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Observation of a Discrete Time Crystal J LETTER doi:10.1038/nature21413 Observation of a discrete time crystal J. Zhang1, P. W. Hess1, A. Kyprianidis1, P. Becker1, A. Lee1, J. Smith1, G. Pagano1, I.-D. Potirniche2, A. C. Potter3, A. Vishwanath2,4, N. Y. Yao2 & C. Monroe1,5 Spontaneous symmetry breaking is a fundamental concept in Here we report the direct observation of discrete time translational many areas of physics, including cosmology, particle physics symmetry breaking and DTC formation in a spin chain of trapped and condensed matter1. An example is the breaking of spatial atomic ions, under the influence of a periodic Floquet many-body translational symmetry, which underlies the formation of crystals localization (MBL) Hamiltonian. We experimentally implement a and the phase transition from liquid to solid. Using the analogy of quantum many-body Hamiltonian with long-range Ising interac- crystals in space, the breaking of translational symmetry in time and tions and disordered local effective fields, using optical control tech- the emergence of a ‘time crystal’ was recently proposed2,3, but was niques19,20. Following the evolution through many Floquet periods, we later shown to be forbidden in thermal equilibrium4–6. However, measure the temporal correlations of the spin magnetization dynamics. non-equilibrium Floquet systems, which are subject to a periodic A DTC requires the ability to control the interplay between three key drive, can exhibit persistent time correlations at an emergent ingredients: strong drive, interactions and disorder. These are reflected subharmonic frequency7–10. This new phase of matter has been in the applied Floquet Hamiltonian H, consisting of the following three 10 dubbed a ‘discrete time crystal’ . Here we present the experimental successive pieces with overall period T =​ t1 +​ t2 +​ t3 (see Fig. 1) (ħ = 1): observation of a discrete time crystal, in an interacting spin chain y of trapped atomic ions. We apply a periodic Hamiltonian to the Hg11=−(1 εσ)t∑ i ime t i system under many-body localization conditions, and observe = = σσx x H HJ22∑i ij i j time t (1) a subharmonic temporal response that is robust to external HD= σx time t perturbations. The observation of such a time crystal opens the door 33∑i i i to the study of systems with long-range spatio-temporal correlations γ and novel phases of matter that emerge under intrinsically non- Here, σi (γ = x, y, z) is the Pauli matrix acting on the ith spin, g is the 7 equilibrium conditions . Rabi frequency with small perturbation ε, 2gt1 =​ π , Jij is the coupling For any symmetry in a Hamiltonian system, its spontaneous breaking strength between spins i and j, and Di is a site-dependent disordered in the ground state leads to a phase transition11. The broken symmetry potential sampled from a uniform random distribution with itself can assume many different forms. For example, the breaking of Di ∈ [0, W]. spin-rotational symmetry leads to a phase transition from paramag- To implement the Floquet Hamiltonian, each of the effective spin-1/2 2 netism to ferromagnetism when the temperature is brought below the particles in the chain is encoded in the S012/ Fm==,0F 〉 and 171 + Curie point. The breaking of spatial symmetry leads to the formation Fm==1, F 0〉 hyperfine ‘clock’ states of a Yb ion, denoted ↓〉z and of crystals, where the continuous translational symmetry of space is ↑〉z and separated by 12.642831 GHz (F and mF denote the hyperfine replaced by a discrete one. and Zeeman quantum numbers, respectively). We store a chain of We now pose an analogous question: can the translational symmetry 2 of time be broken? The proposal of such a ‘time crystal’ for time- H1 12 independent Hamiltonians has led to much discussion , with the Global rotation H2 conclusion that such structures cannot exist in the ground state or y 4–6 H1 = g(1 – H) Vj H any thermal equilibrium state of a quantum mechanical system . 3 A simple intuitive explanation is that quantum equilibrium states have (100 Floquet periods H1 time-independent observables by construction; thus, time transla- H tional symmetry can only be spontaneously broken in non-equilibrium Interactions 2 x x 7–10 H Time systems . In particular, the dynamics of periodically driven Floquet H2 = Jij ı i ı j 3 systems possesses a discrete time translational symmetry governed by the drive period. This symmetry can be further broken into ‘super- lattice’ structures where physical observables exhibit a period larger Disorder H1 ) 13 than that of the drive . Such a response is analogous to commensurate H H = D x 2 charge density waves that break the discrete translational symmetry 3 i ı i 1 H of their underlying lattice . The robust subharmonic synchronization 3 of the many-body Floquet system is the essence of the discrete time crystal (DTC) phase7–10. In a DTC, the underlying Floquet drive should Figure 1 | Floquet evolution of a spin chain. Three Hamiltonians are applied sequentially in time: a global spin rotation of nearly π (H ), long- generally be accompanied by strong disorder, leading to many-body 1 range Ising interactions (H ), and strong disorder (H ) (left). The system localization14 and thereby preventing the quantum system from absorb- 2 3 15–17 evolves for 100 Floquet periods of this sequence (right). On the left, circles ing the drive energy and heating to infinite temperatures . We note with arrows denote spins (that is, ions 1 to 10), where the red colour that under certain conditions, time crystal dynamics can persist for denotes initial magnetization. Curved coloured lines between spins denote rather long times even in the absence of localization before ultimately the spin–spin interactions, and the black trace illustrates the applied being destroyed by thermalization18. disorder. 1Joint Quantum Institute, University of Maryland Department of Physics and National Institute of Standards and Technology, College Park, Maryland 20742, USA. 2Department of Physics, University of California Berkeley, Berkeley, California 94720, USA. 3Department of Physics, University of Texas at Austin, Austin, Texas 78712, USA. 4Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA. 5IonQ, Inc., College Park, Maryland 20742, USA. 9 MARCH 2017 | VOL 543 | NATURE | 217 © 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved. RESEARCH LETTER 10 ions in a linear radio frequency Paul trap, and apply single spin of the rotation angle to a precision < 0.5%. The second evolution rotations using optically driven Raman transitions between the two operator e−iH22t applies the spin–spin Ising interaction, where the spin states. Spin–spin interactions are generated by spin-dependent maximum nearest-neighbour coupling J0 ranges from 2π(0.04 kHz) to optical dipole forces, which give rise to a tunable long-range Ising cou- 2π(0.25 kHz) and decays with distance at a power law exponent α = 1.5. 21 α pling that falls off approximately algebraically as JJij ∝/0 ij− . The duration of the interaction term is set so that J0t2 < 0.04 rad of Programmable disorder among the spins is generated by the ac Stark phase accumulation. The third evolution operator e−iH33t provides shift from a tightly focused laser beam that addresses each spin disorder to localize the system, and is programmed so that the variance 20 z individually . The Stark shift is an effective site-dependent σi field, so of the disorder is set by Wt3 = π . In this regime, MBL is expected to in order to transform it into the x direction of the Bloch sphere we sur- persist even in the presence of long-range interactions23,24. round this operation with π/2 pulses (see Methods). Finally, we measure To observe the DTC, we initialize the spins to the state 1 the magnetization of each spin by collecting the spin-dependent ψ0〉= ↓〉xz=↓〉+↑〉z through optical pumping followed by a 2 fluorescence on a camera for site-resolved imaging. This allows access () γ global π /2 rotation. After many periods of the above Floquet unitary to the single-site magnetization, σi , along any direction with a detec- equation (2), we measure the magnetization of each spin along x, which tion fidelity > 98% per spin. The unitary time evolution under a single gives the time-correlation function: Floquet period is: x x x 〈〉σψi ()tt=〈 00σσi () i (0) ψ 〉 UT()= ee−−iH33tiHt22e−iH11t (2) Figure 2 depicts the measured spin magnetization dynamics, both in The first evolution operator e−iH11t nominally flips all the spins around the time and the frequency domain, up to N =​ 100 Floquet periods. the y-axis of the Bloch sphere by an angle 2gt1 = π , but also includes a A single Floquet period T is set to a value between 74 μs and 75 μ s, controlled perturbation in the angle, επ , where ε <​ 0.15. This critical depending on the parameters in the Hamiltonian. rotation step is susceptible to noise in the Rabi frequency (1% r.m.s.) The global π -pulse e−iH11t rotates the spins roughly half way around from laser intensity instability, and also optical inhomogeneities (<5%) the Bloch sphere, so that we expect a response of the system at twice across the chain due to the shape of the Raman laser beams. In order the drive period 2T, or half the Floquet frequency. The frequency of to accurately control H1, we use the BB1 dynamical decoupling echo this subharmonic response in the magnetization is sensitive to the 22 sequence (see Methods) to suppress these effects, resulting in control precise value of the global rotation in H1 and is therefore expected to ab c d s s s 1.0 1.0 s 1.0 1.0 0.5 0.5 0.5 0.5 netization g 0.0 0.0 0.0 0.0 –0.5 –0.5 –0.5 –0.5 le ion ma g ingle ion magnetization in –1.0 –1.0 –1.0 –1.0 ingle ion magnetization S S Single ion magnetization 0 20 40 60 80 100 0 20 40 60 80 100 S 0 20 40 60 80 100 0 20 40 60 80 100 Time (T) Time (T) Time (T) Time (T) 0.06 0.06 2H 0.06 0.06 0.04 0.04 0.04 0.04 0.02 0.02 0.02 0.02 0 Fourier spectrum 0 Fourier spectrum 0 Fourier spectrum 0 Fourier spectrum 0.4 0.5 0.6 0.4 0.5 0.6 0.4 0.5 0.6 0.4 0.5 0.6 Frequency (1/T) Frequency (1/T) Frequency (1/T) Frequency (1/T) H = 0.03, W t3 = 0 H = 0.03, W t3 = π H = 0.03, 2J0 t2 = 0.072 H = 0.11, 2J0 t2 = 0.072 Interactions off Interactions on e f 1.0 Ion no.
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