Time-optimal navigation through quantum wind

The quantum navigation problem of finding the time-optimal control Hamiltonian that transports a given initial state to a target state through quantum wind, that is, under the influence of external fields or potentials, is analysed. By lifting the problem from the state space to the space of unitary gates realising the required task, we are able to deduce the form of the solution to the problem by deriving a universal quantum speed limit. The expression thus obtained indicates that further simplifications of this apparently difficult problem are possible if we switch to the interaction picture of quantum mechanics. A complete solution to the navigation problem for an arbitrary quantum system is then obtained, and the behaviour of the solution is illustrated in the case of a two-level system

Minimal time for the bilinear control of Schrödinger equations

International audienceWe consider a quantum particle in a potential V (x) (x in R^N) subject to a (spatially homogeneous) time-dependent electric field E(t), which plays the role of the control. Under generic assumptions on V , this system is approximately controllable on the L2(R^N;C)-sphere, in su ffiently large times T, as proved by Boscain, Caponigro, Chambrion and Sigalotti. In the present article, we show that this approximate controllability result is false in small time. As a consequence, the result by Boscain et al. is, in some sense, optimal with respect to the control time T

Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds

2010 Mathematics Subject Classification: Primary: 37C85, 37A17, 37A45; Secondary: 11K36, 11L07.We prove quantitative equidistribution results for actions of Abelian subgroups of the (2g + 1)-dimensional Heisenberg group acting on compact (2g + 1)-dimensional homogeneous nilmanifolds. The results are based on the study of the C∞-cohomology of the action of such groups, on tame estimates of the associated cohomological equations and on a renormalization method initially applied by Forni to surface flows and by Forni and the second author to other parabolic flows. As an application we obtain bounds for finite Theta sums defined by real quadratic forms in g variables, generalizing the classical results of Hardy and Littlewood [25, 26] and the optimal result of Fiedler, Jurkat, and Körner [17] to higher dimension.This work was partially done while L. Flaminio visited the Isaac Newton Institute in Cambridge, UK. He wishes to thank the Institute and the organizers of the program Interactions between Dynamics of Group Actions and Number Theory for their hospitality. L. Flaminio was supported in part by the Labex CEMPI (ANR-11-LABX-07). S. Cosentino was partially supported by CMAT - Centro de Matematica da Universidade do Minho, financed by the Strategic Project PEst-OE/MAT/UI0013/2014

Analysis of a splitting scheme for a class of nonlinear stochastic Schr\uf6dinger equations

We analyze the qualitative properties and the order of convergence of a splitting scheme for a class of nonlinear stochastic Schr\uf6dinger equations driven by additive It\uf4 noise. The class of nonlinearities of interest includes nonlocal interaction cubic nonlinearities. We show that the numerical solution is symplectic and preserves the expected mass for all times. On top of that, for the convergence analysis, some exponential moment bounds for the exact and numerical solutions are proved. This enables us to provide strong orders of convergence as well as orders of convergence in probability and almost surely. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme