Solution of two-mode bosonic Hamiltonians and related physical systems

TL;DR

This paper develops a method to solve two-mode bosonic Hamiltonians by mapping algebraic representations, enabling analysis of nonlinear quantum-optical systems and the construction of special functions for quantum optics.

Contribution

It introduces a novel mapping between Schwinger and Gelfand-Dyson representations for su(2) and su(1, 1), facilitating quasi-exact solutions of two-boson Hamiltonians.

Findings

Derived conditions for quasi-exact solvability.

Mapped algebraic representations to study nonlinear systems.

Constructed special functions for quantum optics analysis.

Abstract

We have constructed the quasi-exactly-solvable two-mode bosonic realizations of su(2) and su(1, 1) algebra. We derive the relations leading to the conditions for quasi-exact solvability of two-boson Hamiltonians by determining a general procedure which maps the Schwinger representations of the su(2) and su(1, 1) algebras to the Gelfand-Dyson representations, respectively. This mapping allows us to study nonlinear quantum-optical systems in the framework of quasi-exact solvability. Our approach also leads to a simple construction of special functions of two variables which are the most appropriate functions to study quasi-probabilities in quantum optics.