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<li id="toc-Time-translation_symmetry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Time-translation_symmetry"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Time-translation symmetry</span> </div> </a> <ul id="toc-Time-translation_symmetry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Broken_symmetry_in_normal_crystals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Broken_symmetry_in_normal_crystals"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Broken symmetry in normal crystals</span> </div> </a> <ul id="toc-Broken_symmetry_in_normal_crystals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Broken_symmetry_in_discrete_time_crystals_(DTC)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Broken_symmetry_in_discrete_time_crystals_(DTC)"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Broken symmetry in discrete time crystals (DTC)</span> </div> </a> <ul id="toc-Broken_symmetry_in_discrete_time_crystals_(DTC)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Thermodynamics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Thermodynamics"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Thermodynamics</span> </div> </a> <ul id="toc-Thermodynamics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Experiments" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Experiments"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Experiments</span> </div> </a> <ul id="toc-Experiments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Academic_articles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Academic_articles"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Academic articles</span> </div> </a> <ul id="toc-Academic_articles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Books" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Books"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Books</span> </div> </a> <ul id="toc-Books-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Press" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Press"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Press</span> </div> </a> <ul id="toc-Press-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> 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<div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Time crystal</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%92%D1%80%D0%B5%D0%BC%D0%B5%D0%B2%D0%B8_%D0%BA%D1%80%D0%B8%D1%81%D1%82%D0%B0%D0%BB" title="Времеви кристал – Bulgarian" lang="bg" hreflang="bg" data-title="Времеви кристал" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Cristall_de_temps" title="Cristall de temps – Catalan" lang="ca" hreflang="ca" data-title="Cristall de temps" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Zeitkristall" title="Zeitkristall – German" lang="de" hreflang="de" data-title="Zeitkristall" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A7%CF%81%CE%BF%CE%BD%CE%BF%CE%BA%CF%81%CF%8D%CF%83%CF%84%CE%B1%CE%BB%CE%BB%CE%BF%CF%82" title="Χρονοκρύσταλλος – Greek" lang="el" hreflang="el" data-title="Χρονοκρύσταλλος" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Cristal_de_espacio_tiempo" title="Cristal de espacio tiempo – Spanish" lang="es" hreflang="es" data-title="Cristal de espacio tiempo" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%A9%D8%B1%DB%8C%D8%B3%D8%AA%D8%A7%D9%84_%D8%B2%D9%85%D8%A7%D9%86" title="کریستال زمان – Persian" lang="fa" hreflang="fa" data-title="کریستال زمان" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Cristal_temporel" title="Cristal temporel – French" lang="fr" hreflang="fr" data-title="Cristal temporel" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%8B%9C%EA%B0%84_%EA%B2%B0%EC%A0%95" title="시간 결정 – Korean" lang="ko" hreflang="ko" data-title="시간 결정" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Kristal_waktu" title="Kristal waktu – Indonesian" lang="id" hreflang="id" data-title="Kristal waktu" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Cristallo_temporale" title="Cristallo temporale – Italian" lang="it" hreflang="it" data-title="Cristallo temporale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%92%D7%91%D7%99%D7%A9_%D7%96%D7%9E%D7%9F" title="גביש זמן – Hebrew" lang="he" hreflang="he" data-title="גביש זמן" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%9F%E0%B5%88%E0%B4%82_%E0%B4%95%E0%B5%8D%E0%B4%B0%E0%B4%BF%E0%B4%B8%E0%B5%8D%E0%B4%B1%E0%B5%8D%E0%B4%B1%E0%B5%BD" title="ടൈം ക്രിസ്റ്റൽ – Malayalam" lang="ml" hreflang="ml" data-title="ടൈം ക്രിസ്റ്റൽ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%99%82%E9%96%93%E7%B5%90%E6%99%B6" title="時間結晶 – Japanese" lang="ja" hreflang="ja" data-title="時間結晶" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Kryszta%C5%82_czasoprzestrzenny" title="Kryształ czasoprzestrzenny – Polish" lang="pl" hreflang="pl" data-title="Kryształ czasoprzestrzenny" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Cristal_do_tempo" title="Cristal do tempo – Portuguese" lang="pt" hreflang="pt" data-title="Cristal do tempo" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BC%D0%BF%D0%BE%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D0%BA%D1%80%D0%B8%D1%81%D1%82%D0%B0%D0%BB%D0%BB" title="Темпоральный кристалл – Russian" lang="ru" hreflang="ru" data-title="Темпоральный кристалл" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Time_crystal" title="Time crystal – Simple English" lang="en-simple" hreflang="en-simple" data-title="Time crystal" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Aikakide" title="Aikakide – Finnish" lang="fi" hreflang="fi" data-title="Aikakide" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A7%D0%B0%D1%81%D0%BE%D0%B2%D0%B8%D0%B9_%D0%BA%D1%80%D0%B8%D1%81%D1%82%D0%B0%D0%BB" title="Часовий кристал – Ukrainian" lang="uk" hreflang="uk" data-title="Часовий кристал" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a 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a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks hlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">Condensed matter physics</a></th></tr><tr><td class="sidebar-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:QuantumPhaseTransition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/QuantumPhaseTransition.svg/220px-QuantumPhaseTransition.svg.png" decoding="async" width="220" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/QuantumPhaseTransition.svg/330px-QuantumPhaseTransition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/QuantumPhaseTransition.svg/440px-QuantumPhaseTransition.svg.png 2x" data-file-width="512" data-file-height="369" /></a></span></td></tr><tr><td class="sidebar-content"> <div class="hlist"><ul><li><a href="/wiki/Phase_(matter)" title="Phase (matter)">Phases</a></li><li><a href="/wiki/Phase_transition" title="Phase transition">Phase transition</a></li><li><a href="/wiki/Quantum_critical_point" title="Quantum critical point">QCP</a></li></ul></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/State_of_matter" title="State of matter">States of matter</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Solid" title="Solid">Solid</a></li><li><a href="/wiki/Liquid" title="Liquid">Liquid</a></li><li><a href="/wiki/Gas" title="Gas">Gas</a></li><li><a href="/wiki/Plasma_(physics)" title="Plasma (physics)">Plasma</a></li><li><a href="/wiki/Bose%E2%80%93Einstein_condensate" title="Bose–Einstein condensate">Bose–Einstein condensate</a></li><li><a href="/wiki/Bose_gas" title="Bose gas">Bose gas</a></li><li><a href="/wiki/Fermionic_condensate" title="Fermionic condensate">Fermionic condensate</a></li><li><a href="/wiki/Fermi_gas" title="Fermi gas">Fermi gas</a></li><li><a href="/wiki/Fermi_liquid_theory" title="Fermi liquid theory">Fermi liquid</a></li><li><a href="/wiki/Supersolid" title="Supersolid">Supersolid</a></li><li><a href="/wiki/Superfluidity" title="Superfluidity">Superfluidity</a></li><li><a href="/wiki/Luttinger_liquid" title="Luttinger liquid">Luttinger liquid</a></li><li><a class="mw-selflink selflink">Time crystal</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Phase phenomena</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Order_parameter" class="mw-redirect" title="Order parameter">Order parameter</a></li><li><a href="/wiki/Phase_transition" title="Phase transition">Phase transition</a></li><li><a href="/wiki/Quantum_critical_point" title="Quantum critical point">QCP</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Electronic phases</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Electronic_band_structure" title="Electronic band structure">Electronic band structure</a></li><li><a href="/wiki/Plasma_(physics)" title="Plasma (physics)">Plasma</a></li><li><a href="/wiki/Insulator_(electricity)" title="Insulator (electricity)">Insulator</a></li><li><a href="/wiki/Mott_insulator" title="Mott insulator">Mott insulator</a></li><li><a href="/wiki/Semiconductor" title="Semiconductor">Semiconductor</a></li><li><a href="/wiki/Semimetal" title="Semimetal">Semimetal</a></li><li><a href="/wiki/Electrical_conductor" title="Electrical conductor">Conductor</a></li><li><a href="/wiki/Superconductivity" title="Superconductivity">Superconductor</a></li><li><a href="/wiki/Thermoelectric_effect" title="Thermoelectric effect">Thermoelectric</a></li><li><a href="/wiki/Piezoelectricity" title="Piezoelectricity">Piezoelectric</a></li><li><a href="/wiki/Ferroelectricity" title="Ferroelectricity">Ferroelectric</a></li><li><a href="/wiki/Topological_insulator" title="Topological insulator">Topological insulator</a></li><li><a href="/wiki/Spin_gapless_semiconductor" title="Spin gapless semiconductor">Spin gapless semiconductor</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Electronic phenomena</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Quantum_Hall_effect" title="Quantum Hall effect">Quantum Hall effect</a></li><li><a href="/wiki/Spin_Hall_effect" title="Spin Hall effect">Spin Hall effect</a></li><li><a href="/wiki/Kondo_effect" title="Kondo effect">Kondo effect</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Magnetic phases</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Diamagnetism" title="Diamagnetism">Diamagnet</a></li><li><a href="/wiki/Superdiamagnetism" title="Superdiamagnetism">Superdiamagnet</a><br /> <a href="/wiki/Paramagnetism" title="Paramagnetism">Paramagnet</a></li><li><a href="/wiki/Superparamagnetism" title="Superparamagnetism">Superparamagnet</a><br /><a href="/wiki/Ferromagnetism" title="Ferromagnetism">Ferromagnet</a></li><li><a href="/wiki/Antiferromagnetism" title="Antiferromagnetism">Antiferromagnet</a><br /><a href="/wiki/Metamagnetism" title="Metamagnetism">Metamagnet</a></li><li><a href="/wiki/Spin_glass" title="Spin glass">Spin glass</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Quasiparticle" title="Quasiparticle">Quasiparticles</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Phonon" title="Phonon">Phonon</a></li><li><a href="/wiki/Exciton" title="Exciton">Exciton</a></li><li><a href="/wiki/Plasmon" title="Plasmon">Plasmon</a><br /><a href="/wiki/Polariton" title="Polariton">Polariton</a></li><li><a href="/wiki/Polaron" title="Polaron">Polaron</a></li><li><a href="/wiki/Magnon" title="Magnon">Magnon</a></li><li><a href="/wiki/Roton" title="Roton">Roton</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Soft_matter" title="Soft matter">Soft matter</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Amorphous_solid" title="Amorphous solid">Amorphous solid</a></li><li><a href="/wiki/Colloid" title="Colloid">Colloid</a></li><li><a href="/wiki/Granular_material" title="Granular material">Granular material</a></li><li><a href="/wiki/Liquid_crystal" title="Liquid crystal">Liquid crystal</a></li><li><a href="/wiki/Polymer" title="Polymer">Polymer</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Johannes_Diderik_van_der_Waals" title="Johannes Diderik van der Waals">Van der Waals</a></li><li><a href="/wiki/Heike_Kamerlingh_Onnes" title="Heike Kamerlingh Onnes">Onnes</a></li><li><a href="/wiki/Max_von_Laue" title="Max von Laue">von Laue</a></li><li><a href="/wiki/William_Henry_Bragg" title="William Henry Bragg">Bragg</a></li><li><a href="/wiki/Peter_Debye" title="Peter Debye">Debye</a></li><li><a href="/wiki/Felix_Bloch" title="Felix Bloch">Bloch</a></li><li><a href="/wiki/Lars_Onsager" title="Lars Onsager">Onsager</a></li><li><a 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title="ISO 8601">ISO 8601</a></li> <li><a href="/wiki/Metric_time" title="Metric time">Metric</a></li> <li><a href="/wiki/Hexadecimal_time" title="Hexadecimal time">Hexadecimal</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:#ddddff;background:#ddf;padding-bottom:0;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist nowrap"><ul><li>Science</li><li><a href="/wiki/Naturalism_(philosophy)" title="Naturalism (philosophy)">Naturalism</a></li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Chronobiology" title="Chronobiology">Chronobiology</a></li> <li><a href="/wiki/Cosmogony" title="Cosmogony">Cosmogony</a></li> <li><a href="/wiki/Evolution" title="Evolution">Evolution</a></li> <li><a href="/wiki/Radiometric_dating" title="Radiometric dating">Radiometric dating</a></li> <li><a href="/wiki/Ultimate_fate_of_the_universe" title="Ultimate fate of the universe"><span class="wrap">Ultimate fate of the universe</span></a></li> <li><a href="/wiki/Time_in_physics" title="Time in physics">Time in physics</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddddff;background:#ddf;padding-bottom:0;color: var(--color-base)"><div class="sidebar-list-title-c">Related topics</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Motion" title="Motion">Motion</a></li> <li><a href="/wiki/Space" title="Space">Space</a></li> <li><a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></li> <li><a href="/wiki/Time_travel" title="Time travel">Time travel</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Time_sidebar" title="Template:Time sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Time_sidebar" title="Template talk:Time sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Time_sidebar" title="Special:EditPage/Template:Time sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">condensed matter physics</a>, a <b>time crystal</b> is a <a href="/wiki/Quantum_system" class="mw-redirect" title="Quantum system">quantum system</a> of particles whose <a href="/wiki/Lowest-energy_state" class="mw-redirect" title="Lowest-energy state">lowest-energy state</a> is one in which the particles are in repetitive motion. The system cannot lose energy to the environment and come to rest because it is already in its quantum <a href="/wiki/Ground_state" title="Ground state">ground state</a>. Time crystals were first proposed theoretically by <a href="/wiki/Frank_Wilczek" title="Frank Wilczek">Frank Wilczek</a> in 2012 as a <a href="/wiki/Time_in_physics" title="Time in physics">time-based</a> analogue to common crystals – whereas the atoms in crystals are arranged periodically in space, the atoms in a time crystal are arranged periodically in both space and time.<sup id="cite_ref-aps2012_1-0" class="reference"><a href="#cite_note-aps2012-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Several different groups have demonstrated matter with stable periodic evolution in systems that are periodically driven.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-aps2017_5-0" class="reference"><a href="#cite_note-aps2017-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> In terms of practical use, time crystals may one day be used as <a href="/wiki/Quantum_memory" title="Quantum memory">quantum computer memory</a>.<sup id="cite_ref-technologyreview.com_6-0" class="reference"><a href="#cite_note-technologyreview.com-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>The existence of <a href="/wiki/Crystal" title="Crystal">crystals</a> in nature is a manifestation of spontaneous <a href="/wiki/Symmetry_breaking" title="Symmetry breaking">symmetry breaking</a>, which occurs when the lowest-energy state of a system is less symmetrical than the equations governing the system. In the crystal ground state, the continuous <a href="/wiki/Translational_symmetry" title="Translational symmetry">translational symmetry</a> in space is broken and replaced by the lower discrete symmetry of the periodic crystal. As the laws of physics are symmetrical under continuous <a href="/wiki/Time-translation_symmetry" title="Time-translation symmetry">translations in time</a> as well as space, the question arose in 2012 as to whether it is possible to break symmetry temporally, and thus create a "time crystal" that is resistant to <a href="/wiki/Entropy" title="Entropy">entropy</a>.<sup id="cite_ref-aps2012_1-1" class="reference"><a href="#cite_note-aps2012-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>If a discrete time-translation symmetry is broken (which may be realized in periodically driven systems), then the system is referred to as a <i>discrete time crystal</i>. A discrete time crystal never reaches <a href="/wiki/Thermal_equilibrium" title="Thermal equilibrium">thermal equilibrium</a>, as it is a type (or phase) of non-equilibrium matter. Breaking of time symmetry can occur only in non-equilibrium systems.<sup id="cite_ref-aps2017_5-1" class="reference"><a href="#cite_note-aps2017-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Discrete time crystals have in fact been observed in physics laboratories as early as 2016. One example of a time crystal, which demonstrates non-equilibrium, broken time symmetry is a constantly rotating ring of charged ions in an otherwise lowest-energy state.<sup id="cite_ref-technologyreview.com_6-1" class="reference"><a href="#cite_note-technologyreview.com-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Concept">Concept</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_crystal&action=edit&section=1" title="Edit section: Concept"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ordinary (non-time) crystals form through spontaneous symmetry breaking related to a spatial symmetry. Such processes can produce materials with interesting properties, such as <a href="/wiki/Diamonds" class="mw-redirect" title="Diamonds">diamonds</a>, <a href="/wiki/Salt_crystals" class="mw-redirect" title="Salt crystals">salt crystals</a>, and <a href="/wiki/Ferromagnetic" class="mw-redirect" title="Ferromagnetic">ferromagnetic</a> metals. By analogy, a time crystal arises through the spontaneous breaking of a time-translation symmetry. A time crystal can be informally defined as a time-periodic self-organizing structure. While an ordinary crystal is periodic (has a repeating structure) in space, a time crystal has a repeating structure in time. A time crystal is periodic in time in the same sense that the pendulum in a pendulum-driven clock is periodic in time. Unlike a pendulum, a time crystal "spontaneously" self-organizes into robust periodic motion (breaking a temporal symmetry).<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Time-translation_symmetry">Time-translation symmetry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_crystal&action=edit&section=2" title="Edit section: Time-translation symmetry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Time-translation_symmetry" title="Time-translation symmetry">Time-translation symmetry</a></div> <p>Symmetries in nature lead directly to conservation laws, something which is precisely formulated by <a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's theorem</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>The basic idea of <i>time-translation symmetry</i> is that a translation in time has no effect on physical laws, i.e. that the laws of nature that apply today were the same in the past and will be the same in the future.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> This symmetry implies the <a href="/wiki/Conservation_of_energy" title="Conservation of energy">conservation of energy</a>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Broken_symmetry_in_normal_crystals">Broken symmetry in normal crystals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_crystal&action=edit&section=3" title="Edit section: Broken symmetry in normal crystals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Crystal_symmetry" class="mw-redirect" title="Crystal symmetry">Crystal symmetry</a> and <a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">spontaneous symmetry breaking</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Phonon_nu_process.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Phonon_nu_process.svg/280px-Phonon_nu_process.svg.png" decoding="async" width="280" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Phonon_nu_process.svg/420px-Phonon_nu_process.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Phonon_nu_process.svg/560px-Phonon_nu_process.svg.png 2x" data-file-width="584" data-file-height="291" /></a><figcaption>Normal process (N-process) and Umklapp process (U-process). While the N-process conserves total <a href="/wiki/Phonon" title="Phonon">phonon</a> momentum, the U-process changes phonon momentum.</figcaption></figure> <p>Common crystals exhibit <i>broken translation symmetry</i>: they have repeated patterns in space and are not invariant under arbitrary translations or rotations. The laws of physics are unchanged by arbitrary translations and rotations. However, if we hold fixed the atoms of a crystal, the dynamics of an electron or other particle in the crystal depend on how it moves relative to the crystal, and particle momentum can change by interacting with the atoms of a crystal — for example in <a href="/wiki/Umklapp_process" class="mw-redirect" title="Umklapp process">Umklapp processes</a>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Quasimomentum" class="mw-redirect" title="Quasimomentum">Quasimomentum</a>, however, is conserved in a perfect crystal.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>Time crystals show a broken symmetry analogous to a discrete space-translation symmetry breaking. For example,<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (May 2019)">citation needed</span></a></i>]</sup> the molecules of a liquid freezing on the surface of a crystal can align with the molecules of the crystal, but with a pattern <i>less</i> symmetric than the crystal: it breaks the initial symmetry. This broken symmetry exhibits three important characteristics:<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (May 2019)">citation needed</span></a></i>]</sup> </p> <ul><li>the system has a lower symmetry than the underlying arrangement of the crystal,</li> <li>the system exhibits spatial and temporal long-range order (unlike a local and intermittent order in a liquid near the surface of a crystal),</li> <li>it is the result of interactions between the constituents of the system, which align themselves relative to each other.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Broken_symmetry_in_discrete_time_crystals_(DTC)"><span id="Broken_symmetry_in_discrete_time_crystals_.28DTC.29"></span>Broken symmetry in discrete time crystals (DTC)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_crystal&action=edit&section=4" title="Edit section: Broken symmetry in discrete time crystals (DTC)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Time crystals seem to break <a href="/wiki/Time-translation_symmetry" title="Time-translation symmetry">time-translation symmetry</a> and have repeated patterns in time even if the laws of the system are invariant by translation of time. The time crystals that are experimentally realized show <i>discrete</i> time-translation symmetry breaking, not the <i>continuous</i> one: they are periodically driven systems oscillating at a <i>fraction</i> of the frequency of the driving force. (According to <a href="/wiki/Philip_Ball" title="Philip Ball">Philip Ball</a>, DTC are so-called because "their periodicity is a discrete, integer multiple of the driving period".<sup id="cite_ref-PB2018_13-0" class="reference"><a href="#cite_note-PB2018-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>) </p><p>The initial symmetry, which is the discrete time-translation symmetry (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\to t+nT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">→</mo> <mi>t</mi> <mo>+</mo> <mi>n</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\to t+nT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0e990b1178155893805c5e88790d5b2b8c218be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.165ex; height:2.343ex;" alt="{\displaystyle t\to t+nT}"></span>) with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}"></span>, is spontaneously broken to the lower discrete time-translation symmetry with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>1}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> is time, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> the driving period, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> an integer.<sup id="cite_ref-else_et_al_2020_14-0" class="reference"><a href="#cite_note-else_et_al_2020-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>Many systems can show behaviors of spontaneous time-translation symmetry breaking but may not be discrete (or Floquet) time crystals: <a href="/wiki/Convection_cells" class="mw-redirect" title="Convection cells">convection cells</a>, <a href="/wiki/Chemical_oscillator" title="Chemical oscillator">oscillating chemical reactions</a>, <a href="/wiki/Aeroelasticity" title="Aeroelasticity">aerodynamic flutter</a>, and subharmonic response to a periodic driving force such as the <a href="/wiki/Faraday_wave" title="Faraday wave">Faraday instability</a>, <a href="/wiki/Nuclear_magnetic_resonance" title="Nuclear magnetic resonance">NMR</a> <a href="/wiki/Spin_echo" title="Spin echo">spin echos</a>, <a href="/wiki/Parametric_down-conversion" class="mw-redirect" title="Parametric down-conversion">parametric down-conversion</a>, and <a href="/wiki/Period-doubling_bifurcation" title="Period-doubling bifurcation">period-doubled</a> nonlinear dynamical systems.<sup id="cite_ref-else_et_al_2020_14-1" class="reference"><a href="#cite_note-else_et_al_2020-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>However, discrete (or Floquet) time crystals are unique in that they follow a strict definition of <b>discrete time-translation symmetry breaking</b>:<sup id="cite_ref-YaoNayak2018_15-0" class="reference"><a href="#cite_note-YaoNayak2018-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <ul><li>it is a broken symmetry –  the system shows oscillations with a period <i>longer</i> than the driving force,</li> <li>the system is in <i>crypto-equilibrium</i> –  these oscillations generate no <a href="/wiki/Entropy" title="Entropy">entropy</a>, and a time-dependent frame can be found in which the system is indistinguishable from an equilibrium when measured stroboscopically<sup id="cite_ref-YaoNayak2018_15-1" class="reference"><a href="#cite_note-YaoNayak2018-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> (which is not the case of convection cells, oscillating chemical reactions and aerodynamic flutter),</li> <li>the system exhibits <i>long-range order</i> –  the oscillations are in phase (synchronized) over arbitrarily long distances and time.</li></ul> <p>Moreover, the broken symmetry in time crystals is the result of <a href="/wiki/Many-body_problem" title="Many-body problem">many-body interactions</a>: the order is the consequence of a <i>collective process</i>, just like in spatial crystals.<sup id="cite_ref-else_et_al_2020_14-2" class="reference"><a href="#cite_note-else_et_al_2020-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> This is not the case for NMR spin echos. </p><p>These characteristics makes discrete time crystals analogous to spatial crystals as described above and may be considered a novel type or phase of nonequilibrium matter.<sup id="cite_ref-else_et_al_2020_14-3" class="reference"><a href="#cite_note-else_et_al_2020-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Thermodynamics">Thermodynamics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_crystal&action=edit&section=5" title="Edit section: Thermodynamics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Time crystals do not violate the <a href="/wiki/Laws_of_thermodynamics" title="Laws of thermodynamics">laws of thermodynamics</a>: energy in the overall system is conserved, such a crystal does not spontaneously convert thermal energy into mechanical work, and it cannot serve as a perpetual store of work. But it may change perpetually in a fixed pattern in time for as long as the system can be maintained. They possess "motion without energy"<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup>—their apparent motion does not represent conventional kinetic energy.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Recent experimental advances in probing discrete time crystals in their periodically driven nonequilibrium states have led to the beginning exploration of novel phases of nonequilibrium matter.<sup id="cite_ref-else_et_al_2020_14-4" class="reference"><a href="#cite_note-else_et_al_2020-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>Time crystals do not evade the Second Law of Thermodynamics,<sup id="cite_ref-googlequantumtc_18-0" class="reference"><a href="#cite_note-googlequantumtc-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> although they spontaneously break "time-translation symmetry", the usual rule that a stable object will remain the same throughout time. In thermodynamics, a time crystal's entropy, understood as a measure of disorder in the system, remains stationary over time, marginally satisfying the second law of thermodynamics by not decreasing.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Google2021_20-0" class="reference"><a href="#cite_note-Google2021-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_crystal&action=edit&section=6" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Frank_Wilczek.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Frank_Wilczek.jpg/220px-Frank_Wilczek.jpg" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Frank_Wilczek.jpg/330px-Frank_Wilczek.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Frank_Wilczek.jpg/440px-Frank_Wilczek.jpg 2x" data-file-width="2956" data-file-height="1971" /></a><figcaption><a href="/wiki/Nobel_laureate" class="mw-redirect" title="Nobel laureate">Nobel laureate</a> <a href="/wiki/Frank_Wilczek" title="Frank Wilczek">Frank Wilczek</a> at <a href="/wiki/University_of_Paris-Saclay" class="mw-redirect" title="University of Paris-Saclay">University of Paris-Saclay</a></figcaption></figure> <p>The idea of a quantized time crystal was theorized in 2012 by <a href="/wiki/Frank_Wilczek" title="Frank Wilczek">Frank Wilczek</a>,<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> a <a href="/wiki/Nobel_laureate" class="mw-redirect" title="Nobel laureate">Nobel laureate</a> and professor at <a href="/wiki/MIT" class="mw-redirect" title="MIT">MIT</a>. In 2013, <a href="/wiki/Xiang_Zhang" title="Xiang Zhang">Xiang Zhang</a>, a nanoengineer at <a href="/wiki/University_of_California,_Berkeley" title="University of California, Berkeley">University of California, Berkeley</a>, and his team proposed creating a time crystal in the form of a constantly rotating ring of charged ions.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>In response to Wilczek and Zhang, Patrick Bruno (<a href="/wiki/European_Synchrotron_Radiation_Facility" title="European Synchrotron Radiation Facility">European Synchrotron Radiation Facility</a>) and Masaki Oshikawa (<a href="/wiki/University_of_Tokyo" title="University of Tokyo">University of Tokyo</a>) published several articles stating that space–time crystals were impossible.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>Subsequent work developed more precise definitions of <a href="/wiki/Time-translation_symmetry" title="Time-translation symmetry">time-translation symmetry</a>-breaking, which ultimately led to the Watanabe–Oshikawa <a href="/wiki/No-go_theorem" title="No-go theorem">"no-go" statement</a> that quantum space–time crystals in equilibrium are not possible.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-WO_2015_28-0" class="reference"><a href="#cite_note-WO_2015-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> Later work restricted the scope of Watanabe and Oshikawa: strictly speaking, they showed that long-range order in both space and time is not possible in equilibrium, but breaking of time-translation symmetry alone is still possible.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-KMS_2019_30-0" class="reference"><a href="#cite_note-KMS_2019-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p><p>Several realizations of time crystals, which avoid the equilibrium no-go arguments, were later proposed.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> In 2014 <a href="/wiki/Krzysztof_Sacha" title="Krzysztof Sacha">Krzysztof Sacha</a> at <a href="/wiki/Jagiellonian_University" title="Jagiellonian University">Jagiellonian University</a> in <a href="/wiki/Krak%C3%B3w" title="Kraków">Kraków</a> predicted the behaviour of discrete time crystals in a periodically driven system with "an ultracold atomic cloud bouncing on an oscillating mirror".<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-TCbook_34-0" class="reference"><a href="#cite_note-TCbook-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p><p>In 2016, research groups at Princeton and at Santa Barbara independently suggested that periodically driven quantum spin systems could show similar behaviour.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> Also in 2016, Norman Yao at <a href="/wiki/University_of_California,_Berkeley" title="University of California, Berkeley">Berkeley</a> and colleagues proposed a different way to create discrete time crystals in spin systems.<sup id="cite_ref-yao_et_al_2017_36-0" class="reference"><a href="#cite_note-yao_et_al_2017-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> These ideas were successful and independently realized by two experimental teams: a group led by <a href="/wiki/Harvard_University" title="Harvard University">Harvard</a>'s <a href="/wiki/Mikhail_Lukin" title="Mikhail Lukin">Mikhail Lukin</a><sup id="cite_ref-choi2017_37-0" class="reference"><a href="#cite_note-choi2017-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> and a group led by <a href="/wiki/Christopher_Monroe" title="Christopher Monroe">Christopher Monroe</a> at <a href="/wiki/University_of_Maryland" class="mw-redirect" title="University of Maryland">University of Maryland</a>.<sup id="cite_ref-zhang2017_38-0" class="reference"><a href="#cite_note-zhang2017-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> Both experiments were published in the same issue of <a href="/wiki/Nature_(journal)" title="Nature (journal)"><i>Nature</i></a> in March 2017. </p><p>Later, time crystals in open systems, so called dissipative time crystals, were proposed in several platforms breaking a discrete <sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> and a continuous<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_44-0" class="reference"><a href="#cite_note-:0-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> time-translation symmetry. A dissipative time crystal was experimentally realized for the first time in 2021 by the group of Andreas Hemmerich at the Institute of Laser Physics at the <a href="/wiki/University_of_Hamburg" title="University of Hamburg">University of Hamburg</a>.<sup id="cite_ref-Keßler_043602_45-0" class="reference"><a href="#cite_note-Keßler_043602-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> The researchers used a <a href="/wiki/Bose%E2%80%93Einstein_condensate" title="Bose–Einstein condensate">Bose–Einstein condensate</a> strongly coupled to a dissipative <a href="/wiki/Optical_cavity" title="Optical cavity">optical cavity</a> and the time crystal was demonstrated to spontaneously break discrete time-translation symmetry by periodically switching between two atomic density patterns.<sup id="cite_ref-Keßler_043602_45-1" class="reference"><a href="#cite_note-Keßler_043602-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_46-0" class="reference"><a href="#cite_note-:1-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:2_47-0" class="reference"><a href="#cite_note-:2-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> In an earlier experiment in the group of <a href="/wiki/Tilman_Esslinger" title="Tilman Esslinger">Tilman Esslinger</a> at <a href="/wiki/ETH_Zurich" title="ETH Zurich">ETH Zurich</a>, limit cycle dynamics<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> was observed in 2019,<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> but evidence of robustness against perturbations and the spontaneous character of the time-translation symmetry breaking were not addressed. </p><p>In 2019, physicists Valerii Kozin and Oleksandr Kyriienko proved that, in theory, a permanent quantum time crystal can exist as an isolated system if the system contains unusual long-range multiparticle interactions. The original "no-go" argument only holds in the presence of typical short-range fields that decay as quickly as <span class="texhtml"><i>r</i><sup>−<i>α</i></sup></span> for some <span class="texhtml"><i>α</i> > 0</span>. Kozin and Kyriienko instead analyzed a <a href="/wiki/Spin-1/2" title="Spin-1/2">spin-1/2</a> many-body <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> with long-range multispin interactions, and showed it broke continuous time-translational symmetry. Certain spin correlations in the system oscillate in time, despite the system being closed and in a <a href="/wiki/Ground_state" title="Ground state">ground energy state</a>. However, demonstrating such a system in practice might be prohibitively difficult,<sup id="cite_ref-science_2019_50-0" class="reference"><a href="#cite_note-science_2019-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> and concerns about the physicality of the long-range nature of the model have been raised.<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> </p><p>In 2022, the Hamburg research team, supervised by Hans Keßler and Andreas Hemmerich, demonstrated, for the first time, a continuous dissipative time crystal exhibiting spontaneous breaking of continuous time-translation symmetry.<sup id="cite_ref-:3_53-0" class="reference"><a href="#cite_note-:3-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:4_54-0" class="reference"><a href="#cite_note-:4-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:5_55-0" class="reference"><a href="#cite_note-:5-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:6_56-0" class="reference"><a href="#cite_note-:6-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> </p><p>In February 2024, a team from Dortmund University in Germany built a time crystal from indium gallium arsenide that lasted for 40 minutes, nearly 10 million times longer than the previous record of around 5 milliseconds. In addition, the lack of any decay suggests the crystal could have lasted even longer, stating that it could last "at least a few hours, perhaps even longer".<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Experiments">Experiments</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_crystal&action=edit&section=7" title="Edit section: Experiments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In October 2016, Christopher Monroe at the <a href="/wiki/University_of_Maryland,_College_Park" title="University of Maryland, College Park">University of Maryland</a> claimed to have created the world's first discrete time crystal. Using the ideas proposed by Yao et al.,<sup id="cite_ref-yao_et_al_2017_36-1" class="reference"><a href="#cite_note-yao_et_al_2017-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> his team trapped a chain of <a href="/wiki/Ytterbium-171" class="mw-redirect" title="Ytterbium-171"><sup>171</sup>Yb<sup>+</sup></a> ions in a <a href="/wiki/Paul_trap" class="mw-redirect" title="Paul trap">Paul trap</a>, confined by radio-frequency electromagnetic fields. One of the two <a href="/wiki/Spin_quantum_number" title="Spin quantum number">spin states</a> was selected by a pair of laser beams. The lasers were pulsed, with the shape of the pulse controlled by an <a href="/wiki/Acousto-optic_modulator" title="Acousto-optic modulator">acousto-optic modulator</a>, using the <a href="/wiki/Tukey_window" class="mw-redirect" title="Tukey window">Tukey window</a> to avoid too much energy at the wrong optical frequency. The <a href="/wiki/Hyperfine" title="Hyperfine">hyperfine</a> electron states in that setup, <sup>2</sup><i>S</i><sub>1/2</sub> <span class="nowrap">|<i>F</i> = 0, <i>m<sub>F</sub></i> = 0⟩</span> and <span class="nowrap">|<i>F</i> = 1, <i>m</i><sub>F</sub> = 0⟩</span>, have very close energy levels, separated by 12.642831 GHz. Ten <a href="/wiki/Doppler_cooling" title="Doppler cooling">Doppler-cooled</a> ions were placed in a line 0.025 mm long and coupled together. </p><p>The researchers observed a subharmonic oscillation of the drive. The experiment showed "rigidity" of the time crystal, where the oscillation frequency remained unchanged even when the time crystal was perturbed, and that it gained a frequency of its own and vibrated according to it (rather than only the frequency of the drive). However, once the perturbation or frequency of vibration grew too strong, the time crystal "melted" and lost this subharmonic oscillation, and it returned to the same state as before where it moved only with the induced frequency.<sup id="cite_ref-zhang2017_38-1" class="reference"><a href="#cite_note-zhang2017-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> </p><p>Also in 2016, <a href="/wiki/Mikhail_Lukin" title="Mikhail Lukin">Mikhail Lukin</a> at Harvard also reported the creation of a driven time crystal. His group used a <a href="/wiki/Diamond" title="Diamond">diamond</a> crystal doped with a high concentration of <a href="/wiki/Nitrogen-vacancy_center" title="Nitrogen-vacancy center">nitrogen-vacancy centers</a>, which have strong dipole–dipole coupling and relatively long-lived spin <a href="/wiki/Coherence_(physics)#Quantum_coherence" title="Coherence (physics)">coherence</a>. This strongly interacting dipolar spin system was driven with microwave fields, and the ensemble spin state was determined with an optical (laser) field. It was observed that the spin polarization evolved at half the frequency of the microwave drive. The oscillations persisted for over 100 cycles. This <a href="/wiki/Subharmonic_function" title="Subharmonic function">subharmonic</a> response to the drive frequency is seen as a signature of time-crystalline order.<sup id="cite_ref-choi2017_37-1" class="reference"><a href="#cite_note-choi2017-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p><p>In May 2018, a group in <a href="/wiki/Aalto_University" title="Aalto University">Aalto University</a> reported that they had observed the formation of a time quasicrystal and its phase transition to a continuous time crystal in a <a href="/wiki/Helium-3" title="Helium-3">Helium-3</a> <a href="/wiki/Superfluid" class="mw-redirect" title="Superfluid">superfluid</a> cooled to within one ten thousandth of a kelvin from absolute zero (0.0001 K). <sup id="cite_ref-time_quasicrystal_62-0" class="reference"><a href="#cite_note-time_quasicrystal-62"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup> On August 17, 2020 <i>Nature Materials</i> published a letter from the same group saying that for the first time they were able to observe interactions and the flow of constituent particles between two time crystals. <sup id="cite_ref-nature_materials_63-0" class="reference"><a href="#cite_note-nature_materials-63"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> </p><p>In February 2021 a team at <a href="/wiki/Max_Planck_Institute_for_Intelligent_Systems" title="Max Planck Institute for Intelligent Systems">Max Planck Institute for Intelligent Systems</a> described the creation of time crystal consisting of <a href="/wiki/Magnon" title="Magnon">magnons</a> and probed them under <a href="/wiki/Scanning_transmission_X-ray_microscopy" title="Scanning transmission X-ray microscopy">scanning transmission X-ray microscopy</a> to capture the recurring periodic magnetization structure in the first known video record of such type.<sup id="cite_ref-64" class="reference"><a href="#cite_note-64"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-65" class="reference"><a href="#cite_note-65"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup> </p><p>In July 2021, a team led by Andreas Hemmerich at the Institute of Laser Physics at the <a href="/wiki/University_of_Hamburg" title="University of Hamburg">University of Hamburg</a> presented the first realization of a time crystal in an open system, a so-called <i>dissipative time crystal</i> using <a href="/wiki/Ultracold_atom" title="Ultracold atom">ultracold atoms</a> coupled to an <a href="/wiki/Optical_cavity" title="Optical cavity">optical cavity</a>. The main achievement of this work is a positive application of dissipation – actually helping to stabilise the system's dynamics.<sup id="cite_ref-Keßler_043602_45-2" class="reference"><a href="#cite_note-Keßler_043602-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_46-1" class="reference"><a href="#cite_note-:1-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:2_47-1" class="reference"><a href="#cite_note-:2-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> </p><p>In November 2021, a collaboration between <a href="/wiki/Google" title="Google">Google</a> and physicists from multiple universities reported the observation of a discrete time crystal on Google's <a href="/wiki/Sycamore_processor" title="Sycamore processor">Sycamore processor</a>, a <a href="/wiki/Quantum_computing" title="Quantum computing">quantum computing</a> device. A chip of 20 qubits was used to obtain a <a href="/wiki/Many-body_localization" title="Many-body localization">many-body localization</a> configuration of up and down spins and then stimulated with a laser to achieve a periodically driven "<a href="/wiki/Floquet_theory" title="Floquet theory">Floquet</a>" system where all <i>up</i> spins are flipped for <i>down</i> and vice-versa in periodic cycles which are multiples of the laser's frequency. While the laser is necessary to maintain the necessary environmental conditions, no energy is absorbed from the laser, so the system remains in a <a href="/wiki/Localization-protected_quantum_order#Localization-protected_order_in_Floquet_systems" title="Localization-protected quantum order">protected eigenstate order</a>.<sup id="cite_ref-Google2021_20-1" class="reference"><a href="#cite_note-Google2021-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-66" class="reference"><a href="#cite_note-66"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> </p><p>Previously in June and November 2021 other teams had obtained virtual time crystals based on floquet systems under similar principles to those of the Google experiment, but on <a href="/wiki/Quantum_simulator" title="Quantum simulator">quantum simulators</a> rather than quantum processors: first a group at the <a href="/wiki/University_of_Maryland,_College_Park" title="University of Maryland, College Park">University of Maryland</a> obtained time crystals on <a href="/wiki/Trapped-ion_simulator" class="mw-redirect" title="Trapped-ion simulator">trapped-ions</a> qubits using high frequency driving rather than many-body localization<sup id="cite_ref-67" class="reference"><a href="#cite_note-67"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-68" class="reference"><a href="#cite_note-68"><span class="cite-bracket">[</span>68<span class="cite-bracket">]</span></a></sup> and then a collaboration between <a href="/wiki/Delft_University_of_Technology" title="Delft University of Technology">TU Delft</a> and <a href="/wiki/Netherlands_Organisation_for_Applied_Scientific_Research" title="Netherlands Organisation for Applied Scientific Research">TNO</a> in the Netherlands called Qutech created time crystals from nuclear spins in carbon-13 <a href="/wiki/Nitrogen-vacancy_center" title="Nitrogen-vacancy center">nitrogen-vacancy (NV) centers</a> on a diamond, attaining longer times but fewer qubits.<sup id="cite_ref-69" class="reference"><a href="#cite_note-69"><span class="cite-bracket">[</span>69<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-70" class="reference"><a href="#cite_note-70"><span class="cite-bracket">[</span>70<span class="cite-bracket">]</span></a></sup> </p><p>In February 2022, a scientist at <a href="/wiki/University_of_California,_Riverside" title="University of California, Riverside">UC Riverside</a> reported a dissipative time crystal akin to the system of July 2021 but all-optical, which allowed the scientist to operate it at room temperature. In this experiment <a href="/wiki/Injection_locking" title="Injection locking">injection locking</a> was used to direct lasers at a specific frequency inside a <a href="/wiki/Optical_microcavity" title="Optical microcavity">microresonator</a> creating a <a href="/wiki/Optical_lattice" title="Optical lattice">lattice trap</a> for <a href="/wiki/Soliton" title="Soliton">solitons</a> at subharmonic frequencies.<sup id="cite_ref-71" class="reference"><a href="#cite_note-71"><span class="cite-bracket">[</span>71<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-72" class="reference"><a href="#cite_note-72"><span class="cite-bracket">[</span>72<span class="cite-bracket">]</span></a></sup> </p><p>In March 2022, a new experiment studying time crystals on a quantum processor was performed by two physicists at the <a href="/wiki/University_of_Melbourne" title="University of Melbourne">University of Melbourne</a>, this time using IBM's <a href="/wiki/List_of_quantum_processors" title="List of quantum processors">Manhattan</a> and Brooklyn quantum processors observing a total of 57 qubits.<sup id="cite_ref-73" class="reference"><a href="#cite_note-73"><span class="cite-bracket">[</span>73<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-74" class="reference"><a href="#cite_note-74"><span class="cite-bracket">[</span>74<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-75" class="reference"><a href="#cite_note-75"><span class="cite-bracket">[</span>75<span class="cite-bracket">]</span></a></sup> </p><p>In June 2022, the observation of a continuous time crystal was reported by a team at the Institute of Laser Physics at the <a href="/wiki/University_of_Hamburg" title="University of Hamburg">University of Hamburg</a>, supervised by Hans Keßler and Andreas Hemmerich. In periodically driven systems, time-translation symmetry is broken into a discrete time-translation symmetry due to the drive. Discrete time crystals break this discrete time-translation symmetry by oscillating at a multiple of the drive frequency. In the new experiment, the drive (pump laser) was operated continuously, thus respecting the continuous time-translation symmetry. Instead of a subharmonic response, the system showed an oscillation with an intrinsic frequency and a time phase taking random values between 0 and 2π, as expected for spontaneous breaking of continuous time-translation symmetry. Moreover, the observed <a href="/wiki/Limit_cycle" title="Limit cycle">limit cycle</a> oscillations were shown to be robust against perturbations of technical or fundamental character, such as quantum noise and, due to the openness of the system, fluctuations associated with dissipation. The system consisted of a <a href="/wiki/Bose%E2%80%93Einstein_condensate" title="Bose–Einstein condensate">Bose–Einstein condensate</a> in an <a href="/wiki/Optical_cavity" title="Optical cavity">optical cavity</a>, which was pumped with an optical standing wave oriented perpendicularly with regard to the cavity axis and was in a <a href="/wiki/Superradiance" title="Superradiance">superradiant</a> phase localizing at two <a href="/wiki/Bistability" title="Bistability">bistable</a> <a href="/wiki/Ground_state" title="Ground state">ground states</a> between which it oscillated.<sup id="cite_ref-:3_53-1" class="reference"><a href="#cite_note-:3-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:4_54-1" class="reference"><a href="#cite_note-:4-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:5_55-1" class="reference"><a href="#cite_note-:5-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:6_56-1" class="reference"><a href="#cite_note-:6-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> </p> <style data-mw-deduplicate="TemplateStyles:r1130092004">.mw-parser-output .portal-bar{font-size:88%;font-weight:bold;display:flex;justify-content:center;align-items:baseline}.mw-parser-output .portal-bar-bordered{padding:0 2em;background-color:#fdfdfd;border:1px solid #a2a9b1;clear:both;margin:1em auto 0}.mw-parser-output .portal-bar-related{font-size:100%;justify-content:flex-start}.mw-parser-output .portal-bar-unbordered{padding:0 1.7em;margin-left:0}.mw-parser-output .portal-bar-header{margin:0 1em 0 0.5em;flex:0 0 auto;min-height:24px}.mw-parser-output .portal-bar-content{display:flex;flex-flow:row wrap;flex:0 1 auto;padding:0.15em 0;column-gap:1em;align-items:baseline;margin:0;list-style:none}.mw-parser-output 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cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFZakrzewski2012" class="citation web cs1">Zakrzewski, Jakub (15 October 2012). <a rel="nofollow" class="external text" href="https://archive.today/20170202102150/http://physics.aps.org/articles/v5/116">"Viewpoint: Crystals of Time"</a>. <i>physics.aps.org</i>. APS Physics. Archived from <a rel="nofollow" class="external text" href="http://physics.aps.org/articles/v5/116">the original</a> on 2 February 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=physics.aps.org&rft.atitle=Viewpoint%3A+Crystals+of+Time&rft.date=2012-10-15&rft.aulast=Zakrzewski&rft.aufirst=Jakub&rft_id=http%3A%2F%2Fphysics.aps.org%2Farticles%2Fv5%2F116&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSacha2015" class="citation journal cs1">Sacha, Krzysztof (2015). "Modeling spontaneous breaking of time-translation symmetry". <i>Physical Review A</i>. <b>91</b> (3): 033617. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1410.3638">1410.3638</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015PhRvA..91c3617S">2015PhRvA..91c3617S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevA.91.033617">10.1103/PhysRevA.91.033617</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1050-2947">1050-2947</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118627872">118627872</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+A&rft.atitle=Modeling+spontaneous+breaking+of+time-translation+symmetry&rft.volume=91&rft.issue=3&rft.pages=033617&rft.date=2015&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118627872%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2015PhRvA..91c3617S&rft_id=info%3Aarxiv%2F1410.3638&rft.issn=1050-2947&rft_id=info%3Adoi%2F10.1103%2FPhysRevA.91.033617&rft.aulast=Sacha&rft.aufirst=Krzysztof&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#CITEREFKhemani_et_al.2016">Khemani et al. (2016)</a></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFElse_et_al.2016">Else et al. (2016)</a>.</span> </li> <li id="cite_note-aps2017-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-aps2017_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-aps2017_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRicherme2017" class="citation journal cs1">Richerme, Phil (January 18, 2017). <a rel="nofollow" class="external text" href="https://physics.aps.org/articles/v10/5">"How to Create a Time Crystal"</a>. <i>Physics</i>. <b>10</b>. American Physical Society: 5. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017PhyOJ..10....5R">2017PhyOJ..10....5R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysics.10.5">10.1103/Physics.10.5</a></span><span class="reference-accessdate">. Retrieved <span class="nowrap">5 April</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics&rft.atitle=How+to+Create+a+Time+Crystal&rft.volume=10&rft.pages=5&rft.date=2017-01-18&rft_id=info%3Adoi%2F10.1103%2FPhysics.10.5&rft_id=info%3Abibcode%2F2017PhyOJ..10....5R&rft.aulast=Richerme&rft.aufirst=Phil&rft_id=https%3A%2F%2Fphysics.aps.org%2Farticles%2Fv10%2F5&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></span> </li> <li id="cite_note-technologyreview.com-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-technologyreview.com_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-technologyreview.com_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.technologyreview.com/2016/10/04/157185/physicists-create-worlds-first-time-crystal/">"Physicists Create World's First Time Crystal"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Physicists+Create+World%27s+First+Time+Crystal&rft_id=https%3A%2F%2Fwww.technologyreview.com%2F2016%2F10%2F04%2F157185%2Fphysicists-create-worlds-first-time-crystal%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSachaZakrzewski2018" class="citation journal cs1">Sacha, Krzysztof; Zakrzewski, Jakub (1 January 2018). "Time crystals: a review". <i>Reports on Progress in Physics</i>. <b>81</b> (1): 016401. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1704.03735">1704.03735</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2018RPPh...81a6401S">2018RPPh...81a6401S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1361-6633%2Faa8b38">10.1088/1361-6633/aa8b38</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/28885193">28885193</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:28224975">28224975</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reports+on+Progress+in+Physics&rft.atitle=Time+crystals%3A+a+review&rft.volume=81&rft.issue=1&rft.pages=016401&rft.date=2018-01-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A28224975%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2018RPPh...81a6401S&rft_id=info%3Aarxiv%2F1704.03735&rft_id=info%3Apmid%2F28885193&rft_id=info%3Adoi%2F10.1088%2F1361-6633%2Faa8b38&rft.aulast=Sacha&rft.aufirst=Krzysztof&rft.au=Zakrzewski%2C+Jakub&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCao2004" class="citation book cs1">Cao, Tian Yu (25 March 2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=d0wS0EJHZ3MC"><i>Conceptual Foundations of Quantum Field Theory</i></a>. Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-60272-3" title="Special:BookSources/978-0-521-60272-3"><bdi>978-0-521-60272-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Conceptual+Foundations+of+Quantum+Field+Theory&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=2004-03-25&rft.isbn=978-0-521-60272-3&rft.aulast=Cao&rft.aufirst=Tian+Yu&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dd0wS0EJHZ3MC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span> See p. 151.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilczek2015" class="citation book cs1">Wilczek, Frank (16 July 2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Oh3ICAAAQBAJ"><i>A Beautiful Question: Finding Nature's Deep Design</i></a>. Penguin Books Limited. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-84614-702-9" title="Special:BookSources/978-1-84614-702-9"><bdi>978-1-84614-702-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Beautiful+Question%3A+Finding+Nature%27s+Deep+Design&rft.pub=Penguin+Books+Limited&rft.date=2015-07-16&rft.isbn=978-1-84614-702-9&rft.aulast=Wilczek&rft.aufirst=Frank&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DOh3ICAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span> See Ch. 3.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFengJin2005" class="citation book cs1">Feng, Duan; Jin, Guojun (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-iuYN5arHwoC"><i>Introduction to Condensed Matter Physics</i></a>. singapore: World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-238-711-0" title="Special:BookSources/978-981-238-711-0"><bdi>978-981-238-711-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Condensed+Matter+Physics&rft.place=singapore&rft.pub=World+Scientific&rft.date=2005&rft.isbn=978-981-238-711-0&rft.aulast=Feng&rft.aufirst=Duan&rft.au=Jin%2C+Guojun&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-iuYN5arHwoC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span> See p. 18.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSólyom2007" class="citation book cs1">Sólyom, Jenö (19 September 2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zn-se2TKv3QC"><i>Fundamentals of the Physics of Solids: Volume 1: Structure and Dynamics</i></a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-72600-5" title="Special:BookSources/978-3-540-72600-5"><bdi>978-3-540-72600-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+the+Physics+of+Solids%3A+Volume+1%3A+Structure+and+Dynamics&rft.pub=Springer&rft.date=2007-09-19&rft.isbn=978-3-540-72600-5&rft.aulast=S%C3%B3lyom&rft.aufirst=Jen%C3%B6&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dzn-se2TKv3QC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span> See p. 193.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSólyom2007" class="citation book cs1">Sólyom, Jenö (19 September 2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zn-se2TKv3QC"><i>Fundamentals of the Physics of Solids: Volume 1: Structure and Dynamics</i></a>. 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"Classical Time Crystals". <i>Physical Review Letters</i>. <b>109</b> (16): 160402. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1202.2537">1202.2537</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2012PhRvL.109p0402S">2012PhRvL.109p0402S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.109.160402">10.1103/PhysRevLett.109.160402</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9007">0031-9007</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/23215057">23215057</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4506464">4506464</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Classical+Time+Crystals&rft.volume=109&rft.issue=16&rft.pages=160402&rft.date=2012&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4506464%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2012PhRvL.109p0402S&rft_id=info%3Aarxiv%2F1202.2537&rft.issn=0031-9007&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.109.160402&rft_id=info%3Apmid%2F23215057&rft.aulast=Shapere&rft.aufirst=Alfred&rft.au=Wilczek%2C+Frank&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">See Li et al. (<a href="#CITEREFLi_et_al.2012a">2012a</a>, <a href="#CITEREFLi_et_al.2012b">2012b</a>).</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWolchover2013" class="citation web cs1">Wolchover, Natalie (25 April 2013). <a rel="nofollow" class="external text" href="https://archive.today/20170202105225/https://www.quantamagazine.org/20130425-perpetual-motion-test-could-amend-theory-of-time/">"Perpetual Motion Test Could Amend Theory of Time"</a>. <i>quantamagazine.org</i>. Simons Foundation. Archived from <a rel="nofollow" class="external text" href="https://www.quantamagazine.org/20130425-perpetual-motion-test-could-amend-theory-of-time/#">the original</a> on 2 February 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=quantamagazine.org&rft.atitle=Perpetual+Motion+Test+Could+Amend+Theory+of+Time&rft.date=2013-04-25&rft.aulast=Wolchover&rft.aufirst=Natalie&rft_id=https%3A%2F%2Fwww.quantamagazine.org%2F20130425-perpetual-motion-test-could-amend-theory-of-time%2F%23&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">See <a href="#CITEREFBruno2013a">Bruno (2013a)</a> and <a href="#CITEREFBruno2013b">Bruno (2013b)</a>.</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThomas2013" class="citation web cs1">Thomas, Jessica (15 March 2013). <a rel="nofollow" class="external text" href="https://archive.today/20170202100552/http://physics.aps.org/articles/v6/31">"Notes from the Editors: The Aftermath of a Controversial Idea"</a>. <i>physics.aps.org</i>. APS Physics. Archived from <a rel="nofollow" class="external text" href="http://physics.aps.org/articles/v6/31">the original</a> on 2 February 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=physics.aps.org&rft.atitle=Notes+from+the+Editors%3A+The+Aftermath+of+a+Controversial+Idea&rft.date=2013-03-15&rft.aulast=Thomas&rft.aufirst=Jessica&rft_id=http%3A%2F%2Fphysics.aps.org%2Farticles%2Fv6%2F31&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text">See <a href="#CITEREFNozières2013">Nozières (2013)</a>, <a href="#CITEREFYao_et_al.2017">Yao et al. (2017)</a>, p. 1 and <a href="#CITEREFVolovik2013">Volovik (2013)</a>.</span> </li> <li id="cite_note-WO_2015-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-WO_2015_28-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWatanabeOshikawa2015" class="citation journal cs1">Watanabe, Haruki; Oshikawa, Masaki (2015). "Absence of Quantum Time Crystals". <i>Physical Review Letters</i>. <b>114</b> (25): 251603. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1410.2143">1410.2143</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015PhRvL.114y1603W">2015PhRvL.114y1603W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.114.251603">10.1103/PhysRevLett.114.251603</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9007">0031-9007</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/26197119">26197119</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:312538">312538</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Absence+of+Quantum+Time+Crystals&rft.volume=114&rft.issue=25&rft.pages=251603&rft.date=2015&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A312538%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2015PhRvL.114y1603W&rft_id=info%3Aarxiv%2F1410.2143&rft.issn=0031-9007&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.114.251603&rft_id=info%3Apmid%2F26197119&rft.aulast=Watanabe&rft.aufirst=Haruki&rft.au=Oshikawa%2C+Masaki&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMedenjakBučaJaksch2020" class="citation journal cs1">Medenjak, Marko; Buča, Berislav; Jaksch, Dieter (2020-07-20). "Isolated Heisenberg magnet as a quantum time crystal". <i>Physical Review B</i>. <b>102</b> (4): 041117. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1905.08266">1905.08266</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2020PhRvB.102d1117M">2020PhRvB.102d1117M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevb.102.041117">10.1103/physrevb.102.041117</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2469-9950">2469-9950</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:160009779">160009779</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+B&rft.atitle=Isolated+Heisenberg+magnet+as+a+quantum+time+crystal&rft.volume=102&rft.issue=4&rft.pages=041117&rft.date=2020-07-20&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A160009779%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2020PhRvB.102d1117M&rft_id=info%3Aarxiv%2F1905.08266&rft.issn=2469-9950&rft_id=info%3Adoi%2F10.1103%2Fphysrevb.102.041117&rft.aulast=Medenjak&rft.aufirst=Marko&rft.au=Bu%C4%8Da%2C+Berislav&rft.au=Jaksch%2C+Dieter&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></span> </li> <li id="cite_note-KMS_2019-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-KMS_2019_30-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKhemaniMoessnerSondhi2019" class="citation arxiv cs1">Khemani, Vedika; Moessner, Roderich; Sondhi, S. L. (23 October 2019). "A Brief History of Time Crystals". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1910.10745">1910.10745</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cond-mat.str-el">cond-mat.str-el</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=A+Brief+History+of+Time+Crystals&rft.date=2019-10-23&rft_id=info%3Aarxiv%2F1910.10745&rft.aulast=Khemani&rft.aufirst=Vedika&rft.au=Moessner%2C+Roderich&rft.au=Sondhi%2C+S.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFUhrichDefenuJafariHalimeh2020" class="citation journal cs1">Uhrich, P.; Defenu, N.; Jafari, R.; Halimeh, J. C. (2020). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevb.101.245148">"Out-of-equilibrium phase diagram of long-range superconductors"</a>. <i>Physical Review B</i>. <b>101</b> (24): 245148. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1910.10715">1910.10715</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2020PhRvB.101x5148U">2020PhRvB.101x5148U</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevb.101.245148">10.1103/physrevb.101.245148</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+B&rft.atitle=Out-of-equilibrium+phase+diagram+of+long-range+superconductors&rft.volume=101&rft.issue=24&rft.pages=245148&rft.date=2020&rft_id=info%3Aarxiv%2F1910.10715&rft_id=info%3Adoi%2F10.1103%2Fphysrevb.101.245148&rft_id=info%3Abibcode%2F2020PhRvB.101x5148U&rft.aulast=Uhrich&rft.aufirst=P.&rft.au=Defenu%2C+N.&rft.au=Jafari%2C+R.&rft.au=Halimeh%2C+J.+C.&rft_id=https%3A%2F%2Fdoi.org%2F10.1103%252Fphysrevb.101.245148&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text">See <a href="#CITEREFWilczek2013b">Wilczek (2013b)</a><span class="error harv-error" style="display: none; font-size:100%"> harvp error: no target: CITEREFWilczek2013b (<a href="/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span> and <a href="#CITEREFYoshii_et_al.2015">Yoshii et al. (2015)</a>.</span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSacha2015" class="citation journal cs1">Sacha, Krzysztof (2015). "Modeling spontaneous breaking of time-translation symmetry". <i>Physical Review A</i>. <b>91</b> (3): 033617. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1410.3638">1410.3638</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015PhRvA..91c3617S">2015PhRvA..91c3617S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevA.91.033617">10.1103/PhysRevA.91.033617</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1050-2947">1050-2947</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118627872">118627872</a>. <q>We show that an ultracold atomic cloud bouncing on an oscillating mirror can reveal spontaneous breaking of a discrete time-translation symmetry</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+A&rft.atitle=Modeling+spontaneous+breaking+of+time-translation+symmetry&rft.volume=91&rft.issue=3&rft.pages=033617&rft.date=2015&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118627872%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2015PhRvA..91c3617S&rft_id=info%3Aarxiv%2F1410.3638&rft.issn=1050-2947&rft_id=info%3Adoi%2F10.1103%2FPhysRevA.91.033617&rft.aulast=Sacha&rft.aufirst=Krzysztof&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></span> </li> <li id="cite_note-TCbook-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-TCbook_34-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSacha2020" class="citation book cs1">Sacha, Krzysztof (2020). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007%2F978-3-030-52523-1"><i>Time Crystals</i></a>. Springer Series on Atomic, Optical, and Plasma Physics. Vol. 114. Springer. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-030-52523-1">10.1007/978-3-030-52523-1</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-52522-4" title="Special:BookSources/978-3-030-52522-4"><bdi>978-3-030-52522-4</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:240770955">240770955</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Time+Crystals&rft.series=Springer+Series+on+Atomic%2C+Optical%2C+and+Plasma+Physics&rft.pub=Springer&rft.date=2020&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A240770955%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-3-030-52523-1&rft.isbn=978-3-030-52522-4&rft.aulast=Sacha&rft.aufirst=Krzysztof&rft_id=https%3A%2F%2Flink.springer.com%2Fbook%2F10.1007%252F978-3-030-52523-1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text">See <a href="#CITEREFKhemani_et_al.2016">Khemani et al. (2016)</a> and <a href="#CITEREFElse_et_al.2016">Else et al. (2016)</a></span> </li> <li id="cite_note-yao_et_al_2017-36"><span class="mw-cite-backlink">^ <a href="#cite_ref-yao_et_al_2017_36-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-yao_et_al_2017_36-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYao_et_al.2017" class="citation journal cs1">Yao, N. Y.; Potter, A. C.; Potirniche, I.-D.; Vishwanath, A. (2017). <a rel="nofollow" class="external text" href="https://physics.aps.org/featured-article-pdf/10.1103/PhysRevLett.118.030401">"Discrete Time Crystals: Rigidity, Criticality, and Realizations"</a>. <i>Physical Review Letters</i>. <b>118</b> (3): 030401. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1608.02589">1608.02589</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017PhRvL.118c0401Y">2017PhRvL.118c0401Y</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.118.030401">10.1103/PhysRevLett.118.030401</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9007">0031-9007</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/28157355">28157355</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:206284432">206284432</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Discrete+Time+Crystals%3A+Rigidity%2C+Criticality%2C+and+Realizations&rft.volume=118&rft.issue=3&rft.pages=030401&rft.date=2017&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A206284432%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2017PhRvL.118c0401Y&rft_id=info%3Aarxiv%2F1608.02589&rft.issn=0031-9007&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.118.030401&rft_id=info%3Apmid%2F28157355&rft.aulast=Yao&rft.aufirst=N.+Y.&rft.au=Potter%2C+A.+C.&rft.au=Potirniche%2C+I.-D.&rft.au=Vishwanath%2C+A.&rft_id=https%3A%2F%2Fphysics.aps.org%2Ffeatured-article-pdf%2F10.1103%2FPhysRevLett.118.030401&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></span> </li> <li id="cite_note-choi2017-37"><span class="mw-cite-backlink">^ <a href="#cite_ref-choi2017_37-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-choi2017_37-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChoi_et_al.2017" class="citation journal cs1">Choi, Soonwon; Choi, Joonhee; Landig, Renate; Kucsko, Georg; Zhou, Hengyun; Isoya, Junichi; Jelezko, Fedor; Onoda, Shinobu; Sumiya, Hitoshi; Khemani, Vedika; von Keyserlingk, Curt; Yao, Norman Y.; Demler, Eugene; Lukin, Mikhail D. 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W.; Kyprianidis, A.; Becker, P.; Lee, A.; Smith, J.; Pagano, G.; Potirniche, I.-D.; Potter, A. C.; Vishwanath, A.; Yao, N. Y.; Monroe, C. (2017). "Observation of a discrete time crystal". <i>Nature</i>. <b>543</b> (7644): 217–220. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1609.08684">1609.08684</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017Natur.543..217Z">2017Natur.543..217Z</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnature21413">10.1038/nature21413</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/28277505">28277505</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4450646">4450646</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Observation+of+a+discrete+time+crystal&rft.volume=543&rft.issue=7644&rft.pages=217-220&rft.date=2017&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4450646%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2017Natur.543..217Z&rft_id=info%3Aarxiv%2F1609.08684&rft_id=info%3Apmid%2F28277505&rft_id=info%3Adoi%2F10.1038%2Fnature21413&rft.aulast=Zhang&rft.aufirst=J.&rft.au=Hess%2C+P.+W.&rft.au=Kyprianidis%2C+A.&rft.au=Becker%2C+P.&rft.au=Lee%2C+A.&rft.au=Smith%2C+J.&rft.au=Pagano%2C+G.&rft.au=Potirniche%2C+I.-D.&rft.au=Potter%2C+A.+C.&rft.au=Vishwanath%2C+A.&rft.au=Yao%2C+N.+Y.&rft.au=Monroe%2C+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIeminiRussomannoKeelingSchirò2018" class="citation journal cs1">Iemini, Fernando; Russomanno, Angelo; Keeling, Jonathan; Schirò, Marco; Dalmonte, Marcello; Fazio, Rosario (16 July 2018). 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Rachel, Stephan (2022-03-04). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8890700">"Realization of a discrete time crystal on 57 qubits of a quantum computer"</a>. <i>Science Advances</i>. <b>8</b> (9): eabm7652. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2105.06632">2105.06632</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2022SciA....8M7652F">2022SciA....8M7652F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fsciadv.abm7652">10.1126/sciadv.abm7652</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2375-2548">2375-2548</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8890700">8890700</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/35235347">35235347</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science+Advances&rft.atitle=Realization+of+a+discrete+time+crystal+on+57+qubits+of+a+quantum+computer&rft.volume=8&rft.issue=9&rft.pages=eabm7652&rft.date=2022-03-04&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC8890700%23id-name%3DPMC&rft_id=info%3Abibcode%2F2022SciA....8M7652F&rft_id=info%3Aarxiv%2F2105.06632&rft.issn=2375-2548&rft_id=info%3Adoi%2F10.1126%2Fsciadv.abm7652&rft_id=info%3Apmid%2F35235347&rft.aulast=Frey&rft.aufirst=Philipp&rft.au=Rachel%2C+Stephan&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC8890700&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></span> </li> <li id="cite_note-75"><span class="mw-cite-backlink"><b><a href="#cite_ref-75">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFreyRachel2022" class="citation web cs1">Frey, Philipp; Rachel, Stephan (2 March 2022). <a rel="nofollow" class="external text" href="https://theconversation.com/an-ever-ticking-clock-we-made-a-time-crystal-inside-a-quantum-computer-178164">"<span class="cs1-kern-left"></span>'An ever-ticking clock': we made a 'time crystal' inside a quantum computer"</a>. <i>The Conversation</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-03-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Conversation&rft.atitle=%27An+ever-ticking+clock%27%3A+we+made+a+%27time+crystal%27+inside+a+quantum+computer&rft.date=2022-03-02&rft.aulast=Frey&rft.aufirst=Philipp&rft.au=Rachel%2C+Stephan&rft_id=http%3A%2F%2Ftheconversation.com%2Fan-ever-ticking-clock-we-made-a-time-crystal-inside-a-quantum-computer-178164&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Academic_articles">Academic articles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_crystal&action=edit&section=9" title="Edit section: Academic articles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-hanging-indents refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoyle_et_al.2016" class="citation journal cs1">Boyle, Latham; Khoo, Jun Yong; Smith, Kendrick (2016). "Symmetric Satellite Swarms and Choreographic Crystals". <i>Physical Review Letters</i>. <b>116</b> (1): 015503. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1407.5876">1407.5876</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016PhRvL.116a5503B">2016PhRvL.116a5503B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.116.015503">10.1103/PhysRevLett.116.015503</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9007">0031-9007</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/26799028">26799028</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17918689">17918689</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Symmetric+Satellite+Swarms+and+Choreographic+Crystals&rft.volume=116&rft.issue=1&rft.pages=015503&rft.date=2016&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17918689%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2016PhRvL.116a5503B&rft_id=info%3Aarxiv%2F1407.5876&rft.issn=0031-9007&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.116.015503&rft_id=info%3Apmid%2F26799028&rft.aulast=Boyle&rft.aufirst=Latham&rft.au=Khoo%2C+Jun+Yong&rft.au=Smith%2C+Kendrick&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBruno2013a" class="citation journal cs1">Bruno, Patrick (2013a). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1184403">"Comment on 'Quantum Time Crystals'<span class="cs1-kern-right"></span>"</a>. <i>Physical Review Letters</i>. <b>110</b> (11): 118901. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1210.4128">1210.4128</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013PhRvL.110k8901B">2013PhRvL.110k8901B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.110.118901">10.1103/PhysRevLett.110.118901</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9007">0031-9007</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/25166585">25166585</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:41459498">41459498</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Comment+on+%27Quantum+Time+Crystals%27&rft.volume=110&rft.issue=11&rft.pages=118901&rft.date=2013&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A41459498%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2013PhRvL.110k8901B&rft_id=info%3Aarxiv%2F1210.4128&rft.issn=0031-9007&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.110.118901&rft_id=info%3Apmid%2F25166585&rft.aulast=Bruno&rft.aufirst=Patrick&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1184403&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBruno2013b" class="citation journal cs1">Bruno, Patrick (2013b). "Comment on "Space-Time Crystals of Trapped Ions"<span class="cs1-kern-right"></span>". <i>Physical Review Letters</i>. <b>111</b> (2): 029301. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1211.4792">1211.4792</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013PhRvL.111b9301B">2013PhRvL.111b9301B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.111.029301">10.1103/PhysRevLett.111.029301</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9007">0031-9007</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/23889455">23889455</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1502258">1502258</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Comment+on+%22Space-Time+Crystals+of+Trapped+Ions%22&rft.volume=111&rft.issue=2&rft.pages=029301&rft.date=2013&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A1502258%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2013PhRvL.111b9301B&rft_id=info%3Aarxiv%2F1211.4792&rft.issn=0031-9007&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.111.029301&rft_id=info%3Apmid%2F23889455&rft.aulast=Bruno&rft.aufirst=Patrick&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFElse_et_al.2016" class="citation journal cs1">Else, Dominic V.; Bauer, Bela; Nayak, Chetan (2016). "Floquet Time Crystals". <i>Physical Review Letters</i>. <b>117</b> (9): 090402. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1603.08001">1603.08001</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016PhRvL.117i0402E">2016PhRvL.117i0402E</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.117.090402">10.1103/PhysRevLett.117.090402</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9007">0031-9007</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/27610834">27610834</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1652633">1652633</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Floquet+Time+Crystals&rft.volume=117&rft.issue=9&rft.pages=090402&rft.date=2016&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A1652633%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2016PhRvL.117i0402E&rft_id=info%3Aarxiv%2F1603.08001&rft.issn=0031-9007&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.117.090402&rft_id=info%3Apmid%2F27610834&rft.aulast=Else&rft.aufirst=Dominic+V.&rft.au=Bauer%2C+Bela&rft.au=Nayak%2C+Chetan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrifoniHänggi1998" class="citation journal cs1">Grifoni, Milena; Hänggi, Peter (1998). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170211080112/https://pdfs.semanticscholar.org/9477/590bf9c4bc44f0aadf036bd6ab45ce76ebc8.pdf">"Driven quantum tunneling"</a> <span class="cs1-format">(PDF)</span>. <i>Physics Reports</i>. <b>304</b> (5–6): 229–354. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1998PhR...304..229G">1998PhR...304..229G</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.65.9479">10.1.1.65.9479</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0370-1573%2898%2900022-2">10.1016/S0370-1573(98)00022-2</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0370-1573">0370-1573</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120738031">120738031</a>. Archived from <a rel="nofollow" class="external text" href="https://pdfs.semanticscholar.org/9477/590bf9c4bc44f0aadf036bd6ab45ce76ebc8.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2017-02-11.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Reports&rft.atitle=Driven+quantum+tunneling&rft.volume=304&rft.issue=5%E2%80%936&rft.pages=229-354&rft.date=1998&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120738031%23id-name%3DS2CID&rft_id=info%3Abibcode%2F1998PhR...304..229G&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.65.9479%23id-name%3DCiteSeerX&rft.issn=0370-1573&rft_id=info%3Adoi%2F10.1016%2FS0370-1573%2898%2900022-2&rft.aulast=Grifoni&rft.aufirst=Milena&rft.au=H%C3%A4nggi%2C+Peter&rft_id=https%3A%2F%2Fpdfs.semanticscholar.org%2F9477%2F590bf9c4bc44f0aadf036bd6ab45ce76ebc8.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuo_et_al.2013" class="citation journal cs1">Guo, Lingzhen; Marthaler, Michael; Schön, Gerd (2013). "Phase Space Crystals: A New Way to Create a Quasienergy Band Structure". <i>Physical Review Letters</i>. <b>111</b> (20): 205303. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1305.1800">1305.1800</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013PhRvL.111t5303G">2013PhRvL.111t5303G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.111.205303">10.1103/PhysRevLett.111.205303</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9007">0031-9007</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/24289695">24289695</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:9337383">9337383</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Phase+Space+Crystals%3A+A+New+Way+to+Create+a+Quasienergy+Band+Structure&rft.volume=111&rft.issue=20&rft.pages=205303&rft.date=2013&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A9337383%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2013PhRvL.111t5303G&rft_id=info%3Aarxiv%2F1305.1800&rft.issn=0031-9007&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.111.205303&rft_id=info%3Apmid%2F24289695&rft.aulast=Guo&rft.aufirst=Lingzhen&rft.au=Marthaler%2C+Michael&rft.au=Sch%C3%B6n%2C+Gerd&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuo_et_al.2020" class="citation journal cs1">Guo, Lingzhen; Liang, Pengfei (2020). 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B. (1967). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170211080425/http://jetp.ac.ru/cgi-bin/dn/e_024_05_1006.pdf">"The quasienergy of a quantum-mechanical system subjected to a periodic action"</a> <span class="cs1-format">(PDF)</span>. <i>Soviet Physics JETP</i>. <b>24</b> (5): 1006–1008. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1967JETP...24.1006Z">1967JETP...24.1006Z</a>. Archived from <a rel="nofollow" class="external text" href="http://jetp.ac.ru/cgi-bin/dn/e_024_05_1006.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2017-02-11<span class="reference-accessdate">. Retrieved <span class="nowrap">2017-02-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Soviet+Physics+JETP&rft.atitle=The+quasienergy+of+a+quantum-mechanical+system+subjected+to+a+periodic+action&rft.volume=24&rft.issue=5&rft.pages=1006-1008&rft.date=1967&rft_id=info%3Abibcode%2F1967JETP...24.1006Z&rft.aulast=Zel%27Dovich&rft.aufirst=Y.+B.&rft_id=http%3A%2F%2Fjetp.ac.ru%2Fcgi-bin%2Fdn%2Fe_024_05_1006.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="Books">Books</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_crystal&action=edit&section=10" title="Edit section: Books"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin refbegin-hanging-indents refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSacha2020" class="citation book cs1">Sacha, Krzysztof (2020). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007%2F978-3-030-52523-1"><i>Time Crystals</i></a>. Springer Series on Atomic, Optical, and Plasma Physics. Vol. 114. Springer. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-030-52523-1">10.1007/978-3-030-52523-1</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-52522-4" title="Special:BookSources/978-3-030-52522-4"><bdi>978-3-030-52522-4</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:240770955">240770955</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Time+Crystals&rft.series=Springer+Series+on+Atomic%2C+Optical%2C+and+Plasma+Physics&rft.pub=Springer&rft.date=2020&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A240770955%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-3-030-52523-1&rft.isbn=978-3-030-52522-4&rft.aulast=Sacha&rft.aufirst=Krzysztof&rft_id=https%3A%2F%2Flink.springer.com%2Fbook%2F10.1007%252F978-3-030-52523-1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="Press">Press</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_crystal&action=edit&section=11" title="Edit section: Press"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin refbegin-hanging-indents refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBall2021" class="citation journal cs1">Ball, Philip (20 September 2021). <a rel="nofollow" class="external text" href="https://physics.aps.org/articles/v14/131">"Focus: Turning a Quantum Computer into a Time Crystal"</a>. <i>Physics</i>. <b>14</b>. APS Physics: 131. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysics.14.131">10.1103/Physics.14.131</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics&rft.atitle=Focus%3A+Turning+a+Quantum+Computer+into+a+Time+Crystal&rft.volume=14&rft.pages=131&rft.date=2021-09-20&rft_id=info%3Adoi%2F10.1103%2FPhysics.14.131&rft.aulast=Ball&rft.aufirst=Philip&rft_id=https%3A%2F%2Fphysics.aps.org%2Farticles%2Fv14%2F131&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBall2016" class="citation web cs1">Ball, Philip (8 January 2016). <a rel="nofollow" class="external text" href="https://archive.today/20170203141844/http://physics.aps.org/articles/v9/4">"Focus: New Crystal Type is Always in Motion"</a>. <i>physics.aps.org</i>. 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Archived from <a rel="nofollow" class="external text" href="https://phys.org/news/2013-08-physicist-impossibility-quantum-crystals.html">the original</a> on 3 February 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=phys.org&rft.atitle=Physicist+proves+impossibility+of+quantum+time+crystals&rft.date=2013-08-22&rft.aulast=Zyga&rft.aufirst=Lisa&rft_id=https%3A%2F%2Fphys.org%2Fnews%2F2013-08-physicist-impossibility-quantum-crystals.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZyga2015" class="citation web cs1">Zyga, Lisa (9 July 2015). <a rel="nofollow" class="external text" href="https://archive.today/20150709134338/http://phys.org/news/2015-07-physicists-definition-crystalsthen-dont.html">"Physicists propose new definition of time crystals—then prove such things don't exist"</a>. <i>phys.org</i>. Science X. Archived from <a rel="nofollow" class="external text" href="https://phys.org/news/2015-07-physicists-definition-crystalsthen-dont.html">the original</a> on 9 July 2015.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=phys.org&rft.atitle=Physicists+propose+new+definition+of+time+crystals%E2%80%94then+prove+such+things+don%27t+exist&rft.date=2015-07-09&rft.aulast=Zyga&rft.aufirst=Lisa&rft_id=https%3A%2F%2Fphys.org%2Fnews%2F2015-07-physicists-definition-crystalsthen-dont.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZyga2016" class="citation web cs1">Zyga, Lisa (9 September 2016). <a rel="nofollow" class="external text" href="https://archive.today/20160911165310/http://phys.org/news/2016-09-crystals.html">"Time crystals might exist after all (Update)"</a>. <i>phys.org</i>. Science X. Archived from <a rel="nofollow" class="external text" href="https://phys.org/news/2016-09-crystals.html">the original</a> on 11 September 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=phys.org&rft.atitle=Time+crystals+might+exist+after+all+%28Update%29&rft.date=2016-09-09&rft.aulast=Zyga&rft.aufirst=Lisa&rft_id=https%3A%2F%2Fphys.org%2Fnews%2F2016-09-crystals.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+crystal" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_crystal&action=edit&section=12" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://www.umdphysics.umd.edu/people/faculty/current/item/348-monroe.html">Christopher Monroe</a> at <a href="/wiki/University_of_Maryland" class="mw-redirect" title="University of Maryland">University of Maryland</a></li> <li><a rel="nofollow" class="external text" href="http://frankawilczek.com/">Frank Wilczek</a></li> <li><a rel="nofollow" class="external text" href="http://lukin.physics.harvard.edu/">Lukin Group</a> at <a href="/wiki/Harvard_University" title="Harvard University">Harvard University</a></li> <li><a rel="nofollow" class="external text" href="http://physics.berkeley.edu/people/faculty/norman-yao">Norman Yao</a> at the <a href="/wiki/University_of_California_at_Berkeley" class="mw-redirect" title="University of California at Berkeley">University of California at Berkeley</a></li> <li><a rel="nofollow" class="external text" href="https://chaos.if.uj.edu.pl/~sacha/index.html">Krzysztof Sacha</a> at <a href="/wiki/Jagiellonian_University_in_Krak%C3%B3w" 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title="Liquid">Liquid</a></li> <li><a href="/wiki/Gas" title="Gas">Gas</a></li> <li><a href="/wiki/Plasma_(physics)" title="Plasma (physics)">Plasma</a></li> <li><a href="/wiki/Bose%E2%80%93Einstein_condensate" title="Bose–Einstein condensate">Bose–Einstein condensate</a></li> <li><a href="/wiki/Fermionic_condensate" title="Fermionic condensate">Fermionic condensate</a></li> <li><a href="/wiki/Fermi_gas" title="Fermi gas">Fermi gas</a></li> <li><a href="/wiki/Supersolid" title="Supersolid">Supersolid</a></li> <li><a href="/wiki/Superfluidity" title="Superfluidity">Superfluid</a></li> <li><a href="/wiki/Luttinger_liquid" title="Luttinger liquid">Luttinger liquid</a></li> <li><a class="mw-selflink selflink">Time crystal</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:QuantumPhaseTransition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/QuantumPhaseTransition.svg/220px-QuantumPhaseTransition.svg.png" decoding="async" width="220" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/QuantumPhaseTransition.svg/330px-QuantumPhaseTransition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/QuantumPhaseTransition.svg/440px-QuantumPhaseTransition.svg.png 2x" data-file-width="512" data-file-height="369" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Phase phenomena</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Order_parameter" class="mw-redirect" title="Order parameter">Order parameter</a></li> <li><a href="/wiki/Phase_transition" title="Phase transition">Phase transition</a></li> <li><a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">Spontaneous symmetry breaking</a></li> <li><a href="/wiki/Critical_phenomena" title="Critical phenomena">Critical phenomena</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Electrons in solids</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hall_effect" title="Hall effect">Hall effect</a></li> <li><a href="/wiki/Quantum_Hall_effect" title="Quantum Hall effect">Quantum Hall effect</a></li> <li><a href="/wiki/Spin_Hall_effect" title="Spin Hall effect">Spin Hall effect</a></li> <li><a href="/wiki/Quantum_spin_Hall_effect" title="Quantum spin Hall effect">Quantum spin Hall effect</a></li> <li><a href="/wiki/Berry_phase" class="mw-redirect" title="Berry phase">Berry phase</a></li> <li><a href="/wiki/Aharonov%E2%80%93Bohm_effect" title="Aharonov–Bohm effect">Aharonov–Bohm effect</a></li> <li><a href="/wiki/Josephson_effect" title="Josephson effect">Josephson effect</a></li> <li><a href="/wiki/Kondo_effect" title="Kondo effect">Kondo effect</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Drude_model" title="Drude model">Drude model</a></li> <li><a href="/wiki/Free_electron_model" title="Free electron model">Free electron model</a></li> <li><a href="/wiki/Nearly_free_electron_model" title="Nearly free electron model">Nearly free electron model</a></li> <li><a href="/wiki/Bloch%27s_theorem" title="Bloch's theorem">Bloch's theorem</a></li> <li><a href="/wiki/Fermi_liquid_theory" title="Fermi liquid theory">Fermi liquid theory</a></li> <li><a href="/wiki/Electronic_band_structure" title="Electronic band structure">electronic band structure</a></li> <li><a href="/wiki/Anderson_localization" title="Anderson localization">Anderson localization</a></li> <li><a href="/wiki/BCS_theory" title="BCS theory">BCS theory</a></li> <li><a href="/wiki/Tight_binding_model" class="mw-redirect" title="Tight binding model">tight binding model</a></li> <li><a href="/wiki/Hubbard_model" title="Hubbard model">Hubbard model</a></li> <li><a href="/wiki/Density_functional_theory" title="Density functional theory">Density functional theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Conduction</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Insulator_(electricity)" title="Insulator (electricity)">Insulator</a></li> <li><a href="/wiki/Mott_insulator" title="Mott insulator">Mott insulator</a></li> <li><a href="/wiki/Semiconductor" title="Semiconductor">Semiconductor</a></li> <li><a href="/wiki/Semimetal" title="Semimetal">Semimetal</a></li> <li><a href="/wiki/Electrical_conductor" title="Electrical conductor">Conductor</a></li> <li><a href="/wiki/Superconductivity" title="Superconductivity">Superconductor</a></li> <li><a href="/wiki/Topological_insulator" title="Topological insulator">Topological insulator</a></li> <li><a href="/wiki/Spin_gapless_semiconductor" title="Spin gapless semiconductor">Spin gapless semiconductor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Couplings</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Thermoelectric_effect" title="Thermoelectric effect">Thermoelectricity</a></li> <li><a href="/wiki/Piezoelectricity" title="Piezoelectricity">Piezoelectricity</a></li> <li><a href="/wiki/Ferroelectricity" title="Ferroelectricity">Ferroelectricity</a></li> <li><a href="/wiki/Flexoelectricity" title="Flexoelectricity">Flexoelectricity</a></li> <li><a href="/wiki/Electrostriction" title="Electrostriction">Electrostriction</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Magnetic phases</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amorphous_magnet" title="Amorphous magnet">Amorphous magnet</a></li> <li><a href="/wiki/Diamagnetism" title="Diamagnetism">Diamagnet</a></li> <li><a href="/wiki/Superdiamagnetism" title="Superdiamagnetism">Superdiamagnet</a></li> <li><a href="/wiki/Paramagnetism" title="Paramagnetism">Paramagnet</a></li> <li><a href="/wiki/Superparamagnetism" title="Superparamagnetism">Superparamagnet</a></li> <li><a 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title="Quantum computing">Quantum computing</a> <ul><li><a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">timeline</a></li></ul></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></li> <li><a href="/wiki/Quantum_simulator" title="Quantum simulator">Quantum simulation</a></li> <li><a href="/wiki/Qubit" title="Qubit">Qubit</a> <ul><li><a href="/wiki/Physical_and_logical_qubits" title="Physical and logical qubits">physical vs. logical</a></li></ul></li> <li><a href="/wiki/List_of_quantum_processors" title="List of quantum processors">Quantum processors</a> <ul><li><a href="/wiki/Cloud-based_quantum_computing" title="Cloud-based quantum computing">cloud-based</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell%27s_theorem" title="Bell's theorem">Bell's</a></li> <li><a href="/wiki/Eastin%E2%80%93Knill_theorem" title="Eastin–Knill theorem">Eastin–Knill</a></li> <li><a href="/wiki/Gleason%27s_theorem" title="Gleason's theorem">Gleason's</a></li> <li><a href="/wiki/Gottesman%E2%80%93Knill_theorem" title="Gottesman–Knill theorem">Gottesman–Knill</a></li> <li><a href="/wiki/Holevo%27s_theorem" title="Holevo's theorem">Holevo's</a></li> <li><a href="/wiki/No-broadcasting_theorem" title="No-broadcasting theorem">No-broadcasting</a></li> <li><a href="/wiki/No-cloning_theorem" title="No-cloning theorem">No-cloning</a></li> <li><a href="/wiki/No-communication_theorem" title="No-communication theorem">No-communication</a></li> <li><a href="/wiki/No-deleting_theorem" title="No-deleting theorem">No-deleting</a></li> <li><a href="/wiki/No-hiding_theorem" title="No-hiding theorem">No-hiding</a></li> <li><a href="/wiki/No-teleportation_theorem" title="No-teleportation theorem">No-teleportation</a></li> <li><a href="/wiki/PBR_theorem" class="mw-redirect" title="PBR theorem">PBR</a></li> <li><a href="/wiki/Quantum_speed_limit_theorems" class="mw-redirect" title="Quantum speed limit theorems">Quantum speed limit</a></li> <li><a href="/wiki/Threshold_theorem" title="Threshold theorem">Threshold</a></li> <li><a href="/wiki/Solovay%E2%80%93Kitaev_theorem" title="Solovay–Kitaev theorem">Solovay–Kitaev</a></li> <li><a href="/wiki/Schr%C3%B6dinger%E2%80%93HJW_theorem" title="Schrödinger–HJW theorem">Purification</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantum<br />communication</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_capacity" title="Classical capacity">Classical capacity</a> <ul><li><a href="/wiki/Entanglement-assisted_classical_capacity" title="Entanglement-assisted classical capacity">entanglement-assisted</a></li> <li><a href="/wiki/Quantum_capacity" title="Quantum capacity">quantum capacity</a></li></ul></li> <li><a href="/wiki/Entanglement_distillation" title="Entanglement distillation">Entanglement distillation</a></li> <li><a href="/wiki/Monogamy_of_entanglement" title="Monogamy of entanglement">Monogamy of entanglement</a></li> <li><a href="/wiki/LOCC" title="LOCC">LOCC</a></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">Quantum channel</a> <ul><li><a href="/wiki/Quantum_network" title="Quantum network">quantum network</a></li></ul></li> <li><a href="/wiki/Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a> <ul><li><a href="/wiki/Quantum_gate_teleportation" title="Quantum gate teleportation">quantum gate teleportation</a></li></ul></li> <li><a href="/wiki/Superdense_coding" title="Superdense coding">Superdense coding</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Quantum_cryptography" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Post-quantum_cryptography" title="Post-quantum cryptography">Post-quantum cryptography</a></li> <li><a href="/wiki/Quantum_coin_flipping" title="Quantum coin flipping">Quantum coin flipping</a></li> <li><a href="/wiki/Quantum_money" title="Quantum money">Quantum money</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a> <ul><li><a href="/wiki/BB84" title="BB84">BB84</a></li> <li><a href="/wiki/SARG04" title="SARG04">SARG04</a></li> <li><a href="/wiki/List_of_quantum_key_distribution_protocols" title="List of quantum key distribution protocols">other protocols</a></li></ul></li> <li><a href="/wiki/Quantum_secret_sharing" title="Quantum secret sharing">Quantum secret sharing</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amplitude_amplification" title="Amplitude amplification">Amplitude amplification</a></li> <li><a href="/wiki/Bernstein%E2%80%93Vazirani_algorithm" title="Bernstein–Vazirani algorithm">Bernstein–Vazirani</a></li> <li><a href="/wiki/BHT_algorithm" title="BHT algorithm">BHT</a></li> <li><a href="/wiki/Boson_sampling" title="Boson sampling">Boson sampling</a></li> <li><a href="/wiki/Deutsch%E2%80%93Jozsa_algorithm" title="Deutsch–Jozsa algorithm">Deutsch–Jozsa</a></li> <li><a href="/wiki/Grover%27s_algorithm" title="Grover's algorithm">Grover's</a></li> <li><a href="/wiki/HHL_algorithm" title="HHL algorithm">HHL</a></li> <li><a href="/wiki/Hidden_subgroup_problem" title="Hidden subgroup problem">Hidden subgroup</a></li> <li><a href="/wiki/Quantum_annealing" title="Quantum annealing">Quantum annealing</a></li> <li><a href="/wiki/Quantum_counting_algorithm" title="Quantum counting algorithm">Quantum counting</a></li> <li><a href="/wiki/Quantum_Fourier_transform" title="Quantum Fourier transform">Quantum Fourier transform</a></li> <li><a href="/wiki/Quantum_optimization_algorithms" title="Quantum optimization algorithms">Quantum optimization</a></li> <li><a href="/wiki/Quantum_phase_estimation_algorithm" title="Quantum phase estimation algorithm">Quantum phase estimation</a></li> <li><a href="/wiki/Shor%27s_algorithm" title="Shor's algorithm">Shor's</a></li> <li><a href="/wiki/Simon%27s_problem" title="Simon's problem">Simon's</a></li> <li><a href="/wiki/Variational_quantum_eigensolver" title="Variational quantum eigensolver">VQE</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum<br />complexity theory</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/BQP" title="BQP">BQP</a></li> <li><a href="/wiki/Exact_quantum_polynomial_time" title="Exact quantum polynomial time">EQP</a></li> <li><a href="/wiki/QIP_(complexity)" title="QIP (complexity)">QIP</a></li> <li><a href="/wiki/QMA" title="QMA">QMA</a></li> <li><a href="/wiki/PostBQP" title="PostBQP">PostBQP</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantum <br /> processor benchmarks</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_supremacy" title="Quantum supremacy">Quantum supremacy</a></li> <li><a href="/wiki/Quantum_volume" title="Quantum volume">Quantum volume</a></li> <li><a href="/wiki/Randomized_benchmarking" title="Randomized benchmarking">Randomized benchmarking</a> <ul><li><a href="/wiki/Cross-entropy_benchmarking" title="Cross-entropy benchmarking">XEB</a></li></ul></li> <li><a href="/wiki/Relaxation_(NMR)" title="Relaxation (NMR)">Relaxation times</a> <ul><li><a href="/wiki/Spin%E2%80%93lattice_relaxation" title="Spin–lattice relaxation"><i>T</i><sub>1</sub></a></li> <li><a href="/wiki/Spin%E2%80%93spin_relaxation" title="Spin–spin relaxation"><i>T</i><sub>2</sub></a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantum<br /><a href="/wiki/Model_of_computation" title="Model of computation">computing models</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adiabatic_quantum_computation" title="Adiabatic quantum computation">Adiabatic quantum computation</a></li> <li><a href="/wiki/Continuous-variable_quantum_information" title="Continuous-variable quantum information">Continuous-variable quantum information</a></li> <li><a href="/wiki/One-way_quantum_computer" title="One-way quantum computer">One-way quantum computer</a> <ul><li><a href="/wiki/Cluster_state" title="Cluster state">cluster state</a></li></ul></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum circuit</a> <ul><li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">quantum logic gate</a></li></ul></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a> <ul><li><a href="/wiki/Quantum_neural_network" title="Quantum neural network">quantum neural network</a></li></ul></li> <li><a href="/wiki/Quantum_Turing_machine" title="Quantum Turing machine">Quantum Turing machine</a></li> <li><a href="/wiki/Topological_quantum_computer" title="Topological quantum computer">Topological quantum computer</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_error_correction" title="Quantum error correction">Quantum<br />error correction</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Codes <ul><li><a href="/wiki/CSS_code" title="CSS code">CSS</a></li> <li><a href="/wiki/Quantum_convolutional_code" title="Quantum convolutional code">quantum convolutional</a></li> <li><a href="/wiki/Stabilizer_code" title="Stabilizer code">stabilizer</a></li> <li><a href="/wiki/Shor_code" class="mw-redirect" title="Shor code">Shor</a></li> <li><a href="/wiki/Bacon%E2%80%93Shor_code" title="Bacon–Shor code">Bacon–Shor</a></li> <li><a href="/wiki/Steane_code" title="Steane code">Steane</a></li> <li><a href="/wiki/Toric_code" title="Toric code">Toric</a></li> <li><a href="/wiki/Gnu_code" title="Gnu code"><i>gnu</i></a></li></ul></li> <li><a href="/wiki/Entanglement-assisted_stabilizer_formalism" title="Entanglement-assisted stabilizer formalism">Entanglement-assisted</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Physical<br />implementations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_optics" title="Quantum optics">Quantum optics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cavity_quantum_electrodynamics" title="Cavity quantum electrodynamics">Cavity QED</a></li> <li><a href="/wiki/Circuit_quantum_electrodynamics" title="Circuit quantum electrodynamics">Circuit QED</a></li> <li><a href="/wiki/Linear_optical_quantum_computing" title="Linear optical quantum computing">Linear optical QC</a></li> <li><a href="/wiki/KLM_protocol" title="KLM protocol">KLM protocol</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Ultracold_atom" title="Ultracold atom">Ultracold atoms</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Neutral_atom_quantum_computer" title="Neutral atom quantum computer">Neutral atom QC</a></li> <li><a href="/wiki/Trapped-ion_quantum_computer" title="Trapped-ion quantum computer">Trapped-ion QC</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Spin_(physics)" title="Spin (physics)">Spin</a>-based</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kane_quantum_computer" title="Kane quantum computer">Kane QC</a></li> <li><a href="/wiki/Spin_qubit_quantum_computer" title="Spin qubit quantum computer">Spin qubit QC</a></li> <li><a href="/wiki/Nitrogen-vacancy_center" title="Nitrogen-vacancy center">NV center</a></li> <li><a href="/wiki/Nuclear_magnetic_resonance_quantum_computer" title="Nuclear magnetic resonance quantum computer">NMR QC</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Superconducting_quantum_computing" title="Superconducting quantum computing">Superconducting</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Charge_qubit" title="Charge qubit">Charge qubit</a></li> <li><a href="/wiki/Flux_qubit" title="Flux qubit">Flux qubit</a></li> <li><a href="/wiki/Phase_qubit" title="Phase qubit">Phase qubit</a></li> <li><a href="/wiki/Transmon" title="Transmon">Transmon</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum<br />programming</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/OpenQASM" title="OpenQASM">OpenQASM</a>–<a href="/wiki/Qiskit" title="Qiskit">Qiskit</a>–<a href="/wiki/IBM_Quantum_Experience" class="mw-redirect" title="IBM Quantum Experience">IBM QX</a></li> <li><a href="/wiki/Quil_(instruction_set_architecture)" title="Quil (instruction set architecture)">Quil</a>–<a href="/wiki/Rigetti_Computing" title="Rigetti Computing">Forest/Rigetti QCS</a></li> <li><a href="/wiki/Cirq" title="Cirq">Cirq</a></li> <li><a href="/wiki/Q_Sharp" title="Q Sharp">Q#</a></li> <li><a href="/wiki/Libquantum" title="Libquantum">libquantum</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">many others...</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Quantum_information_science" title="Category:Quantum information science">Quantum information science</a></li> <li><span class="noviewer" typeof="mw:File"><span title="Template"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/16px-Symbol_template_class_pink.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/23px-Symbol_template_class_pink.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/31px-Symbol_template_class_pink.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Template:Quantum_mechanics_topics" title="Template:Quantum mechanics topics">Quantum mechanics topics</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Statistical_mechanics" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Statistical_mechanics_topics" title="Template:Statistical mechanics topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Statistical_mechanics_topics" title="Template talk:Statistical mechanics topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Statistical_mechanics_topics" title="Special:EditPage/Template:Statistical mechanics topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Statistical_mechanics" style="font-size:114%;margin:0 4em"><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical mechanics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Principle_of_maximum_entropy" title="Principle of maximum entropy">Principle of maximum entropy</a></li> <li><a href="/wiki/Ergodic_theory" title="Ergodic theory">ergodic theory</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="7" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Increasing_disorder.svg/150px-Increasing_disorder.svg.png" decoding="async" width="150" height="42" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Increasing_disorder.svg/225px-Increasing_disorder.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Increasing_disorder.svg/300px-Increasing_disorder.svg.png 2x" data-file-width="464" data-file-height="131" /></span></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Statistical_thermodynamics" class="mw-redirect" title="Statistical thermodynamics">Statistical thermodynamics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Statistical_ensemble_(mathematical_physics)" class="mw-redirect" title="Statistical ensemble (mathematical physics)">Ensembles</a></li> <li><a href="/wiki/Partition_function_(statistical_mechanics)" title="Partition function (statistical mechanics)">partition functions</a></li> <li><a href="/wiki/Equation_of_state" title="Equation of state">equations of state</a></li> <li><a href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">thermodynamic potential</a>: <ul><li><a href="/wiki/Internal_energy" title="Internal energy">U</a></li> <li><a href="/wiki/Enthalpy" title="Enthalpy">H</a></li> <li><a href="/wiki/Helmholtz_free_energy" title="Helmholtz free energy">F</a></li> <li><a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">G</a></li></ul></li> <li><a href="/wiki/Maxwell_relations" title="Maxwell relations">Maxwell relations</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Statistical_model" title="Statistical model">Models</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Spin_model" title="Spin model">Ferromagnetism models</a> <ul><li><a href="/wiki/Ising_model" title="Ising model">Ising</a></li> <li><a href="/wiki/Potts_model" title="Potts model">Potts</a></li> <li><a href="/wiki/Heisenberg_model_(quantum)" class="mw-redirect" title="Heisenberg model (quantum)">Heisenberg</a></li> <li><a href="/wiki/Percolation_theory" title="Percolation theory">percolation</a></li></ul></li> <li>Particles with <a href="/wiki/Force_field_(chemistry)" title="Force field (chemistry)">force field</a> <ul><li><a href="/wiki/Depletion_force" title="Depletion force">depletion force</a></li> <li><a href="/wiki/Lennard-Jones_potential" title="Lennard-Jones potential">Lennard-Jones potential</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mathematical approaches</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boltzmann_equation" title="Boltzmann equation">Boltzmann equation</a></li> <li><a href="/wiki/H-theorem" title="H-theorem">H-theorem</a></li> <li><a href="/wiki/Vlasov_equation" title="Vlasov equation">Vlasov equation</a></li> <li><a href="/wiki/BBGKY_hierarchy" title="BBGKY hierarchy">BBGKY hierarchy</a></li> <li><a href="/wiki/Stochastic_process" title="Stochastic process">stochastic process</a></li> <li><a href="/wiki/Mean-field_theory" title="Mean-field theory">mean-field theory</a> and <a href="/wiki/Conformal_field_theory" title="Conformal field theory">conformal field theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Critical_phenomena" title="Critical phenomena">Critical phenomena</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Phase_transition" title="Phase transition">Phase transition</a></li> <li><a href="/wiki/Critical_exponent" title="Critical exponent">Critical exponents</a> <ul><li><a href="/wiki/Correlation_function" title="Correlation function">correlation length</a></li> <li><a href="/wiki/Scaling_(geometry)" title="Scaling (geometry)">size scaling</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Entropy" title="Entropy">Entropy</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boltzmann%27s_entropy_formula" title="Boltzmann's entropy formula">Boltzmann</a></li> <li><a href="/wiki/Shannon_entropy" class="mw-redirect" title="Shannon entropy">Shannon</a></li> <li><a href="/wiki/Tsallis_entropy" title="Tsallis entropy">Tsallis </a></li> <li><a href="/wiki/R%C3%A9nyi_entropy" title="Rényi entropy">Rényi</a></li> <li><a href="/wiki/Von_Neumann_entropy" title="Von Neumann entropy">von Neumann</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Statistical_field_theory" title="Statistical field theory">Statistical field theory</a> <ul><li><a href="/wiki/Elementary_particle" title="Elementary particle">elementary particle</a></li> <li><a href="/wiki/Superfluidity" title="Superfluidity">superfluidity</a></li></ul></li> <li><a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">Condensed matter physics</a></li> <li><a href="/wiki/Complex_system" title="Complex system">Complex system</a> <ul><li><a href="/wiki/Chaos_theory" title="Chaos theory">chaos</a></li> <li><a href="/wiki/Information_theory" title="Information theory">information theory</a></li> <li><a href="/wiki/Boltzmann_machine" title="Boltzmann machine">Boltzmann machine</a></li></ul></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="States_of_matter_(list)" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:States_of_matter" title="Template:States of matter"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:States_of_matter" title="Template talk:States of matter"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:States_of_matter" title="Special:EditPage/Template:States of matter"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="States_of_matter_(list)" style="font-size:114%;margin:0 4em"><a href="/wiki/State_of_matter" title="State of matter">States of matter</a> (<a href="/wiki/List_of_states_of_matter" title="List of states of matter">list</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">State</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Solid" title="Solid">Solid</a></li> <li><a href="/wiki/Liquid" title="Liquid">Liquid</a></li> <li><a href="/wiki/Gas" title="Gas">Gas</a> / <a href="/wiki/Vapor" title="Vapor">Vapor</a></li> <li><a href="/wiki/Supercritical_fluid" title="Supercritical fluid">Supercritical fluid</a></li> <li><a href="/wiki/Plasma_(physics)" title="Plasma (physics)">Plasma</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="7" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Phase_change_-_en.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Phase_change_-_en.svg/150px-Phase_change_-_en.svg.png" decoding="async" width="150" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Phase_change_-_en.svg/225px-Phase_change_-_en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Phase_change_-_en.svg/300px-Phase_change_-_en.svg.png 2x" data-file-width="493" data-file-height="524" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Low energy</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bose%E2%80%93Einstein_condensate" title="Bose–Einstein condensate">Bose–Einstein condensate</a></li> <li><a href="/wiki/Fermionic_condensate" title="Fermionic condensate">Fermionic condensate</a></li> <li><a href="/wiki/Degenerate_matter" title="Degenerate matter">Degenerate matter</a></li> <li><a href="/wiki/Quantum_Hall_effect" title="Quantum Hall effect">Quantum Hall</a></li> <li><a href="/wiki/Rydberg_matter" title="Rydberg matter">Rydberg matter</a></li> <li><a href="/wiki/Strange_matter" title="Strange matter">Strange matter</a></li> <li><a href="/wiki/Superfluidity" title="Superfluidity">Superfluid</a></li> <li><a href="/wiki/Supersolid" title="Supersolid">Supersolid</a></li> <li><a href="/wiki/Photonic_molecule" title="Photonic molecule">Photonic molecule</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">High energy</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/QCD_matter" title="QCD matter">QCD matter</a></li> <li><a href="/wiki/Quark%E2%80%93gluon_plasma" title="Quark–gluon plasma">Quark–gluon plasma</a></li> <li><a href="/wiki/Color-glass_condensate" title="Color-glass condensate">Color-glass condensate</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other states</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Colloid" title="Colloid">Colloid</a></li> <li><a href="/wiki/Crystal" title="Crystal">Crystal</a></li> <li><a href="/wiki/Liquid_crystal" title="Liquid crystal">Liquid crystal</a></li> <li><a class="mw-selflink selflink">Time crystal</a></li> <li><a href="/wiki/Quantum_spin_liquid" title="Quantum spin liquid">Quantum spin liquid</a></li> <li><a href="/wiki/Exotic_matter" title="Exotic matter">Exotic matter</a></li> <li><a href="/wiki/Programmable_matter" title="Programmable matter">Programmable matter</a></li> <li><a href="/wiki/Dark_matter" title="Dark matter">Dark matter</a></li> <li><a href="/wiki/Antimatter" title="Antimatter">Antimatter</a></li> <li>Magnetically ordered <ul><li><a href="/wiki/Antiferromagnetism" title="Antiferromagnetism">Antiferromagnet</a></li> <li><a href="/wiki/Ferrimagnetism" title="Ferrimagnetism">Ferrimagnet</a></li> <li><a href="/wiki/Ferromagnetism" title="Ferromagnetism">Ferromagnet</a></li></ul></li> <li><a href="/wiki/String-net_liquid" title="String-net liquid">String-net liquid</a></li> <li><a href="/wiki/Superglass" title="Superglass">Superglass</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Phase_transitions" class="mw-redirect" title="Phase transitions">Phase transitions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boiling" title="Boiling">Boiling</a></li> <li><a href="/wiki/Boiling_point" title="Boiling point">Boiling point</a></li> <li><a href="/wiki/Condensation" title="Condensation">Condensation</a></li> <li><a href="/wiki/Critical_line_(thermodynamics)" title="Critical line (thermodynamics)">Critical line</a></li> <li><a href="/wiki/Critical_point_(thermodynamics)" title="Critical point (thermodynamics)">Critical point</a></li> <li><a href="/wiki/Crystallization" title="Crystallization">Crystallization</a></li> <li><a href="/wiki/Deposition_(phase_transition)" title="Deposition (phase transition)">Deposition</a></li> <li><a href="/wiki/Evaporation" title="Evaporation">Evaporation</a></li> <li><a href="/wiki/Flash_evaporation" title="Flash evaporation">Flash evaporation</a></li> <li><a href="/wiki/Freezing" title="Freezing">Freezing</a></li> <li><a href="/wiki/Chemical_ionization" title="Chemical ionization">Chemical ionization</a></li> <li><a href="/wiki/Ionization" title="Ionization">Ionization</a></li> <li><a href="/wiki/Lambda_point" title="Lambda point">Lambda point</a></li> <li><a href="/wiki/Melting" title="Melting">Melting</a></li> <li><a href="/wiki/Melting_point" title="Melting point">Melting point</a></li> <li><a href="/wiki/Plasma_recombination" title="Plasma recombination">Recombination</a></li> <li><a href="/wiki/Regelation" title="Regelation">Regelation</a></li> <li><a href="/wiki/Vapor%E2%80%93liquid_equilibrium" title="Vapor–liquid equilibrium">Saturated fluid</a></li> <li><a href="/wiki/Sublimation_(phase_transition)" title="Sublimation (phase transition)">Sublimation</a></li> <li><a href="/wiki/Supercooling" title="Supercooling">Supercooling</a></li> <li><a href="/wiki/Triple_point" title="Triple point">Triple point</a></li> <li><a href="/wiki/Vaporization" title="Vaporization">Vaporization</a></li> <li><a href="/wiki/Vitrification" title="Vitrification">Vitrification</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantities</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Enthalpy_of_fusion" title="Enthalpy of fusion">Enthalpy of fusion</a></li> <li><a href="/wiki/Enthalpy_of_sublimation" title="Enthalpy of sublimation">Enthalpy of sublimation</a></li> <li><a href="/wiki/Enthalpy_of_vaporization" title="Enthalpy of vaporization">Enthalpy of vaporization</a></li> <li><a href="/wiki/Latent_heat" title="Latent heat">Latent heat</a></li> <li><a href="/wiki/Latent_internal_energy" title="Latent internal energy">Latent internal energy</a></li> <li><a href="/wiki/Trouton%27s_rule" title="Trouton's rule">Trouton's rule</a></li> <li><a href="/wiki/Volatility_(chemistry)" title="Volatility (chemistry)">Volatility</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Baryonic_matter" class="mw-redirect" title="Baryonic matter">Baryonic matter</a></li> <li><a href="/wiki/Binodal" title="Binodal">Binodal</a></li> <li><a href="/wiki/Compressed_fluid" title="Compressed fluid">Compressed fluid</a></li> <li><a href="/wiki/Cooling_curve" title="Cooling curve">Cooling curve</a></li> <li><a href="/wiki/Equation_of_state" title="Equation of state">Equation of state</a></li> <li><a href="/wiki/Leidenfrost_effect" title="Leidenfrost effect">Leidenfrost effect</a></li> <li><a href="/wiki/Macroscopic_quantum_phenomena" title="Macroscopic quantum phenomena">Macroscopic quantum phenomena</a></li> <li><a href="/wiki/Mpemba_effect" title="Mpemba effect">Mpemba effect</a></li> <li><a href="/wiki/Order_and_disorder_(physics)" class="mw-redirect" title="Order and disorder (physics)">Order and disorder (physics)</a></li> <li><a href="/wiki/Spinodal" title="Spinodal">Spinodal</a></li> <li><a href="/wiki/Superconductivity" title="Superconductivity">Superconductivity</a></li> <li><a href="/wiki/Superheated_water" title="Superheated water">Superheated vapor</a></li> <li><a href="/wiki/Superheating" title="Superheating">Superheating</a></li> <li><a href="/wiki/Thermo-dielectric_effect" title="Thermo-dielectric effect">Thermo-dielectric effect</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Major_branches_of_physics" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Branches_of_physics" title="Template:Branches of physics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Branches_of_physics" title="Template talk:Branches of physics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Branches_of_physics" title="Special:EditPage/Template:Branches of physics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Major_branches_of_physics" style="font-size:114%;margin:0 4em">Major <a href="/wiki/Branches_of_physics" title="Branches of physics">branches of physics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Divisions</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Basic_research" title="Basic research">Pure</a></li> <li><a href="/wiki/Applied_physics" title="Applied physics">Applied</a> <ul><li><a href="/wiki/Engineering_physics" title="Engineering physics">Engineering</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Approaches</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Experimental_physics" title="Experimental physics">Experimental</a></li> <li><a href="/wiki/Theoretical_physics" title="Theoretical physics">Theoretical</a> <ul><li><a href="/wiki/Computational_physics" title="Computational physics">Computational</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Classical_physics" title="Classical physics">Classical</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a> <ul><li><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newtonian</a></li> <li><a href="/wiki/Analytical_mechanics" title="Analytical mechanics">Analytical</a></li> <li><a href="/wiki/Celestial_mechanics" title="Celestial mechanics">Celestial</a></li> <li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum</a></li></ul></li> <li><a href="/wiki/Acoustics" title="Acoustics">Acoustics</a></li> <li><a href="/wiki/Classical_electromagnetism" title="Classical electromagnetism">Classical electromagnetism</a></li> <li><a href="/wiki/Classical_optics" class="mw-redirect" title="Classical optics">Classical optics</a> <ul><li><a href="/wiki/Geometrical_optics" title="Geometrical optics">Ray</a></li> <li><a href="/wiki/Physical_optics" title="Physical optics">Wave</a></li></ul></li> <li><a href="/wiki/Thermodynamics" title="Thermodynamics">Thermodynamics</a> <ul><li><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical</a></li> <li><a href="/wiki/Non-equilibrium_thermodynamics" title="Non-equilibrium thermodynamics">Non-equilibrium</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Modern_physics" title="Modern physics">Modern</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Relativistic_mechanics" title="Relativistic mechanics">Relativistic mechanics</a> <ul><li><a href="/wiki/Special_relativity" title="Special relativity">Special</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General</a></li></ul></li> <li><a href="/wiki/Nuclear_physics" title="Nuclear physics">Nuclear physics</a></li> <li><a href="/wiki/Particle_physics" title="Particle physics">Particle physics</a></li> <li><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></li> <li><a href="/wiki/Atomic,_molecular,_and_optical_physics" title="Atomic, molecular, and optical physics">Atomic, molecular, and optical physics</a> <ul><li><a href="/wiki/Atomic_physics" title="Atomic physics">Atomic</a></li> <li><a href="/wiki/Molecular_physics" title="Molecular physics">Molecular</a></li> <li><a href="/wiki/Optics#Modern_optics" title="Optics">Modern optics</a></li></ul></li> <li><a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">Condensed matter physics</a> <ul><li><a href="/wiki/Solid-state_physics" title="Solid-state physics">Solid-state physics</a></li> <li><a href="/wiki/Crystallography" title="Crystallography">Crystallography</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Applied_and_interdisciplinary_physics" title="Category:Applied and interdisciplinary physics">Interdisciplinary</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Astrophysics" title="Astrophysics">Astrophysics</a></li> <li><a href="/wiki/Atmospheric_physics" title="Atmospheric physics">Atmospheric physics</a></li> <li><a href="/wiki/Biophysics" title="Biophysics">Biophysics</a></li> <li><a href="/wiki/Chemical_physics" title="Chemical physics">Chemical physics</a></li> <li><a href="/wiki/Geophysics" title="Geophysics">Geophysics</a></li> <li><a href="/wiki/Materials_science" title="Materials science">Materials science</a></li> <li><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical physics</a></li> <li><a href="/wiki/Medical_physics" title="Medical physics">Medical physics</a></li> <li><a href="/wiki/Physical_oceanography" title="Physical oceanography">Ocean physics</a></li> <li><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td 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Template:Condensed_matter_physics"," 6.66% 115.667 11 Template:Harvp"," 5.68% 98.600 1 Template:Short_description"," 4.44% 77.065 8 Template:Navbox"," 3.38% 58.797 12 Template:Hlist"]},"scribunto":{"limitreport-timeusage":{"value":"1.219","limit":"10.000"},"limitreport-memusage":{"value":9943310,"limit":52428800},"limitreport-logs":"table#1 {\n [\"size\"] = \"tiny\",\n}\nanchor_id_list = table#1 {\n [\"CITEREFAuttiEltsovVolovik2018\"] = 1,\n [\"CITEREFAuttiHeikkinenMäkinenVolovik2021\"] = 1,\n [\"CITEREFBall2016\"] = 1,\n [\"CITEREFBall2018\"] = 1,\n [\"CITEREFBall2021\"] = 2,\n [\"CITEREFBoerkamp2021\"] = 1,\n [\"CITEREFBoyle_et_al.2016\"] = 1,\n [\"CITEREFBruno2013a\"] = 1,\n [\"CITEREFBruno2013b\"] = 1,\n [\"CITEREFBučaJaksch2019\"] = 1,\n [\"CITEREFBučaTindallJaksch2019\"] = 1,\n [\"CITEREFCao2004\"] = 1,\n [\"CITEREFCho2019\"] = 1,\n [\"CITEREFCho2022\"] = 1,\n [\"CITEREFChoi_et_al.2017\"] = 1,\n [\"CITEREFColeman2013\"] = 1,\n [\"CITEREFCowen,_Ron2017\"] = 1,\n [\"CITEREFCowen2012\"] = 1,\n [\"CITEREFCrew\"] = 1,\n [\"CITEREFDograLandiniKroegerHruby2019\"] = 1,\n [\"CITEREFElseMonroeNayakYao2020\"] = 1,\n [\"CITEREFElse_et_al.2016\"] = 1,\n [\"CITEREFFengJin2005\"] = 1,\n [\"CITEREFFilippo_MariaCarolloMarcuzziGarrahan2019\"] = 1,\n [\"CITEREFFreyRachel2022\"] = 2,\n [\"CITEREFGibney2017\"] = 1,\n [\"CITEREFGongHamazakiUeda2018\"] = 1,\n [\"CITEREFGongUeda2021\"] = 1,\n [\"CITEREFGrifoniHänggi1998\"] = 1,\n [\"CITEREFGrossman2012\"] = 1,\n [\"CITEREFGuo_et_al.2013\"] = 1,\n [\"CITEREFGuo_et_al.2020\"] = 1,\n [\"CITEREFHackett2016\"] = 1,\n [\"CITEREFHamburg2022\"] = 1,\n [\"CITEREFHannafordSacha2020\"] = 1,\n [\"CITEREFHewitt2013\"] = 1,\n [\"CITEREFIeminiRussomannoKeelingSchirò2018\"] = 2,\n [\"CITEREFJohnston2016\"] = 1,\n [\"CITEREFJoint_Quantum_Institute2011\"] = 1,\n [\"CITEREFKeßlerKongkhambutGeorgesMathey2021\"] = 1,\n [\"CITEREFKhemaniMoessnerSondhi2019\"] = 1,\n [\"CITEREFKhemaniMoessnerSondhi2020\"] = 1,\n [\"CITEREFKhemani_et_al.2016\"] = 1,\n [\"CITEREFKongkhambutSkulteMatheyCosme2022\"] = 1,\n [\"CITEREFKozinKyriienko2019\"] = 1,\n [\"CITEREFKubotaUniversity\"] = 1,\n [\"CITEREFKyprianidisMachadoMorongBecker2021\"] = 1,\n [\"CITEREFLeBlanc2022\"] = 1,\n [\"CITEREFLi_et_al.2012a\"] = 1,\n [\"CITEREFLi_et_al.2012b\"] = 1,\n [\"CITEREFLindner_et_al.2011\"] = 1,\n [\"CITEREFMedenjakBučaJaksch2020\"] = 1,\n [\"CITEREFMendonçaDodonov2014\"] = 1,\n [\"CITEREFMiIppolitiQuintanaGreene2022\"] = 1,\n [\"CITEREFNozières2013\"] = 1,\n [\"CITEREFOrf2024\"] = 1,\n [\"CITEREFOuellette2017\"] = 1,\n [\"CITEREFPiazzaRitsch2015\"] = 1,\n [\"CITEREFPowell2013\"] = 1,\n [\"CITEREFRandallBradleyvan_der_GrondenGalicia2021\"] = 1,\n [\"CITEREFRicherme2017\"] = 1,\n [\"CITEREFRobicheauxNiffenegger2015\"] = 1,\n [\"CITEREFSacha2015\"] = 3,\n [\"CITEREFSacha2015a\"] = 1,\n [\"CITEREFSacha2020\"] = 2,\n [\"CITEREFSachaZakrzewski2018\"] = 1,\n [\"CITEREFSacha_et_al.2018\"] = 1,\n [\"CITEREFSersBerkeley2021\"] = 1,\n [\"CITEREFShapereWilczek2012\"] = 1,\n [\"CITEREFShirley1965\"] = 1,\n [\"CITEREFSmith_et_al.2016\"] = 1,\n [\"CITEREFStarr2022\"] = 1,\n [\"CITEREFSólyom2007\"] = 2,\n [\"CITEREFTaheriMatskoMalekiSacha2022\"] = 1,\n [\"CITEREFThomas2013\"] = 1,\n [\"CITEREFTrägerGruszeckiLisieckiGroß2021\"] = 1,\n [\"CITEREFUhrichDefenuJafariHalimeh2020\"] = 1,\n [\"CITEREFUniversity_of_California,_Berkeley2017\"] = 1,\n [\"CITEREFVolovik2013\"] = 1,\n [\"CITEREFWang_et_al.2013\"] = 1,\n [\"CITEREFWatanabeOshikawa2015\"] = 1,\n [\"CITEREFWeiner2017\"] = 1,\n [\"CITEREFWilczek2012\"] = 1,\n [\"CITEREFWilczek2013\"] = 1,\n [\"CITEREFWilczek2013a\"] = 1,\n [\"CITEREFWilczek2015\"] = 1,\n [\"CITEREFWilliams2021\"] = 1,\n [\"CITEREFWolchover2013\"] = 1,\n [\"CITEREFWolchover2021\"] = 1,\n [\"CITEREFWood2017\"] = 1,\n [\"CITEREFYaoNayak2018\"] = 1,\n [\"CITEREFYao_et_al.2017\"] = 1,\n [\"CITEREFYirka2012\"] = 1,\n [\"CITEREFYoshii_et_al.2015\"] = 1,\n [\"CITEREFZakrzewski2012\"] = 1,\n [\"CITEREFZel\u0026#039;Dovich1967\"] = 1,\n [\"CITEREFZhangHessKyprianidisBecker2017\"] = 1,\n [\"CITEREFZyga2012\"] = 1,\n [\"CITEREFZyga2013\"] = 1,\n [\"CITEREFZyga2015\"] = 1,\n [\"CITEREFZyga2016\"] = 1,\n [\"CITEREFvon_Keyserlingk_et_al.2016\"] = 1,\n}\ntemplate_list = table#1 {\n [\"!\"] = 2,\n [\"=\"] = 4,\n [\"Citation needed\"] = 2,\n [\"Cite arXiv\"] = 2,\n [\"Cite book\"] = 7,\n [\"Cite journal\"] = 68,\n [\"Cite news\"] = 4,\n [\"Cite web\"] = 35,\n [\"Condensed matter physics\"] = 1,\n [\"Condensed matter physics topics\"] = 1,\n [\"For\"] = 1,\n [\"Harvid\"] = 16,\n [\"Harvp\"] = 11,\n [\"Harvs\"] = 1,\n [\"Ill\"] = 1,\n [\"Main\"] = 2,\n [\"Math\"] = 2,\n [\"Nobr\"] = 2,\n [\"Physics-footer\"] = 1,\n [\"Portal bar\"] = 1,\n [\"Quantum computing\"] = 1,\n [\"Refbegin\"] = 3,\n [\"Refend\"] = 3,\n [\"Reflist\"] = 1,\n [\"Short description\"] = 1,\n [\"Snd\"] = 3,\n [\"State of matter\"] = 1,\n [\"Statistical mechanics topics\"] = 1,\n [\"Technical\"] = 1,\n [\"Time 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