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

TL;DR

This paper investigates how the energy of certain periodically driven quantum systems with shrinking spectral gaps evolves over time, showing that energy diffusion can be minimized by controlling the decay of matrix entries.

Contribution

It establishes conditions under which the energy diffusion exponent can be made arbitrarily small in quantum systems with spectra that grow sublinearly, extending previous results to more general spectral behaviors.

Findings

Energy diffusion can be bounded by a power law with arbitrarily small exponent.

The decay rate of matrix entries influences the energy diffusion rate.

Application to a specific Hamiltonian on the circle demonstrates the theoretical results.

Abstract

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}^{-2\gamma} 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 \Psi\in Dom(H^{1/2}), the diffusion of energy is bounded from above as <H>_\Psi(t)=O(t^\sigma) where \sigma=\alpha/(2\ceil{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,d\theta) which was discussed earlier in the literature by Howland.