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Superposition of Two Squeezed Displaced Fock States with Different Coherent Parameters

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Superposition of Two Squeezed Displaced Fock States with Different Coherent Parameters Appl. Math. Inf. Sci. 11, No. 5, 1399-1406 (2017) 1399 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.18576/amis/110517 Superposition of Two Squeezed Displaced Fock States With Different Coherent Parameters A.-S. F. Obada1, M. M. A. Ahmed1 and Norhan H. Gerish2,∗ 1 Department of Mathematics, Faculty of Science, Al-Azhar University, Cairo, Egypt. 2 Department of Mathematics, Faculty of Science, Suez Canal University, Ismailia, Egypt. Received: 2 Apr. 2017, Revised: 12 Jun. 2017, Accepted: 27 Jun. 2017 Published online: 1 Sep. 2017 Abstract: The s-parametrized characteristic function C(λ,s) for a superposition of two squeezed displaced Fock states (SDFSs ) is presented. The s-parametrized distribution functions for the superposition of SDFSs is investigated for different coherent parameters. The moments are obtained by using this characteristic function. The Gluaber second-order correlation function is calculated.The squeezing properties of this superposition are studied. Analytical and numerical results for the quadrature component distributions are presented. A generation scheme is discussed. The behavior of the above statistical aspectes differ on changing of the coherence parameters. Keywords: SDFSs, Quasiprobability Function, correlation function, Squeezing 1 Introduction been reported experimentally ([20]-[23]).In the above experiments an ion with laser cooled in a Paul trap to the ground harmonic state .Then the ion is set into different The number (Fock) state | ni is the main block in builting quantum states of motion by applications of optical and the electromagnetic field state .It is the eigen state of the † † electric fields. This moved the study of these states from photon number operatorn ˆ = a a where a(a ) is the the academic field to the world of experimentation, that annihilation (creation) operator. There is an alernative encouraged researchers to study the superposition of important state that is the coherent state| αi which is α these states. defined by applying a displacement operator D( ) on the Superposition of the quantum mechanical states of vacuum state. It is the eigen state of the annihilation electromagnetic field have lately recevied more care in operator (a), and also it is defined to be linear quantum optics([24]-[29]),because these states can reveal superposition of all | ni states with coefficients selected non-classical properties of light such as quadrature such that the photon number distribution is Poissonian squeezing and sub Poissonian statistics [5]. ([1]-[4]) .On the other hand the squeezed state is one of The studies of some types of superposition of the non-classical states of the electromagnetic field ,such Glauber(ordinary) coherent states have shown quadrature that certain observables reveal fluctuations less than for squueezing and sub-Poissonian statistics([24]-[29]. the vaccum state ([5]-[8]). It is known as the impact of the Methods for production of superposition of coherent squeezed operator S(z) on the coherent state[5].Squeezed states in experiments have been made by many Displsed Fock states (SDFSs) have been studied and workers([25]-[29]). Superposition of two binomial states various aspects of these states such as squezzing and and two negative binomial states have been studied in photon statistics have been investigated ([9]-[16]). [30]. Properties of the superposition of displaced Fock Two-photon coherent states (squeezed coherent states), states and generation scheme have been discussed [31]. squeezed number states [17] and displaced Fock states Let the class of quantum state | ψi have the form ([18],[19]) are generated as special cases of squeezed displaced Fock states. Lately the creation of no-classical N states of motion of a trapped ion as Fock states, coherent ψ −1/2 α | i = AN ∑ k j | j,z j,m ji (1) states, squeezed states and Schrodinger cat states have j=1 ∗ Corresponding author e-mail: [email protected] c 2017 NSP Natural Sciences Publishing Cor. 1400 A.-S. F. Obada et al.: Superposition of two SDFSs With... withAN as the normalization constant, and the SDFSs where Cn(α j,z j,m j) defined [32] as | α,z,mi defined by Cn(α j,z j,m j)= hn | α j,z j,m ji α z m D α S z m (2) ν α 2 ν∗ | , , i = ( ) ( ) | i n! 1/2 j n/2 | ¯ j | j 2 = ( ) ( ) exp(− + α¯j ) µ jm j! 2µ j 2 2µ j min(n,m j) µ ν i/2 ν∗ 1 1 2 m j (2/ j j) − j (m −i)/2 (8) D α exp αa† α∗a S z exp z∗a2 za† ∑ ( ) j ( )= ( − ) ( )= ( − ) i (n − i)! 2µ 2 2 i=0 j (3) α∗ α α¯ j − j The operator D( ) is the displacement operator, × H ( )H ( ) j = 1,2 n−i µ ν 1/2 m j −i µ ν∗ 1/2 α =| α | exp(iθ), and S(z) is squeeze operator, (2 j j) (−2 j j ) z = rexp(iφ) [5]. Where a(a†) is the annihilation (creation) operator of the boson field. while Hn(x) is the Hermite polynomial given by In section 2 we discuss the construction and properties of superposition of two SDFSs for different coherent [n/2] n!(−1)m(2x)n−2m Hn(x)= ∑ (9) parameter and calculate the photon number distributions. m!(n − 2m)! In section 3 we study the s-parametrized quasiprobability m=0 distribution function. In section 4 we discusse some Then we can cast the state (4) as applications of the charactristic function and the ∞ quadrature component distribution for these states. In | ψi = N−1/2 ∑ [k C (α ,z,m)+ k C (α ,z,m)] | ni section 5 finally we present a generation scheme. 1 n 1 2 n 2 n=0 (10) and the photon number distribution P(n) is given by 2 −1 2 2 Superposition of Two SDFSs With P(n)=| hn | ψi | = N | k1Cn(α1,z,m)+k2Cn(α2,z,m) | Different Coherent Parameters (11) Where Cn(α j,z,m) are defined in equation(8). With probability distribution function is obtained therefore We use the quantum state (1) for the superposition of a some statistical aspects can be calculated and discussed. pair of SDFSs with different coherent parameters but the squeezing parameters and number of photon are the same. ψ The superposition state | i is defined as 3 S-Parametrized Quasiprobability Function −1/2 | ψi = N [k1 | α1,z,mi + k2 | α2,z,mi] (4) Quasiprobability distribution function such as Glaubers P(β) function [2] , the Wigner W(β) [[33],[34]] function with the normalization constant N given by and Husimi Q(β) C[[35],[36]] function have proved to be very useful theoreticaly tools in performing quantum 2 2 ∗ ∗ optical calculations [32]. The s-parametrized N =| k1 | + | k2 | +[k1k + k k2] 2 1 characteristic function C(λ,s) plays an important role in | α¯ − α¯ |2 (5) exp(− 1 2 )L (| α¯ − α¯ |2) the fundamental exposition of the quasiprobability 2 m 1 2 functions. It is defined as the trace of the product of the density operator with the displacement operator as where α¯ = µα + να∗ withµ = coshr , follows: ν φ σ = exp(i )sinhr and r =| z |,while Lm(x) is the s associated Laguerre polynomial which is given by C(λ,s)= Tr[ρD(λ)]exp( | λ |2) (12) 2 m s σ m + σ (−x) with D(λ) defined in equation(3). The s-parametrized L (x)= ∑ (6) m m − s s! quasiprobability distribution function can be defined as a s=0 Fourier transformation of the s-parametrized characteristic function,and it is given by Now we try to obtain the photon statistics for the state of equation(4). First we set 1 F(β,s)= C(λ,s)exp(λ ∗β − λβ ∗)d2λ (13) π2 ∞ Z α α | j,z j,m ji = ∑ hn | j,z j,m ji | ni It can be cast into the form[37] as n=0 (7) ∞ 2 (1 + s)k = ∑ C j(α ,z ,m ) | ni, j = 1,2 F(β,s)= ∑ (−1)k hβ,k | ρ | β,ki (14) n j j j π k+1 n=0 k=0 (1 − s) c 2017 NSP Natural Sciences Publishing Cor. Appl. Math. Inf. Sci. 11, No. 5, 1399-1406 (2017) / www.naturalspublishing.com/Journals.asp 1401 The density operator corresponding to the state (4) is given where by k ν ν∗ k (1 + s) k! k/2 k/2 −1 Ak = (−1) ( k+1 )( )( ) ( ) (19) ρ = N [(k1 | α1,z,mi + k2 | α2,z,mi) (1 − s) µ n! 2µ 2µ ∗ α ∗ α (k1h 1,z,m | +k2h 2,z,m |)] (15) Since we use γ = α1 − β, δ = α2 − β. The general Then the characteristic function can be written in the represtation function F(β,s) reduce to the weight following form functions Q(β), W(β) and P(β) when the order parameter take the values s = −1,zero and +1. exp( s | λ |2) | λ¯ |2 respectively from this characteristic function we can C(λ,s) = ( 2 )[exp(− )L (| λ¯ |)2 N 2 m calculate any expectation value for the field operators. 2 α∗λ α λ ∗ 2 α∗λ α λ ∗ [| k1 | exp( 1 − 1 )+ | k2 | exp( 2 − 2 )] ∗ 1 ¯ 2 ¯ 2 + k k2 exp(− | λ + α¯ 2 − α¯ 1 | )Lm(| λ + α¯ 2 − α¯ 1 | ) 1 2 4 Some Statistical Properties 1 + k k∗ exp(− | λ¯ + α¯ − α¯ |2)L (| λ¯ + α¯ − α¯ |2)], 1 2 2 1 2 m 1 2 (16) After the characteristic function is calculated, we bocus ¯ ∗ Where λ = µλ + νλ , by using equations (14),(15) and on some statistical quantities namely: correlation function, (8) we can obtain the s-parametrized distribution function squeezing and quadrature distribution.
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