Tunneling from an oscillating initial state in quantum mechanics

TL;DR

This paper derives an analytical expression for quantum tunneling decay from oscillating initial states in metastable potentials, validated by numerical comparisons.

Contribution

It provides a closed-form, semiclassical approximation for tunneling probabilities from oscillating states, including a time-dependent decay rate.

Findings

Analytical formula for tunneling probability current in terms of resonant states

Approximate time-dependent decay rate for oscillating initial states

Good agreement between analytical approximation and numerical solutions

Abstract

We study the decay of general initial states out of a metastable potential well in quantum mechanics. We provide a closed-form expression for the probability current that tunnels through the barrier in terms of the resonant states into which the initial state can be decomposed. All ingredients in the equation are computed analytically to first subleading order in the semiclassical limit. Specializing to a coherently-oscillating initial state, we derive an approximation to the time-dependent decay rate and demonstrate its accuracy by comparing it to a numerical solution of the Schr\"odinger equation.