[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 ULiège - University of Liège
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.
Mead C A Truhlar D G 1979 On the determination of Born-Oppenheimer nuclear motion wave functions including complications due to conical intersections and identical nuclei J. Chem. Phys. 70 2284 96 2284-96 10.1063/1.437734
Mead C A 1992 The geometric phase in molecular systems Rev. Mod. Phys. 64 51 85 51-85 10.1103/RevModPhys.64.51
Resta R 1994 Macroscopic polarization in crystalline dielectrics: the geometric phase approach Rev. Mod. Phys. 66 899 915 899-915 10.1103/RevModPhys.66.899
Shapere A Wilczek F 1989 Gauge kinematics of deformable bodies Am. J. Phys. 57 514 8 514-8 10.1119/1.15986
Chruscinski D Jamiołkowski A 2004 Geometric Phases in Classical and Quantum Mechanics Boston, MA Birkhäuser
Cohen E Larocque H Bouchard F Nejadsattari F Gefen Y Karimi E 2019 Geometric phase from Aharonov-Bohm to Pancharatnam-Berry and beyond Nat. Rev. Phys. 1 437 49 437-49 10.1038/s42254-019-0071-1
Zanardi P Rasetti M 1999 Holonomic quantum computation Phys. Lett. A 264 94 99 94-99 10.1016/S0375-9601(99)00803-8
Nielsen M A Chuang I L 2011 Quantum Computation and Quantum Information: 10th Anniversary Edition 10th edn New York Cambridge University Press
Berry M V 1984 Quantal phase factors accompanying adiabatic changes Proc. R. Soc. A 392 45 57 45-57 10.1098/rspa.1984.0023
Wilczek F Zee A 1984 Appearance of gauge structure in simple dynamical systems Phys. Rev. Lett. 52 2111 4 2111-4 10.1103/PhysRevLett.52.2111
Aharonov Y Anandan J 1987 Phase change during a cyclic quantum evolution Phys. Rev. Lett. 58 1593 6 1593-6 10.1103/PhysRevLett.58.1593
Anandan J 1988 Non-adiabatic non-abelian geometric phase Phys. Lett. A 133 171 5 171-5 10.1016/0375-9601(88)91010-9
Samuel J Bhandari R 1988 General setting for Berry’s phase Phys. Rev. Lett. 60 2339 42 2339-42 10.1103/PhysRevLett.60.2339
Mukunda N Simon R 1993 Quantum kinematic approach to the geometric phase: I. General formalism Ann. Phys., NY 228 205 68 205-68 10.1006/aphy.1993.1093
Bitter T Dubbers D 1987 Manifestation of Berry’s topological phase in neutron spin rotation Phys. Rev. Lett. 59 251 4 251-4 10.1103/PhysRevLett.59.251
Zhou Z Margalit Y Moukouri S Meir Y Folman R 2020 An experimental test of the geodesic rule proposition for the noncyclic geometric phase Sci. Adv. 6 eaay8345 10.1126/sciadv.aay8345
Simon B 1983 Holonomy, the quantum adiabatic theorem and Berry’s phase Phys. Rev. Lett. 51 2167 70 2167-70 10.1103/PhysRevLett.51.2167
Fuentes-Guridi I Girelli F Livine E 2005 Holonomic quantum computation in the presence of decoherence Phys. Rev. Lett. 94 45 10.1103/PhysRevLett.94.020503
Sjöqvist E Tong D-M Andersson L M Hessmo B Johansson M Singh K 2012 Non-adiabatic holonomic quantum computation New J. Phys. 14 103035 10.1088/1367-2630/14/10/103035
Xu G F Tong D M Sjöqvist E 2018 Path-shortening realizations of nonadiabatic holonomic gates Phys. Rev. A 98 052315 10.1103/PhysRevA.98.052315
Wang X Allegra M Jacobs K Lloyd S Lupo C Mohseni M 2015 Quantum brachistochrone curves as geodesics: obtaining accurate minimum-time protocols for the control of quantum systems Phys. Rev. Lett. 114 170501 10.1103/PhysRevLett.114.170501
Banyaga A Hurtubise D 2004 Lectures on Morse Homology Dordrecht Springer Science+Business Media
We define geodesics as zero-acceleration curves—alternative characterizations, e.g., as extremals of the length functional, can be shown to be essentially equivalent. A detailed discussion of geodesics in projective space is given in section 4 of [15]
Frankel T 2004 The Geometry of Physics 2nd edn Cambridge Cambridge University Press
Ashtekar A Schilling T 1999 Geometrical formulation of quantum mechanics On Einstein’s Path Harvey A Berlin Springer pp 23 65 pp 23-65
Majorana E 1932 Atomi orientati in campo magnetico variabile Nuovo Cimento 9 43 50 43-50 10.1007/BF02960953
Chryssomalakos C Guzmán-González E Serrano-Ensástiga E 2018 Geometry of spin coherent states J. Phys. A: Math. Theor. 51 165202 10.1088/1751-8121/aab349
Zimba J 2006 Anticoherent spin states via the Majorana representation Electron. J. Theor. Phys. 3 143 56 143-56
Martin J Giraud O Braun P A Braun D Bastin T 2010 Multiqubit symmetric states with high geometric entanglement Phys. Rev. A 81 062347 10.1103/PhysRevA.81.062347
Klyachko A 2007 Dynamic symmetry approach to entanglement Physics and Theoretical Computer Science (Proceedings of the Nato Advanced Study Institute on Emerging Computer Security Gazeau J-P Nešetřil J Rovan B Amsterdam IOS Press
Sawicki A Oszmaniec M Kuś M 2012 Critical sets of the total variance can detect all stochastic local operations and classical communication classes of multiparticle entanglement Phys. Rev. A 86 040304(R) 10.1103/PhysRevA.86.040304
Moroianu A 2007 Lectures on Kähler Geometry Cambridge Cambridge University Press
