  We investigate the effect of a quantised vibrational mode on electron
tunneling through a chain of three quantum dots. The outer dots are coupled to
voltage leads, but the position of the central dot is not rigidly fixed. Motion
of the central dot modulates the size of the tunneling barriers in opposite
ways so that electron tunneling is correlated with the position of the
oscillator. We treat the electronic part of the problem using a simple
Coulomb-blockade picture, and model the vibration of the central dot as a
quantum oscillator. We calculate the eigenspectrum of the system as a function
of the energy level shift between the outer dots. Using a density matrix
method, we include couplings to external leads and calculate the steady-state
current through the device. The current shows marked resonances which
correspond to avoided-level crossings in the eigenvalue spectrum. When the
tunneling length of the electrons is of order the zero-point position
uncertainty of the quantum oscillator, current far from the electronic
resonance is dominated by electrons hopping on and off the central dot
sequentially; the oscillator can be regarded as shuttling electrons across the
system [L.Y. Gorelik, et al., Phys. Rev. Lett. 80, 4526 (1998)]. Damping of the
oscillator can increase the current by preventing electrons from hopping
`backwards'.
