Dynamical system analysis of quantum tunneling in an asymmetric double-well potential

TL;DR

This paper presents a dynamical systems approach based on the Ehrenfest formalism to analyze quantum tunneling in an asymmetric double-well potential, providing insights into tunneling thresholds and regimes of detectability.

Contribution

It introduces a reduced dynamical system incorporating skewness for asymmetric potentials, enabling interpretability and approximation of quantum tunneling phenomena.

Findings

Identifies energy thresholds for tunneling detection.

Reveals regimes where tunneling is theoretically possible but practically undetectable.

Shows the approach reproduces key features of full Schrödinger solutions.

Abstract

We study quantum tunneling in an asymmetric double-well potential using a dynamical systems--based approach rooted in the Ehrenfest formalism. In this framework, the time evolution of a Gaussian wave packet is governed by a hierarchy of coupled equations linking lower- and higher-order position moments. An approximate closure scheme, required to render the system tractable, yields a reduced dynamical system for the mean and variance, with skewness explicitly entering due to the potential's asymmetry. Stability analysis of this system identifies energy thresholds for detectable tunneling across the barrier and reveals regimes where tunneling, though theoretically allowed, remains practically undetectable. Comparison with full numerical solutions of the time-dependent Schr\"odinger equation shows that, beyond reproducing key tunneling features, the dynamical systems approach provides an interpretable description of quantum transport through tunneling in an effective asymmetric two-level system.