  We introduce a response theory for open quantum systems within nonequilibrium
steady-states subject to a Hamiltonian perturbation. Working in the weak
system-bath coupling regime, our results are derived within the
Lindblad-Gorini-Kossakowski-Sudarshan formalism. We find that the response of
the system to a small perturbation is not simply related to a correlation
function within the system, unlike traditional linear response theory in closed
systems or expectations from the fluctuation-dissipation theorem. In limiting
cases, when the perturbation is small relative to the coupling to the
surroundings or when it does not lead to a change of the eigenstructure of the
system, a perturbative expansion exists where the response function is related
to a sum of a system correlation functions and additional forces induced by the
surroundings. Away from these limiting regimes however, the secular
approximation results in a singular response that cannot be captured within the
traditional approach, but can be described by reverting to a microscopic
Hamiltonian description. These findings are illustrated by explicit
calculations in coupled qubits and anharmonic oscillators in contact with
bosonic baths at different temperatures.
