$SU(1,1)\times SU(2)$ approach and the Mandel parameter to the Hamiltonian of two oscillators with weak coupling

TL;DR

This paper employs an algebraic $SU(1,1)\times SU(2)$ approach to analyze the Hamiltonian of two weakly coupled isotropic oscillators, deriving their energy spectrum, eigenfunctions, and photon statistical properties.

Contribution

It introduces a novel algebraic method using group transformations to solve the weakly coupled oscillators' Hamiltonian and compute photon statistics.

Findings

Derived energy spectrum and eigenfunctions for weakly coupled oscillators.

Calculated Mandel Q-parameter indicating photon statistics.

Obtained second-order correlation function for photon numbers.

Abstract

We study the Hamiltonian of two isotropic oscillators with weak coupling from an algebraic approach. We write the Hamiltonian of this problem in terms of the boson generators of the $SU(1,1)$ and $SU(2)$ groups. This allows us to apply two tilting transformations based on both group similarity transformations to obtain its energy spectrum and eigenfunctions. Then, we obtain the Mandel $Q$-parameter and the second-order correlation function $g^2(0)$ of the photon numbers $n_a$ and $n_b$. It is important to note that in our procedure we consider the case of weak coupling.