Reading Out the Geometry From an Atom's Memory
Madhusudhana, Bharath Hebbe
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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.