title: Singularity of the time-energy uncertainty in adiabatic perturbation and cycloids on a Bloch sphere.
abstract: 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 length of the quantum trajectory, calculated by the exact solution is entirely different from the one obtained by the adiabatic path traced by the instantaneous eigenstate. 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 cycloid arcs.
authors.affiliation: Qatar Environment and Energy Research Institute, Hamad Bin Khalifa University, Qatar Foundation, PO Box 5825, Doha, Qatar.
authors.forename: Sangchul
authors.initials: S
authors.lastname: Oh
authors.affiliation: Department of Physics, University at Buffalo, State University of New York, Buffalo, New York 14260-1500, USA.
authors.forename: Xuedong
authors.initials: X
authors.lastname: Hu
authors.affiliation: Center for Emergent Matter Science, RIKEN, Saitama 351-0198, Japan.|Physics Department, The University of Michigan, Ann Arbor, Michigan, 48109-1040, USA.
authors.forename: Franco
authors.initials: F
authors.lastname: Nori
authors.affiliation: Qatar Environment and Energy Research Institute, Hamad Bin Khalifa University, Qatar Foundation, PO Box 5825, Doha, Qatar.|Department of Chemistry, Department of Physics and Birck Nanotechnology Center, Purdue University, West Lafayette, IN 47907 USA.
authors.forename: Sabre
authors.initials: S
authors.lastname: Kais
mesh_terms: D055585:Physical Phenomena; D035501:Uncertainty