Examples of small-time controllable Schr\"odinger equations

TL;DR

This paper presents the first examples of small-time approximate controllability for bilinear Schr"odinger equations, demonstrating how control of quadratic potential frequency enables rapid state evolution and time-contraction.

Contribution

It introduces novel methods for achieving small-time controllability in Schr"odinger equations by controlling quadratic potential frequency and utilizing space-dilations.

Findings

First examples of small-time controllability for bilinear Schr"odinger equations.

Control of quadratic potential frequency enables rapid state evolution.

Space-dilations are used to generate time-contractions.

Abstract

A variety of physically relevant bilinear Schr\"odinger equations are known to be approximately controllable in large times. There are however examples which are approximately controllable in large times, but not in small times. This obstruction happens e.g. in the presence of (sub)quadratic potentials, because Gaussian states are preserved, at least for small times. In this work, we provide the first examples of small-time approximately controllable bilinear Schr\"odinger equations. In particular, we show that a control on the frequency of a quadratic potential permits to construct approximate solutions that evolve arbitrarily fast along space-dilations. Once we have access to space-dilations, we can exploit them to generate time-contractions. In this way, we build on previous results of large-time control, to obtain control in small times.