  We consider a 1D linear Schr{\"o}dinger equation, on a bounded interval, with
Dirichlet boundary conditions and bilinear control. We study its
controllability around the ground state when the linearized system is not
controllable. More precisely, we study to what extent the nonlinear terms of
the expansion can recover the directions lost at the first order.In previous
works, for any positive integer $n$, assumptions have been formulated under
which the quadratic term induces a drift in the nonlinear dynamics, quantified
by the $H^{-n}$ norm of the control. This drift is an obstruction to the
small-time local controllability (STLC) under a smallness assumption on the
controls in regular spaces. In this paper, we prove that for controls small in
less regular spaces, the cubic term can recover the controllability lost at the
linear level, despite the quadratic drift. The proof is inspired by Sussman's
method to prove the sufficiency of the $\mathcal{S}(\theta)$ condition for STLC
of ODEs. However, it uses a different global strategy relying on a new concept
of tangent vector, better adapted to the infinite-dimensional setting of PDEs.
Given a target, we first realize the expected motion along the lost direction
by using control variations for which the cubic term dominates the quadratic
one. Then, we correct the other components exactly, by using a STLC in
projection result, with simultaneous estimates of weak norms of the control.
These estimates ensure that the new error along the lost direction is
negligible, and we conclude with the Brouwer fixed point theorem.
