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Optics and Spectroscopy

Jaynes–Cummings Model: Solutions without Rotating-Wave Approximation

É. A. Tur

Published Oct 1, 2000• 46 citations

Abstract

The Jaynes–Cummings model in the general nonresonant case without rotating-wave approximation is considered. The analysis is carried out using the resolvent formalism. It is shown that one, given the matrix of Hamiltonian resolvent, can easily find all basic physical quantities corresponding to the given model. The matrix of the resolvent of the total Hamiltonian for the given model is found. Matrix elements of the resolvent are expressed in terms of continued fractions. It is shown that these fractions uniformly converge to meromorphic functions, which corresponds to a purely point spectrum of the total Hamiltonian. The time evolution in the case of exact resonance for different coupling constants is numerically studied. It is shown that the rotating- wave approximation is not satisfactory for large coupling constants even in the case of exact resonance. In this case, probabilities of multiphoton transitions increase with increasing coupling constant.

DOI: 10.1134/1.1319919

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The Jaynes–Cummings model in the general nonresonant case without rotating-wave approximation is considered. The analysis is carried out using the resolvent formalism. It is shown that one, given the matrix of Hamiltonian resolvent, can easily find all basic physical quantities corresponding to the given model. The matrix of the resolvent of the total Hamiltonian for the given model is found. Matrix elements of the resolvent are expressed in terms of continued fractions. It is shown that these fractions uniformly converge to meromorphic functions, which corresponds to a purely point spectrum of the total Hamiltonian. The time evolution in the case of exact resonance for different coupling constants is numerically studied. It is shown that the rotating- wave approximation is not satisfactory for large coupling constants even in the case of exact resonance. In this case, probabilities of multiphoton transitions increase with increasing coupling constant.

The eigenenergy spectrum of the Jaynes–Cummings (JC) model with and without the rotating-wave approximation (RWA) is investigated. The numerical analysis indicates that the non-RWA spectrum can only be approximated by the RWA in the range of sufficiently small coupling constant and detuning. In other region, the counter-rotating terms remarkably change the nature of the RWA energy spectrum. A simple expression with high accuracy for ground eigenenergy and eigenvector for non-RWA is available. The ground eigenvector for non-RWA shows that the ground state is not a dark state and very different from that of RWA which is a dark state.

In the semiclassical version of the Jaynes-Cummings1 model without the rotating-wave approximation, an exact periodic solution has been obtained and studied for some specific values of the coupling constant and the detuning. It is shown that when the atomic frequency is equal to three times the cavity frequency and the coupling constant is arbitrary, the oscillation amplitudes are maximal. The Fourier spectrum of the inversion is discrete. As is known (see, for example, Ref. 2) chaos has been found for the semiclassical Jaynes-Cummings model.