  In this paper, we investigate the quantum transport of a double quantum dot
coupled with a nanomechanical resonator at arbitrary strong electron-phonon
coupling regimes. We employ the generalized quantum master equation to study
full counting statistics of currents. We demonstrate the coherent phonon states
method can be applied to decouple the electron-phonon interaction
non-perturbatively. With the help of this non-perturbative treatment of
electron-phonon couplings, we find that the phonon-assisted resonant tunneling
emerges when the excess energy from the left quantum dot to the right one can
excite integer number of phonons and multi-phonon excitations can enhance the
transport in strong electron-phonon coupling regime. Moreover, we find that as
the electron-phonon coupling increases, it first plays a constructive role to
assist the transport, and then plays the role of scattering and strongly
represses the transport.
