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Physical Review A

Exact solution to the von Neumann equation of the quantum characteristic function of the two-level Jaynes-Cummings model

Shun-Jin Wang, Dun Zhao, Hong‐Gang Luo, Li-Xiang Cen, Chenglong Jia

Published Oct 4, 2001• 15 citations
Publisher

Abstract

The equation of motion of the quantum characteristic function for the linear-algebraic dynamical system is derived from the von Neumann equation and reduced to a Schr\"odinger-like equation in a differential adjoint representation. The exact analytical solution to the von Neumann equation of the quantum characteristic function of the two-level Jaynes-Cummings model is obtained.

DOI
10.1103/physreva.64.052102

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The equation of motion of the quantum characteristic function for the linear-algebraic dynamical system is derived from the von Neumann equation and reduced to a Schr"odinger-like equation in a differential adjoint representation. The exact analytical solution to the von Neumann equation of the quantum characteristic function of the two-level Jaynes-Cummings model is obtained.
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.
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.
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