Time crystal

Ordinary (non-time) crystals form through spontaneous symmetry breaking related to spatial symmetry. Such processes can produce materials with interesting properties, such as diamonds, salt crystals, and ferromagnetic 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).{{cite journal |last1=Sacha |first1=Krzysztof |last2=Zakrzewski |first2=Jakub |title=Time crystals: a review |journal=Reports on Progress in Physics |date=1 January 2018 |volume=81 |issue=1 |pages=016401 |doi=10.1088/1361-6633/aa8b38|pmid=28885193 |arxiv=1704.03735 |bibcode=2018RPPh...81a6401S |s2cid=28224975 }}

{{Main|Time-translation symmetry}}

Symmetries in nature lead directly to conservation laws, something which is precisely formulated by Noether's theorem.{{cite book|last1=Cao|first1=Tian Yu|title=Conceptual Foundations of Quantum Field Theory|url=https://books.google.com/books?id=d0wS0EJHZ3MC|date=25 March 2004|publisher=Cambridge University Press|isbn=978-0-521-60272-3|location=Cambridge}} See p. 151.

The basic idea of time-translation symmetry 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.{{cite book|last1=Wilczek|first1=Frank|title=A Beautiful Question: Finding Nature's Deep Design|url=https://books.google.com/books?id=Oh3ICAAAQBAJ|date=16 July 2015|publisher=Penguin Books Limited|isbn=978-1-84614-702-9}} See Ch. 3. This symmetry implies the conservation of energy.{{cite book|last1=Feng|first1=Duan|last2=Jin|first2=Guojun|title=Introduction to Condensed Matter Physics|url=https://books.google.com/books?id=-iuYN5arHwoC|year=2005|publisher=World Scientific|location=singapore|isbn=978-981-238-711-0}} See p. 18.

{{Main|Crystal symmetry|spontaneous symmetry breaking}}

Image:Phonon nu process.svg momentum, the U-process changes phonon momentum.]]

Common crystals exhibit broken translation symmetry: 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 Umklapp processes.{{cite book|last1=Sólyom|first1=Jenö|title=Fundamentals of the Physics of Solids: Volume 1: Structure and Dynamics|url=https://books.google.com/books?id=zn-se2TKv3QC|date=19 September 2007|publisher=Springer|isbn=978-3-540-72600-5}} See p. 193. Quasimomentum, however, is conserved in a perfect crystal.{{cite book|last1=Sólyom|first1=Jenö|title=Fundamentals of the Physics of Solids: Volume 1: Structure and Dynamics|url=https://books.google.com/books?id=zn-se2TKv3QC|date=19 September 2007|publisher=Springer|isbn=978-3-540-72600-5}} See p. 191.

Time crystals show a broken symmetry analogous to a discrete space-translation symmetry breaking. For example,{{citation needed|date=May 2019}} the molecules of a liquid freezing on the surface of a crystal can align with the molecules of the crystal, but with a pattern less symmetric than the crystal: it breaks the initial symmetry. This broken symmetry exhibits three important characteristics:{{citation needed|date=May 2019}}

• the system has a lower symmetry than the underlying arrangement of the crystal,
• the system exhibits spatial and temporal long-range order (unlike a local and intermittent order in a liquid near the surface of a crystal),
• it is the result of interactions between the constituents of the system, which align themselves relative to each other.

Time crystals seem to break time-translation symmetry 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 discrete time-translation symmetry breaking, not the continuous one: they are periodically driven systems oscillating at a fraction of the frequency of the driving force. (According to Philip Ball, DTC are so-called because "their periodicity is a discrete, integer multiple of the driving period".{{cite journal |last= Ball|first=Philip |date= July 17, 2018|title= In search of time crystals|url= https://physicsworld.com/a/in-search-of-time-crystals/|journal= Physics World |volume=31 |issue=7 |page=29 |doi=10.1088/2058-7058/31/7/32 |bibcode=2018PhyW...31g..29B |s2cid=125917780 |quote= The “discrete” comes from the fact that their periodicity is a discrete, integer multiple of the driving period.|access-date=September 6, 2021}})

The initial symmetry, which is the discrete time-translation symmetry ($t\; \backslash to\; t\; +\; nT$) with $n=1$, is spontaneously broken to the lower discrete time-translation symmetry with $n>1$, where $t$ is time, $T$ the driving period, $n$ an integer.

Many systems can show behaviors of spontaneous time-translation symmetry breaking but may not be discrete (or Floquet) time crystals: convection cells, oscillating chemical reactions, aerodynamic flutter, and subharmonic response to a periodic driving force such as the Faraday instability, NMR spin echos, parametric down-conversion, and period-doubled nonlinear dynamical systems.{{cite journal |last1=Else |first1=D. W. |last2=Monroe |first2=C. |last3=Nayak |first3=C. |last4=Yao |first4=N. Y. |title=Discrete Time Crystals |journal=Annual Review of Condensed Matter Physics |date=March 2020 |volume=11 |pages=467–499 |doi=10.1146/annurev-conmatphys-031119-050658 |arxiv=1905.13232 |bibcode=2020ARCMP..11..467E |s2cid=173188223 |url=https://doi.org/10.1146/annurev-conmatphys-031119-050658}}

However, discrete (or Floquet) time crystals are unique in that they follow a strict definition of discrete time-translation symmetry breaking:{{cite journal |last1=Yao |last2=Nayak |title=Time crystals in periodically driven systems |journal=Physics Today |volume=71 |issue=9 |year=2018 |pages=40–47 |issn=0031-9228 |doi=10.1063/PT.3.4020|arxiv=1811.06657 |bibcode=2018PhT....71i..40Y |s2cid=119433979 }}

• it is a broken symmetry{{snd}} the system shows oscillations with a period longer than the driving force,
• the system is in crypto-equilibrium{{snd}} these oscillations generate no entropy, and a time-dependent frame can be found in which the system is indistinguishable from an equilibrium when measured stroboscopically (which is not the case of convection cells, oscillating chemical reactions and aerodynamic flutter),
• the system exhibits long-range order{{snd}} the oscillations are in phase (synchronized) over arbitrarily long distances and time.

Moreover, the broken symmetry in time crystals is the result of many-body interactions: the order is the consequence of a collective process, just like in spatial crystals. This is not the case for NMR spin echos.

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.

Time crystals do not violate the laws of thermodynamics: 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"{{Cite news |url=https://www.sciencealert.com/time-crystals-might-exist-after-all-and-they-could-break-the-symmetry-of-space-and-time |title=Time Crystals Might Exist After All – And They Could Break Space-Time Symmetry |last=Crew |first=Bec |work=ScienceAlert |access-date=2017-09-21 |language=en-gb}}—their apparent motion does not represent conventional kinetic energy.{{Cite news |url=https://www.scientificamerican.com/article/time-crystals-could-be-legitimate-form-perpetual-motion/ |title='Time Crystals' Could Be a Legitimate Form of Perpetual Motion|author=Cowen, Ron |date=2017-02-02 |work=Scientific American |access-date=2023-07-22}} 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.

Time crystals do not evade the second law of thermodynamics,{{cite news |title=Google May Have Created an Unruly New State of Matter: Time Crystals |url=https://www.popularmechanics.com/science/a37211606/google-time-crystal-quantum-computing/ |access-date=4 August 2021 |publisher=Popular Mechanics}} 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.{{Cite web|last1=Kubota|first1=Taylor|last2=University|first2=Stanford|title=Physicists create time crystals with quantum computers|url=https://phys.org/news/2021-11-physicists-crystals-quantum.html|access-date=2021-12-03|website=phys.org|language=en}}{{Cite journal|last1=Mi|first1=Xiao|last2=Ippoliti|first2=Matteo|last3=Quintana|first3=Chris|last4=Greene|first4=Ami|last5=Chen|first5=Zijun|last6=Gross|first6=Jonathan|last7=Arute|first7=Frank|last8=Arya|first8=Kunal|last9=Atalaya|first9=Juan|last10=Babbush|first10=Ryan|last11=Bardin|first11=Joseph C.|year=2022|title=Time-Crystalline Eigenstate Order on a Quantum Processor|journal=Nature|volume=601|issue=7894|language=en|pages=531–536|arxiv=2107.13571|doi=10.1038/s41586-021-04257-w|pmid=34847568|pmc=8791837|bibcode=2022Natur.601..531M|issn=1476-4687}}

File:Frank Wilczek.jpg Frank Wilczek at University of Paris-Saclay]]

The idea of a quantized time crystal was theorized in 2012 by Frank Wilczek,{{cite journal|last1=Wilczek|first1=Frank|title=Quantum Time Crystals|journal=Physical Review Letters|volume=109|issue=16|pages=160401|year=2012|issn=0031-9007|doi=10.1103/PhysRevLett.109.160401|pmid=23215056|arxiv=1202.2539|bibcode=2012PhRvL.109p0401W|s2cid=1312256}}{{cite journal|last1=Shapere|first1=Alfred|last2=Wilczek|first2=Frank|title=Classical Time Crystals|journal=Physical Review Letters|volume=109|issue=16|pages=160402|year=2012|issn=0031-9007|doi=10.1103/PhysRevLett.109.160402|pmid=23215057|bibcode=2012PhRvL.109p0402S|arxiv=1202.2537|s2cid=4506464}} a Nobel laureate and professor at MIT. In 2013, Xiang Zhang, a nanoengineer at University of California, Berkeley, and his team proposed creating a time crystal in the form of a constantly rotating ring of charged ions.See {{harvs|txt|last1=Li et al.|year1=2012a|year2=2012b}}.{{cite web|last1=Wolchover|first1=Natalie|title=Perpetual Motion Test Could Amend Theory of Time|url=https://www.quantamagazine.org/20130425-perpetual-motion-test-could-amend-theory-of-time/#|website=quantamagazine.org|publisher=Simons Foundation|archive-url=https://archive.today/20170202105225/https://www.quantamagazine.org/20130425-perpetual-motion-test-could-amend-theory-of-time/|archive-date=2 February 2017|date=25 April 2013|url-status=dead}}

In response to Wilczek and Zhang, Patrick Bruno (European Synchrotron Radiation Facility) and Masaki Oshikawa (University of Tokyo) published several articles stating that space–time crystals were impossible.See {{harvp|Bruno|2013a}} and {{harvp|Bruno|2013b}}.{{cite web|last1=Thomas|first1=Jessica|title=Notes from the Editors: The Aftermath of a Controversial Idea|url=http://physics.aps.org/articles/v6/31|website=physics.aps.org|publisher=APS Physics|archive-url=https://archive.today/20170202100552/http://physics.aps.org/articles/v6/31|archive-date=2 February 2017|date=15 March 2013|url-status=dead}}

Subsequent work developed more precise definitions of time-translation symmetry-breaking, which ultimately led to the Watanabe–Oshikawa "no-go" statement that quantum space–time crystals in equilibrium are not possible.See {{harvp|Nozières|2013}}, {{harvp|Yao et al.|2017|p=1}} and {{harvp|Volovik|2013}}.{{cite journal|last1=Watanabe|first1=Haruki|last2=Oshikawa|first2=Masaki|title=Absence of Quantum Time Crystals|journal=Physical Review Letters|volume=114|issue=25|year=2015|issn=0031-9007|doi=10.1103/PhysRevLett.114.251603|bibcode=2015PhRvL.114y1603W|arxiv=1410.2143|pmid=26197119|page=251603|s2cid=312538}} 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.{{cite journal|last1=Medenjak|first1=Marko|last2=Buča|first2=Berislav|last3=Jaksch|first3=Dieter|date=2020-07-20|title=Isolated Heisenberg magnet as a quantum time crystal|journal=Physical Review B|volume=102|issue=4|page=041117|arxiv=1905.08266|bibcode=2020PhRvB.102d1117M|doi=10.1103/physrevb.102.041117|issn=2469-9950|s2cid=160009779}}{{cite arXiv |last1=Khemani |first1=Vedika |last2=Moessner |first2=Roderich |last3=Sondhi |first3=S. L. |title=A Brief History of Time Crystals |date=23 October 2019 |class=cond-mat.str-el |eprint=1910.10745}}{{cite journal |last1=Uhrich |first1=P. |last2=Defenu |first2=N. |last3=Jafari |first3=R. |last4=Halimeh |first4=J. C. |date=2020 |title=Out-of-equilibrium phase diagram of long-range superconductors |journal=Physical Review B |volume=101 |issue=24 |page=245148 | doi=10.1103/physrevb.101.245148|arxiv=1910.10715 |bibcode=2020PhRvB.101x5148U | doi-access=free}}

Several realizations of time crystals, which avoid the equilibrium no-go arguments, were later proposed.See {{harvp|Wilczek|2013b}} and {{harvp|Yoshii et al.|2015}}. In 2014 {{ill|Krzysztof Sacha|pl}} at Jagiellonian University in Kraków predicted the behaviour of discrete time crystals in a periodically driven system with "an ultracold atomic cloud bouncing on an oscillating mirror".{{cite journal|last1=Sacha|first1=Krzysztof|title=Modeling spontaneous breaking of time-translation symmetry|journal=Physical Review A|volume=91|issue=3|pages=033617|year=2015|issn=1050-2947|doi=10.1103/PhysRevA.91.033617|bibcode=2015PhRvA..91c3617S|arxiv=1410.3638|s2cid=118627872|quote=We show that an ultracold atomic cloud bouncing on an oscillating mirror can reveal spontaneous breaking of a discrete time-translation symmetry}}{{cite book|last1=Sacha|first1=Krzysztof|title=Time Crystals|series=Springer Series on Atomic, Optical, and Plasma Physics|url=https://link.springer.com/book/10.1007%2F978-3-030-52523-1|date=2020|volume=114|publisher=Springer|doi=10.1007/978-3-030-52523-1|isbn=978-3-030-52522-4|s2cid=240770955}}

In 2016, research groups at Princeton and at Santa Barbara independently suggested that periodically driven quantum spin systems could show similar behaviour.See {{harvp|Khemani et al.|2016}} and {{harvp|Else et al.|2016}} Also in 2016, Norman Yao at Berkeley and colleagues proposed a different way to create discrete time crystals in spin systems.{{cite journal|last1=Yao|first1=N. Y.|last2=Potter|first2=A. C.|last3=Potirniche|first3=I.-D.|last4=Vishwanath|first4=A.|title=Discrete Time Crystals: Rigidity, Criticality, and Realizations|journal=Physical Review Letters|volume=118|issue=3|pages=030401|year=2017|url=https://physics.aps.org/featured-article-pdf/10.1103/PhysRevLett.118.030401|issn=0031-9007|doi=10.1103/PhysRevLett.118.030401|pmid=28157355|arxiv=1608.02589|bibcode=2017PhRvL.118c0401Y|s2cid=206284432|ref={{harvid|Yao et al.|2017}} }} These ideas were successful and independently realized by two experimental teams: a group led by Harvard's Mikhail Lukin{{cite journal|last1=Choi|first1=Soonwon|last2=Choi|first2=Joonhee|last3=Landig|first3=Renate|last4=Kucsko|first4=Georg|last5=Zhou|first5=Hengyun|last6=Isoya|first6=Junichi|last7=Jelezko|first7=Fedor|last8=Onoda|first8=Shinobu|last9=Sumiya|first9=Hitoshi|last10=Khemani|first10=Vedika|last11=von Keyserlingk|first11=Curt|last12=Yao|first12=Norman Y.|last13=Demler|first13=Eugene|last14=Lukin|first14=Mikhail D.|title=Observation of discrete time-crystalline order in a disordered dipolar many-body system|arxiv=1610.08057|bibcode=2017Natur.543..221C|journal=Nature|volume=543|issue=7644|year=2017|pages=221–225|issn=0028-0836|doi=10.1038/nature21426|pmid=28277511|pmc=5349499|ref={{harvid|Choi et al.|2017}} }} and a group led by Christopher Monroe at University of Maryland.{{Cite journal|arxiv=1609.08684|doi=10.1038/nature21413|title=Observation of a discrete time crystal|year=2017|last1=Zhang|first1=J.|last2=Hess|first2=P. W.|last3=Kyprianidis|first3=A.|last4=Becker|first4=P.|last5=Lee|first5=A.|last6=Smith|first6=J.|last7=Pagano|first7=G.|last8=Potirniche|first8=I.-D.|last9=Potter|first9=A. C.|last10=Vishwanath|first10=A.|last11=Yao|first11=N. Y.|last12=Monroe|first12=C.|journal=Nature|volume=543|issue=7644|pages=217–220|pmid=28277505|bibcode=2017Natur.543..217Z|s2cid=4450646}} Both experiments were published in the same issue of Nature in March 2017.

Later, time crystals in open systems, so called dissipative time crystals, were proposed in several platforms breaking a discrete {{cite journal|last1=Iemini|first1=Fernando|last2=Russomanno|first2=Angelo|last3=Keeling|first3=Jonathan|last4=Schirò|first4=Marco|last5=Dalmonte|first5=Marcello|last6=Fazio|first6=Rosario|date=16 July 2018|title=Boundary time crystals|journal=Phys. Rev. Lett.|volume=121|issue=35301|page=035301|arxiv=1708.05014|bibcode=2018PhRvL.121c5301I|doi=10.1103/PhysRevLett.121.035301|pmid=30085780|s2cid=51683292}}{{cite journal|last1=Gong|first1=Zongping|last2=Hamazaki|first2=Ryusuke|last3=Ueda|first3=Masahito|date=25 January 2018|title=Discrete Time-Crystalline Order in Cavity and Circuit QED Systems|journal=Phys. Rev. Lett.|volume=120|issue=40404|page=040404|arxiv=1708.01472|bibcode=2018PhRvL.120d0404G|doi=10.1103/PhysRevLett.120.040404|pmid=29437420|s2cid=206307409}}{{cite journal|last1=Filippo Maria|first1=Gambetta|last2=Carollo|first2=Federico|last3=Marcuzzi|first3=Matteo|last4=Garrahan|first4=Juan P.|last5=Lesanovsky|first5=Igor|date=8 January 2019|title=Discrete Time Crystals in the Absence of Manifest Symmetries or Disorder in Open Quantum Systems|journal=Phys. Rev. Lett.|volume=122|issue=15701|page=015701|arxiv=1807.10161|bibcode=2019PhRvL.122a5701G|doi=10.1103/PhysRevLett.122.015701|pmid=31012672|s2cid=119187766}}{{Cite journal |last1=Buča |first1=Berislav |last2=Jaksch |first2=Dieter |date=2019-12-23 |title=Dissipation Induced Nonstationarity in a Quantum Gas |url=https://link.aps.org/doi/10.1103/PhysRevLett.123.260401 |journal=Physical Review Letters |volume=123 |issue=26 |pages=260401 |doi=10.1103/PhysRevLett.123.260401 |pmid=31951440 |arxiv=1905.12880 |bibcode=2019PhRvL.123z0401B |s2cid=170079211}} and a continuous{{Cite journal|last1=Iemini|first1=F.|last2=Russomanno|first2=A.|last3=Keeling|first3=J.|last4=Schirò|first4=M.|last5=Dalmonte|first5=M.|last6=Fazio|first6=R.|date=2018-07-16|title=Boundary Time Crystals|url=https://link.aps.org/doi/10.1103/PhysRevLett.121.035301|journal=Physical Review Letters|volume=121|issue=3|pages=035301|doi=10.1103/PhysRevLett.121.035301|pmid=30085780|arxiv=1708.05014|bibcode=2018PhRvL.121c5301I|hdl=10023/14492|s2cid=51683292}}{{cite journal |last1=Buča |first1=Berislav |last2=Tindall |first2=Joseph |last3=Jaksch |first3=Dieter |date=2019-04-15 |title=Non-stationary coherent quantum many-body dynamics through dissipation |journal=Nature Communications |volume=10 |issue=1 |page=1730 |arxiv=1804.06744 |bibcode=2019NatCo..10.1730B |doi=10.1038/s41467-019-09757-y |issn=2041-1723 |pmc=6465298 |pmid=30988312}} 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 University of Hamburg.{{Cite journal |last1=Keßler |first1=Hans |last2=Kongkhambut |first2=Phatthamon |last3=Georges |first3=Christoph |last4=Mathey |first4=Ludwig |last5=Cosme |first5=Jayson G. |last6=Hemmerich |first6=Andreas |date=2021-07-19 |title=Observation of a Dissipative Time Crystal |url=https://link.aps.org/doi/10.1103/PhysRevLett.127.043602 |journal=Physical Review Letters |volume=127 |issue=4 |pages=043602 |arxiv=2012.08885 |doi=10.1103/PhysRevLett.127.043602 |pmid=34355967 |bibcode=2021PhRvL.127d3602K |s2cid=229210935}} The researchers used a Bose–Einstein condensate strongly coupled to a dissipative optical cavity and the time crystal was demonstrated to spontaneously break discrete time-translation symmetry by periodically switching between two atomic density patterns.{{Cite journal|last1=Gong|first1=Zongping|last2=Ueda|first2=Masahito|date=2021-07-19|title=Time Crystals in Open Systems|url=https://physics.aps.org/articles/v14/104|journal=Physics|language=en|volume=14|page=104|doi=10.1103/Physics.14.104|bibcode=2021PhyOJ..14..104G|s2cid=244256783|doi-access=free}}{{Cite journal|last=Ball|first=Philip|date=September 2021|title=Quantum time crystals open up|url=https://www.nature.com/articles/s41563-021-01090-4|journal=Nature Materials|language=en|volume=20|issue=9|pages=1172|doi=10.1038/s41563-021-01090-4|pmid=34433935|bibcode=2021NatMa..20.1172B|s2cid=237299508|issn=1476-4660}} In an earlier experiment in the group of Tilman Esslinger at ETH Zurich, limit cycle dynamics{{Cite journal |last1=Piazza |first1=Francesco |last2=Ritsch |first2=Helmut |date=2015-10-15 |title=Self-Ordered Limit Cycles, Chaos, and Phase Slippage with a Superfluid inside an Optical Resonator |url=https://link.aps.org/doi/10.1103/PhysRevLett.115.163601 |journal=Physical Review Letters |volume=115 |issue=16 |pages=163601 |arxiv=1507.08644 |doi=10.1103/PhysRevLett.115.163601 |pmid=26550874 |bibcode=2015PhRvL.115p3601P |s2cid=5080527}} was observed in 2019,{{Cite journal|last1=Dogra|first1=Nishant|last2=Landini|first2=Manuele|last3=Kroeger|first3=Katrin|last4=Hruby|first4=Lorenz|last5=Donner|first5=Tobias|last6=Esslinger|first6=Tilman|date=2019-12-20|title=Dissipation-induced structural instability and chiral dynamics in a quantum gas|url=https://www.science.org/doi/10.1126/science.aaw4465|journal=Science|language=en|volume=366|issue=6472|pages=1496–1499|doi=10.1126/science.aaw4465|pmid=31857481|arxiv=1901.05974|bibcode=2019Sci...366.1496D|s2cid=119283814|issn=0036-8075}} but evidence of robustness against perturbations and the spontaneous character of the time-translation symmetry breaking were not addressed.

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 {{math|r^−α}} for some {{math|α > 0}}. Kozin and Kyriienko instead analyzed a spin-1/2 many-body Hamiltonian 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 ground energy state. However, demonstrating such a system in practice might be prohibitively difficult,{{cite news |last1=Cho |first1=Adrian |title=Back to the future: The original time crystal makes a comeback |url=https://www.science.org/content/article/back-future-original-time-crystal-makes-comeback |access-date=19 March 2020 |journal=Science |date=27 November 2019 |doi=10.1126/science.aba3793}}{{Cite journal|last1=Kozin|first1=Valerii K.|last2=Kyriienko|first2=Oleksandr|date=2019-11-20|title=Quantum Time Crystals from Hamiltonians with Long-Range Interactions|journal=Physical Review Letters|language=en|volume=123|issue=21|pages=210602|doi=10.1103/PhysRevLett.123.210602|pmid=31809146|issn=0031-9007|arxiv=1907.07215|bibcode=2019PhRvL.123u0602K|s2cid=197431242}} and concerns about the physicality of the long-range nature of the model have been raised.{{Cite arXiv | last1=Khemani | first1=Vedika | last2=Moessner | first2=Roderich | last3=Sondhi | first3=S. L. | title=Comment on 'Quantum Time Crystals from Hamiltonians with Long-Range Interactions' | year=2020 | class=cond-mat.str-el | eprint=2001.11037}}

In October 2016, Christopher Monroe at the University of Maryland claimed to have created the world's first discrete time crystal. Using the ideas proposed by Yao et al., his team trapped a chain of ¹⁷¹Yb⁺ ions in a Paul trap, confined by radio-frequency electromagnetic fields. One of the two spin states was selected by a pair of laser beams. The lasers were pulsed, with the shape of the pulse controlled by an acousto-optic modulator, using the Tukey window to avoid too much energy at the wrong optical frequency. The hyperfine electron states in that setup, ²S_1/2 {{nobr|{{!}}F {{=}} 0, m_F {{=}} 0⟩}} and {{nobr|{{!}}F {{=}} 1, m_F {{=}} 0⟩}}, have very close energy levels, separated by 12.642831 GHz. Ten Doppler-cooled ions were placed in a line 0.025 mm long and coupled together.

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.

Also in 2016, Mikhail Lukin at Harvard also reported the creation of a driven time crystal. His group used a diamond crystal doped with a high concentration of nitrogen-vacancy centers, which have strong dipole–dipole coupling and relatively long-lived spin coherence. 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 subharmonic response to the drive frequency is seen as a signature of time-crystalline order.

In May 2018, a group in Aalto University reported that they had observed the formation of a time quasicrystal and its phase transition to a continuous time crystal in a Helium-3 superfluid cooled to within one ten thousandth of a kelvin from absolute zero (0.0001 K).{{cite journal |last1=Autti |first1=S. |last2=Eltsov |first2=V. B. |last3=Volovik |first3=G. E.|title=Observation of a Time Quasicrystal and Its Transition to a Superfluid Time Crystal |journal=Physical Review Letters |date=May 2018 |volume=120 |issue=21 |pages=215301 |doi=10.1103/PhysRevLett.120.215301|pmid=29883148 |arxiv=1712.06877 |bibcode=2018PhRvL.120u5301A |s2cid=46997186 }} On August 17, 2020 Nature Materials 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.{{cite journal |last1=Autti |first1=S. |last2=Heikkinen |first2=P. J. |last3=Mäkinen |first3=J. T. |last4=Volovik |first4=G. E. |last5=Zavjalov |first5=V. V. |last6=Eltsov |first6=V. B. |title=AC Josephson effect between two superfluid time crystals |journal=Nature Materials |date=February 2021 |volume=20 |issue=2 |pages=171–174 |doi=10.1038/s41563-020-0780-y|pmid=32807922 |arxiv=2003.06313 |bibcode=2021NatMa..20..171A |s2cid=212717702 }}

In February 2021, a team at Max Planck Institute for Intelligent Systems described the creation of time crystal consisting of magnons and probed them under scanning transmission X-ray microscopy to capture the recurring periodic magnetization structure in the first known video record of such type.{{Cite journal|last1=Träger|first1=Nick|last2=Gruszecki|first2=Paweł|last3=Lisiecki|first3=Filip|last4=Groß|first4=Felix|last5=Förster|first5=Johannes|last6=Weigand|first6=Markus|last7=Głowiński|first7=Hubert|last8=Kuświk|first8=Piotr|last9=Dubowik|first9=Janusz|last10=Schütz|first10=Gisela|last11=Krawczyk|first11=Maciej|date=2021-02-03|title=Real-Space Observation of Magnon Interaction with Driven Space–Time Crystals|url=https://link.aps.org/doi/10.1103/PhysRevLett.126.057201|journal=Physical Review Letters|volume=126|issue=5|pages=057201|arxiv=1911.13192|doi=10.1103/PhysRevLett.126.057201|pmid=33605763|bibcode=2021PhRvL.126e7201T|s2cid=208512720}}{{Cite web|last=Williams|first=Jon|title=World's first video recording of a space–time crystal|url=https://is.mpg.de/news/world-s-first-video-recording-of-a-space-time-crystal|date=9 February 2021|access-date=2021-08-07|website=Max Planck Institute for Intelligent Systems|language=en}}

In July 2021, a team led by Andreas Hemmerich at the Institute of Laser Physics at the University of Hamburg presented the first realization of a time crystal in an open system, a so-called dissipative time crystal using ultracold atoms coupled to an optical cavity. The main achievement of this work is a positive application of dissipation – actually helping to stabilise the system's dynamics.

In November 2021, a collaboration between Google and physicists from multiple universities reported the observation of a discrete time crystal on Google's Sycamore processor, a quantum computing device. A chip of 20 qubits was used to obtain a many-body localization configuration of up and down spins and then stimulated with a laser to achieve a periodically driven "Floquet" system where all up spins are flipped for down 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 protected eigenstate order.{{Cite web|last=Wolchover|first=Natalie|date=2021-07-30|title=Eternal Change for No Energy: A Time Crystal Finally Made Real|url=https://www.quantamagazine.org/first-time-crystal-built-using-googles-quantum-computer-20210730/|access-date=2021-07-30|website=Quanta Magazine|language=en}}

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 quantum simulators rather than quantum processors: first a group at the University of Maryland obtained time crystals on trapped-ions qubits using high frequency driving rather than many-body localization{{Cite journal |last1=Kyprianidis |first1=A. |last2=Machado |first2=F. |last3=Morong |first3=W. |last4=Becker |first4=P. |last5=Collins |first5=K. S. |last6=Else |first6=D. V. |last7=Feng |first7=L. |last8=Hess |first8=P. W. |last9=Nayak |first9=C. |last10=Pagano |first10=G. |last11=Yao |first11=N. Y. |date=2021-06-11 |title=Observation of a prethermal discrete time crystal |url=https://www.science.org/doi/10.1126/science.abg8102 |journal=Science |language=en |volume=372 |issue=6547 |pages=1192–1196 |arxiv=2102.01695 |bibcode=2021Sci...372.1192K |doi=10.1126/science.abg8102 |issn=0036-8075 |pmid=34112691 |s2cid=231786633}}{{Cite web |last1=S |first1=Robert |last2=ers |last3=Berkeley |first3=U. C. |date=2021-11-10 |title=Creating Time Crystals Using New Quantum Computing Architectures |url=https://scitechdaily.com/creating-time-crystals-using-new-quantum-computing-architectures/ |access-date=2021-12-27 |website=SciTechDaily |language=en-US}} and then a collaboration between TU Delft and TNO in the Netherlands called Qutech created time crystals from nuclear spins in carbon-13 nitrogen-vacancy (NV) centers on a diamond, attaining longer times but fewer qubits.{{Cite journal|last1=Randall|first1=J.|last2=Bradley|first2=C. E.|last3=van der Gronden|first3=F. V.|last4=Galicia|first4=A.|last5=Abobeih|first5=M. H.|last6=Markham|first6=M.|last7=Twitchen|first7=D. J.|last8=Machado|first8=F.|last9=Yao|first9=N. Y.|last10=Taminiau|first10=T. H.|date=2021-12-17|title=Many-body–localized discrete time crystal with a programmable spin-based quantum simulator|url=https://www.science.org/doi/10.1126/science.abk0603|journal=Science|language=en|volume=374|issue=6574|pages=1474–1478|doi=10.1126/science.abk0603|pmid=34735218|arxiv=2107.00736|bibcode=2021Sci...374.1474R|s2cid=235727352|issn=0036-8075}}{{Cite web|date=2021-11-17|title=Physicists create discrete time crystals in a programmable quantum simulator|url=https://physicsworld.com/physicists-create-discrete-time-crystals-in-a-programmable-quantum-simulator/|access-date=2021-12-27|work=Physics World |first=Martijn |last=Boerkamp|language=en-GB}}

In February 2022, a scientist at UC Riverside 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 injection locking was used to direct lasers at a specific frequency inside a microresonator creating a lattice trap for solitons at subharmonic frequencies.{{Cite web |last=Starr |first=Michelle |title=New Breakthrough Could Bring Time Crystals Out of The Lab And Into The Real World |url=https://www.sciencealert.com/time-crystals-have-been-observed-in-a-system-that-isn-t-isolated-from-its-environment |access-date=2022-03-11 |website=ScienceAlert |date=16 February 2022 |language=en-gb}}{{Cite journal |last1=Taheri |first1=Hossein |last2=Matsko |first2=Andrey B. |last3=Maleki |first3=Lute |last4=Sacha |first4=Krzysztof |date=14 February 2022 |title=All-optical dissipative discrete time crystals |journal=Nature Communications |language=en |volume=13 |issue=1 |pages=848 |doi=10.1038/s41467-022-28462-x |issn=2041-1723 |pmc=8844012 |pmid=35165273|bibcode=2022NatCo..13..848T }}

In March 2022, a new experiment studying time crystals on a quantum processor was performed by two physicists at the University of Melbourne, this time using IBM's Manhattan and Brooklyn quantum processors observing a total of 57 qubits.{{Cite journal |date=2022-03-02 |title=Physicists produce biggest time crystal yet |url=https://www.science.org/content/article/physicists-produce-biggest-time-crystal-yet |journal= Science |first=Adrian |last=Cho |language=en |doi=10.1126/science.adb1790}}{{Cite journal |last1=Frey |first1=Philipp |last2=Rachel |first2=Stephan |date=2022-03-04 |title=Realization of a discrete time crystal on 57 qubits of a quantum computer |journal=Science Advances |language=en |volume=8 |issue=9 |pages=eabm7652 |doi=10.1126/sciadv.abm7652 |pmid=35235347 |pmc=8890700 |arxiv=2105.06632 |bibcode=2022SciA....8M7652F |issn=2375-2548}}{{Cite web |last1=Frey |first1=Philipp |last2=Rachel |first2=Stephan |title='An ever-ticking clock': we made a 'time crystal' inside a quantum computer |url=http://theconversation.com/an-ever-ticking-clock-we-made-a-time-crystal-inside-a-quantum-computer-178164 |date=2 March 2022 |access-date=2022-03-08 |website=The Conversation |language=en}}

In June 2022, the observation of a continuous time crystal was reported by a team at the Institute of Laser Physics at the University of Hamburg, 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 limit cycle 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 Bose–Einstein condensate in an optical cavity, which was pumped with an optical standing wave oriented perpendicularly with regard to the cavity axis and was in a superradiant phase localizing at two bistable ground states between which it oscillated.{{Cite journal |last1=Kongkhambut |first1=Phatthamon |last2=Skulte |first2=Jim |last3=Mathey |first3=Ludwig |last4=Cosme |first4=Jayson G. |last5=Hemmerich |first5=Andreas |last6=Keßler |first6=Hans |date=2022-08-05 |title=Observation of a continuous time crystal |url=https://www.science.org/doi/10.1126/science.abo3382 |journal=Science |language=en |volume=377 |issue=6606 |pages=670–673 |arxiv=2202.06980 |doi=10.1126/science.abo3382 |pmid=35679353 |bibcode=2022Sci...377..670K |s2cid=246863968 |issn=0036-8075}}{{Cite journal |last=LeBlanc |first=Lindsay J. |date=2022-08-05 |title=Unleashing spontaneity in a time crystal |url=https://www.science.org/doi/10.1126/science.add2015 |journal=Science |language=en |volume=377 |issue=6606 |pages=576–577 |doi=10.1126/science.add2015 |pmid=35926056 |bibcode=2022Sci...377..576L |s2cid=251349796 |issn=0036-8075}}{{Cite web |title=Researchers observe continuous time crystal |url=https://www.cui-advanced.uni-hamburg.de/en/research/wissenschaftsnews/22-06-10-crystal.html |access-date=2022-08-07 |website=www.cui-advanced.uni-hamburg.de |language=en}}{{Cite web |last=Hamburg |first=University of |date=2022-07-03 |title=Physicists Create Continuous Time Crystal for the First Time |url=https://scitechdaily.com/physicists-create-continuous-time-crystal-for-the-first-time/ |access-date=2022-08-07 |website=SciTechDaily |language=en-us}}

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".{{cite web | url=https://www.popularmechanics.com/science/a46854031/time-crystal-long-lasting/ | title=Scientists Built a Time Crystal That Lasted for 40 Minutes. That's Astonishing | date=24 February 2024 }}{{cite web | url=https://gizmodo.com/a-time-crystal-survived-a-whopping-40-minutes-1851221490 | title=A Time Crystal Survived a Whopping 40 Minutes | date=6 February 2024 }}{{Cite web | url=https://www.msn.com/en-ca/news/technology/scientists-built-a-time-crystal-that-lasted-for-40-minutes-that-s-astonishing/ar-BB1iODrc | title=Scientists Built a Time Crystal That Lasted for 40 Minutes. That's Astonishing | website=www.msn.com | date=2024-02-24 | first=Darren | last=Orf }}{{cite web | url=https://www.sciencealert.com/radical-new-time-crystal-revealed-that-lasts-millions-of-times-longer | title=Radical New Time Crystal Revealed That Lasts Millions of Times Longer | date=5 February 2024 }}{{cite web | url=https://phys.org/news/2024-02-physicists-highly-robust-crystal.html | title=Physicists develop highly robust time crystal }}

In March 2025, researchers at TU Dortmund University observed complex nonlinear behavior in a semiconductor-based time crystal made of indium gallium arsenide. By periodically driving the system with laser pulses, they uncovered transitions from synchronized oscillations to chaotic motion. The system exhibited structures such as the Farey tree sequence and the devil's staircase—patterns never before seen in semiconductor time crystals—offering new insights into dynamic phase transitions and chaos in driven quantum systems.{{cite journal |last=Greilich |first=Alex |display-authors=et al. |title=Exploring nonlinear dynamics in periodically driven time crystal from synchronization to chaotic motion |journal=Nature Communications |volume=16 |issue=2936 |date=26 March 2025 |doi=10.1038/s41467-025-58400-6 |arxiv=2406.06243 |bibcode=2025NatCo..16.2936G }}

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{{reflist|30em}}

{{Refbegin|30em|indent=yes}}

• {{cite journal|last1=Boyle|first1=Latham|last2=Khoo|first2=Jun Yong|last3=Smith|first3=Kendrick|title=Symmetric Satellite Swarms and Choreographic Crystals|journal=Physical Review Letters|volume=116|issue=1|year=2016|issn=0031-9007|doi=10.1103/PhysRevLett.116.015503|arxiv=1407.5876|bibcode=2016PhRvL.116a5503B|ref={{harvid|Boyle et al.|2016}}|pmid=26799028|page=015503|s2cid=17918689}}
• {{cite journal|last1=Bruno|first1=Patrick|title=Comment on 'Quantum Time Crystals'|journal=Physical Review Letters|volume=110|issue=11|year=2013a|issn=0031-9007|doi=10.1103/PhysRevLett.110.118901|arxiv=1210.4128|bibcode=2013PhRvL.110k8901B|pmid=25166585|page=118901|s2cid=41459498|url=https://zenodo.org/record/1184403}}
• {{cite journal|last1=Bruno|first1=Patrick|title=Comment on "Space-Time Crystals of Trapped Ions"|journal=Physical Review Letters|volume=111|issue=2|pages=029301|year=2013b|issn=0031-9007|doi=10.1103/PhysRevLett.111.029301|pmid=23889455|arxiv=1211.4792 |bibcode=2013PhRvL.111b9301B|s2cid=1502258}}
• {{cite journal|last1=Else|first1=Dominic V.|last2=Bauer|first2=Bela|last3=Nayak|first3=Chetan|title=Floquet Time Crystals|journal=Physical Review Letters|volume=117|issue=9|year=2016|issn=0031-9007|doi=10.1103/PhysRevLett.117.090402|arxiv=1603.08001|bibcode=2016PhRvL.117i0402E|ref={{harvid|Else et al.|2016}}|pmid=27610834|page=090402|s2cid=1652633}}
• {{cite journal|last1=Grifoni|first1=Milena|last2=Hänggi|first2=Peter|title=Driven quantum tunneling|journal=Physics Reports|volume=304|issue=5–6|year=1998|pages=229–354|issn=0370-1573|doi=10.1016/S0370-1573(98)00022-2|url=https://pdfs.semanticscholar.org/9477/590bf9c4bc44f0aadf036bd6ab45ce76ebc8.pdf|archive-url=https://web.archive.org/web/20170211080112/https://pdfs.semanticscholar.org/9477/590bf9c4bc44f0aadf036bd6ab45ce76ebc8.pdf|url-status=dead|archive-date=2017-02-11|bibcode=1998PhR...304..229G|citeseerx=10.1.1.65.9479|s2cid=120738031}}
• {{cite journal|last1=Guo|first1=Lingzhen|last2=Marthaler|first2=Michael|last3=Schön|first3=Gerd|title=Phase Space Crystals: A New Way to Create a Quasienergy Band Structure|journal=Physical Review Letters|volume=111|issue=20|pages=205303|year=2013|issn=0031-9007|doi=10.1103/PhysRevLett.111.205303|pmid=24289695|arxiv=1305.1800|bibcode=2013PhRvL.111t5303G|s2cid=9337383|ref={{harvid|Guo et al.|2013}} }}
• {{cite journal|last1=Guo|first1=Lingzhen|last2=Liang|first2=Pengfei|title=Condensed matter physics in time crystals|journal=New Journal of Physics|volume=22|year=2020|issue=7|doi=10.1088/1367-2630/ab9d54|arxiv=2005.03138|ref={{harvid|Guo et al.|2020}}|page=075003|bibcode=2020NJPh...22g5003G|s2cid=218538401}}
• {{cite journal|last1=Khemani|first1=Vedika|last2=Lazarides|first2=Achilleas|last3=Moessner|first3=Roderich|last4=Sondhi|first4=S. L.|title=Phase Structure of Driven Quantum Systems|journal=Physical Review Letters|volume=116|issue=25|pages=250401|year=2016|issn=0031-9007|doi=10.1103/PhysRevLett.116.250401|pmid=27391704|bibcode=2016PhRvL.116y0401K|arxiv=1508.03344|s2cid=883197|ref={{harvid|Khemani et al.|2016}} }}
• {{cite journal|last1=Li|first1=Tongcang|last2=Gong|first2=Zhe-Xuan|last3=Yin|first3=Zhang-Qi|last4=Quan|first4=H. T.|last5=Yin|first5=Xiaobo|last6=Zhang|first6=Peng|last7=Duan|first7=L.-M.|last8=Zhang|first8=Xiang|title=Space-Time Crystals of Trapped Ions|journal=Physical Review Letters|volume=109|issue=16|pages=163001|year=2012a|issn=0031-9007|doi=10.1103/PhysRevLett.109.163001|pmid=23215073|arxiv=1206.4772|bibcode=2012PhRvL.109p3001L|s2cid=8198228|ref={{harvid|Li et al.|2012a}} }}
• {{cite journal|last1=Li|first1=Tongcang|last2=Gong|first2=Zhe-Xuan|last3=Yin|first3=Zhang-Qi|last4=Quan|first4=H. T.|last5=Yin|first5=Xiaobo|last6=Zhang|first6=Peng|last7=Duan|first7=L.-M.|last8=Zhang|first8=Xiang|title=Reply to Comment on "Space–Time Crystals of Trapped Ions"|journal=Unpublished|date=2012|arxiv=1212.6959|bibcode= 2012arXiv1212.6959L

|ref={{harvid|Li et al.|2012b}} }}

• {{cite journal|last1=Lindner|first1=Netanel H.|last2=Refael|first2=Gil|last3=Galitski|first3=Victor|title=Floquet topological insulator in semiconductor quantum wells|journal=Nature Physics|volume=7|issue=6|year=2011|pages=490–495|issn=1745-2473|doi=10.1038/nphys1926|arxiv=1008.1792|bibcode=2011NatPh...7..490L|s2cid=26754031|ref={{harvid|Lindner et al.|2011}} }}
• {{cite journal|last1=Mendonça|first1=J. T.|last2=Dodonov|first2=V. V.|title=Time Crystals in Ultracold Matter|journal=Journal of Russian Laser Research|volume=35|issue=1|year=2014|pages=93–100|issn=1071-2836|doi=10.1007/s10946-014-9404-9|s2cid=122631523|url=https://www.researchgate.net/publication/272040551}}
• {{cite journal|last1=Nozières|first1=Philippe|title=Time crystals: Can diamagnetic currents drive a charge density wave into rotation?|journal=EPL|volume=103|issue=5|year=2013|pages=57008|issn=0295-5075|doi=10.1209/0295-5075/103/57008|arxiv=1306.6229|bibcode=2013EL....10357008N|s2cid=118662499}}
• {{cite journal|last1=Robicheaux|first1=F.|last2=Niffenegger|first2=K.|title=Quantum simulations of a freely rotating ring of ultracold and identical bosonic ions|journal=Physical Review A|volume=91|issue=6|year=2015|pages=063618|issn=2469-9926|doi=10.1103/PhysRevA.91.063618|bibcode=2015PhRvA..91f3618R|doi-access=free}}
• {{cite journal|last1=Sacha|first1=Krzysztof|title=Modeling spontaneous breaking of time-translation symmetry|journal=Physical Review A|volume=91|issue=3|year=2015|issn=2469-9934|doi=10.1103/PhysRevA.91.033617|arxiv=1410.3638|page=033617|bibcode=2015PhRvA..91c3617S|s2cid=118627872}}
• {{cite journal|last1=Sacha|first1=Krzysztof|title=Anderson localization and Mott insulator phase in the time domain|journal=Scientific Reports|volume=5|year=2015|doi=10.1038/srep10787|arxiv=1502.02507|ref={{harvid|Sacha|2015a}}|page=10787|pmid=26074169|pmc=4466589|bibcode=2015NatSR...510787S}}
• {{cite journal|last1=Sacha|first1=Krzysztof|last2=Zakrzewski|first2=Jakub|title=Time Crystals: a review|journal=Reports on Progress in Physics|volume=81|issue=1|year=2018|doi=10.1088/1361-6633/aa8b38|arxiv=1704.03735|ref={{harvid|Sacha et al.|2018}}|page=016401|pmid=28885193|bibcode=2018RPPh...81a6401S|s2cid=28224975}}
• {{cite journal|last1=Shirley|first1=Jon H.|title=Solution of the Schrödinger Equation with a Hamiltonian Periodic in Time|journal=Physical Review|volume=138|issue=4B|year=1965|pages=B979–B987|issn=0031-899X|doi=10.1103/PhysRev.138.B979|bibcode=1965PhRv..138..979S}}
• {{cite journal|last1=Smith|first1=J.|last2=Lee|first2=A.|last3=Richerme|first3=P.|last4=Neyenhuis|first4=B.|last5=Hess|first5=P. W.|last6=Hauke|first6=P.|last7=Heyl|first7=M.|last8=Huse|first8=D. A.|last9=Monroe|first9=C.|title=Many-body localization in a quantum simulator with programmable random disorder|journal=Nature Physics|volume=12|issue=10|year=2016|pages=907–911|issn=1745-2473|doi=10.1038/nphys3783|arxiv=1508.07026|bibcode=2016NatPh..12..907S|s2cid=53408060|ref={{harvid|Smith et al.|2016}} }}
• {{cite journal|last1=Volovik|first1=G. E.|title=On the broken time translation symmetry in macroscopic systems: Precessing states and off-diagonal long-range order|journal=JETP Letters|volume=98|issue=8|year=2013|pages=491–495|issn=0021-3640|doi=10.1134/S0021364013210133|arxiv=1309.1845 |bibcode=2013JETPL..98..491V|s2cid=119100114}}
• {{cite journal|last1=von Keyserlingk|first1=C. W.|last2=Khemani|first2=Vedika|last3=Sondhi|first3=S. L.|title=Absolute stability and spatiotemporal long-range order in Floquet systems|journal=Physical Review B|volume=94|issue=8|pages=085112|year=2016|issn=2469-9950|doi=10.1103/PhysRevB.94.085112|arxiv=1605.00639|bibcode=2016PhRvB..94h5112V|s2cid=118699328|ref={{harvid|von Keyserlingk et al.|2016}} }}
• {{cite journal|last1=Wang|first1=Y. H.|last2=Steinberg|first2=H.|last3=Jarillo-Herrero|first3=P.|last4=Gedik|first4=N.|title=Observation of Floquet-Bloch States on the Surface of a Topological Insulator|journal=Science|volume=342|issue=6157|year=2013|pages=453–457|issn=0036-8075|doi=10.1126/science.1239834|pmid=24159040|arxiv=1310.7563|bibcode=2013Sci...342..453W|ref={{harvid|Wang et al.|2013}} |hdl=1721.1/88434|s2cid=29121373}}
• {{cite journal|last1=Wilczek|first1=Frank|title=Wilczek Reply|journal=Physical Review Letters|volume=110|issue=11|pages=118902|year=2013a|issn=0031-9007|doi=10.1103/PhysRevLett.110.118902|pmid=25166586|url=http://xa.yimg.com/kq/groups/2385221/2027721577/name/WilzcekreplyPhysRevLett.110.118902-1.pdf|bibcode=2013PhRvL.110k8902W|archive-url=https://web.archive.org/web/20170211155500/http://xa.yimg.com/kq/groups/2385221/2027721577/name/WilzcekreplyPhysRevLett.110.118902-1.pdf|archive-date=2017-02-11}}
• {{cite journal|last1=Wilczek|first1=Frank|title=Superfluidity and Space–Time Translation Symmetry Breaking|journal=Physical Review Letters|volume=111|issue=25|year=2013b|page=250402|issn=0031-9007|doi=10.1103/PhysRevLett.111.250402|pmid=24483732|bibcode=2013PhRvL.111y0402W|arxiv=1308.5949|s2cid=7537145}}
• {{cite journal|last1=Yoshii|first1=Ryosuke|last2=Takada|first2=Satoshi|last3=Tsuchiya|first3=Shunji|last4=Marmorini|first4=Giacomo|last5=Hayakawa|first5=Hisao|last6=Nitta|first6=Muneto|title=Fulde-Ferrell-Larkin-Ovchinnikov states in a superconducting ring with magnetic fields: Phase diagram and the first-order phase transitions|journal=Physical Review B|volume=92|issue=22|pages=224512|year=2015|issn=1098-0121|doi=10.1103/PhysRevB.92.224512|arxiv=1404.3519|bibcode=2015PhRvB..92v4512Y|s2cid=118348062|ref={{harvid|Yoshii et al.|2015}} }}
• {{cite journal|last1=Zel'Dovich|first1=Y. B.|title=The quasienergy of a quantum-mechanical system subjected to a periodic action|journal=Soviet Physics JETP|date=1967|volume=24|issue=5|pages=1006–1008|url=http://jetp.ac.ru/cgi-bin/dn/e_024_05_1006.pdf|bibcode=1967JETP...24.1006Z|access-date=2017-02-08|archive-date=2017-02-11|archive-url=https://web.archive.org/web/20170211080425/http://jetp.ac.ru/cgi-bin/dn/e_024_05_1006.pdf|url-status=dead}}

{{refend}}

{{Refbegin|30em|indent=yes}}

• {{cite book|last1=Sacha|first1=Krzysztof|title=Time Crystals|series=Springer Series on Atomic, Optical, and Plasma Physics|url=https://link.springer.com/book/10.1007%2F978-3-030-52523-1|date=2020|volume=114|publisher=Springer|doi=10.1007/978-3-030-52523-1|isbn=978-3-030-52522-4|s2cid=240770955}}

{{refend}}

{{Refbegin|30em|indent=yes}}

• {{Cite journal|last=Ball|first=Philip|date=20 September 2021|title=Focus: Turning a Quantum Computer into a Time Crystal|url=https://physics.aps.org/articles/v14/131|journal=Physics|volume=14|page=131 |publisher=APS Physics|doi=10.1103/Physics.14.131|language=en|doi-access=free}}
• {{cite web|last1=Ball|first1=Philip|title=Focus: New Crystal Type is Always in Motion|url=http://physics.aps.org/articles/v9/4|website=physics.aps.org|publisher=APS Physics|archive-url=https://archive.today/20170203141844/http://physics.aps.org/articles/v9/4|archive-date=3 February 2017|date=8 January 2016|url-status=dead}}
• {{cite journal|last1=Coleman|first1=Piers|title=Quantum physics: Time crystals|journal=Nature|volume=493|issue=7431|date=9 January 2013|pages=166–167|issn=0028-0836|doi=10.1038/493166a|pmid=23302852|bibcode=2013Natur.493..166C|s2cid=205075903}}
• {{cite web|last1=Cowen|first1=Ron|title="Time Crystals" Could Be a Legitimate Form of Perpetual Motion|url=https://www.scientificamerican.com/article/time-crystals-could-be-legitimate-form-perpetual-motion/|website=scientificamerican.com|publisher=Scientific American|archive-url=https://archive.today/20170202101455/https://www.scientificamerican.com/article/time-crystals-could-be-legitimate-form-perpetual-motion/|archive-date=2 February 2017|date=27 February 2012}}
• {{cite journal|last1=Gibney|first1=Elizabeth|title=The quest to crystallize time|journal=Nature|volume=543|issue=7644|year=2017|pages=164–166|issn=0028-0836|doi=10.1038/543164a|pmid=28277535|bibcode=2017Natur.543..164G|s2cid=4460265}}
• {{cite web|last1=Grossman|first1=Lisa|title=Death-defying time crystal could outlast the universe|url=https://www.newscientist.com/article/mg21328484-000-death-defying-time-crystal-could-outlast-the-universe/|website=newscientist.com|publisher=New Scientist|archive-url=https://archive.today/20170202104619/https://www.newscientist.com/article/mg21328484-000-death-defying-time-crystal-could-outlast-the-universe/|archive-date=2 February 2017|date=18 January 2012}}
• {{cite web|last1=Hackett|first1=Jennifer|title=Curious Crystal Dances for Its Symmetry|url=https://www.scientificamerican.com/article/curious-crystal-dances-for-its-symmetry/|website=scientificamerican.com|publisher=Scientific American|archive-url=https://archive.today/20170203152135/https://www.scientificamerican.com/article/curious-crystal-dances-for-its-symmetry/|archive-date=3 February 2017|date=22 February 2016|url-status=dead}}
• {{cite web|last1=Hannaford|first1=Peter|last2=Sacha|first2=Krzysztof|title=Time crystals enter the real world of condensed matter|url=https://physicsworld.com/a/time-crystals-enter-the-real-world-of-condensed-matter/|website=physicsworld.com|publisher=Institute of Physics|date=17 Mar 2020}}
• {{cite web|last1=Hewitt|first1=John|title=Creating time crystals with a rotating ion ring|url=https://phys.org/news/2013-05-crystals-rotating-ion.html|website=phys.org|publisher=Science X|archive-url=https://archive.today/20130704232506/http://phys.org/news/2013-05-crystals-rotating-ion.html|archive-date=4 July 2013|date=3 May 2013|url-status=dead}}
• {{cite web|last1=Johnston|first1=Hamish|title='Choreographic crystals' have all the right moves|url=http://physicsworld.com/cws/article/news/2016/jan/18/choreographic-crystals-have-all-the-right-moves|website=physicsworld.com|publisher=Institute of Physics|archive-url=https://archive.today/20170203142329/http://physicsworld.com/cws/article/news/2016/jan/18/choreographic-crystals-have-all-the-right-moves|archive-date=3 February 2017|date=18 January 2016|url-status=dead}}
• {{cite web|author1=Joint Quantum Institute|title=Floquet Topological Insulators|url=http://jqi.umd.edu/news/floquet-topological-insulators|website=jqi.umd.edu|publisher=Joint Quantum Institute|date=22 March 2011}}
• {{cite web|last1=Ouellette|first1=Jennifer|title=World's first time crystals cooked up using new recipe|url=https://www.newscientist.com/article/2119804-worlds-first-time-crystals-cooked-up-using-new-recipe/|website=newscientist.com|publisher=New Scientist|archive-url=https://archive.today/20170201185926/https://www.newscientist.com/article/2119804-worlds-first-time-crystals-cooked-up-using-new-recipe/|archive-date=1 February 2017|date=31 January 2017|url-status=dead}}
• {{cite journal|last1=Powell|first1=Devin|title=Can matter cycle through shapes eternally?|journal=Nature|year=2013|issn=1476-4687|doi=10.1038/nature.2013.13657|s2cid=181223762|url=http://www.nature.com/news/can-matter-cycle-through-shapes-eternally-1.13657|archive-url=https://archive.today/20170203080014/http://www.nature.com/news/can-matter-cycle-through-shapes-eternally-1.13657|archive-date=3 February 2017}}
• {{cite web|author1=University of California, Berkeley|title=Physicists unveil new form of matter—time crystals|url=https://phys.org/news/2017-01-physicists-unveil-mattertime-crystals.html|website=phys.org|publisher=Science X|archive-url=https://archive.today/20170128214118/https://phys.org/news/2017-01-physicists-unveil-mattertime-crystals.html|archive-date=28 January 2017|date=26 January 2017|url-status=dead}}
• {{cite web|last1=Weiner|first1=Sophie|title=Scientists Create A New Kind Of Matter: Time Crystals|url=http://www.popularmechanics.com/science/a24957/time-crystals/|website=popularmechanics.com|publisher=Popular mechanics|archive-url=https://archive.today/20170203161545/http://www.popularmechanics.com/science/a24957/time-crystals/|archive-date=3 February 2017|date=28 January 2017|url-status=dead}}
• {{cite web|last1=Wood|first1=Charlie|title=Time crystals realize new order of space-time|url=http://www.csmonitor.com/Science/2017/0131/Time-crystals-realize-new-order-of-space-time|website=csmonitor.com|publisher=Christian Science Monitor|archive-url=https://archive.today/20170202110602/http://www.csmonitor.com/Science/2017/0131/Time-crystals-realize-new-order-of-space-time|archive-date=2 February 2017|date=31 January 2017|url-status=dead}}
• {{cite web|last1=Yirka|first1=Bob|title=Physics team proposes a way to create an actual space-time crystal|url=https://phys.org/news/2012-07-physics-team-actual-space-time-crystal.html|website=phys.org|publisher=Science X|archive-url=https://archive.today/20130415190255/http://phys.org/news/2012-07-physics-team-actual-space-time-crystal.html|archive-date=15 April 2013|date=9 July 2012|url-status=dead}}
• {{cite web|last1=Zyga|first1=Lisa|title=Time crystals could behave almost like perpetual motion machines|url=https://phys.org/news/2012-02-crystals-perpetual-motion-machines.html#nRlv|website=phys.org|publisher=Science X|archive-url=https://archive.today/20170203082720/https://phys.org/news/2012-02-crystals-perpetual-motion-machines.html%23nRlv|archive-date=3 February 2017|date=20 February 2012|url-status=dead}}
• {{cite web|last1=Zyga|first1=Lisa|title=Physicist proves impossibility of quantum time crystals|url=https://phys.org/news/2013-08-physicist-impossibility-quantum-crystals.html|website=phys.org|publisher=Space X|archive-url=https://archive.today/20170203084130/https://phys.org/news/2013-08-physicist-impossibility-quantum-crystals.html|archive-date=3 February 2017|date=22 August 2013|url-status=dead}}
• {{cite web|last1=Zyga|first1=Lisa|title=Physicists propose new definition of time crystals—then prove such things don't exist|url=https://phys.org/news/2015-07-physicists-definition-crystalsthen-dont.html|website=phys.org|publisher=Science X|archive-url=https://archive.today/20150709134338/http://phys.org/news/2015-07-physicists-definition-crystalsthen-dont.html|archive-date=9 July 2015|date=9 July 2015|url-status=dead}}
• {{cite web|last1=Zyga|first1=Lisa|title=Time crystals might exist after all (Update)|url=https://phys.org/news/2016-09-crystals.html|website=phys.org|publisher=Science X|archive-url=https://archive.today/20160911165310/http://phys.org/news/2016-09-crystals.html|archive-date=11 September 2016|date=9 September 2016|url-status=dead}}

{{Refend}}

• [https://www.umdphysics.umd.edu/people/faculty/current/item/348-monroe.html Christopher Monroe] at University of Maryland
• [http://frankawilczek.com/ Frank Wilczek]
• [http://lukin.physics.harvard.edu/ Lukin Group] at Harvard University
• [http://physics.berkeley.edu/people/faculty/norman-yao Norman Yao] at the University of California at Berkeley
• [https://chaos.if.uj.edu.pl/~sacha/index.html Krzysztof Sacha] at Jagiellonian University in Kraków

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