Lower Bound on Quantum Tunneling for Strong Magnetic Fields

TL;DR

This paper establishes bounds on quantum tunneling rates for a particle in a double well potential under strong magnetic fields, providing insights into the effects of magnetic fields on quantum tunneling in solid state systems.

Contribution

It introduces new bounds on the eigenvalue splitting related to tunneling probabilities in the presence of strong magnetic fields and deep potential wells.

Findings

Derived upper and lower bounds on eigenvalue splitting.

Provided a lower bound on hopping probability between wells.

Enhanced understanding of magnetic field effects on quantum tunneling.

Abstract

We consider a particle bound to a two-dimensional plane and a double well potential, subject to a perpendicular uniform magnetic field . The energy difference between the lowest two eigenvalues--the eigenvalue splitting--is related to the tunneling probability between the two wells. We obtain upper and lower bounds on this splitting in the regime where both the magnetic field strength and the depth of the wells are large. The main step is a lower bound on the hopping probability between the wells, a key parameter in tight binding models of solid state physics.