Phononic heat transport in the transient regime: An analytic solution

TL;DR

This paper derives an analytic solution for transient phononic heat transport in harmonic systems, enabling detailed analysis of heat current dynamics and relaxation times, with validation against numerical methods.

Contribution

It provides a novel analytic expression for time-dependent phonon transport, bridging transient and steady-state regimes with controlled approximations.

Findings

Analytic solution matches numerical results for weak contacts.

Transient oscillations and relaxation times are characterized.

Good agreement across temperature gradients and energy dispersions.

Abstract

We investigate the time-resolved quantum transport properties of phonons in arbitrary harmonic systems connected to phonon baths at different temperatures. We obtain a closed analytic expression of the time-dependent one-particle reduced density matrix by explicitly solving the equations of motion for the nonequilibrium Green's function. This is achieved through a well-controlled approximation of the frequency-dependent bath self-energy. Our result allows for exploring transient oscillations and relaxation times of local heat currents, and correctly reduces to an earlier known result in the steady-state limit. We apply the formalism to atomic chains, and benchmark the validity of the approximation against full numerical solutions of the bosonic Kadanoff--Baym equations for the Green's function. We find good agreement between the analytic and numerical solutions for weak contacts and baths with a wide energy dispersion. We further analyze relaxation times from low to high temperature gradients.