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Optik 122 (2011) 1058–1060

Contents lists available at ScienceDirect

Optik

journa l homepage: www.e lsev ier .de / i j leo

on-classical properties of superposition of two coherent statesaving phase difference �

ari Prakasha, Pankaj Kumarb,∗

Department of Physics, University of Allahabad, Allahabad (U.P) 211002, IndiaDepartment of Physics, Bhavan’s Mehta Mahavidyalaya (V. S. Mehta College of Science), Bharwari, Kaushambi (U.P) 212201, India

r t i c l e i n f o

rticle history:eceived 24 March 2010ccepted 4 July 2010

ACS:2.50.Dv

a b s t r a c t

Recently Ahmad et al. [Optik 2009;120;68; Optics Commun. 2007;271:162; Chin. Phys. Lett.2006;23:2438] have studied non-classical properties of superposition of two-coherent states of the form,∣∣ ⟩

= K[|˛〉 + ei�∣∣˛ei�⟩] for the special cases with values � =�/2, 3�/2, and�, and for arbitrarily fixed

values of �. We point out that some of their results are special cases of our recently published work[Physica A 319, 305 (2003); Physica A 341, 201(2004)] on the most general superposition of two arbi-trary coherent states of the form ∼(Z1 |˛〉 + Z2

∣∣ˇ⟩), where X1,2, ˛ and ˇ are arbitrary and only restriction

eywords:on-classical features of lightoherent statequeezing

on these is the normalization condition for the superposed state. To make our point we first obtainresults for (i) squeezing of the most general Hermitian operator X� = X1 cos � + X2 sin �, with X1 + iX2 = a,is the annihilation operator, and (ii) sub-Poissonian photon statistics, for the superposed state

∣∣ ⟩with

a general � and, then obtain results of Ahmad et al. for � =�/2, 3�/2, and� and for � = 0 and�/2. It isinteresting to note that the arbitrarily fixed values � = |˛|2 and −|˛|2 for � =�/2 and 3�/2, respectively by

es at

ub-Poissonian photon statisticsisplacement operatorhase shifting operator

Ahmad et al. are the valu

. Introduction

States of light, properties of which cannot be explained on theasis of classical theory are called non-classical states [1]. The non-lassical nature of a state can be manifested in different ways likentibunching, sub-Poissonian photon statistics and various kindsf squeezing, etc. Earlier study of such non-classical effects wasargely in academic interest [2], but now their applications inuantum information theory such as communication [3], quantumeleportation [4], dense coding [5] and quantum cryptography [6]re well realized. It has been demonstrated that non-classicality ishe necessary input for entangled state [7].

A coherent state [8] defined as the eigenstate of annihilationperator, i.e., a |˛〉 = ˛ |˛〉, does not exhibit non-classical effectsut a superposition of coherent states exhibit [9–16] variouson-classical effects such as squeezing, higher-order squeezing,ub-Poissonian statistics and higher-order sub-Poissonian statis-

ics. Buzek et al. [9] and Xia et al. [12] studied such effectsn the superposition of two-coherent states |˛〉 and |−˛〉 andeported that the even coherent state exhibits squeezing but notub-Poissonian statistics while the odd coherent state exhibits sub-

∗ Corresponding author.E-mail addresses: prakash hari123@rediffmail.com (H. Prakash),

ankaj k25@rediffmail.com (P. Kumar).

030-4026/$ – see front matter © 2010 Elsevier GmbH. All rights reserved.oi:10.1016/j.ijleo.2010.07.004

which we get maximum squeezing working in a rigorous way.© 2010 Elsevier GmbH. All rights reserved.

Poissonian statistics but not squeezing. Xia et al. also studied [12]such effects in the displaced even and odd coherent state. Schleichet al. [10] studied such effects in the superposition of two-coherentstates, |˛ 〉 and |˛∗〉, of identical mean photon number but differ-ent phases and reported that such superposition can exhibit bothsqueezing and sub-Poissonian statistics when |˛|2 � 1. Recently,we studied maximum squeezing in the most general superposedcoherent state [13]. Later, we used this result to study [14] max-imum simultaneous squeezing and sub-Poissonian statistics insuperposed coherent states. We also studied [15] maximum fourth-order squeezing in superposition of two arbitrary coherent states.In practice, the superposition of coherent states can be generatedin interaction of coherent state with nonlinear media [16] and inquantum nondemolition techniques [17].

Recently Ahmad et al. [18–20] studied the non-classical prop-erties of the superposition with equal weights of pairs of coherentstates (|˛〉 and

∣∣i˛⟩), (|˛〉 and∣∣−i˛⟩) and (|˛〉 and |−˛〉). We point

out that the results obtained by Ahmad et al. [18–20] are spe-cial cases of our recently published work [13,14] for the mostgeneral superposition of two-coherent states of the form

∣∣ ′⟩ =(Z1 |˛〉 + Z2

∣∣ˇ⟩), where Z1,2, ˛ and ˇ are arbitrary and only restric-

tion on these is the normalization condition for the superposedcoherent state. To make our point, we consider the more generalsuperposed state,

∣∣ ⟩= K[|˛〉 + ei�

∣∣˛ei�⟩], (1)

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H. Prakash, P. Kumar / O

f two-coherent states |˛〉 and∣∣˛ei�⟩, and study squeezing of the

ost general Hermitian Operator X� defined by

� = X1 cos � + X2 sin �, (2)

nd sub-Poissonian photon statistics in the superposed coherenttate

∣∣ ⟩using recent results [13,14]. Here Hermitian operators

1,2 are defined by X1 + iX2 = a, the annihilation operator, and com-lex number ˛ = |˛| ei�˛ , angles � and � are completely arbitraryith the only restriction being the normalization condition for

∣∣ ⟩.

e show that the results of Ahmad et al. are special cases � =�/2,�/2, and�. Also we show that the arbitrarily fixed values � = |˛|2nd −|˛|2 for � =�/2 and 3�/2, respectively by Ahmad et al. are,s a matter of fact, the values obtained for maximum squeezing,orking in a rigorous way.

. Squeezing in superposed coherent state

Using the Hesenberg’s uncertainty relation, the conditionor squeezing of X� in the any state

∣∣ ⟩can be defined as

∣∣ (�X�)

2∣∣ ⟩

< 1/4, where�X� = X� −⟨

∣∣X�∣∣ ⟩

. Recently wetudied [14] squeezing of the hermitian operator X� in the mosteneral superposition state

∣∣ ′⟩ = Z1 |˛〉 + Z2

∣∣ˇ⟩. Using the facthat displacement and phase shift does not change the amount ofqueezing in any state we related

∣∣ ′⟩ to superposition of coher-

nt states∣∣∣˛+ˇ

2

⟩and

∣∣∣−˛+ˇ2

⟩and showed [14] that the maximum

queezing in the superposed state∣∣ ′⟩ with the absolute mini-

um value 0.11077 of variance⟨ ′∣∣ (�X�)

2∣∣ ′⟩ occurs for infinite

umbers of combinations with

− ˇ = 1.59912 exp[± i(�

2

)+ i�],

Z1

Z2= exp

[12

(ˇ˛∗ − ˇ∗˛)], (3)

nd with arbitrary (˛+ˇ) and �.For the state

∣∣ ⟩, we can obtain results for maximum squeezing

y using Eq. (3). For˛ = |˛| ei�˛ , this gives maximum squeezing withhe absolute minimum value 0.11077 of variance

⟨

∣∣ (�X�)2∣∣ ⟩

ccurring for infinite numbers of combinations with

|˛| sin�

2= 1.59912; � = −|˛|2 sin�, (4)

nd � = �˛ +�/2. We can obtain the results of Ahmad et al. forqueezing by putting the values � =�/2, 3�/2, and� which gives˛| = 1.1307, 1.1307, 0.7996 and � = −|˛|2, |˛|2 and 0, respectively.his gives a rigorous derivation of values |˛|2 and –|˛|2 for � for theases � =�/2 and 3�/2, respectively, which have been fixed arbi-rary by Ahmad et al.

. Simultaneous squeezing and sub-Poissonian photontatistics in superposed coherent state

For characterization of the photon statistics of opticaleld in the state

∣∣ ⟩, Mandel [21] introduced a parameter

ased on intensity fluctuations of the field defined by Q =⟨

∣∣ (�N)2∣∣ ⟩

/⟨

∣∣N ∣∣ ⟩] − 1, where �N = N −

⟨

∣∣N ∣∣ ⟩and

+

= a a. When 0 < Q ≤ − 1, the photon statistics is called sub-oissonian and the field is called antibunched. Recently we studied13] simultaneous occurrence of antibunching and squeezing of theermitian operator X� in the superposed state

∣∣ ′⟩ of two-coherent

tates |˛〉 and∣∣ˇ⟩. We investigated the connection between squeez-

[

22 (2011) 1058–1060 1059

ing and antibunching in superposed state∣∣ ′⟩ and found a relation

between Mandel’s Q parameter and variance⟨ ′∣∣ (�X�)

2∣∣ ′⟩,

Q ∼= 4[⟨ ′′∣∣ (�X�)

2∣∣ ′′⟩ − 1

4

]. (10)

for∣∣ ′⟩ = D

[(˛+ ˇ)/2

] ∣∣ ′′⟩ and the condition∣∣˛+ ˇ

∣∣ �∣∣˛− ˇ

∣∣.Using the results [13], i.e., Eq. (3) we find that the state

∣∣ ′⟩exhibits maximum simultaneous squeezing and antibunching withminimum value 0.11077 of

⟨ ′∣∣ (�X�)

2∣∣ ′⟩ and minimum value

−0.55692 of Mandel’s Q parameter for an infinite number ofconditions as mentioned in Eq. (3) and � = arg(˛+ˇ), for thecase

∣∣˛+ ˇ∣∣ �

∣∣˛− ˇ∣∣. Using these results we report that the

superposed coherent state∣∣ ⟩

exhibits maximum simultane-ous squeezing and antibunching with minimum value 0.11077of

⟨

∣∣ (�X�)2∣∣ ⟩

and minimum value −0.55692 of Mandel’s Qparameter for an infinite combination with 2 |˛| sin�/2 = 1.59912,� = �˛ +�/2, � = −|˛|2 sin� under the consideration |˛| � 1.

4. Conclusion

In this paper we consider a more general superposition ofcoherent states,

∣∣ ⟩= K[|˛〉 + ei�

∣∣˛ei�⟩]; ˛ = |˛| ei�˛ and stud-ied photon statistics and ordinary squeezing of the most generalHermitian operator X� defined by X� = X1 cos � + X2 sin �, in thissuperposed coherent state

∣∣ ⟩using our recently reported results

[13,14]. We conclude that maximum squeezing of X� in the state∣∣ ⟩occurs with the absolute minimum value 0.11077 of vari-

ance⟨

∣∣ (�X�)2∣∣ ⟩

for infinite numbers of combinations with,

2 |˛| sin�/2 = 1.59912, � = �˛ +�/2 and � = −|˛|2 sin�. Using theseresults and the properties of displacement operator we alsoconclude that this superposed coherent states also exhibits sub-Poissonian photon statistics for infinite numbers of combinationswith, 2 |˛| sin �/2 = 1.59912, � = �˛ +�/2 and � = −|˛|2 sin�when|˛| � 1. The results of Ahmad et al. [18–20] are special cases(� =�/2, 3�/2 and �) of this general study.

Acknowledgement

We would like to thank Professors N. Chandra and R. Prakash fortheir interest and some critical comments.

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