Abstract
Solidstate nuclear spins surrounding individual, optically addressable qubits^{1,2} are a crucial resource for quantum networks^{3,4,5,6}, computation^{7,8,9,10,11} and simulation^{12}. Although hosts with sparse nuclear spin baths are typically chosen to mitigate qubit decoherence^{13}, developing coherent quantum systems in nuclearspinrich hosts enables exploration of a much broader range of materials for quantum information applications. The collective modes of these dense nuclear spin ensembles provide a natural basis for quantum storage^{14}; however, using them as a resource for singlespin qubits has thus far remained elusive. Here, by using a highly coherent, optically addressed ^{171}Yb^{3+} qubit doped into a nuclearspinrich yttrium orthovanadate crystal^{15}, we develop a robust quantum control protocol to manipulate the multilevel nuclear spin states of neighbouring ^{51}V^{5+} lattice ions. Via a dynamically engineered spinexchange interaction, we polarize this nuclear spin ensemble, generate collective spin excitations, and subsequently use them to implement a quantum memory. We additionally demonstrate preparation and measurement of maximally entangled ^{171}Yb–^{51}V Bell states. Unlike conventional, disordered nuclearspinbased quantum memories^{16,17,18,19,20,21,22,23,24}, our platform is deterministic and reproducible, ensuring identical quantum registers for all ^{171}Yb^{3+} qubits. Our approach provides a framework for utilizing the complex structure of dense nuclear spin baths, paving the way towards building largescale quantum networks using single rareearth ion qubits^{15,25,26,27,28}.
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The data that support the findings of this study are available from the corresponding authors upon request.
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Acknowledgements
This work was funded by the Institute of Quantum Information and Matter, an NSF Physics Frontiers Center (PHY1733907) with support from the Moore Foundation, NSF 1820790, Office of Naval Research award no. N000141912182, Air Force Office of Scientific Research grant no. FA95501810374 and no. FA95502110055, Northrop Grumman, General Atomics, and Weston Havens Foundation. The device nanofabrication was performed in the Kavli Nanoscience Institute at the California Institute of Technology. J.R. acknowledges the support from the Natural Sciences and Engineering Research Council of Canada (NSERC) (PGSD35028442017). A.R. acknowledges the support from the Eddleman Graduate Fellowship. J.C. acknowledges support from the IQIM postdoctoral fellowship. We thank J. Kindem, J. G. Bartholomew, N. Yao, A. Sipahigil, M. Lei and T. Xie for discussion, and M. Shaw for help with superconducting photon detectors.
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A.R., J.C. and A.F. conceived the experiments. J.R. fabricated the device. A.R. and C.J.W. performed the experiments and analysed the data. A.R. and J.C. designed the control sequences. A.R., J.C. and A.F. wrote the manuscript with input from all authors. J.C. and A.F. supervised the project.
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Extended data figures and tables
Extended Data Fig. 1 Experimental setup and sequence detail.
a, Energy level structure of ^{171}Yb^{3+}:YVO_{4} ^{2}F_{7/2}(0) and ^{2}F_{5/2}(0). Initialization into 0_{g}⟩ involves repeated pulses on the F transition combined with consecutive pairs of π pulses applied to the A and f_{e} transitions leading to excitation into 1_{e}⟩. Subsequently, decay via E leads to initialization into 0_{g}⟩. Optical readout relies on repeated optical π pulses on the A transition, each followed by a photon detection window during which we measure cavityenhanced emission via A. b, Experimental setup. Optical control of the A and F transitions is realized via two frequencystabilized lasers, each modulated using acoustooptic modulator (AOM) shutters. Microwave control is divided into two paths: a lowfrequency path consisting of 675 MHz ground state control (f_{g} transition) and RF, both generated using a single arbitrary waveform generator (AWG) channel and a highfrequency path consisting of 3.4 GHz excitedstate microwave control (f_{e} transition). Each path is independently amplified and combined using a diplexer. The device chip and a superconducting nanowire single photon detector (SNSPD) are cooled to ~500 mK in a cryostat. c, Detailed pulse sequence used for quantum state storage and retrieval. First, the ^{51}V register and ^{171}Yb qubit are initialized into 0_{v}⟩ and 0_{g}⟩, respectively, as described in the text. Subsequently, the ^{171}Yb is prepared in a superposition state via a π/2 pulse, which is swapped onto the ^{51}V register using a ZenPol sequence resonant with the 991 kHz ω_{c} ^{51}V transition. After a wait time, t, the state is swapped back to ^{171}Yb and measured in the x basis via a π/2 pulse followed by optical readout.
Extended Data Fig.2 Randomized benchmarking and ^{171}Yb qubit coherence.
a, We measure the average fidelity of singlequbit gates applied to the ^{171}Yb 0_{g}⟩ ↔ 1_{g}⟩ transition. We apply a series of M_{gate} randomly sampled Clifford gates followed by the inverse operation (top inset). When averaged over a sufficiently large number of samples (in our case 100) we can extract an average gate fidelity from the 1/e exponential decay constant, leading to f = 0.99975 ± 0.00004. b, We also measure the coherence time of the qubit transition using an XY8 dynamical decoupling pulse sequence (top inset) with a fixed interπpulse separation of 5.6 μs and variable number of repetitions, M′. This leads to an exponential decay with 1/e time constant T_{2} = 16 ± 2 ms.
Extended Data Fig. 3 Hartmann Hahn spectroscopy.
a, Hartmann Hahn (HH) sequence used to perform spectroscopy of the nuclear spin environment. During the HH pulse (red), the ^{171}Yb 0_{g}⟩ ↔ 1_{g}⟩ qubit transition is driven resonantly for duration t with yphase leading to a pair of dressed states, \(\pm \rangle =\frac{1}{\sqrt{2}}({0}_{{\rm{g}}}\rangle \pm {\rm{i}}{1}_{{\rm{g}}}\rangle )\), separated by energy splitting equal to the Rabi frequency, Ω. An initial −xphase π/2 pulse prepares the ^{171}Yb qubit in the −⟩ dressed state. When the Rabi frequency of the HH pulse is tuned to equal one of the ^{51}V transition frequencies, the ^{171}Yb is transferred into the +⟩ dressed state as a result of resonant population exchange (green arrows). The +⟩ state population is mapped to 1_{g}⟩ with a final xphase π/2 pulse for readout. b, HH spectroscopy experimental results. To identify nuclear spin resonances, both the HH pulse amplitude and duration are varied. The three evenly spaced horizontal resonance features occurring at pulse amplitudes of 0.15, 0.3, and 0.45 (in arbitrary units, a.u.) correspond to interaction with the ω_{a}, ω_{b} and ω_{c} transitions, respectively. In the nodriving (Ω = 0) case, the sequence probes the decoherence dynamics of the prepared −⟩ state; that is, it measures the Ramsey coherence time. c, HH spectroscopy simulation results. Simulation results agree well with the experiment, corroborating that ^{171}Yb–^{51}V interactions are dominant in our system.
Extended Data Fig. 4 ZenPol sequence detail.
a, ZenPol sequence with the togglingframe transformation of the \({\widehat{\tilde{S}}}_{z}\) operator for the ^{171}Yb qubit. The ZenPol sequence consists of a series of π and π/2 pulses about the x and y axes combined with a synchronously applied, squarewave RF magnetic field with period 2τ. The Overhauser and RFinduced interactions are determined by the togglingframe transformations of \({\widehat{\tilde{S}}}_{z}\), which are given by \({\widehat{\tilde{S}}}_{x}{f}_{x}^{{\rm{OH}}}+{\widehat{\tilde{S}}}_{y}{f}_{y}^{{\rm{OH}}}\) and \({\widehat{\tilde{S}}}_{x}{f}_{x}^{{\rm{RF}}}+{\widehat{\tilde{S}}}_{y}{f}_{y}^{{\rm{RF}}}\), respectively (see yellow and purple lines for \({f}_{x,y}^{{\rm{OH}}}\) and \({f}_{x,y}^{{\rm{RF}}}\), respectively). At the resonance condition 1/2τ = ω_{j}/2πk for odd integer k with ^{51}V spin precession frequency ω_{j}, the sequence realizes noiserobust spinexchange interaction with a timeaveraged Hamiltonian that depends only on the RF magnetic field amplitude. b, ZenPol sequence filter functions corresponding to the Fourier transforms of \({f}_{x}^{{\rm{OH}}}\) (yellow) and \({f}_{x}^{{\rm{RF}}}\) (purple). For a sequence with fixed τ, the peak positions determine the resonant frequencies at which ^{171}Yb–^{51}V interactions can occur. Note that the incoherent Overhauserinduced interactions occur at evenk resonances and are spectrally separated from the coherent RFinduced interactions occurring at oddk resonances.
Extended Data Fig. 5 Polarization of multilevel nuclear register spins.
a, Polarization readout by polarization inversion (PROPI) experiments for the ^{51}V register ω_{c} transition. The PROPI sequence performs a repeated swap operation based on the ZenPol sequence, periodically interleaved with ^{171}Yb qubit readout and reinitialization into 1_{g}⟩. A total of 20 polarizing cycles are applied to the ω_{c} transition to polarize the ^{51}V register into ±5/2⟩. As a result of register polarization, the ^{171}Yb population in 1_{g}⟩ increases over time, indicating the accumulation of the ^{51}V population in ±5/2⟩ (left). We observe that the register polarization saturates after approximately 10 cycles. Subsequently, we perform repolarization cycles where ^{171}Yb is initialized into 0_{g}⟩ and ^{51}V register spins are transferred to ±7/2⟩ with similar saturation timescale (right). b, PROPI experiments for the ^{51}V register ω_{b} transition. Applying a ZenPol sequence resonant with the ω_{b} transition, interleaved with ^{171}Yb initialization into 1_{g}⟩ (0_{g}⟩), results in ^{51}V register polarization into ±5/2⟩ (±3/2⟩), as indicated by an increase (decrease) in ^{171}Yb 1_{g}⟩ population. c, Experimental results of ZenPol spinexchange dynamics with varying degree of ^{51}V register polarization. As the number of polarization cycles used to prepare the 0_{v}⟩ state increases, the subsequent spinexchange oscillations become more pronounced. Note that these polarization cycles are interleaved between the ω_{b} and ω_{c} transitions.
Extended Data Fig. 6 Spinexchange dynamics.
a, ZenPol sequence schematic. The squarewave RF magnetic field amplitude B^{RF} determines the ^{171}Yb–^{51}V interaction strength, the pulse spacing τ/4 varies the sequence detuning from a specific ^{51}V nuclear spin transition, and the number of ZenPol periods, M, determines the total interaction time. b, Simulated spinexchange dynamics near the ω_{c} transition at k = 5, probed as a function of sequence resonance frequency ω and the number of ZenPol periods, M. c, Measured spinexchange dynamics showing good agreement with the numerical simulation in b. d, Experimental demonstration of tunable spinexchange rate by varying B^{RF}. When increasing B^{RF} from 0.8 G to 2.0 G, we observe a corresponding linear increase in the spinexchange rate. In all cases, numerical simulations (solid lines), taking into account incomplete register polarization, control pulse imperfections and an exponential phenomenological decay, show reasonable agreement with the experimental data (markers). A simulation result without this phenomenological decay (dashed line) displays a discrepancy, which needs further investigation. See Supplementary Information for simulation details.
Extended Data Fig. 7 Direct ^{51}V nuclear spin driving.
a, Details of ^{51}V nuclear spin driving scheme. To directly drive the ^{51}V nuclear spin ω_{c} transition, a sinusoidal zdirected RF magnetic field, \({B}_{z}^{{\rm{osc}}}\,\sin ({\omega }_{{\rm{c}}}t)\), is applied to the system at a frequency of ω_{c}/2π = 991 kHz after initializing the ^{171}Yb and ^{51}V register into 0_{g}⟩ and 0_{v}⟩ = ↓↓↓↓⟩, respectively (drive protocol 1). This induces an oscillating magnetic dipole moment on the ^{171}Yb qubit, which in turn generates an amplified transverse driving field at each ^{51}V (Methods). Consequently, the four ^{51}V register spins undergo independent Rabi oscillation between the ↑⟩ = ±5/2⟩ and ↓⟩ = ±7/2⟩ states. To probe the nuclear spin Rabi oscillation, the ↓⟩ population is measured by preparing the ^{171}Yb in 1_{g}⟩ via an xphase π pulse, performing a single swap gate and reading out the ^{171}Yb population. b, Decoupling of magnetic field noise originating from the ^{171}Yb Knight field. To improve the nuclear spin control fidelity, a train of equidistant π pulses are applied to the ^{171}Yb during the driving period, thereby cancelling dephasing due to the ^{171}Yb Knight field (drive protocol 2). Each π pulse is accompanied by a π phase shift of the sinusoidal field to ensure phase continuity of the nuclear Rabi driving, and an even number of π pulses ensures the ^{171}Yb state is returned to 0_{g}⟩ at the end of the sequence (Methods). c, Measured ^{51}V register Rabi oscillations using the aforementioned schemes. We observe coherent nuclear Rabi oscillations between the ↓⟩ and ↑⟩ states at a Rabi frequency of Ω_{D}/2π = 7.65 ± 0.05 kHz. An exponential decay is observed with a 1/e time constant of 280 ± 30 μs without decoupling (blue). The additional π pulses applied to the ^{171}Yb qubit lead to an enhancement in control fidelity, giving a 1/e Gaussian decay time of 1040 ± 70 μs (red). The black arrow at t ≈ 69 μs indicates the ^{51}V π pulse used in Fig. 3c.
Extended Data Fig. 8 ^{51}V spin register population relaxation.
a, Measured relaxation timescales, \({T}_{1}^{(W)}\), of the entangled register state, W_{v}⟩, under various conditions. Top, the ^{51}V register is prepared in the W_{v}⟩ state by swapping a single spin excitation from the ^{171}Yb initialized into 1_{g}⟩. After a variable wait time, t, the ^{51}V state is swapped back onto ^{171}Yb and measured (top inset). The resulting Gaussian decay shows a 1/e relaxation time of \({T}_{1}^{(W)}\) = 39.5 ± 1.3 μs (blue trace), limited by dephasing of the entangled W_{v}⟩ state. Middle, the \({T}_{1}^{(W)}\) lifetime can be extended by applying a series of equidistant π pulses to the ^{171}Yb separated by 2t_{w} = 6 μs (middle inset). This decouples the W_{v}⟩ state from dephasing induced by the ^{171}Yb Knight field, equivalent to the coherence time extension in Fig. 3b, leading to an extended 1/e lifetime of \({T}_{1}^{(W)}\) = 127 ± 8 μs (red trace). Bottom, further extension of the \({T}_{1}^{(W)}\) lifetime is achieved by dynamical decoupling whereby additionally two ^{51}V π pulses are applied during the wait time with a variable pulse separation 2t_{D} (bottom inset). This gives rise to a substantially prolonged lifetime of \({T}_{1}^{(W)}\) = 640 ± 20 μs (yellow trace), equivalent to the coherence time extension in Fig. 3c. b, Measured relaxation timescale, \({T}_{1}^{(0)}\), of the polarized register state 0_{v}⟩.The register is initialized in 0_{v}⟩ and after a variable wait time, t, the ^{51}V state is swapped onto ^{171}Yb and measured (inset). We observe an exponential decay with a 1/e relaxation time of \({T}_{1}^{(0)}\) = 0.54 ± 0.08 s, probably limited by incoherent population transfer to the bath. See Supplementary Information for detailed discussion of T_{1} relaxation mechanisms.
Extended Data Fig. 9 Population measurement histograms for register fidelity characterization.
a, Sequential tomography protocol for characterizing ^{171}Yb–^{51}V populations in the basis spanned by {0_{g}0_{v}⟩, 0_{g}W_{v}⟩, 1_{g}0_{v}⟩, 1_{g}W_{v}⟩}. Reconstructing the population probability distribution utilizes readout sequences 1 and 2, each including three consecutive ^{171}Yb state readouts interleaved with singlequbit gate operations and a swap gate. b, Table summarizing the postprocessing criteria for state attribution. Readout sequences 1 and 2 measure the {0_{g}0_{v}⟩, 0_{g}W_{v}⟩} and {1_{g}0_{v}⟩, 1_{g}W_{v}⟩} populations, respectively, conditioned on the three measurement outcomes. See Methods for full details of the postprocessing procedure. c, Reconstructed population distributions for estimating state preparation fidelity. The four basis states, {0_{g}0_{v}⟩, 0_{g}W_{v}⟩, 1_{g}0_{v}⟩, 1_{g}W_{v}⟩}, are independently prepared by applying a combination of ^{171}Yb π pulses and swap gates to the initial 0_{g}0_{v}⟩ state (see insets). Subsequently, the sequential tomography protocol for state readout (RO) is applied iteratively, alternating between readout 1 and 2 sequences to fully reconstruct the population probability distributions. d, Reconstructed population distribution for the ^{171}Yb–^{51}V Bell state (reproduced from Fig. 4c). The maximally entangled Bell state \({\Psi }^{+}\rangle =\frac{1}{\sqrt{2}}({1}_{g}{0}_{{\rm{v}}}\rangle {\rm{i}}{0}_{{\rm{g}}}{{\rm{W}}}_{{\rm{v}}}\rangle )\) is prepared by applying a \(\sqrt{{\rm{swap}}}\) gate to 1_{g}0_{v}⟩ and measured using RO (inset). In c, d, the uncorrected and readoutcorrected measurement results are presented as dashed and solid filled histograms, respectively, with error bars indicating one standard deviation. Populations are corrected by accounting for the swap gate error during the readout sequences (Methods).
Extended Data Fig. 10 Experimental demonstration of deterministic nuclear spin register.
To demonstrate the deterministic nature of the nuclear spin register, we perform the same measurements on two additional ^{171}Yb ion qubits present in the device: ion 2 (red) and ion 3 (yellow). Results for ion 1 (blue) are reproduced from Figs. 2 and 3 for ease of comparison. a, ZenPol spectra near the ω_{c}(k = 5) resonance of the ^{51}V register spins. Note that for all three ions, the bath and register transitions are identified at the same resonance frequencies of \({\omega }_{{\rm{c}}}^{{\rm{bath}}}/2{\rm{\pi }}=1,028{\rm{kHz}}\) and ω_{c}/2π = 991 kHz, respectively. b, Dynamically engineered spinexchange dynamics between the ^{171}Yb qubit and ^{51}V register. Using constant ZenPol squarewave RF amplitude we obtain equal spinexchange rates for all three ions. c, Characterization of ^{51}V register coherence times with decoupling from the ^{171}Yb Knight field. The 1/e coherence times are measured to be 225 ± 9 μs, 273 ± 12 μs and 261 ± 9 μs for ions 1, 2 and 3, respectively. All of these results demonstrate that our platform provides a nearly identical nuclear spin register for every ^{171}Yb qubit in the system.
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Ruskuc, A., Wu, CJ., Rochman, J. et al. Nuclear spinwave quantum register for a solidstate qubit. Nature 602, 408–413 (2022). https://doi.org/10.1038/s41586021042936
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DOI: https://doi.org/10.1038/s41586021042936
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