  We present a theoretical study of time-dependent transport via a quantum
shuttle within the non-equilibrium Green's function technique. An arbitrary
voltage is applied to the tunnel junction and electrons in the leads are
considered to be at zero temperature. The transient and the steady state
behavior of the system is considered here in order to explore the quantum
dynamics of the shuttle device as a function of time and applied bias. The
properties of the phonon distribution of the oscillating dot coupled to the
electrons are investigated using a non-perturbative approach. We derive a
relation for the oscillator momentum charge density correlation function which
is an interesting physical example for the visualization of shuttling
phenomenon. We consider the crossover between the tunneling and shuttling
regimes for different values of the key parameters as a function of applied
bias and time. We also consider the energy transferred from the electrons to
the oscillating dot as a function of time. This will provide useful insight for
the design of experiments aimed at studying the quantum behavior of a shuttling
device.
