Storing quantum coherence in a quantum dot nuclear spin ensemble for over 100 milliseconds

Domain: physics
Published: 2025-12-04
Editorial outcome: minor_revision

Abstract
States with long coherence are a crucial requirement for qubits and quantum memories. Nuclear spins in epitaxial GaAs/AlGaAs quantum dots are a great candidate, offering excellent isolation from external environments and on-demand coupling to optical flying qubits. However, coherence times are limited to ≲1 ms by the dipole-dipole interactions between the nuclei and by the nuclear quadrupolar coupling to inhomogeneous crystal strain. Here, we combine strain engineering of the nuclear spin ensemble and tailored dynamical decoupling sequences to achieve nuclear spin coherence times exceeding 100 ms. Recently, a reversible transfer of quantum information into nuclear spin ensembles has been demonstrated in quantum dots: our results provide a path to develop this concept into a functioning solid-state quantum memory suitable for quantum repeaters in optical quantum communication networks.

Introduction
Quantum memories are indispensable in large scale quantum networks, which are expected to enable long distance communication of quantum information1–4. Quantum memories have several key requirements2, a primary figure of merit is the storage time, which is directly related to quantum repeater communication distance. Millisecond-range storage time allows for improvements over direct transmission through an optical fibre5,6. Another requirement is for the ratio of the storage time and the entanglement generation time, known as quantum link efficiency7, to be as high as possible. Although entanglement generation time is currently the main limitation8, its continuous improvement (reduction) highlights the need for even longer storage times, exceeding 100 ms, in order to achieve worldwide optical communication.

The storage of a quantum state in a memory is limited by the coherence time T2. Several material systems offer long T2, ranging from seconds to hours9, including trapped atomic ensembles10,11, ions12–17, electron spins18,19 and phosphorus nuclear spins20–22 in silicon, as well as electron and nuclear spins of impurities in diamond23,24. However, long T2 are often negated by poor optical properties required for a long-distance quantum network. There are promising hybrid approaches, such as combination of transmon qubits with solid-state quantum memories25, but these often suffer from coupling inefficiencies and bandwidth mismatch9 (see further discussion in Supplementary Note 4).

Epitaxial quantum dots (QDs) in group III-V semiconductors have high qubit entanglement rates26 and are excellent on-demand emitters of single9,27–29 and entangled photons30,31. At the same time, QDs host material qubits: Electron spin qubits can be interfaced with optical photon qubits32–34, but the coherence of the electron spin is limited to ≈100 μs35. The nuclear spins are isolated from external environments, resulting in long lifetimes and coherence times36–38. The recent demonstration of a reversible quantum state transfer between an electron spin qubit and a nuclear spin ensemble (with fidelity of ≈0.68)39,40 offers a route for electron-mediated storage of a photonic quantum state in a nuclear ensemble of a QD. However, since all atoms in group III-V materials have non-zero nuclear spins, the natural nuclear spin coherence is limited to a rather modest ≈1 ms range41. Extending nuclear spin coherence is thus a key task in achieving quantum memories suitable for quantum repeaters42,43.

Here, we achieve nuclear spin coherence of over T2 ≈ 100 ms, made possible by applying two concepts: Firstly, elastic strain is used to engineer the spin-3/2 nuclei and spectrally isolate the subspace with I z = ±1/2 spin projections. The small inhomogeneity of this subspace allows application of thousands of coherent control operations, thus enabling efficient dynamical decoupling. Secondly, a dedicated 40-pulse decoupling sequence cycle is designed to extend nuclear spin ensemble coherence while overcoming the parasitic spin locking effects encountered in previous decoupling experiments44,45. The analysis of the results shows that residual decoherence is dominated by the finite-pulse effects and the effective three-body nuclear spin interactions, which are often overlooked. We predict that even longer T2 values, on the order of ≈1 s, are well within reach through larger strains and further advances in dynamical decoupling. The macroscopically long coherence times achieved here were previously possible only in group IV semiconductor spin qubits21,46, where optical efficiency is limited. Our demonstration of engineered long coherence unlocks the unrivalled optical properties of group III-V materials for applications in quantum memory devices.

Results
Strain-engineered nuclear spin ensemble
We study the nuclear spin coherence of GaAs/AlGaAs QDs grown by molecular beam epitaxy. The right inset of Fig. 1a sketches the QD nuclear spin system of N ≈ 5 × 104 nuclei. The three isotopes 75As, 69Ga, and 71Ga all have nuclear spin I = 3/2. The sample is cooled to ≈4.2 K. A superconducting magnet is used to apply a static magnetic field B z ≈ 5.16 T along the sample growth crystal direction [001], lifting the degeneracy of the four nuclear spin states with spin projections I z = ±1/2, ±3/2 (left inset in Fig. 1a). Optical pumping with circularly polarized light (Faraday geometry) is used to polarize the nuclear spins along the static magnetic field47. The nuclear spin lifetime is typically T1 > 10 s48, significantly longer than the coherence times measured in this work. Nuclear spin polarization is measured via photoluminescence (PL) spectroscopy49, see examples in Fig. 2a. A copper coil generates oscillating magnetic field perpendicular to B z, enabling optically detected nuclear magnetic resonance (ODNMR) measurements. Radio frequency (Rf) bursts with raised cosine envelope are used to transfer coherence in and out of the storage nuclear spin subspace I z = ±1/2 and to perform its dynamical decoupling.

The sample is stressed uniaxially along the [110] crystal direction, perpendicular to the static magnetic field. The resulting anharmonicity37,50, is characterised by the first-order nuclear quadrupolar splitting ν(1) Q. The measured NMR spectrum, shown in Fig. 1a for 75As nuclei in a neutral (0e) GaAs/AlGaAs QD, reveals ν(1) Q ≈ 255.1 kHz. This splitting significantly exceeds the linewidths of the NMR transitions: the full width at half maximum (FWHM) Δν +1/2↔+3/2 ≈ Δν −3/2↔−1/2 ≈ 13.8 kHz of the satellite transitions (STs) −3/2↔−1/2 and +1/2↔+3/2 is dominated by the inhomogeneous quadrupolar broadening, while the FWHM Δν −1/2↔+1/2 ≈ 0.8 kHz of the central transition (CT) −1/2↔+1/2 is controlled by a combination of the second-order quadrupolar inhomogeneity and the dipole-dipole interactions45,51. The small linewidth, combined with strain-induced spectral isolation from STs, makes the CT an ideal spin subspace for coherence storage. Notably, the lattice-matched GaAs/AlGaAs QDs offer a significant advantage over Stranski-Krastanov QDs characterised by the much larger Δν −1/2↔+1/2 ≈ 10–40 kHz45.

Hamiltonian engineering of a nuclear spin ensemble
Dynamical control of spin interactions is a powerful technique in magnetic resonance52,53. The method is based on applying a sequence of Rf pulses that perform fast coherent rotations of the spins, separated by the free evolution intervals. In the interaction picture (the “toggling” frame of reference) the Rf pulses can be viewed as transformations of the spin-interaction Hamiltonian. The π-pulse rotations invert the sign of the frequency shifts, allowing refocusing of the dephasing54, which in QDs is caused primarily by inhomogeneous quadrupolar broadening. However, sequences of π pulses, such as Carr-Purcell55 or XY856, do not recouple the nuclear spin-spin dipolar interactions. Instead, these dipolar interactions can be averaged to zero with a sequence of four phase-shifted π/2-pulses57. The average Hamiltonian is the leading (0th order) term in the Magnus expansion of the entire Hamiltonian in the toggling frame. By introducing more complex sequences of pulses it is possible to eliminate the unwanted interactions to higher orders, thus engineering the spin Hamiltonian to have the desired form53.

Here, we engineer a “time suspension”58 type of sequence, where the Hamiltonian terms are eliminated as much as possible to preserve an arbitrary coherent state of the nuclear spin ensemble for the longest possible time. As a starting point we use a CHASE-10 cyclic sequence45 shown in Fig. 1b. By combining π/2 and π pulses, this sequence eliminates the average (0th order) free-evolution Hamiltonian both for the resonance frequency shifts and the spin-spin interactions. Simultaneous suppression of both types of interactions is crucial for dynamical decoupling in a dense nuclear spin ensemble of GaAs. By symmetrizing the sequence, a CHASE-20 supercycle is formed, which further eliminates all the 1st-order terms in the Hamiltonian. The CHASE-20 sequence has been applied to QDs previously, demonstrating its ability to suppress decoherence even under large inhomogeneous resonance broadenings in Stranski-Krastanov InGaAs/GaAs QDs45. In low-strain GaAs/AlGaAs QDs, nuclear spin coherence times up to T2 ≈ 20 ms have been achieved37. However, the π pulses cause spin locking44,45 which selectively accelerate decoherence of the spin states polarized along a certain equatorial axis of the Bloch sphere in the rotating frame, while artificially enhancing (“locking”) the coherence of the states polarized along the orthogonal equatorial axis. This behaviour is unwanted in quantum memory applications as it may lead to distortion of the state during storage.

Here we use a different approach, where four CHASE-10 cycles are combined into a CHASE-40 supercycle shown in Fig. 1c. The phases of the Rf pulses in each CHASE-10 subcycle are stepped by π/2. While each subcycle causes spin locking, the preferential direction of the “lock”, when viewed in the rotating frame, makes a full rotation about the direction of the static magnetic field (z) over the CHASE-40 supercycle. This four-step “rotating spin lock” eliminates the net spin locking effect for an arbitrary coherent state, as demonstrated through rigorous calculation (See Supplementary Note 5). Furthermore, the leading order residual Hamiltonian of CHASE-40 is twice smaller than in CHASE-20, resulting in extended coherence.

Extended spin coherence under dynamical decoupling
We start by examining experimentally the nuclear spin dynamics under continuous resonant Rf driving. The results shown in Fig. 2b reveal Rabi oscillations, which confirm the coherent nature of spin driving and allow the π/2 and π Rf pulses to be calibrated for dynamical decoupling (See Supplementary Note 2C). We then apply dynamical decoupling to the isolated I z = ±1/2 nuclear spin subspace with two varying parameters: the number of sequence cycles n Cycles and the total free evolution time T FreeEvol. Figure 2c shows nuclear spin coherence decay measured using ODNMR. In the simplest case of free induction decay (FID), without any dynamical decoupling (open black squares), the dephasing time is T2* ≈ 0.46 ms and is a combined effect of quadrupolar inhomogeneity and dipole-dipole interactions. By using a single π pulse as a decoupling sequence (solid squares), we find the Hahn echo54 coherence time THE 2 ≈ 1.38 ms, which is dominated by the dipole-dipole interactions.

The measured effect of dynamical decoupling with one cycle of CHASE-40 is shown by the circles in Fig. 2c. The decay of the transverse nuclear polarization is plotted as a function of the total free evolution time T FreeEvol (i.e. excluding the duration of the Rf pulses), and reveals a significant extension of the coherence time T 1xCHASE-40 2 ≈ 12.3 ms. With the increasing number of CHASE-40 cycles the coherence time is extended further, reaching T 48xCHASE-40 2 ≈ 106.6 ms for n Cycles = 48 (orange triangles), an improvement by 2 orders of magnitude compared to the bare Hahn echo coherence time. The achieved nuclear spin coherence is also 3 orders of magnitude longer than the coherence of a dynamically decoupled electron spin in these QDs35. These results demonstrate the superior properties of the nuclear spins as a coherence-storage medium.

It can be seen from Fig. 2c that under an increasing number of decoupling cycles n Cycles the coherence (the NMR echo signal) is reduced even in the limit of T FreeEvol → 0. Moreover, the dependence of transverse nuclear spin polarization on T FreeEvol becomes non-monotonic. These are the indications of nuclear spin decoherence during the finite (non zero duration) Rf pulses. For quantum memory applications, we are interested in minimising decoherence, whether caused by the Rf control pulses or spin interactions during free evolution. We seek this optimum by replotting the data of Fig. 2c in Fig. 3a, where the normalized nuclear spin coherence is shown as a function of the total evolution time T EvolTot (horizontal axis) and the duration T Cycle of one CHASE-40 cycle (vertical axis). The individual decay plots of Fig. 2c measured at fixed n Cycles now appear along the diagonal lines in Fig. 3a. We fit the decay of coherence with an exponential function of T EvolTot: the resulting decay time, which we denote as spin memory time T M, is distinct from the coherence time T2 and is shown as a function of T Cycle by the single solid line in Fig. 3e. The maximum T M ≈ 136 ms is achieved not under the fastest possible Rf pulsing, but at T Cycle ≈ 2 ms, which is 5 times longer than the minimum T M ≈ 0.8 ms achieved at T Cycle = 20 μs. The non-monotonic dependence T M(T Cycle) also manifests in non-monotonic decay of spin coherence when plotted as a function of T FreeEvol (n Cycles = 32 and 48 shown in Fig. 2c). We conclude that the finite-pulse effects are the main limitation to extending coherence storage through fast dynamical decoupling.

Decoherence during the Rf pulses can in principle be suppressed by reducing T Rf. However, experiments conducted on the I z = ±1/2 spin states with a reduced T Rf = 10 μs yield faster decoherence than under T Rf = 20 μs (see Supplementary Note 3A). This seemingly contradictory result is understood by considering the spectral profiles of the Rf pulses (dashed lines in Fig. 1a). While the unwanted spin decoherence is indeed suppressed under the shorter T Rf = 10 μs pulses, their broader spectral profile results in a stronger overlap with the STs. Such overlap leads to a faster “leakage” of coherence from the storage I z = ±1/2 subspace into the I z = ±3/2 states. Thus, there is an optimal pulse duration that balances the finite-pulse and the leakage effects. For the studied structure with the strain-induced quadrupolar splitting of ν(1) Q ≈ 255.1 kHz, this optimum is close to T Rf = 20 μs. Increasing elastic strain from ≈0.0025 in the studied sample to the ≈0.01 range59 is a promising route for applying shorter Rf pulses and a further significant improvement of the storage time and fidelity in a QD nuclear spin quantum memory.

CHASE decoupling of an inhomogeneous spin ensemble
In order to demonstrate the importance of isolating the homogeneous I z = ±1/2 CT storage subspace, we examine the opposite case of a I z = (−3/2, −1/2) ST subspace. The considerably larger inhomogeneous broadening is characterised by the spectral shape of the ST NMR transitions. For each ST, this is a weighted sum of a peak observed in Fig. 1a with a linewidth of Δν +1/2↔+3/2 ≈ Δν −3/2↔−1/2 ≈ 13.8 kHz (65% weight), and a much broader peak (35% weight), that requires a different NMR technique to be observed. That broad spectral component is not shown in Fig. 1a, but was measured previously35 to stretch to ≈±100 kHz and is caused by the atomic-scale strain of the randomly positioned Al and Ga atoms. The measured ST decoherence under CHASE decoupling is shown in Fig. 3b, and the resulting spin memory time T M is shown by the double solid line in Fig. 3e. Unlike with I z = ±1/2, the best possible dynamical decoupling of the I z = (−3/2, −1/2) subspace is achieved at the shortest possible T Cycle. Despite this fastest possible Rf pulsing, the maximum achieved memory time is T M ≈ 31 ms. Although this storage time is a factor of ≈15 improvement over the simple Hahn echo, it is a factor of ≈4.4 worse than T M achieved for the spectrally narrow I z = ±1/2 subspace.

The inferior spin memory time for an inhomogeneously broadened spin ensemble demonstrates how CHASE-40, as any other dynamical decoupling protocol, reaches the limit of its performance when the interaction that is being decoupled is no longer a small perturbation. The inhomogeneous broadening of the I z = ±1/2 CT subspace is a small perturbation, characterized by Δν −1/2↔+1/2 T Rf ≈ 0.015 ≪ 1. By contrast, the relative broadening of the I z = (−3/2, −1/2) subspace is comparable to unity for the observed part of the ST NMR peak (Δν −3/2↔−1/2 T Rf ≈ 0.28) and violates the perturbative approximation (Δν −3/2↔−1/2 T Rf > 1) for the broad component of the ST. The large inhomogeneity of the ST exacerbates decoherence through spin evolution during the finite Rf pulses. Moreover, Δν −3/2↔−1/2 T Rf ≳ 1 means that Rf control pulses become more “soft” (i.e. not infinitely broad spectrally), resulting in imperfect rotations of the nuclear spins60,61. On the other hand, CHASE-40 shows no sign of spin-locking even for the I z = (−3/2, −1/2) subspace, despite its larger inhomogeneous broadening (See Supplementary Note 3B).

Uniform decoupling of an arbitrary coherent nuclear spin state
An ideal quantum memory must store any given state with equally high fidelity. However, dynamical decoupling can create parasitic spin locking regimes, where storage effectiveness depends on the initial state44,45. We examine the uniformness of the quantum state storage by measuring dynamical decoupling of the homogeneous I z = ±1/2 subspace with different initial states. We use CHASE-40 with n Cycles = 4 cycles, which provides a balance between extending the coherence time by an order of magnitude, while keeping to a minimum the finite-pulse decoherence. The phase ϕ of the initialization Rf pulse is varied to prepare transverse nuclear spin polarization along the different axes in the equatorial xy plane of the rotating frame (ϕ = 0 corresponds to a “+x” Rf pulse and prepares polarization along the −y axis, a “+y” pulse with ϕ = π/2 prepares polarization along the x axis). We further perform a measurement, where the initialization pulse is omitted, corresponding to initial nuclear spin polarization along the strong magnetic field (θ = 0). The measured decay curves are shown in Fig. 4a. The coherence times obtained from fitting are shown in Fig. 4b, and are around T2 ≈ 18 ms, nearly independent of the initial state.

The uniform dynamical decoupling of different initial states confirms experimentally the design principle of the CHASE-40 supercycle. Stepping of the Rf pulse phases between the four constituent CHASE-10 subcycles can be understood intuitively as a “rotary spin lock”: the axis of spin locking is slowly precessing with respect to the rotating frame, with a net effect of removing any preferential spin locking axis over the entire CHASE-40 cycle. Rigorous calculations confirm this result, showing that the symmetry axis of the residual Hamiltonian of CHASE-40 is along the strong static magnetic field (z axis). It is also worth noting that the measurement without any initialization pulse (θ = 0 in Fig. 4b) yields the longitudinal spin relaxation time T1, as opposed to the transverse coherence time T2 measured with an initialization π/2 Rf pulse (θ = π/2). In the absence of dynamical decoupling, the strong static magnetic field imposes a pronounced anisotropy with T1 > 10 s48 much larger than T2 ≈ 1 ms. Under CHASE-40 decoupling T2 decreases and T1 increases, converging to very similar values. This confirms that CHASE-40 is a well-balanced time-suspension sequence that eliminates not only the spin locking anisotropy in the xy transverse plane but also the anisotropy of the strong quantizing magnetic field along the z axis.

The rotary spin lock offers a simple and reliable approach for reusing the dynamical decoupling sequences where spin locking is otherwise present45,62,63. The robustness of CHASE-40 against spin locking is key to achieving long spin memory times T M ≳ 100 ms through repeated cycling (with up to 2400 pulses).

Predicting dynamical decoupling performance through analytical and numerical modelling
For any dynamical decoupling sequence the effective spin Hamiltonian can be calculated as a Magnus expansion series. However, finding the spin dynamics from a known Hamiltonian is still a difficult problem. This problem is simplified by noting that the exact spin dynamics is related to the exact NMR spectral lineshape through Fourier transform. The NMR lineshape can be approximated as a Gaussian, and its linewidth can be approximated in terms of the second moment M2, which in turn can be found from the Hamiltonian through direct calculation. The coherent memory time can then be approximated as T M ≈ 2/M2. The residual Hamiltonian of CHASE-40 has been computed analytically up to second order: it is too bulky to reproduce in full, a more detailed discussion can be found in Supplementary Note 5. The Hamiltonian depends on two parameters: the magnitude of the dipole-dipole interaction and the quadrupolar inhomogeneity. These parameters are derived from FID and Hahn Echo experimental data, allowing T CHASE-40 M to be calculated up to second order in analytical form and without any fitting parameters. The results are shown by the dashed lines in Fig. 3e. For the homogeneous subspace I z = ±1/2 (single dashed line), the analytical model accurately predicts the peak in the spin memory time T M at T Cycle ≈ 2 ms. The actual peak value of T M is underestimated but matches the experiment within a factor of ≈2. By contrast, for the inhomogeneous subspace I z = (−3/2, −1/2), the model underestimates T M by an order of magnitude. This indicates the limited accuracy of the perturbative Magnus expansion, which breaks down when the inhomogeneous broadening (in frequency units) is no longer small in comparison to the reciprocal cycle time 1/T Cycle. In principle, the analytical model could be improved by extending the Magnus expansion. However, the increasing complexity of the high-order terms makes this approach impractical.

The advantage of the analytical model is that it allows insights into the underlying physics. In particular, we find that direct dipole-dipole interactions of the ith and jth spins of the ensemble, characterised by the coupling constant ν ij, is eliminated by CHASE-40 up to second order inclusive. However, the dipolar interaction of the spins i, j remains, but with a coupling constant ∝ ν ik ν kj, where k ≠ i, j is any other spin. Such a term can be interpreted as an effective three-particle coupling, where the interaction of any two spins i and j is mediated by any other spin k. Although often ignored, here we find that the residual decoherence of the homogeneous subspace I z = ±1/2 can be explained only by taking into account this effective three-body interaction. The three-body second-order interaction limits T M under slow dynamical decoupling (long T Cycle). In the opposite limit of fast decoupling (short T Cycle) the memory time T M is limited by the zero-order dipole-dipole term arising from spin evolution during the finite Rf pulses (T Rf > 0). A combination of these two effects results in a non-monotonic dependence T M(T Cycle) with a maximum in T M, as shown by the single lines in Fig. 3e. By contrast, the decoherence in the inhomogeneous subspace I z = (−3/2, −1/2) is dominated by the quadrupolar offset inhomogeneity under finite (T Rf > 0) control pulses.

We analyse the data from an alternative perspective by conducting numerical modelling of the CHASE dynamical decoupling. The exact Schrödinger equation of a system of N = 12 spins is solved using mixed initial spin states with large transverse polarization, mimicking the NMR experiments (see details in Supplementary Note 6). Selected results are shown in Fig. 2d and reproduce well all the main features of experimental results in Fig. 2c. The detailed results for the case of the inhomogeneous subspace I z = (−3/2, −1/2), are shown in Fig. 3d. Since the decoherence rate is dominated by the quadrupolar offsets, which is a single-particle effect, a good quantitative agreement with the experiment (Fig. 3b) is obtained by using realistic values of the inhomogeneous quadrupolar shifts in the numerical model. The results for the I z = ±1/2 subspace, where quadrupolar inhomogeneity is taken to be zero, are shown in Fig. 3c. The numerical model reproduces the main features of the experimental data on the I z = ±1/2 subspace (Fig. 3a), in particular the nonmonotonic dependence of the spin memory time T M on the decoupling sequence cycle time T Cycle. However, the agreement is only within an order of magnitude: the numerically-simulated maximum T M ≈ 2 s occurs at T Cycle ≈ 8 ms, compared to the measured maximum T M ≈ 0.136 s at T Cycle ≈ 2 ms (Fig. 3e). This discrepancy may seem unexpected, given that the numerically-simulated coherence time under simple Hahn echo THE 2 ≈ 1.4 ms is very close to the measured THE 2 ≈ 1.38 ms. However, the decoherence under Hahn echo is governed by the direct (pairwise) dipole-dipole interaction of the nuclear spins, whereas decoherence under CHASE-40 is dominated by the residual effective three-spin interaction. We therefore ascribe the discrepancy in T M to the limited number of spins in the numerical model: the number of three-spin combinations contributing to decoherence in an ensemble with N = 12 is considerably smaller than in a real crystal lattice of a QD (see Supplementary Note 6). The discrepancy is also likely to include the small but nonzero quadrupolar inhomogeneity of the I z = ±1/2 subspace, which reduces CHASE-40 T M in a real QD.

Decoupling of coherent spin wave states
The dynamical decoupling experiments of this work are conducted on nuclear spin states with macroscopic transverse polarization, which corresponds to multiple-quantum coherence. By contrast, recent proposals for nuclear-spin-based quantum memories40 rely on single-quantum spin wave (magnon) nuclear spin states. The applicability of CHASE-40 dynamical decoupling to the spin wave coherent states is a non-trivial question, which we now investigate using numerical modelling on an ensemble of N = 12 spins. We compare two types of initial wavefunction states ψ Init. One is a state with full transverse polarization (macroscopic magnetization), while the other is a superposition of a ground state with full longitudinal polarization and a spin wave excited state. The spin wave is a single-quantum excitation of the ground state64 (see details in Supplementary Note 6). The Schrödinger equation is propagated to find the final state ψ Fin of the spin ensemble, and the overlap probability is calculated as |⟨ψ Init|ψ Fin⟩|2 with results shown in Fig. 5.

The squares in Fig. 5 show the evolution without dynamical decoupling (free induction decay). The state with full transverse polarization exhibits a Gaussian decay on a timescale of THE 2 ≈ 0.9 ms (full squares), which is a close match to THE 2 ≈ 1.4 ms computed above for an initial mixed state with partial transverse polarization. Interestingly, the decoherence of the spin wave superposition (open squares) is considerably slower even without any active dynamical decoupling. Some periodic oscillations and revivals are observed and can be ascribed to the smallness of the spin ensemble (N = 12) combined with the pure nature of the initial state. Next we model spin ensemble evolution as a function of T FreeEvol under dynamical decoupling with a fixed number n Cycles = 4 of CHASE-40 cycles (triangles in Fig. 5). Once again, the decay exhibits periodic partial revivals, both for macroscopically polarized (full triangles) and spin wave (open triangles) coherent states. Most importantly, compared to free induction decay, CHASE-40 decoupling is found to slow down the decoherence both for macroscopically polarized and single-quantum spin wave coherent states. The computed coherence times T 4xCHASE-40 2 ≈ 90 ms for pure initial states are found to be very similar to coherence times calculated above for mixed initial states. Notably, the spin wave state is more robust against the finite-pulse decoherence (limit of short T FreeEvol) than the macroscopically polarized state. These results suggest that numerical modelling and NMR experiments on spin states with macroscopic transverse polarization can be used to predict the dynamics of a single-quantum spin wave excitation. This initial finding confirms the prospect of using dynamical decoupling in spin wave quantum memories, which will be explored further in future work.

Methods

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This paper reports on comprehensive experimental study of coherence of the nuclear spin system in GaAs/AlGaAs quantum dots. The authors combine strain engineering approach to select the 1/2 subspace of the nuclear spins and the tailored protocol for dynamical decoupling. This allows them to get remarkably long nuclear spin coherence time of 100 ms. The developed and used CHASE-40 protocol suppresses leaking effects presented in other approaches and extends the coherence time by a factor of five comparing to the Hahn echo protocol. The authors give analytic and numerical modelling of their results and provide in-depth discussion on this bases. Experiments are made with great care to minor details, being based on long term systematic research of the authors on this problem.\nPaper is written very clearly, all experimental results are well presented and discussed, and conclusions are convincing. I readily recommend this paper for publication in Nature Communications.\nRecommendations:\nTo give material of the studied GaAs/AlGaAs QDs in abstract as the used approach of strain engineering are the reached times are specific to that.", "reviewer_2": "In this work Dyte et al. measure nuclear spin coherence times in a lithographically-defined GaAs quantum dot. By utilizing pulse-based NMR (specifically CHASE pulse sequences) they demonstrate that these coherence times can be extended to ~100ms. The authors claim that these results provide a path towards long-lived quantum memories for quantum networking. The paper is well written and technically sound with thoroughly analyzed experimental results. However, prior to recommending publication, I would ask for some more detail regarding the novelty and impact of this work.\nIn particular, I am concerned that these results constitute a minor advancement compared to [Chekhovich, E. A. et al. “Nuclear spin quantum register in an optically active semiconductor quantum dot”, Nature Nanotechnology, 2020]. This prior work utilized a similar quantum dot sample and operated in a very similar experimental regime. The main differences seem to be a minor change in the pulse sequence (a cyclic permutation of pulse phases mitigating issues associated with spin locking) and a 5x improvement in coherence.\nFurthermore, I do not believe that the pulse sequences and coherence times characterized here are useful for quantum networking applications. Specifically, the authors measure coherence times using un-entangled nuclear spin states which are globally polarized along the X (or Y) directions. However, for quantum memory applications, the coherence times of nuclear magnon states are relevant (such as those explored in [Hayhurst, M. et al. “A many-body quantum register for a spin qubit”, Nature Physics, 2025]). This is a crucial distinction since, I believe, that the CHASE sequences explored here only cancels dipolar interactions for globally polarized states and not magnonic states.\nI also have several more minor/technical queries:\n• On Page 1 the authors claim that “Recently, two-way quantum state transfer between photon and nuclear spins has been demonstrated”, however, I do not believe that the two cited papers demonstrate transfer of photonic qubits onto nuclear spin memories.\n• It would help in the introduction to introduce the “quantum link efficiency” since it is an essential metric for quantum repeaters; it would also help to explain/characterize the trade-off between memory times and entanglement rates.\n• On page 2 the acronym ‘CT’ needs to be defined.\n• Regarding the experimental results in Figure 2c: could the authors please provide a qualitative explanation for why the coherence initially increases with free evolution time for 32 and 48 cycle cases?\n• On page 4 the authors refer to a “broader invisible peak”, could they please be a bit more specific about what they mean by ‘invisible’ here?\n• In Figure 3e it would help to additionally plot the numerical simulation results for ease of comparison.\n• In Figure 4 the authors chose to utilize a pulse sequence with only 4 cycles and 18ms coherence time. Could the authors provide equivalent plots for their 100ms coherence time sequence.\n• On page 6 the authors state that “It is also worth noting that the T decay time of the transverse polarization (measured with 2 an initialization π/2 pulse) is nearly identical to that of the longitudinal T decay time.” I am a bit surprised by this statement 1 since the T time is >10s, whereas the T time is 100ms, could the authors please clarify.\n1 2\n• Also on page 6 the authors state that “We therefore ascribe the discrepancy in T to the limited number of spins in the M numerical model”. It would be helpful to explore the convergence of the simulated coherence time with increasing numbers of nuclear spins. Could the authors plot the simulated coherence times for 2≤N≤12? Have the authors considered using the cluster-correlation expansion (CCE) method to speed up simulations?", "reviewer_3": "This manuscript proposes and demonstrates a novel approach to constructing a long-lived quantum memory by utilizing a subspace of the nuclear spin ensemble in a quantum dot. Notably, the memory is encoded into a homogeneous subspace by exploiting strain-induced perturbations in the nuclear spins. As a result, intrinsically long coherence times are successfully achieved. Furthermore, a practical dynamical decoupling (DD) protocol optimized for this system is designed, leading to an experimental demonstration of the longest coherence time to date. These results are accompanied by highly precise and extensive theoretical analysis, which adds depth to the understanding of the mechanisms and ensures reproducibility. Overall, this is a high-quality study on both the experimental and theoretical sides and is evaluated as an important advancement toward the practical realization of quantum repeaters. The novel design concept, the supporting data, and the consistency of the theoretical framework all contribute to the impact of this paper.\nMinor comments:\n1. The fidelity of the memory initialization remains unclear. When storing or retrieving a photonic quantum state in the quantum memory, the initialization rate and DD control fidelity are important aspects. I would appreciate clarification, as much as possible, regarding these points.\n2. The motivation for using CHASE is not clearly described. DD sequences such as XY8 are widely used in general, so I would like the authors to provide some explanation of the advantages of CHASE. Since the nuclear spin ensemble system may have a structural compatibility with CHASE, I believe the authors’ perspective on this would deepen the readers’ understanding."}, "author_response": "We would like to start by thanking all three Reviewers for the careful reading and the insightful comments that prompted us to conduct some additional studies and improve the manuscript. Below we provide a point-by-point response to all critical comments in their original order. Reviewers’ comments are quoted in italics font. The changes in the revised manuscript are highlighted in red font.\nReviewer #1:\nThis paper reports on comprehensive experimental study of coherence of the nuclear spin system in GaAs/AlGaAs quantum dots. The authors combine strain engineering approach to select the 1/2 subspace of the nuclear spins and the tailored protocol for dynamical decoupling. This allows them to get remarkably long nuclear spin coherence time of 100 ms. The developed and used CHASE-40 protocol suppresses leaking effects presented in other approaches and extends the coherence time by a factor of five comparing to the Hahn echo protocol. The authors give analytic and numerical modelling of their results and provide in-depth discussion on this bases. Experiments are made with great care to minor details, being based on long term systematic research of the authors on this problem.\nPaper is written very clearly, all experimental results are well presented and discussed, and conclusions are convincing. I readily recommend this paper for publication in Nature Communications.\nRecommendations:\nTo give material of the studied GaAs/AlGaAs QDs in abstract as the used approach of strain engineering are the reached times are specific to that.\nWe thank the Reviewer for the positive comments. As per Reviewer’s suggestion, we have modified the abstract to clarify that the work is done on GaAs/AlGaAs quantum dots.\nReviewer #2:\nIn this work Dyte et al. measure nuclear spin coherence times in a lithographically-defined GaAs quantum dot. By utilizing pulse-based NMR (specifically CHASE pulse sequences) they demonstrate that these coherence times can be extended to ~100ms. The authors claim that these results provide a path towards long-lived quantum memories for quantum networking. The paper is well written and technically sound with thoroughly analyzed experimental results. However, prior to recommending publication, I would ask for some more detail regarding the novelty and impact of this work.\nIn particular, I am concerned that these results constitute a minor advancement compared to [Chekhovich, E. A. et al. “Nuclear spin quantum register in an optically active semiconductor quantum dot”, Nature Nanotechnology, 2020]. This prior work utilized a similar quantum dot sample and operated in a very similar experimental regime. The main differences seem to be a minor change in the pulse sequence (a cyclic permutation of pulse phases mitigating issues associated with spin locking) and a 5x improvement in coherence.\nThe aim of the work published previously [Nature Nanotechnol. 15, 999 (2020)] was to establish epitaxial (optically active) GaAs/AlGaAs quantum dots as a system of choice for spin qubit studies in QDs, as well as to identify the range of possible applications and future directions to explore. Indeed, one of the demonstrations was that nuclear spin coherence can be extended to tens of milliseconds. However, this was discussed only briefly (one paragraph) and primarily in the context of coherent control fidelity (i.e. the number of coherent control gates that can be applied before the onset of decoherence). Possible applications for quantum memories were not discussed there at all.\nThe distinct advancements of the present work can be summarized as follows:\n- Indeed, the maximum achieved coherence times have been extended from ~20 ms to ~100 ms. To our knowledge, 100 ms is the longest quantum coherence (of any kind, not just nuclear spin) achieved in a GaAs semiconductor device. While even longer spin coherence is possible in other materials, GaAs quantum dots are unique in their quantum optical properties. We believe that achieving record-long coherence in this material system is by itself an important result.\n- Equally important is our demonstration that the ~100 ms mark is the actual upper limit to nuclear ensemble coherence, at least with the present level of elastic strain (~0.3%). We identify the mechanisms that are responsible for that upper limit (finite-pulse effects, cross-coupling to satellite NMR transitions) and analyse the pathways for increasing coherence further. These conclusions are supported by rigorous first-principles theory (average Hamiltonian theory) and numerical modelling. None of these results have been reported previously.\n- Notably, the coherence is extended by nearly two orders of magnitude (from Hahn echo T ~1.38 ms to a maximum T ~136 ms under dynamical decoupling). In other 2 M words, our dynamical decoupling protocol “switches off” ~99% of interactions in a dense interacting 3D spin lattice. We then reveal the physical mechanisms responsible for the remaining ~1%, in particular identifying the effective three-particle interactions as the residual coupling that is not suppressed by dynamical decoupling. We believe it is a substantial advancement, since understanding how to “switch off” interactions in a many-body system is a stepping stone to engineering an arbitrary Hamiltonian, with potential applications in quantum technologies and magnetic resonance spectroscopy.\n- We demonstrate here that a four-step “rotating spin lock” technique gives a nearly perfect isotropic dynamical decoupling of an arbitrary coherent state: it not only eliminates the preferential spin locking direction in the transverse direction, but actually eliminates the distinction between longitudinal relaxation (T ) and 1 decoherence (T ). Importantly, this concept is proven experimentally (it is not 2 uncommon for dynamical decoupling sequences to show good promise on paper but offer minimal or no advantage in a real-world experimental test). This demonstration has wider applications in dynamical decoupling, beyond quantum dots.\n- We present new results showing that dynamical decoupling is effective not only for coherent states with macroscopic polarization, but also for single-quantum wave states, which are of particular relevance in prospective quantum memory applications. This is discussed further in our response to the next point raised by Reviewer.\nThe scope of the present study is also different. While the 2020 paper focused on identifying GaAs nuclear spins as a promising quantum system, our present work addresses broader questions. These new results will be relevant to dynamical decoupling techniques in systems beyond quantum dots as well as wider applications in magnetic resonance spectroscopy and many-body quantum physics.\nWe hope this clarifies why believe our present work to be a significant advancement compared to prior publications.\nFurthermore, I do not believe that the pulse sequences and coherence times characterized here are useful for quantum networking applications. Specifically, the authors measure coherence times using un-entangled nuclear spin states which are globally polarized along the X (or Y) directions. However, for quantum memory applications, the coherence times of nuclear magnon states are relevant (such as those explored in [Hayhurst, M. et al. “A many-body quantum register for a spin qubit”, Nature Physics, 2025]). This is a crucial distinction since, I believe, that the CHASE sequences explored here only cancels dipolar interactions for globally polarized states and not magnonic states.\nWe thank the Reviewer for this important observation. Stimulated by this question, we have conducted numerical modelling of dynamical decoupling for spin-wave (magnon) coherent states. The results are shown in a new Fig. 5 of the revised manuscript. The key findings are that: CHASE dynamical decoupling is applicable to spin wave states in a sense that it increases the coherence time compared to the no-decoupling case; CHASE decoupling works similarly well for states with multiple-quantum coherence (globally polarized along transverse direction) and for states with single-quantum coherence (spin waves) in a sense that their coherence times under CHASE decoupling are comparable. Based on these observations we conclude that dynamical decoupling of states with macroscopic transverse polarization (performed via numerical modelling or NMR experiments) is a good predictor for dynamical decoupling of single-quantum spin wave excitation. This observation derived from numerical modelling can also be understood from the first principles: dynamical decoupling works through averaging out of the selected interaction terms by switching the Hamiltonian (in a toggling frame of reference) over a period of the pulse sequence cycle. This averaging is a rigorous mathematical procedure (Magnus series expansion summarised in Supplementary Note 5B): it is a genuine alteration of the effective Hamiltonian that does not depend on the nature of the spin state (whether it’s a spin wave or macroscopic polarization). In other words, if a certain interaction term is cancelled through dynamical decoupling, it has no effect on spin evolution, regardless of the nature of the spin state. To sum up, our findings confirm the prospect of using dynamical decoupling in spin wave quantum memories and hopefully answer the Reviewer’s immediate point.\nOur numerical studies on dynamical decoupling of spin waves (magnons) are rather preliminary at this stage. Several interesting observations can be made even from these preliminary results (Fig. 5). For example, we find that the coherence times of the spin wave state is longer than coherence of a macroscopically polarized state, even without any decoupling. Moreover, when CHASE dynamical decoupling is applied, the spin wave state is found to be more robust against the finite-pulse decoherence. This suggests some additional cancelations in the spin interactions, specific to spin waves. There are many outstanding questions that need to be answered (through further numerical studies), and the ultimate experimental verification of dynamical decoupling of spin waves needs to be conducted. This is an ongoing work. We plan to report on these studies in future separate publications to avoid overloading this present manuscript.\nI also have several more minor/technical queries:\n• On Page 1 the authors claim that “Recently, two-way quantum state transfer between photon and nuclear spins has been demonstrated”, however, I do not believe that the two cited papers demonstrate transfer of photonic qubits onto nuclear spin memories.\nWe have corrected the sentence to state that references [38, 39] report on two-way state transfer between an electron spin qubit and the nuclear spin ensemble of a QD. There is a prospect for a two-way transfer of a photonic state, which would require using the electron spin qubit as a mediator.\n• It would help in the introduction to introduce the “quantum link efficiency” since it is an essential metric for quantum repeaters; it would also help to explain/characterize the trade-off between memory times and entanglement rates.\nIn the revised manuscript we briefly introduce the quantum link efficiency and state that increasing the ratio of the memory storage time and the entanglement generation time is one of the key requirements for a quantum memory.\n• On page 2 the acronym ‘CT’ needs to be defined.\nThe CT acronym, which stands for central transition, is now defined at the first use in the main text on page 2.\n• Regarding the experimental results in Figure 2c: could the authors please provide a qualitative explanation for why the coherence initially increases with free evolution time for 32 and 48 cycle cases?\nThe apparent non-monotonic coherence decay in experiments with 32 and 48 cycles is caused by the decoherence that takes place during the finite-duration Rf control pulses. The finite-pulse effects are best demonstrated by replotting the data as a function of the total evolution time T and the decoupling cycle time T ., which we do in Fig. 3. By EvolTot Cycle contrast, Fig. 2c shows the as-measured data where the number of decoupling sequence cycles is kept fixed and only the free evolution time T is varied. Varying T alone FreeEvol FreeEvol (horizontal axis in Fig. 2c) corresponds to varying both T and T , which in essence EvolTot Cycle corresponds to moving along the “diagonal” line in the logarithmic (T , T ) EvolTot Cycle coordinates used in Fig 3. Thus, when T is increased (e.g. along the horizontal axis in FreeEvol Fig. 2c) it increases not only the spin evolution time, but also the T parameter, which Cycle controls the balance between the finite-pulse decoherence and decoherence through free evolution.\nSome complexity in interpreting the data is inevitable here and arises from the duality of the metrics used to characterise dynamical decoupling sequences. When considering the spin memory performance, it does not matter whether spin decoherence happens through finite-pulse effects of through free evolution of the interacting spins. In this case, it is more convenient to use the memory time T as a metric and plot the data in the (T , T ) M EvolTot Cycle coordinates (as in Fig. 3). By contrast, when analysing the fundamental properties of the dynamical decoupling sequence (i.e. the order in which the Hamiltonian is switched in the toggling frame) the interaction of spins during free evolution is most important. In that case it is more convenient to use the coherence time T as a metric and plot the data in the 2 (T , n ) coordinates (as in Fig. 2c). FreeEvol Cycles\nIn the revised manuscript we have added a brief comment to relate the non-monotonic character of the raw data curves in Fig. 2c to the non-monotonic dependence of the spin memory time T (T ), which ultimately stems from the finite-pulse decoherence. M Cycle\n• On page 4 the authors refer to a “broader invisible peak”, could they please be a bit more specific about what they mean by ‘invisible’ here?\nThis point was indeed poorly worded in the original manuscript. The broader peak (arising from random positioning of gallium and aluminium atoms in the crystal lattice) is not invisible of course, but just requires a different NMR spectroscopy technique to be revealed. This technique, dubbed “integral NMR”, was applied previously to a separate piece of the same semiconductor wafer in [Nature Nanotechnol. 18, 257 (2023)]. Since the role of the broad spectral peaks is not central to the topic of the present work, we don’t repeat these integral NMR experiments here, and simply quote the pervious study for the key parameters, i.e. the +/-100 kHz spectral extent and the 35% weight of the broad NMR peak. This point is clarified in the revised experiment.\n• In Figure 3e it would help to additionally plot the numerical simulation results for ease of comparison.\nAs suggested, we have added the spin memory times T derived from numerical modelling M to Fig. 3e (dotted lines). We have also added selected numerical modelling results in Fig. 2d, to simplify comparison with experiments.\n• In Figure 4 the authors chose to utilize a pulse sequence with only 4 cycles and 18ms coherence time. Could the authors provide equivalent plots for their 100ms coherence time sequence.\nThe 100 ms coherence can be achieved by using n = 48 cycles of the decoupling CHASE-40 sequence. As we show in Fig. 2c, for such a large number of cycles the finite duration of the control Rf pulses (which is 20 us in our experiments, currently limited by the magnitude of crystal strain) already leads to substantial (more than a half) loss of transverse spin magnetization, even in the limit of short free evolution time. Thus, experiments with a large number of cycles do not reveal much information about the anisotropy of decoherence during free evolution, because the coherence is lost even before any free evolution takes place. The interest here is in spin dynamics during free evolution because decoherence during the pulses can in principle be removed by shortening the control pulses. By contrast, the pulse control sequence itself (i.e. the order in which the Hamiltonian is switched in the toggling frame) is best characterized by the free evolution between the pulse. Consequently, we choose n = 4 cycles as a compromise between extending the coherence substantially (from ~1.38 ms in Hahn echo to ~18 ms with 4 cycles of CHASE-40) and minimising the finite-pulse effects. We clarify this point in the revised manuscript.\nWe do have a dataset comparing dynamical decoupling of the x- and y-polarized initial states with n=36 cycles (but no data for the two diagonal directions of the initial polarization). Since all the pulses in the CHASE-40 sequence are either x- or y-, experiments with x- and y-polarized initial states are in fact sufficient to detect any preferential direction. This data is now shown in Supplementary Fig. 7. We find a small asymmetry in the decoherence induced by the parasitic finite-pulse effects. However, the decoherence during free evolution, characterised by T ~84 ms, is found to be very similar 2 for both the x- and y-polarized initial states even at n=36 cycles of CHASE-40. This gives further weight to our conclusions about uniform (isotropic) decoupling of arbitrary coherent spin states, and hopefully satisfies the Reviewer’s request.\n• On page 6 the authors state that “It is also worth noting that the T decay time of the transverse 2 polarization (measured with an initialization π/2 pulse) is nearly identical to that of the longitudinal T decay time.” I am a bit surprised by this statement since the T time is >10s, whereas the T time is 1 1 2 100ms, could the authors please clarify.\nThis point was also poorly worded in the original manuscript. We clarify this in the revised manuscript as follows. In the absence of any dynamical decoupling, the strong static magnetic field indeed imposes a pronounced anisotropy with the longitudinal spin relaxation time T > 10 s significantly exceeding the transverse coherence time T ~ 1 ms 1 2 (Hahn echo). T is probed in an NMR measurement without any initialization pulse (theta=0 1 in Fig. 4b), while the transverse coherence time T is measured with an initialization pi/2 Rf 2 pulse (theta=pi/2 in Fig. 4b). Under CHASE-40 decoupling T decreases and T increases, 1 2 converging to very similar values. This is a sign of an optimally balanced time-suspension sequence that eliminates not only the spin locking anisotropy in the xy transverse plane but also the anisotropy of the strong quantising magnetic field along the z axis.\n• Also on page 6 the authors state that “We therefore ascribe the discrepancy in T to the limited M number of spins in the numerical model”. It would be helpful to explore the convergence of the simulated coherence time with increasing numbers of nuclear spins. Could the authors plot the simulated coherence times for 2≤N≤12?\nFollowing the Reviewer’s comment, we have investigated the dependence of the spin ensemble decoherence on the number of spins up to N=14. The results are shown in Supplementary Fig. 8 (Supplementary Note 6) and broadly corroborate our initial conclusions. We compare dynamical decoupling under simple Hahn echo and under 4 cycles of CHASE-40 (160 decoupling pulses). In case of Hahn echo, the coherence time follows a clear trend of reduction with an increasing number of spins N. This is expected, since an increasing N means that each spin interacts with a larger number of neighbours, accelerating the decoherence. In case of CHASE-40 decoupling, the computed coherence time shows no clear dependence on N. Since CHASE-40 eliminates the pairwise spin-spin interactions, the resulting spin dynamics are governed by a delicate balance of weak residual three-particle (and possibly higher order) correlations. Our interpretation is that the available number of spins (up to N=12..14 in exact modelling) is too small to converge to spin dynamics that are measured in a nuclear spin bath with a much larger number of spins (N~10000 in a quantum dot).\nHave the authors considered using the cluster-correlation expansion (CCE) method to speed up simulations?\nThe technical answer to this question is the balance of cost and benefit. Implementing CCE would come at a cost due to the complexity of the setup and data interpretation. Since CCE is an approximation, it comes with an extra burden of carefully checking its convergence. Exact numerical modelling is easier to setup computationally and doesn’t involve any truncation parameters or other “tuning knobs”, allowing for straightforward interpretation. On the other hand, we have access to a very “clean” (strain-engineered) nuclear spin system of a real quantum dot, which yields exact results for a system of N~10000 spins. (One can view the real quantum dot as a specialized quantum simulator for studying dynamical decoupling.) From that perspective, implementing CCE would have limited benefit, at least in our particular research problem.\nFrom a broader perspective, this is an interesting question. We anticipate that application of CCE (or similar methods) may encounter challenges that can be formulated as follows. The main philosophy of the approximate expansion methods is to keep the leading orders of interactions while truncating the small residual interactions. The dynamical decoupling works essentially in reverse: it averages out the leading interaction terms to zero so that the resulting spin dynamics are governed by the residual weak interactions. The better the dynamical decoupling sequence, the more complex is the nature of the residual interaction terms and the larger is the number of such residual terms. Thus, the better the dynamical sequence, the more difficult it would be to achieve convergence in an approximate computation method. There are signs of that issue already reported in the numerical tests of CCE [PRB 78 085315 (2008)]: under certain conditions, even going from free induction decay to a 2-pulse Carr-Purcell decoupling required increasing the cluster size from 2-spin to 6-spin. With the 40-pulse decoupling sequence, one may expect to have to include correlations of much higher orders to achieve convergence. The problem of efficient theoretical modelling of dynamical coupling certainly deserves attention, but is somewhat outside the scope of the present work. We comment on these points in the revised Supplementary Note 6.\nReviewer #3:\nThis manuscript proposes and demonstrates a novel approach to constructing a long-lived quantum memory by utilizing a subspace of the nuclear spin ensemble in a quantum dot. Notably, the memory is encoded into a homogeneous subspace by exploiting strain-induced perturbations in the nuclear spins. As a result, intrinsically long coherence times are successfully achieved. Furthermore, a practical dynamical decoupling (DD) protocol optimized for this system is designed, leading to an experimental demonstration of the longest coherence time to date. These results are accompanied by highly precise and extensive theoretical analysis, which adds depth to the understanding of the mechanisms and ensures reproducibility. Overall, this is a high-quality study on both the experimental and theoretical sides and is evaluated as an important advancement toward the practical realization of quantum repeaters. The novel design concept, the supporting data, and the consistency of the theoretical framework all contribute to the impact of this paper.\nWe thank the Reviewer for the positive comments.\nMinor comments:\n1. The fidelity of the memory initialization remains unclear. When storing or retrieving a photonic quantum state in the quantum memory, the initialization rate and DD control fidelity are important aspects. I would appreciate clarification, as much as possible, regarding these points.\nIn this work we deal specifically with dynamical decoupling of the nuclear spin coherent states. The actual initialization and readout of a prototype nuclear spin quantum memory in a GaAs QD (but without dynamical decoupling of the nuclear spins) has been studied recently [Nature Phys 21, 368 (2025)]. A two-way transfer of a coherent state was demonstrated between a single electron spin qubit and its many-body nuclear spin environment with a transfer fidelity of ~68%. In the revised manuscript we have quoted the fidelity achieved recently in [Nature Phys 21, 368 (2025)].\n2. The motivation for using CHASE is not clearly described. DD sequences such as XY8 are widely used in general, so I would like the authors to provide some explanation of the advantages of CHASE. Since the nuclear spin ensemble system may have a structural compatibility with CHASE, I believe the authors’ perspective on this would deepen the readers’ understanding.\nThe CHASE sequence and sequences such as Carr-Purcell and it’s derivative XY8 belong to different classes. Pulse sequences consisting of pi pulses, including XY8, are efficient in decoupling interactions that are linear in terms of spin operators (e.g. ~I z). These interactions include inhomogeneous chemical shifts and quadrupolar interactions, as well as inhomogeneities and fluctuations of the external magnetic fields. However, sequences of pi pulses are inefficient in decoupling of the spin-spin interactions which are quadratic in terms of spin operators (e.g. ~I I for spins 1 and 2). In many scenarios sequences such z,1 z,2 as XY8 may offer excellent decoupling performance, for example in silicon or diamond-based devices where nuclear spins are dilute and spin-spin interactions are relatively small. However, simultaneous suppression of both the linear and quadratic interactions is crucial for dynamical decoupling in a dense nuclear spin latice of GaAs. CHASE belongs to the class of dynamical decoupling sequences that provides such simultaneous decoupling. In the revise manuscript, we stress this distinctive feature of CHASE dynamical decoupling. Furthermore, in order to verify this point, we have performed numerical modelling of dynamical decoupling in a 12-nucleus dipolar-coupled spin ensemble: as expected, XY8 pulse sequence provides no improvement over Hahn echo, and is considerably inferior to CHASE decoupling."}, {"round": 2, "reviewer_comments": {"reviewer_1": "I am fine with made revisions and recommend this paper for publication.", "reviewer_3": "The authors have fully and clearly addressed the two minor comments raised in my previous review. The revised manuscript has improved substantially in clarity and completeness, and I am satisfied with the revisions. I therefore recommend the manuscript for publication in Nature Communications."}, "author_response": ""}]

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