Controllability of the discrete-spectrum Schrodinger equation driven by an external field

We prove approximate controllability of the bilinear Schr\"odinger equation in the case in which the uncontrolled Hamiltonian has discrete non-resonant spectrum. 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 Galerkin 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, both controlled by suitable potentials

Which notion of energy for bilinear quantum systems?

In this note we investigate what is the best L^p-norm in order to describe the relation between the evolution of the state of a bilinear quantum system with the L^p-norm of the external field. Although L^2 has a structure more easy to handle, the L^1 norm is more suitable for this purpose. Indeed for every p>1, it is possible to steer, with arbitrary precision, a generic bilinear quantum system from any eigenstate of the free Hamiltonian to any other with a control of arbitrary small L^p norm. Explicit optimal costs for the L^1 norm are computed on an example

Efficient finite dimensional approximations for the bilinear Schrodinger equation with bounded variation controls

This the text of a proceeding accepted for the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014). We present some results of an ongoing research on the controllability problem of an abstract bilinear Schrodinger equation. We are interested by approximation of this equation by finite dimensional systems. Assuming that the uncontrolled term $A$ has a pure discrete spectrum and the control potential $B$ is in some sense regular with respect to $A$ we show that such an approximation is possible. More precisely the solutions are approximated by their projections on finite dimensional subspaces spanned by the eigenvectors of $A$. This approximation is uniform in time and in the control, if this control has bounded variation with a priori bounded total variation. Hence if these finite dimensional systems are controllable with a fixed bound on the total variation of the control then the system is approximatively controllable. The main outcome of our analysis is that we can build solutions for low regular controls such as bounded variation ones and even Radon measures

Controllability of the bilinear Schr\"odinger equation with several controls and application to a 3D molecule

We show the approximate rotational controllability of a polar linear molecule by means of three nonresonant linear polarized laser fields. The result is based on a general approximate controllability result for the bilinear Schr\"odinger equation, with wavefunction varying in the unit sphere of an infinite-dimensional Hilbert space and with several control potentials, under the assumption that the internal Hamiltonian has discrete spectrum

Approximate Controllability, Exact Controllability, and Conical Eigenvalue Intersections for Quantum Mechanical Systems

International audienceWe study the controllability of a closed control-affine quantum system driven by two or more external fields. We provide a sufficient condition for controllability in terms of existence of conical intersections between eigenvalues of the Hamiltonian in dependence of the controls seen as parameters. Such spectral condition is structurally stable in the case of three controls or in the case of two controls when the Hamiltonian is real. The spectral condition appears naturally in the adiabatic control framework and yields approximate controllability in the infinite-dimensional case. In the finite-dimensional case it implies that the system is Lie-bracket generating when lifted to the group of unitary transformations, and in particular that it is exactly controllable. Hence, Lie algebraic conditions are deduced from purely spectral properties. We conclude the article by proving that approximate and exact controllability are equivalent properties for general finite-dimensional quantum systems

Small time reachable set of bilinear quantum systems

This note presents an example of bilinear conservative system in an infinite dimensional Hilbert space for which approximate controllability in the Hilbert unit sphere holds for arbitrary small times. This situation is in contrast with the finite dimensional case and is due to the unboundedness of the drift operator

Explicit approximate controllability of the Schr\"odinger equation with a polarizability term

We consider a controlled Schr\"odinger equation with a dipolar and a polarizability term, used when the dipolar approximation is not valid. The control is the amplitude of the external electric field, it acts non linearly on the state. We extend in this infinite dimensional framework previous techniques used by Coron, Grigoriu, Lefter and Turinici for stabilization in finite dimension. We consider a highly oscillating control and prove the semi-global weak $H^2$ stabilization of the averaged system using a Lyapunov function introduced by Nersesyan. Then it is proved that the solutions of the Schr\"odinger equation and of the averaged equation stay close on every finite time horizon provided that the control is oscillating enough. Combining these two results, we get approximate controllability to the ground state for the polarizability system