Quantum response theory for nonequilibrium steady states

Quantum response theory is a cornerstone of statistical physics. However, the standard Kubo formalism is restricted to isolated equilibrium systems. We here generalize Kubo's results to open quantum systems in nonequilibrium steady states. We derive three different, but equivalent, forms of the quantum response function. We discuss for each of them the role of the noncommutativity of quantum operators and introduce a steady-state extension of the Kubo transformation. We show in particular that the equilibrium response vanishes for perturbations that commute with the unperturbed Hamiltonian, while the steady-state response does not, highlighting the profound difference between the two linear response theories.