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Journal of Mathematical Physics

Exact solutions of Heisenberg equations for multiphoton Jaynes–Cummings model

H. Iwasawa, Kuniaki Matsuo

Published Jun 1, 1995• 3 citations
Publisher

Abstract

The Heisenberg operator solutions for the multiphoton Jaynes–Cummings model will be presented. By solving the operator differential equations, the exact solutions for the photon annihilation operator a(t), the atomic dipole-moment operator σ−(t), and the population difference operator σz(t) are obtained. The time dynamical behaviors for the operators are compared with the results obtained by the density operator methods.

DOI
10.1063/1.531061

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The Heisenberg operator solutions for the multiphoton Jaynes–Cummings model will be presented. By solving the operator differential equations, the exact solutions for the photon annihilation operator a(t), the atomic dipole-moment operator σ−(t), and the population difference operator σz(t) are obtained. The time dynamical behaviors for the operators are compared with the results obtained by the density operator methods.
The dynamics of the degenerate two-photon Jaynes-Cummings model in Heisenberg picture is investigated. By solving the Heisenberg equations, the exact solutions of the photon annihilation operator a( t), the atomic dipole moment operator σ-( t), and the population difference operator σ z ( t) are obtained. We demonstrate that the correlation functions for the photon annihilation and dipole moment operators can be directly evaluated from these exact solutions. The solutions of the time-evolution operators are shown to be quite useful in the calculation of two-time correlation functions. Furthermore, some results of the dynamical behavior of the average values for the photon number <n (t)>, the in-phase field component <X 1(t)>, and the population difference <σ z (t)> are demonstrated by computer simulations.
One of the simplest models for the interactions between an atom and a quantized electromagnetic field is known as the Jaynes-Cummings model GCM) [1]. In this model, the atom is assumed to have two levels, and the electromagnetic field is a single mode. For the past several years, we have investigated the model for the singlephoton, the two-photon, and the multi-photon cases. First, we studied the characteristic functions and their time behaviors for a simple JCM[Z] using the density matrix approach, and then, by using the Heisenberg equation method, we exmined the degenerate two-photon JCM[31, where the operator differential equations and their solutions are obtained. We further studied the cases of the non-degenerate two-photon JCM[4] and the mUlti-photon JCM[51. Moreover, the atomic spectrum for mUlti-photon JCM[61 and the field spectrum for multiphoton JCM[71 were explored, and these spectra were presented in the 3-dimensional pictures. We also solved the degenerate two-photon JCM with Stark-shifts [8] , and further we just published the results of the non-degeneratetwo-photon JCM with Stark-shifts[91. In this paper, we show a derivaition of the product operator alZ ( t ) = al ( t ) az ( t ). We explicitly use the idea of the constant motion operator. Since the derivations are quite complicated and it is not streightword, we showed them in detail here. The present paper is organized as follows. In section II, we show the effective non-degenerate Hamiltonians and the constant of motion operators. In section Ill, the second-order differential equation for a product operator alZ ( t ) = al ( t ) az ( t) of Mode 1 and Mode 2, will