A method for solving exact-controllability problems governed by closed quantum spin systems

Abstract

The Liouville–von Neumann master equation models closed quantum spin systems that arise in nuclear magnetic resonance applications. In this paper, an efficient and robust computational framework to solve exact-controllability problems governed by the Liouville–von Neumann master equation is presented. The proposed control framework is based on a new optimisation formulation of exact-controllability quantum spin problems that allows the application of efficient computational techniques. This formulation results in an optimality system with four differential equations and an optimality condition. The differential equations are approximated with an appropriate modified Crank–Nicholson scheme and the resulting discretised optimality system is solved with a matrix-free Krylov–Newton scheme combined with a cascadic nonlinear conjugate gradient initialisation. Results of numerical experiments demonstrate the ability of the proposed framework to solve quantum spin exact-controllability control problems.

Keywords:

• quantum spin systems
• Liouville–von Neumann master equation
• exact-controllability problem
• optimal control theory
• optimality conditions
• modified Crank–Nicholson scheme
• Krylov–Newton scheme

Additional information

Funding

This article is supported in part by the Deutsche Forschungsgemeinschaft (DFG) project ‘Controllability and Optimal Control of Interacting Quantum Dynamical Systems’ (COCIQS) and by the Bayerisch-Französisches Hochschulzentrum, BFHZ Projekt FK-10-12. The second author was also partially supported by the ‘Agence Nationale de la Recherche’ (ANR), Projet Blanc EMAQS [grant number ANR-2011-BS01-017-01].