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Japanese Journal of Applied Physics

Heisenberg-Picture Operators for Two-Photon Jaynes-Cummings Model

Hiroshi Iwasawa, Kuniaki Matsuo

Published Nov 1, 1994• 5 citations
Publisher

Abstract

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.

DOI
10.1143/jjap.33.l1519

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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
We develop a systematic method of operator perturbation theory in the Heisenberg picture, which is the formal analog of time-independent stationary-state perturbation theory in the Schr\"odinger picture. However, we use the eigenoperators and operator eigenfrequencies to calculate the time evolution of the dynamical variables of interest. The spectrum of the unperturbed Hamiltonian is assumed to be discrete. The method is especially well-suited for treating systems where the unperturbed Hamiltonian represents noninteracting fermions and/or bosons, and therefore, we illustrate the method using the exactly solvable model of Jaynes and Cummings: a single two-level atom interacting with a single quantized field mode. The time evolution of these operator solutions is shown to be unitary order by order in powers of the interaction strength. We find that the time evolution of exact operator solutions is well approximated for significantly longer times by the time evolution of this method's operator solutions than by the time evolution of the same operator solutions calculated using the Dyson expansion. We also demonstrate that time-dependent operator solutions are very convenient for computing quantities such as correlation functions.