  A quantum dot molecule doped with a single electron in the presence of
diagonal and off-diagonal carrier-phonon couplings is studied by means of a
non-perturbative quantum kinetic theory. The interaction with acoustic phonons
by deformation potential and piezoelectric coupling is taken into account. We
show that the phonon-mediated relaxation is fast on a picosecond timescale and
is dominated by the usually neglected off-diagonal coupling to the lattice
degrees of freedom leading to phonon-assisted electron tunneling. We show that
in the parameter regime of current electrical and optical experiments, the
microscopic non-Markovian theory has to be employed.
