Hamiltonian identification for quantum systems: well-posedness and numerical approaches

¹ INRIA Rocquencourt, Rocquencourt B.P. 105, 78153 Le Chesnay Cedex, France.
² CERMICS-ENPC, 6 & 8 Av. B. Pascal, 77455 Marne la Vallée Cedex, France; lebris@cermics.enpc.fr
³ École des Mines de Paris, CAS, 60 Bd Saint-Michel, 75272 Paris Cedex 06, France; mazyar.mirrahimi@ensmp.fr
⁴ Department of Chemistry, Princeton University, Princeton, New Jersey 08544-1009; hrabitz@princeton.edu
⁵ CEREMADE, Université Paris Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France; Gabriel.Turinici@dauphine.fr

Received: 23 January 2006

Abstract

This paper considers the inversion problem related to the manipulation of quantum systems using laser-matter interactions. The focus is on the identification of the field free Hamiltonian and/or the dipole moment of a quantum system. The evolution of the system is given by the Schrödinger equation. The available data are observations as a function of time corresponding to dynamics generated by electric fields. The well-posedness of the problem is proved, mainly focusing on the uniqueness of the solution. A numerical approach is also introduced with an illustration of its efficiency on a test problem.