  We consider a natural analogue of Brownian motion on free orthogonal quantum
groups and prove that it exhibits a cutoff at time $N\ln(N)$. Then, we study
the induced classical process on the real line and compute its atoms and
density. This enables us to find the cutoff profile, which involves free
Poisson distributions and the semi-circle law. We prove similar results for
quantum permutations and quantum random transpositions.
