32,897 research outputs found
Time Minimal Trajectories for a Spin 1/2 Particle in a Magnetic Field
In this paper we consider the minimum time population transfer problem for
the -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 be the two
energy levels, and the bound on the field amplitude. For
each couple of values and , we determine the time optimal synthesis
starting from the level 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
, every time optimal trajectory is bang-bang and in particular the
corresponding control is periodic with frequency of the order of the resonance
frequency . On the other side, for , the time optimal
trajectory steering the state one to the state two is bang-bang with exactly
one switching. Fixed we also prove that for the time needed to
reach the state two tends to zero. In the case 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 and 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 , giving a partial proof of a
conjecture formulated in a previous paper.Comment: 31 pages, 10 figures, typos correcte
Geometric optimal control of the contrast imaging problem in Nuclear Magnetic Resonance
The objective of this article is to introduce the tools to analyze the
contrast imaging problem in Nuclear Magnetic Resonance. Optimal trajectories
can be selected among extremal solutions of the Pontryagin Maximum Principle
applied to this Mayer type optimal problem. Such trajectories are associated to
the question of extremizing the transfer time. Hence the optimal problem is
reduced to the analysis of the Hamiltonian dynamics related to singular
extremals and their optimality status. This is illustrated by using the
examples of cerebrospinal fluid / water and grey / white matter of cerebrum.Comment: 30 pages, 13 figur
Optimisation of Quantum Trajectories Driven by Strong-field Waveforms
Quasi-free field-driven electron trajectories are a key element of
strong-field dynamics. Upon recollision with the parent ion, the energy
transferred from the field to the electron may be released as attosecond
duration XUV emission in the process of high harmonic generation (HHG). The
conventional sinusoidal driver fields set limitations on the maximum value of
this energy transfer, and it has been predicted that this limit can be
significantly exceeded by an appropriately ramped-up cycleshape. Here, we
present an experimental realization of such cycle-shaped waveforms and
demonstrate control of the HHG process on the single-atom quantum level via
attosecond steering of the electron trajectories. With our optimized optical
cycles, we boost the field-ionization launching the electron trajectories,
increase the subsequent field-to-electron energy transfer, and reduce the
trajectory duration. We demonstrate, in realistic experimental conditions, two
orders of magnitude enhancement of the generated XUV flux together with an
increased spectral cutoff. This application, which is only one example of what
can be achieved with cycle-shaped high-field light-waves, has farreaching
implications for attosecond spectroscopy and molecular self-probing
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