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Chinese Physics Letters

Unified Solution Formulas to Jaynes-Cummings Models with Field Nonlinearity and Strong Atom-Field Coupling

Xiaoxue Yang, Ying Wu, Kelin Gao

Published Nov 1, 1998• 13 citations
Publisher

Abstract

Unified and standardized solving formulas for eigenvalues, eigenvectors, evolution operators, time-varying atomic inversion and photon numbers are given to single-mode Jaynes-Cummings models with any forms of intensity-dependent atom-field coupling and the nonlinearity of the quantized photonic field itself such as the Kerr effect.
DOI
10.1088/0256-307x/15/11/008

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In this article we present a simple and unified procedure to solve various nonlinear two-level Jaynes-Cummings models. By establishing their similarity to the model describing a spin- particle in a magnetic field, we obtain the standardized forms, appropriate uniformly for all Jaynes-Cummings models, of eigenvalues, eigenstates, evolution, and atomic inversion operators. In this way, we show that the analytical solution of any single-mode and two-mode nonlinear Jaynes-Cummings model can easily be obtained. We also apply this procedure to a three-mode Jaynes-Cummings model.
A generalization of the time-dependent Jaynes–Cummings Hamiltonian is considered including nonlinear terms in the field and/or an intensity-dependent coupling term. Using algebraic methods we find the corresponding time evolution operator of the problem as a product of exponential operators. Also, we study the effect of the atomic motion in the cavity on its quantum state. We found that, when the atom is vibrating classically with a small amplitude A0 (in comparison with the effective length of the cavity L), i.e. A0 ≪ L, the effects of the atom’s motion are negligible but when A0 ∼ L/4, the state of the atom changes drastically. Our study is devoted to a Jaynes–Cummings Hamiltonian with an additional Kerr medium.
The dynamics of the much studied Jaynes–Cummings model, of a single two-level atom interacting with a single quantized field mode, is simple to solve for because the number of quanta in the system is strictly conserved. Including the effects of cavity damping makes the model significantly more difficult to treat. Here we present a simple derivation of the evolution of the atomic inversion in the presence of cavity damping. This will enable us to explain the physical meaning of the solution, first derived more than 20 years ago, and also to show that it is valid for much stronger damping than was appreciated at that time.
The emission spectrum of a single atom in a two-mode Jaynes-Cummings model including any form of nonlinearity of both the two-mode fields and the intensity-dependent atom-field coupling is studied. A general expression for the emission spectrum is given as an explicit function of any nonlinear terms presented in the model. The effects of the nonlinearities on the spectrum are demonstrated for several important examples. It is shown that features of the emission spectrum are influenced significantly by the kinds of intensity-dependent atom-field coupling and the nonlinearities of the two-mode fields. It is also found in an analytic way that the effects on the emission spectrum of both the specific intensity-dependent nonlinear medium and the initial two-mode field photon statistics can be counterbalanced in some special cases.