Controllability of Quantum Systems on the Lie Group SU(1,1)

TL;DR

This paper investigates the controllability of quantum systems with SU(1,1) symmetry, establishing necessary and sufficient conditions based on the Hamiltonian's elliptic property, with implications for related Lie groups.

Contribution

It provides a rigorous characterization of controllability for quantum systems on SU(1,1), including conditions for small-time and strong controllability, extending to related Lie groups.

Findings

Elliptic condition is necessary and sufficient for controllability.

Conditions for small time local controllability are established.

Results apply to systems on SO(2,1) and SL(2,R).

Abstract

This paper examines the controllability for quantum control systems with SU(1,1) dynamical symmetry, namely, the ability to use some electromagnetic field to redirect the quantum system toward a desired evolution. The problem is formalized as the control of a right invariant bilinear system evolving on the Lie group SU(1,1) of two dimensional special pseudo-unitary matrices. It is proved that the elliptic condition of the total Hamiltonian is both sufficient and necessary for the controllability. Conditions are also given for small time local controllability and strong controllability. The results obtained are also valid for the control systems on the Lie groups SO(2,1) and SL(2,R).