Quantum heat transfer in harmonic chains with self-consistent reservoirs: Exact numerical simulations

We describe a numerical scheme for exactly simulating the heat current behavior in a quantum harmonic chain with self-consistent reservoirs. Numerically exact results are compared to classical simulations and to the quantum behavior under the linear-response approximation. In the classical limit or for small temperature biases our results coincide with previous calculations. At large bias and for low temperatures the quantum dynamics of the system fundamentally differs from the close-to-equilibrium behavior, revealing in particular the effect of thermal rectification for asymmetric chains. Since this effect is absent in the classical analog of our model, we conclude that in the quantum model studied here thermal rectification is a purely quantum phenomenon, rooted in the quantum statistics.

Figure 1

Scheme of a harmonic chain of $N=5$ beads, where the inner particles are connected to SC baths. The wiggly lines represent harmonic bonds. The temperatures $T_1$ and $T_5$ set the boundary conditions; the temperatures $T_l$ ($l=2,3,4$) are determined by demanding that the leaking currents vanish, $F_l=0$. The net heat current across the system is given by $F_1=-F_5$.

Figure 2

Convergence of the inner reservoirs’ temperatures with increasing number of iterations. $N=10$, ${\beta }_1=1$, ${\beta }_N=5,$ and ${\gamma }_n=0.2$ for $n=1,\dots ,N$. The main plot shows the temperature at site number 5. The inset displays the corresponding convergence of the net heat current, $F_1$. Results are obtained using the numerical method (12) (square) and the Newton-Raphson method (dotted).

Figure 3

(left) The currents $F_l$, $l=1,2,\dots ,N$ [see Eq. (5)], at each site, for a chain with $N=10$ beads, obtained after applying the iterative procedure $p+q=50$ times, ${\beta }_1=1$, ${\beta }_N=5$, ${\gamma }_l=0.2$. (right) Zooming on the internal currents $F_2$ to $F_9$, zero for perfect SC reservoirs.

Figure 4

Heat current as a function of chain size using the quantum-exact method ($○$), quantum linear response formalism (diamond), and the classical limit ($+$). ${\gamma }_l=0.2$ for all sites. (a) ${\beta }_1=0.1$, ${\beta }_N=0.2$, (b) ${\beta }_1=1$, ${\beta }_N=1.2$, (c) ${\beta }_1=1$, ${\beta }_N=5$, (d) ${\beta }_1=5$, ${\beta }_N=20$. The dotted lines represent the exact quantum behavior when the internal reservoirs are detached from the chain.

Figure 5

Temperature profile of the SC reservoirs at site $n$, $N=10$, using the exact quantum method ($○$), quantum linear response formalism (diamond), and the classical expression ($+$), ${\gamma }_l=0.2$ for all sites. (a) ${\beta }_1=0.1$, ${\beta }_N=0.2$, (b) ${\beta }_1=1$, ${\beta }_N=1.2$, (c) ${\beta }_1=1$, ${\beta }_N=5$, (d) ${\beta }_1=5$, ${\beta }_N=20$.

Figure 6

A scheme of the mass-graded harmonic chain with SC baths assuming $N=4$ beads.

Figure 7

Thermal rectification in a mass graded system: The magnitude of the heat current as a function of chain size for forward temperature bias $T_1>T_N$ ($○$) and for the backward direction, $T_N>T_1$ (square), ${\gamma }_n=0.2$ for all sites, $M_1=0.2$, $M_n=M_1+(n-1)\times 0.2$. In the forward direction we used (a) ${\beta }_1=0.1$, ${\beta }_N=0.2$, (b) ${\beta }_1=1$, ${\beta }_N=1.2$, and (c) ${\beta }_1=1$, ${\beta }_N=5$. The opposite polarities were used in each case to generate $J_{-}$.

Figure 8

Temperature profile for a mass-graded $N=7$ chain. In the forward direction ($○$) we used (a) ${\beta }_1=0.1$, ${\beta }_N=0.2$ and (b) ${\beta }_1=1$, ${\beta }_N=5$. The opposite polarities were used to generate the reversed profile (square). Other parameters are the same as in Fig. 7. The inset zooms on the chain center.

Figure 9

Thermal rectification in systems with contact asymmetry: The magnitude of the heat current as a function of chain size for forward temperate bias $T_1>T_N$ ($○$) and for the backward direction, $T_N>T_1$ (diamond) ${\gamma }_1=0.1$, ${\gamma }_N=0.4$, ${\gamma }_{n\ne 1,N}=0.2$. ${\beta }_1=1$, ${\beta }_N=5$, and the the reversed contact symmetry. (a) Rectification ratio $|J_{+}/J_{-}|$. (b) The forward and backward currents at high temperature ${\beta }_1=0.1$ and ${\beta }_N=0.2$, and the reversed setup, with no significant rectification observed.