In this talk, I will discuss small-time local controllability near the ground state of a bilinear Schrödinger equation with Neumann boundary conditions. I will present a new result stating that if the linearized system is not controllable, then a Lie bracket-like condition ensures that either the nonlinear system is controllable or it exhibits a quadratic obstruction. I will give some ideas of proof. This is joint work with Karine Beauchard and Frédéric Marbach.