Open harmonic chain: Exact versus global and local reduced dynamics

In the following, we study the dissipative time evolution of a quantum chain consisting of three coupled harmonic oscillators, the first and third of which weakly interact quadratically with two independent thermal baths in equilibrium at different temperatures. Due to the quadratic form of the total Hamiltonian, the uni- tary dynamics of the compound system is formally analytically solvable and defines a one-parameter group of Gaussian maps which enables us to solve the exact dynamics of the chain numerically. Following the Gorini-Kossakowski-Sudarshan-Lindblad approach to open quantum systems, one can perform the rotating-wave approximation with respect to the interacting or noninteracting chain Hamiltonian and respectively derive the so-called global and local master equations. The solutions of the ensuing different master equations can then be compared with the exact one, possibly sorting out the two approaches in correspondence to different timescales of the system. We derive the steady states of the open chain quantum dynamics in the two approaches and show that the behavior of fidelity between them versus interoscillator coupling depends on the two bath temperatures, revealing the existence of a temperature-dependent critical interoscillator coupling strength that determines the domain of validity of each approach. When the newly found coupling is less than this critical value, the local approach outperforms the global approach, whereas for larger interoscillator coupling, the global approach is a better approximation of the exact evolution. This critical value of interoscillator coupling depends on the two bath temperatures, which then play a crucial role in deciding the best possible approximating open dynamics.