  We investigate the possibility of an electromechanical which-path
interferometer, in which electrons travelling through an Aharonov-Bohm ring
incorporating a quantum dot in one of the arms are dephased by an interaction
with the fundamental flexural mode of a radio frequency cantilever. The
cantilever is positioned so that its tip lies just above the dot and a bias is
applied so that an electric field exists between the dot and the tip. This
electric field is modified when an additional electron hops onto the dot,
coupling the flexural mode of the cantilever and the microscopic electronic
degrees of freedom. We analyze the transmission properties of this system and
the dependence of interference fringe visibility on the cantilever-dot coupling
and on the mechanical properties of the cantilever. The fringes are
progressively destroyed as the interaction with the cantilever is turned up, in
part due to dephasing arising from the entanglement of the electron and
cantilever states and also due to the thermal smearing that results from
fluctuations in the state of the cantilever. When the dwell time of the
electron on the dot is comparable to or longer than the cantilever period, we
find coherent features in the transmission amplitude. These features are washed
out when the cantilever is decohered by its coupling to the environment.
