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Journal of Physics A Mathematical and Theoretical

Three-mode two-boson Jaynes–Cummings model in trapped ions

Benedetto Militello, Alexander Galkin, A. G. Nikitin, ANTONINO MESSINA

Published Dec 20, 2006• 2 citations
Publisher

Abstract

In this paper, we analyse a two-boson three-mode Jaynes–Cummings model which can be implemented in the context of trapped ions. The symmetries of the Hamiltonian are brought to light and analysed in detail in order to solve the eigenvalue problem. The calculation of the time evolution operator shows the possibility of realizing interesting applications, such as the generation of nonclassical states.

DOI
10.1088/1751-8113/40/3/012

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In this paper, we analyse a two-boson three-mode Jaynes–Cummings model which can be implemented in the context of trapped ions. The symmetries of the Hamiltonian are brought to light and analysed in detail in order to solve the eigenvalue problem. The calculation of the time evolution operator shows the possibility of realizing interesting applications, such as the generation of nonclassical states.
The implementation of a vibronic coupling in the context of trapped ions, and its diagona- lization are reported. The specific spin-boson-like Hamiltonian model is studied exploiting its symmetries and supersymmetries. Applications are discussed. The study of symmetries plays a central role in the analysis of trapped ion dynamical prob- lems. Indeed, in most of the cases, symmetries provide an elegant and effective way for finding Hamiltonian eigensolutions. In this paper we describe a trapped ion physical situation described by a Hamiltonian model possessing interesting and useful symmetries. In detail, conservation of total excitation num- ber and rotational invariance are found. Moreover a Second Order Hidden Supersymmetry is discovered. Symmetry based diagonalization provides the possibility of writing down the time evolution concerning simple initial conditions. Interesting nonclassical behaviors like the gene- ration of GHZ (7) states are predicted.
Starting from the Jaynes-Cummings (J-C) model describing the interaction of a trapped ion with a standing-wave laser field, we derive the interaction Hamiltonian on the nonlinear J-C model of the trapped ion excitated resonantly with 1-quantum vibrational mode. After solving the time-dependent states of this model, we study the evolution of population inversion for the ion numerically. It is shown that the period of the quantum collapse and revival of the population inversionis related with the Lamb-Dick parameter and the position of the ion in the standing-wave laser field. With the increasing of the parameter, it will decrease the period of the collapse and revival. When moving the ion from the node of the standing-wave laser to the antinode, the period of the collapse and revival becomes longer.
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.