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Communications in Theoretical Physics

Exact Solution for Jaynes–Cummings Model with Bosonic Field Nonlinearity and Strong Boson-Fermion Coupling

Yang Jin, Yu Wan‐Lun, Xiang An-ping

Published Jan 1, 2006• 4 citations
Publisher

Abstract

We use Lewis–Riesenfeld invariant approach to treat the modified Jaynes–Cummings models involving any forms of nonlinearty of the bosonic field when strong boson-fermion couplings are nilpotent Grassmann valued. The general state functions, time evolution operator and the time-evolution expressions for both the bosonic number and the fermionic number are presented.

DOI
10.1088/0253-6102/45/1/027

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We use Lewis–Riesenfeld invariant approach to treat the modified Jaynes–Cummings models involving any forms of nonlinearty of the bosonic field when strong boson-fermion couplings are nilpotent Grassmann valued. The general state functions, time evolution operator and the time-evolution expressions for both the bosonic number and the fermionic number are presented.
Based on the construction of supersymmetric generators, we use the Lewis–Riesenfeld invariant method to deduce the exact and explicit eigen-energy spectrum with the time-dependent thermo Jaynes–Cummings model. One of the advantages of this approach is that it can transform the hidden form, related to the chronological product, of the time evolution operator into an explicit expression. Moreover, the dynamical and statistics properties of physical quantities are obtained for the given initial states in the thermo Jaynes–Cummings system.
Phase space distribution functional methods utilising c-number and Grassmann fields for bosons and fermions respectively are being developed for theoretical treatments of degenerate quantum gases with large fermionic and bosonic atom numbers. The Jaynes-Cummings model where a single mode bosonic system and a two mode fermionic system are coupled provides a simple testbed for phase space methods involving Grassmann variables. The Fokker-Planck equation for the distribution function is shown to be equivalent to six coupled equations for c-number bosonic functions. Numerical solutions enable well-known effects such as collapses/revivals of the atomic population difference to be demonstrated.
An approach to exact quantum solutions of the time-dependent two energy level Jaynes–Cummings model with an imaginary photon process is represented in this work. The Lewis–Riesenfeld invariant treatment and the unitary transformation method are used for this purpose. The original Schrödinger equation is reduced to an equivalent solvable one through unitary transformations by using suitable unitary operators. The reduced equation corresponds to a simpler Hamiltonian which is written as a linear combination of the generators of the reduced-dimensional SU(2) algebra. A Hermitian invariant operator is constructed based on the same algebraic formulation and its instantaneous eigenfunctions are obtained. By utilizing such eigenfunctions, the complete quantum wave functions of the system are evaluated. Such wave functions are necessary when we analyze the quantum characteristics of the system.