Using Lie Algebras to Extract Non-Classical Evolution in Optomechanics

The task of finding the time evolution of quantum systems governed by time-dependent, noncommuting Hamiltonians [H (t), H (t')] $\ne $ 0, is generally quite complex. Factorization of the evolution operator into time-dependent exponential functions of the time-independent basis elements of the Lie algebra constructed from the Hamiltonian, makes it possible to separately resolve the issues of operator ordering and time-dependence. We apply this method to oscillator dynamics and obtain analytic results. We then consider optomechanical systems, consisting of coupled optical and mechanical oscillator modes to study the generation of non-classical states. We also show how the same method allows for the inclusion of dissipative effects.