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In this paper we study the dynamics of the atomic inversion, von Neumann entropy and entropy squeezing for moving and non-moving two-level atoms interacting with a Perelomov coherent state. The final state of the system using specific initial conditions is obtained. The effects of Perelomov and detuning parameters are examined in the absence and presence of the atomic motion. Important phenomena such as the collapse and revival are shown to be very sensitive to the variation of the Perelomov parameter in the presence of detuning parameter. The results show that the Perelomov parameter is very useful in generating a high amount of entanglement due to variation of the detuning parameter.

Quantum entanglement is one of the most outstanding features of quantum mechanics. Quantum entangled states, as a fundamental physical resource of quantum information processing [

In fact, these different kinds of interactions have been classified from the point of view of Lie algebra depending on the nature of the interaction. For example, the Hamiltonian which represents the interaction between two fields is described in the form of the parametric frequency converter which is of S U ( 2 ) Lie algebra type. While the Hamiltonian which represents the non-degenerate parametric amplifier is of S U ( 1 , 1 ) Lie algebra type. In this context a system describes the interaction between S U ( 2 ) and S U ( 1 , 1 ) Lie algebra, in which a Hamiltonian of the following form was treated

H = ω K ^ z + ω 0 2 σ ^ z + g f ( z ) [ σ ^ + K ^ − + K ^ + σ ^ − ] , (1)

where ω and ω 0 are the field and atomic frequencies, respectively, σ ^ z and σ ^ ± are the atomic pseudospin operators that obey the commutation relation

[ σ ^ + , σ ^ − ] = σ ^ z , [ σ ^ z , σ ^ ± ] = ± 2 σ ^ ± (2)

while g is the atom field coupling constant, f ( z ) denotes a shape function of cavity field mode.

We restrict our study to the atomic motion along the z-axis so that the z-dependence of the field-mode function should be taken into account. The atomic motion can be incorporated in the usual way, i.e.

f ( z ) → f ( v t ) = p 1 + s i n ( p 2 π v t L ) , (3)

where v denotes the atomic motion velocity, p 1 and p 2 are the atomic motion parameters well, if we put p 1 = 1 and p 2 = 0 , then the shape function takes the form

Ω ( t ) = ∫ 0 t f ( v t ′ ) d t ′ = t (4)

which means, there is no atomic motion inside the cavity, but if p 1 = 0 and p 2 = p , where p represents the number of half-wave lengths of the field mode inside a cavity of the length L, the shape function for a particular choice of the

atomic motion velocity v = g L π will be

Ω ( t ) = ∫ 0 t f ( v t ′ ) d t ′ = 1 p g [ 1 − cos ( p g t ) ] (5)

Over the last two decades much attention has been focused on the properties of the Jaynes-Cummings model JCM for a moving atom. The theoretical efforts have been stimulated by experimental progress in cavity QED. Besides the experimental drive, there also exists a theoretical motivation to include atomic motion effect to JCM because its dynamics becomes more interesting. A number of authors have treated the JCM in the presence of atomic motion by the use of analytic approximations [

The solution in the presence of atomic motion is not only of theoretical interest, but also important from a practical point of view for cold atoms. The important demonstration of the quantum collapse and revival phenomena was observed in a one-atom maser by Rempe et al. [

In this article, we consider the extension of the problem by considering the two-level interaction with S U ( 1,1 ) quantum system. We focus on the effect of the Perelomov parameter, field-mode structure parameter and detuning parameter on the evolution of the atomic inversion, von Neumann entropy in the case of absence and presence of the atomic motion effect.

We organize the material of this paper as follows: in Section 2 we introduce our Hamiltonian model which represents the interaction between S U ( 1,1 ) and S U ( 2 ) , Then we derive the effective two-level atom Hamiltonian model, and we use the evolution operator method to find an exact expression of the wave function at the time t > 0 . We devote Section 3 to discuss the atomic inversion in order to see the change that would occur in its behavior during interaction. While in Section 4 we discuss the degree of entanglement for the atomic system via von Neumann entropy and entropy squeezing. Finally, we draw a summary in Section 5.

The Hamiltonian which describe the interaction between a single two-level atom and S U ( 1,1 ) quantum system take the following form

H = ω K ^ z + ω 0 2 σ ^ z + g f ( z ) [ σ ^ + K ^ − + K ^ + σ ^ − ] , (6)

while K ^ ± and K ^ z satisfy the following commutation relation

[ K ^ z , K ^ ± ] = ± K ^ ± , [ K ^ − , K ^ + ] = 2 K ^ z (7)

The Heisenberg equation of motion for any operator O ^ is given by

i d O ^ d t = [ O ^ , H ] , ( ℏ = 1 ) (8)

thus, the equations of motion for σ ^ z and K ^ z are given by

d σ ^ z d t = − i [ σ ^ z , H ] = 2 i g f ( v t ) ( K ^ + σ ^ − − σ ^ + K ^ − ) . (9)

d K ^ z d t = − i [ K ^ z , H ] = i g f ( v t ) ( − K + σ ^ − + σ ^ + K ^ − ) . (10)

d K ^ z d t + 1 2 d σ ^ z d t = 0 (11)

from the above equation, we can see that N = K ^ z + 1 2 σ ^ z is constant of motion, therefore, the Hamiltonian takes the following form

H = ω N + H I , (12)

where

H I = Δ 2 σ ^ z + g f ( v t ) ( σ ^ + K ^ − + K ^ + σ ^ − ) , (13)

where Δ = ω 0 − ω is the detuning parameter. We note that, [ N , H I ] = 0 , therefore [ N , H ] = [ H , H I ] = 0 , i.e. N and H I are the constants of motion. Where the time evolution operator is defined as

U ( t ) = ( u 11 u 12 u 21 u 22 ) (14)

where

u 11 = cos ( μ ^ 1 t ) − i Δ 2 μ 1 sin ( μ ^ 1 t ) , u 12 = − i g μ ^ 1 sin ( μ ^ 1 t ) K ^ − u 21 = − i g μ ^ 2 sin ( μ ^ 2 t ) K ^ + , u 22 = cos ( μ ^ 2 t ) + i Δ 2 μ ^ 2 sin ( μ 2 t ) (15)

and

μ ^ j 2 = Δ 2 4 + v j , j = 1 , 2 , v 1 = g 2 K ^ − K ^ + , v 2 = g 2 K ^ + K ^ − (16)

Let us assume the initial state of the atom in excited state | 1 〉 and the cavity field mode is in the Perelomov coherent state [

| μ ; k 〉 = ∑ m = 0 ∞ Q m | m , k 〉 , Q m = ( 1 − | μ | 2 ) k ( Γ ( 2 k + m ) m ! Γ ( 2 k ) ) 1 2 μ m , (17)

where, μ the Perelomov parameter and

K ^ z | m , k 〉 = ( m + k ) | m , k 〉 , K ^ + | m , k 〉 = ( m + 1 ) ( m + 2 k ) | m + 1 , k 〉 , K ^ − | m , k 〉 = m ( m + 2 k − 1 ) | m − 1 , k 〉 , (18)

Therefore, we can write the wave function at the time t > 0 in the form

| ψ ( t ) 〉 = | C ( t ) 〉 | 1 〉 + | S ( t ) 〉 | 2 〉 , (19)

where,

| C ( t ) 〉 = ∑ m = 0 ∞ X 1 ( m , t ) | m , k 〉 , | S ( t ) 〉 = ∑ m = 0 ∞ X 2 ( m , t ) | m + 1 , k 〉 , (20)

here,

X 1 ( m , t ) = Q m [ cos ( μ 1 t ) − i Δ 2 μ 1 sin ( μ 1 t ) ] , X 2 ( m , t ) = − i Q m g ( m + 1 ) ( m + 2 k ) μ 1 sin ( μ 1 t ) , (21)

where,

μ 1 = Δ 2 4 + g 2 ( m + 1 ) ( m + 2 k ) , μ 2 = Δ 2 4 + g 2 m ( m + 2 k − 1 ) . (22)

The atomic reduced density operator for the system is given by

ρ A ( t ) = T r f | ψ ( t ) 〉 〈 ψ ( t ) | = ( ρ 11 ρ 12 ρ 21 ρ 22 ) (23)

where

ρ 11 = ∑ m = 0 ∞ | X 1 ( m , t ) | 2 , ρ 22 = ∑ m = 0 ∞ | X 2 ( m , t ) | 2 ρ 12 = ∑ m = 0 ∞ X 1 ( m + 1 , t ) X 2 * ( m , t ) , ρ 21 = ρ 12 * (24)

Atomic inversion can be considered as the simplest important quantity to be calculated. It is related to the difference between the probabilities of finding the atom in the upper state | 1 〉 and in the ground state | 2 〉 . Using Equation (20), we can calculate the time evolution of the atomic inversion as

W ( t ) = ∑ m = 0 ∞ [ | X 1 ( m , t ) | 2 − | X 2 ( m , t ) | 2 ] (25)

In

When we increase the value of μ to 0.7 a dramatic change can be seen in the function behavior. In this case the function fluctuates around the value zero and exhibits periods of collapses and revivals, however the revival periods start to spread out as time increases. In the meantime we observe a decrease in the

amplitude of each period of the revival (slow decay) and we observe somewhat a compress in the fluctuations during the revival periods see

In order to discuss effects of atomic motion and field mode structure on the atomic inversion, we have plotted in

periodical function on the scale time g t with period 2π p .

In this section we study the evaluation of the von Neumann entropy, defined as

S = − T r { ρ ln ρ } (26)

where ρ is the density operator for a given quantum system. For a pure state, S = 0 , but when S ≠ 0 the system is in a mixed state. The entropies of the atomic and the field systems can be defined through the corresponding reduced

operators [

entropy = − ( λ ( + ) ln λ ( + ) + λ ( − ) ln λ ( − ) ) (27)

To discuss the entanglement of the atomic system with the field we plot

when atomic motion is taken into account. From these figures, we can conclude that: 1) the atomic motion leads to the periodic evolution of the field entropy; 2) an increase of the parameter p leads to the shortening of the evolution periodicity of the von Neumann entropy.

Entropy Squeezing PropertiesNow we study the squeezing of the atomic entropy, where we can express the entropy squeezing of the two-level atom by using the quantum information entropy theory. The information entropy of the Pauli operators 〈 σ ^ α 〉 ( α = x , y , z )

〈 σ ^ x 〉 = 2 Re ( ρ 12 ) , 〈 σ ^ y 〉 = 2 Im ( ρ 12 ) , 〈 σ ^ z 〉 = ρ 11 − ρ 22 (28)

for a two-level atom system is

H ( σ ^ α ) = − ∑ i = 1 2 P i ( σ ^ α ) ln P i ( σ ^ α ) , (29)

where P i ( σ ^ α ) = 〈 ψ α i | ρ | ψ α i 〉 ( i = 1 , 2 ), which are the probability distributions for two possible measurements of an operator σ ^ α . H ( σ ^ x ) , H ( y ) , and H ( z ) satisfy

H ( σ ^ x ) + H ( σ ^ y ) ≥ 2 ln 2 − H ( σ ^ z ) , (30)

which may also be rewritten as

δ H ( σ ^ x ) δ H ( σ ^ y ) ≥ 4 δ H ( σ ^ z ) , (31)

where δ H ( σ ^ α ) = exp ( H ( σ ^ α ) ) . The squeezing of the atom is determined by using the EUR Equation (31) named entropy squeezing. The fluctuation in the component ( σ ^ α , α = x , y ) of the atomic dipole is said to be squeezed in entropy if the information entropy H ( σ ^ α ) satisfies the following condition:

E ( σ ^ α ) = δ H ( σ ^ α ) − 2 | δ H ( σ ^ z ) | < 0 , ( α = x , y ) . (32)

By using ρ A ( t ) , we can obtain the information entropies of the atomic operators σ ^ x , σ ^ y and σ ^ z as follows:

H ( σ ^ x ) = − ( ξ 1 ln ξ 1 + ξ 2 ln ξ 2 ) , H ( σ ^ y ) = − ( ξ 3 ln ξ 3 + ξ 4 ln ξ 4 ) , H ( σ ^ z ) = − ( ρ 22 ln ρ 22 + ρ 11 ln ρ 11 ) , (33)

where

ξ 1 = 1 2 + Re ( ρ 12 ) , ξ 2 = 1 2 − Re ( ρ 12 ) , ξ 3 = 1 2 + Im ( ρ 12 ) , ξ 4 = 1 2 − Im ( ρ 12 ) , (34)

We discuss the effects of the detuning parameter, the Perelomov parameter, the atomic motion and the field-mode structure on the properties of the entropy squeezing. The time evolution of the squeezing factors, E ( σ ^ x ) , E ( σ ^ y ) are shown in

E ( σ ^ x ) and E ( σ ^ y ) disappears quickly and will not reappear anymore. While in

We have introduced the problem of the interaction between a two-level atom and a quantum system. We have used the Perelomov generalized coherent state as the initial state for the S U ( 1,1 ) quantum system, while we considered the atom to be initial in the excited state. The wave function is calculated via the evolution operator and the result is used to obtain the atomic density operator. The effects of the different parameters such as detuning parameter, atomic motion and Perelomov parameter on the atomic inversion, von Neumann entropy and Entropy squeezing have been studied. It is shown that some new features such as: 1) the Perelomov parameter is very useful in generating a high amount of entanglement in the absence and presence of the detuning parameter; 2) the atomic inversion is sensitive to the variation of the Perelomov parameter in the presence of detuning parameter; 3) the atomic motion and field-mode structure parameter play an important role on the time evolution of the atomic inversion, entropy squeezing and von Neumann entropy.

Ismail, M.Kh. and El-Shahat, T.M. (2017) Entanglement of Moving and Non-Moving Two-Level Atoms. Journal of Quantum Information Science, 7, 172-184. https://doi.org/10.4236/jqis.2017.74014