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Physical Review A

Exact solution in the semiclassical Jaynes-Cummings model without the rotating-wave approximation

Magda Jeleńska-Kuklińska, Marek Kuś

Published Mar 1, 1990• 18 citations
Publisher

Abstract

Following the lines sketched by Kujawski [Phys. Rev. A 37, 1386 (1988)] for the semiclassical version of the Jaynes-Cummings model without the rotating-wave approximation, we have obtained an alternative analytic periodic solution given in terms of elliptic functions. We discuss domains of values of the coupling constant and the detuning for which known yet exact periodic solutions exist.

DOI
10.1103/physreva.41.2889

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Following the lines sketched by Kujawski [Phys. Rev. A 37, 1386 (1988)] for the semiclassical version of the Jaynes-Cummings model without the rotating-wave approximation, we have obtained an alternative analytic periodic solution given in terms of elliptic functions. We discuss domains of values of the coupling constant and the detuning for which known yet exact periodic solutions exist.
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.
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.
This report presents an approach to the exact solutions of the time-dependent Jaynes-Cummings (J-C) model without the rotating wave approximation (RWA). It is shown that there is a squeezing-operator unitary transformation for relating the J-C model without RWA and the one with RWA. Thus by using an appropriate squeezing unitary transformation, the time-dependent J-C model without RWA can be transformed into the one with RWA that has been exactly solved previously, and based on this one can readily obtain the exact solutions of the time-dependent Jaynes-Cummings (J-C) model without RWA. The approach presented here also shows that we can treat these two kinds of time-dependent J-C models (both with and without RWA) in a unified way.
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.