Charged particle motion in a time-dependent flux-driven ring: an exactly solvable model

TL;DR

This paper presents an exactly solvable model for a charged particle in a time-dependent flux-driven quantum ring, revealing how wave packets evolve along classical trajectories and highlighting topological effects on quantum dynamics.

Contribution

It introduces a non-Hermitian invariant approach to solve the time-dependent quantum problem, linking classical trajectories with quantum wave packet evolution in a topologically nontrivial system.

Findings

Wave packets follow classical angular trajectories.

The wave packet distribution depends on initial conditions.

Peak positions do not correspond to expectation values, challenging Ehrenfest's theorem.

Abstract

We consider a charged particle driven by a time-dependent flux threading a quantum ring. The dynamics of the charged particle is investigated using classical treatment, Fourier expansion technique, time-evolution method, and Lewis-Riesenfeld approach. We have shown that, by properly managing the boundary conditions, a time-dependent wave function can be obtained using a general non-Hermitian time-dependent invariant, which is a specific linear combination of initial angular-momentum and azimuthal-angle operators. It is shown that the linear invariant eigenfunction can be realized as a Gaussian-type wave packet with a peak moving along the classical angular trajectory, while the distribution of the wave packet is determined by the ratio of the coefficient of the initial angle to that of the initial canonical angular momentum. From the topologically nontrivial nature as well as the classical trajectory and angular momentum, one can determine the dynamical motion of the wave packet. It should be noted that the peak position is no longer an expectation value of the angle operator, and hence the Ehrenfest theorem is not directly applicable in such a topologically nontrivial system.