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.

Algebraic approach and Berry phase of a Hamiltonian with a general $SU(1,1)$ symmetry

“…and considering the commutation relations of the Sp(4, R) Lie algebra [39], we can find the following results…”

“…where H 0 is the Hamiltonian of the two-dimensional harmonic oscillator, u i , u, u ′ , s, v are real constants, and ω 1 , ω 2 are the oscillation frequencies. By introducing the bosonic operators a i = (mω i x i + ip i )/ √ 2mω i and using the realizations ( 6), ( 7) and (11), the above Hamiltonian can be rewritten as [39]…”

“…We recently have applied the theory of the SU (1, 1) and SU (2) groups to obtain the energy spectrum, eigenfunctions and the Berry phase of some of these Quantum Optics models [34][35][36][37][38]. In reference [39], an algebraic method based on the Sp(4, R) group (and contains the SU (1, 1) and SU (2) groups) was introduced to solve exactly the interaction part of the most general Hamiltonian of a two-level system in two-dimensional geometry.…”