Beyond Bilinear Controllability: Applications to Quantum Control
Quantum control is traditionally expressed through bilinear models and their associated Lie algebra controllability criteria. But, the first order approximation are not always sufficient and higher order developments are used in recent works. Motivated by
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eyond Bilinear Controllability: Applications to Quantum Control Gabriel Turinici Abstract. Quantum control is traditionally expressed through bilinear models and their associated Lie algebra controllability criteria. But, the first order approximation are not always sufficient and higher order developments are used in recent works. Motivated by these applications, we give in this paper a criterion that applies to situations where the evolution operator is expressed as sum of possibly non-linear real functionals of the same control that multiplies some time independent (coupling) operators. Mathematics Subject Classification (2000). Primary 34H05, 93B05; Secondary 35Q40. Keywords. Controllability, bilinear controllability, quantum control, laser control.
Contents 1 Background on quantum control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Background on controllability criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Infinite-dimensional bilinear control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Finite-dimensional bilinear control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Criteria for non linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Applications to quantum control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Density matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Wave function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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This work was partially supported by INRIA-Rocquencourt and CERMICS-ENPC, Champs sur Marne, France. The author acknowledges an ACI-NIM grant from the Minist`ere de la Recherche, France.
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1. Background on quantum control Controlling the evolution of molecular systems at quantum level has been envisioned from the very beginnings of the laser technology. However, approaches based on designing laser pulses based on intuition alone did not succeed in general situations due to the very complex interactions that are at work between the laser and the molecules to be controlled, which results, e.g., in the redistribution of the incoming laser energy to the whole molecule. Even if this circumstance initially slowed down investigations in this area, the realization that this inconvenient can be recast and attacked with the tools of (optimal) control theory [18] greatly contributed to the first positive experimental results [2, 21, 33, 6, 5, 17, 20]. The regime that is relevant for this work is related to time scales of the order of the femtosecond (10−15 ) up to picoseconds (10−12 ) and the space scales from the size of one or two atoms to large polyatomic molecules. Historically,
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