Floquet–Lyapunov approach to optomechanical systems

In optomechanics we are often dealing with squeezing and backaction-evasion schemes that use two-tone driving. In general the time development of this kind of time-dependent Hamiltonian has to be solved from the master equation. Optomechanical system typically have large thermal noise which makes solving the master equation difficult, especially for
steady state.

We introduce a method to solve the dynamics of a time-periodic system, by using the Floquet approach to write the dynamics in the form of a Lyapunov equation. With the Floquet method, we can transform a periodic linear differential equation into a time-independent linear system. After which, for the time-independent Hamiltonian that is quadratic in the
canonical operators, the time development can be expressed in the form of a Lyapunov equation, which can be solved efficiently.

We demonstrate this method in an optomechanical system with dissipative generation of squeezing. We show that we can include the counter-rotating terms ignored in the rotating-wave approximation. The method is also useful with levitating particles in an amplitude modulated trapping field.