The main motivation of this article is to derive sufficient conditions for dynamical stability of periodically driven quantum systems described by a Hamiltonian H(t), i.e. conditions under which it holds true sup t ∈ ℝ|〈ψt, H(t)ψt〉| < ∞ where ψt denotes a trajectory at time t of the quantum system under consideration. We start from an analysis of the domain of the quasi-energy operator. Next, we show, under certain assumptions, that if the spectrum of the monodromy (Floquet) operator U(T, 0) is pure point then there exists a dense subspace of initial conditions for which the mean value of the energy is uniformly bounded in the course of time. Further, we show that if the propagator admits a differentiable Floquet decomposition then ‖H(t)ψt‖ is bounded in time for any initial condition ψ0, and one employs the quantum KAM algorithm to prove the existence of this type of decomposition for a fairly large class of H(t). In addition, we derive bounds uniform in time on transition probabilities between different energy levels, and we also propose an extension of this approach to the case of a higher order of differentiability of the Floquet decomposition. The procedure is demonstrated on a solvable example of the periodically time-dependent harmonic oscillator.
Phase space formulation of density operator for non-Hermitian Hamiltonians and its application in quantum theory of decay
