Observation of a discrete time crystal

Journal name:
Nature
Volume:
543,
Pages:
217–220
Date published:
DOI:
doi:10.1038/nature21413
Received
Accepted
Published online

Spontaneous symmetry breaking is a fundamental concept in many areas of physics, including cosmology, particle physics and condensed matter1. An example is the breaking of spatial translational symmetry, which underlies the formation of crystals and the phase transition from liquid to solid. Using the analogy of crystals in space, the breaking of translational symmetry in time and the emergence of a ‘time crystal’ was recently proposed2, 3, but was later shown to be forbidden in thermal equilibrium4, 5, 6. However, non-equilibrium Floquet systems, which are subject to a periodic drive, can exhibit persistent time correlations at an emergent subharmonic frequency7, 8, 9, 10. This new phase of matter has been dubbed a ‘discrete time crystal’10. Here we present the experimental observation of a discrete time crystal, in an interacting spin chain of trapped atomic ions. We apply a periodic Hamiltonian to the system under many-body localization conditions, and observe a subharmonic temporal response that is robust to external perturbations. The observation of such a time crystal opens the door to the study of systems with long-range spatio-temporal correlations and novel phases of matter that emerge under intrinsically non-equilibrium conditions7.

At a glance

Figures

  1. Floquet evolution of a spin chain.
    Figure 1: Floquet evolution of a spin chain.

    Three Hamiltonians are applied sequentially in time: a global spin rotation of nearly π (H1), long-range Ising interactions (H2), and strong disorder (H3) (left). The system evolves for 100 Floquet periods of this sequence (right). On the left, circles with arrows denote spins (that is, ions 1 to 10), where the red colour denotes initial magnetization. Curved coloured lines between spins denote the spin–spin interactions, and the black trace illustrates the applied disorder.

  2. Spontaneous breaking of discrete time translational symmetry.
    Figure 2: Spontaneous breaking of discrete time translational symmetry.

    Time-evolved magnetizations of each spin and their Fourier spectra are displayed, showing the subharmonic response of the system to the Floquet Hamiltonian. a, When only the H1 spin flip is applied, the spins oscillate with a subharmonic response that beats owing to the perturbation ε = 0.03 from perfect π pulses, with a clear splitting in the Fourier spectrum. Wt3 denotes the maximum phase accumulated by the disorder, which is fixed to π throughout the experiment. b, With both the H1 spin flip and the disorder H3, the spins precess with various Larmor rates in the presence of different individual fields. c, Finally, adding the spin–spin interaction term H2 (shown with the largest interaction 2J0t2 = 0.072), the spins lock to the subharmonic frequency of the drive period. Here the Fourier spectrum merges into a single peak even in the face of perturbation ε on the spin flip H1. d, When the perturbation is too strong (ε = 0.11), we cross the boundary from the DTC into a symmetry unbroken phase10. e, Spin magnetization for all 10 spins corresponding to the case of b, indicated by blue box. f, Spins of ions 3 and 8 corresponding to the case of c, indicated by red box. Each point is the average of 150 experimental repetitions. Error bars are computed from quantum projection noise and detection infidelities.

  3. Variance of the subharmonic peak amplitude as a signature of the DTC transition.
    Figure 3: Variance of the subharmonic peak amplitude as a signature of the DTC transition.

    a, Variances of the central peak height, computed over the 10 sites and averaged over 10 instances of disorder, for four different strengths of the long-range interaction term J0. The crossover from a symmetry unbroken state to a DTC is observed as a peak in the measured variance of the subharmonic system response. Dashed lines, numerical results, scaled vertically to fit the experimental data (see Methods for detailed analysis procedures and possible sources of decoherence). Experimental error bars, s.e.m.; a.u., arbitrary units. b, Crossover determined by a fit to the variance peak location (filled circles). Dashed line, numerically determined phase boundary with experimental long-range coupling parameters10. Grey shaded region indicates 90% confidence level of the DTC to symmetry unbroken phase boundary. Interaction strengths are normalized to be unitless, referencing to the fixed disorder accumulated phase π (ref. 10).

  4. Subharmonic peak height as a function of the drive perturbation.
    Figure 4: Subharmonic peak height as a function of the drive perturbation.

    Main panel, the central subharmonic peak height in the Fourier spectrum as a function of the perturbation ε, averaged over the 10 sites and 10 disorder instances, for four different interaction strengths (see key at top). Solid lines are guides to the eye. The height decreases across the phase boundary and eventually diminishes as the single peak is split into two. Error bars, ±1 s.d. Inset, numerical simulations given experimental parameters.

  5. Experimental pulse sequence.
    Extended Data Fig. 1: Experimental pulse sequence.

    Bottom left, we initialize the spins via optical pumping plus a spin rotation, and then start the Floquet evolution. Each period includes the three parts of the Hamiltonian as described in the main text and Methods (top left), and is repeated for 100 times. We then perform an analysis rotation to the desired direction on the Bloch sphere, and then perform spin state detection (bottom right). Inset at top right, all pulses in the Floquet evolution are shaped with a ‘Tukey’ window. See text for detailed explanations.

  6. Build-up of a DTC.
    Extended Data Fig. 2: Build-up of a DTC.

    From a to b to c, we fix the disorder instance (that is, a single sample of the random disorder realizations) and the perturbation ε, while gradually increasing the interactions (2J0t2) for different experimental runs. The temporal oscillations are synchronized with increasing interactions, and the Fourier subharmonic peak is enhanced. The top panel shows the time-traces of the individual spin magnetizations, and the bottom panel shows the Fourier spectrum.

  7. Comparing the ions’ dynamics with and without disorder.
    Extended Data Fig. 3: Comparing the ions’ dynamics with and without disorder.

    a, Without disorder (W0t3 set to 0), the spin–spin interactions suppress the beatnote imposed by the external perturbation ε. Left, experimental data; right, exact numerics calculated under Floquet time-evolution. b, With disorder (W0t3 = π), the time crystal is more stable. Left, experimental result for a single disorder instance; right, numerical simulations. The top panels show the individual spin magnetizations as a function of time, and the bottom panels shows the Fourier transforms to the frequency domain.

  8. Finite size scaling.
    Extended Data Fig. 4: Finite size scaling.

    Shown are subharmonic central peak heights as a function of the perturbation, for different numbers of ions, N = 10, 12, and 14. We observe a sharpening, that is, that the curvature of the slope is increasing, as the chain size grows, consistent with expectations. Each curve contains one disorder realization, which is a single sample of random instance with W0t3 = π, but the data are averaged over all the ions (yielding the error bars shown). Inset, phenomenological scaling collapse of the three curves10, with the following parameters: εc = 0.041, v = 0.33, β = 1.9, which are critical exponents following the analysis in ref. 10. The extracted value of εc is consistent with the interaction strength, which is fixed at 2J0t2 = 0.048 throughout this dataset.

  9. Different initial states.
    Extended Data Fig. 5: Different initial states.

    Starting with the left half of the chain initialized in the opposite direction, we observe that the time crystal is still persistent in the presence of perturbations. Top left, time-dependent magnetizations for ions in the first half of the chain. Bottom left, magnetizations for the second half. Notice that the spins on the two halves oscillate with opposite phases throughout, until the end of the evolution. Right, average peak height as a function of perturbation. Inset, FFT spectrum for 2% perturbation. 2J0t2 was fixed at 0.048 throughout this dataset.

  10. A random sampling from numerical evolutions.
    Extended Data Fig. 6: A random sampling from numerical evolutions.

    Shown are averages of 10 disorder instances from numerical evolution under H for 2J0t2 = 0.072. Left, an example random numerical dataset (points) and the fit to equation (3) in Methods (dashed line). Right, the normalized probability distribution (PDF) of peak fit centres εp is shown in yellow, and a normal distribution is overlaid in red. The normal distribution is fitted using only the mean and standard deviation of the sample, showing excellent agreement with Gaussian statistics. For this value of J0 the mean and the standard deviation σεp = 0.006.

References

  1. Chaikin, P. & Lubensky, T. Principles of Condensed Matter Physics Vol. 1 (Cambridge Univ. Press, 1995)
  2. Wilczek, F. Quantum time crystals. Phys. Rev. Lett. 109, 160401 (2012)
  3. Wilczek, F. Superfluidity and space-time translation symmetry breaking. Phys. Rev. Lett. 111, 250402 (2013)
  4. Bruno, P. Comment on “quantum time crystals”. Phys. Rev. Lett. 110, 118901 (2013)
  5. Bruno, P. Impossibility of spontaneously rotating time crystals: a no-go theorem. Phys. Rev. Lett. 111, 070402 (2013)
  6. Watanabe, H. & Oshikawa, M. Absence of quantum time crystals. Phys. Rev. Lett. 114, 251603 (2015)
  7. Khemani, V., Lazarides, A., Moessner, R. & Sondhi, S. L. Phase structure of driven quantum systems. Phys. Rev. Lett. 116, 250401 (2016)
  8. Else, D. V., Bauer, B. & Nayak, C. Floquet time crystals. Phys. Rev. Lett. 117, 090402 (2016)
  9. von Keyserlingk, C. W., Khemani, V. & Sondhi, S. L. Absolute stability and spatiotemporal long-range order in Floquet systems. Phys. Rev. B 94, 085112 (2016)
  10. Yao, N. Y., Potter, A. C., Potirniche, I.-D. & Vishwanath, A. Discrete time crystals: rigidity, criticality, and realizations. Phys. Rev. Lett. 118, 030401 (2017)
  11. Sachdev, S. Quantum Phase Transitions (Cambridge Univ. Press, 1999)
  12. Li, T. et al. Space-time crystals of trapped ions. Phys. Rev. Lett. 109, 163001 (2012)
  13. Sacha, K. Modeling spontaneous breaking of time-translation symmetry. Phys. Rev. A 91, 033617 (2015)
  14. Nandkishore, R. & Huse, D. A. Many-body localization and thermalization in quantum statistical mechanics. Annu. Rev. Condens. Matter Phys. 6, 1538 (2015)
  15. D’Alessio, L. & Rigol, M. Long-time behavior of isolated periodically driven interacting lattice systems. Phys. Rev. X 4, 041048 (2014)
  16. Lazarides, A., Das, A. & Moessner, R. Equilibrium states of generic quantum systems subject to periodic driving. Phys. Rev. E 90, 012110 (2014)
  17. Ponte, P., Chandran, A., Papic, Z. & Abanin, D. A. Periodically driven ergodic and many-body localized quantum systems. Ann. Phys. 353, 196204 (2015)
  18. Else, D. V., Bauer, B. & Nayak, C. Pre-thermal time crystals and Floquet topological phases without disorder. Preprint at http://arXiv.org/abs/1607.05277 (2016)
  19. Smith, J. et al. Many-body localization in a quantum simulator with programmable random disorder. Nat. Phys. 12, 907911 (2016)
  20. Lee, A. C. et al. Engineering large Stark shifts for control of individual clock state qubits. Phys. Rev. A 94, 042308 (2016)
  21. Porras, D. & Cirac, J. I. Effective quantum spin systems with trapped ions. Phys. Rev. Lett. 92, 207901 (2004)
  22. Brown, K. R., Harrow, A. W. & Chuang, I. L. Arbitrarily accurate composite pulse sequences. Phys. Rev. A 70, 052318 (2004)
  23. Burin, A. L. Localization in a random XY model with long-range interactions: intermediate case between single-particle and many-body problems. Phys. Rev. B 92, 104428 (2015)
  24. Yao, N. Y. et al. Many-body localization in dipolar systems. Phys. Rev. Lett. 113, 243002 (2014)
  25. Bordia, P., Luschen, H., Schneider, U., Kanp, M. & Bloch, I. Periodically driving a many-body localized quantum system. Preprint at http://arXiv.org/abs/1607.07868 (2016)
  26. von Keyserlingk, C. W. & Sondhi, S. L. Phase structure of one-dimensional interacting Floquet systems. I. Abelian symmetry-protected topological phases. Phys. Rev. B 93, 245145 (2016)
  27. Potter, A. C., Morimoto, T. & Vishwanath, A. Classification of interacting topological Floquet phases in one dimension. Phys. Rev. X 6, 041001 (2016)
  28. Jotzu, G. et al. Experimental realization of the topological Haldane model with ultracold fermions. Nature 515, 237240 (2014)
  29. Else, D. V. & Nayak, C. Classification of topological phases in periodically driven interacting systems. Phys. Rev. B 93, 201103(R) (2016)
  30. von Keyserlingk, C. W. & Sondhi, S. L. Phase structure of one-dimensional interacting Floquet systems. II. Symmetry-broken phases. Phys. Rev. B 93, 245146 (2016)
  31. Olmschenk, S. et al. Manipulation and detection of a trapped Yb+ hyperfine qubit. Phys. Rev. A 76, 052314 (2007)
  32. Hayes, D. et al. Entanglement of atomic qubits using an optical frequency comb. Phys. Rev. Lett. 104, 140501 (2010)
  33. Kim, K. et al. Entanglement and tunable spin-spin couplings between trapped ions using multiple transverse modes. Phys. Rev. Lett. 103, 120502 (2009)

Download references

Author information

Affiliations

  1. Joint Quantum Institute, University of Maryland Department of Physics and National Institute of Standards and Technology, College Park, Maryland 20742, USA

    • J. Zhang,
    • P. W. Hess,
    • A. Kyprianidis,
    • P. Becker,
    • A. Lee,
    • J. Smith,
    • G. Pagano &
    • C. Monroe
  2. Department of Physics, University of California Berkeley, Berkeley, California 94720, USA

    • I.-D. Potirniche,
    • A. Vishwanath &
    • N. Y. Yao
  3. Department of Physics, University of Texas at Austin, Austin, Texas 78712, USA

    • A. C. Potter
  4. Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

    • A. Vishwanath
  5. IonQ, Inc., College Park, Maryland 20742, USA

    • C. Monroe

Contributions

J.Z., P.W.H., A.K., P.B., A.L., J.S., G.P. and C.M. all contributed to experimental design, construction, data collection and analysis. I.-D.P., A.C.P., A.V. and N.Y.Y. all contributed to the theory for the experiment. All work was performed under the guidance of N.Y.Y. and C.M. All authors contributed to this manuscript.

Competing financial interests

C.M. is a founding scientist of ionQ, Inc.

Corresponding author

Correspondence to:

Reviewer Information Nature thanks H. Haeffner 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: Experimental pulse sequence. (177 KB)

    Bottom left, we initialize the spins via optical pumping plus a spin rotation, and then start the Floquet evolution. Each period includes the three parts of the Hamiltonian as described in the main text and Methods (top left), and is repeated for 100 times. We then perform an analysis rotation to the desired direction on the Bloch sphere, and then perform spin state detection (bottom right). Inset at top right, all pulses in the Floquet evolution are shaped with a ‘Tukey’ window. See text for detailed explanations.

  2. Extended Data Figure 2: Build-up of a DTC. (292 KB)

    From a to b to c, we fix the disorder instance (that is, a single sample of the random disorder realizations) and the perturbation ε, while gradually increasing the interactions (2J0t2) for different experimental runs. The temporal oscillations are synchronized with increasing interactions, and the Fourier subharmonic peak is enhanced. The top panel shows the time-traces of the individual spin magnetizations, and the bottom panel shows the Fourier spectrum.

  3. Extended Data Figure 3: Comparing the ions’ dynamics with and without disorder. (199 KB)

    a, Without disorder (W0t3 set to 0), the spin–spin interactions suppress the beatnote imposed by the external perturbation ε. Left, experimental data; right, exact numerics calculated under Floquet time-evolution. b, With disorder (W0t3 = π), the time crystal is more stable. Left, experimental result for a single disorder instance; right, numerical simulations. The top panels show the individual spin magnetizations as a function of time, and the bottom panels shows the Fourier transforms to the frequency domain.

  4. Extended Data Figure 4: Finite size scaling. (184 KB)

    Shown are subharmonic central peak heights as a function of the perturbation, for different numbers of ions, N = 10, 12, and 14. We observe a sharpening, that is, that the curvature of the slope is increasing, as the chain size grows, consistent with expectations. Each curve contains one disorder realization, which is a single sample of random instance with W0t3 = π, but the data are averaged over all the ions (yielding the error bars shown). Inset, phenomenological scaling collapse of the three curves10, with the following parameters: εc = 0.041, v = 0.33, β = 1.9, which are critical exponents following the analysis in ref. 10. The extracted value of εc is consistent with the interaction strength, which is fixed at 2J0t2 = 0.048 throughout this dataset.

  5. Extended Data Figure 5: Different initial states. (331 KB)

    Starting with the left half of the chain initialized in the opposite direction, we observe that the time crystal is still persistent in the presence of perturbations. Top left, time-dependent magnetizations for ions in the first half of the chain. Bottom left, magnetizations for the second half. Notice that the spins on the two halves oscillate with opposite phases throughout, until the end of the evolution. Right, average peak height as a function of perturbation. Inset, FFT spectrum for 2% perturbation. 2J0t2 was fixed at 0.048 throughout this dataset.

  6. Extended Data Figure 6: A random sampling from numerical evolutions. (96 KB)

    Shown are averages of 10 disorder instances from numerical evolution under H for 2J0t2 = 0.072. Left, an example random numerical dataset (points) and the fit to equation (3) in Methods (dashed line). Right, the normalized probability distribution (PDF) of peak fit centres εp is shown in yellow, and a normal distribution is overlaid in red. The normal distribution is fitted using only the mean and standard deviation of the sample, showing excellent agreement with Gaussian statistics. For this value of J0 the mean and the standard deviation σεp = 0.006.

Additional data