Permuting quantum eigenmodes by a quasi-adiabatic motion of a potential wall

TL;DR

This paper demonstrates how quasi-adiabatic manipulation of high, localized potential walls in a Schrödinger equation can permute eigenmodes and achieve approximate controllability of quantum states.

Contribution

It introduces a method to permute eigenmodes and control quantum states via slow, quasi-adiabatic variation of potential walls, combining adiabatic and non-adiabatic dynamics.

Findings

Eigenmode permutations achieved through potential wall movements.

Approximate controllability of quantum states using slow potential variations.

Demonstration of non-trivial eigenstate transformations despite slow parameter changes.

Abstract

We study the Schr\"odinger equation $i\partial_t\psi=-\Delta\psi+V\psi$ on $L^2((0,1),\mathbb{C})$ where $V$ is a very high and localized potential wall. We aim to perform permutations of the eigenmodes and to control the solution of the equation. We consider the process where the position and the height of the potential wall change as follows. First, the potential increases from zero to a very large value, so a narrow potential wall is formed that almost splits the interval into two parts; then the wall moves to a different position, after which the height of the wall decays to zero again. We show that even though the rate of the variation of the potential's parameters can be arbitrarily slow, this process alternates adiabatic and non-adiabatic dynamics, leading to a non-trivial permutation of the eigenstates. Furthermore, we consider potentials with several narrow walls and we show how an arbitrarily slow motion of the walls can lead the system from any given state to an arbitrarily small neighborhood of any other state, thus proving the approximate controllability of the above Schr\"odinger equation by means of a soft, quasi-adiabatic variation of the potential.