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Applied Mechanics and Materials

Solution of Generalized Jaynes-Cummings Model for many Particle Systems

NICKOLAI N. BOGOLUBOV, M. Yu. Rasulova, Ilhom Tishabaev

Published Dec 1, 2013• 0 citations
Publisher

Abstract

We consider the dynamics of a system consisting of N two-level atoms interacting with a multi-mode cavity field. For the given system, the generalized kinetic equation is obtained and conditions are given under which its solution is reduced to solution of a linear equation, and of the one-dimensional nonlinear Schrodinger equation, respectively.

DOI
10.4028/www.scientific.net/amm.481.272

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•Jan 1, 2005•0 citations
We consider the dynamics of a system consisting of N two-level atoms interacting with a multi-mode cavity field. For the given system, the generalized kinetic equation is obtained and conditions are given under which its solution is reduced to solution of a linear equation, and of the one-dimensional nonlinear Schrodinger equation, respectively.
The paper is devoted to the investigation of dynamics of system, consisting of interacting two-level atoms and cavity field. Such a system also covers the Jaynes–Cummings model. This paper consists of an introduction, Secs. 2 and 3 and conclusion. In the introduction, a short history of the problem is presented and the problem is formulated. In Sec. 2, the generalized kinetic equation for the system is derived. In Sec. 3, the generalized kinetic equation is solved. In conclusion, the result is formulated.
The paper is devoted to the investigation of dynamics of system, consisting of interacting two-level atoms and field (cavity field). To such system covers also Jaynes-Cummings model. In this paper, evolution of two-level many bosons system interacting with a field is investigated by means of generalized kinetic equation. On base of the Bogolyubov's method (method of elimination of bosons), solution of the generalized kinetic equation is determined and explicit expression of average value of physical quantities, characterizing present system is defined. Moreover, solution of the generalized kinetic equation is defined for the system, consisting of a single two-level atom, interacting with a single cavity field mode. Thereby, offered in the work, statistical approach to the problem allows to generalize well-known theoretical and applied results of Jaynes-Cummings model for the systems, consisting of many two-level atoms and cavity field. This paper consists of an introduction, sections 2, 3 and conclusion. In the introduction, a short history of the problem is presented and the problem is formulated. In section 2, the generalized kinetic equation for the system is derived. In section 3, the generalized kinetic equation is solved. In conclusion the result is formulated.
Due to the improvement within cavity quantum electrodynamics experiments during the last decades, what was former seen as 'toy models' are today realized in laboratories. A controlled isolated coherent evolution of one or a few atoms coupled to a single mode inside a cavity is achievable. Such systems are well suited for studying purely quantum mechanical effects, and also for performing quantum gates, necessary for quantum computing. The Jaynes-Cummings model has served as a theoretical description of the interaction. However, as the experimental techniques are improved, for example, atom cooling, the use of multi-level atoms or multi-modes and driving of atoms or elds by external lasers, extensions of the original Jaynes-Cummings model are needed. In this thesis we study some of these extended models, and in particular multi-level models, time-dependent models and quantized motion models. Both analytical and numerical analysis are considered. The two-level structure of the Jaynes-Cummings model leads to applications of known solvable time-dependent two-level Schrödinger equations. In other cases, di erent forms of adiabatic approximate solutions are used, and with the analytically solvable models, the amplitudes of non-adiabatic contributions may be estimated. For higher dimensional systems, STIRAP and multi-STIRAP methods are applied. It is shown how the time-dependent models may be used for preparation of various kinds of non-classical states, and also to generate universal sets of quantum gates, both on atomic and eld qubits. When the atoms are cooled to very low temperatures, their velocities must be treated quantum mechanically, and we have studied the dynamics of such cases for di erent coupling shapes. Again numerical and analytical approaches have been used and compared, wave-packet propagations of the atom, approaching and traversing the cavity, have been performed. For periodic couplings, standing wave cavity modes, the dynamics has been described by e ective parameters; group velocity or atomic index of refraction and effective mass. Tunneling resonances for ultra cold atoms have been exhibited in the STIRAP models for certain initial conditions.