Entanglement of multiphoton supercoherent states

Journal Pre-proof Entanglement of multiphoton supercoherent states Amin Motamedinasab, Somayeh Mehrabankar

PII:

S0030-4026(19)31743-7

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https://doi.org/10.1016/j.ijleo.2019.163845

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IJLEO 163845

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Optik

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9 September 2019

Accepted Date:

20 November 2019

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Entanglement of multiphoton supercoherent states Amin Motamedinasaba, Somayeh Mehrabankar*b

Department of physics, Faculty of Sorush, Isfehan branch, Technical and Vocational University (TVU), Isfehan, Iran

b

Physics Department, Shahid Chamran University of Ahvaz, Ahvaz, Iran

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a

*

E-mail address: [email protected] Corresponding author at: Physics Department, Shahid Chamran University of Ahvaz, Ahvaz, Iran

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E-mail address: [email protected]

Abstract:

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Multiphoton supercoherent states are defined as eigenstates of m-th power of supersymmetric annihilation operator, where m is a positive integer. In this paper, we study the entanglement of

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such states. To this purpose, we use concurrence as measure of entanglement. The results show that multiphoton supercoherent states are entangled for all values of m. Also, the entanglement does not depend on the coherency parameter for m  1 . Moreover, the entanglement for other

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values of m ( m  1 ) depends on the coherency parameters. In these cases, the entanglement for arbitrary m coincide with the entanglement of states with m  1 for large values of coherency parameter.

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Keywords: Supercoherent states, entanglement, coherency parameter

1. Introduction

Supersymmetric harmonic oscillator is composed of bosonic and fermionic oscillators where it’s Hamiltonian is equivalent to the Hamiltonian of charged particle interacting with a uniform magnetic field [1]. For this system, coherent states may obtain as eigenstates of supersymmetric

annihilation operator (named supercoherent states) [2-3]. Supercoherent states may be useful in the quantum electronics with graphene [4-6]. Also, they have important role in generalized JaynesCummings model and constructing super algebra [7-8]. The existing algebra between ladders operators and Hamiltonian fort the bosonic harmonic oscillator is Heisenberg algebra and for the supersymmetric harmonic oscillator is Lie algebra (consisting both commuting and anti-commuting relations) [9]. A generalization of Heisenberg algebra is known as the polynomial Heisenberg , in which the ladder operators are the m-th power

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of the ordinary ladder operators ( aˆ  aˆ m , aˆ †  aˆ † m ); also, the commutator of the creation and annihilation operators of the system, is a polynomial function of the m-th order of the Hamiltonian.

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Eigenstates of generalized annihilation operator ( aˆ m ), are called multiphoton coherent states [10]. Considering some nonclassical indicator such as Mandel parameter, Wigner function and

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squeezing show that these states may present nonclassical effects such as sub-Poissonian statistics and squeezing; However, for the large values of coherency parameter, their statistics is Poissonian

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(like coherent states). In addition, these states are cyclic quantum states with a geometric phase [10-11].

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Recently, similar to the multiphoton coherent states, (generalizing the annihilation operator), multi supercoherent states are introduced as eigenstates of m-th order supersymmetric annihilation operator; [11]. Application of supersymmetry in the generalized Jaynes-Cummings model, matter-

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radiation interaction in the exact resonance of field and atom, let us to find multiphoton supercoherent states as instrument to explanation of absorption or emission of multiphoton [12]. Like to the multiphoton coherent states, these states are cyclic with non-classical features such as sub-Poissonian statistics [11].

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On the other hand, the entanglement is a property of composite correlated systems that can be exist between two spatially separated objects (called quantum entanglement) or two degrees of freedom of a single object (called classical entanglement) [13-14]. Recently, based on classical entanglement a teleportation protocol has been implemented [15]. In the framework of supersymmetry, Spin and spatial variables of electron may be considered as the two degrees of freedom [16-17].

In this paper, we study the entanglement of supercoherent states. To this end, in the second and third sections, we study multiphoton supercoherent states of the supersymmetric harmonic oscillator and their entanglement, respectively. Finally, Sec.4 is devoted to the conclusion.

2. Multiphoton supercoherent states

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Kornbluth and Zypman introduced the generalized annihilation operator of the supersymmetric harmonic oscillator as follows [3]:

(1)

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k2   k1aˆ Aˆ   2 , ˆ ˆ k a k a 3   3

where ki ’s coefficients are arbitrary numbers and aˆ is the bosonic annihilation operator. In a

supercoherent as follows [11]: f

  2 Z , m, j

s

, 1 ,  2 

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Z , m, j  1 Z , m , j

eigenstates of Aˆ m are given as multiphoton

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special case ( k1  k2  k4  1 and k3  0 ),

 ,  

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* 1  z z , m, s j  z , m, s j Z , m, j s  2 z , m, s j   z, m, s j  Z , m, j f   ,  0    s j   j  1 1   ( m 1), j  , z , m, s j  aˆ †

(2)

z , m, s j

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where aˆ † is bosonic creation operator and z, m, j   n 0

z mn  j

 mn  j !

mn  j .

(3)

In such a case, the Hilbert space of a harmonic supersymmetric is decomposed as the sum of m orthogonal subspaces where each subspace consists of another j subspaces with j  0,1,...m  1 .

3. Entanglement of multiphoton supercoherent states In order to analyze the entanglement of multiphoton supercoherent states, we use the concurrence. For a pure two qubit state as follows:

 AB  a 0 A ,0B  b 0 A ,1B  c 1A ,0B  d 1A ,1B ,

(4)

the concurrence is given by [18]:





C  AB  2 ad  bc .

(5)

Now, we rewrite Eq. (2) as follows:

z 2

2



*

 1   z, m, s j  z, m, s j     2 z, m, s j  0  2



 0    , 1  

(6)

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

Z , m, j  N  1 z, m, s j 

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where N is a normalization constant. Considering the condition Z , m, j Z , m, j  1 and using Eq.

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(4), N is obtained as follows:

1

(7)

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2 mn  s 2 mn  j  2 mn  s j      2  2 2   2 2  1  mn  s j  z  j  z z 2 2   .   1   N   z    2  2  n  n 0  mn  j !  mn  s ! mn  s ! n     j j     

1

1b

1

z, m, j

F G

z

1     0  0 , 1 f   , 1 

, 0



f

(8)

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0b

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Now, we define the new bosonic and fermionic bases as follows:

*

z, m, j  z , m, s j

where F and G are given by: 2 mn  j 

 mn  j !

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F 

z

n

 1  mn  s  z  G  mn  s !

2 mn  s j



z

j

n

2

 n

j

z



2 mn  s j

(9)



.

 mn  s ! j

Using the Eqs. (8) and (9), we rewrite the Eq. (6) as follows: Z , m, j  N  1 F 0 

f

0

b



2 2

G 0

f

1b

2 2

F 1

f

 0 b . 

(10)

Finally, with the use of Eq. (5), we calculate the entanglement as follows:

C  2

2

N

2

(11)

FG .

It is clear that the entanglement in Eq. (11) depends on the m and j indices; just in a particular case of m  1 (supercoherent states), it is shown that the entanglement is independent of the coherency parameter, in this case we have: 2

 2  1 2

2

.

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C

2

(12)

lim  2

N

2

FG 

2

2

 2  1 2

2

.

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z 

2

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In other cases of m , the concurrence will be changed for different values of j . Also, for large values of coherency parameter, z , in any arbitrary m and j , the entanglement goes to its value for m  1 ; it means: (13)

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In Figures 1,2 and 3, assuming 1   2 , the concurrence is plotted as function of coherency parameter for multiphoton supercoherent states with m=1, 2 and 3.

Fig.1. Entanglement of multiphoton supercoherent states as function of

z for m  1, m  2, j  0, m  2, j  1 .

z for m  3, j  0, m  3, j  1, m  3, j  2 .

Fig.3.

Entanglement

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Fig.2. Entanglement of multiphoton supercoherent states as function of

multiphoton

supercoherent

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m  4, j  0, m  4, j  1, m  4, j  2, m  4, j  3 .

states

as

function

of

z

for

As it is evident from the plots, the most fluctuations are occurred in the small values of coherency parameters, and for large values of z , the entanglement converges to C  0.5 ( lim C  0.5 ). Also, z 

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for different values of m , there is at least one j in such a way that the entanglement will be higher than C  0.5 ; For other values of j , the entanglement will be less than this value.

4. Conclusion

Using supersymmetric quantum mechanics, some recently interesting researches are done on Dirac electrons in graphene on the presence of external magnetic or electric field; So, the entanglement in this system can be interpreted as entanglement between two coordinates of an electron. For this

system, using a generalization (m-th power) of annihilation operator, multiphoton supercoherent states could be obtained. In this paper, the entanglement of such states has been investigated. We have shown that, these states are entangled. Also, we have observed that the entanglement of states with any arbitrary m for the large values of coherency parameter is coincide with the entanglement of m=1 states. It is also observed that in the case of j  m  1 , in some ranges of parameter the entanglement could be higher than its boundary value; However, in other cases ( j  m  1 ), it is

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less than its boundary value for all values of z .

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