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

dc.contributor.areaMathematicsen_US
dc.contributor.authorBoscain, Ugoen_US
dc.contributor.authorMason, Paoloen_US
dc.contributor.departmentFunctional Analysis and Applicationsen_US
dc.date.accessioned2005en_US
dc.date.accessioned2011-09-07T20:27:43Z
dc.date.available2005en_US
dc.date.available2011-09-07T20:27:43Z
dc.date.issued2005en_US
dc.description.abstractIn 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.extent550597 bytesen_US
dc.format.mimetypeapplication/pdfen_US
dc.identifier.citationJournal of Mathematical Physics 47, 062101 (2006)en_US
dc.identifier.urihttps://openscience.sissa.it/handle/1963/1734en_US
dc.language.isoen_USen_US
dc.relation.ispartofseriesSISSA;82/2005/Men_US
dc.relation.ispartofseriesarXiv.org;quant-ph/0512074en_US
dc.relation.uri10.1063/1.2203236en_US
dc.titleTime Minimal Trajectories for a Spin 1/2 Particle in a Magnetic fielden_US
dc.typePreprinten_US
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