Curves in quantum state space, geometric phases, and the brachistophase

TL;DR

This paper explores the relationship between the geometry of curves in quantum state space and the geometric phase, introducing a new expression for phase derivatives and solving the brachistophase problem for spin systems.

Contribution

It derives a general formula for geometric phase derivatives in quantum state space and provides an analytical solution to the brachistophase problem for small evolution times.

Findings

Geodesics do not accumulate geometric phase in Fubini-Study metric.

Derived a formula linking phase derivatives to covariant derivatives of curves.

Solved the brachistophase problem analytically for all spin values at small times.

Abstract

Given a curve in quantum spin state space, we inquire what is the relation between its geometry and the geometric phase accumulated along it. Motivated by Mukunda and Simon's result that geodesics (in the standard Fubini-Study metric) do not accumulate geometric phase, we find a general expression for the derivatives (of various orders) of the geometric phase in terms of the covariant derivatives of the curve. As an application of our results, we put forward the brachistophase problem: given a quantum state, find the (appropriately normalized) hamiltonian that maximizes the accumulated geometric phase after time $\tau$ - we find an analytical solution for all spin values, valid for small $\tau$. For example, the optimal evolution of a spin coherent state consists of a single Majorana star separating from the rest and tracing out a circle on the Majorana sphere.