Reading Out the Geometry From an Atom's Memory

Abstract

Geometric phase in classical and quantum mechanical systems has its origin in the geometry of the path traversed by the system in the phase space or the Hilbert space. As a non-relativistic analogue of Wilson loop operators and as a key tool to explore the deep relationship between geometry and physics, geometric phase remains an active area of research.Here we formulate a non-abelian geometric phase for spin systems. When the spin vector of a quantum system is transported along a closed loop inside the solid spin sphere (i.e., the unit ball), the tensor of second moments picks up a geometric phase in the form of an SO(3) operator. Considering spin1 quantum systems, we formulate this phase. Geometrically interpreting this holonomy is tantamount to defining a steradian angle for loops inside a unit ball, including the ones that pass through the center. We accomplish this by projecting the loop onto the real projective plane. We show that the SO(3) holonomy of a loop inside the unit ball is equal to the steradian angle of the projected path in the real projective plane. This can be generalized to any spin system.