  Continuing our work on the nature and existence of fluctuation-dissipation
relations (FDR) in linear and nonlinear open quantum systems [1-3], 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 inter-oscillator coupling strength. We further
show that this term is related to the steady heat flow between the two baths
when the system is in NESS. The ability to know the real-time development of
the inter-heat exchange (between the baths and the end-oscillators) and the
intra-heat 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.
