E. M. Khalil

Pair entanglement of two-level atoms in the presence of a nondegenerate parametric amplifier

The problem of two two-level atoms in interaction with a nondegenerate parametric amplifier is considered. The analytical solution of the wavefunction is obtained and used to derive the density matrix operator. The temporal evolution of the atomic inversion, the degree of entanglement, as well as the variance and entropy squeezing are discussed. In our analysis, we assumed that the atomic systems are in the excited states and the fields in the squeezed-pair coherent state. It has been shown that the coupling parameter, λ3 (the coupling between the two fields), gets more effective for the case in which the q-parameter is not equal to zero. Also for a strong coupling parameter λ3 the superstructure phenomenon can be observed. In the meantime, as we increase the value of the coupling parameter, the entanglement between the atoms and the fields gets stronger. Also it has been shown that in the presence of the parametric amplifier term, the system never reaches the pure state except during revival periods.

Generation of a nonlinear two-mode Stark shift through the adiabatic elimination method

In this contribution a three-level atom in interaction with a two-mode quantized electromagnetic field, initially prepared in an entangled two-mode coherent state is considered. Through the elimination of an intermediate level by using the adiabatic elimination method a nonlinear Stark shift is introduced. Exact solution of the wave function in the Schrödinger picture is obtained. Some statistical aspects through the effective two-level atom interacting with the mode through multiphotons with the Stark shift are presented. The results are employed to perform a careful investigation of the temporal evolution of the atomic inversion, entropy squeezing. It has been shown that the system is sensitive to any change in the parameter representing the Stark shift. General conclusions reached are illustrated by numerical results.

Quantum Mechanics of Two-Field Parametric Frequency Converter Interacting with a Single Atom

In this communication we deal with a new model Hamiltonian representing the interaction between a two-level atom and two electromagnetic field modes in a cavity. The interaction between the modes has been taken into account and considered to be of a parametric frequency converter type. The model can be regarded as a generalization of two different systems: the Jaynes–Cummings model (atom–field interaction) and the two-mode frequency converter model (field–field interaction). Under a certain condition an exact solution for the equations of motion in the Heisenberg picture is given. The wavefunction in the Schrödinger picture is also constructed and used to discuss some statistical properties related to the model. We assume that the fields are initially in coherent states. We discuss atomic inversion, photon number distribution, squeezing and other phenomena. We show in all cases that the system is very sensitive to any variation in the mean photon numbers.

QUANTUM TREATMENT OF TWO COUPLED OSCILLATORS IN INTERACTION WITH A TWO-LEVEL ATOM: PARAMETRIC AMPLIFIER MODEL

In this communication we handle a modified model representing the interaction between a two-level atom and two modes of the electromagnetic field in a cavity. The interaction between the modes is assumed to be of a parametric amplifier type. The model consists of two different systems, one represents the Jaynes–Cummings model (atom–field interaction) and the other represents the two mode parametric amplifier model (field–field interaction). After some canonical transformations the constants of the motion have been obtained and used to derive the time evolution operator. The wave function in the Schrodinger picture is constructed and employed to discuss some statistical properties related to the system. Further discussion related to the statistical properties of some physical quantities is given where we have taken into account an initial correlated pair-coherent state for the modes. We concentrate in our examination on the system behavior that occurred as a result of the variation of the parametric amplifier coupling parameter as well as the detuning parameter. It has been shown that the interaction of the parametric amplifier term increases the revival period and consequently longer period of strong interaction between the atom and the fields.

The influence of phase damping on a two-level atom in the presence of the classical laser field

In this paper we consider the influence of phase damping on the Jaynes?Cummings model (JCM) in the presence of the classical laser field. It is shown that for the temporal evolution of the atomic inversion a detuning parameter plays a role in delaying the effect of the damping. Our consideration is also extended to discuss the atomic Wehrl entropy and entropy squeezing. For the case of the marginal distribution, it is noted that the damping factor plays a considerable role in reducing the number of the fluctuations in the function behavior. On the other hand the damping factor removes the phenomenon of squeezing from both quadratures of the entropy squeezing.

Nonclassical properties of the integrability condition of the time-dependent SU(2) quantum system

In this paper, we treat the problem of the multi-photon Tavis–Cummings classically, where the effect of time-dependence takes place. Under certain integrability conditions the time-dependent wave function in the Schrodinger picture is obtained. Some statistical properties of the system are discussed based on the solution for the equations of motion in the Heisenberg picture. Using the atomic coherent state as a basis, the phenomenon of squeezing is observed in the quadrature variances Fx and Fy; however, it was pronounced in the quadrature Fy. The phenomenon of squeezing is also reported for the usual atomic state case where squeezing is only observed in the first quadrature Fx. An examination of the correlation function using the atomic state showed us that the system is sensitive to variation in the integrability parameter β. The nonclassical effect is also found to be pronounced for small values of the parameter β.

Linear entropy and squeezing of the interaction between two quantum system described by su (1, 1) and su(2) Lie group in presence of two external terms

A Hamiltonian, that describes the interaction between a two-level atom (su(2) algebra) and a system governed by su(1,1) Lie algebra besides two external interaction, is considered. Two canonical transformations are used, which results into removing the external terms and changing the frequencies of the interacting systems. The solution of the equations of motion of the operators is obtained and used to discuss the atomic inversion, entanglement, squeezing and correlation functions of the present system. Initially the atom is considered to be in the excited state while the other systems is in the Perelomov coherent state. Effects of the variations in the coupling parameters to the external systems are considered. They are found to be sensitive to changing entanglement, variance and entropy squeezing.

Time-dependent interaction between a two-level atom and a su(1,1) Lie algebra quantum system

The problem of the interaction between a two-level atom and a two-mode field in the parametric amplifier-type is considered. A similar problem appears in an ion trapped in a two-dimensional trap. The problem is transformed into an interaction governed by su(1,1) Lie algebraic operators with phase and coupling parameter depending on time. Under an integrability condition, that relates phase and coupling, a solution to the wavefunction is obtained using the Schrodinger equation. The effects of the functional dependence of the coupling and the initial state of the two-level atom on atomic inversion, the degree of entanglement, the fidelity and the Glauber second-order correlation function are investigated. It is shown that the acceleration term plays an important role in controlling the function behavior of the considered quantities.

ENTANGLEMENT OF A TWO-LEVEL ATOM INTERACTING WITH A NEW STRUCTURE OF A GENERALIZED NONLINEAR STARK SHIFT VIA Ξ CONFIGURATION

The problem of a two-level atom interacting with single mode cavity field is considered, however, the optical cavity is filled with new structure of a generalized nonlinear Stark shift via Ξ configuration. One starts with a three-level trapped atom interacting with the quantized field of center of mass motion thus a Hamiltonian for one-phonon process with nonlinearities is derived. Through the elimination of the intermediate level by using the adiabatic elimination method, we generate a new structure of effective Hamiltonian for a two-level atom with a nonlinear Stark shift. The temporal evolution of the atomic inversion is studied, we introduce that in the presence of the Stark shift parameter the atom leaves in a maximal entangled sate. We use the von Neuman entropy to measure the degree of entanglement between the atom and the field. After adding the nonlinear Stark shift the system never reaches the pure state. Also we study the Q-function for obtaining more information in phase space for this system. These aspects are sensitive to changes in the Stark shift parameter. The results shows that the effect of the nonlinearity in the Stark shift changes the quasiperiod of the field entropy and hence the entanglement between the particle and the field.