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The heat current across a quantum harmonic system connected to reservoirs atdifferent temperatures is given by the Landauer formula, in terms of anintegral over phonon frequencies \omega, of the energy transmittance T(\omega).There are several different ways to derive this formula, for example using theKeldysh approach or the Langevin equation approach. The energy transmittanceT({\omega}) is usually expressed in terms of nonequilibrium phonon Green'sfunction and it is expected that it is related to the transmission coefficient{\tau}({\omega}) of plane waves across the system. In this paper, for aone-dimensional set-up of a finite harmonic chain connected to reservoirs whichare also semi-infinite harmonic chains, we present a simple and directdemonstration of the relation between T({\omega}) and {\tau}({\omega}). Ourapproach is easily extendable to the case where both system and reservoirs arein higher dimensions and have arbitrary geometries, in which case the meaningof {\tau} and its relation to T are more non-trivial.

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