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Open quantum systems can display periodic dynamics at the classical leveleither due to external periodic modulations or to self-pulsing phenomenatypically following a Hopf bifurcation. In both cases, the quantum fluctuationsaround classical solutions do not reach a quantum-statistical stationary state,which prevents adopting the simple and reliable methods used for stationaryquantum systems. Here we put forward a general and efficient method to computetwo-time correlations and corresponding spectral densities of time-periodicopen quantum systems within the usual linearized (Gaussian) approximation fortheir dynamics. Using Floquet theory we show how the quantum Langevin equationsfor the fluctuations can be efficiently integrated by partitioning the timedomain into one-period duration intervals, and relating the properties of eachperiod to the first one. Spectral densities, like squeezing spectra, arecomputed similarly, now in a two-dimensional temporal domain that is treated asa chessboard with one-period x one-period cells. This technique avoidscumulative numerical errors as well as efficiently saves computational time. Asan illustration of the method, we analyze the quantum fluctuations of a dampedparametrically-driven oscillator (degenerate parametric oscillator) belowthreshold and far away from rotating-wave approximation conditions, which is arelevant scenario for modern low-frequency quantum oscillators. Our methodreveals that the squeezing properties of such devices are quite robust againstthe amplitude of the modulation or the low quality of the oscillator, althoughoptimal squeezing can appear for parameters that are far from the onespredicted within the rotating-wave approximation.

physical review. a/physical review, a

Floquet theory for temporal correlations and spectra in time-periodic  open quantum systems: Application to squeezed parametric oscillation beyond  the rotating-wave approximation