$Sp(4,R)$ algebraic approach of the most general Hamiltonian of a two-level system in two-dimensional geometry

Abstract

In this paper we introduce the bosonic generators of the $sp(4,R)$ algebra and study some of their properties, based on the $SU(1,1)$ and $SU(2)$ group theory. With the developed theory of the $Sp(4,R)$ group, we solve the interaction part of the most general Hamiltonian of a two-level system in two-dimensional geometry in an exact way. As particular cases of this Hamiltonian, we reproduce the solution of earlier problems as the Dirac oscillator and the Jaynes-Cummings model with one and two modes of oscillation.