An exact algebraic solution of two harmonic modes coupled through the angular momentum

Starting from the Hamiltonian of two oscillators coupled through the angular momentum, we obtain the equations of motion involving the field operators for two harmonic modes. These operator differential equations of motions are found coupled to each other. The noncommuting nature of the operators are on the way for getting the exact analytical solution directly. To get rid of these problems and on the basis of physical and mathematical considerations, the solutions are assumed in terms of some constant (independent of time) coefficients. The coupled differential equations involving these coefficients are finally decoupled at the cost of fourth order differential equations. Finally, we obtain the exact analytical solutions of these coefficients and hence the field operators involving the oscillators. As an application of these solutions, we investigate the well known squeezing effects of the input coherent light interacting with the oscillators coupled through the angular momentum. It is to be remembered that we retain the nonconserving energy terms for investigating the dynamical behaviour of the oscillators coupled through the angular momentum.