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, 543 (7644), 221-225

Observation of Discrete Time-Crystalline Order in a Disordered Dipolar Many-Body System


Observation of Discrete Time-Crystalline Order in a Disordered Dipolar Many-Body System

Soonwon Choi et al. Nature.


Understanding quantum dynamics away from equilibrium is an outstanding challenge in the modern physical sciences. Out-of-equilibrium systems can display a rich variety of phenomena, including self-organized synchronization and dynamical phase transitions. More recently, advances in the controlled manipulation of isolated many-body systems have enabled detailed studies of non-equilibrium phases in strongly interacting quantum matter; for example, the interplay between periodic driving, disorder and strong interactions has been predicted to result in exotic 'time-crystalline' phases, in which a system exhibits temporal correlations at integer multiples of the fundamental driving period, breaking the discrete time-translational symmetry of the underlying drive. Here we report the experimental observation of such discrete time-crystalline order in a driven, disordered ensemble of about one million dipolar spin impurities in diamond at room temperature. We observe long-lived temporal correlations, experimentally identify the phase boundary and find that the temporal order is protected by strong interactions. This order is remarkably stable to perturbations, even in the presence of slow thermalization. Our work opens the door to exploring dynamical phases of matter and controlling interacting, disordered many-body systems.

Conflict of interest statement

The authors declare no competing financial interests.


Extended Data Figure 1
Extended Data Figure 1. Effect of rotary echo sequence.
a Experimental sequence: during the interaction interval τ1, the phase of the microwave driving along is inverted after τ1/2. b Comparison of time traces of P(nT) in the presence (left) and absence (right) of an /- rotary echo sequence at similar τ1 and θ (left: τ1 = 379 ns, θ = 0.979π; right: τ1 = 384 ns, θ = 0.974π). The rotary echo leads to more pronounced 2T-periodic oscillations at long time. Microwave frequencies used in the rotary echo sequence: Ωx = 2π × 52.9 MHz, Ωy = 2π × 42.3 MHz.
Fig. 1
Fig. 1. Experimental setup for observing time-crystalline order.
a, NV centers in a nanobeam fabricated from black diamond are illuminated by a focused green laser beam and irradiated by a microwave source. Spins are prepared in the (|ms=0+|ms=1)/2 state using a microwave (−π/2)-pulse along the y^ axis. Subsequently, within one Floquet cycle, the spins evolve under a dipolar interaction and microwave field Ωx aligned along the x^ axis for duration τ1, immediately followed by a global microwave θ-pulse along the y^ axis. After n repetitions of the Floquet cycle, the spin polarization the x^ axis is read out. We choose τ1 as an integer multiple of 2πx to minimize accidental dynamical decoupling. b-d, Representative time traces of the normalized spin polarization P(nT) and respective Fourier spectra, |S(ν)|2, for different values of interaction time τ1 and θ: (b) τ1 = 92 ns, θ = π, (c) τ1 = 92 ns, θ = 1.034π, and (d) τ1 = 989 ns, θ = 1.034π. Dashed lines in c indicate ν = 1/2 ± (θπ)/2π. Data are averaged over more than 2 ⋅104 measurements.
Fig. 2
Fig. 2. Long-time behavior of time-crystalline order.
a Representative time trace of the normalized spin polarization P (nT) in the crystalline phase (τ1 = 790 ns and θ = 1.034π). The time-dependent intensity of the ν = 1/2 peak is extracted from a short-time Fourier transformation with a time window of length m = 20 shifted from the origin by nsweep. b Peak height at ν = 1/2 as a function of nsweep for different pulse imperfections at τ1 = 790 ns. Lines indicate fits to the data using a phenomenological double exponential function. The noise floor corresponds to 0.017, extracted from the mean value plus the standard deviation of ∑ν|S(ν)|2 excluding the ν = 1/2 peak. c Extracted lifetime of the time-crystalline order as a function of the interaction time τ1, for θ = 1.034π. Shaded region indicates the spin life-time T1ρ=60±2μs (extracted from a stretched exponential27) due to coupling with the external environment. d Extracted decay rate of the time-crystalline order as a function of θ for different interaction times, τ1 = 385 ns (circle), 586 ns (square) and 788 ns (triangle). Only very weak dependence on θπ is observed within the DTC, contrary to a dephasing model (Methods). In c, d, vertical error bars display the statistical error (s. d.) from the fit and empty symbols mark data near the time-crystalline phase boundary.
Fig. 3
Fig. 3. Phase diagram and transition.
a Crystalline fraction f as a function of θ obtained from a Fourier transform at late times (50 < n ≤ 100). Vertical error bars are limited by the noise floor (see Methods), horizontal error bars indicate the pulse uncertainty of 1%. Grey lines denote a super-Gaussian fit to extract the phase boundary (see Methods). In a, b, red diamonds mark the phenomenological phase boundary, identified as a 10% crystalline fraction. Horizontal error bars denote the statistical error (s. d.) from the fit. The colors of the round data points in b represent the extracted crystalline fraction at the associated parameter set. The dashed line corresponds to a disorder-averaged theoretical prediction for the phase boundary. Asymmetry in the boundary arises from an asymmetric distribution of rotation angles (see Methods). c Evolution of the Fourier spectra as a function of θ for two different interaction times, τ1 = 385 ns (top) and τ1 = 92 ns (bottom). d Bloch sphere indicating a single spin trajectory of the 2T-periodic evolution under the long-range dipolar Hamiltonian (red) and global rotation (blue).
Fig. 4
Fig. 4. ℤ3 time-crystalline order.
a Experimental sequence to demonstrate a 3T-periodic discrete time-crystalline order. A single Floquet cycle is composed of three operations: time evolution under long-range dipolar Hamiltonian and rapid microwave pulses for two different transitions. b Visualization of the 3T-periodicity in the polarization dynamics for the case of θ = π. c Fourier spectra of the polarization dynamics for two different interaction times and for three different rotation angles θ: 1.00π (red), 1.086π (blue) and 1.17π (yellow). Dashed lines indicate ν = 1/3, 2/3.

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