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International Journal of Modern Physics B

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

Eied Mahmoud Khalil, M. Sebawe Abdalla, A.‐S. F. Obada

Published Jun 30, 2004• 10 citations
Publisher

Abstract

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 Schrödinger 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.

DOI
10.1142/s0217979204025361

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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 Schrödinger 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 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.
In this paper, we study the interaction between the time-dependent field and a two-level atom with one mode electromagnetic field. We consider that the field of photons is assumed to be coupled with modulated coupling parameter which depends explicitly on time. It is shown that the considered model can be reduced to a well-known form of the time-dependent generalized Jaynes–Cummings model. Under special initial conditions, in which the atom and the field are prepared in the excited and the coherent states, respectively, the explicit time evolution of the wave function of the entire system is analytically obtained. Our proposal has many advantages over the previous optical schemes and can be realized in several multiple experiments, such as trapped ions and quantum electrodynamics cavity. The influence of the time-dependent field parameter on the collapses-revivals, the normal squeezing of the radiation, the anti-bunching of photons and the entanglement phenomena for the considered atomic system is examined. The linear entropy, the von Neumann entropy are used to quantify entanglement in the quantum systems. We noticed that these phenomena are affected by the existence of both the time-dependent coupling field and detuning parameters.
We study the quantum dynamics of a two-mode two-photon Jaynes–Cummings model of a two-level atom interacting with the correlated two-mode SU(1, 1) coherent states. The collapse and revival patterns in this case are noted to be quite different from those in the case of uncorrelated coherent states and correlated pair-coherent states. As an example of the time evolution of the field statistics we show that violation of the Cauchy–Schwartz inequality by the initial field states may be nullified. We also discuss the evolution of the variance of the two-mode field quadrature operator.
Squeezed states of the Jaynes-Cummings (JC) model are constructed and the statistical properties of the field oscillator in these states are studied. The presence of interaction between the field oscillator and the atom results in the appearance of slow temporal variations in the average values of the dynamic quantities. Two cases essentially differ: When only one of two modes of the JC model is excited, the variations in time, in particular, of the average value of the coordinate are governed by the mode anharmonicity. In the case when both modes of the JC model with close frequencies are excited, the temporal variations represent collapse and revival of oscillations with the frequency 2kn+12k\sqrt {n + 1} 2kn+1​ , where k is the constant of the atom-field oscillator interaction and n is the number of photons. The constructed squeezed vacuum states are not two-photon states, because the probability of finding an odd number of photons is not equal to zero and varies with time.
We consider the interaction between a two-level atom and a quantum system that consists of three electromagnetic fields. An analytic solution is provided for the wave function of a pairwise mutual interaction between a two-level atom and three modes using a frequency converter. SU(2) group generators are used to describe these field mode interactions. In addition, a canonical transformation is employed in order to convert the Hamiltonian model into a Jaynes–Cumming-like model, which is used to solve the Schrödinger equation. Statistical properties related to the atomic inversion, entanglement, and squeezing phenomena are discussed. Superstructure patterns and partial disentanglement, as well as squeezing swapping between quadratures, are displayed for selected parameters.
In this manuscript, the phenomenon of time dependence is treated using the integral property of a model consisting of interaction of an atom with a cavity in which a bi-mode field of an amplifier type. The exact solution is obtained through the Schrodinger equation, followed by the wave function characteristic of the proposed system. The effect of both the deformed pair coherent states and the time dependence on some statistical quantities are studied. The phenomena of revival and collapse are studied by computing the atomic inversion. Moreover, the entanglement between the parts of the quantum system is studied, and the periods of squeezing are determined by entropy squeezing and variance squeezing. The results are also compared and statistical quantities linked together.