Approximate controllability of the Schr\"{o}dinger Equation with a polarizability term in higher Sobolev norms

TL;DR

This paper investigates the approximate controllability of quantum systems governed by the Schrödinger equation, incorporating a polarizability term, and provides explicit control laws for state transfers, extending to unbounded potentials.

Contribution

It introduces sufficient conditions for controllability with a polarizability term and explicitly constructs control laws for eigenstate transfers, broadening the scope of quantum control theory.

Findings

Sufficient conditions for approximate controllability are established.

Explicit control laws are derived for transfers between eigenstates.

Results apply to systems with unbounded or non-regular potentials.

Abstract

This analysis is concerned with the controllability of quantum systems in the case where the standard dipolar approximation, involving the permanent dipole moment of the system, is corrected with a polarizability term, involving the field induced dipole moment. Sufficient conditions for approximate controllability are given. For transfers between eigenstates of the free Hamiltonian, the control laws are explicitly given. The results apply also for unbounded or non-regular potentials.