Some Features on Entropy Squeezing for Two-Level System with a New Nonlinear Coherent State ()
1. Introduction
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 [1] , who introduced the entropy (Shannon entropy) into communications theory. Recently it has been shown that nonlinear coherent states are useful in the description of the motion of a trapped ion and various non-classical properties of such states have also been studied [2] . We note that in Refs [2] and [3] nonlinear coherent states have been defined as the right eigenstate of a generalized annihilation operator 
(which emerges from the Hamiltonian describing the dynamics) this is because in the case of nonlinear algebras the commutator 
is not a constant or a linear function of the generators of the algebra but nonlinear in the in the generators. As a consequence it is difficult to obtain an explicit form of nonlinear coherent state constructed via the displacement operator technique. The nonlinear coherent state (NCS) was introduced as a new state of the source of the coherent field to describe some of the non-classical properties like squeezing and Sub-Poissnian behavior [4] . There are some previous studies on the entropy squeezing of a two-level atom, such as one photon transition [5] , the nonlinear Kerr medium [6] and degenerate two-photon process [7] . The authors of these papers have focused only on the initial coherent state of the field.
Recently, much attention has been drawn to squeezing in an ensemble of atoms illuminated with light, involving quantum noise and atomic spin polarization measurement [8] , and quantum-controlled few-photon states generated by squeezed atoms [9] . These studies of atomic squeezing are based on the Heisenberg uncertainty relation (HUR), which is regarded as the standard limitation on measurements of quantum fluctuations. HUR is formulated in terms of the variances or standard deviations of the system observable. As an alternative to the HUR, Hirschman [10] studied quantum uncertainty by using quantum entropy theory, and obtained an entropic uncertainty relation for position and momentum which can overcome the limitations of the HUR.
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 [11] . Also, the similarities and differences of both reservoirs for the two different models have been explained through some calculations, such as the atomic inversion and the von Neumann entropy. It is to be noted that considering a mode structure plays a role like the squeezing parameter in the case of a squeezed vacuum reservoir. Also, the entropy squeezing of a two-level atom driven by a strong classical field and damped into a modeled reservoir with non-flat density of modes has been investigated [12] . On the other hand, the dynamics of a single atom entropy squeezing of the two-qubit system, in the presence of local squeezed reservoirs, has been discussed. Our aim in the present paper is to investigate the entropy squeezing of a two level atom when the initial state of the field is taken to be NCS and discuss different features of entropy squeezing in the case of NCS. These features are connected with the Lamb-Dick parameter. Here we also examine the influence of a nonlinear medium and the detuning parameter on the squeezing parameter of the atomic operators 
and
. The organization of the paper is arranged as follows. In Section 2, we present a brief review of the Hamiltonian model and give an exact expression for the density matrix
. In Section 3, we employ the density matrix to investigate the properties of the entropy squeezing. Finally, we give our discussion in Section 4.
2. Dynamics of One Photon JCM
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 
is the field frequency, 
is the atomic frequency, 
and 
are the annihilation and the creation operators for the mode of the cavity field satisfying 
and 
and 
are the atomic spin operators defined by
(2)
We denoted by 
and 
the upper and lower states of the atom, respectively and 
is the effective coupling constant. Also, we denoted by 
the dispersive part of the third-order nonlinearity of the Kerr-like medium, with the detuning parameter 
Therefore, we employ the unites of 
The effective Hamiltonian can be written as
where,
(3)
For convenience, we take, 
, 
and assume that the atom is initially in the superposition state
. Also, we assume that the field is initially in the new nonlinear coherent state (NCS),
(4)
where, 
is the distribution function of the NCS. The NCS is defined as 
i.e., it is a coherent state (CS) corresponding to the second algebra [13] . One can write 
in the following form [4] -[7] [13]
where 
is a normalization constant, which can be determined from the condition 
and is given by
While 
and the coefficients 
it is clear that for different choices of the nonlinearity function, we shall get different nonlinear coherent states. In the present case we choose a nonlinearity function, which has been used in the description of the motion of a trapped ion [14] .
where 
is known as the Lamb-Dick parameter and 
are generalized Lagurre polynomials given by
(5)
Clearly 
when 
and in this case nonlinearity coherent states become the standard coherent states. However, when 
nonlinearity starts developing with degree of depending on the magnitude of 
[14] . Then the initial state of the atom-field coupling system reads as
(6)
where, 
, 
and
. Here 
denotes the initial coherence of the two-level atom and 
is the relative phase between the upper and lower states of the two-level atom. Thus the initial density operator of the system is given by 
where, 
and
, 
describes the initial values for the field-atom density operator.
At any time 
the solution of the Schrödinger equation
(7)
for the state vector 
with the initial condition (6) is
(8)
and the density matrix of system is
(9)
where the coefficient 
and 
are given by
(10)
where, 
, 
with 
is the Rabi frequency which depends on the detuning parameter and nonlinear medium parameter.
3. Atomic Inversion
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 
and 
represent the probabilities that at time 
the field has 
photons present and the atom is in level 
and 
respectively. The probability 
that there is 
photons in the field at time 
is therefore obtained by taking the trace over the atomic states, i.e.,
(11)
where 
is the probability that there are 
photons present the field at time
, which is given for a NCS for the field by
(12)
Another important quantity one may consider is the atomic inversion 
which is related to the probability amplitudes 
and 
by the expression
(13)
Thus from Equation (13) and after some rearrangements, we can obtain
(14)
4. Entropy Squeezing
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 [13] . For instance, for a two-level atom, characterized by Pauli operator 
and
, the uncertainty relation is given by
where the Pauli operators 
and
, satisfying the commutation 
and 
. In this way, the fluctuation in the component 
of the atomic dipole is said to be squeezed if 
satisfies the condition
(15)
An optimal entropic uncertainty relation for sets of 
complementary observables with non-degenerate eigenvalues in an even N-dimensional Hilbert space has been recently investigated using quantum entropy theory [5] . It takes the form
(16)
where 
represents the information entropy of the variable
. The aim of this paper is to use entropy uncertainty relation EUR (16) as a general criterion for the squeezing in terms of information entropy for a two-level atom in the Jaynes-Cumming model with one-photon process in a non-linear Kerr medium.
The probability distribution for N possible outcomes of measurements for an arbitrary quantum state of an operators 
is
, where 
is an eigenvector of the operator 
such that
. The corresponding Shannon information entropies are then defined as
(17)
To obtain the Shannon information entropies of the atomic operators
, 
and 
for a two-level atom, with
, one can use the reduced atomic density operator
, thus we have the following expression,
so that
(18)
(19)
(20)
where 
and 
are given by
(21)
(22)
(23)
(24)
(25)
(26)
(27)
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
, we have 
and the Shannon information entropies of the operators 
will satisfy the inequality
(27)
In other words, if we define 
then we can write
(28)
It is evident that 
corresponds to the atom being in a pure state and 
corresponds to the atom being in a mixed state. The EUR (16) shows the impossibility of simultaneously having complete information about the observables 
and 
where 
and 
respectively measure the uncertainties of the polarization components 
and 
Now, we define the squeezing of the atom using EUR (16), named squeezing entropy [5] . The fluctuation of the component 
of the atom dipole are said to be “squeezed in entropy” if the information entropy 
of 
satisfies the condition
(29)
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.
5. Numerical Computation
On the basis of the analytical solution presented in the previous section, we shall study numerically the dependence of the 
entropy squeezing and the atomic inversion
, for various parameters of the one photon model. We recall that time 
has been scaled; one unit of time is given by the inverse of the coupling constant
. In all our plots we take 
to represent the Lamb-Dicke parameter and the parameter
. For the case of 
(Figure 1), we investigate the influence of the mean photon number on the entropy squeezing
factor
, and population inversion
. We notice that no squeezing occurs on the atomic variable 
as the parameter increased (see Figures 1(a)-(c)).
But Figures 1(d)-(f), present that with increasing the parameter
, the entropy squeezing on the atomic variables 
is increased. Also the quantum revival is increased but the collapse phenomenon is decreased as 
increase (see Figures 1(g)-(i)). These results agree with the case of coherent field but the difference between the coherent state and the nonlinear coherent state appear on the collapse and revival phenomena. For fixed values of the detuning 
the detuning parameter and the case of absence the nonlinear medium parameter 
the entropy squeezing factor 
and 
are plotted as a function of the scaled time
where we set three different values of 
i.e.
. We noticed no entropy squeezing on the atomic variable 
as the detuning parameter increases; see Figures 2(a), (c). Although there is great entropy squeezing on when
, see Figure 2(d) and no squeezing on 
when 
see Figures 2(c) and (f).
This is because 
then
(30)
where 
Also one can see as the detuning parameter increased the period of evolution increased. Figure 3 explains the effect of the detuning parameter when the atom is initially in the superposition state of the upper and lower atomic states and all the other parameters are the same as Figure 2. It is observed that the situation is completely different between the excited and superposition states (see Figures 2 and 3) First we see that when the detuning increases, the entropy squeezing on the atomic variables 
increases.
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 
when 
but there is entropy squeezing on 
When 
there is an optimal entropy squeezing on both the atomic variables 
and 
For large values of 
as 
one can see there are more and more optimal entropy squeezing on both of the atomic variables 
and 
Also the entropy squeezing factors 
and 
are periodic functions, which have a period 
and 
respectively (see Figures 4(c) and (f)).
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 
When
, then the field is initially in the coherent state, but when 
the field is initially in the nonlinear coherent state. Figures 5(a), (d) and (g) show the entropy squeezing in the case of coherent state, which has been examined in many papers [2] [5] [7] . But Figures 5(b), (e) and (h) show the difference between the coherent and new nonlinear coherent states, where we set 
while we set 
in Figures 5(d), (f) and (i). One can see there is strong entropy squeezing on the atomic variables 
and there is an entropy squeezing and the period of collapse also increases.
6. Conclusions
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 
and Kerr medium nonlinearity through parameter
. The first decreases the entropy squeezing but the second increases entropy squeezing.