On Reachable and Controllability Sets for Minimum-Time Control of an Open Two-Level Quantum System

Oleg V. Morzhin ; Alexander N. Pechen

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematics of Quantum Technologies, Tome 313 (2021), pp. 161-177

Résumé

We consider a two-level open quantum system whose dynamics is governed by the Gorini–Kossakowski–Sudarshan–Lindblad equation with Hamiltonian and dissipation superoperator depending, respectively, on coherent and incoherent controls. Results about reachability, controllability, and minimum-time control are obtained in terms of the Bloch parametrization. First, we consider the case when the zero coherent and incoherent controls satisfy the Pontryagin maximum principle in the class of piecewise continuous controls. Second, for zero coherent control and for incoherent control lying in the class of constant functions, the reachability and controllability sets of the system are exactly described and some analytical results on the minimum-time control are found. Third, we consider a series of increasing values of the final time and the corresponding classes of controls with zero incoherent control and with coherent control equal to zero until a switching time instant and to a cosine function after it. The corresponding reachable points in the Bloch ball are numerically obtained and visualized. Fourth, a known method for estimating reachable sets is adapted and used to analyze the situation where the zero coherent and incoherent controls satisfy the Pontryagin maximum principle in the class of piecewise continuous controls while, as shown numerically, are not optimal.

Détail

Keywords: quantum control, open quantum system, coherent control, incoherent control, reachable sets, controllability sets, minimum-time control, optimization.