Frank WILCZEK, et al.
Time Crystals
Time Crystals
https://link.aps.org/doi/10.1103/PhysRevLett.109.160401
Phys. Rev. Lett. 109, 160401 – Published 15 October, 2012
DOI: https://doi.org/10.1103/PhysRevLett.109.160401
Quantum Time Crystals
Frank Wilczek
Abstract -- Some subtleties and apparent difficulties associated with the notion of spontaneous breaking of time-translation symmetry in quantum mechanics are identified and resolved. A model exhibiting that phenomenon is displayed. The possibility and significance of breaking of imaginary time-translation symmetry is discussed.
https://en.wikipedia.org/wiki/Time_crystal
Time Crystal
In condensed matter physics, a time crystal is a quantum system of particles whose lowest-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 quantum ground state. Time crystals were first proposed theoretically by Alfred Shapere and Frank Wilczek in 2012 as a time-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. Several different groups have demonstrated matter with stable periodic evolution in systems that are periodically driven. In terms of practical use, time crystals may one day be used as quantum computer memory.
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."
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 occur only in non-equilibrium systems.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.
Concept
Ordinary (non-time) crystals form through spontaneous symmetry breaking related to spatial symmetry. Such processes can produce materials with interesting properties, such as diamonds, salt crystals, and ferromagnetic metals. By analogy, a time crystal arises through the spontaneous breaking of a time-translation symmetry. A time crystal can be informally defined as a time-periodic self-organizing structure. While an ordinary crystal is periodic (has a repeating structure) in space, a time crystal has a repeating structure in time. A time crystal is periodic in time in the same sense that the pendulum in a pendulum-driven clock is periodic in time. Unlike a pendulum, a time crystal "spontaneously" self-organizes into robust periodic motion (breaking a temporal symmetry).
Time-translation symmetry
Symmetries in nature lead directly to conservation laws, something which is precisely formulated by Noether's 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
Normal process (N-process) and Umklapp process (U-process). While the N-process conserves total phonon momentum, the U-process changes phonon momentum.
Common crystals exhibit broken translation symmetry: they have repeated patterns in space and are not invariant under arbitrary translations or rotations. The laws of physics are unchanged by arbitrary translations and rotations. However, if we hold fixed the atoms of a crystal, the dynamics of an electron or other particle in the crystal depend on how it moves relative to the crystal, and particle momentum can change by interacting with the atoms of a crystal—for example in Umklapp processes. Quasimomentum, however, is conserved in a perfect crystal.
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:
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 (DTC)
Time crystals seem to break time-translation symmetry and have repeated patterns in time even if the laws of the system are invariant by translation of time. The time crystals that are experimentally realized show discrete time-translation symmetry breaking, not the continuous one: they are periodically driven systems oscillating at a fraction of the frequency of the driving force. (According to Philip Ball, DTC are so-called because "their periodicity is a discrete, integer multiple of the driving period".
The initial symmetry, which is the discrete time-translation symmetry ( t → t + n T {\displaystyle t\to t+nT}) with n = 1 {\displaystyle n=1}, is spontaneously broken to the lower discrete time-translation symmetry with n > 1 {\displaystyle n>1}, where t {\displaystyle t} is time, T {\displaystyle T} the driving period, n {\displaystyle n} an integer.
Many systems can show behaviors of spontaneous time-translation symmetry breaking but may not be discrete (or Floquet) time crystals: convection cells, oscillating chemical reactions, aerodynamic flutter, and subharmonic response to a periodic driving force such as the Faraday instability, NMR spin echos, parametric down-conversion, and period-doubled nonlinear dynamical systems.
However, discrete (or Floquet) time crystals are unique in that they follow a strict definition of discrete time-translation symmetry breaking:
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-dependent frame can be found in which the system is indistinguishable from an equilibrium when measured stroboscopically (which is not the case of convection cells, oscillating chemical reactions and aerodynamic flutter),
the system exhibits long-range order – the oscillations are in phase (synchronized) over arbitrarily long distances and time.
Moreover, the broken symmetry in time crystals is the result of many-body interactions: the order is the consequence of a collective process, just like in spatial crystals. This is not the case for NMR spin echos.
These characteristics makes discrete time crystals analogous to spatial crystals as described above and may be considered a novel type or phase of nonequilibrium matter.
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. Recent experimental advances in probing discrete time crystals in their periodically driven nonequilibrium states have led to the beginning exploration of novel phases of nonequilibrium matter.
Time crystals do not evade the second law of thermodynamics, 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 a system, stays constant over time, which barely satisfies the second law of thermodynamics by not decreasing.
History
The idea of a quantized time crystal was theorized in 2012 by Alfred Shapere and 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 (European Synchrotron Radiation Facility) and Masaki Oshikawa (University of Tokyo) published several articles stating that space–time crystals were impossible.
Subsequent work developed more precise definitions of time-translation symmetry-breaking, which ultimately led to the Watanabe–Oshikawa "no-go" statement that quantum space–time crystals in equilibrium are not possible. 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.
Several realizations of time crystals, which avoid the equilibrium no-go arguments, were later proposed. In 2014 Krzysztof Sacha at Jagiellonian University in Kraków predicted the behaviour of discrete time crystals in a periodically driven system with "an ultracold atomic cloud bouncing on an oscillating mirror".
In 2016, research groups at Princeton and at Santa Barbara independently suggested that periodically driven quantum spin systems could show similar behaviour.[36] Also in 2016, Norman Yao at Berkeley and colleagues proposed a different way to create discrete time crystals in spin systems. These ideas were successful 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.[39] Both experiments were published in the same issue of Nature in March 2017.
Later, time crystals in open systems, so-called "dissipative time crystals," were proposed in several platforms breaking a discrete and a continuous[44][45] time-translation symmetry. A dissipative time crystal was experimentally realized for the first time in 2021 by the group of Andreas Hemmerich at the Institute of Laser Physics at the University of Hamburg. The researchers used a Bose–Einstein condensate strongly coupled to a dissipative optical cavity and the time crystal was demonstrated to spontaneously break discrete time-translation symmetry by periodically switching between two atomic density patterns. In an earlier experiment in the group of Tilman Esslinger at ETH Zurich, limit cycle dynamics was observed in 2019, but evidence of robustness against perturbations and the spontaneous character of the time-translation symmetry breaking were not addressed.
In 2019, physicists Valerii Kozin and Oleksandr Kyriienko proved that, in theory, a permanent quantum time crystal can exist as an isolated system if the system contains unusual long-range multiparticle interactions. The original "no-go" argument only holds in the presence of typical short-range fields that decay as quickly as 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, and concerns about the physicality of the long-range nature of the model have been raised.
The first experimental observation of a continuous time crystal was reported in 2022 by Hans Keßler, Andreas Hemmerich, and collaborators at the University of Hamburg, using a continuous pump laser to drive a Bose–Einstein condensate in an optical cavity, where the system responded with an intrinsic oscillation between two bistable ground states. In 2024, Alejandro Fainstein, Paulo Santos, and collaborators at the Instituto Balseiro and Paul Drude Institute reported a solid-state continuous time crystal in a driven-dissipative exciton–polariton condensate in a semiconductor microcavity, with its time-crystalline oscillations synchronized to self-sustained mechanical vibrations of the cavity. The oscillation frequency also exhibited period doubling relative to the self-induced phonons, corresponding to a discrete time-crystal transition.
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.,[37] 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.[citation needed]
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.[39]
Also in 2016, Mikhail Lukin at Harvard also reported the creation of a driven time crystal. His group used a diamond crystal doped with a high concentration of nitrogen-vacancy centers, which have strong dipole–dipole coupling and relatively long-lived spin coherence. This strongly interacting dipolar spin system was driven with microwave fields, and the ensemble spin state was determined with an optical (laser) field. It was observed that the spin polarization evolved at half the frequency of the microwave drive. The oscillations persisted for over 100 cycles. This subharmonic response to the drive frequency is seen as a signature of time-crystalline order.[38]
In May 2018, a group in Aalto University reported that they had observed the formation of a time quasicrystal and its phase transition to a continuous time crystal in a Helium-3 superfluid cooled to within one ten thousandth of a kelvin from absolute zero (0.0001 K). On August 17, 2020 Nature Materials published a letter from the same group saying that for the first time they were able to observe interactions and the flow of constituent particles between two time crystals.
In February 2021, a team at Max Planck Institute for Intelligent Systems described the creation of time crystal consisting of magnons and probed them under scanning transmission X-ray microscopy to capture the recurring periodic magnetization structure in the first known video record of such type.
In July 2021, a team led by Andreas Hemmerich at the Institute of Laser Physics at the University of Hamburg presented the first realization of a time crystal in an open system, a so-called dissipative time crystal using ultracold atoms coupled to an optical cavity. The main achievement of this work is a positive application of dissipation – actually helping to stabilise the system's dynamics.
In November 2021, a collaboration between Google and physicists from multiple universities reported the observation of a discrete time crystal on Google's Sycamore processor, a quantum computing device. A chip of 20 qubits was used to obtain a many-body localization configuration of up and down spins and then stimulated with a laser to achieve a periodically driven "Floquet" system where all up spins are flipped for down and vice-versa in periodic cycles which are multiples of the laser's frequency. While the laser is necessary to maintain the necessary environmental conditions, no energy is absorbed from the laser, so the system remains in a protected eigenstate order.
Previously in June and November 2021 other teams had obtained virtual time crystals based on floquet systems under similar principles to those of the Google experiment, but on quantum simulators rather than quantum processors: first a group at the University of Maryland obtained time crystals on trapped-ions qubits using high frequency driving rather than many-body localization[65][66] and then a collaboration between TU Delft and TNO in the Netherlands called Qutech created time crystals from nuclear spins in carbon-13 nitrogen-vacancy (NV) centers on a diamond, attaining longer times but fewer qubits.
In February 2022, a scientist at UC Riverside reported a dissipative time crystal akin to the system of July 2021 but all-optical, which allowed the scientist to operate it at room temperature. In this experiment injection locking was used to direct lasers at a specific frequency inside a microresonator creating a lattice trap for solitons at subharmonic frequencies.
In March 2022, a new experiment studying time crystals on a quantum processor was performed by two physicists at the University of Melbourne, this time using IBM's Manhattan and Brooklyn quantum processors observing a total of 57 qubits.
In June 2022, the observation of a continuous time crystal was reported by a team at the Institute of Laser Physics at the University of Hamburg, supervised by Hans Keßler and Andreas Hemmerich. In periodically driven systems, time-translation symmetry is broken into a discrete time-translation symmetry due to the drive. Discrete time crystals break this discrete time-translation symmetry by oscillating at a multiple of the drive frequency. In the new experiment, the drive (pump laser) was operated continuously, thus respecting the continuous time-translation symmetry. Instead of a subharmonic response, the system showed an oscillation with an intrinsic frequency and a time phase taking random values between 0 and 2π, as expected for spontaneous breaking of continuous time-translation symmetry. Moreover, the observed limit cycle oscillations were shown to be robust against perturbations of technical or fundamental character, such as quantum noise and, due to the openness of the system, fluctuations associated with dissipation. The system consisted of a Bose–Einstein condensate in an optical cavity, which was pumped with an optical standing wave oriented perpendicularly with regard to the cavity axis and was in a superradiant phase localizing at two bistable ground states between which it oscillated.
In May 2024, a team led by Alejandro Fainstein (Instituto Balseiro and Centro Atómico Bariloche, Argentina) and Paulo Santos (Paul Drude Institute, Germany) reported a solid-state continuous time crystal in a driven-dissipative microcavity exciton-polariton condensate with a built-in mechanical clock. The platform consisted of a (Ga,Al)As semiconductor microcavity with GaAs quantum wells in the spacer, pumped by a non-resonant continuous-wave laser that created an incoherent particle bath which relaxed into a Bose–Einstein condensate of lower polaritons. For increasing power of a non-circularly polarized excitation, the condensate pseudo-spin spontaneously underwent Larmor-like precession without an external magnetic field or pulsed drive, a signature of a continuous time crystal. The precession frequency could be controlled by the excitation power and locked to self-sustained coherent breathing vibrations of the microcavity at about 20 GHz, stabilizing the time crystal. At higher drive powers, the system exhibited period doubling relative to the phonon frequency, realizing a discrete time crystal phase under continuous excitation in the same device.
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".
In March 2025, researchers at TU Dortmund University observed complex nonlinear behavior in a semiconductor-based time crystal made of indium gallium arsenide. By periodically driving the system with laser pulses, they uncovered transitions from synchronized oscillations to chaotic motion. The system exhibited structures such as the Farey tree sequence and the devil's staircase — patterns never before seen in semiconductor time crystals — offering new insights into dynamic phase transitions and chaos in driven quantum systems.
s
See World’s First Video of a Space-Time Crystal
By Max-Planck-GesellschaftFebruary 24, 2021
A team of researchers has succeeded in creating a micrometer-sized space-time crystal consisting of magnons at room temperature. With the help of an ultra-precise X-ray microscope, they were able to capture the recurring periodic magnetization structure in a movie.
Periodic pattern consisting of magnons is formed at room temperature.
https://www.youtube.com/watch?v=kUY3TglEUCU
Space-Time Crystal // MaxPlanckSociety
World's first video of a space-time crystal -- Periodic pattern consisting of magnons is formed at room temperature
A team of researchers has succeeded in creating a micrometer-sized space-time crystal consisting of magnons at room temperature. With the help of an ultra-precise X-ray microscope, they were able to capture the recurring periodic magnetization structure in a movie.
A
space time crystal formed by magnons filmed with the
scanning transmission X-ray microscope Maxymus.
https://www.mpg.de/16401528/world-s-first-video-recording-of-a-space-time-crystal
A German-Polish research team has succeeded in creating a micrometer-sized space-time crystal consisting of magnons at room temperature. With the help of the scanning transmission X-ray microscope Maxymus at Bessy II at Helmholtz Zentrum Berlin, they were able to film the recurring periodic magnetization structure in a crystal. Published in the Physical Review Letters, the research project was a collaboration between scientists from the Max Planck Institute for Intelligent Systems in Stuttgart, Germany, the Adam Mickiewicz University and the Polish Academy of Sciences in Poznań in Poland.
Order in space and a periodicity in time
A crystal is a solid whose atoms or molecules are regularly arranged in a particular structure. If one looks at the arrangement with a microscope, one discovers an atom or a molecule always at the same intervals. It is similar with space-time crystals: however, the recurring structure exists not only in space, but also in time. The smallest components are constantly in motion until, after a certain period, they arrange again into the original pattern.
In 2012, the Nobel Prize winner in physics Frank Wilczek discovered the symmetry of matter in time. He is considered the discoverer of these so-called time crystals, although as a theorist he predicted them only hypothetically. Since then, several scientists have searched for materials in which the phenomenon is observed. The fact that space-time crystals actually exist was first confirmed in 2017. However, the structures were only a few nanometers in size and formed only at very cold temperatures below minus 250 degrees Celsius. The fact that the German-Polish scientists have now succeeded in imaging relatively large space-time crystals of a few micrometers in a video at room temperature is therefore considered groundbreaking. But also because they were able to show that their space-time crystal, which consists of magnons, can interact with other magnons that encounter it.
An exceptional experiment succeeded
"We took the regularly recurring pattern of magnons in space and time, sent more magnons in, and they eventually scattered. Thus, we were able to show that the time crystal can interact with other quasiparticles. No one has yet been able to show this directly in an experiment, let alone in a video," says Nick Träger, a doctoral student at Max Planck Institute for Intelligent Systems who, together with Pawel Gruszecki, is first author of the publication.
In their experiment, Gruszecki and Träger placed a strip of magnetic material on a microscopic antenna through which they sent a radio-frequency current. This microwave field triggered an oscillating magnetic field, a source of energy that stimulated the magnons in the strip – the quasiparticle of a spin wave. Magnetic waves migrated into the strip from left and right, spontaneously condensing into a recurring pattern in space and time. Unlike trivial standing waves, this pattern was formed before the two converging waves could even meet and interfere. The pattern, which regularly disappears and reappears on its own, must therefore be a quantum effect.
Gisela Schütz, Director at Max Planck Institute for Intelligent Systems who heads the Modern Magnetic Systems Department, points out the uniqueness of the X-ray camera: "Not only can it make the wavefronts visible with very high resolution, which is 20 times better than the best light microscope. It can even do so at up to 40 billion frames per second and with extremely high sensitivity to magnetic phenomena as well."
"We were able to show that such space-time crystals are much more robust and widespread than first thought," says Pawel Gruszecki, a scientist at the Faculty of Physics of the Adam Mickiewicz University in Poznań. "Our crystal condenses at room temperature and particles can interact with it – unlike in an isolated system. Moreover, it has reached a size that could be used to do something with this magnonic space-time crystal. This may result in many potential applications."
Joachim Gräfe, former research group leader in the Department of Modern Magnetic Systems and last author of the publication, concludes: "Classical crystals have a very broad field of applications. Now, if crystals can interact not only in space but also in time, we add another dimension of possible applications. The potential for communication, radar or imaging technology is huge."
https://www.sciencedirect.com/topics/physics-and-astronomy/magnon
Magnons : A spin wave is defined as a collective motion of magnetic moments in magnetically ordered materials, playing a crucial role in spintronics as a carrier of spin current, which is the flow of spin angular momentum.
https://pubmed.ncbi.nlm.nih.gov/33605763/
journals.aps.org/prl/abstract/10.1103/PhysRevLett.126.057201
Phys. Rev. Lett. 126, 057201 – Published 3 February, 2021
Real-Space Observation of Magnon Interaction with Driven Space-Time Crystals
Nick Träger, et al
Abstract -- The concept of space-time crystals (STC), i.e., translational symmetry breaking in time and space, was recently proposed and experimentally demonstrated for quantum systems. Here, we transfer this concept to magnons and experimentally demonstrate a driven STC at room temperature. The STC is realized by strong homogeneous microwave pumping of a micron-sized permalloy (Py) stripe and is directly imaged by scanning transmission x-ray microscopy (STXM). For a fundamental understanding of the formation of the STC, micromagnetic simulations are carefully adapted to model the experimental findings. Beyond the mere generation of a STC, we observe the formation of a magnonic band structure due to back folding of modes at the STC’s Brillouin zone boundaries. We show interactions of magnons with the STC that appear as lattice scattering, which results in the generation of ultrashort spin waves (SW) down to 100-nm wavelengths that cannot be described by classical dispersion relations for linear SW excitation. We expect that room-temperature STCs will be useful to investigate nonlinear wave physics, as they can be easily generated and manipulated to control their spatial and temporal band structures.
DISCRETE SPACE-TIME CRYSTAL DEVICE AND METHOD OF FORMING SAME -- WO2026174186 (PDF )
Excerpts :
[0011] SUMMARY OF THE DISCLOSURE
...[0012] Various embodiments of the present disclosure relate to classical discrete space-time crystals (DSTCs) in a chiral nematic liquid crystal (LC) system. Electrical switching of LCs is at the heart of the modern LC-enabled industries, including information displays and electrooptic devices; however, advantages of time-crystal emergence in these soft matter systems were never analyzed and have not been utilized. As set forth in more detail below, by applying a Floquet electrical signal to a chiral nematic LC sandwiched between parallel electrodes, both the spatial and temporal symmetries of emergent LC's structure revealed by optical images can be broken discretely and spontaneously, and the internal temporal periodicity of the system doubles in relation to the external drive. Both 1+1 dimensional (1 + ID) and 2+1 dimensional (2 + ID) discrete space-time crystals are observed, with the discrete time crystallization phases depending on temperature and external driving schemes, as illustrated by constructing a phase diagram. The rigidity (robustness) of the time crystals against temporal perturbation and spatial defects has been verified, with the DSTC phase maintaining order locally for a remarkably long time. Moreover, in analogy to defects in crystals, we also observe defects and their dynamics within the DSTCs.
[0013] In accordance with examples of the disclosure, a device includes a first substrate having a first electrical contact thereon, a second substrate having a second electrical contact thereon, a doped chiral nematic liquid crystal material between the first substrate and the second substrate, and a power supply coupled to one or more of the first electrical contact and the second electrical contact. The power supply can be configured to apply an applied signal having a first period, wherein, in response to the applied signal, a period of an optical signal of light emitted from the device is an integer multiple of the first period. In accordance with examples of these embodiments, the doped chiral nematic liquid crystal materialcomprises an ionic dopant. The ionic dopant can be or include a solvable ionic dopant that increases the electrical instability of the nematic liquid crystal material. The doped chiral nematic liquid crystal material can include greaterthan 0 and less than 0.2 wt% of the ionic dopant or greater than 0.05 wt% and less than 0.2 wt% of the ionic dopant...
[0045] As set forth in more detail below, various examples of the disclosure provide methods and devices that utilize 1+1-dimensional and 2+1-dimensional discrete space-time crystals in a liquid crystal system driven by a Floquet electrical signal. These classical time crystals comprise particle-like structural features and exist over a wide range of temperatures and electrical driving conditions. The phenomenon-enabling period-doubling effect in 1+1-dimensional discrete space-time crystals comes from their topological Majorana-like quasiparticle features, where periodic inter transformations of co-existing topological solitons and disclinations emerge in response to external stimuli and play pivotal roles. Exemplary discrete space-time crystals exhibit robustness against temporal perturbations and spatial defects. Our findings show that the simultaneous symmetry breaking in time and space can be a widespread occurrence in numerous open systems, not only in quantum but also in a classical soft matter context.
[0046] Electrical switching of liquid crystals (LCs) is at the heart of the modern LC-enabled industries, including information displays and electro-optic devices. By applying a Floquet electrical signal to a chiral nematic LC sandwiched between (e.g., parallel) electrodes, both the spatial and temporal symmetries of the emergent LC's structure revealed by opticalimages can be broken discretely and spontaneously, and the internal temporal periodicity of the system doubles in relation to the external drive. Both 1+1 dimensional (1 + ID) and 2+1 dimensional (2 + ID) discrete space-time crystals are observed, with the time crystallization phases depending on temperature and external driving parameters, as illustrated by constructing a phase diagram. Computer simulations demonstrate that the period-doubling effect is intimately related to the inter-transformations, generations, and annihilations of coexisting topological solitons and singular disclinations. The different states of these topological objects can be viewed as the particle and anti-particle states of the observed Majorana-like quasiparticles (a classical analogue of Majorana particles) forming our 1 + ID space-time crystals. The rigidity (robustness) of the time crystals against temporal perturbation and spatial defects, with the discrete space-time crystals (DSTCs) phase maintaining order locally for a remarkably long time. Moreover, a candidate for a fractional discrete space-time crystal is observed when changing the sample 22 thickness. Our findings provide a new paradigm of time-crystalline LC meta matter, with technological utility....
[0050] Doped chiral nematic liquid crystal material 110 can be any suitable doped chiral nematic liquid crystal material that exhibits discrete space-time crystal as described herein. In accordance with examples, the doped chiral nematic liquid crystal material comprises 1 + ID discrete space-time crystals. In accordance with further examples, the doped chiral nematic liquid crystal material comprises 2 + ID discrete space-time crystals. In accordance with yet further examples, the doped chiral nematic liquid crystal material comprises one or more of 5CB(4-Cyano-4'-pentylbiphenyl) and E7 (a mixture of 4-Cyano-4'-pentylbiphenyl, 4'-Heptyl-4-biphenylcarbonitrile, 4'-Octyloxy-4-biphenylcarbonitrile or 4-Cyano-4'-Pentylterphenyl). In accordance with further examples, the doped chiral nematic liquid crystal material comprises an ionic dopant. The ionic dopant can be, for example, a solvable ionic dopant that increases the electrical instability of the nematic liquid crystal material. By way of particular examples, the ionic dopant can be or include cetyltrimethylammonium bromide, other cetyl alkyl ammonium compounds, or the like. The doped chiral nematic liquid crystal material can include greater than 0 and less than 0.2 wt% of the ionic dopant or greater than 0.05 wt% and less than 0.2 wt% of the ionic dopant...
[0056] A sample is prepared by sandwiching a chiral nematic LC between two electrically conductive transparent substrates (e.g., substrates 102, 106), where the LC is doped with ionic substances or an ionic dopant. In response to a Floquet electrical signal (e.g., as shown in FIG. 2), the confined LC can spontaneously form spatially periodic configurations, which can be captured by a camera of an optical microscope system (e.g., analyzer 120) or using the system illustrated in FIG. 3.. The spatially varying optical phase retardation pattern is produced by the LC with complex structure of director orientation driven by the field, which can be vividly revealed by inserting an additional first-order full-wave retardation plate (e.g., retardation plate 116, FIGS. 1 and 3). The alternating blue and purple spatial regions shown in FIG. 2 indicate spatial variations in the LC's three-dimensional 10 (3D) structures represented by the locally averaged molecular orientation direction n (dubbed the "director")...
[0065] As noted above, observations of classical discrete space-time crystals reveal the generality of time crystallization dynamics. In our classical LC-based discrete space-time crystals, the time symmetry is discretely broken while being accompanied with space symmetry breaking, yielding 1 + ID and 2 + ID space-20 time crystals. These DSTCs can be described as comprising arrays of spatially and temporarily localized quasiparticles interacting with each other within the overall out-of-equilibrium setting (FIG. 12 (a) and (b)) observed experimentally. These classical DSTCs may offer a new route to creating various forms of meta matter, where the basic building blocks are localized not only in space, but also in time, as well as have the emergent topological nature. They may allow designing spatially or temporally localized structures as versatile reconfigurable beam deflectors, steerers, and lasing elements. The examined rigidity of our classical DSTC allows for maintaining order locally over times much longer than the discrete time crystals in quantum systems, which is because, although the classical system cannot enjoy the benefits of many-body localization, there is no quantum coherence, and the relative noise from thermal fluctuations is much smaller for soft matter systems when considering the system's internal elasticity-mediated interactions.
CONTINUOUS SPACE-TIME CRYSTAL DEVICE AND METHOD OF FORMING AND USING SAME -- WO2026183497 (PDF )
Excerpts:
[0016] Embodiments of the disclosure relate to devices that exhibit continuous spacetime crystal behavior. Exemplary devices include a first substrate, a second substrate, a photo-responsive dye layer, and a nematic liquid crystal material between the first substrate and the second substrate, wherein, in response to an applied light, a continuous space-time crystallization phase forms within the nematic liquid crystal material. In accordance with examples of the disclosure, the nematic liquid crystal material comprises one or more of 5CB(4-Cyano-4'-pentylbiphenyl) or E7 (a mixture of 4-Cyano-4'-pentylbiphenyl, 4'-Heptyl-4-biphenylcarbonitrile, 4'-Octyloxy-4-biphenylcarbonitrile and 4-Cyano-4'-Pentylterphenyl).
[0017] In accordance with further examples, the photo-responsive dye layer is coated onto an inner surface of one or more of the first substrate orthe second substrate. The photo-responsive dye layer can be or include an azo compound, such as azobenzene or 2-(4-dimethylamino-phenylazo)-N-(3-triethoxysilane-propyl)-benzamide (dMR). In accordance with examples of the disclosure, the photo-responsive dye layer is sensitive to certain (e.g., visible) wavelengths and insensitive to other (e.g., visible) wavelengths of light. The applied light can be or include ambient light, light from a light source, and/or a polarized light source. Exemplary devices can exhibit 1 + ID continuous space-time crystal behavior, 0 + ID continuous space-time crystal behavior, 2 + ID continuous space-time crystal behavior, 3 + ID continuous space-time crystal behavior...
[0083] 6 (a)-(c)). This "time watermark" can be fabricated at low cost, since a 1 cm x 1 cm x 2 pm sample requires only ~2 x 10-14 g of LC and the surface monolayer with <10-14 g of the azobenzene dye that can be sandwiched between glass or other surfaces. Due to the spontaneous temporal symmetry breaking, the time crystals can be exploited in pseudorandom number generators. By combining multiple CSTCs, the synthetic systems generate unique, fingerprint-like states corresponding to the spatiotemporal topological soliton arrays (FIG. 6 (d),(e)) each time they emerge, maintaining order for a remarkably long time (FIG. 8). Additionally, the phases of CSTCs can be tuned by smoothly switching the driving light intensity (FIG. 6 (f)), allowing for the creation of a 2 + ID barcode via superimposing multiple CSTCs (FIG. 6 (g)). As the 2D barcode can store over 100 times more bits than a ID barcode, the capacity of storing information with proper data coding in higher-dimensional barcodes like 2 + ID is effectively unbounded due to the extra temporal coordinate (>100,000 bits per second). The intrinsic robustness of the space-time order, supported by the time-crystallinity of topological solitons, could further enhance error correction capabilities of 2 + ID barcodes. For the third level of the anti-counterfeiting uses, the temporal periodicity of CSTCs can be utilized as keys in cryptographic systems. For example, with two CSTCs having temporal periodicities Ti and T2 (assuming TI<T2), an identical spatial pattern only recurs after a time interval of (TIXT2)/(T2-TI) (FIG. 6 (h)), which could be used to check authenticity. CSTCs with different temporal periodicities can be introduced by incorporating pre-programmed light intensity filters. The entire system may display disorder-like spatiotemporal behavior, however, within each CSTC, the time-crystalline order with a specific temporal periodicity can be maintained for a longtime. Overall, the entire fingerprint-like CSTC states can be precisely predicted if the information about the pre-programmed light intensity filters (the system's keys) is known (FIG. 6 (i)) and can be utilized for anti-counterfeiting purposes.
[0084] Exemplary Materials and sample preparation
[0085] The glass substrates were coated with photo-responsive material 2-(4-dimethylamino-phenylazo)-N-(3-triethoxysilane-propyl)-benzamide (dMR)<3561>, which is sensitive to the blue and violet light and insensitive to the red light. To coat monolayer dMR on the glass surfaces, we submerge the glass plates into a 1 wt% solution of dMR in toluene at a temperature of 45 °C. After a 90-min submersion, the dMR molecules are bonded to the glass surfaces; we then wash away the excess dMR by a toluene rinse, followed by blowing the glass plates with dry nitrogen and curing them at 115 °C for 2 hours. The LC cells are constructed using two glass substrates coated with monolayers of dMR, where the cell thickness d = 2-4 pm is defined by glass spheres mixed with a methanol-diluted epoxy. Once the epoxy has cured, we fill the cell via capillary forces with nematic 4-cyano-4'-pentylbiphenyl (5CB, EM Chemicals).
[0086] Quasi-long-range order and relative time phases of the CSTC
[0087] To obtain the time order of CSTCs, we calculate the correlation function G in time coordinate, which is a common tool for analyzing the spatial order of crystals and liquid crystals. For crystals, the spatial correlation function G(r) is a constant, where r is the distance between the two measured positions. For smectic liquid crystals, the spatial correlation function G(r) decays as ~r (<(<0.15) along the smectic layers, which is a quasi-long-range order??. For CSTCs, we measure the normalized digital signal <t>i(t) of each pixel at different times, where subscript / denotes spatial coordinates. The correlation function G(t)= Ii (t)= ^( lOlH^ fO)) is calculated with 2200 spatial pixels and 9000 temporal frames, showing a quasi-long-range order in time.
[0088] In experiment, the relative time phases are measured from 100 experimental realizations. In each realization, the driving light is blocked with red colour filter at first(allowing only red color light to pass through). Subsequently, we remove the red color filter, and the CSTC spontaneously emerges. After a time interval At (At = 60s), we start measuring the light signals from the recorded video, and calculate the phase using Fast Fourier Transform analysis function in MATLAB (MathWorks).
...For example, the photo-responsive dye layer can be or include azobenzene or 2-(4-dimethylamino-phenylazo)-N-(3-triethoxysilane-propyl)-benzamide (dMR)...
[0090] As described above, device 1500 can exhibit 1 + ID continuous space-time crystal behavior, 0 + ID continuous space-time crystal behavior, 2 + ID continuous space-time crystal behavior, or 3 + ID continuous space-time crystal behavior. In accordance with further examples, an output of light from the device is dependent on the wavelength(s) and polarization of an input light to the device. In some cases, the nematic liquid crystal material includes anisotropic particles 1510 therein. Exemplary anisotropic particles include 4-Cyano-4'-pentylbiphenyl having a first dimension of between about 0.1 and 10 nm or about 0.5 nm and a second dimension of between about 0.5 and 5 nm or about 2 nm. In some cases, the second dimension is at least about 3 to about 10 times greater than the first dimension.[0074] In accordance with further examples, device 1500 forms part of a telecommunications device, cryptography device, optical device, photonic time crystal generator, anti-counterfeiting system, or barcode device, or the like.
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Given a system in which the forces are periodic—such as a pendulum under a periodic driving force, or an oscillating circuit driven by alternating current—the overall behavior of the system is not necessarily fully periodic. For instance, consider a child being pushed on a swing: although the motion is driven by regular, periodic pushes, the swing can gradually reach greater heights while still oscillating to and fro. This results in a combination of underlying periodicity and growth.
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Scientists discovered hopfion crystals – which are flying in spacetime
Figure 1 (a) Concept of a 1D space-time hopfion crystal. (b) Spin textures on x-y transverse cross-section and (c) that on selective longitudinal cross-sections through the center of a unit hopfion lattice. (d) The arrow indicator of all states of spin on a parametric sphere. (e) The isospin lines of selected spins on the equator of the parametric sphere form a torus-knot configuration. (f) All isospin lines of a hopfion lattice show a layer-by-layer torus-knot fibration configuration.
An internationally joint research group between Singapore and Japan has unveiled a blueprint for arranging exotic, knot-like patterns of light into repeatable crystals that extend across both space and time. The work lays out how to build and control “hopfion” lattices using structured beams at two different colors, pointing to future systems for dense, robust information processing in photonics.
Hopfions are three-dimensional topological textures whose internal “spin” patterns weave into closed, interlinked loops. They have been observed or theorized in magnets and light fields, but previously they were mainly produced as isolated objects. The authors show how to assemble them into ordered arrays that repeat periodically, much like atoms in a crystal, only here the pattern repeats in time as well as in space.
Starting from this one-dimensional chain, the researchers then describe how to sculpt higher-order versions whose topological strength can be dialed up or down. In their scheme, one can tune an integer that counts how many times the internal loops wind and even flip its sign by swapping the two wavelengths. In simulations, the resulting fields show near-ideal topological quality when integrated over a full period.
Beyond time-only repetition, the paper outlines a route to true three-dimensional hopfion crystals: a far-field lattice formed by an array of tiny emitters with tailored phase and polarization, all driven at two close colors. The lattice naturally divides into subcells with opposite local topology, yet preserves a clean, alternating pattern across the whole structure. The authors sketch practical layouts using dipole arrays, grating couplers, or microwave antennas to realize the source arrangement.
Unlike earlier optical hopfions that relied on beam diffraction along the propagation axis, this design works in the joint space-time domain at a fixed plane, with periodic beating doing the heavy lifting. The team also discusses when the structures can “fly” some distance while maintaining their topology, and when diffraction undermines their integrity.
Why it matters: topological textures like skyrmions have already reshaped ideas for dense, low-error data storage and signal routing. Extending that toolkit to hopfion crystals in light could unlock high-dimensional encoding schemes, resilient communications, atom trapping strategies, and new light-matter interactions. “The birth of space-time hopfion crystals,” the authors write, opens a path to condensed, robust topological information processing across optical, terahertz, and microwave domains.
https://en.wikipedia.org/wiki/Hopfion
Hopfion
A hopfion is a topological soliton. It is a stable three-dimensional localised configuration of a three-component field n → = ( n x , n y , n z ) {\displaystyle {\vec {n}}=(n_{x},n_{y},n_{z})} of unit length with a knotted topological structure. They are the three-dimensional counterparts of 2D skyrmions, which exhibit similar topological properties in 2D. Hopfions are widely studied in many physical systems over the last half century.
The soliton is mobile and stable: i.e. it is protected from a decay by an energy barrier. It can be deformed but always conserves an integer Hopf topological invariant. It is named after the German mathematician, Heinz Hopf...
https://arxiv.org/html/2504.03981v1
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Abstract -- Hopfions, three-dimensional topological solitons characterized by nontrivial Hopf indices, represent a fundamental class of field configurations that emerge across diverse areas of physics. Despite extensive studies of isolated hopfions, a framework for constructing spatially ordered arrays of hopfions, i.e., hopfion crystals, has been lacking. Here, we present a systematic approach for generating hopfion crystals with cubic symmetry by combining the Hopf map with rational mapping techniques. By superposing helical waves in ℝ4, we construct hopfion crystals with tunable Hopf indices and controllable topology. We demonstrate simple cubic, face-centered cubic, and body-centered cubic hopfion crystals, and extend our framework to create crystals of more complex topological structures, including axially symmetric tori, torus links, and torus knots with higher Hopf indices. Our results provide a foundation for searching hopfions in real materials and studying their collective phenomena.