  We analyze a simple microscopic model to pump heat from a cold to a hot
reservoir in a nanomechanical system. The model consists of a one-dimensional
chain of masses and springs coupled to a back gate through which a
time-dependent perturbation is applied. The action of the gate is to modulate
the coupling of the masses to a substrate via additional springs that introduce
a moving phononic barrier. We solve the problem numerically using
non-equilibrium Green function techniques. For low driving frequencies and for
sharp traveling barriers, we show that this microscopic model realizes a phonon
refrigerator.
