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]

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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)

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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. as follows

2 (1 + s)k F(β,s)= ∑−1k( ) π k+1 4.1 The auto-correlation function k (1 − s) 2 1 γ 2 2 2 δ 2 [| k1 | | Ck ( ,z,m) | + | k2 | | Ck ( ,z,m) | ∗ We present the moments of the photon operators for the ∗ 1 γ 2 δ + k1k2Ck ( ,z,m)Ck ( ,z,m) superposition of two SDFSs in order to calculate different ∗ 2 δ 1∗ γ + k1k2Ck ( ,z,m)Ck ( ,z,m)] statistical quantities. The s-parametrized averge value of a (17) and a† are presented as and also F(β,s) in a mathematics form is: ∂ k ∂ l †k l ρ †k l λ h[a a ]si = Tr[ {a a }s]= C( ,s) |λ =λ ∗=0 2 min(n,k) n n (2/µν)i/2(2/µν∗) j/2 ∂λ k ∂(−λ ∗)l F(β,s)= ∑AK ∑ (20) π i j (k − i)!(k − j)! k i= j=0 i.e. the average values of power of creation and γ 2 ν∗ γ¯ annihilation operators are derived by differentiating the | | γ2 [exp(− + )H(k−i)( ) λ λ ∗ 2 2µ (2νµ)1/2 characteristic function with respect to and − as γ∗ γ 2 ν shown above. Also by using equation (8) we can − 2 | | γ∗2 calculatethe averge values for any different α ,m and z H(n−i)( )[| k1 | exp(− + ( ) ) j j j (−2ν∗µ)1/2 2 2µ as −γ γ¯∗ H ( )H ( ) (k− j) νµ 1/2 (n− j) ν∗µ 1/2 (−2 ) (2 ) k s!(s + l − k)! h[a† al] i = ∑ C∗(α ,z ,m ) δ 2 ν δ¯∗ s s j j j ∗ | | ∗2 s=0 s (s − k)! (21) + k1k exp(− + ( )δ )H ( ) 2 2 2µ (k− j) (2ν∗µ)1/2 Cs+l−k(α j,z j,m j) s ≥ k −δ | δ |2 ν∗ H ( )] + exp(− + ( )δ 2) (n− j) (−2νµ)1/2 2 2µ therefor, by equation (20) we clculate; δ¯ −δ ∗ † −1 2 ∗ 2 ∗ ∗ H(k−i)( )H(n−i)( ) ha i = N [| k | α + | k | α + k k (2νµ)1/2 (−2ν∗µ)1/2 1 1 2 2 1 2 δ 2 ν δ 1 2 1 2 2 | | ∗2 − exp(− | α¯1 − α¯2 | )(A1 − B1)[− Lm(| α¯1 − α¯2 | ) [| k2 | exp(− + ( )δ )H ( ) 2 2 2 2µ (k− j) (−2νµ)1/2 1 α α 2 ∗ 1 α α 2 δ¯∗ γ 2 ν + Lm 1(| ¯1 − ¯2 | )] + k1k2 exp(− | ¯2 − ¯1 | ) ∗ | | ∗2 − 2 H ( )+ k k2 exp(− + ( )γ ) (n− j) (2ν∗µ)1/2 1 2 2µ 1 (A − B )[− L (| α¯ − α¯ |2)+ L1 (| α¯ − α¯ |2)]] γ¯∗ −γ 2 2 2 m 2 1 m−1 2 1 H ( )H ( )]], (k− j) (2ν∗µ)1/2 (n− j) (−2νµ)1/2 = hai∗ (18) (22)

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g 2 and 3.0 †2 −1 2 α∗2 2 α∗2 2 2 2.5 ha i = N [| k1 | 1 + | k2 | 2 − (| k1 | + | k2 | ) 2.0 ∗ ∗ ∗ 1 2 [µν + 2µν m]+ k k exp(− | α¯ − α¯ | ) 1.5 1 2 2 1 2 1.0 1 2 2 [[(−µν − (A1 − B1) )]Lm(| α¯1 − α¯2 | ) 4 0.5

2 µν 1 α α 2 r − [(A1 − B1)) − 2 ]Lm−1(| ¯1 − ¯2 | ) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 2 2 2 A B L α¯ α¯ (2) + ( 1 − 1) m−2(| 1 − 2 | )] Fig. 1: Correlation functiong against the squeezed parameter r,with α1 = α 1 1 1, 2 = 2 for k1 = 1,k2 = 1. The squeeze parameter is assumed to be real and runs ∗ α α 2 µν 2 from 0 to 3. The photon number has the value m = 0(line),m = 1(dashed),m = + k1k2 exp(− | ¯2 − ¯1 | )[[(− − (A2 − B2) )] 2 4 2(thick)and m = 3(dot dashed). 2 2 Lm(| α¯2 − α¯1 | ) − [(A2 − B2) − 2µν] 1 2 2 2 2 L α¯ α¯ A B L α¯ α¯ g 2 m−1(| 2 − 1 | ) + ( 2 − 2) m−2(| 2 − 1 | )]] 3.0 2 ∗ = ha i 2.5 (23) 2.0 While the average number of photons can be calculated analogously as follows 1.5

1.0 † −1 2 α 2 2 α 2 2 2 ha ai = N [| k1 | | 1 | + | k2 | | 2 | +(| k1 | + | k2 | ) 0.5 1 µ 2 ν 2 ∗ α α 2 r [m(| | + | | ) −C]+ k k exp(− | ¯ − ¯ | ) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 1 2 2 1 2 −1 (2) α 2 µ 2 ν 2 2 Fig. 2: Correlation functiong against the squeezed parameter r, with 1 = [ (A1 − B1) ) − [((| | + | | ) −C) − (A1 − B1) ] α 4 1, 2 = −2 for k1 = k2 = 1. The squeeze parameter is assumed to be real and 1 α α 2 2 α α 2 runs from 0 to 3. The photon number has the value m = 0 (line), m = 1(dashed), Lm−1(| ¯1 − ¯2 | ) − (A1 − B1)B1Lm−2(| ¯1 − ¯2 | )] m = 2(thick) and m = 3(dot dashed). 1 −1 + k∗k exp(− | α¯ − α¯ |2)[ (A − B )2) 1 2 2 2 1 4 2 2 2 2 2 − [((| µ | + | ν | ) −C) − (A2 − B2) ] In figure 2 we plot the displacement parameters α1 = 1 1 α α 2 2 α α 2 Lm−1(| ¯2 − ¯1 | ) − (A2 − B2)B2Lm−2(| ¯2 − ¯1 | )], and α2 = −2 and k1 = 1,k2 = 1. We note that for (24) m = 0,1,2,3 there is super-Possionian distribution , super where thermal is observed for m = 0, r > 2. While a small amount of sub-possionian is noted for m = 3 for a very A = −A = ν∗(α¯ − α¯ ), B = −B = µ(α¯ ∗ − α¯ ∗) 1 2 1 2 1 2 1 2 short period of r 0.1 < r < 0.3 The results are different 1 α α C = [s − (| µ |2 + | ν |2)] from earlier studies ( 1 = 1, 2 = −1)see[32]. 2 (25) The Gluaber second-order correlation function is defined by 4.2 Squeezing 2 ha† a2i g(2) = (26) We discuss the squeezing for the superposition of SDFSs ha†ai2 (4). The quadrature operators of the one mode field are The light with g(2) < 1 has a sub-Possionian distribution, defined by (2) the light with 1 < g < 2 has a super-Possianion 1 1 (2) X = (a + a†) X = (a − a†)[X ,X ]= i/2 (27) distribution, and the light with g > 2 has a 1 2 2 2i 1 2 super-thermal distribution. Coherent light has g(1) = 1 while thermal light has g(2) = 2. which satisfy the uncertainty relation (2) ∆ 2 ∆ 2 1 In figure 1 and 2 we plot the correlation function g ,h( X1) ih( X2) i≥ 16 with the variance against the squeeze parameter r, and we take the values ∆ 2 2 2 m = 0,1,2,3, with different value of the displacement h( Xj) i = hXj i − hXji j = 1,2 (28) parametersα1,α2 and the direction of squeezing is zero φ ∆ 2 1 = 0. The field is said to be squeezed if ( Xj) ≤ 4 for j=1,2. We plot figure 1 for α1 = 1,α2 = 2, and k1 = 1,k2 = 1. The average values hX1iand hX2i of the quadrature field 2 There is thermal distribution for m = 0 ,sub-Possionian is operators are directly computed, the variances h(∆X1) i 2 observed for m = 1 in the range 0.5 < r < 1, and for and h(∆X2) i of the quadrature field operators are m = 2,3 there is super-Possionian distribution ∀r. presented from equations (22),(23)and(24).

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The squeezing is best parametrized by the q parameter deﬁned as

2 h(∆Xj) i− 0.25 q j = , j = 1,2 (29) 0.25

q1 5

r 0.5 1.0 1.5 2.0 2.5 3.0

-5 Fig. 4: Quadrature distribution P(x,ϕ) of the state | ψi consisting of the superposition ofSDFSs, with m = 0,α1 = 1,α2 = 2,r = 1, and k1 = k2 = 1 -10

-15

-20

Fig. 3: Squeezing parameter q1 with the squeeze parameter r for k1 = k2 = 1 and pHx,0L α1 = 1,α2 = 2 with different values for m = 0(line), m = 1(dashed), m = 2(thick)and m = 3(dot dashed) 3.0

2.5

2.0 In ﬁgure 3 we plot q1 with the squeeze parameter r for 1.5 k1 = k2 = 1 , and α1 = 1,α2 = 2 with different values for m = 0,1,2,3. We observe that the squeezing exists −20 < 1.0 0.5 q j < 0 in all r,i.e., the squeezing condition reads q j < 0 x and it increases with r. The results are different from the -5 5 (a) α α pHx,HΠ2LL earlier studies ( 1 = 1, 2 = −1 ) see[32]. 1.0

0.8

4.3 Quadrature Distributions 0.6

0.4 The quadrature component distribution for the superposition state (4) is deﬁned as 0.2

x 2 -5 5 (b) P(x,ϕ)=| hx,ϕ | ψmi | (30) Fig. 5: Quadrature distribution P(x,ϕ) against x (a) at ϕ = 0,(b) at ϕ = π/2 We ﬁrst expand the eigenstate of quadrature component 1 x(ϕ)= [exp(−iϕ)a + exp(iϕ)a†] (31) 21/2 in the photon number basis as

∞ 1 x2 exp(iϕ j) | x,ϕi = exp(− ) ∑ H (x) | ji (32) π1/4 j 1/2 j 2 j=0 (2 j!) 3

Then by using equations (29) the quadrature component 5 1 distribution becomes 0 0 0 x 2 ∞ ϕ 2 ϕ 1 x exp(i ( j − l) P(x, )= exp(− ) ∑ -5 π1/2 j+l 1/2 j 4 N 2 j,l=0 (2 l! j!) (33) 6 s α s α [k1Cj( 1,z,m)+ k2Cj( 2,z,m)] Fig. 6: Quadrature distribution P(x,ϕ) of the state | ψi consisting of the ∗ s α ∗ s α α α φ [k1Cl ( 1,z,m)+ k2Cl ( 2,z,m)]Hj(x)Hl (x) superposition ofSDFSs,with m = 0, 1 = 1, 2 = −2,r = 1, = 0 and k1 = k2 = 1

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pHx,0L

3.0

2.5

2.0

1.5

1.0

0.5

x -5 5 (a) pHx,HΠ2LL

1.0

0.8 Fig. 10: Quadrature distribution P(x,ϕ) of the state | ψi consisting of the 0.6 superposition of SDFSs, with m = 0,α1 = 3 = −α2,r = 1,φ = 0 and k1 = k2 = 1

0.4

0.2 pHx,0L

x -5 5 (b) 40

Fig. 7: Quadrature distribution P(x,ϕ) against x (a)at ϕ = 0,(b)at ϕ = π/2 30

x -5 5 (a) pHx,HΠ2LL

x -5 5 (b) Fig. 8: Quadrature distribution P(x,ϕ) of the state | ψi consisting of the superposition of SDFSs, with m = 0,α1 = 2 = −α2,r = 1,φ = 0 and k1 = k2 = 1 Fig. 11: Quadrature distribution P(x,ϕ) against x (a)at ϕ = 0,(b)at ϕ = π/2

pHx,0L

3.0 In figures 4, 6, 8 and 10 we plot the 2.5 phase-parametrized field strength distribution(quadrature 2.0 component) P(x,ϕ) with m = 0,r = 1 , and different value 1.5 of α1,α2. 1.0 In figure 4 in case α1 = 1,α2 = 2, k1 = k2 = 1 we note ϕ π 0.5 that two symmetric peaks at = 0,2 , the small peak is ϕ x at x ≈ 0.7 and the large peak at x ≈ 3. As increase they -4 -2 2 4 (a) pHx,HΠ2LL becomes three peaks with small heights compared to the 2.0 original ones. The middle is larger while the side peaks are small but as ϕ reaches to π ,they combine into 2 peaks 1.5 ,and then they are swing the small peak is around

1.0 x ≈−0.7 and the large peak is around x ≈−3 see ﬁgure 5.The oscillation is repeated with a period 2π and also we

0.5 ﬁnd that it is bounded. Incidentally we have also the same results if we take k1 = −k2 = 1.

x -4 -2 2 4 (b) In figure 6 we take α1 = 1,α2 = −2, k1 = k2 = 1 we Fig. 9: Quadrature distribution P(x,ϕ) against x (a) at ϕ = 0,(b) at ϕ = π/2 note that the motion of the peak in the (x,ϕ) plan is the same as figure 4 but in opposite directions as may be seen from figure 7.

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In figure 8 we take α1 = 2,α2 = −2 we found that the Hamiltonian by applying a π/2 carrier pulse will stay in figure is symmetric around x = 0 and ϕ = π/2. It is this state and will be left unchanged ([37]-[40]) .Thus the observed that there are two peaks only for ϕ = 0,π and dynamics are reduced to those of the motional degrees of with height x ≈ 3 . Also we observe that the height of freedom only.Under this Hamiltonian the motional peak is increasing with increasing the value of α as dynamics evolve towards the SDFSs | α,z,mi by intially shown in figure 10 where α1 = 3,α2 = −3 where the preparing the Fock state | mi, then applying the linear part height reche ≈ 40 . Thus behavior is different from the first, and after that the quadratic part. The Fock state | mi case α1 = 1 = −α2 see[32]. can be prepared with very high efficiency according to In figure 5,7,9 and 11 we plot Quadrature distribution recent experiments ([20]-[23]). In this case ϕ ϕ α ∗ ∗ P(x, ) against x with different values of , We find = 2i(g1 + g2) and z = −2(g3 + g4). The prepartion of another structures and oscillation between the peaks and superposition of these states can be done according to the the number of them increase with increasing α as show in scheme described in by applying a further displacement the case ϕ = π/2 see figures 5(b),7(b),9(b)and 11(b). operator adjusted in a way to effect the | ei state alone, producing (1/21/2)(expiφ | β,z,mi | ei + k | α,z,mi | gi), then after a carrier π/2 pulse is applied to give finally the superposition of states. On detecting the ion any one of its 5 Generation Scheme internal states (| ei or | gi) we get the desired superposition (| α,z,mi + k | β,z,mi). After the discussion of the properties of the superposition of the SDFSs, we wish to consider the production of such state. We learn Let a two-level ion of 6 Conclusion mass M move in a harmonic potential of frequency ωxin † the x-direction. Let a(a ) stand for the annihilation We have discussed some properties and generation (creation) operator of the vibrational quanta in the scheme of superposition of Squeezed Displaced Fock x-direction. Then the position operator is given by States with different cohernt parameters (SDFSs). We ∆ † ∆ ω −1/2 x = x0(a + a ) with x0 = (2 xM) the width of the have calculated the photon number distribution, harmonic ground state. In this scheme, four beams of characteristic function and quasiprobability distribution laser applied along the x-axis are required to manipulate functions. moments have been presented through the motion of the atom; they are detuned by ±ωx and charactristic function. The second-order correlation ±2ωx. In the rotating-wave approximation the function g2 have been calculated numerically.The Hamiltonian for this system is given by squeezeing properties of this state which is presented as ω analytical and numerical results are increasing by † 0 − H = ωxaa + ( )σz − [µE (x,t)σ+ + hc] (34) changing the coherent parameters. We have found the 2 basic features of two of superposition of SDFSs, such as The first two terms describe the external and internal the appearance of increased number of separted peaks in free motions of the ion and the last term stands for the the quadrature distribution by increasing the coherent atom-field interaction. µ is the dipole matrix element and parameters. ω0 the transition frequency of the two-level ion , and the operators σz = |eihe| − |gihg| , σ+ = |eihg| , σ − = |gihe| where |ei and|gi are the atomic excited and References ground states respectively. We follow the same scheme as in [32] ,and we keep the [1] R. J. Glauber, 1963, Phys. Rev., 130, 2529. first and second order term of the Hamiltonian as: [2] R. J. Glauber, 1963, Phys. Rev., 131, 2766. [3] J. Perina, 1984, Quantum Statistics of Linear Optical Phenomena (Dordrecht: Rediel). ¯ ∗ † ∗ ∗ †2 H1 = −[2(g1 + g2)a + 2(g1 + g2)a + 2(g3 + g4)a [4] D. F. Walls, and Milburn, G. J., 1994, Quantum Optics ∗ 2 σ σ (Berlin: Springer). + 2(g3 + g4)a ]( − + +) (35) [5] D. Stoler, 1970, Phys. Rev. D, 1, 3217. [6] D. Stoler, 1971 Phys. Rev .D, 4, 1925. where [7] H. P. Yuen, 1976, Phys. Rev. A, 13 2226. η2 [8] R. Loudon and Knight, P. L., 1987, J. Mod. Optics, 34, 709. Ω η2 φ j g j(t)= i j j exp(i j)exp(− ), j = 1,2 [9] M . V. Satayanarayan, Phys. D, 32, 400. 2 (36) [10] P. Kral, 1990, J. mod. Optics, 37, 889; 1990, Phys. Rev. A, 2 η 42, 4177. g (t)= −Ω η2 exp(iφ )exp(− l ),l = 3,4 j l l l 2 [11] V. Perinova, and J. Krepelka, 1993, Phys. Rev. A 48, 3881. [12] I. Mendus and D. B. Propovic, 1995, Phys. Rev. A 52, 4356. Any atom prepared in the state (1/21/2)(| ei+ | gi) [13] I. Foldesi, P. Adam and J. Janszky, 1995, Phys. Lett. A which can be generated from the ground state under this ,20416.

c 2017 NSP Natural Sciences Publishing Cor. 1406 A.-S. F. Obada et al.: Superposition of two SDFSs With...

[14] Z. Z. Xin, Y. B. Duan, H. M. Zhang, M. Hirayama and K. Abdel-Shafy F. Obada, Matumoto, 1996, J. Phys. B ,29 ,4493. received D. Sc. and Ph. D. [15] K. B. Moller, T. G. Jorgensen, and K. Dahl, 1996, Phys. Rev. from Manchester University, A ,54 ,5378. 1994 and 1964. His current [16] V. I. Manko, and A. Wunsche, 1997, Qunta. semiclass. research interests include Optics, 9, 381. quantum resources, optical [17] M. S. Kim, F. A. M. De Oliveira and P. L. Knight, 1989, and atomic implementations Phys. Rev. A, 40, 2494. of quantum information [18] F. A. M.De Oliveira., Kim, M.S.,Knight,P.L.,and tasks and generations. Buzek,V.,1990, Phys.Rev.A,41,2645. Membership of Scientific [19] N. Moya-Cessa and P. L. Knight, 1993, Phys. Rev. A, 48, Societies: Institute of Physics (London), International 2479. Association of Mathematical Physics, International [20] D. Leibfried, D. M. Meekhof, B. E. king, ,C. Monroe, W.M.Itano and D. J. Wineland, 1996, Phys. Rev. Lett, 77, Center for Theoretical Physics (ICTP) Trieste, Italy, and 4281. Mathematical and Physical Society of Egypt. He is the [21] D. M.Meekhof, C. Monroe,B. E. King, W. M. Itano and D. Chairman of Egyptian Mathematical Society, member of J. Wineland, 1996, Phy. Rev. Lett, 76, 1796. American Mathematical Society, Egyptian Academy of [22] C. Monroe, D. M. Meekhof, B. E. King and D. J. Wineland, Sciences, New York Academy of Sciences and African 1996, Science, 272, 1131. Academy of Sciences. He has awarded several prizes, [23] W. M. Itano, C.Monroe, D. M. Meekhof, D. Leibfried, B. E. such as, State prize for scientific publications and citation King and D. J. Wineland, 1997, Broc. SPIE, 2995, 43. 2008, State Prize for Scientific Innovation 2004, State [24] V. V. Dodonov, I. A. Malkin and V. I. Manko, 1974 ,Physica. Recognition Prize in Basic Sciences for 2005. He has A, 72, 597. published over 200 papers in international refereed [25] B. Yurke and D. Stoler, 1986, phys. Lett. A, 57, 13. journals, some books and book chapters. [26] J. Janszky and A. V. Vinogradov, 1990, Phys. Rev. Lett. A, 64, 2771. Mohammed M. A. [27] V. Buzek and P. L. Kinght, 1991, Optics Commun, 81, 331. Ahmed Professor of applied [28] V. Buzek, A. Vidiella-Barranco and P. L. Kinght, 1992, Mathematical and Head of Phys. Rev. A, 45, 6570. Department of Mathematics, [29] M. Ban, 1994, Phys. Lett. A, 193, 121, 1995, Phys. Rev. A, Faculty of Science, Al-Azhar 51, 1604. University, Cairo (Egypt). He [30] A. Joshi and A. S. F .Obada, 1988, J. Phy, A30, 81. got his Ph.D. degree in 1987 [31] S. B. Zhang and C. G. Guo, 1996, Quantum Semiclass. Opt, for some studies in Nonlinear 8, 951. [32] A. S. F. Obada and G. M. Al-Kader, 1999, J. Mod. Optics, Optics (Quantum optics). His 46, 263. research interest is Quantum [33] E. Wigner, 1932, Phys. Rev, 40, 749. Optics. [34] E. Wigner, 1932, Z. Phys.Chem. B, 19, 203. [35] K . Husimi, 1940, Proc. Phys. Math. Soc. Japan, 22, 264. Norhan H. Gerish [36] Y. Kano, 1965, J. Math. Phys, 6, 1913. is a M. Sc. student at [37] Hector Moya-Cessa and P. L.Knight, 1993, Phys. Rev, 48, 3. Department of Mathematics, [38] V. V. Dodonov, Y. A. Korennoy, V. I and Y. A. Moukhin, Faculty of Science, Suez 1996, Quant. Semiclass. Optics, 8, 413. Canal University, Ismailia [39] W. Vogel and R. L. De Matos Filho, 1995, Phys. Rev. A, 52, (Egypt). She works as 4214. Demonstrator of applied [40] J. Steinbach, J. Twamley and P. L. Kinght, 1997, Phys. Rev. Mathematics. Her research A, 56, 4815. interest is Quantum optics .

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