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Thermalization in parametrically driven coupled oscillators

We consider a system of two coupled oscillators one of which is driven parametrically and investigate both classical and quantum dynamics within Floquet description. Characteristic changes in the time evolution of the quantum fluctuations are observed for dynamically stable and unstable regions. Dynamical instability generated by the parametrically driven oscillator leads to infinite temperature thermalization of the undriven oscillator which is evident from the equi-partitioning of energy, reduced density matrix and saturation of entanglement entropy. We also confirm that the classical Lyapunov exponent is correctly captured from the growth rate of `unequal time commutator' of dynamical variables which indicates thermalization stems from the instability in quantum system.