Abstract
Platforms that enable the study and control of quantum many-body interactions are fundamentally important in quantum science and related emerging technologies. Optically addressable solid-state spins offer scalability to a large Hilbert space but suffer from large on-site disorder and undesired couplings to the environment. Here we investigated a strongly interacting ensemble of millions of optically addressable ytterbium-171 ions in a crystal. This platform features a clock transition that is first-order insensitive to magnetic fluctuations, thus exhibiting superior coherence and small disorder. Notably, the clock transition also gives rise to pure spin-exchange interactions, realizing the dipolar XY model, which is difficult to access in other solid-state spin systems. We exploited this feature to investigate quantum thermalization by varying the relative ratio of interaction strength to disorder, dynamically engineering the XY model into other many-body Hamiltonian models and realizing a time-crystalline phase of matter through periodic driving. Our results demonstrated that an ensemble of rare earth ions serves as a versatile test bed for many-body physics and developing quantum technologies.
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Acknowledgements
We thank S. L. N. Hermans and T. Xie for reading the paper and providing useful feedback, and A. Ruskuc and A. Beckert for useful discussions. This study was supported by the US Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-Design Center for Quantum Advantage (C2QA) under contract no. DE-SC0012704; Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (PHY-1733907) with support from the Moore Foundation EPI program; NSF QuIC-TAQS award no. 2137984. The initial device and experimental setup development was supported by the Office of Naval Research award nos. N00014-19-1-2182 and N00014-22-1-2422. The device nanofabrication was performed at the Kavli Nanoscience Institute of the California Institute of Technology. A.F. and J.C. were supported by AFOSR under grant no. FA9550-23-1-0625. R.F. acknowledges support from the Quad Fellowship. E.B. acknowledges support from the National Science Foundation (grant no. 1847078). J.C. acknowledges support from the Terman Faculty Fellowship at Stanford University.
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M.L., J.C. and A.F. conceived the idea and experiment. M.L. built the experimental setup, performed the measurements and analysed the data. R.F. performed numerical simulations. C.-J.W. obtained the data for a reference sample. M.L., R.F., E.B., S.E.E., J.C. and A.F. interpreted the results. M.L., J.C. and A.F. wrote the paper with inputs from all authors. All work was supervised by J.C. and A.F.
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Extended data
Extended Data Fig. 1 Experimental setup and device.
a, The red line indicates the beam path of a 984 nm laser. The laser is locked to the frequency of the A transition. A frequency sideband at the F transition is generated by an EOM. Optical pulses are generated using AOMs. Both the EOM and AOMs are driven by gated RF sources. The light passes through a circulator to the device, and the reflected light is directed to a SNSPD for time-resolved photon counting. The blue lines indicate microwave signal delivery. RF pulses for the ground state spin transition (fg ≈ 0.675 GHz) and excited state spin transition (fe ≈ 3.37 GHz) are generated using frequency up-conversion from the local oscillator signal mixed with the output of the AWG. The signals after the mixers pass through band-pass filters and amplifiers, and are then combined using a diplexer before being sent to the device. b, Scanning electron microscope image of the chip in the dilution fridge, surrounded by a coplanar waveguide. c, Scanning electron microscope image of the nanophotonic device.
Extended Data Fig. 2 Experimental sequences.
a, The simplified energy level diagram for a 171Yb3+ ion in a YVO4 crystal. Both the ground and optically excited states exhibit fine structures labeled as {\(\left\vert 0\right\rangle ,\left\vert 1\right\rangle ,\left\vert {\rm{Aux}}\right\rangle\)} and {\({\left\vert 0\right\rangle }_{e},{\left\vert 1\right\rangle }_{e},{\left\vert {\rm{Aux}}\right\rangle }_{e}\)}, respectively. Microwave spin transitions occur at frequencies fg ≈ 0.675 GHz and fe ≈ 3.37 GHz for the ground and excited states, respectively. The optical transitions A and F occur near a wavelength of 984 nm, with a separation of 6.118 GHz. Note that transition A is coupled to the cavity mode used for fast spin state readout, while transition F is not coupled to the cavity mode and is driven for optical initialization. b, Experimental sequences include state initialization, spin dynamics control, and optical readout. c, Control of the average spin-spin interaction strength J within the qubit manifold {\(\left\vert 0\right\rangle ,\left\vert 1\right\rangle\)} (orange shaded area). Adjusting the driving amplitude on the F transition during initialization effectively controls the transfer of population from the auxiliary state \(\left\vert {\rm{Aux}}\right\rangle\) to the qubit manifold {\(\left\vert 0\right\rangle ,\left\vert 1\right\rangle\)} (blue arrows). Subsequently, a combination of optical pulses driving the A transition and microwave pulses driving the fg transition is applied to polarize the spin state to \(\left\vert 0\right\rangle\) (red arrows). A higher (smaller) population in the qubit manifold corresponds to a larger (smaller) average interaction strength J.
Extended Data Fig. 3 Decoherence dominated by spin-spin interactions in spin-echo measurements.
a, Normalized coherence decay of the spin ensemble as a function of evolution time in the spin-echo sequence. The early-time decay within 1 μs accelerates with increasing J, achieved by increasing the driving amplitude on the F transition during state initialization. Error bars represent the standard deviation of the experimental measurements (see Methods for details), while the solid lines are simple exponential fits used to extract the 1/e decoherence times. b, The fitted decoherence times monotonically decrease with increasing drive amplitude on the F transition, implying that a larger population within the qubit manifold leads to stronger spin-spin interactions and thus faster decoherence.
Extended Data Fig. 4 Coherence characterization in spin-echo measurements.
a, The spin-echo sequence with a final π/2 analyzer pulse with a variable phase angle θ. The phase angle θ defines the rotation axis of the π/2 pulse relative to the initial π/2 pulse. While maintaining a fixed free evolution time τ, θ is varied from 0 to 2π to quantify the residual coherence of the spin ensemble. b, Photoluminescence signals from the spin system exhibit a sinusoidal oscillation as θ is swept, from which coherence is extracted as the contrast of the oscillation. c, Coherence decay profiles as a function of evolution time τ for the cases of large J ≈ 2π × 0.35 MHz and small J ≈ 2π × 0.19 MHz. d, Comparison of long-time decoherence dynamics between experiment and simulation under the spin-echo dynamics. The numerical simulations with calibrated parameters (solid lines) show excellent agreement with the experiments (markers) over extended long times for both the small J (orange) and large J (blue) cases.
Extended Data Fig. 5 Numerical simulation results of the spin-echo decoherence dynamics.
a, The crystal structure of YVO4. b, System size scaling of the spin-echo decoherence profiles for different numbers of simulated spins from N = 2 to N = 10 ions. The decay profiles start to converge approximately when N > 8 ions. c, Effects of on-site disorder on spin-echo decoherence dynamics, simulated for different disorder strengths W. After the early-time transient (t < 1 μs) where the decay rate is independent of on-site disorder, spin decoherence slows down, indicating a crossover from an interaction-dominated to a disorder-dominated regime. d, Effects of initial spin polarization, ηpol, on spin-echo decoherence dynamics. e, Normalized coherence for the decay time traces shown in d. Upon rescaling, the decay profiles overlap for different polarization values of ηpol, indicating that the decoherence dynamics are independent of initial spin polarization.
Extended Data Fig. 6 Interaction-picture-based toggling-frame transformation of the \({\hat{\boldsymbol{S}}}_{\boldsymbol{z}}\) operator for ϵ-CPMG sequences.
a, ϵ = − π/2 is a preferable choice for decoupling spin-spin interactions in a strongly interacting spin system. b, ϵ = 0 corresponds to the conventional CPMG sequence, which decouples on-site disorder through periodic π pulses. These base sequences are repeated in time stroboscopically with fixed pulse spacing τ in dynamic Hamiltonian engineering.
Extended Data Fig. 7 Coherence characterization in ϵ-CPMG measurements.
a, b, We compare coherence decay profiles as a function of total interrogation time T for two different ϵ-CPMG measurements at ϵ ≈ 0 (yellow) and ϵ ≈ − π/2 (purple). For both cases, the corresponding base sequence has a periodicity of τ + tp, where τ is the free evolution period and tp is the pulse duration, and is repeated k times to advance in time (see the insets). We measure coherence at stroboscopic times T = k(τ + tp). a, τ/2 = 200 ns, tp = 45 ns for ϵ ≈ 0; tp = 25 ns for ϵ ≈ − π/2. b, τ/2 = 50 ns, tp = 43 ns for ϵ ≈ 0; tp = 22 ns for ϵ ≈ − π/2. Experimental data are obtained under the large J condition, and solid lines represent phenomenological stretched exponential fits. To facilitate comparison between the two different ϵ cases, the coherence decay profiles are normalized by their respective maximum coherence values. c, Dependence of ϵ-CPMG sequence coherence on ϵ for the long τ (blue) and short τ (red) cases. When sweeping ϵ, fixed base sequence parameters of (τ/2 = 200 ns, k = 5; blue) and (τ/2 = 50 ns, k = 20; red) are used respectively. Here, all experimental data across different ϵ values are globally normalized by the initial state polarization, ηpol (see Methods and Extended Data Fig. 5d, e). The solid lines denote numerical simulation results using the experimentally calibrated system parameters. The simulated results were globally rescaled to facilitate comparison with the experimental data.
Extended Data Fig. 8 WAHUHA-echo sequence measurement.
a, WAHUHA-echo sequence. The base pulse sequence consists of π/2 and π pulses with a periodicity of 6τ, where τ denotes the time separation between the centers of two adjacent π/2 pulses. We maintain a spacing of τ − tp between each pulse, where tp is the duration of the π/2 pulse. Note that the duration of a π pulse is 2tp. The base sequence is repeated k times to evolve over a total interrogation time T. The final π/2 analyzer pulse with a variable phase angle θ is employed for coherence extraction. b, Comparison of the WAHUHA-echo coherence dynamics between numerical simulation (red) and experiment (blue). The experiment is carried out under the small J condition, with error bars representing the standard deviation of the data (see Methods for details). The simulation is conducted using the experimentally calibrated interaction and disorder strengths. The blue line is a single exponential fit to extract the 1/e decay time. c, The 1/e decay times of the WAHUHA-echo coherence measurements are shown as a function of the pulse separation τ for both the large J (blue) and small J (orange) cases.
Extended Data Fig. 9 Spin-locking sequence measurement.
a, Spin-locking sequence. After initializing the spin ensemble along the y-axis via the initial π/2 pulse, a continuous drive along the y-axis with strength Ωy is applied for duration T to lock the spin orientation. The final π/2 analyzer pulse with a variable phase angle θ is employed for coherence extraction. b, Decoherence dynamics of the spin ensemble under the spin-locking sequence for large J (blue) and small J (orange) cases. Error bars represent the standard deviation of the experimental data (see Methods for details). The solid lines are single exponential fits used to extract the 1/e decay times, T1/e, yielding T1/e ≈ 50 μs and T1/e ≈ 73 μs for the large J and small J cases, respectively.
Extended Data Fig. 10 Robustness of DTC phases to initial spin states.
a, Floquet control pulse sequence for probing DTC phases with different initial spin orientations, controlled by the first pulse with a variable rotation angle ϕ; otherwise, the sequence is the same as in Fig. 5a of the main text. b-f, DTC phase diagrams for different initial rotation angles ranging from ϕ = 0 to ϕ = π/2. See the main text for details on the phase diagram reconstruction. g, Phase boundaries for d-f where the initial spin orientations are close to the y-axis. The phase boundaries for each interaction time τ are determined by identifying a threshold in the value of ϵ where the subharmonic peak intensity at frequency ν = 0.5, ∣S(ν = 0.5)∣2, falls below 0.4, that is, ∣S(ν = 0.5)∣2 < 0.4. We note a lateral 1% offset in the swept angle ϵ at the center position of the DTC phase diagram, that is, ϵ/π ≈ 0.01, attributed to an experimental calibration error of the rotation pulse.
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Lei, M., Fukumori, R., Wu, CJ. et al. Quantum thermalization and Floquet engineering in a spin ensemble with a clock transition. Nat. Phys. 21, 1196–1202 (2025). https://doi.org/10.1038/s41567-025-02943-4
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DOI: https://doi.org/10.1038/s41567-025-02943-4
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