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# Nuclear spin-wave quantum register for a solid-state qubit

## Abstract

Solid-state nuclear spins surrounding individual, optically addressable qubits1,2 are a crucial resource for quantum networks3,4,5,6, computation7,8,9,10,11 and simulation12. Although hosts with sparse nuclear spin baths are typically chosen to mitigate qubit decoherence13, developing coherent quantum systems in nuclear-spin-rich 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 storage14; however, using them as a resource for single-spin qubits has thus far remained elusive. Here, by using a highly coherent, optically addressed 171Yb3+ qubit doped into a nuclear-spin-rich yttrium orthovanadate crystal15, we develop a robust quantum control protocol to manipulate the multi-level nuclear spin states of neighbouring 51V5+ lattice ions. Via a dynamically engineered spin-exchange 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 171Yb–51V Bell states. Unlike conventional, disordered nuclear-spin-based quantum memories16,17,18,19,20,21,22,23,24, our platform is deterministic and reproducible, ensuring identical quantum registers for all 171Yb3+ qubits. Our approach provides a framework for utilizing the complex structure of dense nuclear spin baths, paving the way towards building large-scale quantum networks using single rare-earth ion qubits15,25,26,27,28.

## Relevant articles

• ### Harnessing many-body spin environment for long coherence storage and high-fidelity single-shot qubit readout

Nature Communications Open Access 13 July 2022

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## Data availability

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 (PHY-1733907) with support from the Moore Foundation, NSF 1820790, Office of Naval Research award no. N00014-19-1-2182, Air Force Office of Scientific Research grant no. FA9550-18-1-0374 and no. FA9550-21-1-0055, 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) (PGSD3-502844-2017). 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.

## Author information

Authors

### Contributions

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.

### Corresponding authors

Correspondence to Joonhee Choi or Andrei Faraon.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

Nature thanks Fedor Jelezko and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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## Extended data figures and tables

### Extended Data Fig. 1 Experimental set-up and sequence detail.

a, Energy level structure of 171Yb3+:YVO4 2F7/2(0) and 2F5/2(0). Initialization into |0g involves repeated pulses on the F transition combined with consecutive pairs of π pulses applied to the A and fe transitions leading to excitation into |1e. Subsequently, decay via E leads to initialization into |0g. Optical readout relies on repeated optical π pulses on the A transition, each followed by a photon detection window during which we measure cavity-enhanced emission via A. b, Experimental set-up. Optical control of the A and F transitions is realized via two frequency-stabilized lasers, each modulated using acousto-optic modulator (AOM) shutters. Microwave control is divided into two paths: a low-frequency path consisting of 675 MHz ground state control (fg transition) and RF, both generated using a single arbitrary waveform generator (AWG) channel and a high-frequency path consisting of 3.4 GHz excited-state microwave control (fe 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 51V register and 171Yb qubit are initialized into |0v and |0g, respectively, as described in the text. Subsequently, the 171Yb is prepared in a superposition state via a π/2 pulse, which is swapped onto the 51V register using a ZenPol sequence resonant with the 991 kHz ωc 51V transition. After a wait time, t, the state is swapped back to 171Yb and measured in the x basis via a π/2 pulse followed by optical readout.

### Extended Data Fig.2 Randomized benchmarking and 171Yb qubit coherence.

a, We measure the average fidelity of single-qubit gates applied to the 171Yb |0g ↔ |1g transition. We apply a series of Mgate 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 XY-8 dynamical decoupling pulse sequence (top inset) with a fixed inter-π-pulse separation of 5.6 μs and variable number of repetitions, M′. This leads to an exponential decay with 1/e time constant T2 = 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 171Yb |0g ↔ |1g qubit transition is driven resonantly for duration t with y-phase 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 −x-phase π/2 pulse prepares the 171Yb qubit in the |− dressed state. When the Rabi frequency of the HH pulse is tuned to equal one of the 51V transition frequencies, the 171Yb is transferred into the |+ dressed state as a result of resonant population exchange (green arrows). The |+ state population is mapped to |1g with a final x-phase 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 ωa, ωb and ωc transitions, respectively. In the no-driving (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 171Yb–51V interactions are dominant in our system.

### Extended Data Fig. 4 ZenPol sequence detail.

a, ZenPol sequence with the toggling-frame transformation of the $${\widehat{\tilde{S}}}_{z}$$ operator for the 171Yb 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, square-wave RF magnetic field with period 2τ. The Overhauser- and RF-induced interactions are determined by the toggling-frame 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 = ωj/2πk for odd integer k with 51V spin precession frequency ωj, the sequence realizes noise-robust spin-exchange interaction with a time-averaged 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 171Yb–51V interactions can occur. Note that the incoherent Overhauser-induced interactions occur at even-k resonances and are spectrally separated from the coherent RF-induced interactions occurring at odd-k resonances.

### Extended Data Fig. 5 Polarization of multi-level nuclear register spins.

a, Polarization readout by polarization inversion (PROPI) experiments for the 51V register ωc transition. The PROPI sequence performs a repeated swap operation based on the ZenPol sequence, periodically interleaved with 171Yb qubit readout and reinitialization into |1g. A total of 20 polarizing cycles are applied to the ωc transition to polarize the 51V register into |±5/2. As a result of register polarization, the 171Yb population in |1g increases over time, indicating the accumulation of the 51V population in |±5/2 (left). We observe that the register polarization saturates after approximately 10 cycles. Subsequently, we perform repolarization cycles where 171Yb is initialized into |0g and 51V register spins are transferred to |±7/2 with similar saturation timescale (right). b, PROPI experiments for the 51V register ωb transition. Applying a ZenPol sequence resonant with the ωb transition, interleaved with 171Yb initialization into |1g (|0g), results in 51V register polarization into |±5/2 (|±3/2), as indicated by an increase (decrease) in 171Yb |1g population. c, Experimental results of ZenPol spin-exchange dynamics with varying degree of 51V register polarization. As the number of polarization cycles used to prepare the |0v state increases, the subsequent spin-exchange oscillations become more pronounced. Note that these polarization cycles are interleaved between the ωb and ωc transitions.

### Extended Data Fig. 6 Spin-exchange dynamics.

a, ZenPol sequence schematic. The square-wave RF magnetic field amplitude BRF determines the 171Yb–51V interaction strength, the pulse spacing τ/4 varies the sequence detuning from a specific 51V nuclear spin transition, and the number of ZenPol periods, M, determines the total interaction time. b, Simulated spin-exchange dynamics near the ωc transition at k = 5, probed as a function of sequence resonance frequency ω and the number of ZenPol periods, M. c, Measured spin-exchange dynamics showing good agreement with the numerical simulation in b. d, Experimental demonstration of tunable spin-exchange rate by varying BRF. When increasing BRF from 0.8 G to 2.0 G, we observe a corresponding linear increase in the spin-exchange 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 51V nuclear spin driving.

a, Details of 51V nuclear spin driving scheme. To directly drive the 51V nuclear spin ωc transition, a sinusoidal z-directed RF magnetic field, $${B}_{z}^{{\rm{osc}}}\,\sin ({\omega }_{{\rm{c}}}t)$$, is applied to the system at a frequency of ωc/2π = 991 kHz after initializing the 171Yb and 51V register into |0g and |0v = |↓↓↓↓, respectively (drive protocol 1). This induces an oscillating magnetic dipole moment on the 171Yb qubit, which in turn generates an amplified transverse driving field at each 51V (Methods). Consequently, the four 51V 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 171Yb in |1g via an x-phase π pulse, performing a single swap gate and reading out the 171Yb population. b, Decoupling of magnetic field noise originating from the 171Yb Knight field. To improve the nuclear spin control fidelity, a train of equidistant π pulses are applied to the 171Yb during the driving period, thereby cancelling dephasing due to the 171Yb 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 171Yb state is returned to |0g at the end of the sequence (Methods). c, Measured 51V register Rabi oscillations using the aforementioned schemes. We observe coherent nuclear Rabi oscillations between the |↓ and |↑ states at a Rabi frequency of ΩD/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 171Yb 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 51V π pulse used in Fig. 3c.

### Extended Data Fig. 8 51V spin register population relaxation.

a, Measured relaxation timescales, $${T}_{1}^{(W)}$$, of the entangled register state, |Wv, under various conditions. Top, the 51V register is prepared in the |Wv state by swapping a single spin excitation from the 171Yb initialized into |1g. After a variable wait time, t, the 51V state is swapped back onto 171Yb 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 |Wv state. Middle, the $${T}_{1}^{(W)}$$ lifetime can be extended by applying a series of equidistant π pulses to the 171Yb separated by 2tw = 6 μs (middle inset). This decouples the |Wv state from dephasing induced by the 171Yb 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 51V π pulses are applied during the wait time with a variable pulse separation 2tD (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 |0v.The register is initialized in |0v and after a variable wait time, t, the 51V state is swapped onto 171Yb 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 T1 relaxation mechanisms.

### Extended Data Fig. 9 Population measurement histograms for register fidelity characterization.

a, Sequential tomography protocol for characterizing 171Yb–51V populations in the basis spanned by {|0g0v, |0gWv, |1g0v, |1gWv}. Reconstructing the population probability distribution utilizes readout sequences 1 and 2, each including three consecutive 171Yb state readouts interleaved with single-qubit gate operations and a swap gate. b, Table summarizing the post-processing criteria for state attribution. Readout sequences 1 and 2 measure the {|0g0v, |0gWv} and {|1g0v, |1gWv} populations, respectively, conditioned on the three measurement outcomes. See Methods for full details of the post-processing procedure. c, Reconstructed population distributions for estimating state preparation fidelity. The four basis states, {|0g0v, |0gWv, |1g0v, |1gWv}, are independently prepared by applying a combination of 171Yb π pulses and swap gates to the initial |0g0v 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 171Yb–51V 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 |1g0v and measured using RO (inset). In c, d, the uncorrected and readout-corrected 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 171Yb 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 ωc(k = 5) resonance of the 51V 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 ωc/2π = 991 kHz, respectively. b, Dynamically engineered spin-exchange dynamics between the 171Yb qubit and 51V register. Using constant ZenPol square-wave RF amplitude we obtain equal spin-exchange rates for all three ions. c, Characterization of 51V register coherence times with decoupling from the 171Yb 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 171Yb qubit in the system.

## Supplementary information

### Supplementary Information

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Ruskuc, A., Wu, CJ., Rochman, J. et al. Nuclear spin-wave quantum register for a solid-state qubit. Nature 602, 408–413 (2022). https://doi.org/10.1038/s41586-021-04293-6

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