  A quantum shuttle is an archetypical nanoelectromechanical device, where the
mechanical degree of freedom is quantized. Using a full-scale numerical
solution of the generalized master equation describing the shuttle, we have
recently shown [Novotn\'{y} {\it et al.}, Phys. Rev. Lett. {\bf 92}, 248302
(2004)] that for certain limits of the shuttle parameters one can distinguish
three distinct charge transport mechanisms: (i) an incoherent tunneling regime,
(ii) a shuttling regime, where the charge transport is synchronous with the
mechanical motion, and (iii) a coexistence regime, where the device switches
between the tunneling and shuttling regimes. While a study of the cross-over
between these three regimes requires the full numerics, we show here that by
identifying the appropriate time-scales it is possible to derive vastly simpler
equations for each of the three regimes. The simplified equations allow a clear
physical interpretation, are easily solved, and are in good agreement with the
full numerics in their respective domains of validity.
