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Cavity optomechanics and electromechanics form an established field ofresearch investigating the interactions between electromagnetic fields and themotion of quantum mechanical resonators. In many applications, linearised formof the interaction is used, which allows for the system dynamics to be fullydescribed using a Lyapunov equation for the covariance matrix of the Wignerfunction. This approach, however, is problematic in situations where theHamiltonian becomes time dependent as is the case for systems driven atmultiple frequencies simultaneously. This scenario is highly relevant as itleads to dissipative preparation of mechanical states or backaction-evadingmeasurements of mechanical motion. The time-dependent dynamics can be solvedwith Floquet techniques whose application is, nevertheless, notstraightforward. Here, we describe a general method for combining the Lyapunovapproach with Floquet techniques that enables us to transform the initialtime-dependent problem into a time-independent one, albeit in a larger Hilbertspace. We show how the lengthy process of applying the Floquet formalism to theoriginal equations of motion and deriving a Lyapunov equation from theirtime-independent form can be simplified with the use of properly definedFourier components of the drift matrix of the original time-dependent system.We then use our formalism to comprehensively analyse dissipative generation ofmechanical squeezing beyond the rotating wave approximation. Our method isapplicable to various problems with multitone driving schemes in cavityoptomechanics, electromechanics, and related disciplines.

Combining Floquet and Lyapunov techniques for time-dependent problems in optomechanics and electromechanics