Repetitive Nuclear-Assisted Spin Readout

• Repetitive nuclear-assisted spin readout is a measurement protocol that uses robust nuclear states as persistent memories to amplify weak electron or optical signals.
• It alternates conditional mapping, high-gain auxiliary readout, and reset steps to maximize signal accumulation while minimizing back-action on the nuclear state.
• Implementations across silicon donors, NV centers, and quantum dots demonstrate high fidelity and scalability for advanced quantum memory, sensing, and error correction applications.

Repetitive nuclear-assisted spin readout denotes a class of measurement protocols in which a nuclear spin, nuclear qudit, or nuclear-spin ensemble acts as an ancilla memory, a QND-like meter, or a signal-amplifying reservoir for the readout of a more weakly observable spin degree of freedom. The central idea is to exploit the long lifetime, relatively weak environmental coupling, and hyperfine addressability of nuclear states, while repeatedly interfacing them with an electron spin, an optical cycling transition, a charge sensor, or a macroscopic NMR channel. Across platforms, the protocol usually alternates a conditional mapping step, a high-gain readout of an auxiliary degree of freedom, and a reset of that auxiliary degree of freedom while attempting to leave the nuclear state unchanged. This architecture has been realized in single-donor silicon, diamond and silicon-carbide defect centers, neutral-atom arrays, semiconductor quantum dots, bulk nuclear-spin amplifiers, and single-molecule magnets (Pla et al., 2013, Holzgrafe et al., 2018, Huie et al., 2023).

1. Conceptual structure and protocol families

At the most general level, repetitive nuclear-assisted spin readout has three recurring forms. In the first, a single nuclear spin stores a qubit state while an electron ancilla is repeatedly measured and reinitialized. In the second, repeated electron measurements act directly on a nuclear bath through a nonunitary conditional propagator, driving measurement-induced polarization or purification. In the third, information associated with a rare spin is deposited into a larger nuclear-spin reservoir and read out only after macroscopic amplification, or is inferred from nuclear-state-dependent transport statistics. These forms share the same asymmetry of roles: the nuclear subsystem is the slow, persistent degree of freedom, whereas the electron, optical, charge, or transport channel is the fast, dissipative one (Wu, 2010, 1105.4740).

Platform	Nuclear resource	Repeated readout channel
Silicon $^{31}\mathrm{P}$ donor	Single $^{31}\mathrm{P}$ nucleus	ESR, spin-to-charge conversion, SET current
NV center / nanodiamond	Host $^{14}\mathrm{N}$ nucleus	Repetitive optical electron readout
V2 center in 4H-SiC	Nearby $^{29}\mathrm{Si}$ nucleus	Resonant optical fluorescence
$^{171}\mathrm{Yb}$ tweezers	Nuclear spin qubit itself	Near-cycling fluorescence
Quantum dot	Nuclear-spin bath	Repeated electron-spin measurements
Triplet-DNP bulk solid	Abundant nuclear-spin bath	Inductive NMR after amplification
$^{163}\mathrm{DyPc}_2$	$^{163}\mathrm{Dy}$ nuclear qudit	Telegraph noise and split Kondo conductance

A common misconception is that all repetitive nuclear-assisted readout protocols are variations of a single QND ancilla measurement. The literature is broader than that. In silicon, the nuclear spin itself is electrically measured in a QND-like manner (Pla et al., 2013). In nanodiamond and SiC, the nuclear spin is used as a robust optical memory under repeated electron interrogation (Holzgrafe et al., 2018, Hesselmeier et al., 2024). In quantum dots, the repeated measurement does not merely read the nucleus; it deforms the nuclear wavefunction and can polarize the bath (Wu, 2010). In bulk solids, repeated mapping into many abundant nuclei is explicitly an amplification protocol rather than a single-spin ancilla protocol (1105.4740).

2. Hamiltonians, conditional mappings, and measurement operators

The microscopic description is platform-dependent, but the operative ingredients are hyperfine coupling, conditional control, and a readout channel whose signal depends on the nuclear state. For a single $^{31}\mathrm{P}$ donor in silicon, the coupled electron-nuclear Hamiltonian is

$$H = A \mathbf{S}\cdot\mathbf{I} + \gamma_e B_0 S_z - \gamma_n B_0 I_z,$$

with $S=1/2$ and $^{31}\mathrm{P}$0. In the regime $^{31}\mathrm{P}$1, the hyperfine interaction splits the ESR and NMR spectra into nuclear-state-dependent branches,

$^{31}\mathrm{P}$2

$^{31}\mathrm{P}$3

so a selective ESR inversion maps the nuclear projection $^{31}\mathrm{P}$4 onto the electron spin, after which spin-to-charge conversion and SET current detection complete the measurement (Pla et al., 2013).

In defect-spin platforms, the same logic appears with different level structures. For the NV-$^{31}\mathrm{P}$5 system, the ground-state Hamiltonian is written as

$^{31}\mathrm{P}$6

with $^{31}\mathrm{P}$7 and $^{31}\mathrm{P}$8, enabling hyperfine-selective MW control and conditional RF rotations (Holzgrafe et al., 2018). For a V2 center in 4H-SiC coupled to a nearby $^{31}\mathrm{P}$9, the effective Hamiltonian in ground and excited manifolds is

$^{14}\mathrm{N}$0

with $^{14}\mathrm{N}$1 and $^{14}\mathrm{N}$2; a $^{14}\mathrm{N}$3 implemented by two frequency-selective microwave $^{14}\mathrm{N}$4 pulses maps the nuclear state onto electron brightness, and a short resonant optical pulse both measures and resets the electron (Hesselmeier et al., 2024).

Quantum-dot treatments formalize the measurement back-action through Kraus operators acting on the nuclear Hilbert space. If the joint evolution over a finite interval $^{14}\mathrm{N}$5 is $^{14}\mathrm{N}$6, then the conditional nuclear propagator after an electron-spin measurement outcome $^{14}\mathrm{N}$7 is

$^{14}\mathrm{N}$8

and after $^{14}\mathrm{N}$9 successful outcomes,

$^{29}\mathrm{Si}$0

In this formulation, repeated measurements act as a nonunitary gate on the nuclear bath, suppressing components whose eigenvalues under $^{29}\mathrm{Si}$1 have modulus smaller than unity (Wu, 2010).

A distinct mapping principle appears in spin-amplification protocols. In the $^{29}\mathrm{Si}$2–$^{29}\mathrm{Si}$3 system of rare $^{29}\mathrm{Si}$4 nuclei embedded in an abundant $^{29}\mathrm{Si}$5 bath, heteronuclear flip-flops are switched on or off by magnetic-field cycling. At low field, $^{29}\mathrm{Si}$6 turns on the $^{29}\mathrm{Si}$7 terms and allows heteronuclear spin diffusion; at high field, $^{29}\mathrm{Si}$8 freezes them, so repeated cycles convert a single-spin operation on $^{29}\mathrm{Si}$9 into a macroscopic change of the $^{171}\mathrm{Yb}$0-bath magnetization (1105.4740).

3. Implementations across solid-state, atomic, and molecular platforms

In silicon donor devices, repetitive nuclear-assisted readout is most explicitly a QND electrical measurement. A single $^{171}\mathrm{Yb}$1 donor in the $^{171}\mathrm{Yb}$2 state is tunnel-coupled to a silicon MOS SET, while a nearby transmission line delivers both ESR and NMR control. The nuclear readout uses three conditional stages: a fast adiabatic ESR inversion chirp around one of the two ESR lines, spin-to-charge conversion via energy-selective tunneling, and SET-current acquisition. Repeating the electron readout at $^{171}\mathrm{Yb}$3 and $^{171}\mathrm{Yb}$4 yields $^{171}\mathrm{Yb}$5 for each line, and the discriminant $^{171}\mathrm{Yb}$6 assigns the nuclear state. With optimized measurement time $^{171}\mathrm{Yb}$7 ms, the resulting nuclear single-shot fidelity lies between $^{171}\mathrm{Yb}$8 and $^{171}\mathrm{Yb}$9, while $^{163}\mathrm{DyPc}_2$0 ms and the experimental lower bound on the 1-qubit gate fidelity is $^{163}\mathrm{DyPc}_2$1 (Pla et al., 2013).

Diamond-based realizations divide into several regimes. In nanodiamond NV centers, the host $^{163}\mathrm{DyPc}_2$2 nuclear spin stores the electron-spin state during repeated optical interrogation, but the optical cycle induces longitudinal nuclear bit-flip errors through excited-state electron-nuclear flip-flops. A coherent-feedback correction step, implemented by a hyperfine-selective MW $^{163}\mathrm{DyPc}_2$3 pulse and an RF $^{163}\mathrm{DyPc}_2$4 pulse realizing an effective SWAP on the $^{163}\mathrm{DyPc}_2$5 subspace, reverses the dominant $^{163}\mathrm{DyPc}_2$6 error channel in the moderate-field regime (Holzgrafe et al., 2018). In room-temperature electrical NV measurements, spin-dependent photoionization replaces fluorescence collection, and the $^{163}\mathrm{DyPc}_2$7 nuclear orientation is preserved across $^{163}\mathrm{DyPc}_2$8 charge conversion, enabling repetitive, QND-like nuclear readout compatible with nanoscale electrodes (Gulka et al., 2021). In ensemble NV gyroscope readout beyond the electron $^{163}\mathrm{DyPc}_2$9, an optical repump pulse before the CNOT restores electron polarization and allows readout of the longitudinal nuclear component even after the electron has relaxed to a thermal state (Kuan et al., 31 Jan 2025). In a mesoscopic NV ensemble at $^{163}\mathrm{Dy}$0 T, the same nuclear-assisted repetition suppresses photon shot noise below the thermal projection-noise floor and yields direct QND readout of the collective nuclear variable $^{163}\mathrm{Dy}$1 (Maier et al., 15 Sep 2025).

Silicon carbide implements the same ancilla-memory concept under very different optical constraints. For a single V2 center in 4H-SiC at cryogenic temperature, direct electron single-shot readout is infeasible because the optical cyclicity is low and metastable states interrupt fluorescence. Repetitive nuclear-assisted readout therefore uses a nearby $^{163}\mathrm{Dy}$2 as a stable memory: each cycle applies a $^{163}\mathrm{Dy}$3, then a 15 $^{163}\mathrm{Dy}$4s resonant A2 laser pulse, which produces fluorescence for the bright manifold and resets the electron into $^{163}\mathrm{Dy}$5. The scheme reaches an average nuclear-readout fidelity of $^{163}\mathrm{Dy}$6 at $^{163}\mathrm{Dy}$7, up to $^{163}\mathrm{Dy}$8 with $^{163}\mathrm{Dy}$9 postselected success, and $^{31}\mathrm{P}$0 initialization by measurement (Hesselmeier et al., 2024).

Neutral-atom arrays supply an atomic version of repetitive nuclear-spin readout. In $^{31}\mathrm{P}$1, the qubit is encoded in the $^{31}\mathrm{P}$2 nuclear spin-$^{31}\mathrm{P}$3 manifold, while a Zeeman-resolved $^{31}\mathrm{P}$4 state at $^{31}\mathrm{P}$5 G provides a near-cycling optical transition. The bright state cycles on $^{31}\mathrm{P}$6, whereas decay to the dark state is forbidden by dipole selection rules. The resulting bright/dark contrast is $^{31}\mathrm{P}$7, the single-tweezer discrimination fidelity is $^{31}\mathrm{P}$8, and the readout survival is $^{31}\mathrm{P}$9 for a single tweezer and $$H = A \mathbf{S}\cdot\mathbf{I} + \gamma_e B_0 S_z - \gamma_n B_0 I_z,$$0 averaged over the array (Huie et al., 2023).

Other implementations broaden the definition of the field. In electron-spin spectral mapping, repeated chirped MW sweeps transfer the NV electronic spectral density into surrounding $$H = A \mathbf{S}\cdot\mathbf{I} + \gamma_e B_0 S_z - \gamma_n B_0 I_z,$$1 polarization; a pulsed nuclear spin-lock then reads out the nuclear polarization with single-shot SNR $$H = A \mathbf{S}\cdot\mathbf{I} + \gamma_e B_0 S_z - \gamma_n B_0 I_z,$$2 and $$H = A \mathbf{S}\cdot\mathbf{I} + \gamma_e B_0 S_z - \gamma_n B_0 I_z,$$3 s (Pillai et al., 2021). In hBN, ENDOR and stimulated-spin-echo readout of the three nearest-neighbor $$H = A \mathbf{S}\cdot\mathbf{I} + \gamma_e B_0 S_z - \gamma_n B_0 I_z,$$4 nuclei around a $$H = A \mathbf{S}\cdot\mathbf{I} + \gamma_e B_0 S_z - \gamma_n B_0 I_z,$$5 center establish the control primitives needed for future repetitive protocols (Murzakhanov et al., 2021). In $$H = A \mathbf{S}\cdot\mathbf{I} + \gamma_e B_0 S_z - \gamma_n B_0 I_z,$$6, the $$H = A \mathbf{S}\cdot\mathbf{I} + \gamma_e B_0 S_z - \gamma_n B_0 I_z,$$7 nuclear qudit is inferred repetitively from fixed-field telegraph-noise statistics and nuclear-state-dependent modulation of split Kondo peaks, rather than from optical or charge-cycling repetition (Chen et al., 13 Mar 2026).

4. QND character, back-action, and the limits of repetition

The term QND is used frequently in this literature, but it is rarely exact. In the silicon donor case, the sufficient condition for QND is $$H = A \mathbf{S}\cdot\mathbf{I} + \gamma_e B_0 S_z - \gamma_n B_0 I_z,$$8 with an ideal interaction $$H = A \mathbf{S}\cdot\mathbf{I} + \gamma_e B_0 S_z - \gamma_n B_0 I_z,$$9. The physical hyperfine interaction is predominantly isotropic $S=1/2$0, which contains off-diagonal terms $S=1/2$1 that do not commute with $S=1/2$2; small anisotropic terms such as $S=1/2$3 similarly break strict QND. The experimentally relevant statement is therefore weaker: the nuclear lifetimes $S=1/2$4 s and $S=1/2$5 s greatly exceed the optimized measurement time, so the readout is effectively QND on the timescale of a shot (Pla et al., 2013).

Optically mediated platforms exhibit a different failure mode. In nanodiamond NV centers, repeated optical readout cycles drive unwanted electron-nuclear flip-flops in the excited state, with single-cycle probabilities $S=1/2$6. At moderate fields, $S=1/2$7, so the dominant nuclear error is an incrementing bit-flip $S=1/2$8; coherent feedback can then correct it provided $S=1/2$9. This is why the protocol improves fidelity substantially at $^{31}\mathrm{P}$00 mT but only weakly at high field, where both back-action rates are already small (Holzgrafe et al., 2018). In the V2 center, finite-demolition switching rates of $^{31}\mathrm{P}$01 Hz for the bright state and $^{31}\mathrm{P}$02 Hz for the dark state set the practical limit on how many optical cycles can be accumulated before nuclear back-action dominates (Hesselmeier et al., 2024). In $^{31}\mathrm{P}$03, the dominant residual errors are off-resonant Raman and spontaneous scattering through other $^{31}\mathrm{P}$04 sublevels and tweezer-induced state mixing, both scaling as $^{31}\mathrm{P}$05 (Huie et al., 2023).

A second misconception is that repetition can be increased indefinitely and must always improve fidelity. The silicon donor experiment makes the counterexample explicit. If the single-shot error is $^{31}\mathrm{P}$06, the majority-vote error after $^{31}\mathrm{P}$07 independent shots is

$^{31}\mathrm{P}$08

with exponential bounds from Hoeffding and Chernoff, but the optimal $^{31}\mathrm{P}$09 still balances SNR gain against the increased chance of a nuclear jump; experimentally, $^{31}\mathrm{P}$10 was near-optimal (Pla et al., 2013). The same logic reappears in SiC, where hundreds of cycles are feasible but not arbitrary, and in NV gyroscope readout beyond electron $^{31}\mathrm{P}$11, where short repumps preserve the longitudinal nuclear component but repeated repump exposure increases nuclear re-polarization and destroys the transverse phase (Kuan et al., 31 Jan 2025).

The strongest objective evidence for QND-like behavior is consistency across repeated outcomes. Silicon donor time traces show long intervals of fixed nuclear assignment punctuated by rare quantum jumps (Pla et al., 2013). $^{31}\mathrm{P}$12 supports repeated projective measurements, Zeno experiments, and real-time feedforward over many rounds (Huie et al., 2023). In mesoscopic NV ensembles, the collective nuclear variable $^{31}\mathrm{P}$13 can be read more than $^{31}\mathrm{P}$14 times at $^{31}\mathrm{P}$15 T before nuclear $^{31}\mathrm{P}$16-induced averaging limits the usefulness of further repetition (Maier et al., 15 Sep 2025).

5. Quantitative performance and scaling laws

Reported performance metrics span several operational regimes and should not be treated as directly interchangeable, but they establish the range of what repetitive nuclear-assisted readout can achieve. In single-donor silicon, the nuclear single-shot readout fidelity is better than $^{31}\mathrm{P}$17, up to $^{31}\mathrm{P}$18 depending on nuclear state and $^{31}\mathrm{P}$19, with SNR-limited misassignment as low as $^{31}\mathrm{P}$20 for a 260 ms acquisition (Pla et al., 2013). In SiC, the same logic yields $^{31}\mathrm{P}$21 average fidelity at unit success probability and $^{31}\mathrm{P}$22 at $^{31}\mathrm{P}$23, with $^{31}\mathrm{P}$24 measurement-based initialization and $^{31}\mathrm{P}$25 ms for the nuclear memory (Hesselmeier et al., 2024). In $^{31}\mathrm{P}$26, the central metrics are discrimination fidelity $^{31}\mathrm{P}$27, bright/dark contrast $^{31}\mathrm{P}$28, and state-averaged readout survival $^{31}\mathrm{P}$29 (Huie et al., 2023).

Nanodiamond NV experiments quantify improvement relative to conventional optical readout. At $^{31}\mathrm{P}$30 mT, repetitive nuclear-assisted readout raises the metric $^{31}\mathrm{P}$31 from $^{31}\mathrm{P}$32 to $^{31}\mathrm{P}$33 at $^{31}\mathrm{P}$34 cycles, a 13-fold enhancement (Holzgrafe et al., 2018). At $^{31}\mathrm{P}$35 mT, coherent feedback every $^{31}\mathrm{P}$36 cycles improves the maximum fidelity from $^{31}\mathrm{P}$37 to $^{31}\mathrm{P}$38, a $^{31}\mathrm{P}$39 increase, and approximately doubles the cumulative signal (Holzgrafe et al., 2018). In bulk nuclear-spin amplification, the readout gain

$^{31}\mathrm{P}$40

reaches an overall signal gain of $^{31}\mathrm{P}$41 relative to direct $^{31}\mathrm{P}$42 detection after accounting for species-dependent detection factors; the same experiment observed amplification of the polarization difference by $^{31}\mathrm{P}$43 at $^{31}\mathrm{P}$44 and $^{31}\mathrm{P}$45 at $^{31}\mathrm{P}$46 relative to $^{31}\mathrm{P}$47 (1105.4740).

The dominant scaling law depends on what is being repeated. When the same nuclear memory is interrogated repeatedly with statistically independent photon-counting shots, the SNR increases as $^{31}\mathrm{P}$48 or $^{31}\mathrm{P}$49, as emphasized for V2 centers, $^{31}\mathrm{P}$50, and repeated optical/electrical NV readout (Hesselmeier et al., 2024, Huie et al., 2023). When repeated successful electron measurements apply a contractive nonunitary map $^{31}\mathrm{P}$51 to the nuclear bath, the relevant scaling is spectral: components with $^{31}\mathrm{P}$52 are suppressed as $^{31}\mathrm{P}$53, and polarization emerges from repeated conditioning (Wu, 2010). When repeated mapping occurs before a single final NMR detection, as in spin amplification, the signal adds coherently and the paper explicitly contrasts $^{31}\mathrm{P}$54 with the usual $^{31}\mathrm{P}$55 scaling of repeated direct measurements (1105.4740).

The ensemble NV result introduces a further regime in which repetitive nuclear-assisted readout is used not merely to improve state discrimination but to approach the intrinsic fluctuations of the sensor itself. At $^{31}\mathrm{P}$56 T, with $^{31}\mathrm{P}$57 to $^{31}\mathrm{P}$58 repetitions and more than $^{31}\mathrm{P}$59 feasible, the photon shot noise is reduced $^{31}\mathrm{P}$60 dB below the observed projection noise, and fitting the projection-noise plateau yields $^{31}\mathrm{P}$61 active centers in the focal spot (Maier et al., 15 Sep 2025). This suggests that repeated nuclear-assisted readout can transition from a fidelity-enhancement tool into a metrological interface to collective quantum noise.

6. Applications, misconceptions, and future directions

The most immediate applications are in quantum memories, ancilla-based syndrome extraction, and high-fidelity measurement in architectures where electron-spin readout is either destructive or too weak in a single attempt. Silicon donor devices already combine readout fidelity exceeding $^{31}\mathrm{P}$62, gate fidelity $^{31}\mathrm{P}$63, and $^{31}\mathrm{P}$64 ms in a CMOS-compatible nanostructure, which the source text explicitly connects to fault-tolerant syndrome extraction, ancilla-based measurement, and scalable silicon quantum information processing (Pla et al., 2013). In SiC, repetitive nuclear readout enables measurement-based initialization, coherent nuclear control, and ENDOR sensing of weakly coupled bath spins (Hesselmeier et al., 2024). In NV-based sensing, the same principle supports low-field quadrupolar NMR in nanodiamonds, electrical readout in dense arrays, and nuclear-spin gyroscope readout beyond the electron $^{31}\mathrm{P}$65 time (Holzgrafe et al., 2018, Gulka et al., 2021, Kuan et al., 31 Jan 2025).

Another misconception is that nuclear-assisted readout is inherently optical. The surveyed implementations are electrically mediated in silicon donors and room-temperature NV photoelectric detection (Pla et al., 2013, Gulka et al., 2021), inductive in nuclear-spin amplification and ESR-via-NMR spectral mapping (1105.4740, Pillai et al., 2021), and transport-based in fixed-field STM readout of $^{31}\mathrm{P}$66 (Chen et al., 13 Mar 2026). Nor is the nuclear resource always a single nearby ancilla. It may be a host nucleus, a proximal $^{31}\mathrm{P}$67, a three-spin $^{31}\mathrm{P}$68 register in hBN, an abundant bulk bath, or an ensemble-wide stabilized nuclear degree of freedom (Hesselmeier et al., 2024, Murzakhanov et al., 2021, Maier et al., 15 Sep 2025).

Open directions follow directly from the existing demonstrations. The quantum-dot formalism implies that repeated measurement can function as a nuclear-state engineering primitive, not only a readout primitive, by driving purification toward the fully polarized state under generic $^{31}\mathrm{P}$69 (Wu, 2010). Bulk spin amplification suggests a route to repetitive readout without individual addressing or tailored spin networks, provided field-cycled heteronuclear diffusion and long nuclear $^{31}\mathrm{P}$70 are available (1105.4740). The hBN results establish coherent coupling, quadrupolar structure, and ENDOR observability for the three nearest $^{31}\mathrm{P}$71 nuclei around $^{31}\mathrm{P}$72, implying that single-defect repetitive nuclear-assisted readout in a 2D host is a plausible next step (Murzakhanov et al., 2021). The NV ensemble work points toward projection-noise-limited solid-state sensing, spin squeezing, and direct observation of correlated spin states using repeated QND collective readout (Maier et al., 15 Sep 2025).

Taken together, the literature shows that repetitive nuclear-assisted spin readout is not a single protocol but a measurement paradigm. Its defining feature is the use of a nuclear degree of freedom to separate memory from measurement: the nuclear system holds the information, while a faster auxiliary channel repeatedly extracts it. The principal technical problem is always the same—maximize accumulated signal before back-action erases the nuclear record—but the solutions now range from SET-based spin-to-charge conversion and coherent feedback, to near-cycling fluorescence, field-cycled spin diffusion, ENDOR-mediated mapping, and fixed-field transport statistics. This diversity is a sign not of conceptual fragmentation but of generality.