Quantum response of dephasing open systems

We develop a theory of adiabatic response for open systems governed by Lindblad evolutions. The theory determines the dependence of the response coefficients on the dephasing rates and allows for residual dissipation even when the ground state is protected by a spectral gap. We give the quantum response a geometric interpretation in terms of Hilbert space projections: for a two-level system and, more generally, for systems with a suitable functional form of the dephasing, the dissipative and non-dissipative parts of the response are linked to a metric and to a symplectic form. The metric is the Fubini-Study metric and the symplectic form is the adiabatic curvature. When the metric and symplectic structures are compatible, the non-dissipative part of the inverse matrix of response coefficients turns out to be immune to dephasing. We give three examples of physical systems whose quantum states induce compatible metric and symplectic structures on control space: qubit, coherent states and a model of the integer quantum Hall effect.