Quantum response of dephasing open systems

We develop a theory of adiabatic response for open systems gov-erned by Lindblad evolutions. The theory determines the dependence of the response coefficients on the dephasing rates and allows for resid-ual dissipation even when the ground state is protected by a spectral gap. We give quantum response a geometric interpretation in terms of Hilbert space projections: For a two level system and, more generally, for systems with suitable functional form of the dephasing, the dissi-pative 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 de-phasing. We give three examples of physical systems whose quantum states induce compatible metric and symplectic structures on control space: The qubit, coherent states and a model of the integer quantum Hall effect. Two frameworks that provide insight and understanding of transport coef-ficients are Kubo’s theory of linear response [1, 2] and the theory of adiabatic response [3, 4, 5]. Both have a quantum version and a classical version, agree when there is overlap, and endow the non-dissipative transport coefficients with the geometric meaning of the adiabatic (Berry’s) curvature [6]. A notable success of this approach to “geometrization of quantum re-sponse ” has been its application to the integer quantum Hall effect where 1 a