Time crystal

In condensed matter physics, a time crystal refers to a system or subsystem whose lowest-energy states evolve periodically. This name was proposed theoretically by Frank Wilczek in 2012 as a temporal analog to common crystals, which are periodic spatially. Experimental realization of matter with stable periodic evolution was demonstrated in 2016-2017, but not in the way theorized in 2012. In terms of practical use, time crystals may one day be used as quantum memories.
The existence of crystals in nature is a manifestation of spontaneous symmetry breaking, which occurs when the lowest-energy state of a system is less symmetrical than the equations governing the system. In the crystal ground state, the continuous translational symmetry in space is broken and replaced by the lower discrete symmetry of the periodic crystal. As the laws of physics are symmetrical under continuous translations in time as well as space, the question arose in 2012 as to whether it is possible to break symmetry temporally and thus create a "time crystal" resistant to entropy.
In 2015 a theorem purported to rule out systems that would break both space and time translation simultaneously in equilibrium. Although questioned this no-go theorem motivated generalizations of Wilczek's original proposal. Most notable of these are periodically driven nonequilibrium systems that have discrete time translation symmetry which may be spontaneously broken. 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 (or phase) of non-equilibrium matter. Breaking of time symmetry can only occur in non-equilibrium systems.
Discrete time crystals have in fact been observed in physics laboratories as early as 2016 (published in 2017). One example of a time crystal which demonstrates non-equilibrium, broken time symmetry is a constantly rotating ring of charged ions in an otherwise lowest-energy state.
History.
The idea of a quantized time crystal was theorized in 2012 by Frank Wilczek 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.
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.
Subsequent work developed more precise definitions of time translation symmetry-breaking, which ultimately led to the Watanabe-Oshikawa "no-go" proof that quantum space-time crystals in equilibrium are not possible. Note, however, that Khemani et al. pointed out that Watanabe and Oshikawa's excited state proof has a subtle error which renders it invalid. See Appendix A of Ref. Moreover, Watanabe and Oshikawa strictly speaking 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.
Several realizations of time crystals, which avoid the equilibrium no-go arguments, were later proposed. In 2014 Krzysztof Sacha at Jagiellonian University in Krakow predicted the behaviour of discrete time crystals in a periodically driven system of ultracold atoms. Later works suggested that periodically driven quantum spin systems could show similar behaviour.
In 2016 Norman Yao at Berkeley and colleagues proposed a different way to create discrete time crystals in spin systems. His ideas were successfully and independently realized by two experimental teams: a group led by Harvard's Mikhail Lukin and a group led by Christopher Monroe at University of Maryland. Both experiments were published in the same issue of "Nature" in March 2017.
In 2019 dissipative time crystals, which are many body quantum systems that break continuous time translation symmetry by coupling to an external bath have also been proposed and realized experimentally in a Bose-Einstein condensate in strongly dissipative optical cavity by a group lead by Tilman Esslinger at ETH Zurich. 
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 for some . 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, and concerns about the physicality of the long-range nature of the model have been raised.
Time translation symmetry.
Symmetries in nature lead directly to conservation laws, something which is precisely formulated by the Noether theorem.
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. This symmetry implies the conservation of energy.
Broken symmetry in normal crystals.
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. Quasimomentum, however, is conserved in a perfect crystal.
Time crystals show a broken symmetry analogous to a discrete space-translation symmetry breaking. For example, 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:
Broken symmetry in discrete time crystals.
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. The initial symmetry, which is the discrete time-translation symmetry (formula_1) with formula_2, is spontaneously broken to the lower discrete time-translation symmetry with formula_3, where formula_4 is time, formula_5 the driving period, formula_6 an integer.
Many systems can show behaviors of spontaneous time translation symmetry breaking but may not be discrete (or Floquet) time crytals: 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, discrete (or Floquet) time crystals are unique in that they follow a strict definition of discrete time-translation symmetry breaking:
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.
Thermodynamics.
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"—their apparent motion does not represent conventional kinetic energy.
It has been proven that a time crystal cannot exist in thermal equilibrium. Note, however, that Khemani et al. pointed out that this proof has a subtle error which renders it invalid. See Appendix A of Ref. 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.
Experiments.
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, 2"S"1/2 and , 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.
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.
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 (0.0001 K or -273.15 °C).
A clever application of nuclear magnetic resonance techniques offered researchers at MIT a first measured observation of Floquet prethermalization — a quasi-steady state that persists for a very long time.