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Time Minimal Trajectories for a Spin 1/2 Particle in a Magnetic field

Show simple item record Boscain, Ugo en_US Mason, Paolo en_US 2005 en_US 2011-09-07T20:27:43Z 2005 en_US 2011-09-07T20:27:43Z 2005 en_US
dc.identifier.citation Journal of Mathematical Physics 47, 062101 (2006) en_US
dc.identifier.uri en_US
dc.description.abstract In this paper we consider the minimum time population transfer problem for the z-component of the spin of a (spin 1/2) particle driven by a magnetic field, controlled along the x axis, with bounded amplitude. On the Bloch sphere (i.e. after a suitable Hopf projection), this problem can be attacked with techniques of optimal syntheses on 2-D manifolds. Let (-E,E) be the two energy levels, and |omega (t)| ≤ M the bound on the field amplitude. For each couple of values E and M, we determine the time optimal synthesis starting from the level -E and we provide the explicit expression of the time optimal trajectories steering the state one to the state two, in terms of a parameter that can be computed solving numerically a suitable equation. For M/E << 1, every time optimal trajectory is bang-bang and in particular the corresponding control is periodic with frequency of the order of the resonance frequency wR = 2E. On the other side, for M/E > 1, the time optimal trajectory steering the state one to the state two is bang-bang with exactly one switching. Fixed E we also prove that for M → ∞ the time needed to reach the state two tends to zero. In the case M/E > 1 there are time optimal trajectories containing a singular arc. Finally we compare these results with some known results of Khaneja, Brockett and Glaser and with those obtained by controlling the magnetic field both on the x and y directions (or with one external field, but in the rotating wave approximation). As byproduct we prove that the qualitative shape of the time optimal synthesis presents different patterns, that cyclically alternate as M/E → 0, giving a partial proof of a conjecture formulated in a previous paper. en_US
dc.format.extent 550597 bytes en_US
dc.format.mimetype application/pdf en_US
dc.language.iso en_US en_US
dc.relation.ispartofseries SISSA;82/2005/M en_US
dc.relation.ispartofseries;quant-ph/0512074 en_US
dc.relation.uri 10.1063/1.2203236 en_US
dc.title Time Minimal Trajectories for a Spin 1/2 Particle in a Magnetic field en_US
dc.type Preprint en_US
dc.contributor.department Functional Analysis and Applications en_US
dc.contributor.area Mathematics en_US

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