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Entropy squeezing is an important feature in performing different tasks in quantum information processing such as quantum cryptography and superdense coding. These quantum information tasks depend on finding the states in which squeezing can be created. In this article, a new feature on entropy squeezing for a two level system with a class of nonlinear coherent state (NCS) is observed. An interesting result on the comparison between the coherent state (CS) and NCS is explored. The influence of the Lamb-Dick parameter in both absence and presence of the Kerr medium is examined. A rich feature of entropy squeezing in the case of NCS, which is observed to describe the motion of the trapped ion, has been obtained.

It is known that quantum entangled state plays an important role in the fields of quantum information theory as well as quantum teleportation and computation. The entropy which automatically includes all moments of the density operator has been shown to be a very useful operational measure of the purity of the quantum state. The most important and interesting work to understand relation between entropy and information was done by Shannon [

Recently, much attention has been drawn to squeezing in an ensemble of atoms illuminated with light, involving quantum noise and atomic spin polarization measurement [

The entropy squeezing and variance squeezing for the entangled sate of a single two-level atom interacting with a single electromagnetic field in a squeezed vacuum a broad bandwidth are studied [

In this paper, we focus our attention to a quantum optical model, where a single two-level atom via one photon process, interacts with a single quantized cavity mode of the radiation field. Then the Hamiltonian of the above system of interest may be written as

(1)

where

We denoted by

where,

For convenience, we take,

where,

where

While

where

Clearly

where,

At any time

for the state vector

and the density matrix of system is

where the coefficient

where,

We mainly devote the present section to considering the atomic inversion, from which the phenomenon of collapses and revivals can be observed. However, we shall first introduce some expressions for the probability amplitude. The expressions

where

Another important quantity one may consider is the atomic inversion

Thus from Equation (13) and after some rearrangements, we can obtain

In this paper, we use the Heisenberg uncertainty relation (HUR) to study the squeezing information entropy. It has been pointed out that the Heisenberg uncertainty relation (HUR) cannot give sufficient information on the atomic squeezing in some cases [

where the Pauli operators

An optimal entropic uncertainty relation for sets of

where

The probability distribution for N possible outcomes of measurements for an arbitrary quantum state of an operators

To obtain the Shannon information entropies of the atomic operators

so that

where

Since the uncertainty relation of the entropy can be used as a general criterion for the squeezing of an atom, therefore for a two-level atom. For

In other words, if we define

It is evident that

Now, we define the squeezing of the atom using EUR (16), named squeezing entropy [

n what follows we shall consider the effect of detuning parameter and the nonlinear Kerr like medium on the dynamical behavior of the squeezing entropy of the system under consideration.

On the basis of the analytical solution presented in the previous section, we shall study numerically the dependence of the

factor

But Figures 1(d)-(f), present that with increasing the parameter

This is because

where

Also there is a great entropy squeezing on the

In order to see how the entropy squeezing influenced by the nonlinear medium parameter, we set two different values of the Kerr-medium parameter. Figures 4(a), (d) and (g) show the absence of the nonlinear parameter but

the other figures show the influence of the nonlinear parameter. It is remarkable that the nonlinear parameter leads to the following effects, no entropy squeezing occurs on

Also the entropy squeezing factors

In order to explain the difference between the sources of the radiation fields of the coherent state and the nonlinear coherent state, we must set different values of the Lamb-Diche parameter

We have treated the entropy squeezing of a two-level atom when the field is initially prepared in the NCS. New results can be explored as follows.

1) When the field is initially in an NCS, then rich features of the entropy squeezing can be observed, then the entropy squeezing is a good measurement of the information concerning the case of trapped ion.

2) There is a great difference between the influence of the nonlinearity function f(n), which is used in the de-

scription of the motion of a trapped ion through Lamb-Dicke parameter