On the stability of periodically time-dependent quantum systems

TL;DR

This paper establishes conditions for the dynamical stability of periodically driven quantum systems, focusing on spectral properties, Floquet decompositions, and energy bounds, with applications to a time-dependent harmonic oscillator.

Contribution

It introduces new criteria linking spectral properties and Floquet theory to stability, employing quantum KAM methods for a broad class of systems.

Findings

Pure point spectrum of monodromy operator implies bounded energy for many initial states.

Existence of differentiable Floquet decomposition ensures bounded energy norm over time.

Bounds on transition probabilities between energy levels are derived.

Abstract

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.