Incondensed matter physics, atime crystal is aquantum system of particles whoselowest-energy state is one in which the particles are in repetitive motion. The system cannot lose energy to the environment and come to rest because it is already in its quantumground state. Time crystals were first proposed theoretically by Alfred Shapere andFrank Wilczek in 2012 as atime-based analogue to common crystals – whereas the atoms in crystals are arranged periodically in space, the atoms in a time crystal are arranged periodically in both space and time.[1][2][3] Several different groups have demonstrated matter with stable periodic evolution in systems that are periodically driven.[4][5][6][7] In terms of practical use, time crystals may one day be used asquantum computer memory.[8]
The existence ofcrystals in nature is a manifestation of spontaneoussymmetry 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 continuoustranslational symmetry in space is broken and replaced by the lower discrete symmetry of the periodic crystal. As the laws of physics are symmetrical under continuoustranslations 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"[3]
If a discrete time-translation symmetry is broken (which may be realized in periodically driven systems), then the system is referred to as adiscrete time crystal. A discrete time crystal never reachesthermal equilibrium, as it is a type (or phase) of non-equilibrium matter. Breaking of time symmetry can occur only in non-equilibrium systems.[7] Discrete time crystals have in fact been observed in physics laboratories as early as 2016. One example of a time crystal, which demonstrates non-equilibrium, broken time symmetry is a constantly rotating ring of charged ions in an otherwise lowest-energy state.[8]
Ordinary (non-time) crystals form through spontaneous symmetry breaking related to spatial symmetry. Such processes can produce materials with interesting properties, such asdiamonds,salt crystals, andferromagnetic 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).[9]
Symmetries in nature lead directly to conservation laws, something which is precisely formulated byNoether's theorem.[10]
The basic idea oftime-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.[11] This symmetry implies theconservation of energy.[12]
Normal process (N-process) and Umklapp process (U-process). While the N-process conserves totalphonon momentum, the U-process changes phonon momentum.
Common crystals exhibitbroken 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 inUmklapp processes.[13]Quasimomentum, however, is conserved in a perfect crystal.[14]
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 patternless 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.
Time crystals seem to breaktime-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 showdiscrete time-translation symmetry breaking, not thecontinuous one: they are periodically driven systems oscillating at afraction of the frequency of the driving force. (According toPhilip Ball, DTC are so-called because "their periodicity is a discrete, integer multiple of the driving period".[15])
The initial symmetry, which is the discrete time-translation symmetry () with, is spontaneously broken to the lower discrete time-translation symmetry with, where is time, the driving period, an integer.[16]
However, discrete (or Floquet) time crystals are unique in that they follow a strict definition ofdiscrete time-translation symmetry breaking:[17]
it is a broken symmetry – the system shows oscillations with a periodlonger than the driving force,
the system is incrypto-equilibrium – these oscillations generate noentropy, and a time-dependent frame can be found in which the system is indistinguishable from an equilibrium when measured stroboscopically[17] (which is not the case of convection cells, oscillating chemical reactions and aerodynamic flutter),
the system exhibitslong-range order – the oscillations are in phase (synchronized) over arbitrarily long distances and time.
Moreover, the broken symmetry in time crystals is the result ofmany-body interactions: the order is the consequence of acollective process, just like in spatial crystals.[16] 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.[16]
Time crystals do not violate thelaws 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"[18]—their apparent motion does not represent conventional kinetic energy.[19] 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.[16]
Time crystals do not evade the second law of thermodynamics,[20] 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 increasing.[21][22]
Subsequent work developed more precise definitions oftime-translation symmetry-breaking, which ultimately led to the Watanabe–Oshikawa"no-go" statement that quantum space–time crystals in equilibrium are not possible.[29][30] 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.[31][32][33]
Several realizations of time crystals, which avoid the equilibrium no-go arguments, were later proposed.[34] In 2014Krzysztof Sacha atJagiellonian University inKraków predicted the behaviour of discrete time crystals in a periodically driven system with "an ultracold atomic cloud bouncing on an oscillating mirror".[35][36]
In 2016, research groups at Princeton and at Santa Barbara independently suggested that periodically driven quantum spin systems could show similar behaviour.[37] Also in 2016, Norman Yao atBerkeley and colleagues proposed a different way to create discrete time crystals in spin systems.[38] These ideas were successful and independently realized by two experimental teams: a group led byHarvard'sMikhail Lukin[39] and a group led byChristopher Monroe atUniversity of Maryland.[40] Both experiments were published in the same issue ofNature in March 2017.
Later, time crystals in open systems, so-called "dissipative time crystals," were proposed in several platforms breaking a discrete[41][42][43][44] and a continuous[45][46] 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 theUniversity of Hamburg.[47] The researchers used aBose–Einstein condensate strongly coupled to a dissipativeoptical cavity and the time crystal was demonstrated to spontaneously break discrete time-translation symmetry by periodically switching between two atomic density patterns.[47][48][49] In an earlier experiment in the group ofTilman Esslinger atETH Zurich, limit cycle dynamics[50] was observed in 2019,[51] 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 asr−α for someα > 0. Kozin and Kyriienko instead analyzed aspin-1/2 many-bodyHamiltonian 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 aground energy state. However, demonstrating such a system in practice might be prohibitively difficult,[52][53] and concerns about the physicality of the long-range nature of the model have been raised.[54]
In October 2016, Christopher Monroe at theUniversity of Maryland claimed to have created the world's first discrete time crystal. Using the ideas proposed by Yao et al.,[38] his team trapped a chain of171Yb+ ions in aPaul trap, confined by radio-frequency electromagnetic fields. One of the twospin states was selected by a pair of laser beams. The lasers were pulsed, with the shape of the pulse controlled by anacousto-optic modulator, using theTukey window to avoid too much energy at the wrong optical frequency. Thehyperfine 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. TenDoppler-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.[40]
Also in 2016,Mikhail Lukin at Harvard also reported the creation of a driven time crystal. His group used adiamond crystal doped with a high concentration ofnitrogen-vacancy centers, which have strong dipole–dipole coupling and relatively long-lived spincoherence. 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. Thissubharmonic response to the drive frequency is seen as a signature of time-crystalline order.[39]
In May 2018, a group inAalto University reported that they had observed the formation of a time quasicrystal and its phase transition to a continuous time crystal in aHelium-3superfluid cooled to within one ten thousandth of a kelvin from absolute zero (0.0001 K).[55] On August 17, 2020Nature 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.[56]
In July 2021, a team led by Andreas Hemmerich at the Institute of Laser Physics at theUniversity of Hamburg presented the first realization of a time crystal in an open system, a so-calleddissipative time crystal usingultracold atoms coupled to anoptical cavity. The main achievement of this work is a positive application of dissipation – actually helping to stabilise the system's dynamics.[47][48][49]
In November 2021, a collaboration betweenGoogle and physicists from multiple universities reported the observation of a discrete time crystal on Google'sSycamore processor, aquantum computing device. A chip of 20 qubits was used to obtain amany-body localization configuration of up and down spins and then stimulated with a laser to achieve a periodically driven "Floquet" system where allup spins are flipped fordown 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 aprotected eigenstate order.[22][59]
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 onquantum simulators rather than quantum processors: first a group at theUniversity of Maryland obtained time crystals ontrapped-ions qubits using high frequency driving rather than many-body localization[60][61] and then a collaboration betweenTU Delft andTNO in the Netherlands called Qutech created time crystals from nuclear spins in carbon-13nitrogen-vacancy (NV) centers on a diamond, attaining longer times but fewer qubits.[62][63]
In February 2022, a scientist atUC 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 experimentinjection locking was used to direct lasers at a specific frequency inside amicroresonator creating alattice trap forsolitons at subharmonic frequencies.[64][65]
In March 2022, a new experiment studying time crystals on a quantum processor was performed by two physicists at theUniversity of Melbourne, this time using IBM'sManhattan and Brooklyn quantum processors observing a total of 57 qubits.[66][67][68]
In June 2022, the observation of a continuous time crystal was reported by a team at the Institute of Laser Physics at theUniversity 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 observedlimit 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 aBose–Einstein condensate in anoptical cavity, which was pumped with an optical standing wave oriented perpendicularly with regard to the cavity axis and was in asuperradiant phase localizing at twobistableground states between which it oscillated.[69][70][71][72]
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".[73][74][75][76][77]
In March 2025, researchers atTU Dortmund University observed complex nonlinear behavior in asemiconductor-based time crystal made ofindium gallium arsenide. By periodically driving the system with laser pulses, they uncovered transitions from synchronizedoscillations to chaotic motion. The system exhibited structures such as theFarey tree sequence and thedevil's staircase—patterns never before seen in semiconductor time crystals—offering new insights into dynamic phase transitions and chaos in driven quantum systems.[78]
^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].
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:1404.3519.Bibcode:2015PhRvB..92v4512Y.doi:10.1103/PhysRevB.92.224512.ISSN1098-0121.S2CID118348062.
