Section: New Results

Quantum control: new results

Let us list here our new results in quantum control theory.

• In [5] we consider a quantum particle in a potential $V\left(x\right)$ ($x\in {ℝ}^N$) in a time-dependent electric field $E\left(t\right)$ (the control). Boscain, Caponigro, Chambrion and Sigalotti proved in [83] that, under generic assumptions on $V$, this system is approximately controllable on the $L^2({ℝ}^N,ℂ)$-sphere, in sufficiently large time $T$. In the present article we show that approximate controllability does not hold in arbitrarily small time, no matter what the initial state is. This generalizes our previous result for Gaussian initial conditions. Moreover, we prove that the minimal time can in fact be arbitrarily large.

• In [11] we consider the bilinear Schrödinger equation with discrete-spectrum drift. We show, for $n\in \mathbb{N}$ arbitrary, exact controllability in projections on the first $n$ given eigenstates. The controllability result relies on a generic controllability hypothesis on some associated finite-dimensional approximations. The method is based on Lie-algebraic control techniques applied to the finite-dimensional approximations coupled with classical topological arguments issuing from degree theory.

• In [14] we consider the one dimensional Schrödinger equation with a bilinear control and prove the rapid stabilization of the linearized equation around the ground state. The feedback law ensuring the rapid stabilization is obtained using a transformation mapping the solution to the linearized equation on the solution to an exponentially stable target linear equation. A suitable condition is imposed on the transformation in order to cancel the non-local terms arising in the kernel system. This conditions also insures the uniqueness of the transformation. The continuity and invertibility of the transformation follows from exact controllability of the linearized system.

• In [33] we discuss how to control a parameter-dependent family of quantum systems. Our technique is based on adiabatic approximation theory and on the presence of curves of conical eigenvalue intersections of the controlled Hamiltonian. As particular cases, we recover chirped pulses for two-level quantum systems and counter-intuitive solutions for three-level stimulated Raman adiabatic passage (STIRAP). The proposed technique works for systems evolving both in finite-dimensional and infinite-dimensional Hilbert spaces. We show that the assumptions guaranteeing ensemble controllability are structurally stable with respect to perturbations of the parametrized family of systems.