Quantum Transport on KAM Tori

TL;DR

This paper investigates quantum tunneling on KAM tori and shows that in the semiclassical limit, quantum group velocity distributions align with classical velocities, with tunneling effects being suppressed.

Contribution

It demonstrates the suppression of quantum tunneling on KAM tori in the semiclassical limit and connects quantum velocity distributions to classical ones.

Findings

Quantum tunneling between phase space tori is suppressed as 7 7 0.

Quantum group velocity distribution converges to classical distribution near energy E.

The convergence rate is of order 7(1/\u221aE).

Abstract

Although quantum tunneling between phase space tori occurs, it is suppressed in the semiclassical limit $\hbar\searrow 0$ for the Schr\"{o}dinger equation of a particle in $\bR^d$ under the influence of a smooth periodic potential. In particular this implies that the distribution of quantum group velocities near energy $E$ converges to the distribution of the classical asymptotic velocities near $E$, up to a term of the order $\cO(1/\sqrt{E})$.