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arXiv:quant-ph/0207151 · Journal of Optics B Quantum and Semiclassical Optics

Evolution of theN-ion Jaynes–Cummings model beyond the standard rotating wave approximation

Paolo Aniello, A. Porzio, S. Solimeno

Published Jun 1, 2003• 15 citations
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Abstract

A unitary transformation of the N -ion Jaynes–Cummings Hamiltonian is proposed. It is shown that any approximate expression of the evolution operator associated with the transformed Hamiltonian retains its validity independent of the intensity of the external driving field. In particular, using the rotating wave approximation, one obtains a solution for the N -ion Jaynes–Cummings model which improves the standard rotating wave approximation solution.

DOI
10.1088/1464-4266/5/3/353

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A unitary transformation of the N -ion Jaynes–Cummings Hamiltonian is proposed. It is shown that any approximate expression of the evolution operator associated with the transformed Hamiltonian retains its validity independent of the intensity of the external driving field. In particular, using the rotating wave approximation, one obtains a solution for the N -ion Jaynes–Cummings model which improves the standard rotating wave approximation solution.
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.
In this paper, we investigate various dynamical properties of the Jaynes–Cummings (JC) model beyond the rotating wave approximation (RWA). We first show that in the absence of the RWA, the JC Hamiltonian can be transformed into an intensity-dependent Hamiltonian. Corrections produced by the counter-rotating terms (CRTs) appear in the first order as the intensity-dependent detuning, i.e. the dynamical Stark shift and in the second order as the intensity-dependent atom–field coupling. Then, by determining the atom–field wavefunction evolution, we study the effects of CRTs on various dynamical properties of the JC model, including the atomic population inversion, atomic dipole squeezing, atomic entropy squeezing, photon counting statistics, quadrature field squeezing, field entropy squeezing and quantum phase properties of the cavity field.
In this paper we define, in the context of dynamical algebras, a set of operators that are suitable for studying any relevant quantity related to the two-level Jaynes–Cummings model (JCM). We study the usual JCM with and without the rotating wave approximation (RWA), and then we add the presence of an external field. In this last case we find that in the strong driving regime there are significant differences between the results obtained with the RWA and without it. This holds even for the case in which the usual requirements for applying the RWA without an external driving are fulfilled.
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 interaction between an atom and a one mode external driving field is an ubiquitous problem in many branches of physics and is often modeled using the Rabi Hamiltonian. In this paper we present a series of analytically solvable Hamiltonians that approximate the Rabi Hamiltonian and compare our results to the Jaynes-Cummings model which neglects the so-called counter-rotating term in the Rabi Hamiltonian. Through a unitary transformation that diagonalizes the Jaynes-Cummings model, we transform the counter-rotating term into separate terms representing several different physical processes. By keeping only certain terms, we can achieve an excellent approximation to the exact dynamics within specified parameter ranges.