On the energy growth of some periodically driven quantum systems with shrinking gaps in the spectrum

We consider quantum Hamiltonians of the form H(t)=H+V(t) where the spectrum of H is semibounded and discrete, and the eigenvalues behave as E_n~n^alpha, with 0<alpha<1. In particular, the gaps between successive eigenvalues decay as n^{alpha-1}. V(t) is supposed to be periodic, bounded, continuously differentiable in the strong sense and such that the matrix entries with respect to the spectral decomposition of H obey the estimate |V(t)_{m,n}|<=epsilon*|m-n|^{-p}max{m,n}^{-2gamma} for m!=n where epsilon>0, p>=1 and gamma=(1-alpha)/2. We show that the energy diffusion exponent can be arbitrarily small provided p is sufficiently large and epsilon is small enough. More precisely, for any initial condition Psiin Dom(H^{1/2}), the diffusion of energy is bounded from above as <H>_Psi(t)=O(t^sigma) where sigma=alpha/(2ceil{p-1}gamma-1/2). As an application we consider the Hamiltonian H(t)=|p|^alpha+epsilon*v(theta,t) on L^2(S^1,dtheta) which was discussed earlier in the literature by Howland.

On the stability of periodically time-dependent quantum systems

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 sup_{t in R} | (psi(t),H(t) psi(t)) |<infty where psi(t) 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 operator U(T,0) is pure point then there exists a dense subspace of initial conditions for which the mean value of energy is uniformly bounded in the course of time. Further we show that if the propagator admits a differentiable Floquet decomposition then || H(t) psi(t) || is bounded in time for any initial condition psi(0), 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.