Nonclassicality of New Class of States Produced by Superposition of Two Nonlinear Squeezed States with Respective Phase φ

REPORTS ON MATHEMATICAL PHYSICS

Vol. 75 (2015)

No. 2

NONCLASSICALITY OF NEW CLASS OF STATES PRODUCED BY SUPERPOSITION OF TWO NONLINEAR SQUEEZED STATES WITH RESPECTIVE PHASE ϕ S. AL. M ODARES VAMEGH1,2 and M. K. TAVASSOLY1,3 1 Atomic

and Molecular Group, Faculty of Physics, Yazd University, Yazd, Iran 2 Administration of Education of Fars Province, Shiraz, Iran 3 The Laboratory of Quantum Information Processing, Yazd University, Yazd, Iran (e-mail: [email protected]) (Received September 24, 2013 – Revised August 18, 2014) In this paper, we consider two classes of nonlinear squeezed states: squeezed vacuum and squeezed one-photon states, which according to their generation processes, essentially include respectively even and odd bases of Fock space. Then, we produce their general superposition with respective phase ϕ which consists of all bases of Fock space. As a physical realization of the presented approach, we will use the q-deformation nonlinearity function to produce the corresponding nonlinear squeezed states and their superposition. In the continuation, due to extensive interest in nonclassicality features, we discuss about their main nonclassical properties such as sub-Poissonian statistics, normal and amplitude squared squeezing, number and phase squeezing and Husimi quasi-probability function of the obtained states and compare with those of the original constituent components. Meanwhile, we also pay attention to linear squeezed states (vacuum and one-photon) and particularly their superposition, too. Accordingly, the nonclassicality features of the introduced states are evidently established. At last, simple schemes for the generation of such states are presented. Keywords:nonlinear squeezed state, superposition of nonlinear squeezed states, nonclassical properties. PACS: 42.50.Dv, 42.50.-p

1.

Introduction

In recent years, nonclassical states of light have attracted much attention in many areas of physics, in particular, in quantum optics [1–4], quantum information processing [5], quantum communication [6], quantum computation [7], quantum cryptography [8, 9] and quantum teleportation [10, 11]. Many of these states were produced and detected in laboratories [12–15]. Also, these states have been proposed as good candidates for accurate measurements like detecting gravitational waves of weak tidal forces [16]. Squeezed states are one of the most important classes of nonclassical states [17]. Different generalizations are given to squeezed states [18–21]. In this paper, we will deal with a particular generalization of squeezed states, known as nonlinear squeezed states. Indeed, we will proceed with [149]

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the superposition of two particular classes of these states and investigate their nonclassical properties. On the other hand, the importance of superposition of quantum states is a clear subject in quantum mechanics generally and specifically in quantum optics [23, 24]. The role of this phenomenon is easily characterized, especially in the construction of standard coherent state |α, i.e. the state which is composed from special superposition of eigenstates of harmonic oscillator {|n}∞ n=0 . In this example, while the number of states which sometimes are known as the most nonclassical quantum states (due to the fact that their Mandel QM parameter [25] is equal to −1), i.e. their quantum statistics is highly sub-Poissonian, standard coherent state destroys all of nonclassical properties. But, the latter mentioned outcome is not always correct. Occasionally, unlike the above example, quantum superposition may reveal some nonclassical features for the superposed states. As is well known, the most elementary superposition in the quantum optics field is symmetric and anti-symmetric superposition of standard coherent states (|α ± | − α), the so-called even and odd coherent states which, each of them, possesse a few nonclassical properties [26]. Since then, superposition of nonlinear coherent states [27], superposition of coherent states with arbitrary phase [28], superposition of nonlinear coherent states with phase π/2 [23], superposition of the dual family of coherent states [24] etc. have been introduced. In part of the mentioned literature, some classes of states have been obtained which include only even or odd bases of Fock space. In contrast, in this article, our purpose is to produce a superposition of two “nonlinear squeezed states” which according to their production processes essentially include only even and odd bases of Fock space. In fact, we want to produce superposition of these even and odd states which will consist of all bases of Fock space (of course with certain coefficients). As is clear, this perspective is unlike the common list of the above superpositions. By this approach, we have reached several classes of superposed states with distinguishable nonclassicality signs, depending on the chosen nonlinearity functions. In the continuation, after choosing a particular nonlinearity function, we investigate some of the nonclassical features of the resultant state and compare those with the properties of primary components. To show that the introduced states are not far to be produced, simple schemes for the generation of such states are presented. To the best of our knowledge, this work has not been done even for the (linear) squeezed states, too. Henceforth, we added a discussion in this respect at the end of the paper. 2.

Introducing superposition of two distinct classes of nonlinear squeezed states with respective phase ϕ One of the ways for the construction of nonlinear squeezed states is using the ˆ C operator which is defined as [18, 21]  − 1 Cˆ = 1 − |ξ |2 2 (Aˆ − ξ Bˆ † ), (1)

where

ξ=

z tanh |z| = eiϕ tanh r |z|

NONCLASSICALITY OF NEW CLASS OF STATES PRODUCED BY SUPERPOSITION. . .

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ˆ Aˆ † are defined as and A, Aˆ = af ˆ (n), ˆ

ˆ aˆ † , Aˆ † = f † (n)

(2)

with a( ˆ aˆ † ) as the bosonic annihilation (creation) operator. Also, auxiliary operators ˆ B and Bˆ † are defined respectively as the conjugate operators of Aˆ and Aˆ † , Bˆ = aˆ

1 f † (n) ˆ

,

Bˆ † =

1 † aˆ . f (n) ˆ

(3)

Interestingly, these f -deformed operators also satisfy the canonical communication relation ˆ Aˆ † ] = Iˆ, ˆ Bˆ † ] = Iˆ, [B, (4) [A, where Iˆ is the unity operator. In the above relations f (n) is an operator-valued ˆ The nonlinear squeezed vacuum states which include only the function of nˆ = aˆ † a. ˆ 1 = 0 even bases of Fock space were introduced with the eigenvalue equation C|ψ with the explicit form ∞ √  (2n)! ξ n |2n ≡ |ξ, f even , (5) |ψ1  = Nξ,f 2n n! f (2n)! n=0 where the normalization coefficient Nξ,f is determined as −1/2  ∞ (2n)! |ξ |2n . Nξ,f = 22n (n!)2 f (2n)! 2 n=0

(6)

Also, nonlinear squeezed one-photon states which only consist of odd bases of Fock ˆ space have been defined by the eigenvalue equation of the square of C-operator as 2 Cˆ |ψ2  = 0 with the explicit form ∞ √  (2n + 1)! ξ n  |2n + 1 ≡ |ξ, f odd , (7) |ψ2  = Nξ,f 2n n! f (2n + 1)! n=0  where the normalization coefficient Nξ,f is determined as −1/2  ∞ (2n + 1)! |ξ |2n  Nξ,f = . 22n (n!)2 f (2n + 1)! 2 n=0

(8)

Notice that by definition f (n)! ˆ =f ˙ (n)f ˆ (nˆ − 1) . . . f (1) with f (0)!=1. ˙ In addition, the dual of nonlinear squeezed states (5) and (7) are constructed [18–21] which are out of the scope of our work. Now, following the main purpose of the paper, we superpose the nonlinear squeezed vacuum (5) and nonlinear squeezed one-photon states (7) with respective phase ϕ in a general regime as |ξ, f sup = Nξ,f,s (|ξ, f even + eiϕ |ξ, f odd ).

(9)

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Equivalently, one can write these states as  ∞ √  (2n)! ξ n |2n |ξ, f sup = Nξ,f,s Nξ,f 2n n! f (2n)! n=0  ∞ √  (2n + 1)! ξ n iϕ  + e Nξ,f |2n + 1 , 2n n! f (2n + 1)! n=0 where the normalization constant Nξ,f,s reads as  ∞  (2n)! |ξ |2n 2 Nξ,f,s = Nξ,f 22n (n!)2 f (2n)! 2 n=0 2

+ Nξ,f

∞  n=0

(2n + 1)! |ξ |2n 22n (n!)2 f (2n + 1)! 2

(10)

−1/2 ,

(11)

which is independent of ϕ. We can rewrite the superposed states (10) in the new form 

∞     n iϕ  − n (Nξ,f − e Nξ,f ) + Nξ,f 2 |ξ, f sup = Nξ,f,s 2 n=0 √ n n! ξ [ 2 ] |n (12) × [n] n 2 2 [ 2 ]! f (n)! and accordingly    ∞    n iϕ  Nξ,f,s = − n (Nξ,f − e Nξ,f ) + Nξ,f 2 2 n=0      n  × 2 − n (Nξ,f − e−iϕ Nξ,f ) + Nξ,f 2

−1/2 n n! |ξ |2[ 2 ] . (13) × 2[ n ] n 2 2 [ 2 ]! 2 f (n)! 2 It is worth mentioning that, although the phase ϕ remains in (13), it will disappear after further simplification. In Eqs. (12) and (13), [x] represents the integer part of the real number x. Inserting f (n) ˆ = 1 in (10) we can readily obtain the superposition of (linear) vacuum and one-photon squeezed states as  ∞ √  1 (2n)! ξ n 2 1/4 |ξ sup = √ (1 − |ξ | ) |2n 2n n! 2 n=0  ∞ √  (2n + 1)! ξ n iϕ 2 3/4 + e (1 − |ξ | ) |2n + 1 . (14) 2n n! n=0 n

Notice that we can easily replace (−1)2 −1 = 2[ n2 ] − n in Eqs. (12) and (13). One can clearly observe that superposition of the states in (12) and (14) include all basis

NONCLASSICALITY OF NEW CLASS OF STATES PRODUCED BY SUPERPOSITION. . .

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elements of Hilbert space. In addition, the introduced superposed state does not stand in neither of classes of primary squeezed states. This means that we cannot attribute a deformation function for this state or reproduce it from any eigenvalue equation of a deformed annihilation operator. Indeed, the resultant state is more complicated to be categorized in f -deformed (nonlinear) coherent states or squeezed states. We end this section with mentioning that, while the state (14) is unique, as is clear from the state (12), choosing different nonlinearity functions in (10) leads to different superposed states. Various nonlinearity functions may be found in literature. As some examples we may refer to nonlinearity functions corresponding to harmonious states [29], center of mass motion of trapped ion [30–32], Penson–Solomon coherent states [33], SU(1, 1) Barut–Girardello and Gilmore–Perelomov coherent states [34, 35] and so on. Therefore, we can apply the presented approach for every nonlinearity function f (n) corresponding to any class of nonlinear physical system or nonlinear coherent states and obtain their relevant superposition of associated squeezed states. Henceforth, a vast class of quantum states may be produced. Altogether, as a physical appearance of our formalism, in the continuation of the paper, we will work with the well-known nonlinearity has been called as q-deformation [36]  q n − q −n fq (n) = , (15) n(q − q −1 ) where q ∈ R. Setting the chosen function in the states (5), (7) and (12) yields respectively the explicit forms of nonlinear squeezed vacuum, nonlinear squeezed one-photon and the superposed state associated with such a nonlinear physical system. In the continuation, we will use these particular states for our further numerical calculations and investigate their nonclassical properties. 3.

Nonclassical properties of q-deformed superposed squeezed states In this section we want to investigate and compare some of the nonclassical properties of the obtained superposition state (12) with those for the initial components (5) and (7). In order to do this we consider some criteria such as Mandel parameter, quadrature squeezing, amplitude squared squeezing, number and phase squeezing and Husimi Q-function. Each criterion is sufficient, but is not a necessary condition for a state to behave as nonclassical state. For the determination of nonclassical criteria of states (5), (7) and (12) we need the expectation values of different powers of the operators aˆ † , a. ˆ In this respect, the following relations are useful for our further calculations: ∞  p−q †q p 2  a ˆ a ˆ  = N (2n)! (2n + p − q)! |ξ |2n ξ 2 even even ξ,f 

n=◦

  p−q n! n + ! (2n − q)! 2 −1 × f (2n)! f (2n + p − q)! , × 22n+

p−q 2

(16)

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ˆ odd  a



2 aˆ odd = Nξ,f

†q p

∞ 

(2n + 1)! (2n + p − q + 1)! |ξ |2n ξ

p−q 2

n=◦

  p−q ! (2n − q + 1)! n! n + 2 −1 × f (2n + 1)! f (2n + p − q + 1)! , 

× 22n+

p−q 2



∞     n −iϕ  − n (Nξ,f − e Nξ,f ) + Nξ,f 2 ˆ aˆ sup = sup  a 2 n=◦     n+p−q − (n + p − q) × 2 2

n+p−q n iϕ  × (Nξ,f − e Nξ,f ) + Nξ,f ξ ∗[ 2 ] ξ [ 2 ]      n+p−q n n+p−q n × n!(n + p − q)! 2[ 2 ]+[ 2 ] ! ! 2 2 −1 × (n − q)!f (n)! f (n + p − q)! . †q p

(17)

2 Nξ,f,s

(18)

It is noticeable that Eqs. (16) and (17) are true with the constraint that p − q gets even values, otherwise the expectation values of the operators aˆ † aˆ are zero. Also notice that the above general relations are useful for the necessary expectation values of linear squeezed states which will be introduced and discussed in Appendix, simply by setting f (n) = 1. 3.1.

Mandel parameter

One of the criterion for expressing the nonclassical effects is Mandel parameter which gives information about photon statistics distribution of quantum states. This parameter has been defined as [25] QM =

nˆ 2  − n ˆ 2 − 1. n ˆ

(19)

The values QM > 0, QM = 0 and QM < 0 correspond to super-Poissonian (classical), Poissonian (standard coherent state) and sub-Poissonian (nonclassical) statistics, respectively. In Fig. 1 we have plotted the Mandel parameter for a superposed state, in addition to its constituent components. We realize that |ξ, f even has a superPoissonian (classical) behaviour while |ξ, f odd and |ξ, f sup show sub-Poissonian (nonclassical) features. The leakage of this nonclassicality sign from the odd states to the superposed states is interesting.

NONCLASSICALITY OF NEW CLASS OF STATES PRODUCED BY SUPERPOSITION. . .

155

1.0 Ξ, f even Ξ, f odd

QM

0.5

Ξ, f s 0.0

0.5

1.0

0.5

0.0

0.5

Ξ

Fig. 1. Mandel parameter in terms of ξ with fixed value q = 0.5 for the q-deformed squeezed states.

3.2.

Squeezing properties

Squeezing property occurs when fluctuations in one quadrature of field fall below the vacuum or coherent state with preserving the validity of Heisenberg uncertainty relation [17]. In the following, we proceed briefly with the normal (first-order) and amplitude squared squeezing. √ • Normal squeezing: We consider two Hermitian operators as xˆ = (a+ ˆ aˆ † )/ 2 and √ pˆ = (a− ˆ aˆ † )/i 2. Normal squeezing parameters are defined as sr = ((ˆr )2 −0.5)/0.5 where (ˆri )2  = ˆri2  − ˆri 2 , rˆi = xˆ or p. ˆ In detail, the x-squeezing parameter reads as sx = 2aˆ † a ˆ + aˆ 2  + aˆ †2  − a ˆ 2 − aˆ † 2 − 2a ˆ aˆ † ,

(20)

with a similar definition for p as sp = 2aˆ † a ˆ − aˆ 2  − aˆ †2  + a ˆ 2 + aˆ † 2 − 2a ˆ aˆ † .

(21)

Normal squeezing happens in x or p when the inequalities −1 < sx < 0 or −1 < sp < 0 occur, respectively. In Fig. 2 (Fig. 3) we have plotted squeezing in x (p) for considered states and their superposition with respective phases ϕ = 0, π/2, π, 3π/2, respectively. It is seen that |ξ, f even and |ξ, f sup for all of considered values of phase factor are squeezed in x and p components (of course in different regions). It should be noted that in Figs. 2 and 3 the superposition states |ξ, f sup corresponding to coincide with each other. This nonclassicality sign streams ϕ = 0, π and ϕ = π2 , 3π 2 from even states to the supposed states. • Amplitude squared squeezing: We define two Hermitian operators as [37, 38] 2 2 follows Xˆ = (aˆ 2 + aˆ † )/2 and Pˆ = (aˆ 2 − aˆ † )/2i with the commutation relation ˆ Pˆ ] = i(2nˆ + 1). The associated squeezing parameters may be defined as SR = [X, ˆ 2 /nˆ + 1  − 1 where Rˆ = Xˆ or Pˆ . The squeezing parameter for X can be (R) 2

156

S. AL. MODARES VAMEGH and M. K. TAVASSOLY Ξ, f even 6 Ξ, f odd Ξ, f s ,  = 0 4

Ξ, f s ,  =

Π 2

sx

Ξ, f s ,  = Π 2

Ξ, f s , =

3Π 2

0 1.0

0.5

0.0

0.5

1.0

Ξ

Fig. 2. Normal ordered squeezing in x in terms of ξ with fixed value q = 0.5 for the q-deformed squeezed states. Ξ, f even Ξ, f odd

6

Ξ, f s ,  = 0 Ξ, f s ,  =

4

Π 2

sp

Ξ, f s ,  = Π Ξ, f s , =

3Π 2

2

0 1.0

0.5

0.0

0.5

1.0

Ξ

Fig. 3. Normal ordered squeezing in p in terms of ξ with fixed value q = 0.5 for the q-deformed squeezed states. 0.1

0.0

SX

Ξ, f even Ξ, f  odd

0.1

Ξ, f  s 0.2

0.3

4

2

0

2

4

Ξ

Fig. 4. Amplitude squared squeezing in X in terms of ξ with fixed value q = 0.5 for the q-deformed squeezed states.

NONCLASSICALITY OF NEW CLASS OF STATES PRODUCED BY SUPERPOSITION. . .

157

1.0 0.8

SP

0.6 0.4 Ξ, f even 0.2 0.0

Ξ, f  odd Ξ, f  s 4

2

0

2

4

Ξ

Fig. 5. Amplitude squared squeezing in P in terms of ξ with fixed value q = 0.5 for the q-deformed squeezed states.

deduced as SX =

1 4aˆ † a ˆ

+2

 4  aˆ  + aˆ †4  + 2aˆ †2 aˆ 2  − aˆ 2 2 − aˆ †2 2 − 2aˆ 2 aˆ †2  ,

and similarly for P as   1 − aˆ 4  − aˆ †4  + 2aˆ †2 aˆ 2  + aˆ 2 2 + aˆ †2 2 − 2aˆ 2 aˆ †2  . SP = † 4aˆ a ˆ +2

(22)

(23)

The amplitude squared squeezing occurs in Xˆ or Pˆ respectively if −1 < SX < 0 or −1 < SP < 0. We have plotted the amplitude squared squeezing for Xˆ and Pˆ in Figs. 4 and 5, ˆ respectively. As it can be seen |ξ, f even , |ξ, f odd and |ξ, f sup are squeezed in X, however in Pˆ only |ξ, f even is squeezed in a finite range. While the amplitude squared squeezing is observed in Xˆ component, this does not occur for Pˆ component. 3.3.

Quantum phase properties

Quantum phase is generally an important topic in physics, especially in the quantum optics area. It attracted a great deal of attention for a long time, either theoretically [39–49] or experimentally [50–52]. In these lines, searching for a Hermitian phase operator of (single-mode) radiation field started from the beginning of quantum electrodynamics. There exist several approaches to the notion of quantum phase. It seems that the initial attempts for the construction of the explicit form of a quantum phase operator (as a conjugate operator of the number operator) were made by Dirac [39]. Then, Susskind and Glogower [40] proposed an exponential phase operator based on the idea od Dirac, with an extra condition related to vacuum. Unfortunately, although the latter formalism permits a definition for Hermitian operators, it is nonunitary which is indeed a serious problem. In another

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attempt Holevo introduced a new approach [41]. In this approach the mathematical representation of the concept of quantum observable is extended from a self-adjoint operator to a normalized positive operator measure (POM) and it is assumed that any quantum phase observable is a phase shift covariant POM in the interval [0, 2π) as the range of its possible measurements. Later Barnett and Pegg [42–45] as well as Pegg and Barnett [46–48] in a series of papers first introduced a unitary operator in an extended Hilbert space. Their first proposal was problematic due to the fact that it included unphysical negative number states. Along their attempts for completing this concept, a very important development in this field has been made by them. They have defined a unitary and Hermitian phase operator and a phase state but in a finite (although arbitrarily large) subspace, whose dimension is allowed to tend to infinity after the calculation of expectation values of observable quantities and moments. The above three formalisms have been used along each other in various studies of phase properties and the phase fluctuations of different physical systems. A deep investigation of the three suggestions mentioned above shows that neither of them is free from difficulties and therefore the exact description of phase operator is still an open problem. Altogether, in the contents of quantum optics, the Pegg–Barnett [46–48] formalism has been extensively employed in the study of the phase properties of a wide variety of quantum systems in recent literature (as a few examples see the Refs. in [49]). Accordingly, in this subsection we investigate the phase distribution and numberphase squeezing following the Pegg–Barnett formalism. According to this approach a complete set of s + 1 orthonormal phase states |θm  is defined as |θm  = √

s 

1 s+1

exp(inθm )|n,

m = 0, 1, . . . , s,

(24)

n=0

m where θm = θ0 + 2π and θ0 gets an arbitrary value. The Pegg–Barnett Hermitian s+1 phase operator is defined on the basis of phase states (24) as

ˆθ =

s 

θm |θm θm |.

(25)

m=0

Of course, in this approach the upper bound will tend to infinity after calculating the expectation values. Due to θm |θm  = δm,m the k-th power of the Pegg–Barnett phase operator can be written as ˆ kθ =

s 

θmk |θm θm |.

(26)

m=0

So, with the help of the above relation we can write ˆ kθ |ψ = ψ|

s  m=0

θmk |θm |ψ|2 .

(27)

NONCLASSICALITY OF NEW CLASS OF STATES PRODUCED BY SUPERPOSITION. . .

If the upper bound s tends to infinity, we will have θ0 +2π k ˆ θ |ψ = dθ θ k P (θ ), ψ|

159

(28)

θ0

in which the continuous-phase distribution P (θ ) is introduced as s+1 P (θ ) = lim (29) |θm |ψ|2 , s→∞ 2π is the density of states. After some straightforward but lengthy calculations, and s+1 2π the Pegg–Barnett phase probability distribution for the states (5), (7) and (12) can be evaluated respectively as follows:  1 2 Peven (θ) = 1 + 2Nξ,f 2π √   (2n)! (2m)! ξ n ξ ∗m cos[2(n − m)θ ]  , (30) × 2n+m n! m! f (2n)! f (2m)! n n>m  1  1 + 2Nξ,f2 Podd (θ) = 2π √   (2n + 1)! (2m + 1)! ξ n ξ ∗m cos[2(n − m)θ ]  , (31) × 2n+m n! m! f (2n + 1)! f (2m + 1)! n n>m        m  1 2 −iϕ  1 + 2Nξ,f,s − m (Nξ,f − e Nξ,f ) + Nξ,f 2 Psup (θ) = 2π 2 n n>m      n iϕ  − n (Nξ,f − e Nξ,f ) + Nξ,f × 2 2 √  n m n! m! ξ [ 2 ] ξ ∗[ 2 ] cos[(n − m)θ] . (32) × [ n ]+[ m ] n m 2 2 2 [ 2 ]! [ 2 ]! f (n)! f (m)! By setting f (n) = 1 in (30), (31) and (32) one can readily obtain the phase distribution for (linear) squeezed vacuum, squeezed one-photon and their superposition. We have plotted Pegg–Barnett phase probability distribution for states (5), (7) and (12) with different relative phases in each of the Figs. 6–9. According to Fig. 6 we see that the variation of phase distribution for states |ξ, f even , |ξ, f odd and |ξ, f sup with ϕ = 0 are nearly similar to each other and all of them have a central peak at θ = 0. Also, the peak of |ξ, f sup is higher than the others. From Fig. 7 we see that for ϕ = π2 the superposed state |ξ, f sup has two peaks, both of them do not occur at θ = 0, while the states |ξ, f even , |ξ, f odd have a central peak at θ = 0. In Fig. 8 we see that the state |ξ, f sup with ϕ = π has two peaks in θ = ±π, while the states |ξ, f even , |ξ, f odd have a central peak at θ = 0. According to Fig. 8 we see that for ϕ = 3π/2 the state |ξ, f sup has two peaks, both of them do not occur at θ = 0, while the states |ξ, f even , |ξ, f odd have a central peak at θ = 0.

160

S. AL. MODARES VAMEGH and M. K. TAVASSOLY Ξ, f even 0.5 Phase_distribution

Ξ, f odd 0.4 Ξ, f s 0.3 0.2 0.1 0.0 3

2

1

0

1

2

3

Θ

Fig. 6. Phase probability distribution in terms of θ with fixed values ξ = 0.5 and q = 0.5 for the q-deformed squeezed states with phase as ϕ = 0. 0.35

Ξ, f even

Phase_distribution

0.30 Ξ, f odd

0.25

Ξ, f s

0.20 0.15 0.10 0.05 0.00 3

2

1

0

1

2

3

Θ

Fig. 7. Phase probability distribution in terms of θ with fixed values ξ = 0.5 and q = 0.5 for the q-deformed squeezed states with phase as ϕ = π/2. Ξ, f even 0.5 Phase_distribution

Ξ, f odd 0.4 Ξ, f s 0.3 0.2 0.1 0.0 3

2

1

0

1

2

3

Θ

Fig. 8. Phase probability distribution in terms of θ with fixed values ξ = 0.5 and q = 0.5 for the q-deformed squeezed states with phase as ϕ = π .

NONCLASSICALITY OF NEW CLASS OF STATES PRODUCED BY SUPERPOSITION. . . 0.35

161

Ξ, f even

Phase_distribution

0.30 Ξ, f odd

0.25

Ξ, f s

0.20 0.15 0.10 0.05 0.00 3

2

1

0

1

2

3

Θ

Fig. 9. Phase probability distribution in terms of θ with fixed values ξ = 0.5 and q = 0.5 for the q-deformed squeezed states with phase as ϕ = 3π/2.

• Number and phase squeezing: In order to study the number and phase ˆ θ which satisfy the uncertainty squeezing, we consider two conjugate operators nˆ and relation 1 ˆ θ )2 (n) ˆ θ ]|2 , ( (33) ˆ ˆ 2  ≥ |[n, 4 ˆ θ ] = i(1−2π P (θ0 )). If we define the number and phase squeezing paramewhere [n, ˆ f ˆ θ ]|−1 and S f = 2( ˆθ )2 /|[n, ˆ θ ]|−1, ˆ 2 /|[n, ˆ ˆ ters respectively as Sn = 2(n) f f when we have Sn < 0 (S < 0) the associated state is number (phase) squeezed. The number and phase variances with the help of relation (28) can be obtained as (n) ˆ 2  = (aˆ † a) ˆ 2  − aˆ † a ˆ 2, and ˆ θ)  = ( 2



θ0 +2π

θ0

respectively. distributions states. After (5), (7) and

 θ P (θ )dθ − 2

(34) 2

θ0 +2π

θP (θ)dθ

,

(35)

θ0

Eq. (35) is a general relation which may be used for the phase of the states (30), (31) and (32) corresponding to nonlinear squeezed a lengthy but straightforward calculations, the phase variances of states (12) are derived as follows: √ 2 (2n)! (2m)! ξ n ξ ∗m π 2 Nξ,f   2 ˆ ( θ ) even = + 3 2π n n>m 2n+m n! m! f (2n)! f (2m)!  × 2π(m − n) cos[2(m − n)π ]  (36) + (−1 + 2π 2 (m − n)2 ) sin[2(m − n)π] (m − n)−3 ,

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√ 2 π 2 Nξ,f   (2n + 1)! (2m + 1)! ξ n ξ ∗m + 3 2π n n>m 2n+m n! m! f (2n + 1)! f (2m + 1)!  × 2π(m − n) cos[2(m − n)π]  (37) + (−1 + 2π 2 (m − n)2 ) sin[2(m − n)π ] (m − n)−3 ,      2 m π 2 Nξ,f,s    ˆ θ )2 sup = + − m (Nξ,f − e−iϕ Nξ,f 2 ) + Nξ,f ( 3 π 2 n n>m     n  − n (Nξ,f − eiϕ Nξ,f ) + Nξ,f ) × 2 2 √ n m n! m! ξ [ 2 ] ξ ∗[ 2 ] × [ n ]+[ m ] n m 2 2 2 [ 2 ]! [ 2 ]! f (n)! f (m)!  × 4π(m − n) cos[(m − n)π ]  (38) + 2(−2 + π 2 (m − n)2 ) sin[(m − n)π ] (m − n)−3 .

ˆ θ )2  = ( odd

It is worth noticing that by setting f (n) = 1 in (36), (37) and (38) one simply obtains the phase uncertainty relations for the (linear) squeezed states and their superpositions. In Figs. 10–15 we have plotted the number and phase squeezing for |ξ, f even , |ξ, f odd and |ξ, f sup with different chosen phases. From Figs. 10 and 11 we realize that |ξ, f even and |ξ, f odd are squeezed only in number, while both of them do not possess squeezing in phase. Based on Figs. 12–15 we see that |ξ, f sup for ϕ = 0 is squeezed in both the number and phase in different ranges, for ϕ = π, |ξ, f sup is squeezed only in the number and for ϕ = π/2, 3π/2 is not squeezed in the number and phase. It is clearly seen that the interference effect of the superposition states generates the squeezing in the number and phase operator

S nf

Squeezing  Parameters

4

S f

3 2 1 0 1 0

2

4

6

8

10

Ξ

Fig. 10. Phase and number squeezing in terms of ξ with fixed value q = 0.5 for | ξ, f even .

NONCLASSICALITY OF NEW CLASS OF STATES PRODUCED BY SUPERPOSITION. . .

163

S nf

4 Squeezing  Parameters

S f 3 2 1 0 1 0

2

4

6

8

10

Ξ

Fig. 11. Phase and number squeezing in terms of ξ with fixed value q = 0.5 for | ξ, f odd . 4

Squeezing  Parameters

S nf S f

3

2

1

0

0

2

4

6

8

10

Ξ

Fig. 12. Phase and number squeezing in terms of ξ with fixed value q = 0.5 for | ξ, f sup with phase ϕ = 0. 5

Squeezing  Parameters

S nf S f

4

3

2

1 0

2

4

6

8

10

Ξ

Fig. 13. Phase and number squeezing in terms of ξ with fixed value q = 0.5 for | ξ, f sup with phase ϕ = π/2.

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S. AL. MODARES VAMEGH and M. K. TAVASSOLY

S nf

Squeezing  Parameters

4

S f

3 2 1 0 0

2

4

6

8

10

Ξ

Fig. 14. Phase and number squeezing in terms of ξ with fixed value q = 0.5 for | ξ, f sup with phase ϕ = π . 5

Squeezing  Parameters

S nf S f

4

3

2

1 0

2

4

6

8

10

Ξ

Fig. 15. Phase and number squeezing in terms of ξ with fixed value q = 0.5 for | ξ, f sup with phase ϕ = 3π/2.

for particular relative phases ϕ. While their constituent states do not possess none of these nonclassical properties. 3.4.

Q-distribution function

In quantum optics, the Husimi Q-function is an important tool for statistical description of quantum systems and nonclassical effects of radiation field. This is a positive function defined as 1 α|ρ|α, α = x + iy, (39) π where |α is the standard coherent state and ρ = |ψψ| is the density matrix of any arbitrary quantum state. As stated in [53–55], if the contour curves of Q-function are circles (curves), no squeezing (squeezing) on the number of photons will occurr. Q(α) =

NONCLASSICALITY OF NEW CLASS OF STATES PRODUCED BY SUPERPOSITION. . .

For the states (5) and (7), the Q-functions are calculated as

∞  (α ∗ )2n ξ n

2 1 −|α|2

, Qeven (α) = e

Nξ,f π 2 n n! [f (2n)]! n=0

2 ∞

1 −|α|2

  (α ∗ )2n+1 ξ n

, Qodd (α) = e Nξ,f

n π 2 n! [f (2n + 1)]! n=0 and finally for the superposed state (10) one has

∞  (α ∗ )2n ξ n 1 −|α|2

Qsup (α) = e N (N ξ,f,s ξ,f

π 2 n n! [f (2n)]! n=0

2 ∞ 

(α ∗ )2n+1 ξ n iϕ  )

, + e Nξ,f n 2 n! [f (2n + 1)]! n=0

165

(40) (41)

(42)

or equivalently

  ∞     n 1 −|α|2

iϕ  Qsup (α) = e N − n (N 2 − e N ) + N ξ,f ξ,f ξ,f

ξ,f,s π 2 n=0

n (α ∗ )n ξ [ 2 ]

2 × [n] n

. 2 2 [ 2 ]! [f (n)]!

(43)

By setting f (n) = 1 in the above relations, one can obtain the results for (linear) squeezed states. In Figs. 16–21 we have plotted Q-function for |ξ, f even , |ξ, f odd and |ξ, f sup with different phases. In all figures we see that, in some regions,

Q(x,y) Qx,y

0.2 2

0.1 0.0 0 2 0

y

2

x 2

Fig. 16. Three-dimensional plot of Q-function in phase space with fixed values ξ = 0.5 and q = 0.5 for |ξ, f even .

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S. AL. MODARES VAMEGH and M. K. TAVASSOLY

0.10 Qx,y Q(x,y) 2

0.05 0.00 0 2 0

y

2

x 2

Fig. 17. Three-dimensional plot of Q-function in phase space with fixed values ξ = 0.5 and q = 0.5 for |ξ, f odd .

0.3 Qx,y Q(x,y)

0.2 2

0.1 0.0 2

0

y

0 2

x 2

Fig. 18. Three-dimensional plot of Q-function in phase space with fixed values ξ = 0.5 and q = 0.5 for |ξ, f sup with ϕ = 0.

0.20 0.15

Qx,y Q(x,y)

0.10

2

0.05 0.00 0 2 0

y

2

x 2

Fig. 19. Three-dimensional plot of Q-function in phase space with fixed values ξ = 0.5 and q = 0.5 for |ξ, f sup with ϕ = π/2.

NONCLASSICALITY OF NEW CLASS OF STATES PRODUCED BY SUPERPOSITION. . .

167

0.3 0.2 Q(x,y) Qx,y 2

0.1 0.0 0 2 2

0

x

y

2

Fig. 20. Three-dimensional plot of Q-function in phase space with fixed values ξ = 0.5 and q = 0.5 for |ξ, f sup with ϕ = π.

0.20 0.15

Q(x,y) Qx,y

0.10

2

0.05 0.00 0 2 0

y

2

x 2

Fig. 21. Three-dimensional plot of Q-function in phase space with fixed values ξ = 0.5 and q = 0.5 for |ξ, f sup with ϕ = 3π/2.

Q-functions become zero which means that the Glauber–Sudarshan P -distribution function becomes negative in some areas, showing the nonclassicality of the considered states [56]. In addition, due to the non-circle cross-section of the three-dimensional plots, it is clearly seen that the squeezing effect occurs. 4.

Theoretical schemes for generation of the introduced states In this section we try to propose appropriate theoretical schemes for generation of the introduced superposed states. We recall that the vacuum squeezed state is generally generated by utilizing some sort of parametric processes with the help of

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different types of nonlinear optical devices. For instance, the “degenerate parametric down-conversion” and also the “degenerate four-wave mixing” are two well-known nonlinear processes; both of which can give rise to the squeezed light, arrive from the Hamiltonian of the form [57] 2

∗ 2 † Hint = i h(η ¯ a − ηa )

(44)

with η = χ β (η = χ β ) as the (second-) and third-order nonlinear susceptibilities correspond to the two mentioned above nonlinear processes. Now by the action of the time evolution operator U (t) = exp (−iHint /h) ¯ (with the help of the Hamiltonian (44)) on the following initial field state (2)

(3) 2

1 |ψ(0) = √ (|0 + |1) (45) 2 one readily arrives at the superposed state introduced in (14). It is worth mentioning that the above initial state can be prepared by appropriate atom-field interactions (one of us proposed a method for generation of such a state: see Eq. (13) in [58] and the explanation after it). Notice that setting Hint in U (t) yields the unitary squeezed operator with appropriate involved parameters. And about the f -deformed superposed states: as illustrated in [21, 22] the nonlinear squeezed vacuum and one-photon states can be produced by the action of f -deformed squeezed operator Sf (z) = exp(zK+ − z∗ K− ) on the vacuum |0 and one-photon |1 states, where the authors have defined the operators K+ = 12 (f (n)† a † )2 , K− = 12 (af (n))2 and K3 = 12 (N + 12 ), if f −1 = f † . Notice that under these conditions the operators K+ , K− , K3 are the generators of the SU(1, 1) coherent states. Therefore, in an extension of the above explained approach for achieving the superposition of linear squeezed states (14), in terms of the group theoretical language the superposed state in (10) can be obtained by the action of Sf (z) (the generalized unitary squeezed z operator) on the initial superposed state in (45) (with z = |z| tanh |z|). Therefore, in principle, there exist approaches for producing the nonlinear superposed states, too. 5.

Summary and conclusion

We considered two nonlinear squeezed states, nonlinear squeezed vacuum and nonlinear one-photon squeezed states which, according to their production procedures, essentially included even and odd bases of Fock space, respectively. Then, we constructed their superposition with respective phase ϕ which consisted of all bases in its relevant Hilbert space. Even though the presented approach has the potential application to any arbitrary nonlinearity function f (n), we applied our formalism for q-deformation nonlinearity function and established a few nonclassicality features of the obtained states, such as sub-Poissonian statistics, normal squeezing, amplitude squared squeezing, phase probability distribution, the number and phase squeezing based on the Pegg–Barnett approach and finally the Husimi Q-function. Through these, we have shown that nonclassical features exist in all cases. We mentioned

NONCLASSICALITY OF NEW CLASS OF STATES PRODUCED BY SUPERPOSITION. . .

169

that the presented structure can be applied for every nonlinearity function f (n). In other words, we can construct, investigate and compare the properties of different classes of superposed states which clearly depend on the function f (n). It should be emphasized that, to the best of our knowledge, the idea presented in this paper has not been used even for the (linear) vacuum and one-photon squeezed states. The case that will be achieved simply by setting f (n) = 1 in all the formulae obtained by us. We have analyzed the nonclassically of these states, too (see the appendix). Finally, to explain that the introduced superposed states are not so far from experiment, simple schemes for their generation are presented. 6.

Appendix As we mentioned in the main body of this paper, one can investigate the presented approach for (linear) squeezed states and their superposition by setting f (n) = 1 in all obtained results. In Fig. 22 we have plotted the Mandel parameter for the states |ξ, f even , |ξ, f odd and |ξ, f sup with f (n) = 1. (Henceforth, for simplicity, we have shown these states respectively with |ξ even , |ξ odd and |ξ sup ). We see that |ξ odd and |ξ sup possess sub-Poissonian (nonclassical) behaviour, while |ξ even has a super-Poissonian (classical) behaviour. 4

Ξ, f even Ξ, f odd

2

Ξ, f s

QM

3

1

0 1

0.5

0.0

0.5

Ξ

Fig. 22. Mandel parameter in terms of ξ for (linear) squeezed states.

In Figs. 23 and 24 we have plotted normal squeezing in x and p with f (n) = 1 for the states |ξ even , |ξ odd and |ξ sup , respectively. We realize from these figures that, |ξ even , |ξ odd and |ξ sup are squeezed in x and p (of course in different regions). In Fig. 23 we see that the graphs of |ξ even and |ξ sup coincide for ϕ = 0, π . Also, |ξ sup for ϕ = π/2, 3π/2 coincide with each other in this figure. In Fig. 24, |ξ sup for ϕ = 0, π coincide with each other and |ξ even coincides with |ξ sup for ϕ = π/2, 3π/2. In Figs. 25 and 26 we have plotted respectively the amplitude squared squeezing in X and P with f (n) = 1 for the states |ξ even , |ξ odd and |ξ sup . All these states

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S. AL. MODARES VAMEGH and M. K. TAVASSOLY 14 12

Ξ, f even Ξ, f odd

10 Ξ, f s ,  = 0

sx

,= 8 Ξ, f s 6 4

Π 2

Ξ, f s ,  = Π Ξ, f s , =

3Π 2

2 0 0.5

0.0

0.5

Ξ

Fig. 23. Normal ordered squeezing in x in terms of ξ for (linear) squeezed states.

Ξ, f even

12

Ξ, f odd Ξ, f s ,  = 0

sp

10

Π

8

Ξ, f s ,  =

6

Ξ, f s ,  = Π Ξ, f s , =

4

2

3Π 2

2 0 0.5

0.0

0.5

Ξ

Fig. 24. Normal ordered squeezing in p in terms of ξ for (linear) squeezed states.

are squeezed only in P and coincide with each other. Also, in Fig. 25 |ξ even and |ξ odd coincide with each other. In Figs. 27–30 we have plotted the phase probability distribution for |ξ even , |ξ odd and |ξ sup with different values of phase ϕ. We see from the figures that the peaks of |ξ even and |ξ odd occur at the same θ and both have a central peak at θ = 0. Also, the phase distribution |ξ sup for ϕ = 0 has a high and sharp peak at θ = 0, and for ϕ = π/2 we see that the peaks of |ξ sup do not occur at θ = 0. Also, for ϕ = π the peaks occur in θ = ±π , while the peaks of |ξ even and |ξ odd are at θ = 0. For the case ϕ = 3π/2 the two peaks of even and odd states occur in θ = 0 and the peak of superposed states is seen at θ = 0. In Figs. 31–36 we have plotted the number and phase squeezing for |ξ even , |ξ odd and |ξ sup with different phases ϕ. From Figs. 31 and 32 it is realized that |ξ even and |ξ odd are squeezed only in the number. Based on the Figs. 33–36 we see that

NONCLASSICALITY OF NEW CLASS OF STATES PRODUCED BY SUPERPOSITION. . .

171

Ξ, f even

8

Ξ, f  odd 6

SX

Ξ, f  s 4

2

0 0.5

0.0

0.5

Ξ

Fig. 25. Amplitude squared squeezing in X in terms of ξ for (linear) squeezed states. 0.0

0.2

SP

0.4 Ξ, f even

0.6

Ξ, f  odd

0.8

Ξ, f  s 0.5

0.0

0.5

Ξ

Fig. 26. Amplitude squared squeezing in P in terms of ξ for (linear) squeezed states. 1.0 Ξ, f even

Phase_distribution

0.8 Ξ, f odd 0.6

Ξ, f s

0.4

0.2

0.0

3

2

1

0

1

2

3

Θ

Fig. 27. Phase probability distribution in terms of θ with fixed value ξ = 0.5 for (linear) squeezed states with phase as ϕ = 0.

172

S. AL. MODARES VAMEGH and M. K. TAVASSOLY 0.6

Ξ, f even

Phase_distribution

0.5

Ξ, f odd

0.4

Ξ, f s

0.3 0.2 0.1 0.0 3

2

1

0

1

2

3

Θ

Fig. 28. Phase probability distribution in terms of θ with fixed value ξ = 0.5 for (linear) squeezed states with phase as ϕ = π/2. 1.0

Phase_distribution

Ξ, f even 0.8

Ξ, f odd

0.6

Ξ, f s

0.4

0.2

0.0

3

2

1

0

1

2

3

Θ

Phase_distribution

Fig. 29. Phase probability distribution in terms of θ with fixed value ξ = 0.5 for (linear) squeezed states with phase as ϕ = π . 0.6

Ξ, f even

0.5

Ξ, f odd

0.4

Ξ, f s

0.3 0.2 0.1 0.0 3

2

1

0

1

2

3

Θ

Fig. 30. Phase probability distribution in terms of θ with fixed value ξ = 0.5 for (linear) squeezed states with phase as ϕ = 3π/2.

NONCLASSICALITY OF NEW CLASS OF STATES PRODUCED BY SUPERPOSITION. . . 10 S nf Squeezing  Parameters

8

S f

6 4 2 0 0.0

0.2

0.4

0.6

0.8

Ξ

Fig. 31. Phase and number squeezing in terms of ξ for |ξ even . 10 S nf Squeezing  Parameters

8

S f

6 4 2 0 0.0

0.2

0.4

0.6

0.8

Ξ

Fig. 32. Phase and number squeezing in terms of ξ for |ξ odd . 5 S nf

Squeezing  Parameters

4

S f

3 2 1 0 1 0.0

0.2

0.4

0.6

0.8

Ξ

Fig. 33. Phase and number squeezing in terms of ξ for |ξ sup with ϕ = 0.

173

174

S. AL. MODARES VAMEGH and M. K. TAVASSOLY 5 S nf

Squeezing  Parameters

4

S f

3 2 1 0 1 0.0

0.2

0.4

0.6

0.8

Ξ

Fig. 34. Phase and number squeezing in terms of ξ for |ξ sup with ϕ = π/2. 5 S nf

Squeezing  Parameters

4

S f

3 2 1 0 1 0.0

0.2

0.4

0.6

0.8

Ξ

Fig. 35. Phase and number squeezing in terms of ξ for |ξ sup with ϕ = π . 5 S nf

Squeezing  Parameters

4

S f

3 2 1 0 1 0.0

0.2

0.4

0.6

0.8

Ξ

Fig. 36. Phase and number squeezing in terms of ξ for |ξ sup with ϕ = 3π/2.

NONCLASSICALITY OF NEW CLASS OF STATES PRODUCED BY SUPERPOSITION. . .

175

0.2 Qx,y Q(x,y) 2

0.1 0.0 0 2 0

y

2

x 2

Fig. 37. Three-dimensional plot of Q-function in phase space for fixed value ξ = 0.5 for |ξ even .

0.15 0.10

Qx,y Q(x,y)

2

0.05 0.00 0 2 0

y

2

x 2

Fig. 38. Three-dimensional plot of Q-function in phase space for fixed value ξ = 0.5 for |ξ odd .

0.2

Qx,y Q(x,y)

2

0.1 0.0 0

2

y

0 2

x 2

Fig. 39. Three-dimensional plot of Q-function in phase space for fixed value ξ = 0.5 for |ξ sup with ϕ = 0.

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S. AL. MODARES VAMEGH and M. K. TAVASSOLY

0.20 0.15

Q(x,y) Qx,y0.10

2

0.05 0.00 0 2 0

y

2

x 2

Fig. 40. Three-dimensional plot of Q-function in phase space for fixed value ξ = 0.5 for |ξ sup with ϕ = π/2.

0.2

Qx,y Q(x,y)

2

0.1 0.0 0

y

2 2

0

x

2

Fig. 41. Three-dimensional plot of Q-function in phase space for fixed value ξ = 0.5 for |ξ sup with ϕ = π .

0.20 0.15 Qx,y0.10 Q(x,y)

2

0.05 0.00 0 2 0

y

2

x 2

Fig. 42. Three-dimensional plot of Q-function in phase space for fixed value ξ = 0.5 for |ξ sup with ϕ = 3π/2.

NONCLASSICALITY OF NEW CLASS OF STATES PRODUCED BY SUPERPOSITION. . .

177

the state |ξ sup is squeezed only in the number for ϕ = 0 and is phase-squeezed for ϕ = π/2, 3π/2, while squeezing in both the number and phase (in different ranges) is seen for ϕ = π. In Figs. 37–42 we have plotted the Q-function for |ξ even , |ξ odd and |ξ sup with different values of phase. In all figures we see that in some regions the Q-functions become zero which mean that the Glauber–Sudarshan P -distribution function becomes negative in some areas of phase space. Due to the non-circle cross-section of the three dimensional plots, it is clearly seen that the squeezing effect has occurred. Acknowledgements The authors would like to thank the referees for their useful suggestions which caused to improve the contents of the paper. REFERENCES [1] J-P. Gazeau: Coherent States in Quantum Physics, Wiley-VCH, Berlin 2009. [2] S. T. Ali, J-P. Antoine and J-P. Gazeau: Coherent States, Wavelets and Their Generalization, Springer, New York 2000. [3] J. R. Klauder and B-S. Skagerstam: Coherent States, Applications in Physics and Mathematical Physics, Word Scientific, Singapore 1985. [4] A. Perelomov: Generalized Coherent States and their Applications, Texts and Monographs in Physics, Spinger, Berlin 1986. [5] H. P. Yuen and J. H. Shapiro: IEEE Trans, Inf. Theory 24 (1987), 657. [6] R. E. Slusher and B. Yurke: J. Lightwave Technol. 8 (1990), 466. [7] T. Morimae: Phys. Rev. A 81 (2010), 060307. [8] J. Kempe: Phys. Rev. A 60 (1999), 910. [9] C. H. Bennett, G. Brassard and N. D. Mermin: Phys. Rev. Lett. 68 (1992), 557. [10] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres and W. K. Wootters: Phys. Rev. Lett. 70 (1993), 1895. [11] M. B. Plenio and V. Vedral: Contemp. Phys. 39 (1998), 431. [12] K. Fesseha: Fundamentals of Quantum Optics, Lulu, USA 2008. [13] S. M. Barnett and P. M. Radmore: Methods in Theoretical Quantum Optics, Clarendon Press, Oxford 1997. [14] A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble and E. J. Polzik: Science 282 (1998), 706. [15] G. J. Milburn and S. L. Braunstein: Phys. Rev. A 60 (1999), 937. [16] H. J. Kimble and D. F. Walls: J. Opt. Soc. Am. B 4 (1987), 1449. [17] D. F. Walls: Nature 306 (1983), 141. [18] A-S. F. Obada and M. Darwish: J. Opt. B 7 (2005), 57. [19] M. Darwish: Int. J. Phys. B 19 (2005), 715. [20] M. K. Tavassoly: J. Phys. A: Math. Gen. 39 (2006), 11583. [21] A-S. F. Obada and G. M. Abd. Al-Kader: Eur. Phys. J. D 41 (2007), 189. [22] A-S. F. Abd. Al-Kader and A-S. F. Obada: Physica Scripta 78 (2008), 035401. [23] O. Abbasi and M. K. Tavassoly: Opt. Commun. 282 (2009), 3737. [24] O. Abbasi and M. K. Tavassoly: Opt. Commun. 283 (2010), 2566. [25] L. Mandel: Opt. Lett. 4 (1979), 205. [26] V. V. Dodonov, I. A. Malkin and V. I. Man’ko: Physica 72 (1974), 597. [27] S. Sivakumar: J. Phys. A: Math. Theor. 33 (2000), 2289.

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