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

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.