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

## TL;DR

This paper introduces an algebraic framework using the $sp(4,R)$ algebra to exactly solve the most general two-level system Hamiltonian in two-dimensional geometry, unifying various models like the Dirac oscillator and Jaynes-Cummings.

## Contribution

It develops the $sp(4,R)$ algebraic approach and applies it to solve the general two-level Hamiltonian exactly, encompassing several known models as special cases.

## Key findings

- Exact solutions for the general two-level Hamiltonian in 2D geometry.
- Unified algebraic framework for models like Dirac oscillator and Jaynes-Cummings.
- Demonstrates the versatility of $sp(4,R)$ algebra in quantum systems.

## 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.