩ in any state |ψ⟩ does not change on operationof the displacement operator, for studying maximum 2nth-ordersqueezing of Xθ in the state |ψ⟩ we minimize ⟨ψ2|(ΔXδ)2n|ψ2⟩ withrespect to all variables A, χ, δ and ϕ. When we minimize ⟨ψ2|(ΔXδ)2n|ψ2⟩ analytically with respect to χ, we get χ=0. At χ=0, this isindependent of ϕ and hence we can write this minimum value as

hψ2 j ΔXδð Þ2n jψ2iminχ;ϕ

= ∑n−1

i=0

2n! 12

� �3i

2n−2ið Þ!i!A2n−2i cos δð Þ2n−2i + −1ð Þn−i sin δð Þ2n−2i e−2A2h i

2 cosh A2 e−A2 +2n−1ð Þ!!

22n :

ð18Þ

We further minimize ⟨ψ2|(ΔXδ)2n|ψ2⟩min χ,ϕ with respect to δ andfind equal minima at δ = � π

2 given by

hψ2 j ΔXδð Þ2n jψ2iminχ;ϕ;δ= ∑

n−1

i=0

2n! 12

� �3i

2n−2ið Þ!i!A2n−2i −1ð Þn−i e−2A2

2 cosh A2 e−A2 +2n−1ð Þ!!

22n :

ð19Þ

For simplicity we define squeezing factor,

S2n =hψ2 j ΔXδð Þ2n jψ2i−2−2n 2n−1ð Þ!!

2−2n 2n−1ð Þ!! : ð20Þ

This should be noted that 2nth order squeezing occurs if−1≤S2nb0.For S2nb0, we can define the degree of squeezing by D2n=−S2n. Alsowecancall 100D2n aspercentageof squeezing.Using computerprogram-ming, we get maximum higher-order squeezing of Xδ by minimizing(S2n)min χ,ϕ,δwith respect to the parameter A using computer program-ming. Variations of squeezing factor (S2n)min χ,ϕ,δ for ordinary, fourth-order, sixth-order, eighth-order and tenth-order squeezing, i.e., 2n=2, 4, 6, 8 and 10 with the parameter A near the minima are shown inFig. 1. We note that higher-order squeezing increases with increasingthe order 2n of squeezing. Values of (i) Smin, the minimum value of(S2n)min χ,ϕ,δ, and (ii) A2n, the values of A at which Smin occurs areshown in Fig. 2. Variation of values of against the order 2n of squeezingis shown in Fig. 3. We note that A2n decreases with increasing the orderof squeezing, i.e., we get large higher-order squeezing at low intensityof the optical field. Variation of Smin with the order 2n of squeezing isshown in Fig. 4. We note that Smindecreases with increasing the order2n of squeezing. We get any large amount of higher-order squeezingcan be obtained by choosing suitably a large 2n. We also note that weget more than 99% higher-order squeezing for orders greater than 128.

4. Conclusion

In the present paper,we investigatedmaximumhigher-orderHong–Mandel squeezing of the operator Xθ=X1 cosθ+X2 sinθ, in thesuperposed coherent state|ψ⟩=Z1|α⟩+Z2|β⟩, of two coherent states|α⟩ and |β⟩. Here, complex numbers Z1, Z2,α and β are all-arbitrary andthe only restriction on these is normalization condition of thesuperposed state |ψ⟩. We finally concluded using the results of previoussection and Eqs. (5)–(9) that maximum 2nth order squeezing of Xθ inthe superposed coherent state |ψ⟩ occurs for an infinite combinations

with α−β = A2n exp � i12π + iθ

h i, Z2 = Z1 = exp 1

2βα*−β*αð Þ

h iand

with arbitrary values of (α+β) and θ. Here, A2n is a constant whichdepends on the order 2n of squeezing. It has been concluded that anylarge amount of higher-order squeezing can be obtained by choosingsuitably a large2n.Wealsonote thatwegetmore than99%higher-ordersqueezing for the orders greater than 128. Since the action ofdisplacement operator on any state does not affect squeezing but can

changephotonnumbers, it is obvious that this large squeezing can occurat arbitrary large intensities also.

Acknowledgements

We would like to thank Prof. N. Chandra and Prof. R. Prakash forstimulating discussions. One of the authors (RK) is grateful to theUniversity Grants Commission, New Delhi, India for financial support.

References

[1] For excellent review on non-classical phenomena see. D.F. Walls, Nature 306(1983) 141;R. Loudon, P.L. Knight, J. Mod. Opt. 34 (1987) 709;V.V. Dodonov, J. Opt. B 4 (2002) R1.

[2] R.J. Glauber, Phys. Rev. Lett. 10 (1963) 84.[3] E.C.G. Sudarshan, Phys. Rev. Lett. 10 (1963) 277.[4] See, e.g., B. R. Mollow and R. J. Glauber, Phys. Rev. 160, (1967) 1076, 1097

N. Chandra, H. Prakash, Phys. Rev. 1 (1970) 1696;N. Chandra, H. Prakash, Lett. Nuovo. Cim 4 (1970) 1196;N. Chandra and H.Prakash, Indian J. Pure Appl. Phys. 9, (1971) 409, 677, 688; H. Prakash, N. Chandraand Vachaspati, Indian J. Pure Appl. Phys. 13, (1975) 757, 763; H. Prakash, N.Chandra and Vachaspati, Indian J. Pure Appl. Phys. 14, (1976) 41, 48H. Prakash, N. Chandra, Vachaspati, Phys. Rev. A 9 (1974) 2167.

[5] See, e.g.. H.P. Yuen, J.H. Shapiro, IEEE Trans. Inform. Theory IT 24 (1978) 657;J.H. Shapiro, H.P. Yuen, J.A. MachadoMata, IEEE Trans. Inform. Theory IT 25 (1979)179;H.P. Yuen, J.H. Shapiro, IEEE Trans. Inform. Theory T 26 (1980) 78.

[6] See, e.g.. C.H. Bennett, P.W. Shor, J.A. Smolin, A.V. Thapliyal, Phys. Rev. Lett. 83(1999) 3081;B. Schumacher, Phys. Rev. A 54 (1996) 2614;B. Schumacher, M.A. Nielsen, Phys. Rev A 54 (1996) 2629.

[7] See, e.g.. S.L. Braunstein, G.M. D’ Ariano, G.J. Milburn, M.F. Sacchi, Phys. Rev. Lett.84 (2000) 3486;S.L. Braunstein, H.J. Kimble, Phys. Rev. Lett. 80 (1998) 869;C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W.K. Wootters, Phys. Rev.Lett. 70 (1993) 1895.

[8] S.L. Braunstein, H.J. Kimble, Phys. Rev. A 61 (2000) 042302.[9] See, e.g.,. C.H. Bernnett, G. Brassard, N.D. Mermin, Phys. Rev. Lett. 68 (1992) 557;

J. Kempe, Phys. Rev. A 60 (1999) 910.[10] See, e.g.,. J.N. Hollenhorst, Phys. Rev. D 19 (1979) 1669;

C.M. Caves, Phys. Rev. D 23 (1981) 1693.[11] R.J. Glauber, Phys. Rev. 131 (1963) 2766.[12] See, e.g.,. C.K. Hong, L. Mandel, Phys. Rev. A 32 (1985) 974;

C.K. Hong, L. Mandel, Phys. Lett. A 54 (1985) 323.[13] See, e.g.,. R. Lynch, Phys. Rev. A 33 (1986) 4431;

C.K. Hong, L. Mandel, Phys. Rev. A 33 (1986) 4432;R. Jackiw, J. Math. Phys. 9 (1986) 339;R. Lynch, Phys. Rev. A 49 (1994) 2800.

[14] M. Hillery, Phys. Rev. A 36 (1986) 3796.[15] M. Hillery, Phys. Rev. A 40 (1989) 3147.[16] Z.M. Zhang, L. Xu, J.L. Cai, F.L. Li, Phys. Lett. A 150 (1990) 27.[17] H.P. Yuen, J.H. Shapiro, Opt. Lett. 4 (1979) 334;

D.F. Walls, P. Zoller, Phys. Rev. Lett. 47 (1981) 709;G. Milburn, D.F. Walls, Opt. Comm. 39 (1981) 401;W. Becker, M.O. Scully, M.S. Zubairy, Phys. Rev. Lett. 48 (1982) 475;B. Yurke, Phys. Rev. A 29 (1984) 408.

[18] See. e.g.,. L. Mandel, Opt. Commun. 42 (1982) 437;M. Kozierowski, S. Kielich, Phys. Lett. A 94 (1983) 213;P. Meystre, M.S. Zubairy, Phys. Lett. A 89 (1982) 390.

[19] See. e.g.,. F.L. Kien, M. Kozierowski, T. Quang, Phys. Rev. A 38 (1988) 263;L. Xizeng, S. Ying, Phys. Rev. A 40 (1989) 7384;J.J. Gong, P.K. Aravind, Phys. Rev. A 46 (1992) 1586.

[20] See, e.g.,. G.J. Millburn, Phys. Rev. A 33 (1986) 674;C.G. Gerry, Phys. Rev. A 35 (1987) 2146;V. Buzek, Phys. Lett. 136 (1989) 188;C.C. Gerry, S. Rodrigues, Phys. Rev. A 35 (1987) 4440;J. Bajer, A. Miranowicz, R. Tanas, Czech. J. Phys. 52 (2002) 1313;C.C. Gerry, E.R. Vrscay, Phys. Rev. A 37 (1988) 1779;R. Tanas, Phys. Lett. A 141 (1989) 217;H. Prakash, P. Kumar, Int. J. Mod. Phys. B 20 (2006) 1458.

[21] See, e.g.,. M. Hillery, Opt. Commun. 62 (1987) 135;D. Yu, Phys. Rev. A 45 (1992) 2121;M.H. Mahran, A.S.F. Obada, Phys. Rev. A 40 (1989) 4476;H. Prakash, R. Kumar, Int. J. Mod. Phys. B 21 (2006) 3621.

[22] H. Prakash, P. Kumar, J. Opt. B Quantum Semiclass. Opt. 7 (2005) S786;H. Prakash, D.K. Mishra, J. Phys. B 39 (2006) 2291;H. Prakash, D.K. Mishra, J. Phys. B At. Mol. Opt. Phys. 40 (2007) 2531 (c).

[23] See, e.g.,. K. Kim, Phys. Lett. A 245 (1997) 40;D. Erenso, R. Vyas, S. Singh, J. Opt. Soc. Am. B 19 (2002) 1471.

[24] V. Bužek, A. Vidiella-Barranco, P.L. Knight, Phys. Rev. A 45 (1992) 6570.[25] W. Schleich, M. Pernigo, Fam Le Kien, Phys. Phys. Rev. A 44 (1991) 2172.[26] J. Janszky, An.V. Vinogradov, Phys. Rev. Lett. 64 (1990) 2771.

293H. Prakash et al. / Optics Communications 284 (2011) 289–293

[27] X.I.A. Yunjie, G.U.O. Guangcan, Phys. Lett. A 136 (1989) 281.[28] Si.-De. Du, Chang-De Gong, Phys. Lett. A 168 (1992) 296.[29] H. Prakash, P. Kumar, Physica A 319 (2003) 305.H. Prakash and P. Kumar, Optik,

Int. J. Light Electron Opt. (in press), doi:10.1016/j.ijleo.2010.07.004.[30] H. Prakash, P. Kumar, Physica A 341 (2004) 201;

R. Kumar, H. Prakash, Can. J. Phys. 88 (2010) 181.[31] H. Prakash, P. Kumar, Acta Phys. Polon. B 34 (2003) 2769.[32] H. Prakash, P. Kumar, Eur. Phys. J. D 46 (2008) 359.[33] B. Yurke, D. Stoler, Phys. Rev. Lett. 57 (1986) 13;

G.J. Milburn, C.A. Holmes, Phys. Rev. Lett. 56 (1986) 2237;A. Mecozzi, P. Tombesi, Phys. Rev. Lett. 58 (1987) 1055;P. Tombesi, A. Mecozzi, J. Opt. Soc. Am. B 4 (1987) 1700;B.C. Sanders, Phys. Rev. A 39 (1989) 4284;J.J. Slosser, P. Meystre, E.M. Wright, Opt. Lett. 15 (1990) 233.

[34] See, e.g.,. C.M. Caves, B. Yurke, Phys. Rev A 41 (1990) 5261;B. Yurke, W. Schleich, D.F. Walls, Phys. Rev. A 42 (1991) 1703;A. La Porta, R.E. Slusher, B. Yurke, Phys. Rev. Lett. 62 (1989) 26.