Fluctuation-dissipation relation for open quantum systems in a nonequilibrium steady state

Continuing our work on the nature and existence of fluctuation-dissipation relations (FDR) in linear and nonlinear open quantum systems [J.-T. Hsiang, B. L. Hu, and S.-Y. Lin, Phys. Rev. D 100, 025019 (2019)10.1103/PhysRevD.100.025019; J.-T. Hsiang, B. L. Hu, S.-Y. Lin, and K. Yamamoto, Phys. Lett. B 795, 694 (2019)PYLBAJ0370-269310.1016/j.physletb.2019.06.062; J.-T. Hsiang and B. L. Hu, Physics (Utrecht) 1, 430 (2019)PHYSGM1943-287910.3390/physics1030031; 27J.-T. Hsiang and B. L. Hu, Phys. Rev. D 101, 125003 (2020)PRVDAQ2470-001010.1103/PhysRevD.101.125003], here we consider such relations when a linear system is in a nonequilibrium steady state (NESS). With the model of two-oscillators (considered as a short harmonic chain with the two ends) each connected to a thermal bath of different temperatures we find that when the chain is fully relaxed due to interaction with the baths, the relation that connects the noise kernel and the imaginary part of the dissipation kernel of the chain in one bath does not assume the conventional form for the FDR in equilibrium cases. There exists an additional term we call the "bias current"that depends on the difference of the bath's initial temperatures and the interoscillator coupling strength. We further show that this term is related to the steady heat flow between the two baths when the system is in an NESS. The ability to know the real-time development of the interheat exchange (between the baths and the end-oscillators) and the intraheat transfer (within the chain) and their dependence on the parameters in the system offers possibilities for quantifiable control, and in the design of quantum heat engines, or thermal devices.