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2011, 2011(Special): 198-208. doi: 10.3934/proc.2011.2011.198

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

Bernard Bonnard 1, , Jean-Baptiste Caillau 1, and  Olivier Cots 1,

1. 

Math. Institute, Bourgogne Univ. & CRNS, 9 avenue Savary, F-21078 Dijon, France, France, France

Received  August 2010 Revised  March 2011 Published  October 2011

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: conjugate and cut loci, Lorentzian metrics, elliptic functions, Optimal control, Lindblad equation.
Mathematics Subject Classification: Primary: 49K15, 81Q05.
Citation: Bernard Bonnard, Jean-Baptiste Caillau, Olivier Cots. Energy minimization in two-level dissipative quantum control: Th e integrable case. Conference Publications, 2011, 2011 (Special) : 198-208. doi: 10.3934/proc.2011.2011.198
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