Time-dependent treatment of tunneling and Time's Arrow problem

TL;DR

This paper introduces a novel time-dependent approach to quantum tunneling that avoids Laplace transforms, applicable to various potentials, and offers insights into the time's arrow problem and periodically driven tunneling phenomena.

Contribution

It presents a new method for analyzing tunneling dynamics without Laplace transforms, applicable to time-dependent potentials and non-Markovian reservoirs, and extends the Tien-Gordon approach.

Findings

Derived simple expressions for tunneling dynamics in various reservoirs.

Provided a new perspective on the origin of the time's arrow in quantum mechanics.

Extended the Tien-Gordon approach to oscillating tunneling barriers.

Abstract

New time-dependent treatment of tunneling from localized state to continuum is proposed. It does not use the Laplace transform (Green's function's method) and can be applied for time-dependent potentials, as well. This approach results in simple expressions describing dynamics of tunneling to Markovian and non-Markovian reservoirs in the time-interval $-\infty<t<\infty$. It can provide a new outlook for tunneling in the negative time region, illuminating the origin of the time's arrow problem in quantum mechanics. We also concentrate on singularity at $t=0$, which affects the perturbative expansion of the evolution operator. In addition, the decay to continuum in periodically modulated tunneling Hamiltonian is investigated. Using our results, we extend the Tien-Gordon approach for periodically driven transport, to oscillating tunneling barriers.