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International Journal of Modern Physics B

DIAGONALIZATION OF A HAMILTONIAN DESCRIBING A SINGLE TWO-LEVEL ATOM INTERACTING WITH A TWO MODE AMPLIFIER

Paul Bracken

Published Jan 20, 2007• 1 citation
Publisher

Abstract

A Hamiltonian which describes the interaction of a single atom with two photon modes is introduced. It is shown that the Hamiltonian can be diagonalized in a particular basis. The energies and an eigenvector basis set are obtained. Some quasi-probability densities are calculated using amplitudes determined with respect to the rotated basis. Some of the physical phenomena which are manifested in the calculations are discussed.

DOI
10.1142/s0217979207035960

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A Hamiltonian which describes the interaction of a single atom with two photon modes is introduced. It is shown that the Hamiltonian can be diagonalized in a particular basis. The energies and an eigenvector basis set are obtained. Some quasi-probability densities are calculated using amplitudes determined with respect to the rotated basis. Some of the physical phenomena which are manifested in the calculations are discussed.
Abstract In the present paper we show that the Hamiltonian describing the resonant interaction of N two-level systems with a single-mode electromagnetic quantum field in the Coulomb gauge can be diagonalized with a high degree of accuracy using a simple basis set of states. This allows one to find an analytical approximation for the eigenvectors and eigenvalues of the system, which interpolates the numerical solution in a broad range of the coupling constant values. For the first time, we show that the energy spectrum of Dicke model (DM) as a function of N (with N ≫ 1 ) satisfies the simple scaling relation. In addition, the introduced basis states provide a regular way of calculating the corrections and estimating the convergence to the exact numerical solution. Analytical approximations and numerical results describing the energy spectrum are valid for both quantum Rabi model ( N = 1) and DM for N ⩾ 2 atoms.
A diagonalization scheme for the Rabi Hamiltonian, which describes a qubit interacting with a single-mode radiation field via a dipole interaction, is proposed. It is shown that the Rabi Hamiltonian can be solved almost exactly using a progressive scheme that involves a finite set of one variable polynomial equations. The scheme is especially efficient for the lower part of the spectrum. Some low-lying energy levels of the model with several sets of parameters are calculated and compared to those provided by the recently proposed generalized rotating-wave approximation and a full matrix diagonalization.
In the present paper we consider a nonlinear interaction between two electromagnetic field modes and a two-level atom. A new Hamiltonian model is introduced to represent such an interaction, where three coupling parameters are involved. We assume that the two-level atom interacts with each mode separately as well as with the two modes nonlinearly. Thus degenerate and non-degenerate processes are considered. Using a suitable canonical transformation and integrability condition the system is reduced to a single mode case. The density matrix is calculated and used to study the photon distribution, the atomic occupation probabilities, the von Neumann entropy and the entropy squeezing.
The problem of a single two level atom in its excited state, placed in a cavity in which no photons are present initially, and interacting with N modes of the electromagnetic field is solved exactly in the electric dipole, rotating wave approximation. Expressions for the energy eigenvalues and eigenvectors are derived, and an exact representation for the time dependent wavefunction of the system is given. These quantities are evaluated analytically for N=1,2 and 3, and the time dependent properties of the system for these values of N are investigated and discussed.