Heat transport through lattices of quantum harmonic oscillators in arbitrary dimensions

In $d$-dimensional lattices of coupled quantum harmonic oscillators, we analyze the heat current caused by two thermal baths of different temperatures, which are coupled to opposite ends of the lattice, with a focus on the validity of Fourier's law of heat conduction. We provide analytical solutions of the heat current through the quantum system in the nonequilibrium steady state using the rotating-wave approximation and bath interactions described by a master equation of Lindblad form. The influence of local dephasing in the transition of ballistic to diffusive transport is investigated.

Figure 1

A linear array of coupled harmonic oscillators with thermal environments attached to the endpoints.

Figure 2

Left: Heat current vs chain length in a log-log plot for different values of the dephasing rate. Right: Mean excitation number $E_j=\langle a_j^{†}{a}_j\rangle$ of all oscillators for a chain of length $N=20$ in the steady state under dephasing. Parameter values: ${\Gamma }_{1,2}=0.1,V=0.1$, and $\omega =10$.

Figure 3

A two-dimensional harmonic lattice whose boundary oscillators of opposite edges are coupled to local heat baths of different temperatures.