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Optics and Spectroscopy

Energy spectrum of the Hamiltonian of the Jaynes-Cummings model without rotating-wave approximation

É. A. Tur

Published Dec 1, 2001• 11 citations
Publisher

Abstract

The energy spectrum of the Hamiltonian of the Jaynes-Cummings model without a rotating-wave approximation is studied. The trajectories of eigenvalues as functions of the dimensionless coupling constant are constructed. It is shown that the spectrum approaches the equidistant spectrum, i.e., the oscillator spectrum, as the coupling constant increases. As a result, each energy level becomes doubly degenerate.

DOI
10.1134/1.1429703

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The energy spectrum of the Hamiltonian of the Jaynes-Cummings model without a rotating-wave approximation is studied. The trajectories of eigenvalues as functions of the dimensionless coupling constant are constructed. It is shown that the spectrum approaches the equidistant spectrum, i.e., the oscillator spectrum, as the coupling constant increases. As a result, each energy level becomes doubly degenerate.
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.
In this paper we have investigated the eigen energy spectrum of the k-photon Jaynes-Cummings (JC) model without the rotating-wave approximation (RWA). Our analysis shows that for k > 2 the k-photon JC model without the RWA does not have eigenstates in the Hilbert space spanned by the photon number states, i.e. the model becomes ill-defined. Hence, the k-photon JC model without the RWA is qualitatively different from the RWA counterpart which is valid for all values of k and the coupling parameter.
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.
We have investigated the eigenenergy spectrum of the two-photon Jaynes-Cummings (JC) model with and without the rotating-wave approximation (RWA). Our analysis has indicated that the counter-rotating term dramatically changes the nature of the RWA energy spectrum and that the non-RWA spectrum can be approximated by the RWA spectrum only in the range of a sufficiently small coupling constant. Furthermore, unlike the one-photon counterpart, the two-photon JC model without the RWA is well defined only if the coupling parameter is below a certain critical value. As a result, the dynamics of the two-photon JC model without the RWA is significantly different from its RWA counterpart. For instance, the counter-rotating term can dramatically enhance the field squeezing effect. Besides, we would expect that the quantum dynamics of the two-photon JC model without the RWA is qualitatively different from that of the usual one-photon case.
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.