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Energy minimization in two-level dissipative quantum control: the integrable case. (English) Zbl 1306.81050

Summary: The aim of this contribution is to refine some of the computations of B. Bonnard et al. [“The energy minimization problem for two-level dissipative quantum systems”, J. Math. Phys. 51, No. 9, Article ID 092705, 44 p. (2010)]. 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 effect of a magnetic field 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 defines outside singularities a pseudo-Riemannian metric in dimension five. Complete quadratures are given for this subcase by means of Weierstraßelliptic functions. Preliminary computations of cut and conjugate loci are also provided for a two-dimensional restriction using [J. B. Caillau et al., Optim. Methods Softw. 27, No. 2, 177–196 (2012; Zbl 1248.49025)].

MSC:

81Q93 Quantum control
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics

Citations:

Zbl 1248.49025
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