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Journal of Physics A Mathematical and General

Exact isolated solutions for the class of quantum optical systems

Marek Kuś, Maciej Lewenstein

Published Feb 1, 1986• 74 citations
Publisher

Abstract

The authors present a method of obtaining exact isolated solutions for the class of quantum optical systems without the use of a rotating wave approximation. The method generalises the results known from the literature to the case of multilevel atomic systems. The analytical properties of the solutions in the Bargmann representation for the radiation field mode are discussed. The analogues of these solutions for atomic systems interacting with an external field are constructed.

DOI
10.1088/0305-4470/19/2/023

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The authors present a method of obtaining exact isolated solutions for the class of quantum optical systems without the use of a rotating wave approximation. The method generalises the results known from the literature to the case of multilevel atomic systems. The analytical properties of the solutions in the Bargmann representation for the radiation field mode are discussed. The analogues of these solutions for atomic systems interacting with an external field are constructed.
A class of quasi-exact solutions of the Rabi Hamiltonian, which describes a two-level atom interacting with a single-mode radiation field via a dipole interaction without the rotating-wave approximation, are obtained by using a wavefunction ansatz. Exact solutions for part of the spectrum are obtained when the atom-field coupling strength and the field frequency satisfy certain relations. As an example, the lowest exact energy level and the corresponding atom-field entanglement at the quasi-exactly solvable point are calculated and compared to results from the Jaynes–Cummings and counter-rotating cases of the Rabi Hamiltonian.
A quantum model of N two-level atoms coupled by a single-mode radiation field under a restricted rotating-wave approximation is studied, assuming a dipole-interaction approximation and that the total number of excitations of the system is conserved. A solution is found for the case of spontaneous emission when the atoms are initially prepared in the state of complete inversion. This solution is valid for all times. A general solution of the model is also found for any initial configuration of the system, in the cases of both spontaneous and stimulated emission. This solution is in a closed form and valid for times tau < tau max .
One of the simplest models involving the atom-field interaction is the coupling of a single two-level atom with single-mode optical field. Under the rotating wave approximation, this problem is reduced to a form that can be solved exactly. But the approximation is only valid when the two levels are resonant or nearly resonant with the applied electromagnetic radiation. Here we present an analytical solution without the rotating wave approximation and applicable to general atom-field interaction far away from the resonance. We find that there exists remarkable influence of the initial phase of optical field on the Rabi oscillations and Rabi splitting, and this issue cannot be explored in the context of the rotating wave approximation. Due to the retention of the counter-rotating terms, higher-order harmonic appears during the Rabi splitting. The analytical solution suggests a way to regulate and control the quantum dynamics of a two-level atom and allows for exploring more essential features of the atom-field interaction.