Solution of the eigenvalue problem of a N-level system coupled to a bosonic degree of freedom without using RWA

Abstract

The eigenvalue problem of a N-level system coupled to a bosonic degree of freedom is solved without using RWA. For that purpose, the bosonic degree of freedom is transformed to Bargmann's Hilbert space of analytical functions. In this representation the Schrodinger equation is a system of N coupled linear differential equations of first order. Using a discrete symmetry, these equations are simplified by a suitable transformation of the independent variable. Starting from the simplified equations, the authors develop a method to solve the eigenvalue problem of the N-level system. In addition, they present a simple approximate treatment and compare it with the exact results. The approximation turns out to be quite good up to outer level resonance and can be used to explain the differently structured regions in the energy spectra.