Landauer formula for phonon heat conduction: relation between energy transmittance and transmission coefficient

The heat current across a quantum harmonic system connected to reservoirs at different temperatures is given by the Landauer formula, in terms of an integral over phonon frequencies ω , of the energy transmittance \hbox{$\mathcal{T}(\omega)$}T( ω ). There are several different ways to derive this formula, for example using the Keldysh approach or the Langevin equation approach. The energy transmittance \hbox{$\mathcal{T}(\omega)$} T( ω ) is usually expressed in terms of nonequilibrium phonon Green’s function and it is expected that it is related to the transmission coefficient τ ( ω ) of plane waves across the system. In this paper, for a one-dimensional set-up of a finite harmonic chain connected to reservoirs which are also semi-infinite harmonic chains, we present a simple and direct demonstration of the relation between \hbox{$\mathcal{T}(\omega)$}T( ω ) and τ ( ω ). Our approach is easily extendable to the case where both system and reservoirs are in higher dimensions and have arbitrary geometries, in which case the meaning of τ and its relation to \hbox{$\mathcal{T}$}Tare more non-trivial.