Small time reachable set of bilinear quantum systems

TL;DR

This paper demonstrates that certain infinite-dimensional bilinear quantum systems can be approximately controlled in arbitrarily small times, contrasting finite-dimensional cases due to unbounded operators.

Contribution

It provides an example of infinite-dimensional bilinear quantum systems with small-time approximate controllability, highlighting differences from finite-dimensional systems.

Findings

Approximate controllability in small times for infinite-dimensional systems.

Contrast with finite-dimensional systems where small-time controllability does not hold.

Unbounded drift operators enable this controllability.

Abstract

This note presents an example of bilinear conservative system in an infinite dimensional Hilbert space for which approximate controllability in the Hilbert unit sphere holds for arbitrary small times. This situation is in contrast with the finite dimensional case and is due to the unboundedness of the drift operator.