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Energy minimization in two-level dissipative quantum control: Th e integrable case

Pages: 198 - 208, Issue Special, September 2011

 Abstract        Full Text (2810.0K)              

Bernard Bonnard - Math. Institute, Bourgogne Univ. & CRNS, 9 avenue Savary, F-21078 Dijon, France (email)
Jean-Baptiste Caillau - Math. Institute, Bourgogne Univ. & CRNS, 9 avenue Savary, F-21078 Dijon, France (email)
Olivier Cots - Math. Institute, Bourgogne Univ. & CRNS, 9 avenue Savary, F-21078 Dijon, France (email)

Abstract: The aim of this contribution is to re ne some of the computations of [6]. The Lindblad equation modelling a two-level dissipative quantum system is investigated. The control can be interpretated as the action of a laser to rotate a molecule in gas phase, or as the e ect of a magnetic eld on a spin 1=2 particle. For the energy cost, normal extremals of the maximum principle are solution to a three-dimensional Hamiltonian with parameters. The analysis is focussed on an integrable submodel which de nes outside singularities a pseudo-Riemannian metric in dimension ve. Complete quadratures are given for this subcase by means of Weierstra elliptic functions. Preliminary computations of cut and conjugate loci are also provided for a two-dimensional restriction using [9].

Keywords:  Optimal control, Lindblad equation, Lorentzian metrics, elliptic functions, conjugate and cut loci
Mathematics Subject Classification:  Primary: 49K15, 81Q05

Received: August 2010;      Revised: March 2011;      Published: October 2011.