[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-157008-105":59,"doc-detail-157008-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","differences-between-quantum-and-classical-adiabatic-evolution-research-paper","Differences between quantum and classical adiabatic evolution - Research paper","","The study investigates when classical adiabatic evolution can accurately mirror its quantum counterpart in time-modulated metamaterials. It shows that equivalence fails whenever quantum evolution includes effects such as zero modes, because classical mode coupling fundamentally differs from quantum dynamics. The work derives classical adiabaticity conditions, demonstrating that only under these distinct constraints do geometric quantities like the single-band Berry phase and Wilczek-Zee matrix remain meaningful. For general multiband systems, a correction term appears in the classical non-Abelian gauge potential.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/differences-between-quantum-and-classical-adiabatic-evolution-research-paper/157008/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/differences-between-quantum-and-classical-adiabatic-evolution-research-paper/157008.png","ImageObject",300,407,{"name":92,"@type":93},"Skyler","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-29","2026-08-29",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",16,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"Why can’t quantum adiabatic evolution always be reproduced in classical systems?","Question",{"text":113,"@type":114},"Because mode coupling in classical mechanics differs fundamentally from quantum dynamics, so classical models generally do not satisfy the same requirements as quantum adiabatic evolution.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"What determines whether classical evolution yields meaningful Berry-phase and Wilczek-Zee geometry?",{"text":118,"@type":114},"The paper derives classical adiabaticity conditions; only when those conditions (distinct from the quantum ones) hold do the Berry phase and Wilczek-Zee matrix encode the geometry of classical adiabatic evolution.",{"name":120,"@type":111,"acceptedAnswer":121},"What additional effect appears in the classical non-Abelian gauge potential?",{"text":122,"@type":114},"For general multiband systems, the work uncovers a correction term in the non-Abelian gauge potential for classical systems.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},157008,1787971131,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":39,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":130,"read_time":46,"preview_page_count":4},2336464648746,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Differences between quantum and classical adiabatic evolution  \nCyrill Bösch  , 1, 2 , * Andreas Fichtner  , 1 and Marc Serra-Garcia 3  \n1 Institute of Geophysics, ETH Zurich, 8092 Zurich, Switzerland  \n2 Department of Computer Science, Princeton University, Princeton, New Jersey 08540, USA  \n3AMOLF, Science Park 104, 1098 XG Amsterdam, The Netherlands  \n (Received 22 October 2023; revised 24 June 2024; accepted 8 July 2024; published 8 August 2024)  \nAdiabatic evolution is an emergent design principle for time modulated metamaterials, often inspired by insights from topological quantum computing such as braiding operations. However, the pursuit of classical adiabatic metamaterials is rooted in the assumption that classical and quantum adiabatic evolution are equivalent. We show that this is only true in the limit where the frequencies of all the bands are at infinite distance from 0 and some instances of quantum adiabatic evolution, such as those containing zero modes, cannot be reproducedin classical systems. This is because mode coupling is fundamentally different in classical mechanics. We derive classical conditions to ensure adiabaticity and demonstrate that only under these conditions—which are different from quantum adiabatic conditions—do the single band Berry phase and Wilczek-Zee matrix for everywhere degenerate bands emerge as meaningful quantities encoding the geometry of classical adiabatic evolution. Finally, for general multiband systems we uncover a correction term in the non-Abelian gauge potential for classical systems.  \nDOI: 10.1103/PhysRevB.110.064307  \nI. INTRODUCTION  \nQuantum-classical analogies have grown into a vibrant research domain, particularly with the emergence of topological photonic [1] and phononic [2] metamaterials. More recently, classical metamaterials with time and space modulation [3–5] achieve topological pumping [6–15] and non-Abelian physics [16–18], inspiring both fundamental research and novel information processing concepts. At the heart of these advancements lies quantum adiabatic evolution and the corresponding geometric quantities [19–24] . Until now, studies have assumed that classical adiabatic evolution is subject to gap conditions analogous to those in the quantum case and the geometric quantities arising are equivalent [3,6,9, 14–16,25– 29] . Our article challenges this assumption by demonstrating fundamental differences in the mode-coupling equations arising from the fact that a time-dependent Hermitian classical system is mapped to a time-dependent non-Hermitian quantum system. This has two consequences: (i) it introduces a correction term in the non-Abelian gauge potential and (ii) it implies an additional condition to ensure adiabaticity in classical systems, rendering concepts such as braiding of zero modes untenable. Nevertheless, we identify a limit where these differences vanish and classical adiabatic evolution approaches the quantum case.  \nA. Quantum adiabatic evolution  \nIn the context of a slowly changing quantum Hamiltonian Hq (t ) with instantaneous eigenstates |n(t )􀀂 and instantaneous eigenvalues λn (t ), a quantum state initially prepared  \n* Contact author: [cb7454@cs.princeton.edu](cb7454@cs.princeton.edu)  \nas |ψq (0)􀀂 = |n(0)􀀂, and evolving under the Schrödinger equation i|ψ˙q 􀀂 = Hq (t ) |ψq 􀀂, remains in the nth eigenstate, approximately described as |ψqadia (t )􀀂 ≈ exp[−i􀀄(t ) −γ (t )]|n(t )􀀂 [19] . Here, 􀀄 n (t ) := 􀀂t0 dt 􀀄 λn (t 􀀄 ) represents the dynamic phase and γn (t ) = 􀀂t0 dt 􀀄 􀀅n(t 􀀄 ) |n˙(t 􀀄 )􀀂 corresponds to the geometric phase, known as the Berry phase. In what follows we choose a parallel transport gauge, i.e.,􀀅n(t ) |n˙(t )􀀂 = 0. Then the Berry phase can be computed as γ (t ) = Im ln 􀀅n(0)|n(t )􀀂 [30,31] . To obtain |ψqadia (t )􀀂, we expand |ψq (t )􀀂 in terms of the instantaneous eigenstates as |ψq (t )􀀂 = 􀀃n qm (t )exp[−i􀀄m (t )]|m(t )􀀂 . Substituting this into the Schrödinger equation and contracting with 􀀅n|, we integrate from 0 t","cbCaiiUMCpMpbEjO","https://ap.wps.com/l/cbCaiiUMCpMpbEjO","pdf",579120,"English","# Introduction\n## Quantum adiabatic evolution\n## Classical adiabatic evolution\n## Problem statement and structure of the paper","[{\"question\":\"Why can’t quantum adiabatic evolution always be reproduced in classical systems?\",\"answer\":\"Because mode coupling in classical mechanics differs fundamentally from quantum dynamics, so classical models generally do not satisfy the same requirements as quantum adiabatic evolution.\"},{\"question\":\"What determines whether classical evolution yields meaningful Berry-phase and Wilczek-Zee geometry?\",\"answer\":\"The paper derives classical adiabaticity conditions; only when those conditions (distinct from the quantum ones) hold do the Berry phase and Wilczek-Zee matrix encode the geometry of classical adiabatic evolution.\"},{\"question\":\"What additional effect appears in the classical non-Abelian gauge potential?\",\"answer\":\"For general multiband systems, the work uncovers a correction term in the non-Abelian gauge potential for classical systems.\"}]","Differences between quantum and classical adiabatic evolution - Research paper | PDF"]