Long-term effect of inter-mode transitions in quantum Markovian process

We study the Markovian process of a multi-mode open system connecting with a non-equilibrium environment, which consists of several heat baths with different temperatures. As an illustration, we study the steady state of three linearly coupled harmonic oscillators in long time evolution, two of which contact with two independent bosonic heat baths with different temperatures respectively. We show that the inter-mode transitions mediated by the environment is responsible for the long time behavior of the dynamics evolution, which is usually considered to take effect only in short time dynamics of the system immersed in a equilibrium heat bath with a single temperature. These inter-mode transitions are essential to the non-equilibrium flux between subsystems, thus they cannot be neglected.