TY - JOUR T1 - Controllability of the discrete-spectrum Schrodinger equation driven by an external field JF - Ann. Inst. H. Poincare Anal. Non Lineaire 26 (2009) 329-349 Y1 - 2009 A1 - Thomas Chambrion A1 - Paolo Mason A1 - Mario Sigalotti A1 - Ugo Boscain AB - We prove approximate controllability of the bilinear Schrodinger equation in the case in which the uncontrolled Hamiltonian has discrete nonresonant\\nspectrum. The results that are obtained apply both to bounded or unbounded domains and to the case in which the control potential is bounded or unbounded. The method relies on finite-dimensional techniques applied to the\\nGalerkin approximations and permits, in addition, to get some controllability properties for the density matrix. Two examples are presented: the harmonic oscillator and the 3D well of potential controlled by suitable potentials. UR - http://hdl.handle.net/1963/2547 U1 - 1572 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - An estimation of the controllability time for single-input systems on compact Lie Groups JF - ESAIM Control Optim. Calc. Var. 12 (2006) 409-441 Y1 - 2006 A1 - Andrei A. Agrachev A1 - Thomas Chambrion AB - Geometric control theory and Riemannian techniques are used to describe the reachable set at time t of left invariant single-input control systems on semi-simple compact Lie groups and to estimate the minimal time needed to reach any point from identity. This method provides an effective way to give an upper and a lower bound for the minimal time needed to transfer a controlled quantum system with a drift from a given initial position to a given final position. The bounds include diameters of the flag manifolds; the latter are also explicitly computed in the paper. UR - http://hdl.handle.net/1963/2135 U1 - 2108 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Nonisotropic 3-level quantum systems: complete solutions for minimum time and minimum energy JF - Discrete Contin. Dyn. Syst. Ser. B 5 (2005) 957-990 Y1 - 2005 A1 - Ugo Boscain A1 - Thomas Chambrion A1 - Grégoire Charlot AB - We apply techniques of subriemannian geometry on Lie groups and of optimal synthesis on 2-D manifolds to the population transfer problem in a three-level quantum system driven by two laser pulses, of arbitrary shape and frequency. In the rotating wave approximation, we consider a nonisotropic model i.e. a model in which the two coupling constants of the lasers are different. The aim is to induce transitions from the first to the third level, minimizing 1) the time of the transition (with bounded laser amplitudes),\\n2) the energy of lasers (with fixed final time). After reducing the problem to real variables, for the purpose 1) we develop a theory of time optimal syntheses for distributional problem on 2-D-manifolds, while for the purpose 2) we use techniques of subriemannian geometry on 3-D Lie groups. The complete optimal syntheses are computed. UR - http://hdl.handle.net/1963/2259 U1 - 1988 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - On the K+P problem for a three-level quantum system: optimality implies resonance JF - J.Dynam. Control Systems 8 (2002),no.4, 547 Y1 - 2002 A1 - Ugo Boscain A1 - Thomas Chambrion A1 - Jean-Paul Gauthier PB - SISSA Library UR - http://hdl.handle.net/1963/1601 U1 - 2517 U2 - Mathematics U3 - Functional Analysis and Applications ER -