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REPORTS ON MATHEMATICAL PHYSICS

No. 2

ENTANGLEMENT OF PHOTON-ADDED NONLINEAR COHERENT STATES VIA A BEAM SPLITTER G HOLAMREZA H ONARASA Department of Physics, Shiraz University of Technology, P. O. Box:71555-313, Shiraz, Iran (e-mail: [email protected])

A LIREZA BAGHERI and A BDOLRASOUL G HARAATI Department of Physics, Payame Noor University, Shiraz, Iran (e-mail: [email protected], [email protected]) (Received August 31, 2015 – Revised April 29, 2016) Nonlinear coherent states, photon-added coherent states and photon-added nonlinear coherent states are three of the important generalizations of standard coherent states. In this article, a photon-added nonlinear coherent state and a vacuum state are injected on two input modes of a beam splitter and the entanglement of the output state is investigated using linear entropy as the measure. Then, the impact of nonclassicality of the photon-added nonlinear coherent state on entanglement of the output state is studied. Keywords: beam splitter, entanglement, photon-added nonlinear coherent states, Mandel parameter.

1.

Introduction

Coherent states as the eigenstates of the annihilation operator of the radiation field are important states in quantum optics [1]. In 1996 and 1997, the nonlinear coherent states have been introduced by Filho and Manko et al. due to their nonclassical properties of quantum optics and modern physics [2, 3]. Nonclassical states are quantum states with no classical counterpart. Nonclassical states can be recognized by various criteria such as quadrature squeezing, number-phase squeezing, sub-Poissonian photon statistics, Mandel parameter and second-order correlation function. The production and detection of nonclassical states have received considerable attention because of their applications in quantum cryptography and quantum communication [1, 4]. The nonlinear coherent states may be defined as the right eigenstates of a deformed annihilation operator A = af (n) where a and n are bosonic annihilation and number operator, respectively. The nonlinear coherent states associated with the quantum system with degenerate and nondegenerate spectrum were introduced in [5, 6]. Photon-added coherent states or excited coherent states as a class of nonlinear coherent state were constructed in 1991 and then generated in 2004 [7, 8]. A theoretical scheme for generation of any class of nonlinear coherent [245]

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states in a micromaser using the intensity dependent Jaynes-Cummings model and their physical properties has been introduced in [9]. Photon-added nonlinear coherent states were introduced by Safaeian et al. [10]. Recently, the entangled states received considerable attention [11, 12]. To generate quantum entanglement, different devices like beam splitters [13, 14], dissipative cavities [15–18] etc. have been used experimentally. A beam splitter can generate an entangled state when at least one of the input modes is a nonclassical state [13]. The entangled states as the output states of a beam splitter when the input mode is a product of a vacuum state and a photon-added coherent state have been studied by Berrada et al. [19]. In this paper, the entanglement of the output state of a beam splitter is investigated using linear entropy when a photon-added nonlinear coherent state and a vacuum state are injected into the beam splitter. Using the photon-added nonlinear coherent state as the input state instead of excited coherent state gives more control over the entanglement of the output state. In this case, the behaviour of entanglement depends on chosen nonlinearity function and the maximum entanglement occurs in difference values of the amplitude. 2.

Photon-added nonlinear coherent states in a beam splitter

Photon-added nonlinear coherent states have been defined by iterated actions (k times) of A† on the coherent states |αi [10]: |α, f, ki = A†k |αi,

(1)

where A† = f (n)a † is the deformed creation operator where f (n) is a function of the number operator and a † is the harmonic oscillator creation operator. The photon-added nonlinear coherent states in the number states bases are expressed by √ ∞ X α n [f (n + k)]! (n + k)! |n + ki, (2) |α, f, ki = Nα,f,k n![f (n)]! n=0 where

Nα,f,k =

X ∞ |α|2n (n + k)![f 2 (n + k)]! −1/2

(3) (n!)2 [f 2 (n)]! . is the normalization constant and [f (n)]! = f (n) f (n − 1) . . . f (1). The beam splitters can combine two input ports and create entanglement between two quantum states. The beam splitter operator is given by [19, 20] θ B = exp (a1† a2 + a1 a2† ) , (4) 2 n=0

where a1 and a2 (a1† and a2† ) are the boson-annihilation (creation) operators for two input fields. The output state of a beam splitter with an arbitrary input state

ENTANGLEMENT OF PHOTON-ADDED NONLINEAR COHERENT STATES. . .

247

|ψint i is given by |ψout i = B|ψint i. If a number state |ni and a vacuum state are considered as the two input states, the output state is obtained by [19] 1/2 n X n! t p r (n−p) |pi|n − pi, (5) B|ni|0i = p!(n − p)! p=0 where r and t with |r|2 + |t|2 = 1 are the beam splitter’s reflection and transmission coefficients, respectively. Now, we consider a photon-added nonlinear coherent state |α, f, ki and a vacuum state |0i in two input ports, i.e. |ψint i = |α, f, ki ⊗ |0i. The output state of the beam splitter is as follows |ψout i = B|α, f, ki|0i √ ∞ X α n [f (n + k)]! (n + k)! = Nα,f,k B|n + ki|0i. n![f (n)]! n=0

(6)

One can rewrite Eqs. (6) using (5) as follows |outi =

3.

∞ X n+k Nα,f,k X (n + k)![f (n + k)]!α p t p α (n+k−p) r (n+k−p) |pi|n + k − pi. (7) √ √ α k n=0 p=0 n![f (n)]! p! (n + k − p)!

Entanglement of the output states

There are several entanglement measures such as entanglement of formation, negativity, von Neumann entropy and linear entropy [21–23]. Here the linear entropy of the output state is considered as the measure for entanglement. The linear entropy is the upper bound of the von Neumann entropy. The range of linear entropy is between zero (no entanglement) and one (maximum entanglement). The density operator of the output state is given by ρ = |ψout ihψout |.

(8)

With the help of Eq. (7), the reduced density operator can be obtained as: ρa =

2 ∞ X ∞ X Nα,f,k

|α|2k

×

(m + p)!(m + p ′ )![f (m + p)]![f (m + p′ )]! (m + p − k)!(m + p ′ − k)![f (m + p − k)]![f (m + p ′ − k)]! p,p′ =k m=0

|α|2m |t|2m α p α p′ r p r p′ |pihp ′ |. √ m! p!p′ !

(9)

Then, the linear entropy is [24] S = 1 − Tr(ρa2 ). Using the states (7), the linear entropy can be expressed as

(10)

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G. HONARASA, A. BAGHERI and A. GHARAATI

S =1− ×

4 ∞ X Nα,f,k

|α|4k

∞ X

(m + p)!(m + p′ )! (m′ + p)!(m′ + p ′ )! (m + p − k)!(m + p′ − k)! (m′ + p − k)!(m′ + p ′ − k)! p,p′ =k m,m′ =0

[f (m + p)]![f (m + p′ )]! [f (m′ + p)]![f (m′ + p ′ )]! [f (m + p − k)]![f (m + p′ − k)]! [f (m′ + p − k)]![f (m′ + p ′ − k)]! ′

′

|α|2(m+m +p+p ) 2(m+m′ ) 2(p+p′ ) |t| |r| . × m!m′ !p!p ′ !

(11)

4. Entanglement generation by two special photon-added nonlinear coherent states We can apply the presented formalism to any nonlinear coherent states with known nonlinearity functions f (n). Also, the nonlinearity function √ of arbitrary quantum system with the known spectra en has been introduced as en /n [6]. For the first case we will deal with nonlinear coherent states introduced by Penson and Solomon. While for solvable quantum systems, the Poschl–Teller potential will be considered. In all calculations the beam splitter is considered as 50 : 50, then 1 i t=√ . r=√ , 2 2 The Penson–Solomon coherent states are given by [25] ∞ X q n(n−1)/2 n |q, αiPS = N (q, |α|2 ) α |ni, (12) √ n! n=0 where N (q, |α|2 ) is an appropriate normalization constant and 0 ≤ q ≤ 1. These states can be considered as the nonlinear coherent states with the nonlinearity function [6] fPS (n) = q 1−n . (13) The explicit form of the Poshl–Teller potential is given by [26]: λ(λ − 1) κ(κ − 1) 1 + , V (x) = V0 2 cos2 x/2a sin2 x/2a

(14)

where 0 ≤ x ≤ πa, λ, κ > 1 and V0 is a coupling constant. The nonlinear function associated with Poshl–Teller potential is given by [6]: √ (15) fPT (n) = n + ν,

where ν = λ + κ > 2. In Fig. 1, the linear entropy of the output states has been plotted versus |α| for various values of k when the photon-added nonlinear coherent states associated with the Penson–Solomon coherent states and vacuum states are injected into the beam splitter. The figure shows entanglement (S > 0) in the output state for k 6= 0 and for all values of the amplitude |α|. It can be seen that all curves (for different values

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249

Fig. 1. Linear entropy S of photon-added nonlinear coherent states associated with the Penson–Solomon coherent states (q = 0.5) versus |α| for k = 3 (solid line), k = 4 (dashed line) and k = 6 (dash-dotted line).

of k) have minima but that values of |α| which minimize the linear entropy depend on k. The maximum value of entanglement, i.e. S = 1, occurs in the case of high values of the amplitude |α|. Fig. 2 shows the linear entropy of the output state versus |α| for a few values of k when a photon-added nonlinear coherent state associated with the Poshl–Teller potential and a vacuum state are injected into the beam splitter. Again, the behaviour of the linear entropy is the same for different values of k. The linear entropy has its

Fig. 2. Linear entropy S of photon-added nonlinear coherent states associated with the Poshl–Teller potential (ν = 3) versus |α| for k = 3 (solid line), k = 4 (dashed line) and k = 6 (dash-dotted line).

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maximum value at small values of |α| (depends on the photon excitation number k) and decreases with increasing |α|. The entanglement of this output state reaches to the lower bound S = 0 for large enough values of |α|. Also, it can be seen that the degree of entanglement grows with increasing photon excitation number while the entanglement decreases as the amplitude |α| increases. Now, the relation between the entanglement of the output state and the nonclassical properties of the input field can be studied using the Mandel parameter. The Mandel parameter Q characterizes the quantum statistics of the field and it is positive, negative and vanishes for super-Poissonian (classical), sub-Poissonian (nonclassical) and Poissonian statistics, respectively. The Mandel parameter Q is defined as [27] Q=

hn2 i − hni2 − 1. hni

(16)

Using Eq. (2) for photon-added nonlinear coherent states, hni and hn2 i are obtained as ∞ X |α|2n (n + k)![f 2 (n + k)]! 2 hni = Nα,f,k (n + k), (17) (n!)2 [f 2 (n)]! n=0 2 hn2 i = Nα,f,k

∞ X |α|2n (n + k)![f 2 (n + k)]! n=0

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

(n + k)2 .

(18)

Figs. 3 and 4 display the Mandel parameter Q of photon-added nonlinear coherent states associated with the Penson–Solomon coherent states and the Poshl–Teller potential, respectively, versus |α| for some fixed values of the excitation number k. In both figures, the Mandel parameter is negative for all values of |α| and shows

Fig. 3. The Mandel parameter Q of photon-added nonlinear coherent states associated with the Penson–Solomon coherent states (q = 0.5) versus |α| for k = 3 (solid line), k = 4 (dashed line) and k = 6 (dash-dotted line).

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251

Fig. 4. The Mandel parameter Q of photon-added nonlinear coherent states associated with the Poshl–Teller potential (ν = 3) versus |α| for k = 3 (solid line), k = 4 (dashed line) and k = 6 (dash-dotted line).

sub-Poissonian statistics. From the figures, it is observed that the states become more negative and more nonclassical as k increases. The results show that the larger value of entanglement occurs where the input photon-added nonlinear coherent state is more nonclassical. 5.

Summary and conclusion

In summary, the entanglement generated by a 50 : 50 beam splitter has been investigated when a photon-added nonlinear coherent state is injected into one input beam and the vacuum is injected into the other one. The linear entropy has been used as the measure of entanglement. The results show that the photon excitation number increases the entanglement of the output state and the behaviour of the linear entropy can be controlled by nonlinearity function of the input photon-added nonlinear coherent state. Also, the effect of nonclassical properties of input field, the photon-added nonlinear coherent state, on entanglement of the output state has been studied through the Mandel parameter. It is found that the photon-added nonlinear coherent states show more nonclassical behaviour as the excitation number increases and the maximum value of entanglement occurs when the Mandel parameter tends to −1. REFERENCES [1] S. T. Ali, J-P. Antoine and J-P. Gazeau: Coherent States, Wavelets and Their Generalizations, Springer, New York 2000. [2] R. L. de Matos Filho and W. Vogel: Phys. Rev. A 54 (1996), 4560. [3] V. I. Manko, G. Marmo, E. C. G. Sudarshan and F. Zaccaria: Phys. Ser. 55 (1997), 528. [4] J. R. Klauder and B. S. Skagerstam: Coherent States, Applications in Physics and Mathematical Physics, World Scientific, Singapore 1985.

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