Time crystal

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

A time crystal or space-time crystal is a state of matter that repeats in time, as well as in space. Normal three-dimensional crystals have a repeating pattern in space, but remain unchanged as time passes. Time crystals repeat themselves in time as well, leading the crystal to change from moment to moment.

If a discrete time translation symmetry is broken (which may be realized in periodically driven systems), then the system is referred to as a discrete time crystal. A discrete time crystal never reaches thermal equilibrium, as it is a type of non-equilibrium matter, a form of matter proposed in 2012, and first observed in 2017.

The idea of a quantized time crystal was first described by Nobel laureate Frank Wilczek in 2012. In 2014 Krzysztof Sacha predicted the behavior of discrete time crystals in a periodically-driven many-body system[1] and in 2016, Norman Yao et al. proposed a different way to create discrete time crystals in spin systems. From there, Christopher Monroe and Mikhail Lukin independently confirmed this in their labs. Both experiments were published in Nature in 2017. In 2019 it was theoretically proven that a quantum time crystal can be realized in isolated systems with long range multi-particle interactions.[2]


The idea of a space-time crystal was first put forward by Frank Wilczek, a professor at MIT and Nobel laureate, in 2012.[3]

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.[4]

In response to Wilczek and Zhang, Patrick Bruno, a theorist at the European Synchrotron Radiation Facility in Grenoble, France, published several articles in 2013 claiming to show that space-time crystals were impossible. Also later Masaki Oshikawa from the University of Tokyo showed that time crystals would be impossible at their ground state; moreover, he implied that any matter cannot exist in non-equilibrium in its ground state.[5][6]

Subsequent work developed more precise definitions of time translation symmetry-breaking, which ultimately led to a "no-go" proof that quantum time crystals in equilibrium are not possible.[7][8]

Several realizations of time crystals, which avoid the equilibrium no-go arguments, were later proposed.[9] Krzysztof Sacha at Jagiellonian University in Krakow predicted the behaviour of discrete time crystals in a periodically driven system of ultracold atoms.[10] Later works[11] suggested that periodically driven quantum spin systems could show similar behaviour.

Norman Yao at Berkeley studied a different model of time crystals.[12] His ideas were successfully used by two teams: a group led by Harvard's Mikhail Lukin[13] and a group led by Christopher Monroe at University of Maryland.[14]

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 r−α for some 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,[2][15] and concerns about the physicality of the long-range nature of the model have been raised.[16]

Time translation symmetry[edit]

Symmetries in nature lead directly to conservation laws, something which is precisely formulated by the Noether theorem.[17]

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.[18] This symmetry implies the conservation of energy.[19]

Broken symmetry in normal crystals[edit]

Normal process (N-process) and Umklapp process (U-process). While the N-process conserves total phonon momentum, the U-process changes phonon momentum.

Normal 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.[20] Quasimomentum, however, is conserved in a perfect crystal.[21]

Time crystals show a broken symmetry analogous to a discrete space-translation symmetry breaking. For example,[citation needed] 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]

  • 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.

Broken symmetry in discrete time crystals[edit]

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. Actually, studied time crystals show discrete time-translation symmetry breaking: they are periodically driven systems oscillating at a fraction of the frequency of the driving force. The initial symmetry is already a discrete time-translation symmetry (), not a continuous one (), which are instead described by magnetic space groups.[citation needed]

Many systems can show behaviors of spontaneous time translation symmetry breaking: 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.

However, Floquet time crystals are unique in that they follow a strict definition of discrete time-translation symmetry breaking:[22]

  • it is a broken symmetry – the system shows oscillations with a period longer than the driving force,
  • the system is in crypto-equilibrium – these oscillations generate no entropy, and a time-dependant frame can be found in which the system is indistinguishable from an equilibrium when measured stroboscopically[citation needed] (which is not the case of convection cells, oscillating chemical reactions and aerodynamic flutter),
  • the system exhibits long-range order – 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.[citation needed] This is not the case for NMR spin echos.

Fields or particles may change their energy by interacting with a time crystal, just as they can change their momentum by interacting with a spatial crystal.[citation needed]

These characteristics makes time crystals analogous to spatial crystals as described above.


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"[23]—their apparent motion does not represent conventional kinetic energy.[24]

It has been proven that a time crystal cannot exist in thermal equilibrium. Recent years have seen more studies of non-equilibrium quantum fluctuations.[25]


In October 2016, Christopher Monroe at the University of Maryland claimed to have created the world's first discrete time crystal. Using the idea from Yao's proposal, 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, 2S1/2 |F = 0, mF = 0⟩ and |F = 1, mF = 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.[14]

Later 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.[13]

On August 17, 2020 Nature Materials published a letter from Aalto University saying that for the first time they were able to observe interactions and the flow of constituent particles between two time crystals in a Helium-3 superfluid cooled to within one ten thousandth of a degree from absolute zero (0.0001K or -273.15°C)[26]

Related concepts[edit]

A similar idea called a choreographic crystal has been proposed.[27]

By relaxing additional restrictions on the definition of time crystals, continuous time-translation symmetry breaking can be achieved in exceptional cases. For instance, if one allows the system to be open to an environment, but undriven, many-body systems with the appropriate algebraic structure can be time crystals [28]. Likewise, if one drops the requirement of long-range order in space, purely time-translation symmetry breaking is possible [29].


  1. ^ See Sacha (2015).
  2. ^ a b Cho, Adrian (27 November 2019). "Back to the future: The original time crystal makes a comeback". Science. doi:10.1126/science.aba3793. Retrieved 19 March 2020.
  3. ^ See Wilczek (2012) and Shapere & Wilczek (2012).
  4. ^ See Li et al. (2012a, 2012b), Wolchover 2013.
  5. ^ See Bruno (2013a) and Bruno (2013b).
  6. ^ Thomas (2013).
  7. ^ See Nozières (2013), Yao et al. (2017), p. 1 and Volovik (2013).
  8. ^ See Watanabe & Oshikawa (2015).
  9. ^ See Wilczek (2013b) and Yoshii et al. (2015).
  10. ^ See Sacha (2015).
  11. ^ See Khemani et al. (2016) and Else et al. (2016).
  12. ^ See Yao et al. (2017), Richerme (2017).
  13. ^ a b See Choi et al. (2017).
  14. ^ a b See Zhang et al. (2017).
  15. ^ Kozin, Valerii K.; Kyriienko, Oleksandr (2019-11-20). "Quantum Time Crystals from Hamiltonians with Long-Range Interactions". Physical Review Letters. 123 (21): 210602. arXiv:1907.07215. Bibcode:2019PhRvL.123u0602K. doi:10.1103/PhysRevLett.123.210602. ISSN 0031-9007. PMID 31809146. S2CID 197431242.
  16. ^ Khemani, Vedika; Moessner, Roderich; Sondhi, S. L. (2020). "Comment on "Quantum Time Crystals from Hamiltonians with Long-Range Interactions"". arXiv:2001.11037 [cond-mat.str-el].
  17. ^ Cao 2004, p. 151.
  18. ^ Wilczek 2015, ch. 3.
  19. ^ Feng & Jin 2005, p. 18.
  20. ^ Sólyom 2007, p. 193.
  21. ^ Sólyom 2007, p. 191.
  22. ^ Yao; Nayak (2018). "Time crystals in periodically driven systems". Physics Today. 71 (9): 40–47. arXiv:1811.06657. Bibcode:2018PhT....71i..40Y. doi:10.1063/PT.3.4020. ISSN 0031-9228. S2CID 119433979.
  23. ^ Crew, Bec. "Time Crystals Might Exist After All – And They Could Break Space-Time Symmetry". ScienceAlert. Retrieved 2017-09-21.
  24. ^ ""Time Crystals" Could Be a Legitimate Form of Perpetual Motion". archive.is. 2017-02-02. Archived from the original on 2017-02-02. Retrieved 2017-09-21.CS1 maint: BOT: original-url status unknown (link)
  25. ^ See Esposito et al. (2009) and Campisi et al. (2011) for academic review articles on non-equilibrium quantum fluctuations.
  26. ^ See Autti, S., Heikkinen, P.J., Mäkinen, J.T. et al. AC Josephson effect between two superfluid time crystals. Nat. Mater. (2020). https://doi.org/10.1038/s41563-020-0780-y
  27. ^ See Boyle et al. (2016).
  28. ^ Buča, Berislav; Tindall, Joseph; Jaksch, Dieter (2019-04-15). "Non-stationary coherent quantum many-body dynamics through dissipation". Nature Communications. 10 (1): 1730. arXiv:1804.06744. Bibcode:2019NatCo..10.1730B. doi:10.1038/s41467-019-09757-y. ISSN 2041-1723. PMC 6465298. PMID 30988312.
  29. ^ Medenjak, Marko; Buča, Berislav; Jaksch, Dieter (2020-07-20). "Isolated Heisenberg magnet as a quantum time crystal". Physical Review B. 102 (4): 041117. arXiv:1905.08266. Bibcode:2020PhRvB.102d1117M. doi:10.1103/physrevb.102.041117. ISSN 2469-9950. S2CID 160009779.

Academic articles[edit]

Beck, Christian; Mackey, Michael C. (2005). "Could dark energy be measured in the lab?". Physics Letters B. 605 (3–4): 295–300. arXiv:astro-ph/0406504v2. Bibcode:2005PhLB..605..295B. doi:10.1016/j.physletb.2004.11.060. ISSN 0370-2693. S2CID 17235133.CS1 maint: ref=harv (link)
Boyle, Latham; Khoo, Jun Yong; Smith, Kendrick (2016). "Symmetric Satellite Swarms and Choreographic Crystals". Physical Review Letters. 116 (1): 015503. arXiv:1407.5876v2. Bibcode:2016PhRvL.116a5503B. doi:10.1103/PhysRevLett.116.015503. ISSN 0031-9007. PMID 26799028. S2CID 17918689.
Bruno, Patrick (2013a). "Comment on "Quantum Time Crystals"". Physical Review Letters. 110 (11): 118901. arXiv:1210.4128v1. Bibcode:2013PhRvL.110k8901B. doi:10.1103/PhysRevLett.110.118901. ISSN 0031-9007. PMID 25166585. S2CID 41459498.CS1 maint: ref=harv (link)
Bruno, Patrick (2013b). "Comment on "Space-Time Crystals of Trapped Ions"". Physical Review Letters. 111 (2): 029301. arXiv:1211.4792v1. Bibcode:2013PhRvL.111b9301B. doi:10.1103/PhysRevLett.111.029301. ISSN 0031-9007. PMID 23889455. S2CID 1502258.CS1 maint: ref=harv (link)
Campisi, Michele; Hänggi, Peter; Talkner, Peter (2011). "Colloquium: Quantum fluctuation relations: Foundations and applications". Reviews of Modern Physics. 83 (3): 771–791. arXiv:1012.2268v5. Bibcode:2011RvMP...83..771C. CiteSeerX doi:10.1103/RevModPhys.83.771. ISSN 0034-6861. S2CID 119200058.
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. (2017). "Observation of discrete time-crystalline order in a disordered dipolar many-body system". Nature. 543 (7644): 221–225. arXiv:1610.08057v1. Bibcode:2017Natur.543..221C. doi:10.1038/nature21426. ISSN 0028-0836. PMC 5349499. PMID 28277511.
Chernodub, M. N. (2012). "Permanently rotating devices: extracting rotation from quantum vacuum fluctuations?". arXiv:1203.6588v1. Bibcode:2012arXiv1203.6588C. Cite journal requires |journal= (help)CS1 maint: ref=harv (link)
Chernodub, M. N. (2013a). "Zero-point fluctuations in rotation: Perpetuum mobile of the fourth kind without energy transfer". Nuovo Cimento C. 5 (36): 53–63. arXiv:1302.0462v1. Bibcode:2013arXiv1302.0462C. doi:10.1393/ncc/i2013-11523-5. S2CID 118617367.CS1 maint: ref=harv (link)
Chernodub, M. N. (2013b). "Rotating Casimir systems: Magnetic-field-enhanced perpetual motion, possible realization in doped nanotubes, and laws of thermodynamics". Physical Review D. 87 (2): 025021. arXiv:1207.3052v2. Bibcode:2013PhRvD..87b5021C. doi:10.1103/PhysRevD.87.025021. ISSN 1550-7998. S2CID 56430144.CS1 maint: ref=harv (link)
Copeland, Edmund J.; Sami, M.; Tsujikawa, Shinji (2006). "Dynamics of dark energy". International Journal of Modern Physics D. 15 (11): 1753–1935. arXiv:hep-th/0603057. Bibcode:2006IJMPD..15.1753C. doi:10.1142/S021827180600942X. ISSN 0218-2718. S2CID 119434524.
Dillenschneider, R.; Lutz, E. (2009). "Energetics of quantum correlations". EPL. 88 (5): 50003. arXiv:0803.4067. Bibcode:2009EL.....8850003D. doi:10.1209/0295-5075/88/50003. ISSN 0295-5075. S2CID 119262651.CS1 maint: ref=harv (link)
Else, Dominic V.; Bauer, Bela; Nayak, Chetan (2016). "Floquet Time Crystals". Physical Review Letters. 117 (9): 090402. arXiv:1603.08001v4. Bibcode:2016PhRvL.117i0402E. doi:10.1103/PhysRevLett.117.090402. ISSN 0031-9007. PMID 27610834. S2CID 1652633.
Esposito, Massimiliano; Harbola, Upendra; Mukamel, Shaul (2009). "Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems". Reviews of Modern Physics. 81 (4): 1665–1702. arXiv:0811.3717v2. Bibcode:2009RvMP...81.1665E. doi:10.1103/RevModPhys.81.1665. ISSN 0034-6861. S2CID 56003679.
Grifoni, Milena; Hänggi, Peter (1998). "Driven quantum tunneling" (PDF). Physics Reports. 304 (5–6): 229–354. Bibcode:1998PhR...304..229G. CiteSeerX doi:10.1016/S0370-1573(98)00022-2. ISSN 0370-1573. S2CID 120738031.CS1 maint: ref=harv (link)
Guo, Lingzhen; Marthaler, Michael; Schön, Gerd (2013). "Phase Space Crystals: A New Way to Create a Quasienergy Band Structure". Physical Review Letters. 111 (20): 205303. arXiv:1305.1800v3. Bibcode:2013PhRvL.111t5303G. doi:10.1103/PhysRevLett.111.205303. ISSN 0031-9007. PMID 24289695. S2CID 9337383.
Hasan, M. Z.; Kane, C. L. (2010). "Colloquium: Topological insulators". Reviews of Modern Physics. 82 (4): 3045–3067. arXiv:1002.3895v2. Bibcode:2010RvMP...82.3045H. doi:10.1103/RevModPhys.82.3045. ISSN 0034-6861. S2CID 16066223.CS1 maint: ref=harv (link)
Horodecki, Ryszard; Horodecki, Paweł; Horodecki, Michał; Horodecki, Karol (2009). "Quantum entanglement". Reviews of Modern Physics. 81 (2): 865–942. arXiv:quant-ph/0702225v2. Bibcode:2009RvMP...81..865H. doi:10.1103/RevModPhys.81.865. ISSN 0034-6861. S2CID 59577352.
Jaffe, R. L. (2005). "Casimir effect and the quantum vacuum". Physical Review D. 72 (2): 021301. arXiv:hep-th/0503158. Bibcode:2005PhRvD..72b1301J. doi:10.1103/PhysRevD.72.021301. S2CID 13171179.CS1 maint: ref=harv (link)
Jarzynski, Christopher (2011). "Equalities and Inequalities: Irreversibility and the Second Law of Thermodynamics at the Nanoscale" (PDF). Annual Review of Condensed Matter Physics. 2 (1): 329–351. Bibcode:2011ARCMP...2..329J. doi:10.1146/annurev-conmatphys-062910-140506. ISSN 1947-5454.CS1 maint: ref=harv (link)
Jetzer, Philippe; Straumann, Norbert (2006). "Josephson junctions and dark energy". Physics Letters B. 639 (2): 57–58. arXiv:astro-ph/0604522. Bibcode:2006PhLB..639...57J. CiteSeerX doi:10.1016/j.physletb.2006.06.020. ISSN 0370-2693. S2CID 16120742.CS1 maint: ref=harv (link)
Khemani, Vedika; Lazarides, Achilleas; Moessner, Roderich; Sondhi, S. L. (2504). "Phase Structure of Driven Quantum Systems". Physical Review Letters. 116 (25): 250401. arXiv:1508.03344v3. Bibcode:2016PhRvL.116y0401K. doi:10.1103/PhysRevLett.116.250401. ISSN 0031-9007. PMID 27391704. S2CID 883197. Check date values in: |year= (help)
Lees, J. P. (2012). "Observation of Time-Reversal Violation in the B0 Meson System". Physical Review Letters. 109 (21): 211801. arXiv:1207.5832v4. Bibcode:2012PhRvL.109u1801L. doi:10.1103/PhysRevLett.109.211801. ISSN 0031-9007. PMID 23215586. S2CID 3554721.
Li, Tongcang; Gong, Zhe-Xuan; Yin, Zhang-Qi; Quan, H. T.; Yin, Xiaobo; Zhang, Peng; Duan, L.-M.; Zhang, Xiang (2012a). "Space-Time Crystals of Trapped Ions". Physical Review Letters. 109 (16): 163001. arXiv:1206.4772v2. Bibcode:2012PhRvL.109p3001L. doi:10.1103/PhysRevLett.109.163001. ISSN 0031-9007. PMID 23215073. S2CID 8198228.
Li, Tongcang; Gong, Zhe-Xuan; Yin, Zhang-Qi; Quan, H. T.; Yin, Xiaobo; Zhang, Peng; Duan, L.-M.; Zhang, Xiang (2012b). "Reply to Comment on "Space-Time Crystals of Trapped Ions"". arXiv:1212.6959v2. Bibcode:2012arXiv1212.6959L. Cite journal requires |journal= (help)
Lindner, Netanel H.; Refael, Gil; Galitski, Victor (2011). "Floquet topological insulator in semiconductor quantum wells". Nature Physics. 7 (6): 490–495. arXiv:1008.1792v2. Bibcode:2011NatPh...7..490L. doi:10.1038/nphys1926. ISSN 1745-2473. S2CID 26754031.
Nadj-Perge, S.; Drozdov, I. K.; Li, J.; Chen, H.; Jeon, S.; Seo, J.; MacDonald, A. H.; Bernevig, B. A.; Yazdani, A. (2014). "Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor". Science. 346 (6209): 602–607. arXiv:1410.0682v1. Bibcode:2014Sci...346..602N. doi:10.1126/science.1259327. ISSN 0036-8075. PMID 25278507. S2CID 206561257.
Nozières, Philippe (2013). "Time crystals: Can diamagnetic currents drive a charge density wave into rotation?". EPL. 103 (5): 57008. arXiv:1306.6229v1. Bibcode:2013EL....10357008N. doi:10.1209/0295-5075/103/57008. ISSN 0295-5075. S2CID 118662499.CS1 maint: ref=harv (link)
Sacha, Krzysztof (2015). "Modeling spontaneous breaking of time-translation symmetry". Physical Review A. 91 (3): 033617. arXiv:1410.3638v3. Bibcode:2015PhRvA..91c3617S. doi:10.1103/PhysRevA.91.033617. ISSN 1050-2947. S2CID 118627872.CS1 maint: ref=harv (link)
Schwinger, Julian (1975). "Casimir effect in source theory". Letters in Mathematical Physics. 1 (1): 43–47. Bibcode:1975LMaPh...1...43S. doi:10.1007/BF00405585. S2CID 126297065.CS1 maint: ref=harv (link)
Schwinger, Julian; DeRaad, Lester L.; Milton, Kimball A. (1978). "Casimir effect in dielectrics". Annals of Physics. 115 (1): 1–23. Bibcode:1978AnPhy.115....1S. doi:10.1016/0003-4916(78)90172-0.
Scully, Marlan O. (2001). "Extracting Work from a Single Thermal Bath via Quantum Negentropy". Physical Review Letters. 87 (22): 220601. Bibcode:2001PhRvL..87v0601S. doi:10.1103/PhysRevLett.87.220601. ISSN 0031-9007. PMID 11736390.CS1 maint: ref=harv (link)
Scully, Marlan O.; Zubairy, M. Suhail; Agarwal, Girish S.; Walther, Herbert. (2003). "Extracting Work from a Single Heat Bath via Vanishing Quantum Coherence". Science. 299 (5608): 862–864. Bibcode:2003Sci...299..862S. doi:10.1126/science.1078955. ISSN 0036-8075. PMID 12511655. S2CID 120884236.
Seifert, Udo (2012). "Stochastic thermodynamics, fluctuation theorems and molecular machines". Reports on Progress in Physics. 75 (12): 126001. arXiv:1205.4176v1. Bibcode:2012RPPh...75l6001S. doi:10.1088/0034-4885/75/12/126001. ISSN 0034-4885. PMID 23168354. S2CID 782930.CS1 maint: ref=harv (link)
Senitzky, I. R. (1960). "Dissipation in Quantum Mechanics. The Harmonic Oscillator". Physical Review. 119 (2): 670–679. Bibcode:1960PhRv..119..670S. doi:10.1103/PhysRev.119.670. ISSN 0031-899X.CS1 maint: ref=harv (link)
Shapere, Alfred; Wilczek, Frank (2012). "Classical Time Crystals". Physical Review Letters. 109 (16): 160402. arXiv:1202.2537v2. Bibcode:2012PhRvL.109p0402S. doi:10.1103/PhysRevLett.109.160402. ISSN 0031-9007. PMID 23215057. S2CID 4506464.CS1 maint: ref=harv (link)
Shirley, Jon H. (1965). "Solution of the Schrödinger Equation with a Hamiltonian Periodic in Time". Physical Review. 138 (4B): B979–B987. Bibcode:1965PhRv..138..979S. doi:10.1103/PhysRev.138.B979. ISSN 0031-899X.CS1 maint: ref=harv (link)
Smith, J.; Lee, A.; Richerme, P.; Neyenhuis, B.; Hess, P. W.; Hauke, P.; Heyl, M.; Huse, D. A.; Monroe, C. (2016). "Many-body localization in a quantum simulator with programmable random disorder". Nature Physics. 12 (10): 907–911. arXiv:1508.07026v1. Bibcode:2016NatPh..12..907S. doi:10.1038/nphys3783. ISSN 1745-2473. S2CID 53408060.
Maruyama, Koji; Nori, Franco; Vedral, Vlatko (2009). "Colloquium: The physics of Maxwell's demon and information". Reviews of Modern Physics. 81 (1): 1–23. arXiv:0707.3400. Bibcode:2009RvMP...81....1M. doi:10.1103/RevModPhys.81.1. ISSN 0034-6861. S2CID 18436180.
Mendonça, J. T.; Dodonov, V. V. (2014). "Time Crystals in Ultracold Matter". Journal of Russian Laser Research. 35 (1): 93–100. doi:10.1007/s10946-014-9404-9. ISSN 1071-2836. S2CID 122631523.CS1 maint: ref=harv (link)
Modi, Kavan; Brodutch, Aharon; Cable, Hugo; Paterek, Tomasz; Vedral, Vlatko (2012). "The classical-quantum boundary for correlations: Discord and related measures". Reviews of Modern Physics. 84 (4): 1655–1707. arXiv:1112.6238. Bibcode:2012RvMP...84.1655M. doi:10.1103/RevModPhys.84.1655. ISSN 0034-6861. S2CID 119698121.
Ray, M. W.; Ruokokoski, E.; Kandel, S.; Möttönen, M.; Hall, D. S. (2014). "Observation of Dirac monopoles in a synthetic magnetic field". Nature. 505 (7485): 657–660. arXiv:1408.3133v1. Bibcode:2014Natur.505..657R. doi:10.1038/nature12954. ISSN 0028-0836. PMID 24476889. S2CID 918213.
Ray, M. W.; Ruokokoski, E.; Tiurev, K.; Mottonen, M.; Hall, D. S. (2015). "Observation of isolated monopoles in a quantum field" (PDF). Science. 348 (6234): 544–547. Bibcode:2015Sci...348..544R. doi:10.1126/science.1258289. ISSN 0036-8075. PMID 25931553. S2CID 43491454.
Reimann, Peter; Grifoni, Milena; Hänggi, Peter (1997). "Quantum Ratchets" (PDF). Physical Review Letters. 79 (1): 10–13. Bibcode:1997PhRvL..79...10R. doi:10.1103/PhysRevLett.79.10. ISSN 0031-9007. S2CID 14640168.
Robicheaux, F.; Niffenegger, K. (2015). "Quantum simulations of a freely rotating ring of ultracold and identical bosonic ions". Physical Review A. 91 (6): 063618. Bibcode:2015PhRvA.91063618R. doi:10.1103/PhysRevA.91.063618. ISSN 2469-9926.CS1 maint: ref=harv (link)
Roßnagel, J.; Abah, O.; Schmidt-Kaler, F.; Singer, K.; Lutz, E. (2014). "Nanoscale Heat Engine Beyond the Carnot Limit". Physical Review Letters. 112 (3): 030602. arXiv:1308.5935. Bibcode:2014PhRvL.112c0602R. doi:10.1103/PhysRevLett.112.030602. ISSN 0031-9007. PMID 24484127. S2CID 1826585.
Roßnagell, J.; Dawkins, S. T.; Tolazzi, K. N.; Abah, O.; Lutz, E.; Schmidt-Kaler, F.; Singer, K. (2016). "A single-atom heat engine". Science. 352 (6283): 325–329. arXiv:1510.03681. Bibcode:2016Sci...352..325R. doi:10.1126/science.aad6320. ISSN 0036-8075. PMID 27081067. S2CID 44229532.
Tatara, Gen; Kikuchi, Makoto; Yukawa, Satoshi; Matsukawa, Hiroshi (1998). "Dissipation Enhanced Asymmetric Transport in Quantum Ratchets". Journal of the Physical Society of Japan. 67 (4): 1090–1093. arXiv:cond-mat/9711045. Bibcode:1998JPSJ...67.1090T. doi:10.1143/JPSJ.67.1090. ISSN 0031-9015. S2CID 11253455.
Volovik, G. E. (2013). "On the broken time translation symmetry in macroscopic systems: Precessing states and off-diagonal long-range order". JETP Letters. 98 (8): 491–495. arXiv:1309.1845v2. Bibcode:2013JETPL..98..491V. doi:10.1134/S0021364013210133. ISSN 0021-3640. S2CID 119100114.CS1 maint: ref=harv (link)
von Keyserlingk, C. W.; Khemani, Vedika; Sondhi, S. L. (2016). "Absolute stability and spatiotemporal long-range order in Floquet systems". Physical Review B. 94 (8): 085112. arXiv:1605.00639v3. Bibcode:2016PhRvB..94h5112V. doi:10.1103/PhysRevB.94.085112. ISSN 2469-9950. S2CID 118699328.
Wang, Y. H.; Steinberg, H.; Jarillo-Herrero, P.; Gedik, N. (2013). "Observation of Floquet-Bloch States on the Surface of a Topological Insulator". Science. 342 (6157): 453–457. arXiv:1310.7563v1. Bibcode:2013Sci...342..453W. doi:10.1126/science.1239834. hdl:1721.1/88434. ISSN 0036-8075. PMID 24159040. S2CID 29121373.
Watanabe, Haruki; Oshikawa, Masaki (2015). "Absence of Quantum Time Crystals". Physical Review Letters. 114 (25): 251603. arXiv:1410.2143v3. Bibcode:2015PhRvL.114y1603W. doi:10.1103/PhysRevLett.114.251603. ISSN 0031-9007. PMID 26197119. S2CID 312538.CS1 maint: ref=harv (link)
Wilczek, Frank (2012). "Quantum Time Crystals". Physical Review Letters. 109 (16): 160401. arXiv:1202.2539v2. Bibcode:2012PhRvL.109p0401W. doi:10.1103/PhysRevLett.109.160401. ISSN 0031-9007. PMID 23215056. S2CID 1312256.CS1 maint: ref=harv (link)
Wilczek, Frank (2013a). "Wilczek Reply" (PDF). Physical Review Letters. 110 (11): 118902. Bibcode:2013PhRvL.110k8902W. doi:10.1103/PhysRevLett.110.118902. ISSN 0031-9007. PMID 25166586.CS1 maint: ref=harv (link)
Wilczek, Frank (2013). "Superfluidity and Space-Time Translation Symmetry Breaking". Physical Review Letters. 111 (25): 250402. arXiv:1308.5949v1. Bibcode:2013PhRvL.111y0402W. doi:10.1103/PhysRevLett.111.250402. ISSN 0031-9007. PMID 24483732. S2CID 7537145.CS1 maint: ref=harv (link)
Willett, R. L.; Nayak, C.; Shtengel, K.; Pfeiffer, L. N.; West, K. W. (2013). "Magnetic-Field-Tuned Aharonov-Bohm Oscillations and Evidence for Non-Abelian Anyons atν=5/2". Physical Review Letters. 111 (18): 186401. arXiv:1301.2639v1. Bibcode:2013PhRvL.111r6401W. doi:10.1103/PhysRevLett.111.186401. ISSN 0031-9007. PMID 24237543. S2CID 22780228.
Yao, N. Y.; Potter, A. C.; Potirniche, I.-D.; Vishwanath, A. (2017). "Discrete Time Crystals: Rigidity, Criticality, and Realizations". Physical Review Letters. 118 (3): 030401. arXiv:1608.02589v2. Bibcode:2017PhRvL.118c0401Y. doi:10.1103/PhysRevLett.118.030401. ISSN 0031-9007. PMID 28157355. S2CID 206284432.
Yoshii, Ryosuke; Takada, Satoshi; Tsuchiya, Shunji; Marmorini, Giacomo; Hayakawa, Hisao; Nitta, Muneto (2015). "Fulde-Ferrell-Larkin-Ovchinnikov states in a superconducting ring with magnetic fields: Phase diagram and the first-order phase transitions". Physical Review B. 92 (22): 224512. arXiv:1404.3519v2. Bibcode:2015PhRvB..92v4512Y. doi:10.1103/PhysRevB.92.224512. ISSN 1098-0121. S2CID 118348062.
Yukawa, Satoshi; Kikuchi, Macoto; Tatara, Gen; Matsukawa, Hiroshi (1997). "Quantum Ratchets". Journal of the Physical Society of Japan. 66 (10): 2953–2956. arXiv:cond-mat/9706222. Bibcode:1997JPSJ...66.2953Y. doi:10.1143/JPSJ.66.2953. ISSN 0031-9015. S2CID 16578514.
Yukawa, Satoshi (2000). "A Quantum Analogue of the Jarzynski Equality". Journal of the Physical Society of Japan. 69 (8): 2367–2370. arXiv:cond-mat/0007456. Bibcode:2000JPSJ...69.2367Y. doi:10.1143/JPSJ.69.2367. ISSN 0031-9015. S2CID 119097589.CS1 maint: ref=harv (link)
Zel'Dovich, Y. B. (1967). "The quasienergy of a quantum-mechanical system subjected to a periodic action" (PDF). Soviet Physics JETP. 24 (5): 1006–1008. Bibcode:1967JETP...24.1006Z.CS1 maint: ref=harv (link)
Zhang, J.; Hess, P. 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". Nature. 543 (7644): 217–220. arXiv:1609.08684v1. Bibcode:2017Natur.543..217Z. doi:10.1038/nature21413. ISSN 0028-0836. PMID 28277505. S2CID 4450646.


Bordag, M.; Mohideen, U.; Mostepanenko, V.M. (2001). "New developments in the Casimir effect". Physics Reports. 353 (1–3): 1–205. arXiv:quant-ph/0106045. Bibcode:2001PhR...353....1B. doi:10.1016/S0370-1573(01)00015-1. ISSN 0370-1573. S2CID 119352552.CS1 maint: ref=harv (link)
Bordag, M.; Mohideen, U.; Mostepanenko, V.M.; Klimchitskaya, G. L (28 May 2009). Advances in the Casimir Effect. Oxford: Oxford University Press. ISBN 978-0-19-157988-2.CS1 maint: ref=harv (link)
Cao, Tian Yu (25 March 2004). Conceptual Foundations of Quantum Field Theory. Cambridge: Cambridge University Press. ISBN 978-0-521-60272-3.CS1 maint: ref=harv (link)
Enz, Charles P. (1974). "Is the Zero-Point Energy Real?". In Enz, C. P.; Mehra, J. (eds.). Physical Reality and Mathematical Description. Dordrecht: D. Reidel Publishing Company. pp. 124–132. doi:10.1007/978-94-010-2274-3_8. ISBN 978-94-010-2274-3.CS1 maint: ref=harv (link)
Greiner, Walter; Müller, B.; Rafelski, J. (2012). Quantum Electrodynamics of Strong Fields: With an Introduction into Modern Relativistic Quantum Mechanics. Springer. doi:10.1007/978-3-642-82272-8. ISBN 978-3-642-82274-2.
Lee, T. D. (15 August 1981). Particle Physics. CRC Press. ISBN 978-3-7186-0033-5.CS1 maint: ref=harv (link)
Feng, Duan; Jin, Guojun (2005). Introduction to Condensed Matter Physics. singapore: World Scientific. ISBN 978-981-238-711-0.CS1 maint: ref=harv (link)
Milonni, Peter W. (1994). The Quantum Vacuum: An Introduction to Quantum Electrodynamics. London: Academic Press. ISBN 978-0-124-98080-8.CS1 maint: ref=harv (link)
Pade, Jochen (2014). Quantum Mechanics for Pedestrians 2: Applications and Extensions. Undergraduate Lecture Notes in Physics. Dordrecht: Springer. doi:10.1007/978-3-319-00813-4. ISBN 978-3-319-00813-4. ISSN 2192-4791.CS1 maint: ref=harv (link)
Schwinger, Julian (1998a). Particles, Sources, And Fields, Volume 1: v. 1 (Advanced Books Classics). Perseus. ISBN 978-0-738-20053-8.CS1 maint: ref=harv (link)
Schwinger, Julian (1998b). Particles, Sources, And Fields, Volume 2: v. 2 (Advanced Books Classics). Perseus. ISBN 978-0-738-20054-5.CS1 maint: ref=harv (link)
Schwinger, Julian (1998c). Particles, Sources, And Fields, Volume 3: v. 3 (Advanced Books Classics). Perseus. ISBN 978-0-738-20055-2.CS1 maint: ref=harv (link)
Sólyom, Jenö (19 September 2007). Fundamentals of the Physics of Solids: Volume 1: Structure and Dynamics. Springer. ISBN 978-3-540-72600-5.CS1 maint: ref=harv (link)
Wilczek, Frank (16 July 2015). A Beautiful Question: Finding Nature's Deep Design. Penguin Books Limited. ISBN 978-1-84614-702-9.CS1 maint: ref=harv (link)


Aalto University (30 April 2015). "Physicists discover quantum-mechanical monopoles". phys.org. Science X. Archived from the original on 30 April 2015.CS1 maint: ref=harv (link)
Aitchison, Ian (19 November 1981). "Observing the Unobservable". New Scientist. 92 (1280): 540–541. ISSN 0262-4079.CS1 maint: ref=harv (link)
Amherst College (29 January 2014). "Physicists create synthetic magnetic monopole predicted more than 80 years ago". phys.org. Science X. Archived from the original on 29 January 2014.CS1 maint: ref=harv (link)
Aron, Jacob (6 July 2012). "Computer that could outlive the universe a step closer". newscientist.com. New Scientist. Archived from the original on 2 February 2017.CS1 maint: ref=harv (link)
Ball, Philip (8 January 2016). "Focus: New Crystal Type is Always in Motion". physics.aps.org. APS Physics. Archived from the original on 3 February 2017.CS1 maint: ref=harv (link)
Ball, Philip (8 July 2004). "Scepticism greets pitch to detect dark energy in the lab". Nature. 430 (6996): 126. Bibcode:2004Natur.430..126B. doi:10.1038/430126b. ISSN 0028-0836. PMID 15241374.CS1 maint: ref=harv (link)
Cartlidge, Edwin (21 October 2015). "Scientists build heat engine from a single atom". sciencemag.org. Science Magazine. Archived from the original on 1 February 2017.CS1 maint: ref=harv (link)
Chandler, David (24 October 2014). "Topological insulators: Persuading light to mix it up with matter". phys.org. Science X. Archived from the original on 8 February 2017.CS1 maint: ref=harv (link)
Coleman, Piers (9 January 2013). "Quantum physics: Time crystals". Nature. 493 (7431): 166–167. Bibcode:2013Natur.493..166C. doi:10.1038/493166a. ISSN 0028-0836. PMID 23302852. S2CID 205075903.CS1 maint: ref=harv (link)
Cowen, Ron (27 February 2012). ""Time Crystals" Could Be a Legitimate Form of Perpetual Motion". scientificamerican.com. Scientific American. Archived from the original on 2 February 2017.CS1 maint: ref=harv (link)
Daghofer, Maria (29 April 2013). "Viewpoint: Toward Fractional Quantum Hall Physics with Cold Atoms". physics.aps.org. APS Physics. Archived from the original on 7 February 2017.CS1 maint: ref=harv (link)
Gibney, Elizabeth (2017). "The quest to crystallize time". Nature. 543 (7644): 164–166. Bibcode:2017Natur.543..164G. doi:10.1038/543164a. ISSN 0028-0836. PMID 28277535. S2CID 4460265.CS1 maint: ref=harv (link)
Grossman, Lisa (18 January 2012). "Death-defying time crystal could outlast the universe". newscientist.com. New Scientist. Archived from the original on 2 February 2017.
Hackett, Jennifer (22 February 2016). "Curious Crystal Dances for Its Symmetry". scientificamerican.com. Scientific American. Archived from the original on 3 February 2017.CS1 maint: ref=harv (link)
Hewitt, John (3 May 2013). "Creating time crystals with a rotating ion ring". phys.org. Science X. Archived from the original on 4 July 2013.
Johnston, Hamish (18 January 2016). "'Choreographic crystals' have all the right moves". physicsworld.com. Institute of Physics. Archived from the original on 3 February 2017.CS1 maint: ref=harv (link)
Johannes Gutenberg Universitaet Mainz (3 February 2014). "Prototype of single ion heat engine created". sciencedaily.com. ScienceDaily. Archived from the original on 11 February 2014.
Joint Quantum Institute (22 March 2011). "Floquet Topological Insulators". jqi.umd.edu. Joint Quantum Institute.CS1 maint: ref=harv (link)
Morgan, James (30 January 2014). "Elusive magnetic 'monopole' seen in quantum system". bbc.co.uk. BBC. Archived from the original on 30 January 2014.CS1 maint: ref=harv (link)
Moskowitz, Clara (2 October 2014). "New Particle Is Both Matter and Antimatter". scientificamerican.com. Scientific American. Archived from the original on 9 October 2014.CS1 maint: ref=harv (link)
Ouellette, Jennifer (31 January 2017). "World's first time crystals cooked up using new recipe". newscientist.com. New Scientist. Archived from the original on 1 February 2017.CS1 maint: ref=harv (link)
Pilkington, Mark (17 July 2003). "Zero point energy". theguardian.com. The Guardian. Archived from the original on 7 February 2017.CS1 maint: ref=harv (link)
Powell, Devin (2013). "Can matter cycle through shapes eternally?". Nature. doi:10.1038/nature.2013.13657. ISSN 1476-4687. S2CID 181223762. Archived from the original on 3 February 2017.CS1 maint: ref=harv (link)
Rao, Achintya (21 November 2012). "BaBar makes first direct measurement of time-reversal violation". physicsworld.com. Institute of Physics. Archived from the original on 24 March 2015.CS1 maint: ref=harv (link)
Richerme, Phil (18 January 2017). "Viewpoint: How to Create a Time Crystal". physics.aps.org. APS Physics. Archived from the original on 2 February 2017.
Thomas, Jessica (15 March 2013). "Notes from the Editors: The Aftermath of a Controversial Idea". physics.aps.org. APS Physics. Archived from the original on 2 February 2017.CS1 maint: ref=harv (link)
Qi, Xiao-Liang; Zhang, Shou-Cheng (2010). "The quantum spin Hall effect and topological insulators" (PDF). Physics Today. 63 (1): 33–38. arXiv:1001.1602. Bibcode:2010PhT....63a..33Q. doi:10.1063/1.3293411. ISSN 0031-9228. S2CID 35957977.CS1 maint: ref=harv (link)
University of California, Berkeley (26 January 2017). "Physicists unveil new form of matter—time crystals". phys.org. Science X. Archived from the original on 28 January 2017.
Weiner, Sophie (28 January 2017). "Scientists Create A New Kind Of Matter: Time Crystals". popularmechanics.com. Popular mechanics. Archived from the original on 3 February 2017.CS1 maint: ref=harv (link)
Wolchover, Natalie (25 April 2013). "Perpetual Motion Test Could Amend Theory of Time". quantamagazine.org. Simons Foundation. Archived from the original on 2 February 2017.CS1 maint: ref=harv (link)
Wolchover, Natalie (15 May 2014). "Forging a Qubit to Rule Them All". quantamagazine.org. Simmons Foundation. Archived from the original on 15 March 2016.CS1 maint: ref=harv (link)
Wood, Charlie (31 January 2017). "Time crystals realize new order of space-time". csmonitor.com. Christian Science Monitor. Archived from the original on 2 February 2017.CS1 maint: ref=harv (link)
Yirka, Bob (9 July 2012). "Physics team proposes a way to create an actual space-time crystal". phys.org. Science X. Archived from the original on 15 April 2013.
Zakrzewski, Jakub (15 October 2012). "Viewpoint: Crystals of Time". physics.aps.org. APS Physics. Archived from the original on 2 February 2017.
Zeller, Michael (19 November 2012). "Viewpoint: Particle Decays Point to an Arrow of Time". physics.aps.org. APS Physics. Archived from the original on 4 February 2017.CS1 maint: ref=harv (link)
Zyga, Lisa (20 February 2012). "Time crystals could behave almost like perpetual motion machines". phys.org. Science X. Archived from the original on 3 February 2017.
Zyga, Lisa (22 August 2013). "Physicist proves impossibility of quantum time crystals". phys.org. Space X. Archived from the original on 3 February 2017.
Zyga, Lisa (27 January 2014). "Nanoscale heat engine exceeds standard efficiency limit". phys.org. Science X. Archived from the original on 4 April 2015.CS1 maint: ref=harv (link)
Zyga, Lisa (9 July 2015). "Physicists propose new definition of time crystals—then prove such things don't exist". phys.org. Science X. Archived from the original on 9 July 2015.
Zyga, Lisa (9 September 2016). "Time crystals might exist after all (Update)". phys.org. Science X. Archived from the original on 11 September 2016.

External links[edit]