Singularity of the time-energy uncertainty in adiabatic perturbation and cycloids on a Bloch sphere

TL;DR

This paper reveals the singular behavior of time-energy uncertainty in adiabatic processes, showing it relates to cycloid trajectories on a Bloch sphere, and explores the geometric phase's smooth transition from non-adiabatic to adiabatic regimes.

Contribution

It demonstrates the singular nature of time-energy uncertainty in adiabatic perturbation through exact solutions and links it to cycloid trajectories on a Bloch sphere, connecting geometric phases and transitionless driving.

Findings

Time-energy uncertainty diverges from adiabatic predictions due to cycloid trajectories.

Geometric phase approaches the adiabatic Berry phase smoothly despite singular uncertainty.

Cycloid arcs visualize non-adiabatic resonance and transitionless driving.

Abstract

The adiabatic perturbation is shown to be singular from the exact solution of a spin-1/2 particle in a uniformly rotating magnetic field. Due to a non-adiabatic effect, its quantum trajectory on a Bloch sphere is a cycloid traced by a circle rolling along an adiabatic path. As the magnetic field rotates more and more slowly, the time-energy uncertainty, proportional to the distance of the quantum trajectory, calculated by the exact solution is entirely different from the one obtained by the adiabatic path traced by the instantaneous state. However, the non-adiabatic Aharonov-Anandan geometric phase, measured by the area enclosed by the exact path, approaches smoothly the adiabatic Berry phase, proportional to the area enclosed by the adiabatic path. The singular limit of the time-energy uncertainty and the regular limit of the geometric phase are associated with the arc length and arc area of the cycloid on a Bloch sphere, respectively. Prolate and curtate cycloids are also traced by different initial states outside and inside of the rolling circle, respectively. The axis trajectory of the rolling circle, parallel to the adiabatic path, is shown to be an example of transitionless driving. The non-adiabatic resonance is visualized by the number of complete cycloid arcs.