Quantum geometry and topological bounds on dissipation in slowly driven quantum systems

We show that the dissipation of energy in nearly adiabatically driven quantum systems admits a geometric interpretation associated with trajectories on a manifold characterized by the quantum geometry of the problem. This geometric picture allows to devise optimal driving protocols – i.e., minimizing the dissipation rate - for a particular task by following geodesics on the dissipation manifold. Furthermore, the quantum nature of the dissipation manifold implies a relation between the dissipation rate and the Berry curvature of the system. We demonstrate that for a system slowly driven by a two-tone incommensurate drive, the dissipation rate has a lower bound proportional to a topological number describing the energy conversion between the two tones, implying the impossibility of protocols with arbitrarily low dissipation for drives with non-trivial topology. Our results pave the way towards developing optimal driving protocols for topological drives.