Response theory for nonequilibrium steady-states of open quantum systems

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.