Colloquium: Quantum and classical discrete time crystals

The spontaneous breaking of time-translation symmetry has led to the discovery of a new phase of matter: the discrete time crystal. Discrete time crystals exhibit rigid subharmonic oscillations that result from a combination of many-body interactions, collective synchronization, and ergodicity breaking. This Colloquium reviews recent theoretical and experimental advances in the study of quantum and classical discrete time crystals. The breaking of ergodicity is focused upon as the key to discrete time crystals and the delaying of ergodicity as the source of numerous phenomena that share many of the properties of discrete time crystals, including the ac Josephson effect, coupled map lattices, and Faraday waves. Theoretically, there is a diverse array of strategies to stabilize time-crystalline order in both closed and open systems, ranging from localization and prethermalization to dissipation and error correction. Experimentally, many-body quantum simulators provide a natural platform for investigating signatures of time-crystalline order; recent work utilizing trapped ions, solid-state spin systems, and superconducting qubits are reviewed. Finally, this Colloquium concludes by describing outstanding challenges in the field and a vision for new directions on both the experimental and theoretical fronts.

Figure 1

Schematic depicting strategies for stabilizing time-crystalline order in periodically driven, closed systems evolving via deterministic dynamics. In few-body classical systems such as a parametrically driven nonlinear oscillator, stable subharmonic responses are ubiquitous. This stability can be understood as a time-dependent extension of the Kolmogorov-Arnold-Moser (KAM) theorem ([143]; [193]; [96]), which proves that quasiperiodic orbits of dynamical systems remain robust to small perturbations (Sec. 3b4). There is no quantum analog to this classical few-body strategy. For many-body systems, both classical and quantum dynamics can exhibit Floquet prethermalization in the limit of large driving frequencies (Sec. 4d). During an intermediate prethermal window of time (i.e., before Floquet heating occurs), the system can exhibit discrete time-crystalline order. The lifetime of this order scales exponentially with the frequency of the drive ($\tau \sim e^{{\omega }_D/J}$). In strongly disordered, quantum many-body systems, a phenomenon known as many-body localization can occur. This prevents Floquet heating (see Sec. 3c) and also provides a mechanism for the many-body system to avoid becoming ergodic. By breaking ergodicity, time-crystalline order can persist to infinite times in the thermodynamic limit. Since many-body localization relies upon the discreteness of quantum mechanical levels, there is no classical analog to this strategy. Left panel inset: adapted from [112]. Middle panel inset: adapted from [154]. Right panel inset: adapted from [301].

Figure 2

Schematic depicting strategies for stabilizing time-crystalline order in periodically driven, open systems evolving via stochastic dynamics. Dissipative nonlinear dynamical systems have long been known to exhibit stable, many-body time-translation symmetry breaking. Indeed, coupled map lattices can satisfy all of the stated requirements for being a discrete time crystal. The key ingredient for the stability of $\mathrm{S}\tau \mathrm{B}$ in this setting is a generalized version of dissipation; effectively, one can think of the microscopic dynamics in such systems as being coupled to a zero-temperature bath (alternatively, one can say that the microscopic dynamics are not information or measure preserving). This ensures that a finite volume of initial conditions contracts toward a period-doubled fixed point (Sec. 2a). The stability of time-crystalline order in many-body systems coupled to a finite-temperature bath is significantly more subtle. For example, in classical Langevin dynamics or quantum Lindbladian dynamics, at any $T>0$ dissipation always comes with noise. In such systems (for instance, finite-temperature, parametrically driven, coupled nonlinear pendula), at low temperatures time-crystalline order can survive for an “activated” timescale $\sim e^{\mathrm{\Delta }/k_{B}T}$ (Sec. 5a). Somewhat surprisingly, by considering more generic stochastic dynamics (i.e., probabilistic cellular automata), it is possible to realize finite-temperature, time-crystalline order with an infinite lifetime (Sec. 5e). Left panel inset: adapted from [128]. Middle panel inset: adapted from [120]. Right panel inset: adapted from [279].

Figure 3

(a) Schematic depiction of a time-dependent rotating frame, which transforms the original time-dependent $H(t)$ into the time-independent Floquet Hamiltonian $H_F$. Time-translation symmetry breaking (here with a period $m=2$) can be understood as spontaneous symmetry breaking of an internal ${\mathbb{Z}}_m$ symmetry of the Floquet Hamiltonian. Adapted from [299]. (b) Contours of equal “quasienergy” $H_F(Q,P)$ for a nonlinear parametric oscillator near a $2:1$ resonance. The original coordinates of the oscillator $q$ and $p$ are related to the coordinates $Q$ and $P$ through a time-dependent canonical transformation given approximately by Eq. (7). The two minima of $H_F$, at which $(Q,P)$ are time independent, map back to the two exactly period-doubled orbits, $q_{*}(t)=-q_{*}(t+T_0)$. As predicted by Eq. (11), time-translation symmetry manifests as a ${\mathbb{Z}}_2$ symmetry $H_F(Q+\pi ,P)=H_F(Q,P)$. Adapted from [317].

Figure 4

(a) Signatures of time-crystalline order were observed in a one-dimensional chain of ytterbium ions. In this system, the effective spin degree of freedom is encoded in two hyperfine “clock” states of a ${}^{171}{\mathrm{Yb}}^{+}$ ion. (b) By varying the interaction strength ($x$ axis), which is effectively parametrized by an interaction timescale, and the strength of the perturbation on the $\pi$ pulse ($y$ axis), [306] were able to observe the system’s behavior change from a discrete time crystal phase to a time-translation symmetry-unbroken phase. Adapted from [306]. (c) [51] utilized a diamond sample containing a high density of nitrogen-vacancy color centers ($\sim 15\mathrm{ppm}$) ([49]). In this system, the spin consists of two $m_s$ sublevels, and N-$V$ centers interact with one another via magnetic dipole-dipole interactions. The interplay between the long-range dipolar interaction and the three-dimensional nature of the sample is conjectured to lead to critically slow thermalization ([113]; [149]), during which discrete time-crystalline order can be observed. (d) Experimentally measured phase boundary (red dashed line through diamond data points added as a guide) of the critical discrete time crystal as a function of the effective interaction strength and the $\pi$-pulse imperfection. Adapted from [51].

Figure 5

(a) Schematic image depicting the nine-spin subset of the 27 precharacterized nuclear spins surrounding the single N-$V$ center. (b) Each nuclear spin can be individually initialized and read out via coupling to the N-$V$ center. The Floquet sequence consists of $U_{\mathrm{int}}$, which includes a disordered long-range Ising interaction and a disordered longitudinal field followed by an approximate $\pi$ pulse $R_x(\theta )$. The disorder is naturally inherited from the random positioning of the nuclear spins within the diamond lattice. (c) In the MBL discrete time-crystal phase, the system exhibits robust subharmonic oscillations lasting $\sim {10}^3$ Floquet cycles. The various colors are associated with different initial states, which are characterized by their quasienergy in the right panel. The time-crystalline behavior is independent of the initial state, as expected for a MBL time crystal. (d) Schematic depicting the 53 transmon qubit Sycamore chip. [184] isolated one-dimensional chains of $L=8$, 12, 16, and 20 qubits from this two-dimensional grid. (e) Digital Floquet sequence consisting of random longitudinal fields, disordered nearest-neighbor Ising interactions, and approximate $\pi$ pulses are applied to random “bit-string” initial states. (f) In the thermal phase (left panel), the disorder-averaged autocorrelation function quickly decays to zero. In the MBL time-crystal phase, the same correlation function exhibits subharmonic oscillations lasting $\sim {10}^2$ Floquet cycles. [184] also implemented a benchmarking “echo” sequence $U_{\mathrm{echo}}$, which demonstrated that the time crystal’s lifetime is consistent with being limited by experimental imperfections and decoherence. (a)–(c) Adapted from [220]. (d)–(f) Adapted from [12], and [184].

Figure 6

Schematic depicting prethermal time-crystalline order identified via the stroboscopic behavior of the magnetization as a function of time. Left panel: in the non-time-crystal prethermal phase, the magnetization during even and odd Floquet cycles becomes identical after a short time (corresponding to local thermalization). During the intermediate prethermal regime, this magnetization may remain small, but nonzero, until it decays to zero at the Floquet heating timescale. Right panel: in the prethermal discrete time-crystal phase, the magnetization oscillates throughout the prethermal regime, leading to stroboscopic curves that exhibit plateau behavior. The prethermal time crystal ultimately melts at exponentially late times, controlled by the frequency of the driving field. Adapted from [73].

Figure 7

Prethermal discrete time crystal in a disorder-free chain of $L=25$ ${}^{171}{\mathrm{Yb}}^{+}$ ions. (a) For initial states whose energy density (measured with respect to $H_{\mathrm{eff}}$) lies in the trivial phase of the effective Hamiltonian (top panel), the dynamics of the magnetization are largely independent of the drive frequency and quickly equilibrate to zero. Despite the rapid decay of the magnetization, the energy density exhibits slow Floquet heating. For initial states whose energy density lies in the symmetry-broken phase of $H_{\mathrm{eff}}$ (bottom panel), the stroboscopic dynamics of the magnetization exhibit subharmonic oscillations (solid curves correspond to even periods, while dashed curves correspond to odd periods), whose lifetime extends with increasing frequency. The lifetime of this prethermal time-crystalline order is consistent with being cut off by the slow Floquet heating timescale. (b) Experimentally measured phase diagram of a prethermal discrete time crystal as a function of the energy density of the initial state. States near the edge of the spectrum (i.e., which order with respect to $H_{\mathrm{eff}}$) exhibit PDTC order, while states in the trivial phase of $H_{\mathrm{eff}}$ do not exhibit prethermal time-crystalline order. (a),(b) Adapted from [154].

Figure 8

(a) Schematic depicting $N$ atoms bouncing off an oscillating mirror ([233]). (b) Numerical simulations of the scenario in (a) shows that the time evolution of the atomic density exhibits $\mathrm{S}\tau \mathrm{B}$. At each time shown, the positions of ${10}^2$ atoms (out of ${10}^4$) are measured and a histogram of those positions is depicted. (c) A Bose-Einstein condensate in a transversely pumped high-finesse optical cavity can realize signatures of $\mathrm{S}\tau \mathrm{B}$ in an open system. In particular, the atom-cavity system can exhibit a period-doubled oscillation between distinct density wave patterns ([136]). (a)–(c) Adapted from [233], [235], and [136].

Figure 9

Arnold tongues of the damped Matthieu equation. The $x$ axis is the drive frequency expressed via the dimensionless ratio $a=(2{\omega }_0{)}^2/{\omega }_D^2$, while the $y$ axis is the drive amplitude $\delta$. Contours indicate the critical drive amplitude required for resonance when the coefficient of friction takes a given value $c$. When $c>0$, the period-doubled solution onsets only at finite $\delta$. Adapted from [206].

Figure 10

Domain walls between different period-doubled solutions of the parametrically driven 1D Frenkel-Kontorova model [Eq. (12)] in contact with a finite-temperature Langevin bath. Colors indicate the amplitude of the $j$th oscillator observed at even stroboscopic times $q_j(t=2nT)$, with time running vertically. Inset: oscillations at stroboscopic times $q_j(nT)$, which reveal the period doubling. Adapted from [300].

Figure 11

Differential resistance measurements $dV/dI$ in the charge density wave system ${\mathrm{NbSe}}_3$. (a) A bias voltage $V(t)=V_{\mathrm{dc}}+V_{\mathrm{rf}}\cos ({\omega }_{D}t)$ is applied, and the dc component of the resistance $d{V}_{\mathrm{dc}}/d{I}_{\mathrm{dc}}$ is measured, holding $V_{\mathrm{rf}}$ and ${\omega }_D$ fixed. Sharp peaks in the resistance correspond to plateaus in the $I\text{−}V$ curve. The location of these peaks can be attributed to a motion of the CDW in which it shifts by $p/q$ wave vectors per driving period, resulting in a subharmonic response between the ac components of the voltage and current. (b) In the absence of the ac drive, the subharmonic response is absent. Adapted from [37].

Figure 12

Phase diagram of the Toom model. After each CA update, spins randomly flip up or down with rate ${ϵ}_p$ or ${ϵ}_q$, respectively, with “bias” $({ϵ}_p-{ϵ}_q)/({ϵ}_p+{ϵ}_q)$ and “amplitude” ${ϵ}_p+{ϵ}_q$. Within the two-phase region, there are two distinct steady-state distributions and the dynamics are not ergodic. Unlike the Ising model, the system can thus remember 1 bit of information even in the presence of biased noise. Adapted from [19].

Figure 13

Depiction of a prethermal time quasicrystal emerging from a quasiperiodically driven quantum system. During the prethermal regime, local observables exhibit discrete time quasicrystalline order, which ultimately melts at late times as the system Floquet heats to a featureless infinite-temperature state. Adapted from [74].