Time crystal

File:Time crystal phase transition.png

Phase diagram of a discrete time crystal as function of Ising interaction strength and spin-echo pulse imperfections. (Yao et al. 2017)

A time crystal or space-time crystal is an open system in non-equilibrium with its environment that exhibits time translation symmetry breaking (TTSB). It is impossible for a time crystal to be in equilibrium with its environment. The idea of a time crystal was first put forward by Nobel laureate and professor at MIT Frank Wilczek in 2012. Time crystals extend the ordinary three-dimensional symmetry seen in crystals to include the fourth dimension of time; a time crystal spontaneously breaks the symmetry of time translation. The crystal's pattern repeats not in space, but in time, which remarkably allows for the crystal to be in perpetual motion. Time crystals are closely related to the concepts of zero-point energy and the dynamical Casimir effect

In 2016 Norman Yao and his colleagues from the University of California, Berkeley put forward a concrete proposal that would allow time crystals to be created in a laboratory environment. Yao's blueprint was then used by two teams, a group led by Christopher Monroe at the University of Maryland and a group led by Mikhail Lukin at Harvard university, who were both able to successfully create a time crystal. Both experiments have been accepted for publication in peer reviewed journals.

Time crystals are thought to exhibit topological order, an emergent phenomena, in which nonlocal correlations encoded in the whole wave-function of a system allow for fault tolerance against perturbations, thus allowing quantum states to stabilize against decoherence effects that usually limit their useful lifetime. Preventing decoherence has a wide range of implications: The efficiency of some information theory and quantum thermodynamic tasks can be greatly enhanced when using quantum correlated states. It is also that time crystals could also give deeper understanding of the theory of time.

History

Nobel laureate Frank Wilczek at University of Paris-Saclay

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

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.

In response to Wilczek and Zhang, Patrick Bruno, a theorist at the European Synchrotron Radiation Facility in Grenoble, France, published several papers claiming to show that time crystals were impossible.

Subsequent work developed more precise definitions of time translation symmetry breaking (TTSB) which ultimately led to a proof that quantum time crystals in equilibrium are not possible.

Several realizations of time crystals, which avoid the equilibrium no-go arguments, were proposed.

The no-go theorem leaves the door open to TTSB in an non-equilibrium system, and pioneering work has demonstrated that quantum systems subject to periodic driving can indeed exhibit discrete TTSB.

Jakub Zakrzewski and his team at Jagiellonian University in Krakow, Poland have attempted to predict the behaviour of time crystals with numerical simulations.

Using Wilczek's idea, Norman Yao and his colleagues from the University of California, Berkeley studied a different model that would allow the existence of time crystals.

Yao's blueprint was then used by two teams: a group led by Mikhail Lukin at Harvard university and a group led by Christopher Monroe at University of Maryland both of whom were able to independantly create a time crystal successfully.

Time translation symmetry

Symmetry

Main article: Symmetry (physics)

Symmetries are of prime importance in physics and are closely related to the hypothesis that certain physical quantities are only relative and unobservable. Symmetries apply to the equations that govern the physical laws rather then the initial conditions or ts themselves and state that the laws remain unchanged under a transformation. If a symmetry is preserved under a transformation it is said to be invariant. Symmetries in nature lead directly to conservation laws, something which is precisely formulated by the Noether theorem.

Symmetries in condensed matter physics

Symmetry	Transformation	Unobservable	Conservation law
Space-translation	$\mathbf {r} \rightarrow \mathbf {r} +\Delta$	absolute position in space	momentum
Time-translation	$t\rightarrow t+\tau$	absolute time	energy
Rotation	$\mathbf {r} \rightarrow \mathbf {r} '$	absolute direction in space	angular momentum
Space inversion	$\mathbf {r} \rightarrow -\mathbf {r}$	absolute left or right	parity
Time-reversal	$t\rightarrow -t$	absolute sign of time	Krammers degeneracy
Sign reversion of charge	$e\rightarrow -e$	absolute sign of electric charge	charge conjugation
Particle substitution		distinguishability of identical particles	Bose or Fermi statistics
Gauge transformation	$\psi \rightarrow e^{iN\theta }\psi$	relative phase between different normal states	particle number

The basic idea of time-translation symmetry is that a translation in time has no effect on physical laws, that the laws of nature that apply today were the same in the past and will be the same in the future. For example if we measure the energy levels of hydrogen today, tomorrow or in ten years it makes no difference - we will always observe the same energy. Moreover, when we look at distant stars we are really looking back in time, so the fact that the energy levels of hydrogen are the same in all stars we ever looked at tells us something about the symmetry of both space and time. Invariance of time-translation implies absolute time is unobservable and a direct consequence is the conservation of energy. A violation in time-translation symmetry means that under certain conditions or select cases energy is not a conserved quantity and that laws of nature themselves are variable with time.

Broken symmetry

Main article: Symmetry breaking

For a long time physicists believed that symmetries in the laws nature were absolute, but deviations do occur. In 1957 scientists confirmed experimentally a broken symmetry in space inversion (a right-left asymmetry) in weak interactions and that therefore parity was not a conserved quantity (known as a P violation). This lead to the 1957 Nobel Prize in physics being awarded to Tsung-Dao Lee and Chen-Ning Yang who had put forward the original idea in 1956. It was also established in 1957 that there was not only a right-left asymmetries but also asymmetries between the positive and negative signs of electric charge (a charge conjugation or C violation). At around the same time questions of possible asymmetries under time-reversal T and CP violations (the product of C and P transformations) were also raised, though actual experimental confirmation did not come until quite a few years later.

A consequence of these asymmetries in nature is that it is therefore possible to determine an absolute left or right, absolute charge or absolute direction of time in the universe; such terms are not merely relative or a subjective naming convention. If two advanced civilisations were separated on different sides of the universe with no possiblity of physical contact but could somehow send signals to communicate with each other, they would be able to convey the results of experiments that will lead them to agree on the definitions of what direction is left and right, whether they are made of particle or antiparticles and whether time was flowing in the same direction.

Broken symmetry in normal crystals

Main articles: Crystal symmetry and Spontaneous symmetry breaking

Figure 2. Normal process (N-process) and Umklapp process (U-process). While the N-process conserves total phonon momentum, the U-process changes phonon momentum.

Different states of matter can be classified by the symmetries they spontaneously break. In a magnet, for example, spins are limited to a few possible orientations along a common direction chosen spontaneously from ones of less orientation but greater freedom and symmetry; thus a ferromagnet breaks symmetry as this process occurs. Normal crystals exhibit broken translation symmetry. For example, a gas is said to have translational symmetry as its atoms can move freely to occupy any point in a given area. A crystal by contrast does not have the same degree of symmetry; only certain spacial points are permitted and there is a requirement that the atoms have a particular structure or order. If a gas cools to form a crystal the symmetry is said to be continuously broken as the crystal gains order.

Since crystals are not invariant under arbitrary translations, strictly speaking, momentum is not conserved. No strict conservation law can be applied but a discrete translation symmetry may sometimes be achieved. The crystal momentum is called quasimomentum which determines the crystals Bloch state and is the cause of Umklapp processes. This technical violation of the conservation of momentum is important in establishing some of the properties of crystals, for example thermal conductivity of crystals cannot be understood without taking into consideration Umklapp proccesses. The violation in the conservation of momentum can be accounted for as a transfer to the vacuum state (i.e the zero-point field).

Quasienergy

Ordinary crystals break spatial translational symmetries leading to repeated spacial patterns, time crystals spontaneously break time-translation symmetry (TTS) and have repeated patterns in time. Fields or particles in the presence of a time crystal background will appear to violate the conservation of energy, analogous to the apparent violation of the conservation of momentum in crystalline Umklapp processes. In either case the apparent non-conservation is in reality a transfer to the vacuum field (i.e. zero-point field). The term quasienergy has been coined to explain some of the predicted properties of time crystals.

Topological order

Time crystals are thought to exhibit topological order, an emergent phenomena, in which nonlocal correlations encoded in the whole wave-function of the system allow for fault tolerance against perturbations, thus allowing quantum states to stabilize against decoherence effects that usually limit their useful lifetime.

Topological states

Albert Einstein insisted that all fundamental laws of nature could be understood in terms of geometry and symmetry. Before 1980 all states of matter could be classified by the principle of broken symmetry. The quantum Hall state provided the first example of a quantum state that had no spontaneously broken symmetry. Its behaviour depends only on its topology and not its specific geometry. The quantum Hall effect earned von Klitzing the Nobel Prize in Physics for 1985 and though it was not understood at the time, the quantum Hall effect is an example of Topological order.. Topological order violates the long-held belief that ordering requires symmetry breaking. Fundamental laws can be studied under the context of topological field theory.

Recently a new class of topological states has emerged called quantum spin Hall (QSH) states or topological insulators. Inside a topological insulator Maxwell's equations of electromagnetism are dramatically altered to include an extra topological terms which gives rise to novel new physics. A dipole such as an electron above the surface of a topological insulator induces an emergent quasi-particle image magnetic monopole, known as a dyon, which is a composite of electric and magnetic charges. This new particle obeys neither Bose nor Fermi statistics but behave like a so called anyon named as such because it is governed by with "any possible" statistics. When a superconducter is close to the surface of a topological insulator, Majorana fermions occur inside vortices. These particles are governed by non-abelian statistics and could have radical applications in a new form of electronics called spintronics and topological quantum computers. Non-local effects analogous to the Aharonov Bohm effect have been observed in topological insulators, and certain conditions are expected to give rise to the ability of Majorana fermions to teleport, a test of which has been proposed.

Floquent topological states

Floquent topological states combine ideas from photonics and condensed matter physics. A system that is driven by a periodic external field shows a discrete time-translation symmetry. In the framework of the Floquet theory the concepts of quasienergy and Floquet states were introduced to account for this time periodicity: the term quasienergy reflects the formal analogy with the quasimomentum characterizing the Bloch eigenstates in a periodic solid. Recently it has been shown that the topological properties can be "tuned" by applying a time-dependant electromagnetic field. For example when microwaves periodically drive a crystalline material (i.e. a combined spacial and time periodicity) it may become a Floquet topological insulator. The crystal's quasienergy spectrum causes the emergence of new forms of topological order. It is hoped that many topological properties may be transmuted into the material at will simply by using low energy electromagnetic fields, acting like a topological switch.

Floquent time crystals

Time crystals can extend the idea of Floquet topological insulators still further, by enabling entirely new non-equilibrium dynamical phases. These dynamical phases are characterised by properties forbidden in the thermal equilibrium, such as spontaneous time-translation symmetry breaking or dynamical topological order. The latter opens the door to a new realm of quantum topological phenomena, which has only barely begun to be explored.

Fault-tolerance against decoherence

Preventing decoherence via topological order has a wide range of implications: The efficiency of some computing and information theory tasks can be greatly enhanced when using quantum correlated states; quantum correlations are an equally valuable resource in the realm of quantum thermodynamics New types of quantum devices in non-equilibrium states function very differently to their classical counterparts: For example, it has been theoretically shown that non-equilibrium quantum ratchet systems function far more efficiently then that predicted by classical thermodynamics. It has also been shown that quantum coherence can be used to enhance the efficiency of systems beyond the classical Carnot limit. This is because it could be possible to extract work, in the form of photons, from a single heat bath. Quantum coherence can be used in effect to play the role of Maxwell's demon allowing a hypothetical bypassing of the second law of thermodynamics

Thermodynamics

Compatibility with the laws of thermodynamics

File:Qualitative explanation of time translation symmetry breaking.png

Figure 1: We can imagine a situation where a permanently rotating object can be either (a) spontaneous or (b) explicit. In normal situations we must do work to cause an object to rotate; in (a) we must do work to cause the object to stop rotating. (Chernodub 2012)

A time crystal is a driven (i.e. open) quantum system that is in perpetual motion, it does not violate the laws of thermodynamics:

• A time crystal does not produce work as it rotates in its ground state, energy is conserved so that the first law of thermodynamics is not violated. (otherwise such a device would be a perpetuum mobile of the first kind).

• A time crystal does not spontaneously convert thermal energy into mechanical work so that the second law of thermodynamics is not violated. (otherwise such a device would be a perpetuum mobile of the second kind).

• A time crystal cannot serve as a perpetual store of work, so that the third law of thermodynamics is not violated (otherwise the device would be a perpetuum mobile of the third kind).

A time crystal has been said to be a perpetuum mobile of the fourth kind: it does not produce work and it cannot serve as a perpetual energy storage. But it rotates perpetually.

Zero point energy

Main article: zero-point energy

Zero-point radiation continually imparts random impulses on an electron, so that it never comes to a complete stop. Zero-point radiation gives the oscillator an average energy equal to the frequency of oscillation multiplied by one-half of Planck's constant ${\tfrac {1}{2}}\hbar \omega$

Time crystals are closely related to concepts of zero-point energy and the dynamical Casimir effect As temperature is reduced to absolute zero, it might be thought that all motion ceases and particles come completely to rest. In fact, however, kinetic energy is retained by particles even at the lowest possible temperature. The random fluctuation corresponding to this zero-point energy never vanishes as a consequence of the uncertainty principle of quantum mechanics. According to modern physics (i.e. quantum field theory) the universe is made up of matter fields whose quanta are fermions (i.e. leptons and quarks) and force fields, whose quanta are bosons (e.g. photons and gluons). All these fields have zero-point energy. The predicted zero-point energy contained in the vacuum is very large: physicists John Wheeler and Richard Feynman calculated that there is enough energy in the vacuum inside a single light bulb to boil all the world's oceans.

Zero-point energy has many observed physical consequences such as spontaneous emission, Casimir force, Lamb shift and the magnetic moment of the electron. According to the fluctuation-dissipation theorem, fluctuations and dissipation go hand in hand; we cannot have one without the other, and the vacuum therefore dissipates energy. It is relatively easy to show that zero-point motion of a particle is in fact sustained by the driving zero-point field, or vacuum state. Zero-point energy may even be the cause of dark energy and the current acceleration of the universe, though this idea has been disputed.

The zero-point field is reminiscent of the discredited aether theory prevalent before the advent of Einstein's relativity, all broken symmetries can be attributed to the influence of this all pervading vacuum state. If we imagine the entire universe was immersed in a vast magnet, the presence of this magnet might cause the background to be (In the words of Wolfgang Pauli) "weakly left handed" i.e. it might cause a preference of left over right and account for the broken symmetry we observe; the idea of a complex vacuum state that has a rich structure to account for all broken symmetries in nature is in the same spirit of this idea.

No-go theorem in equilibrium

There is a proof that quantum time crystals in thermal equilibrium are not possible.

Non-equilibrium systems

Non-equilibrium quantum fluctuations have been studied for some time and the past few years have seen a surge of interest in this topic. A comprehensive definition of energy and work in these contexts is yet to be formulated and an open area of research.

Experiments

University of Maryland

In October 2016, researchers at the University of Maryland, College Park, claimed to have created the world's first discrete time crystal. Using the idea from the March proposal, they trapped a chain of ¹⁷¹Yb⁺ (ytterbium) 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 are called ²S_1/2 |F=0, m_F = 0⟩ and |F = 1, m_F = 0⟩. The different energy levels of these are very close, separated by 12.642831 GHz. Ten Doppler cooled ions were used in a line 0.025 mm long. The ions were coupled together. The researchers observed a subharmonic oscillation of the drive. The experiment also showed "rigidity" of the time crystal, where the oscillation frequency remained unchanged even when the time crystal was perturbed. However, if the perturbation drive was too great, the time crystal "melted" and lost its oscillation.

Harvard University

Mikhail Lukin led a group at Harvard University who also replicated to creation of a driven time crystal. The group used black diamond dipolar spin impurities and observed sub-harmonics of the drive frequency. Nitrogen vacancies in the diamond exposed to a magnetic field provide sites to store information in the form of spin direction. The diamond is exposed to a green laser and simultaneously to alternating pulses of radiowaves polarized perpendicular to each other. When the spin state is read out it is modulated at a one half frequency of the drive. The oscillations persist for over 100 cycles.

Related Concepts

Choreographic Crystals

A similar idea called a choreographic crystal has been proposed.

Dynamical Casimir effect

Time Crystals are closely related to the dynamical Casimir effect or Unruh effect. These effects are basically an instability of the quantum vacuum, which leads to an exponential growth of emitted boson pairs (known as Superradiance in the form of photons or phonons) when the oscillating frequency of the medium is equal to twice the boson frequency.

See also

• Anisotropy
• Casimir effect
• Chaos theory
• Condensed matter physics
• Constructal theory
• Correlation does not imply causation
• Crystallography
• Dissipative system
• Dynamical Casimir effect
• Electron magnetic moment
• Emergence
• Entropy production
• Entropic gravity
• Error detection and correction
• ER=EPR
• Extremal principles in non-equilibrium thermodynamics
• Free energy principle
• Gauge theory
• Information theory
• Lamb shift
• Laws of thermodynamics
• Green–Kubo relations
• Green's function (many-body theory)
• Hawking radiation
• Information entropy
• Materials science
• Maxwell's demon
• Noether's theorem
• Non-equilibrium thermodynamics
• Nonlinear Schrödinger equation
• Nonlinear system
• Pair production
• Perpetual motion
• Photonics
• Philosophy of information
• Philosophy of physics
• Quantum decoherence
• Quantum entanglement
• Quantum gravity
• Quantum information
• Quantum teleportation
• Self-organization
• Spontaneous symmetry breaking
• Stochastic electrodynamics
• Symmetry breaking
• Symmetry in quantum mechanics
• Topological order
• Unruh effect
• Virtual particle
• Zero-point energy
• Zero-point field

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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" (PDF). Nature Physics. 12 (10): 907–911. arXiv:1508.07026v1. Bibcode:2016NatPh..12..907S. doi:10.1038/nphys3783. ISSN 1745-2473.

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Modi, Kavan; Brodutch, Aharon; Cable, Hugo; Paterek, Tomasz; Vedral, Vlatko (2012). "The classical-quantum boundary for correlations: Discord and related measures" (PDF). Reviews of Modern Physics. 84 (4): 1655–1707. arXiv:1112.6238. Bibcode:2012RvMP...84.1655M. doi:10.1103/RevModPhys.84.1655. ISSN 0034-6861.

Ray, M. W.; Ruokokoski, E.; Kandel, S.; Möttönen, M.; Hall, D. S. (2014). "Observation of Dirac monopoles in a synthetic magnetic field" (PDF). Nature. 505 (7485): 657–660. arXiv:1408.3133v1. Bibcode:2014Natur.505..657R. doi:10.1038/nature12954. ISSN 0028-0836.

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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.

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Wilczek, Frank (2012). "Quantum Time Crystals" (PDF). Physical Review Letters. 109 (16). arXiv:1202.2539v2. Bibcode:2012PhRvL.109p0401W. doi:10.1103/PhysRevLett.109.160401. ISSN 0031-9007. {{cite journal}}: Invalid |ref=harv (help)

Wilczek, Frank (2013a). "Wilczek Reply:" (PDF). Physical Review Letters. 110 (11). Bibcode:2013PhRvL.110k8902W. doi:10.1103/PhysRevLett.110.118902. ISSN 0031-9007. {{cite journal}}: Invalid |ref=harv (help)

Wilczek, Frank (2013b). "Superfluidity and Space-Time Translation Symmetry Breaking" (PDF). Physical Review Letters. 111 (25). arXiv:1308.5949v1. Bibcode:2013PhRvL.111y0402W. doi:10.1103/PhysRevLett.111.250402. ISSN 0031-9007. {{cite journal}}: Invalid |ref=harv (help)

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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. (2016). "Observation of a Discrete Time Crystal" (PDF). arXiv:1609.08684v1. Bibcode:2016arXiv160908684Z. {{cite journal}}: Cite journal requires |journal= (help)

Books

Bordag, M.; Mohideen, U.; Mostepanenko, V.M. (2001). "New developments in the Casimir effect" (PDF). Physics Reports. 353 (1–3): 1–205. arXiv:quant-ph/0106045. doi:10.1016/S0370-1573(01)00015-1. ISSN 0370-1573. {{cite journal}}: Invalid |ref=harv (help)

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. {{cite book}}: Invalid |ref=harv (help)

Cao, Tian Yu (25 March 2004). Conceptual Foundations of Quantum Field Theory. Cambridge: Cambridge University Press. ISBN 978-0-521-60272-3. {{cite book}}: Invalid |ref=harv (help)

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. {{cite book}}: Invalid |ref=harv (help)

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. {{cite book}}: Invalid |ref=harv (help)

Feng, Duan; Jin, Guojun (2005). Introduction to Condensed Matter Physics. singapore: World Scientific. ISBN 978-981-238-711-0. {{cite book}}: Invalid |ref=harv (help)

Milonni, Peter W. (1994). The Quantum Vacuum: An Introduction to Quantum Electrodynamics. London: Academic Press. ISBN 978-0-124-98080-8. {{cite book}}: Invalid |ref=harv (help)

Pade, Jochen (2014). Quantum Mechanics for Pedestrians 2: Applications and Extensions. Dordrecht: Springer. doi:10.1007/978-3-319-00813-4. ISBN 978-3-319-00813-4. ISSN 2192-4791. {{cite book}}: Invalid |ref=harv (help)

Schwinger, Julian (1998a). Particles, Sources, And Fields, Volume 1: v. 1 (Advanced Books Classics). Perseus. ISBN 978-0-738-20053-8. {{cite book}}: Invalid |ref=harv (help)

Schwinger, Julian (1998b). Particles, Sources, And Fields, Volume 2: v. 2 (Advanced Books Classics). Perseus. ISBN 978-0-738-20054-5. {{cite book}}: Invalid |ref=harv (help)

Schwinger, Julian (1998c). Particles, Sources, And Fields, Volume 3: v. 3 (Advanced Books Classics). Perseus. ISBN 978-0-738-20055-2. {{cite book}}: Invalid |ref=harv (help)

Sólyom, Jenö (19 September 2007). Fundamentals of the Physics of Solids: Volume 1: Structure and Dynamics. Springer. ISBN 978-3-540-72600-5. {{cite book}}: Invalid |ref=harv (help)

Wilczek, Frank (16 July 2015). A Beautiful Question: Finding Nature's Deep Design. Penguin Books Limited. ISBN 978-1-84614-702-9. {{cite book}}: Invalid |ref=harv (help)

Press

Aalto University (30 April 2015). "Physicists discover quantum-mechanical monopoles". phys.org. Science X. Archived from the original on 30 Apr 2015. {{cite web}}: Invalid |ref=harv (help)

Aitchison, Ian (19 November 1981). "Observing the Unobservable". New Scientist. 92 (1280): 540–541. ISSN 0262-4079. {{cite journal}}: Invalid |ref=harv (help)

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 Jan 2014. {{cite web}}: Invalid |ref=harv (help)

Aron, Jacob (6 July 2012). "Computer that could outlive the universe a step closer". newscientist.com. New Scientist. Archived from the original on 2 Feb 2017. {{cite web}}: Invalid |ref=harv (help)

Ball, Philip (8 January 2016). "Focus: New Crystal Type is Always in Motion". physics.aps.org. APS Physics. Archived from the original on 3 Feb 2017. {{cite web}}: Invalid |ref=harv (help)

Ball, Philip (8 July 2004). "Scepticism greets pitch to detect dark energy in the lab". Nature. 430 (6996): 126–126. Bibcode:2004Natur.430..126B. doi:10.1038/430126b. ISSN 0028-0836. {{cite journal}}: Invalid |ref=harv (help)

Cartlidge, Edwin (21 October 2015). "Scientists build heat engine from a single atom". sciencemag.org. Science Magazine. Archived from the original on 1 Feb 2017. {{cite web}}: Invalid |ref=harv (help)

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 Feb 2017. {{cite web}}: Invalid |ref=harv (help)

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. {{cite journal}}: Invalid |ref=harv (help)

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 Feb 2017. {{cite web}}: Invalid |ref=harv (help)

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 Feb 2017. {{cite web}}: Invalid |ref=harv (help)

Grossman, Lisa (18 January 2012). "Death-defying time crystal could outlast the universe". newscientist.com. New Scientist. Archived from the original on 2 Feb 2017.

Hackett, Jennifer (22 February 2016). "Curious Crystal Dances for Its Symmetry". scientificamerican.com. Scientific American. Archived from the original on 3 Feb 2017. {{cite web}}: Invalid |ref=harv (help)

Hewitt, John (3 May 2013). "Creating time crystals with a rotating ion ring". phys.org. Science X. Archived from the original on 4 Jul 2013.

Johnston, Hamish (18 January 2016). "'Choreographic crystals' have all the right moves". physicsworld.com. Institute of Physics. Archived from the original on 3 Feb 2017. {{cite web}}: Invalid |ref=harv (help)

Johannes Gutenberg Universitaet Mainz (3 February 2014). "Prototype of single ion heat engine created". sciencedaily.com. ScienceDaily. Archived from the original on 1 Feb 2017.{{cite web}}: CS1 maint: ref duplicates default (link)

Joint Quantum Institute (22 March 2011). "Floquet Topological Insulators". jqi.umd.edu. Joint Quantum Institute. {{cite web}}: Invalid |ref=harv (help)

Morgan, James (30 January 2014). "Elusive magnetic 'monopole' seen in quantum system". bbc.co.uk. BBC. Archived from the original on 30 Jan 2014. {{cite web}}: Invalid |ref=harv (help)

Moskowitz, Clara (2 October 2014). "New Particle Is Both Matter and Antimatter". scientificamerican.com. Scientific American. Archived from the original on 9 Oct 2014. {{cite web}}: Invalid |ref=harv (help)

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 Feb 2017. {{cite web}}: Invalid |ref=harv (help)

Pilkington, Mark (17 July 2003). "Zero point energy". theguardian.com. The Guardian. Archived from the original on 7 Feb 2017. {{cite news}}: Invalid |ref=harv (help)

Powell, Devin (2013). "Can matter cycle through shapes eternally?". Nature. doi:10.1038/nature.2013.13657. ISSN 1476-4687. Archived from the original on 3 Feb 2017. {{cite journal}}: Invalid |ref=harv (help)

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 Mar 2015. {{cite web}}: Invalid |ref=harv (help)

Richerme, Phil (18 January 2017). "Viewpoint: How to Create a Time Crystal". physics.aps.org. APS Physics. Archived from the original on 2 Feb 2017. {{cite web}}: Invalid |ref=harv (help)

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Qi, Xiao-Liang; Zhang, Shou-Cheng (2010). "The quantum spin Hall effect and topological insulators" (PDF). Physics Today. 63 (1): 33–38. doi:10.1063/1.3293411. ISSN 0031-9228. {{cite journal}}: Invalid |ref=harv (help)

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 Jan 2017.

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Wolchover, Natalie (25 April 2013). "Perpetual Motion Test Could Amend Theory of Time". quantamagazine.org. Simons Foundation. Archived from the original on 2 Feb 2017. {{cite web}}: Invalid |ref=harv (help)

Wolchover, Natalie (15 May 2014). "Forging a Qubit to Rule Them All". quantamagazine.org. Simmons Foundation. Archived from the original on 15 Mar 2016. {{cite web}}: Invalid |ref=harv (help)

Wood, Charlie (31 January 2017). "Time crystals realize new order of space-time". csmonitor.com. Christian Science Monitor. Archived from the original on 2 Feb 2017. {{cite web}}: Invalid |ref=harv (help)

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 Apr 2013.

Zakrzewski, Jakub (15 October 2012). "Viewpoint: Crystals of Time". physics.aps.org. APS Physics. Archived from the original on 2 Feb 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 Feb 2017. {{cite web}}: Invalid |ref=harv (help)

Zyga, Lisa (20 February 2012). "Time crystals could behave almost like perpetual motion machines". phys.org. Science X. Archived from the original on 3 Feb 2017.

Zyga, Lisa (22 August 2013). "Physicist proves impossibility of quantum time crystals". phys.org. Space X. Archived from the original on 3 Feb 2017.

Zyga, Lisa (27 January 2014). "Nanoscale heat engine exceeds standard efficiency limit". phys.org. Science X. Archived from the original on 4 Apr 2015. {{cite web}}: Invalid |ref=harv (help)

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 Jul 2015.

Zyga, Lisa (9 September 2016). "Time crystals might exist after all (Update)". phys.org. Science X. Archived from the original on 11 Sep 2016.

External Links

• Christopher Monroe Bio, Maryland
• The Frank Wilczek Web Site
• Lukin Group, Harvard
• Norman Yao Bio, Berkeley