Quantum geometry: from adiabatic transformations to quantum speed limits

Synopsis

In this talk I will first introduce the notion of gauge potentials - generators of adiabatic transformations between nearby eigenstates of quantum systems. In the familiar cases of translations and rotations they reduce to the momentum and the angular momentum operators respectively. These gauge potentials naturally lead to the notion of the so called quantum geometric tensor defining the Berry curvature and the Fubini-Study metric tensor, which in turn are used to classify topological phases of matter and to define quantum information geometry. I will discuss how these gauge potentials define generalized Galilean transformation if we go to the moving frame Using ordinary perturbation theory with respect to the Galilean term I will recover various non-adiabatic forces like the Coriolis force acting on the Foucault pendulum, the Lorentz force acting on charged moving particles, or the inertia force acting on a particle in the accelerated box. I will also discuss how gauge potentials can be used to engineer counter-diabatic (dissipationless) driving protocols and formulate a conjecture relating the geodesic distance between states and quantum speed limits.