Scaling of localization lengths in quantum dynamics.

摘要

In this thesis, we study the behavior of a quantum mechanical system whose classical counterpart exhibits a transition to chaos. In particular, we focus on the KAM regime, when the classical dynamics are dominated by impermeable and slightly permeable invariant curves. We center our attention on the momentum space wavefunction, and especially on the rate of exponential decay of the momentum space probability. Following in a tradition of reasoning by analogy with the classical physics, and using arguments based on a quantum renormalization map, we predict that the momentum space decay rate is a "scaling function" under the renormalization. We perform an extensive set of numerical experiments to verify our reasoning. We investigate the effect of variation of Planck's constant upon the scaling of the decay rate. We use the notion of "accessible phase space structures" to explain the variation of the scaling multiplier with {dollar}hbar{dollar} which we observe numerically. Finally, we study the effect on our model of noise from spontaneous emission. We argue that the noise effectively washes out small phase space structures, and show that this implies the scaling multiplier for the noisy system should be increased relative to that of the non-noisy system. We perform numerical simulations of the noisy system which corroborate our reasoning.