Phononic heat transport in the transient regime: An analytic solution

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.

Figure 1

Heat transport setup where a central system of interest is connected to two reservoirs of different temperatures. The internal structures and couplings are defined by the force constant matrices $K$. The coupling refers to partitioned approach where the different systems are uncoupled at times $t<0$ and coupled at times $t\ge 0$ when the system is driven out of equilibrium.

Figure 2

Frequency dependency of the retarded embedding self-energy for the infinite coupled spring model: The solid red line is the real part and the long-dashed green line is the imaginary part. The respective approximations are the dash-dotted blue line and the short-dashed magenta line. The axes are scaled with the interatom force constant $k_{\lambda }$.

Figure 3

Time-dependent heat current through a dimer molecule (in units of appropriate energy scale) connected to two reservoirs with (a) strong coupling and broad spectrum and (b) weak coupling and narrow spectrum. Thick solid red line is the numerical solution to Eq. (12), long-dashed green, dash-dotted blue, short-dashed magenta, and thin solid cyan lines describe different levels of approximation, see text. The inset shows the difference between the approximate and full solutions.

Figure 4

Time-dependence of the commutator between the displacement and momentum operators at site 1 of a dimer molecule. The parameters of the calculation are the same as in Fig. 3.

Figure 5

Time-dependent energy in a single site connected to two reservoirs with (a) strong coupling and broad spectrum, and (b) weak coupling and narrow spectrum. Solid red line is the numerical solution to Eq. (12), dashed green, blue and magenta lines are obtained from Eq. (35) with different cutoff frequencies. The insets show the $uu$ component of $f_L\left(\omega \right){\mathbf{B}}_L\left(\omega \right)$ (in logarithmic scale) with vertical lines indicating the integration cutoff frequency.

Figure 6

Time-dependent local heat current (in appropriate units) in the middle of a four-site chain connected to two reservoirs with (a) strong coupling and broad spectrum, (b) intermediate coupling and spectrum, and (c) weak coupling and narrow spectrum. Solid red line is the numerical solution to Eq. (12) and dashed green line is obtained from Eq. (35) by cutoff frequency at the phonon bandwidth, ${\omega }_{c,\lambda }=2\sqrt{k_{\lambda }}$. The insets in (b) and (c) show the long-time behavior of the heat current.

Figure 7

Time dependence of the individual density matrix elements (real parts) when computing the heat current between sites 2 and 3 in a four-site chain, cf. Fig. 6. The solid lines are the full numerical solutions and the markers with corresponding color are evaluated from Eq. (35).

Figure 8

Relaxation times (color map) for atomic chains of varying length (horizontal axis) and coupling strength to the reservoirs (vertical axis) at (a) low and (b) high temperature.