Slow decay of temporal correlations in quantum systems with Cantor spectra

Abstract

We prove that the temporal autocorrelation function C(t) for quantum systems with Cantor spectra has an algebraic decay C(t)\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$, where \ensuremath{\delta} equals the generalized dimension ${\mathit{D}}_{2}$ of the spectral measure and is bounded by the Hausdorff dimension ${\mathit{D}}_{0}$\ensuremath{\ge}\ensuremath{\delta}. We study various incommensurate systems with singular continuous and absolutely continuous Cantor spectra and find extremely slow correlation decays in singular continuous cases (\ensuremath{\delta}=0.14 for the critical Harper model and 0\ensuremath{\le}0.84 for the Fibonacci chains). In the kicked Harper model we deomonstrate that the quantum mechanical decay is unrelated to the existence of classical chaos.