Energy transport in the integrable system in contact with various types of phonon reservoirs

We study how energy transport in an integrable system is affected by the spectral densities of heat reservoirs. The model investigated here is the quantum harmonic chain with both ends in contact with two heat reservoirs at different temperatures. The master equation for the reduced density matrix is derived on the assumption that the reservoirs are composed of an infinite number of independent harmonic oscillators. We evaluate temperature profile and energy flux in the stationary state for the master equation and discuss how they depend on the types of spectral densities. When we attach the reservoirs of the same type of spectral density, we find that the temperature profile is independent of the types. On the other hand, when the two reservoirs have different types of spectral densities, the energy profile near the ends of the chain depends on the types. When the coupling is finite, the temperature profile near the ends shows a wide variation of behavior dependent on spectral densities and temperatures of reservoirs. This dependence is discussed with the Fokker-Planck equations obtained in the classical limit.