Quantum-mechanical harmonic chain attached to heat baths. II. Nonequilibrium properties

We study nonequilibrium properties of a one-dimensional harmonic chain to whose ends independent heat baths are attached which are kept at different temperatures. Using the quantum Langevin equation approach, we determine the stationary nonequilibrium state for arbitrary temperatures and coupling strength to the heat baths. This allows us to discuss several typical nonequilibrium properties. We find that the heat flux through the chain is finite as the length of the chain goes to infinity, i.e., we recover the well-known fact that the lattice thermal conductivity of the perfect harmonic chain is infinite. In the quantal case, the heat flux ${\mathit{j}}_{\mathrm{qm}}$ is reduced compared with its classical value ${\mathit{j}}_{\mathrm{cl}}$, ${\mathit{j}}_{\mathrm{qm}}$∝(T¯/${\mathit{FTHETA}}_{\mathit{D}}$ ${)}^3$ ${\mathit{j}}_{\mathrm{cl}}$, where T¯ is the average temperature of the heat baths, and where ${\mathit{FTHETA}}_{\mathit{D}}$ is the Debye temperature of the chain. Furthermore, we investigate the variance of the displacement operators and the temperature profile along the chain. In accordance with the infinite thermal conductivity we find a vanishing temperature gradient in the chain except in boundary layers at its ends.