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Physical Review A

Simple algebraic method to solve a coupled-channel cavity QED model

Ying Wu

Published Nov 1, 1996• 117 citations
Publisher

Abstract

In this paper, we present a simple purely algebraic method to solve a coupled-channel cavity QED model with three nondegenerate quantized field modes proposed by Wang, Puri, and Eberly [Phys. Rev. A 46, 7192 (1992)]. By transforming it into one describing a spin-1/2 particle in a magnetic field and utilizing the relations between field variables and an orbital angular momentum, we can easily obtain energy eigenvalues without the need to know concrete expressions for eigenstates in terms of the usual Fock states, and analytical expressions of evolution and atomic inversion operators. The eigenstates of energy and orbital angular momentum are also explicitly expressed in terms of Fock states. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.4534

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In this paper, we present a simple purely algebraic method to solve a coupled-channel cavity QED model with three nondegenerate quantized field modes proposed by Wang, Puri, and Eberly [Phys. Rev. A 46, 7192 (1992)]. By transforming it into one describing a spin-1/2 particle in a magnetic field and utilizing the relations between field variables and an orbital angular momentum, we can easily obtain energy eigenvalues without the need to know concrete expressions for eigenstates in terms of the usual Fock states, and analytical expressions of evolution and atomic inversion operators. The eigenstates of energy and orbital angular momentum are also explicitly expressed in terms of Fock states. \textcopyright{} 1996 The American Physical Society.
We present an alternative cavity QED model with three nondegenerate quantized field modes. The atom-field interaction is a cavity analog of Raman scattering, and the three field modes can be associated with pump, Stokes, and anti-Stokes frequencies. Unlike systems based on the Jaynes-Cummings model, this one has two independent channels connecting the two active atomic levels, leading to the possibility of quantum interferences. Despite its complexity, the model has eigenstates and eigenvalues that can be found exactly and fully analytically in finite terms. The most interesting eigenstates have a ``chain'' character unknown in previous cavity QED models. Both Fock-state and coherent-state solutions are presented for a number of quantities, including complex three-mode variables. Heisenberg operator solutions are demonstrated in a few cases, and an example of fourth-order quantum interferences due to the coupled channels is worked out in full.
We study a cavity QED model where a single three-level atom interacts with four quantized cavity fields in a Raman-type interaction, forming two transition channels for the atom. We show that this system is equivalent to the coupling of three angular momenta, and give its exact eigenenergies and eigenvectors in terms of Clebsch-Gordan coefficients, even if the ac Stark shift is included. Some Heisenberg operators are solved exactly as well as approximately. Dynamical evolution of average photon numbers and the atomic inversion are discussed for several initial conditions and a simple example of model interference is examined.
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.