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Journal of Physics A Mathematical and General

Algebraic structure and analytic solutions of generalized three-level Jaynes - Cummings models

Quanlin Jie, Shun-Jin Wang, Lian-Fu Wei

Published Sep 7, 1997• 15 citations
Publisher

Abstract

A generalized three-level Jaynes - Cummings model (JCM) which includes various ordinary JCMs is shown explicitly to have an SU(3) structure: the Hamiltonian can be treated as a linear function of the generators of the SU(3) group. Based on this algebraic structure, the exact algebraic solutions of the Schrödinger equation, as well as eigenvalues and eigenstates of the Hamiltonian, are obtained by an algebraic method. Thus the three-level JCM is completely solved algebraically. The SU(N) structure of the N-level JCM is also constructed explicitly and can be solved by the same method.

DOI
10.1088/0305-4470/30/17/021

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A generalized three-level Jaynes - Cummings model (JCM) which includes various ordinary JCMs is shown explicitly to have an SU(3) structure: the Hamiltonian can be treated as a linear function of the generators of the SU(3) group. Based on this algebraic structure, the exact algebraic solutions of the Schrödinger equation, as well as eigenvalues and eigenstates of the Hamiltonian, are obtained by an algebraic method. Thus the three-level JCM is completely solved algebraically. The SU(N) structure of the N-level JCM is also constructed explicitly and can be solved by the same method.
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.
By utilizing the property of the supersymmetric structure in the two-level multiphoton Jaynes-Cummings model, an invariant is constructed in terms of the supersymmetric generators by working in the sub-Hilbert-space corresponding to a particular eigenvalue of the conserved supersymmetric generators. We obtain the exact solutions of the time-dependent Schrödinger equation which describes the time-dependent supersymmetric two-level three-photon Jaynes-Cummings model (TLTJCM) by using the invariant-related unitary transformation formulation. The case under the adiabatic approximation is also discussed. Keywords: Supersymmetric Jaynes-Cummings model; exact solutions; invariant theory; geometric phase factor; adiabatic approximation
The modified Jaynes-Cummings model of a single two-level atom placed in the common domain of two cavities or interacting with two quantized modes is studied by a dynamical algebraic method. With the help of an SU(2) algebraic structure, we then obtain the eigenvalues, eigenstates, time evolution operator and atomic inversion operator for the system. We managed to investigate the modified Jaynes-Cummings model governed by the Milburn equation and present the exact solution of the Milburn equation. PACS: 42. 50. -p, 32. 80. -t, 02. 20. -a, 03. 65. Fd
An approach to exact quantum solutions of the time-dependent two energy level Jaynes–Cummings model with an imaginary photon process is represented in this work. The Lewis–Riesenfeld invariant treatment and the unitary transformation method are used for this purpose. The original Schrödinger equation is reduced to an equivalent solvable one through unitary transformations by using suitable unitary operators. The reduced equation corresponds to a simpler Hamiltonian which is written as a linear combination of the generators of the reduced-dimensional SU(2) algebra. A Hermitian invariant operator is constructed based on the same algebraic formulation and its instantaneous eigenfunctions are obtained. By utilizing such eigenfunctions, the complete quantum wave functions of the system are evaluated. Such wave functions are necessary when we analyze the quantum characteristics of the system.