We experimentally study the response of star-shaped clusters of initially unentangled N=4, 10, and 37 nuclear spin-1/2 moments to an inexact π-pulse sequence and show that an Ising coupling between the center and the satellite spins results in robust period-2 magnetization oscillations. The period is stable against bath effects, but the amplitude decays with a timescale that depends on the inexactness of the pulse. Simulations reveal a semiclassical picture in which the rigidity of the period is due to a randomizing effect of the Larmor precession under the magnetization of surrounding spins. The timescales with stable periodicity increase with net initial magnetization, even in the presence of perturbations, indicating a robust temporal ordered phase for large systems with finite magnetization per spin.
We experimentally study the response of star-shaped clusters of initially unentangled N=4, 10, and 37 nuclear spin-1/2 moments to an inexact π-pulse sequence and show that an Ising coupling between the center and the satellite spins results in robust period-2 magnetization oscillations. The period is stable against bath effects, but the amplitude decays with a timescale that depends on the inexactness of the pulse. Simulations reveal a semiclassical picture in which the rigidity of the period is due to a randomizing effect of the Larmor precession under the magnetization of surrounding spins. The timescales with stable periodicity increase with net initial magnetization, even in the presence of perturbations, indicating a robust temporal ordered phase for large systems with finite magnetization per spin.
We study prethermal time quasicrystalline (TQC) order in a quasiperiodically driven chain of non-interacting spin-1/2 particles. The drive consists of two parts, switched on and off periodically with frequency ωd: (i) disordered Ising interactions, with exchange couplings chosen from a symmetric interval [−J/2,J/2], allowing random antiferromagnetic or ferromagnetic nearest-neighbor couplings, together with a random transverse field; and (ii) a rotating transverse magnetic field with frequency Ω. The ratio ωd/Ω is chosen to be irrational, producing multiple incommensurate frequencies and yielding quasiperiodic dynamics beyond Floquet theory. Using exact diagonalization, we analyze the time autocorrelation function, dynamical structure factor, and entanglement entropy (EE). In the high-frequency regime, robust spectral peaks at incommensurate frequencies (not integer multiples of the fundamental drives) signal quasiperiodic time-translation symmetry breaking (QTTSB). The EE exhibits sublinear power-law growth followed by a prethermal plateau, indicating suppressed resonant heating due to an energy scale mismatch. The nonequilibrium lifetime increases rapidly with driving frequency. Unlike symmetric disorder sampling, an asymmetric distribution of the Ising exchange couplings induces collective spin rigidity, enhancing the system's resistance to heating. The TQC phase remains stable against next-nearest-neighbor (NNN) exchange perturbations and rotational imperfections, with robustness comparable to discrete time crystals (TCs) under periodic driving. Our results establish this quasiperiodically driven system as a platform for long-lived nonequilibrium temporal order, revealing the interplay of disorder, collective rigidity, and quasiperiodic driving.
Motivated by the search for non-equilibrium phases of matter beyond the Floquet paradigm, we propose a route to realize perfect revivals and time quasicrystals (TQCs) in a Fibonacci-driven central-spin model with Ising interactions between the central and satellite spins in the absence of disorder. We analytically demonstrate that when the interaction strength, λ is tuned to 2mπ, no entanglement is generated between the central and satellite spins at stroboscopic times, independent of the strength of the transverse driving field and the drive sequence. This leads to indefinitely long-lived quasiperiodic oscillations in satellite spin magnetization. Furthermore, for initial states with a definite bath magnetization, no entanglement is generated when the transverse driving field, θ is tuned to π for any value of $ λ$, and the satellite spin magnetization oscillations exhibit period-3 revivals at Fibonacci times. Examining the robustness of these oscillations away from this fine-tuned limit, we identify a parameter regime where the quasiperiodic response does not decay over the timescales probed, giving rise to a long-lived π-pulse-proximate TQC regime. We further show that at special Clifford points, the system exhibits perfect quasiperiodic oscillations of the satellite spin magnetization, with a parity-dependent Fourier response. This response persists over the timescales probed, even away from the Clifford points for an even number of satellite spins, leading to the formation of a Clifford-proximate TQC. Finally, we demonstrate that at this Clifford points, the system is steered along an entangled trajectory, leading to the dynamical generation of Bell-cat and spin-cat states. These states exhibit magnetization of collective spin fluctuates quadratically with the number of satellite spins, thereby enabling Heisenberg-limited metrology.
Time-crystalline behavior has been predicted and observed in quantum central-spin systems with periodic driving and Ising interactions. Here, we theoretically show that it can also arise in central-spin systems with Heisenberg interactions. We present two methods to achieve this: application of a sufficiently large Zeeman splitting on the central spin compared to the satellite spins, or else by applying additional pulses to the central spin every Floquet period. In both cases, we show that the system exhibits a subharmonic response in spin magnetizations in the presence of disorder for both pure Heisenberg and XXZ interactions. Our results pertain to any XXZ central-spin system, including hyperfine-coupled electron-nuclear systems in quantum dots or color centers.
Abstract Discrete time-translational symmetry in a periodically driven many-body system can be spontaneously broken to form a discrete time crystal, an exotic new phase of matter. We present observations characteristic of discrete time crystalline order in a driven system of paramagnetic P-donor impurities in isotopically enriched 28Si cooled below 10 K. The observations exhibit a stable subharmonic peak at half the drive frequency which remains pinned even in the presence of pulse error, a signature of discrete time crystalline order. This signal has a finite lifetime of ∼100 Floquet periods, but this effect is long-lived relative to coherent spin–spin interaction timescales, lasting ∼104 times longer. We present simulations of the system based on the paradigmatic central spin model and show good agreement with experiment. We investigate the role of dissipation and interactions within this model, and show that both are capable of giving rise to discrete time crystal-like behaviour.
We show that stronger interactions protect periodically driven time-crystal order against dephasing but destabilise order under random multipolar drives, and that coordination matters beyond the total interaction strength. We studied a disordered kicked Ising model of N=12 and 16 spins with imperfect π pulses and per-site dephasing by exact diagonalisation, comparing a periodic drive, random multipolar drives of order n=0,1,2 and the Thue-Morse sequence, and separated coupling strength from coordination by comparing a grid with a ring at equal total interaction strength zJ. For the periodic drive the coherence time scales approximately as 1/γ, and stronger coupling reduces the transverse tilt on which dephasing acts. For multipolar drives of order n≥1, doubling the coupling on a ring shortens the coherence time to 0.54 (n=1), 0.37 (n=2) and 0.44 (Thue-Morse) at N=16, while random signs (n=0) are unaffected. A disorder scan supports a toggling-frame picture in which interactions spoil the block cancellation of the pulse error. At equal zJ the grid outlives the ring by 2.15 (N=12) and 2.78 (N=16) for the periodic drive despite an identical transverse tilt, a coordination effect beyond this picture. All ratios hold under three definitions of the coherence time.
We investigate a disorder-free quantum Ising chain subject to a time-periodic drive that rotates each spin by an angle $\ensuremath{\pi}(1\ensuremath{-}{\ensuremath{\epsilon}}_{i})$. In case all spins experience the same deviation $\ensuremath{\epsilon}$ and the system is initialized in a fully polarized state, the dynamics is known to be time crystalline: the magnetization of the system exhibits period-doubled oscillations for timescales that grow exponentially with the length of the chain. In this work, we study the effect of a deviation $\ensuremath{\epsilon}$ that differs between spins. We find that reducing $\ensuremath{\epsilon}$ for a single spin drastically enhances the lifetime of spatiotemporal order, suggesting the name metronome spin. Employing perturbative arguments in an average Hamiltonian picture, we explain this observation for initial states with macroscopic bulk magnetization. Furthermore, in the case of random bit-string initial states, we report the enhancement of the lifetime of a topological edge mode, which can also be understood in the same picture. Finally, we discuss an altered geometry in which the metronome spin is not directly part of the chain, affecting the dynamics in different ways in the two scenarios considered. Our findings unveil the intricate dynamics that emerge in Floquet systems under the influence of a spatially varying drive, thereby uncovering new avenues for Floquet engineering.