Observability and controllability for the Schrödinger equation on quotients of groups of Heisenberg type

We give necessary and sufficient conditions for the controllability of a Schr{o}dinger equation involving a subelliptic operator on a compact manifold. This subelliptic operator is the sub-Laplacian of the manifold that is obtained by taking the quotient of a group of Heisenberg type by one of its discrete subgroups. This class of nilpotent Lie groups is a major example of stratified Lie groups of step 2. The sub-Laplacian involved in these Schr{o}dinger equations is subelliptic, and, contrarily to what happens for the usual elliptic Schr{o}dinger equation for example on flat tori or on negatively curved manifolds, there exists a minimal time of controllability. The main tools used in the proofs are (operator-valued) semi-classical measures constructed by use of representation theory and a notion of semi-classical wave packets that we introduce here in the context of groups of Heisenberg type.