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Physical Review A

Unified and standardized procedure to solve various nonlinear Jaynes-Cummings models

Xiaoxue Yang, Ying Wu, Yuanjie Li

Published Jun 1, 1997• 80 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.55.4545

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Physical Review A•Oct 1, 1995•35 citations
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.
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 unitary transformation method is utilized for the closed form solution of the nonlinear generalization of the Jaynes–Cummings (JC) model, which includes arbitrary multiphoton and intensity-dependent interactions between radiation and matter. Specifically, the unitary transformation leading to the diagonalization of the relevant Hamiltonian is established and energy levels and eigenstates are derived. It is also shown that the results for the JC model can be extended to the anti-Jaynes–Cummings (AJC) model through the map between the two models.
A so far most generalized Jaynes-Cummings (JC) model, which includes not only various ordinary JC models but also a special model describing spin-orbit interaction, is investigated from an algebraic point of view. We find that an alternative representation of su(2) algebra arises naturally from the system. This structure plays an essential role in the generalized system. As one result, the system behaves like a spin-1/2 system moving in a magnetic field. So it is rather easy to obtain the energy levels, eigenstates, and the time evolution operator, or any other quantities of physical interest. Two special cases are considered. Finally, a possible generalization to the three-level system is discussed.