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Journal of Physics A Mathematical and Theoretical

Time evolution of the atomic inversion for the generalized Tavis–Cummings model—QIM approach

N. M. Bogoliubov, Igor Ermakov, Andrei V. Rybin

Published Oct 23, 2017• 8 citations
Publisher

Abstract

Abstract A model describing the interaction between two-level atoms and a single mode field in an optical cavity enclosed by a medium with Kerr nonlinearity is considered in our paper. We study the model within the framework of the analytical solution obtained by application of the quantum inverse method (QIM). Time evolution of the atomic inversion is calculated and transition elements of photons are represented in the determinant form. The obtained answers depend on the solutions of the Bethe equations. We provide the numerical solutions of these equations for the different parameters of the model and study the behaviour of certain dynamical correlation functions.

DOI
10.1088/1751-8121/aa8c6a

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Abstract A model describing the interaction between two-level atoms and a single mode field in an optical cavity enclosed by a medium with Kerr nonlinearity is considered in our paper. We study the model within the framework of the analytical solution obtained by application of the quantum inverse method (QIM). Time evolution of the atomic inversion is calculated and transition elements of photons are represented in the determinant form. The obtained answers depend on the solutions of the Bethe equations. We provide the numerical solutions of these equations for the different parameters of the model and study the behaviour of certain dynamical correlation functions.
The model describing interaction between two-level atoms and a single mode field in an optical cavity enclosed by a medium with Kerr nonlinearity is considered in our paper. We study the model within the framework of the analytical solution obtained by application of the quantum inverse method (QIM). The dynamic correlation functions are calculated and transition elements of photons are represented in the determinant form. The obtained answers depend on the solutions of Bethe equations. We provide the numerical solutions of these equations for the different parameters of the model and study the behaviour of certain dynamical correlation functions.
Quantum inverse methods are developed for the exact solution of models which describe N two-level atoms interacting with one mode of the quantized electromagnetic field containing an arbitrary number of excitations M. Either a Kerr-type nonlinearity or a Stark-shift term can be included in the model, and it is shown that these two cases can be mapped from one to the other. The method of solution provides a general framework within which many related problems can similarly be solved. Explicit formulae are given for the Rabi splitting of the models for some N and M, on- and off-resonance. It is also shown that the solution of the pure Tavis - Cummings model can be reduced to solving a homogeneous ordinary differential equation of second order. Generalization of the method to the case of several cavity modes is indicated.
We extend the standard Jaynes-Cummings model to include transient effects due to a modulation of the atom-field coupling. The exact solution for such a generalized model is found in terms of Bessel functions for the atom-cavity mode interaction modulated by the exponential function. This function is incorporated into the basic Hamiltonian phenomenologically for modelling the processes of adiabatic switching on or switching off the interaction of a two-level atom (at rest) with a quantized mode. The dynamical behaviour of the atomic inversion for such processes is plotted and compared with the atomic dynamics for the standard model. If we take into account that, in micromaser experiments, Rydberg atoms move through a cavity with a spatially inhomogenous field, we deal with the Jaynes-Cummings model with a spatial modulation the form of which is defined by the mode-shape function depending, in turn, on the geometrical form of resonator. In the cases of a rectangular and a spherical cavity the analytical solutions for atomic inversion are found in the explicit form in the weak-coupling limit. We consider in detail the time evolution of the atomic inversion while the atom passes through a rectangular cavity. Contrary to the standard model the initial inversion can be exactly restored during the atomic transit depending on the relation between the detuning, the atomic velocity and the number of half-wavelengths of the chosen mode. The numerical solutions of the basic equations are also obtained and plotted for rectangular and spherical resonators.
Dynamics of entanglement due to intensity-dependent interaction between a two-level atom and a single-mode electromagnetic field in a Kerr medium is studied. The form of the interaction is such that the Hamiltonian evolution is exactly solvable. The Hamiltonian is shown to be a deformed Jaynes-Cummings model admitting a closed, symmetric algebra. Dynamics of population inversion and atom-field entanglement are studied taking the initial state of the field to be either a coherent state or a squeezed vacuum. Analysis is extended to the case of a two-mode cavity field interacting with a two-level atom. For the two-mode case, the initial field is a pair coherent state or a two-mode squeezed vacuum. Effects due to nonlinearity, intensity-dependent interaction and detuning on the dynamics are discussed and compared with those of the single-mode case.
The collapse-and-revival phenomenon in the population inversion and the time evolution of the quasiprobability Q function of an atom undergoing a two-photon process are investigated when the cavity is supposed to be filled with a Kerr-like medium. An exact expression for the atomic population inversion has been given for a field initially in the coherent state. The closed form of this expression as well as that of the Q function are obtained under the high-field approximation. Also, the tendency toward inhibited decay of the excited state has been explicitly shown. This study reveals how the simultaneous occurrence of two nonlinear processes affects the atomic dynamics.