Dynamical control of quantum systems in the context of mean ergodic theorems

TL;DR

This paper explores how different pulse control schemes in quantum systems relate to mean ergodic theorems, showing conditions under which they converge to the same evolution and deriving inequalities to support these results.

Contribution

It establishes the conditions for convergence of equidistant and non-equidistant pulse controls to the same quantum evolution, using projections and inequalities.

Findings

Both control schemes converge to the same unitary evolution under certain conditions.

The generator of the limiting evolution can be obtained via projection onto the commutant.

Optimized inequalities are derived for non-equidistant pulse intervals.

Abstract

Equidistant and non-equidistant single pulse "bang-bang" dynamical controls are investigated in the context of mean ergodic theorems. We show the requirements in which the limit of infinite pulse control for both the equidistant and the non-equidistant dynamical control converges to the same unitary evolution. It is demonstrated that the generator of this evolution can be obtained by projecting the generator of the free evolution onto the commutant of the unitary operator representing the pulse. Inequalities are derived to prove this statement and in the case of non-equidistant approach these inequalities are optimised as a function of the time intervals.