In this talk, I will present a new result about small-time local controllability near the ground state for a bilinear Schrödinger equation with Neumann boundary conditions, for which the linearized system is not controllable. I will prove that a Lie bracket–type condition ensures that either the nonlinear system exhibits a quadratic obstruction or, remarkably, recovers controllability at the quadratic order. This is a joint work with Karine Beauchard and Frédéric Marbach.