Tunnel catch from potential wells and energy detection

TL;DR

This paper analyzes the tunnel catch effect in a double-well Schrödinger system, deriving formulas for tunneling probabilities and energy levels, with implications for energy detection and quantum state analysis.

Contribution

It introduces a detailed asymptotic analysis of the tunnel catch effect in a tunable double-well potential, providing formulas for tunneling probabilities and energy level determination.

Findings

Asymptotic formula for tunneling probability in the tunnel catch effect.

Method to determine initial state energy from tunneling observations.

Calculation of tunneling splitting in the double-well potential.

Abstract

We consider the one-dimensional Schr\"{o}dinger operator in the semiclassical regime assuming that its double-well potential is the sum of a finite "physically given" well and a square shape probing well whose width or depth can be varied (tuned). We study the dynamics of initial state localized in the physical well. It is shown that if the probing well is not too close to the physical one and if its parameters are specially tuned, then the {\it tunnel catch effect} appears, i.e. the initial state starts tunneling oscillations between the physical and probing wells. The asymptotic formula for the probability of finding the state in the probing well is obtained. We show that the observation of the tunnel catch effect can be used to determine the energy level of the initial state, and we obtain the corresponding asymptotic formula for the initial state energy. We also calculate the leading term of the tunneling splitting of energy levels in this double well potential.