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Journal of Physics B Atomic Molecular and Optical Physics

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

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

Published Apr 15, 2010• 45 citations
Publisher

Abstract

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.

DOI
10.1088/0953-4075/43/9/095507

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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.
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.
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.
We introduce the problem of three types of interaction between an N-level quantum system and a two-level atom where three coupling parameters are involved. The system can be deduced from the Heisenberg chain. The canonical transformation is used to remove two coupling parameters from the system and consequently it is reduced to atom–atom interaction. The wave function is calculated using the evolution operator and hence we have managed to obtain the expectation value of some dynamical operators. During our study of the atomic inversion we noted that the collapses period is shifted up when we take the effect of λ2 into consideration. While it is shifted up and down in the presence of λ3. The atomic angle plays a crucial role for controlling the degree of entanglement. For the variance squeezing we noted that the coupling parameter λ2 shows amounts of squeezing more than the case of λ3. Similar behavior is noted for the entropy squeezing.
In this paper, we study the interaction between a two-level atom and three types of interaction of three quantized modes of a quantized field, namely: two parametric amplifiers and a frequency converter. The SU(1, 1) algebra is used to represent the combination of the interacting modes. A canonical transformation is used to cast the Hamiltonian into a tangible form. The solution of the Schrodinger equation for the wave function is given analytically. Using this solution we discuss numerically the atomic inversion, the degree of entanglement through the linear entropy and the variance entropy for chosen values of the detuning and coupling parameters. It is shown that the atomic inversion can be controlled through the rotation angle α and the atomic angle θ as well as the Bargmann index k. The degree of entanglement is affected by both α and θ in addition to the ratio ϵ. Variance squeezing is sensitive to changes in the atomic and the phase angles besides the parameter k. For entropy squeezing, the effective parameters are α, θ and k.
We investigate the evolution of the atomic quantum entropy and the atom–field entanglement in a system of a Λ-configuration three-level atom interacting with a two-mode field with additional forms of nonlinearities of both the field and the intensity-dependent atom–field coupling. With the derivation of the unitary operator within the frame of the dressed state and the exact results for the state of the system we perform a careful investigation of the temporal evolution of the entropy. A factorization of the initial density operator is assumed, considering the field to be initially in a non-correlated two-mode squeezed coherent or binomial state. The effects of the mean photon number, detuning, Kerr-like medium and the intensity-dependent coupling functional on the entropy are analysed.