Time crystal

History

The idea of a space-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 space-time crystals were impossible.

Subsequent work developed more precise definitions of time translation symmetry breaking 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 time translation symmetry breaking in a non-equilibrium system, and pioneering work has demonstrated that quantum systems subject to periodic driving can indeed exhibit discrete time translation symmetry breaking.

Krzysztof Sacha at Jagiellonian University in Krakow, Poland, has predicted the behaviour of time crystals in a periodically driven many-body system.

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 independently create a time crystal successfully.

Symmetry

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 than the initial conditions or to 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.

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

For a long time physicists believed that symmetries in the laws of 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 led 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 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 possibility 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 objectively 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

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 the process of magnetisation 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 spatial 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 crystal's 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 processes. 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

While ordinary crystals break spatial translational symmetries leading to repeated spatial 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 phenomenon, 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 term 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 behaves like a so-called anyon, named as such because it is governed by "any possible" statistics. When a superconductor 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.

Floquet topological states

Floquet 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-dependent 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.

Floquet 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 than their classical counterparts: For example, it has been theoretically shown that non-equilibrium quantum ratchet systems function far more efficiently than 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.

Compatibility with the laws of thermodynamics

Because 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 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

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 is an open area of research.

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 the 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 radio waves 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.

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.