  We studied the phononic heat transfer through an atomic dielectric wire with
both infinite and finite lengths by using a model Hamiltonian approach. At low
temperature under ballistic transport, the thermal conductance contributed by
each phonon branch of a uniform and harmonic chain cannot exceed the well-known
value which depends linearly on temperature but is material independent. We
predict that this ballistic thermal conductance will exhibit stepwise behavior
as a function of temperature. By performing numerical calculations on a more
realistic system, where a small atomic chain is placed between two reservoirs,
we also found resonance modes, which should also lead to the stepwise behavior
in the thermal conductance.
