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Acta Physica Sinica

Exact calculations of the energy spectra and the dynamical properties of a two-level system

Xuezao Ren, Jiang Dao-Lai, Cong Hong-Lu, Xu Liao, 西南科技大学理学院，绵阳 621010

Published Jan 1, 2009• 5 citations
Publisher

Abstract

Applying the method of coherent-state orthogonalization expansion，the energy spectra and the dynamical properties of the Jaynes-Cummings model with non-rotating wave approximation are calculated exactly. By comparison with the approximate analytical solutions, it is found that the approximate analytical solutions agree with our numerical exact solutions well both in weak coupling regime and in strong coupling regime, but not so well in intermediate strength coupling regime.

DOI
10.7498/aps.58.5406

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Applying the method of coherent-state orthogonalization expansion，the energy spectra and the dynamical properties of the Jaynes-Cummings model with non-rotating wave approximation are calculated exactly. By comparison with the approximate analytical solutions, it is found that the approximate analytical solutions agree with our numerical exact solutions well both in weak coupling regime and in strong coupling regime, but not so well in intermediate strength coupling regime.
The Jaynes-Cummings model (JCM) is studied in the absence of the rotating-wave approximation (RWA) by a coherent-state expansion technique. In comparison with the previous paper in which the coherent-state expansion was performed only to the third order, we carry out in this paper a complete expansion to demonstrate exactly the dynamics of the JCM without the RWA. Our study gives a systematic method to solve the non-RWA problem, which would be useful in various physical systems, e.g., in a system with an ultracold trapped ion experiencing the running waves of lasers.
The Jaynes-Cummings model without the rotating-wave approximation can be solved exactly by extended Swain's ansatz with the conserved parity. The analytical approximations are then performed at different levels. The well-known rotating-wave approximation is naturally covered in the present zero and first approximations. The effect of the counter rotating-wave term emerges clearly in the second order approximation. The concise analytical expressions are given explicitly and can be applicable up to the ultra-strong coupling regime. The preliminary application to the vacuum Rabi splitting is shown to be very successful.
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.