Non-integrability of the optimal control problem for n-level quantum systems

We study the problem of optimal laser-induced population transfer in n-level quantum systems. This problem can be represented as an energy-optimal control problem, related to a sub-Riemannian problem on SO(n), and it is known (Boscain, Chambrion, Charlot, Gauthier) that for n  =  2 and n  =  3 the H...

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Veröffentlicht in:Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2017-04, Vol.50 (17), p.175202
Hauptverfasser: Duval, Guillaume, Maciejewski, Andrzej, Respondek, Witold
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creator Duval, Guillaume
Maciejewski, Andrzej
Respondek, Witold
description We study the problem of optimal laser-induced population transfer in n-level quantum systems. This problem can be represented as an energy-optimal control problem, related to a sub-Riemannian problem on SO(n), and it is known (Boscain, Chambrion, Charlot, Gauthier) that for n  =  2 and n  =  3 the Hamiltonian system associated with the Pontryagin Maximum Principle PMP is integrable. We will show that this changes completely for n 4. Specifically, for n  =  4, we will prove that the adjoint equation of the PMP does not possess any meromorphic first integral independent of the Hamiltonian on the levels of the Casimir functions. This implies that the system is not B-integrable for any n 4. In proving our nonintegrability results we will use differential Galois theory.
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subjects B-integrability
control of quantum systems
differential Galois theory
Hamiltonian systems on Lie Groups
Liouville integrability
Mathematics
optimal control
Pontryagin Maximum Principle
title Non-integrability of the optimal control problem for n-level quantum systems
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