Quantum Precursor of Shuttle Instability

The effects of a coupling between the quantized mechanical vibrations of a quantum dot and coherent tunneling of electrons through a single level in the dot are studied. The equation of motion for the reduced density operator describing the vibrational degree of freedom is obtained. It is shown that that the expectation value of the displacement is an oscillating function of time with an exponentially increasing amplitude, which is the signature of a quantum shuttle instability.