[en] 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 τ—we find an analytical solution for all spin values, valid for small τ. 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.
Disciplines :
Physics
Author, co-author :
Chryssomalakos, C. ; Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, CDMX, Mexico
Flores-Delgado, A.G. ; Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, CDMX, Mexico
Guzmán-González, E. ; Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, CDMX, Mexico
Hanotel, L. ; Tikhonov Moscow Institute of Electronics and Mathematics, HSE University, Moscow, Russian Federation
Serrano Ensástiga, Eduardo ; Université de Liège - ULiège > Complex and Entangled Systems from Atoms to Materials (CESAM) ; Centro de Nanociencias y Nanotecnología, Universidad Nacional Autónoma de México, Ensenada, Mexico
Language :
English
Title :
Curves in quantum state space, geometric phases, and the brachistophase
Publication date :
14 July 2023
Journal title :
Journal of Physics. A, Mathematical and Theoretical
UNAM - Universidad Nacional Autónoma de México [MX] ULiège - University of Liège [BE]
Funding text :
C C would like to acknowledge financial support from the DGAPA-UNAM Project IN111920. E S E acknowledges financial support from postdoctoral fellowships by DGAPA-UNAM and the IPD-STEMA Program of the University of Liége.
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