  In this paper, we propose a scheme for generating steady-state entanglement
of remote micromechanical oscillators in unidirectionally-coupled cavities. For
the system of two mechanical oscillators, we show that when two cavity modes in
each cavity are driven at red- and blue-detuned sidebands, respectively, a
stationary two-mode squeezed vacuum state of the two mechanical oscillators can
be generated with the help of the cavity dissipation. The degree of squeezing
is controllable by adjusting the relative strength of the pump lasers. Our
calculations also show that the achieved mechanical entanglement is robust
against thermal fluctuations of phononic environments. For the case of multiple
mechanical oscillators, we find that the steady-state genuine multipartite
entanglement can also be built up among the remote mechanical oscillators by
the cavity dissipation. The present scheme does not require nonclassical light
input or conditional quantum measurements, and it can be realized with current
experimental technology.
