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International Journal of Modern Physics B

Ground state and thermal entanglement between two two-level atoms interacting with a nondegenerate parametric amplifier: Different sub-spaces

B. Mojaveri, Alireza Dehghani, M. A. Fasihi, T. Mohammadpour

Published Feb 25, 2019• 7 citations

Abstract

In this paper, a Hamiltonian model that includes interaction of two coupled two-level atoms with a nondegenerate parametric amplifier in a cavity is introduced. By using the two-mode squeezing operator and under a certain condition, the introduced Hamiltonian is reduced to a generalized Jaynes–Cummings Hamiltonian. The constants of motion of system imply the existence of a decomposition of the system’s Hilbert space [Formula: see text] into a direct sum of three infinite dimensional sub-spaces, as [Formula: see text]. This decomposition enables us to study ground and thermally induced entanglement between the atoms in each of the sub-spaces as well as whole Hilbert space. The effect of atom–atom and atom–photon couplings on the degree of ground state and thermal entanglement are also investigated using the concurrence measure. It is found that in the subspaces [Formula: see text] and [Formula: see text] for all experimental values of parameters of the system, the atoms are disentangled. It is also observed that in the total space [Formula: see text] the ground state entanglement is always zero, while in the subspace [Formula: see text] it is at its maximum value. Moreover, it is found that in the subspace [Formula: see text] thermal entanglement between the atoms is robust against temperature.

DOI: 10.1142/s0217979219500358

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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.

The analytical solution of the problem of two two-level atoms with degenerate two-photon transitions interacting with a single-mode radiation field in the presence of a parametric amplifier term is presented. The purity of the atomic state has been used to measure the degree of entanglement between the atom and the field. The temporal evolution of variance and entropy squeezing as well as atomic inversion for the single-atom case are studied. It has been shown that maximum squeezing for the variance and entropy squeezing occurs when the ratio between the amplifier coupling λ3 and the field frequency ω equals 0.26. Increasing the value of the ratio λ3/ω further leads to the vanishing of squeezing from the system. It is also noted that the existence of the coupling parameter results in the system never reaching the pure state except at the points of revival times. The Q function has been also considered to give more information in the phase space about the system. These aspects are sensitive to changes in the amplifier parameter.

The detailed analysis of the two-mode quadrature squeezing and statistical properties of light generated by a nondegenerate three-level laser which has a parametric amplifier and coupled with a thermal reservoir is executed. The combination of the master equation and the stochastic differential equation is presented to study the nonclassical features of the light generated by the quantum system. Moreover, with the aid the resulting solutions together with the correlation properties of noise operators, we calculated the quadrature squeezing, entanglement, and mean number of photon pairs of the cavity light. It is found that the external small-amplitude driving radiation induces a strong correlation between the top and bottom states of three-level atoms to produce a high degree of squeezing. Moreover, the presence of a parametric amplifier is found to enhance the degree of squeezing of the cavity light. We have also established that an increase in the mean thermal photon number appears to degrade the squeezing, but enhances the mean number of photon pairs of the cavity light.

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.

We investigate the entanglement and thermal entanglement between moving atom and Single-mode cavity in two-photon model by means of concurrence.It is shown that initial photons number and field-mode structure parameter affect entanglement and thermal entanglement in a certain degree.The evolution period of thermal entanglement doesn't change when the coupling constant between atom and cavity field has different values,while,the time of the system in maximum entanglement decreases when coupling constant reduces.Thermal entanglement decreases with the increase of system temperature when the temperature is under critical temperature.

In this work we investigate the thermal entanglement between two-level atoms and photons in a nonlinear cavity. We consider intensity-dependent couplings and calculate the negativity, as a measure of atom-photonentanglement. The cavity is assumed to be at a temperature T, so that all number of photons, and at the same time, both atomic states, with definite probabilities, are present. We then demonstrate a condition under which the intensity-dependent coupling leads to entanglement. It is also shown that, as in the case of linear Jaynes- Cummings model, the thermal states of atoms and photons are never separable.