Approximation of diffeomorphisms for quantum state transfers

Abstract: In this paper, we seek to combine two emerging standpoints in control theory. On the one hand, recent advances in infinite-dimensional geometric control have unlocked a method for controlling (with arbitrary precision and in arbitrarily small times) state transfers for bilinear Schr\"odinger PDEs posed on a Riemannian manifold $M$. In particular, these arguments rely on controllability results in the group of the diffeomorphisms of $M$. On the other hand, using tools of $\Gamma$-convergence, it has been proved that we can phrase the retrieval of a diffeomorphism of $M$ as an ensemble optimal control problem. More precisely, this is done by employing a control-affine system for \emph{simultaneously} steering a finite swarm of points towards the respective targets. Here we blend these two theoretical approaches and numerically find control laws driving state transitions (such as eigenstate transfers) in small time in a bilinear Schr\"odinger PDE posed on the torus. Such systems have experimental relevance and are currently used to model rotational dynamics of molecules, and cold atoms trapped in periodic optical lattices.