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In this paper, we have presented the numerical investigation of the geometric phase and field entropy squeezing for a two-level system interacting with coherent field under decoherence effect during the time evolution. The effects of the initial state setting and atomic dissipation damping parameter on the evolution of the geometric phase and entropy squeezing have been examined. We have reported some new results related to the periodicity and regularity of geometric phase and entropy squeezing.

The interaction between the radiation field and matter is an important quantum optical problem that lies at the heart of many problems in laser physics and quantum optics [1,2]. A solution for this problem has been presented through the well known the Jaynes-Cummings Model (JCM) [

Recently, much research attention on the quantum phases such as the Pancharatnam phase which was introduced in 1956 by Pancharatnam [

Presently, the models of quantum computation in which a state is an operator of density matrix are developed [19,20]. It is shown that the geometric phase shift can be used for generation fault tolerance phase shift gates in quantum computation [

It is well known that the study of the field-atom interaction in the presence of decoherence is an important topic in quantum optics and information. In this present contribution, our main interest is to investigate and discuss in detail the time evolution of the field entropy squeezing (FES) and geometric phase. Furthermore, we present the relationship between them in terms of the parameters involved in the system under consideration. This leads to address the question: can the FES and GP be used as a parameter dynamical properties of the system in the presence of decoherence? Also, what is the effect of the cavity damping parameter and initial atomic state position on the behavior of geometric phase and entropy squeezing?

The outline of the paper is as follows: In the next section, the physical model is described; the field density matrix is defined. The geometric phase and field entropy squeezing of the system under consideration are presented in Section 3, 4 respectively. In Section 5 we will examine in detail the effect of damping and initial state setting the geometric phases and field entropy squeezing of the present system. The paper ends with the conclusions in Section 6.

In this section, we introduce the numerical investigation of the master equation for the interaction between a twolevel and coherent field in the presence of damping such as cavity damping and energy dissipation.

In this case, the master equation for the density under the phase damping of the cavity field and at a zero temperature bath, can be written as

where γ_{C} is the cavity -damping constant, are the annihilation (creation) operator of the field mode and is the interaction Hamiltonian between a single twolevel atom and field mode which is given by

where is the decay coefficient from the upper level of the two-level atom i.e. is the dissipation coefficient or damping effect. by substituting from Equation (2) into equation then we have the system of deferential equation as follows:

The information about the system is involved in the field density matrix, which can be written as

In the following section, we will study the dynamical properties of the GP and FES based on two values of the initial atomic position with and without decay effect.

For the quantum system evolving from an initial wave function to a final wave function, if the final wave function cannot be obtained from the initial wave function by a multiplication with a complex number, the initial and final states are distinct and the evolution is noncyclic. Suppose state evolves to a state after a certain time t. If the scalar product [21,28]

the equation (5) can be written as

where is a real number, then the noncyclic phase due to the evolution from to is the angle This noncyclic phase generalizes the cyclic geometric phase since the latter can be regarded as a special case of the former in which. Determination of the phase between the two states for such an evolution is nontrivial. Pancharatnam prescribed the phase acquired during an arbitrary evolution of a wave function from the vector to

as

Subtracting the dynamical phase from the Pancharatnam phase, we obtain the geometric phase. Here, for the time-dependent interaction and considering the resonant case, an exact expression of the geometric phase can be obtained as

Important tools have developed in recent years for the systematic exploration of the squeezing of quantum systems. In this regard, the relationship between squeezing and entangled state transformations has discussed [

has been presented [

The position and momentum entropy of the field are defined as [

where for the position and momentum respectively.

The Fock state of the field can be written in terms of the position and the momentum representation as follows (Equation (8)):

where are the Hermite polynomials. The entropy uncertainty relation of position and momentum is given by [

In this considered case, the FES in terms of the variable is given by

where, the position (momentum) of the field is squeezed in entropy.

Based on Equations (6) and (10), we present some important results for the effect of different parameters on the dynamical properties of GP and FES in the presence of dissipation. We recall that the time t has been scaled; one unit of time is given by the inverse of the coupling constant λ. Here, our primary aim is to provide that the geometric phase, as a goal feature of quantum evolution, is capable to reflect the information on the character of interaction between a two-level atom and its environment. In

–π to π when when θ = π/4 and the other parameters are the same as in

Now, we are in a position to discuss the dynamics of GP as a function of the initial atomic position parameter θ and for fixed value of the scaled time. It is noticed that GP has a periodic behavior with period which equals the scaled time. On the other hand GP is affected by the atomic dissipation when, while the GP does not affected by the atomic dissipation when (see

The time evolution of the field entropy squeezing components shown in

Now, we shed some light to the effect of dissipation of the evolution of. It is noticed that when the effect of dissipation is considered the situation is partially changed where the amplitude of are deceased gradually and thereafter (see

In this paper, we have investigated the geometric phase and entropy squeezing for a two level atom interacting with field in the presence of dissipation. We have examined the effects of initial atomic state conditions and cavity damping parameter on the evolution and properties of geometric phase and entropy squeezing. Our results show

that some new important and interesting features regularity and periodicity of the geometric phase. On the other hand, we have found that there a monotonic relation the geometric phase and entropy squeezing.

It is well known that the study of the physical properties of the atom-field interaction is an important topic in quantum optics and information. In this way, our results show that the interaction between two-level with coherent field in the presence of dissipation provide a much richer structure than the absence the dissipation effect.

We have shown that the geometric phase plays a crucial role in a variety of physical problems and has observable consequences in a wide range of quantum systems by testing the fundamentals of quantum dynamics and details of interactions modeled by Hamiltonians. Also, we observed a very close connection between the structure of the entropy squeezing and geometric phase due to the dissipation effect for different values the initial state parameter, θ.

Due to its simplicity and clear geometric structure the identified definition the mixed state geometric phase may be important for quantum information and quantum computing applications. Also, an important future investigation will be the study of the effect of the both of cavity and atomic damping on the evolution of geometric phase and field entropy squeezing.