Time crystals in periodically driven systems Free

Discrete time-translation symmetry breaking manifests itself in three key ways:

► Broken symmetry: After a possible initial transient period, the system exhibits late-time oscillations with a period longer than that of the drive.

► Crypto-equilibrium: No entropy is generated by the late-time oscillations.

► Rigid long-range order: The oscillations remain in phase over arbitrarily long distances and times.

Making those notions more precise is a bit subtle. When talking about states of matter, one typically starts with a preferred state, the ground state, but not in discrete time crystals: Periodically driven systems do not have a ground state. Again, a comparison to spatial crystals helps. Consider a Floquet system that’s driven at period $T$⁠, so that for any time $t$ the Hamiltonian $Ht$ satisfies $Ht + T=Ht$⁠. The discrete spatial-translation symmetry of a one-dimensional spatial crystal leads to electron states having a quasi-momentum that is only defined modulo $2π/a$⁠, where $a$ is the period of the crystal lattice. Analogously, the discrete time-translation symmetry of a periodically driven system leads to eigenstates having a quasi-energy that is only defined modulo $Ω≡2π/T$⁠, and there is no preferred state with a minimum value of the quasi-energy. Thus any definition of TTSB in periodically driven, nonequilibrium systems cannot be cast in terms of ground-state or low-energy properties.

But if we don’t restrict ourselves to ground-state properties, then oscillations with a frequency $ω≠Ω$−a requirement for TTSB—can be realized fairly easily. Even for a simple harmonic oscillator, if our initial state is a superposition of two eigenstates then the system will naturally exhibit oscillations at a frequency given by the difference between the eigenstate energies. But in most systems, a generic initial state will not lead to late-time oscillations with $ω≠Ω$⁠. So any good definition of TTSB in Floquet systems must generalize the ground-state or equilibrium notion of spontaneous symmetry breaking in such a way that the oscillatory behavior does not depend on the choice of initial state.

That requirement—and thus a precise definition of TTSB in periodically driven systems—can be stated in a remarkably compact form: A discrete time crystal is a state of matter in which the Floquet eigenstates are necessarily “cat states,” that is, entangled superpositions of macroscopically distinct states. One immediate corollary is that because any initial physical state we can prepare must be a superposition of such Floquet cat states, all preparable initial states will exhibit oscillations.

Let us further unpack that definition. Since the Floquet eigenstates of a discrete time crystal aren’t preparable, we can readily distinguish the discrete time crystal from, for example, a simple harmonic oscillator. As mentioned above, whether the harmonic oscillator displays time-periodic behavior depends strongly on the choice of initial state one prepares, and most importantly, nothing prohibits us from preparing a harmonic oscillator in its ground state. By stark contrast, in a discrete time crystal every physically preparable initial state will exhibit oscillations at late times.

The key features of discrete time crystals explicitly distinguish them from a multitude of other systems (see the table on page 46) that exhibit oscillations with unexpected periods.¹⁰ The box above offers some insights into those features through a spatial analogue: methane adsorbed on the surface of graphite.

In principle, an oscillation with any frequency $ω≠Ω$ would break the discrete time-translation symmetry. However, we will focus on the subharmonic case, $ω=Ω/n$ for some integer $n>1$⁠; figure 1 depicts such a system with $n=4$⁠.

Suppose that we measure a nonequilibrium Floquet system stroboscopically—that is, at regular fixed intervals, like frames of a movie reel. When viewed at multiples of the drive period, $t=kT$ for $k=1,2,…$⁠, then as depicted in figure 2, there is some time-dependent frame of reference in which we will be unable to tell that the system is not in static equilibrium. Instead of being observed in the system, the oscillations are subsumed into the frame’s time dependence.

We say that a periodically driven system is in crypto-equilibrium if there exists some reference frame, possibly time dependent, in which the system is indistinguishable from a system in thermal equilibrium, if measured stroboscopically. (In fact, if we measure the system at $t=knT$ for positive integers $k$⁠, then there’s no need to go into a moving frame at all; even in the fixed lab frame, the system looks like an equilibrium system.) In many of the cases we discuss below, the appropriate frame is similar to the rotating frame that is routinely used to simplify the analysis of NMR experiments. For a discrete time crystal to be in crypto-equilibrium, however, the rotating frame must play a more powerful role than just simplifying the analysis: It must transform the periodically driven system into a stationary, equilibrium one.

Crypto-equilibrium requires that the periodic drive add no entropy, and it enables a discrete time crystal to exhibit rigid long-range order. By contrast, oscillating chemical reactions and convection are inescapably nonequilibrium: They are irreversible processes that generate entropy and require a constant flux of incoming reactants or a maintained thermal gradient to drive the oscillatory behavior. While the oscillations appear periodic at first sight, they generally fall out of lockstep when probed over longer times and at distant locations. That behavior distinguishes discrete time crystals from other oscillatory nonequilibrium phenomena as a matter of principle.

The long-range order of discrete time crystals is a remarkable phenomenon. Since they are not in thermal equilibrium, their characteristic rigidity is not reliant on low energy or temperature; it depends instead on emergent features that control the strength of energy and quantum fluctuations. To fully appreciate those features, we must delve deeper into the physics of isolated, periodically driven quantum systems.

That periodically driven systems can be in crypto-equilibrium is rather counterintuitive: A generic periodically driven, isolated system will absorb energy until it looks, locally, like an infinite-temperature state.¹¹ If that were the full story, it would preclude time crystals, since Floquet systems would all simply end up as featureless infinite-temperature states in the thermodynamic limit. Luckily, there are two situations (at least) in which that discouraging conclusion is not correct.

First, if the drive frequency is very large compared with the local energy scales of the system, then the system can only absorb energy from the drive by spreading it out over many excitations.^12–14 Consequently, heating occurs very slowly, and there is a long-lived quasi-steady state—a so-called “pre-thermal” state—in which ordered phases of matter can occur.

Second, if a system has a high density of frozen-in impurities, then a phenomenon called many-body localization can occur.¹⁵ MBL prevents the spreading of energy because the impurities trap excitations, even highly excited ones. For periodically driven Floquet–MBL systems, the energy spreading and thermalization necessary for drive-induced heating cannot occur, so nonequilibrium ordered states of matter can survive indefinitely. Those two loopholes—pre-thermal states and MBL—both basically rely on the discreteness of quantum mechanical energy levels, and they open the door for stabilizing various driven systems, including discrete time crystals.

The quantum mechanical nature of pre-thermal states and MBL raises a natural question: Does time-crystalline order require quantum mechanics? In general, broken-symmetry states have a classical flavor. At times longer than a scale set by the inverse temperature $1/kBT$ and at distances longer than a related length scale, thermal fluctuations tend to dominate quantum fluctuations, and the system admits an effective classical description. One can thus ask about the specific role played by quantum mechanics. Is it only important insofar as it prevents the system from heating up? Or is quantum mechanics essential for enabling the subharmonic synchronization characteristic of a discrete time crystal?

Recent work suggests the former. A dissipative, coupled chain of classical nonlinear pendula, it turns out, can exhibit a phase transition between a time-crystalline phase and a phase with unbroken symmetry.¹⁶ That suggests a third strategy for suppressing the order-ravaging, drive-induced heating of Floquet systems: coupling to a dissipative bath.³ The system would not be in crypto-equilibrium, since entropy is created in the bath. But when the drive is noisy, such a system may be smoothly connected in some precise way to a crypto-equilibrium system. It is natural to ask, given their departure from equilibrium, whether there could be time crystals that break continuous time-translation symmetry. In the pre-thermal regime, the answer is in fact yes: Their experimental signatures have been observed¹⁷ and recently clarified.

A simple set of spin models can host discrete time crystals.^3–6 In particular, consider a Floquet Hamiltonian with $Ht + T=Ht$ and total evolution time $T=t1+t2+t3$⁠:

Here, $H1$⁠, $H2$⁠, and $H3$⁠, illustrated in figure 3, are time-independent Hamiltonians given by

where $σix$⁠, $σiy$⁠, and $σiz$ represent the Pauli spin operators for spin $i$⁠. For a 1D system, many-body localization occurs, and this model supports a period-$2T$ discrete time crystal provided that the local magnetic field vector $hi$ is small and there is sufficient disorder. In more than one dimension, pre-thermal states occur if the energy scales of $H2$ and $H3$ are small compared to the drive frequency $ω$⁠, and if $g≈1$ there will again be a period-$2T$ discrete time-crystal phase for a finite fraction of possible initial states. Similar, slightly more complicated models admit period-$nT$ discrete time crystals; the simplest of them feature an $n$-level system with clock-like transitions between the levels.⁸ 

The above model has a few simple limits. For $g=1$ and $J=hx=hy=0$⁠, the first part of the Floquet sequence applies an exact $π$-pulse, which flips all the spins, and the third part conserves the $z$-component of each spin. As a result, the spins are reversed after one period and return to their initial configuration after two. That behavior is completely trivial, though, since the spins do not interact with one another. Consequently, much like the period doubling of individual nonlinear oscillators,¹⁰ the behavior does not survive the addition of perturbations and fluctuations.

Now suppose we turn on a finite spin–spin coupling $J$⁠. $H1$ still applies an exact $π$-pulse, and $H3$ conserves the spins’ $z$-components. But the net behavior is no longer trivial—two key differences emerge on closer inspection. First, the Floquet eigenstates are now superpositions of two states that differ by a flip of all spins; hence, they are macroscopic cat states. Second, and most remarkably, the period-doubled behavior of the system is qualitatively unchanged by small perturbations such as $g≠1$⁠. That is precisely a manifestation of the rigidity that we expect of an ordered, broken-symmetry phase. Although the perturbations lead to fluctuations, those fluctuations do not cause the discrete time crystal’s oscillations to fall out of phase at distant locations or distant times. The system is long-range ordered!

Two essential features unify spin models that support discrete time crystals: strong interactions and an effective magnetic field that implements an approximate but not necessarily exact $π$-pulse. Those two attributes are realizable in a wide range of quantum optical systems ranging from trapped ions and solid-state spin defects to Rydberg atoms and polar molecules. The signatures of time-crystalline order were reported last year in two wildly disparate platforms: a 1D array of trapped ytterbium ions⁷ (figure 4a) and a 3D ensemble of nitrogen–vacancy spin defects (NV centers) in diamond (figure 4b).⁸ We consider each in turn.

► Trapped ions. Chris Monroe’s group at the University of Maryland⁷ worked with 1D chains of up to 14 ¹⁷¹Yb⁺ ions. The interplay between the trapping forces and the ions’ natural Coulomb repulsion produced the (spatial) crystalline configuration seen in figure 4a. For each ion, the researchers constructed from two hyperfine states an effective spin-½ degree of freedom that could be optically prepared, manipulated, and observed.

The researchers repeatedly cycled their ion chain through the three-Hamiltonian sequence of figure 3. One pair of lasers generated approximate $π$-pulses to flip the spins. A second set of lasers coupled the ions’ hyperfine spin states to the motional vibrations of the crystalline configuration and produced long-range spin–spin interactions that fell off as a tunable power law. Lastly, a tightly focused laser beam that was scanned along the ion chain addressed each spin in turn and generated a site-dependent, disordered longitudinal field.

Long-range interactions, however, tend to disfavor the MBL needed to realize time-crystalline behavior. So the researchers worked in a parameter regime in which, according to numerical simulations, the disorder is strong enough and the long-range power law weak enough that the system could exist in the putative MBL phase.

With the ions’ Floquet evolution in hand, the experimenters observed two of the key features predicted for time-crystalline order: interaction-stabilized period doubling (figure 4e) and a phase transition to a non-time-crystalline state as $π$-pulse deviations increased.

► Nitrogen–vacancy centers. Mikhail Lukin’s group at Harvard University observed signatures of time-crystalline order in ensembles of NV centers.⁸ Each NV center behaved as a spin-1 magnetic impurity in the surrounding diamond lattice (figure 4b). By applying a small magnetic field along the NV center’s axis, the researchers lifted the degeneracy of the $ms=±1$ states and isolated an effective two-level spin system. The resulting spin system could be initialized, manipulated, and detected using a combination of optical and microwave radiation. The NV centers interacted with one another via long-range magnetic dipole–dipole interactions, and disorder was naturally present in the system because of other randomly distributed paramagnetic impurities inside the lattice.

Those features all seem quite reminiscent of the trapped-ion system, and the systems’ signatures of time-crystalline order were similar. But essential differences exist between the trapped-ion and NV platforms, which makes the similarity of their observations all the more surprising. In particular, NV systems are 3D, so many-body localization is not expected. Moreover, it does not appear that pre-thermal states occur in NV systems, either; the energy of the initial state is too high. Yet within a comparable number of Floquet evolution cycles, the NV experiments—like the trapped ions—demonstrated that interactions help stabilize the subharmonic Fourier response of a discrete time crystal (figure 4f). Even more recently, Sean Barrett’s group at Yale University has reported similar observations in NMR experiments focusing on the spin-½ phosphorus nuclei in ammonium dihydrogen phosphate (ADP).¹⁸ 

Whereas the trapped-ion experiments represent a more canonical model of a discrete time crystal, the NV and ADP experiments suggest the possibility that a broader class of time-crystalline behavior can be observed, so long as a mechanism exists for slowing down thermalization and heating. It may be that the interplay between dimensionality and dipolar interactions inherently results in a relatively slow approach to thermal equilibrium and that the observed time-crystalline order is merely manifest in that transient regime.

The recent experiments present preliminary evidence for a discrete time crystal. But improved coherence times are needed if experiments are to truly demonstrate long-range temporal ordering—that is, the observed spin oscillations remain in phase over extended times.

The discrete time crystals discussed here are not the only phases of matter of periodically driven systems. Such systems can have other phases, including so-called topological phases and symmetry-protected topological (SPT) phases. The most remarkable of them share a property with discrete time crystals: They exhibit nonstationary behavior, even when observed stroboscopically. That behavior is more subtle in topological or SPT phases, however, than in discrete time crystals.

Unlike topological and SPT phases, discrete time crystals have the additional property that they are genuine spontaneous symmetry-breaking phases. Moreover, the symmetry that they break, time-translation symmetry, cannot be broken in a stable phase in thermal equilibrium. Thus discrete time crystals lie at the intersection between nonequilibrium and exotic spontaneous symmetry-breaking phases of matter. Although it is too early to say, the greatest long-term impact of time crystals may well be that they have opened our eyes to the new world of nonequilibrium phases of matter.

Norman Yao is an assistant professor in the physics department at the University of California, Berkeley. Chetan Nayak is the director of Microsoft Station Q in Santa Barbara, California, and a professor in the physics department at the University of California, Santa Barbara.