  We propose a new simple construction of a coupling at a fixed time of two
sub-Riemannian Brownian motions on the Heisenberg group and on the free step 2
Carnot groups. The construction is based on a Legendre expansion of the
standard Brownian motion and of the L{\'e}vy area. We deduce sharp estimates
for the decay in total variation distance between the laws of the Brownian
motions. Using a change of probability method, we also obtain the log-Harnack
inequality, a Bismut type integration by part formula and reverse Poincar\'e
inequalities for the associated semi-group.
