Smooth and oscillatory geometric phase corrections for driven spins

Abstract

For a quantum spin driven cyclically by a slowly-rotated magnetic field, geometric phases are well understood. If the cycle takes a long time T, the leading-order (dynamical) phase is proportional to T and the geometric phase is the contribution independent of T. The dynamical and geometric phases are the first two terms of a series in slowness 1/T. Here it is shown with an exactly solvable example that the corrections are of two types: smooth, proportional to powers of slowness, and oscillatory: essential singularities in 1/T, in the form of trigonometric functions of T divided by powers of T. The calculations are elementary and therefore suitable for presentation in graduate quantum theory courses.