Observation of discrete time-crystalline order in a disordered dipolar many-body system

Journal name:
Nature
Volume:
543,
Pages:
221–225
Date published:
DOI:
doi:10.1038/nature21426
Received
Accepted
Published online

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 transitions1, 2. More recently, advances in the controlled manipulation of isolated many-body systems have enabled detailed studies of non-equilibrium phases in strongly interacting quantum matter3, 4, 5, 6; for example, the interplay between periodic driving, disorder and strong interactions has been predicted to result in exotic ‘time-crystalline’ phases7, in which a system exhibits temporal correlations at integer multiples of the fundamental driving period, breaking the discrete time-translational symmetry of the underlying drive8, 9, 10, 11, 12. 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 temperature13, 14, 15. 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 thermalization16, 17. Our work opens the door to exploring dynamical phases of matter and controlling interacting, disordered many-body systems18, 19, 20.

At a glance

Figures

  1. Experimental set-up and observation of discrete time-crystalline order.
    Figure 1: Experimental set-up and observation of discrete time-crystalline order.

    a, Nitrogen–vacancy centres (blue spheres) 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 state using a microwave −π/2 pulse along the axis. Subsequently, within one Floquet cycle, the spins evolve under a dipolar interaction and microwave field Ωx aligned along the axis for duration τ1, immediately followed by a global microwave θ pulse along the axis. After n repetitions of the Floquet cycle, the spin polarization along the axis is read out. We choose τ1 to be an integer multiple of 2π/Ωx to minimize accidental dynamical decoupling14. bd, Representative time traces of the normalized spin polarization P(nT) measured at even (green) and odd (blue) integer multiples of T, and respective Fourier spectra for different values of the interaction time τ1 and θ: τ1 = 92 ns, θ = π (b); τ1 = 92 ns, θ = 1.034π (c); and τ1 = 989 ns, θ = 1.034π (d). Dashed lines in c indicate ν = 1/2 ± (θ − π)/(2π). Data are averaged over more than 2 × 104 measurements.

  2. Long-time behaviour of discrete time-crystalline order.
    Figure 2: Long-time behaviour of discrete 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 (inset) 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, which is extracted from the mean value plus the standard deviation of , 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. The shaded region indicates the spin lifetime (extracted from a stretched exponential28) due to coupling with the external environment. d, Extracted decay rate of the time-crystalline order (in Floquet units) as a function of θ for different interaction times: τ1 = 385 ns (circles), 586 ns (squares) and 788 ns (triangles). Only very weak dependence on θ − π is observed within the DTC phase, contrary to a dephasing model (Methods). In c and d, the vertical error bars display the statistical error (s.d.) from the fit and empty symbols mark data near the time-crystalline phase boundary.

  3. Phase diagram and transition.
    Figure 3: Phase diagram and transition.

    a, b, Crystalline fraction f (a) and its associated phase diagram (b) as a function of θ and τ1 obtained from a Fourier transform at late times (50 < n ≤ 100). The red diamonds mark the phenomenological phase boundary, identified as a 10% crystalline fraction; horizontal error bars denote the statistical error (s.d.) from a super-Gaussian fit. In a, vertical error bars of data points (circles) are limited by the noise floor (see Methods) and horizontal error bars indicate the pulse uncertainty of 1%. Grey lines denote the fit to extract the phase boundary (see Methods). In b, the colours of the data points (circles) 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).

  4. Z3 discrete time-crystalline order.
    Figure 4: Z3 discrete time-crystalline order.

    a, Experimental sequence demonstrating 3T-periodic DTC order. A single Floquet cycle is composed of three operations: time evolution under a 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), θ = 1.086π (blue) and θ = 1.17π (yellow). Dashed lines indicate ν = 1/3 and ν = 2/3.

  5. Effect of rotary echo sequence.
    Extended Data Fig. 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), measured at even (green) and odd (blue) integer multiples of T, 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. The microwave frequencies used in the rotary echo sequence are Ωx = 2π × 52.9 MHz and Ωy = 2π × 42.3 MHz.

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Author information

  1. These authors contributed equally to this work.

    • Soonwon Choi,
    • Joonhee Choi &
    • Renate Landig

Affiliations

  1. Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

    • Soonwon Choi,
    • Joonhee Choi,
    • Renate Landig,
    • Georg Kucsko,
    • Hengyun Zhou,
    • Vedika Khemani,
    • Eugene Demler &
    • Mikhail D. Lukin
  2. School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA

    • Joonhee Choi
  3. Research Centre for Knowledge Communities, University of Tsukuba, Tsukuba, Ibaraki 305-8550, Japan

    • Junichi Isoya
  4. Institut für Quantenoptik and Center for Integrated Quantum Science and Technology, Universität Ulm, 89081 Ulm, Germany

    • Fedor Jelezko
  5. Takasaki Advanced Radiation Research Institute, National Institutes for Quantum and Radiological Science and Technology, 1233 Watanuki, Takasaki, Gunma 370-1292, Japan

    • Shinobu Onoda
  6. Sumitomo Electric Industries Ltd, Itami, Hyougo 664-0016, Japan

    • Hitoshi Sumiya
  7. Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA

    • Curt von Keyserlingk
  8. Department of Physics, University of California Berkeley, Berkeley, California 94720, USA

    • Norman Y. Yao

Contributions

S.C. and M.D.L. developed the idea for the study. J.C., R.L. and G.K. designed and conducted the experiment. H.S., S.O., J.I. and F.J. fabricated the sample. S.C., H.Z., V.K., C.v.K., N.Y.Y. and E.D. conducted the theoretical analysis. All authors discussed the results and contributed to the manuscript.

Competing financial interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to:

Reviewer Information Nature thanks D. A. Huse and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Author details

Extended data figures and tables

Extended Data Figures

  1. Extended Data Figure 1: Effect of rotary echo sequence. (227 KB)

    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), measured at even (green) and odd (blue) integer multiples of T, 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. The microwave frequencies used in the rotary echo sequence are Ωx = 2π × 52.9 MHz and Ωy = 2π × 42.3 MHz.

Additional data