Section: New Results

New results: quantum control

New results have been obtained for the control of the bilinear Schrödinger equation.

• In [16] we obtained a sufficient condition for approximate controllability of the bilinear discrete-spectrum Schrödinger equation exploiting the use of more than one control. The controllability result extends to simultaneous controllability, approximate controllability in $H^s$, and tracking in modulus. The result is more general than those present in the literature even in the case of one control and permits to treat situations in which the spectrum of the uncontrolled operator is very degenerate (e.g. multiple eigenvalues or presence of equal gaps among eigenvalues). These results are applied to the case of a rotating polar linear molecule in the space, driven by three external fields. A remarkable property of this model is the presence of infinitely many degeneracies and resonances in the spectrum preventing the application of the results in the literature.

• In [19] we present a constructive method to control the bilinear Schrödinger equation by means of two or three controlled external fields. The method is based on adiabatic techniques and works if the spectrum of the Hamiltonian admits eigenvalue intersections, with respect to variations of the controls, and if the latter are conical. We provide sharp estimates of the relation between the error and the controllability time.

• In [18] we consider the minimum time population transfer problem for a two level quantum system driven by two external fields with bounded amplitude. The controls are modeled as real functions and we do not use the Rotating Wave Approximation. After projection on the Bloch sphere, we tackle the time-optimal control problem with techniques of optimal synthesis on 2-D manifolds. Based on the Pontryagin Maximum Principle, we characterize a restricted set of candidate optimal trajectories. Properties on this set, crucial for complete optimal synthesis, are illustrated by numerical simulations. Furthermore, when the two controls have the same bound and this bound is small with respect to the difference of the two energy levels, we get a complete optimal synthesis up to a small neighborhood of the antipodal point of the starting point.