Nature  Letter
Observation of a discrete time crystal
 J. Zhang^{1}^{, }
 P. W. Hess^{1}^{, }
 A. Kyprianidis^{1}^{, }
 P. Becker^{1}^{, }
 A. Lee^{1}^{, }
 J. Smith^{1}^{, }
 G. Pagano^{1}^{, }
 I.D. Potirniche^{2}^{, }
 A. C. Potter^{3}^{, }
 A. Vishwanath^{2, 4}^{, }
 N. Y. Yao^{2}^{, }
 C. Monroe^{1, 5}^{, }
 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 matter^{1}. 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 proposed^{2, 3}, but was later shown to be forbidden in thermal equilibrium^{4, 5, 6}. However, nonequilibrium Floquet systems, which are subject to a periodic drive, can exhibit persistent time correlations at an emergent subharmonic frequency^{7, 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 manybody 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 longrange spatiotemporal correlations and novel phases of matter that emerge under intrinsically nonequilibrium conditions^{7}.
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References
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Author information
Affiliations

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

Department of Physics, University of California Berkeley, Berkeley, California 94720, USA
 I.D. Potirniche,
 A. Vishwanath &
 N. Y. Yao

Department of Physics, University of Texas at Austin, Austin, Texas 78712, USA
 A. C. Potter

Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
 A. Vishwanath

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.
Reviewer Information Nature thanks H. Haeffner and the other anonymous reviewer(s) for their contribution to the peer review of this work.
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J. Zhang
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P. W. Hess
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I.D. Potirniche
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Extended data figures and tables
Extended Data Figures
 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.
 Extended Data Figure 2: Buildup 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 (2J_{0}t_{2}) for different experimental runs. The temporal oscillations are synchronized with increasing interactions, and the Fourier subharmonic peak is enhanced. The top panel shows the timetraces of the individual spin magnetizations, and the bottom panel shows the Fourier spectrum.
 Extended Data Figure 3: Comparing the ions’ dynamics with and without disorder. (199 KB)
a, Without disorder (W_{0}t_{3} 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 timeevolution. b, With disorder (W_{0}t_{3} = 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.
 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 W_{0}t_{3} = π, but the data are averaged over all the ions (yielding the error bars shown). Inset, phenomenological scaling collapse of the three curves^{10}, with the following parameters: ε_{c} = 0.041, v = 0.33, β = 1.9, which are critical exponents following the analysis in ref. 10. The extracted value of ε_{c} is consistent with the interaction strength, which is fixed at 2J_{0}t_{2} = 0.048 throughout this dataset.
 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, timedependent 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. 2J_{0}t_{2} was fixed at 0.048 throughout this dataset.
 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 2J_{0}t_{2} = 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 ε_{p} 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 J_{0} the mean and the standard deviation σε_{p} = 0.006.
Additional data

Extended Data Figure 1: Experimental pulse sequence.Hover over figure to zoom

Extended Data Figure 2: Buildup of a DTC.Hover over figure to zoom

Extended Data Figure 3: Comparing the ions’ dynamics with and without disorder.Hover over figure to zoom

Extended Data Figure 4: Finite size scaling.Hover over figure to zoom

Extended Data Figure 5: Different initial states.Hover over figure to zoom

Extended Data Figure 6: A random sampling from numerical evolutions.Hover over figure to zoom