  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.
