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A variety of dynamics in nature and society can be approximately treated as adriven and damped parametric oscillator. An intensive investigation of thistime-dependent model from an algebraic point of view provides a consistentmethod to resolve the classical dynamics and the quantum evolution in order tounderstand the time-dependent phenomena that occur not only in the macroscopicclassical scale for the synchronized behaviors but also in the microscopicquantum scale for a coherent state evolution. By using a FloquetU-transformation on a general time-dependent quadratic Hamiltonian, we exactlysolve the dynamic behaviors of a driven and damped parametric oscillator toobtain the optimal solutions by means of invariant parameters of $K$s tocombine with Lewis-Riesenfeld invariant method. This approach can discriminatethe external dynamics from the internal evolution of a wave packet by producingindependent parametric equations that dramatically facilitate the parametriccontrol on the quantum state evolution in a dissipative system. In order toshow the advantages of this method, several time-dependent models proposed inthe quantum control field are analyzed in details.

Lie transformation method on quantum state evolution of a general time-dependent driven and damped parametric oscillator