  Feasible setup for pulsed quantum non-demolition interaction between two
distant mechanical oscillators through optical or microwave mediator is
proposed. The proposal uses homodyne measurement of the mediator and
feedforward control of the mechanical oscillators to reach the interaction. To
verify quantum nature of the interaction, we investigate the Gaussian
entanglement generated in the mechanical modes. We evaluate it under influence
of mechanical bath and propagation loss for the mediator and propose ways to
optimize the interaction. Finally, both currently available optomechanical and
electromechanical platforms are numerically analyzed. The analysis shows that
implementation is already feasible with current technology.
