Quantum physics

Time crystals

Physicists have come up with the mind-boggling concept of a time crystal. This intriguing proposal, which is based on the notion of broken time-translation symmetry, might open up a whole new field of research.

Philosophers, physicists and writers have long pondered the perpetual periodicities in time that make up our lives — from a heartbeat to the motions of the planets. Isaac Newton and Leonardo da Vinci both considered perpetual motion, concluding that it is the stuff of alchemy. Curiously, we now know that perpetual motion is possible, but perpetual oscillation, that's another thing. There is a new debate afoot involving the connection between time and 'broken symmetry'. A snowflake epitomizes this concept, because as it crystallizes, it lowers its energy by breaking the homogeneity and isotropy of water. Such 'spontaneously broken symmetry' permeates the world we live in: for example, the orientation of the liquid crystals in a television screen and the tiny range of forces between subatomic particles both rely on broken symmetry. These cases of broken symmetry leave the homogeneity of time unchallenged, but the world we live in is also full of clocks — of periodicities and oscillations that break time-translation symmetry.

A series of three papers1,2,3 published in Physical Review Letters proposes and explores the evocative new concept of a time crystal. The authors of these papers ask: can temporal periodicities spontaneously develop in a fashion analogous to the spatial order of crystals? Naturally oscillating physical systems abound: for example, the sound of an organ pipe and the workings of clocks in all their variety. But these systems require an input of energy to drive the oscillation. The idea behind a crystal in the time domain is that it requires no energy input. Like its spatial cousin, it will spontaneously form without a driving force.

It turns out that the concept of a time crystal requires grappling with various subtle issues in quantum mechanics. In quantum mechanics, states of definite energy are called stationary states because the rate of change of any observable quantity associated with the state is zero. Paradoxically, this does not preclude states of perpetual motion: a famous example is superconductivity, whereby certain metals, when cooled, support perpetual electric currents. It is these currents that drive the magnetic fields in a magnetic resonance imaging scanner. If a tiny magnetic field is passed through a metal ring, which is then cooled into the superconducting state, the ring spontaneously develops a persistent current. But whereas the spontaneously developed motion is perpetual, the current is homogeneous and constant, so time-translation symmetry is still unbroken.

In physics, the idea of broken symmetry is associated with the concept of an 'order parameter': in a crystal, this is the periodic modulation in its density. Broken symmetry manifests itself through the development of infinite-range correlations of the order parameter. Thus, in a crystal, the density modulates coherently over infinite distances. In quantum physics, broken symmetry is not immediately evident from a system's wavefunction, because the vertices of the crystal are equally likely to occupy any point in space. For example, you cannot see the broken symmetry in a crystal until you measure the position of one of its atomic constituents. The corresponding crystal wavefunction is completely homogeneous in space, because it is a superposition of broken-symmetry states; once you observe a small piece of the crystal, the modulated density becomes visible. The development of these entangled, infinite-range spatial correlations in an otherwise spatially homogeneous wavefunction is called off-diagonal long-range order4.

In the first of the three papers, Wilczek introduces1 the concept of a time crystal, arguing that such a state would exhibit off-diagonal long-range order in the time domain. Wilczek proposes a thought experiment involving the supercurrent spontaneously induced in a superconducting ring threaded by a tiny magnetic field. Suppose, he argues, one were to induce a weak attraction between the particles in the superconductor, causing them to bunch together. Such a state is called a soliton. Provided that the soliton forms without eliminating the superconductivity, then in a small magnetic field a supercurrent will spontaneously emerge, carrying the soliton periodically around the ring to form a simple example of a time crystal (Fig. 1).

Figure 1: Periodicities in space and time.
figure1

a, The wavefunction, Ψ, of a crystal is spatially homogeneous. But when observed at one corner, it is revealed to have infinite-range spatial correlations associated with broken symmetry, exhibiting a type of order called off-diagonal long-range order4. b, A tiny magnetic field, B, introduced into a metal ring induces a persistent 'supercurrent' in the ring, but time-translation symmetry is unbroken. c, Inducing a weak attraction between the particles that carry the supercurrent causes them to bunch together, leading to the development of a 'soliton' in the particle density. Wilczek proposes1 that, if this soliton is carried around with the superflow in the presence of a small magnetic field, time-translation symmetry is broken and the rotating soliton will form a periodic structure in time — a time crystal.

Although crystals owe their microscopic physics to quantum mechanics, their macroscopic behaviour is governed by classical mechanics: time crystals are surely no different. In the second paper, Shapere and Wilczek show2 that, to understand time crystals, one needs an advanced version of classical mechanics based on Lagrange's 'least action' principle, in which the Newtonian motion of particles is understood as the path that minimizes a combination of kinetic and potential energy called the action. They find that time crystals are classically feasible, but discover that, curiously, once a time crystal develops, the concept of energy becomes ambiguous and the action principle takes its place.

But if time crystals exist, how do we detect them? In the last of the three papers, Li and collaborators3 suggest a method to engineer a time crystal using an ion trap. When ions are trapped in a circular ring, their repulsive interactions cause them to arrange themselves into a spatial crystal. Following Wilczek's proposal1, Li et al. suggest that, when cooled in a small magnetic field, the crystal will spontaneously rotate. This idea is closely related to the idea of 'supersolids', itself a subject of great recent interest and controversy5.

Perhaps the most important aspect of the time-crystal notion is that it envisages broken time-translation symmetry as an equilibrium phenomenon, rather than a non-equilibrium response to a driving force. This is an exciting yet controversial6 development, which if right will open up a whole new field of research. With the discovery of superconductors, the debate on perpetual motion was brought to an end. With time crystals, a new debate has just begun.

References

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    Shapere, A. & Wilczek, F. Phys. Rev. Lett. 109, 160402 (2012).

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    Li, T. et al. Phys. Rev. Lett. 109, 163001 (2012).

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    Bruno, P. Preprint at http://arxiv.org/abs/1210.4128 (2012).

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Correspondence to Piers Coleman.

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Coleman, P. Time crystals. Nature 493, 166–167 (2013). https://doi.org/10.1038/493166a

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