Abstract
Coherence of solid state spin qubits is limited by decoherence and random fluctuations in the spin bath environment. Here we develop spin bath control sequences which simultaneously suppress the fluctuations arising from intrabath interactions and inhomogeneity. Experiments on neutral selfassembled quantum dots yield up to a fivefold increase in coherence of a bare nuclear spin bath. Numerical simulations agree with experiments and reveal emergent thermodynamic behaviour where fluctuations are ultimately caused by irreversible conversion of coherence into manybody quantum entanglement. Simulations show that for homogeneous spin baths our sequences are efficient with nonideal control pulses, while inhomogeneous bath coherence is inherently limited even under idealpulse control, especially for strongly correlated spin9/2 baths. These results highlight the limitations of selfassembled quantum dots and advantages of strainfree dots, where our sequences can be used to control the fluctuations of a homogeneous nuclear spin bath and potentially improve electron spin qubit coherence.
Introduction
The excellent spin–photon interface of confined charges in III–V semiconductor quantum dots (QDs) has recently attracted a lot of attention for potential applications in photonmediated quantum networks^{1,2,3}. The large optical dipole moment of QDs makes ultrafast optical spin control feasible and permits unrivalled entanglement generation rates^{4,5,6}. On the other hand, the coherence properties of the electron or hole spin qubit are strongly affected by hyperfine interaction with the fluctuating spin bath of the ~10^{5} constituent nuclei of the QD^{7,8}. Although hyperfineinduced qubit dephasing can be reduced using dynamical decoupling^{9,10}, this approach requires a significant number of additional qubit manipulations and rapidly loses efficiency with increasing spin bath inhomogeneity, induced, e.g. by nuclear quadrupolar effects^{11,12,13,14,15}.
There is a complementary pathway of controlling the nuclear spin bath itself with pulsed nuclear magnetic resonance (NMR)^{16,17}. The system of a central (electron) spin qubit coupled to a nuclear spin bath is characterized by a complex hierarchy of interactions and dynamics timescales, meaning that achieving optimal control of the combined electron–nuclear spin coherence is a difficult problem. A simpler starting point is the problem of tailoring the coherence of a bare manybody nuclear spin bath (without a central spin). Examples for controlling spin–spin interactions are found in NMR spectroscopy, where sequences such as WAHUHA^{18} and MREV^{19,20} are used to average out dipolar couplings selectively. However, these solid echo sequences do not remove inhomogeneous broadening. By contrast, dynamical decoupling sequences based on πpulse trains suppress inhomogeneous dephasing^{21,22,23}, but exacerbate dipolar dephasing through the parasitic effect of instantaneous diffusion^{24,25,26}. Thus, a different class of control sequences is required to remove both the effect of inhomogeneity and dipolar interactions.
In this work, we introduce pulse sequences which combine the features of dynamical decoupling with those of solid echoes and are designed to preserve coherence of an arbitrary quantum state of a spin bath. These combined Hahn and solid echo (CHASE) pulse sequences are tested using firstprinciple quantum mechanical simulations and experiments on neutral InGaAs/GaAs selfassembled QDs, where different isotopes offer access to spin baths with distinct regimes of inhomogeneous broadening and correlations in a manybody system. Even without the electron, the evolution of a bare spin bath reveals unexpected phenomena: in strongly disordered spin baths, where inhomogeneous broadening significantly exceeds the intrabath interactions, the ability to preserve coherence via global control is inherently limited—this case applies to selfassembled QDs, where spin9/2 indium nuclei with their strong flipflop coupling impose the ultimate limit on the maximum achievable spin bath coherence. In the case of homogeneous spin baths, we find that cyclic application of CHASE sequences can efficiently suppress spin–spin entanglement, leading to significant extension of coherence times even under nonideal control pulses. This highlights the advantages of spin qubits with homogeneous spin environments, such as dilute donor spins^{26,27}, defect centres^{28}, or strainfree GaAs/AlGaAs quantum dots^{9,10,29}. We show that manybody decoherence emerges naturally under unitary evolution^{30}: similar to the second law of thermodynamics, where useful energy is irreversibly dissipated into wasteful heat, the coherence is irreversibly converted into multipartite spin–spin entanglement. While it is not possible to eliminate spin bath dynamics completely, CHASE control sequences can transform random fluctuations into more deterministic evolution, which can in principle be decoupled from the qubit using standard control schemes^{17}. More broadly, CHASE sequences may be used in quantum thermodynamic applications to preserve coherences, which play an important role in extracting work from nanoscale systems^{31,32,33}.
Results
Design of CHASE pulse sequences
An intuitive approach used previously to extend nuclear spin lifetimes in silicon and diamond^{27,28} is to combine solid echo π/2pulse cycles with refocusing πpulses in order to suppress both the inhomogeneous dephasing and dipolar couplings. Here, we employ a rigorous average Hamiltonian theory (AHT)^{34}, which is a form of perturbation theory based on Magnus expansion^{35}. Using AHT as a benchmark tool (see details in Supplementary Note 1), we systematically analyse various combinations of π and π/2pulses to find those that maximise the spin bath coherence while minimising the pulse sequence length. The evolution of a nuclear spin bath I_{i} is analysed under a given pulse cycle in a strong external magnetic field B_{z}. We take into account a dipolar coupling term \({\cal{H}}_{\mathrm{d}}^{{\mathrm{zz}}}\) as well as a generic resonance offset Hamiltonian \({\cal{H}}_0^{\mathrm{z}}\), which describes inhomogeneous resonance broadening due to, e.g. chemical shifts or static quadrupolar interaction
where ν_{ij} is the dipolar coupling constant between two spins I_{i} and I_{j} and Δν_{i} denotes the resonance frequency offset of the ith nuclear spin. A larger spread in the Δν_{i} values describes a spin bath with a larger disorder. The free induction decay (FID) of transverse magnetisation under this Hamiltonian is described by a rate \({\mathrm{\Gamma }} \propto 1/T_2^ \ast \propto \sqrt {\langle {\mathrm{\Delta }}\nu _i^2\rangle }\).
The shortest cycle (CHASE5) giving a noticeable coherence increase contains only five pulses and is illustrated in Fig. 1a. Assuming infinitely short pulses (t_{π} → 0), the zerothorder average Hamiltonian ∝ Γ vanishes. The leading residual contribution to decoherence is a firstorder term ∝ ℏt_{c}Γ^{2} mixing contributions from the inhomogeneous broadening Hamiltonian and the dipolar interaction^{36,37}:
where t_{c} is the full cycle time and \({\cal{H}}_{\mathrm{d}}^{{\mathrm{xx}}}\), \({\cal{H}}_0^{\mathrm{y}}\) are the dipolar and inhomogeneous broadening Hamiltonians acting along orthogonal equatorial axes \(\hat e_{\mathrm{x}}\) and \(\hat e_{\mathrm{y}}\).
Under realistic experimental conditions, the t_{π} → 0 assumption is often not justified. For finite pulse durations t_{π}, the zerothorder average Hamiltonian does not vanish under CHASE5. However, we can obtain an average Hamiltonian of the form of Eq. (2) even for finite t_{π} by extending the cycle to CHASE10 (Fig. 1b). Furthermore, by adding the pulse block shown in Fig. 1c, we can symmetrise the cycle to CHASE20 and remove the firstorder mixing term, condensing the average Hamiltonian to \(\bar {\cal{H}}_{{\mathrm{CHASE}}  {\mathrm{20}}} \propto {\cal{O}}(\hbar t_{\mathrm{c}}^2{\mathrm{\Gamma }}^3)\) independent of the pulse duration t_{π}. Finally, we identify the CHASE34 sequence (Fig. 1d), which reduces the average Hamiltonian to a secondorder mixing term for t_{π} → 0 but has nonvanishing lowerorder terms for finite pulse durations. A comprehensive overview of the AHT calculations and residual Hamiltonians can be found in Supplementary Note 1.
CHASE NMR experiments on selfassembled quantum dots
We study the performance of these sequences experimentally on individual chargefree InGaAs QDs with ~10^{5} nuclear spins. The average fractions of the isotopes in a dot are^{11}: 50% (^{75}As), ~24% (^{69}Ga), ~16% (^{71}Ga), ~9.5% (^{115}In) and <0.5% (^{113}In). We follow the ODNMR pump–probe scheme used in ref. ^{38}: the QD sample is kept at low temperature (T = 4.2 K) and is subjected to a strong magnetic field B_{z} = 8 T. Using a confocal microscopy setup in Faraday configuration, we prepare the nuclear spin bath optically through polarisationselective pumping of an exciton transition (dynamic nuclear polarisation). In this way, we achieve hyperfinemediated spin bath polarisation degrees of up to 65%^{39,40}. Radio frequency (rf) fields are coupled to the QD via a multiwinding copper coil in close proximity to the sample. Changes in the final bath polarisation are probed with a weak optical pulse measuring the energy splitting of the neutral exciton Zeeman doublet^{11}.
We perform resonant pulsed NMR measurements on the inhomogeneously broadened central spin transition −1/2 \(\leftrightarrow\) +1/2 of the spin3/2 ^{75}As (inhomogeneous width of \(\Delta \nu _{{\mathrm{inh}}} \sim 40\,{\mathrm{kHz}}\)), spin3/2 \(^{71}\)Ga \(({\mathrm{\Delta }}\nu _{{\mathrm{inh}}} \sim 10\,{\mathrm{kHz}})\) and spin9/2 ^{115}In \((\Delta \nu _{{\mathrm{inh}}} \sim 30\,{\mathrm{kHz}})\) nuclear ensembles^{11,38}. In each experiment, the population difference of the I_{z} = ±1/2 states is enhanced via adiabatic rapid passage^{38,41} and only one isotope is manipulated with rf pulses, while the spins in the I_{z} ≥ 3/2 states as well as other isotopes are left to evolve freely during the pulse sequence. The phases of the πpulses in all sequences are chosen to produce spin rotations around the \(\hat e_{\mathrm{x}}\) axis of the rotating frame. In each experiment, a π/2pulse is applied prior to the multipulse cycle to initialise the transverse spin polarisation by converting energy into coherence^{31}. We conduct experiments with initial π/2rotation around the \(\hat e_{\mathrm{x}}\) axis (Carr–Purcell or CPlike sequences^{42}, denoted as −X) and around the \(\hat e_{\mathrm{y}}\) axis (Carr–Purcell–Meiboom–Gill or CPMGlike sequences^{43}, −Y): in this way, we distinguish between a genuine improvement of the spin coherence and spin locking effects^{44,45,46}, which only stabilise spin magnetisation along a certain direction. A final π/2pulse is an inverse of the initialisation pulse and projects the spin echo polarisation onto the \(\hat e_{\mathrm{z}}\) axis for optical readout^{38}.
Figure 2 shows representative experimental dependencies of the spin echo amplitude on the total free evolution time τ_{evol} over n sequence cycles for ^{71}Ga (a), ^{75}As (b) and ^{115}In (c) nuclei. The decay of the echo measured in terms of hyperfine shift ΔE_{hf} is modeled by a compressed exponential decay function
where β is a compression factor^{47}, T_{2} describes decoherence of the spin bath during free evolution, while reduction of the echo amplitude ΔE_{hf}(τ_{evol} → 0) at short free evolution compared with the initial spin polarisation ΔE_{hf}(t = 0) quantifies the imperfections of pulse spin rotations. The fitted values of T_{2} and ΔE_{hf}(τ_{evol} → 0) are plotted in Fig. 3a–f for the three studied isotopes at different n expressed in terms of the total control gate time in units of t_{π}. The compression factor β depends on the spectral properties of the random process responsible for the decoherence^{24}. From fitting, we find β≈1.0–2.0 for ^{71}Ga and ^{75}As, agreeing with the values typically found for various spin systems, whereas β≈0.7–1.0 in the case of ^{115}In due to the nonexponential echo decay.
Control of nuclear spin coherence in quantum dots
We start by analysing the common behaviour of all the isotopes under CPX and CPY sequences with an alternating pulse carrier phase (sequence cycle −τ/2 − π_{x} − τ − π_{−x} − τ/2−). For increasing n, the echo amplitude is robust and T_{2} increases under CPX (black squares in Fig. 3a–f), whereas under CPY (blue circles), the echo amplitude is rapidly reduced, owing to the limited available rf power resulting in deviation from the ideal hard pulses (see Supplementary Note 3). The contrasting behaviour of T_{2} under alternating phase CPX/Y has been observed in other systems^{44,45,46,48} and has been attributed variably to spin locking^{48,49} or stimulated echoes^{50}. Here, we ascribe the increase of T_{2} under CPX to a form of pulsed spin locking arising from dipolar evolution during the finiteduration πpulses^{49}: our interpretation is based on the observation that the spin lock disappears for small pulsetocycle time ratios t_{π}/t_{c} (see Supplementary Note 2).
We now examine the spin bath coherence under CHASE10/20 and CHASE34. Figure 3a–c shows an increase of T_{2} under CHASE, compared with the Hahn echo T_{2} for all isotopes. This increase occurs both under −X and −Y initialisation signifying a genuine increase of nuclear spin coherence, as opposed to pulsed spin locking under CPX. However, the effect of CHASE sequence cycling (increasing n, and correspondingly increasing the total rf gate time) depends strongly on the isotope. For ^{71}Ga, there is a steady increase in T_{2} up to \(T_2^{{\mathrm{CHASE}}  {\mathrm{Y}}  {\mathrm{20}}} \approx 6.0\,{\mathrm{ms}}\) at n = 16 (compared with Hahn echo \(T_2^{{\mathrm{HE}}  {\mathrm{X}}/{\mathrm{Y}}} \approx 1.0\,{\mathrm{ms}}\), Fig. 3a), revealing the expected convergence of the average Hamiltonian to zero with reducing cycle time t_{c}. The growth of T_{2} with increasing n is accompanied by a gradual reduction of the echo amplitude (Fig. 3d)—the result of the nonideal (finiteduration) control pulses. By contrast, T_{2} is nearly constant for ^{75}As (Fig. 3b), and the echo amplitude reduction is more pronounced, owing to the larger inhomogeneous broadening Δν_{i} and limited rf pulse amplitude. While there is no difference for ^{71}Ga, in the case of ^{75}As, CHASE34 somewhat improves coherence over CHASE20, showing that with growing inhomogeneous broadening, higherorder Hamiltonian terms have a stronger impact on decoherence than the evolution during nonideal control pulses.
A rather different picture is observed for the spin9/2 ^{115}In: beyond one cycle of CHASE10, sequence cycling actually reduces T_{2} (Fig. 3c). Notably, this is not related to imperfect spin rotations, as reduction in echo amplitudes (Fig. 3f) is less pronounced than for ^{75}As (Fig. 3f), suggesting a fundamentally different mechanism.
Firstprinciple numerical simulations
In order to get a better insight into the underlying mechanisms of the manybody spin decoherence, we conduct firstprinciple quantum mechanical simulations of the nuclear spin bath evolution. We consider an ensemble of 12 dipolarcoupled nuclei with spin I and study the evolution of the I_{z} = ±1/2 subspace under rf driving of the central transition −1/2↔ +1/2 in the limits of large (Δν_{i} ≫ ν_{ij}) and vanishing (Δν_{i} ≪ ν_{ij}) inhomogeneous resonance broadening. Figure 4 shows the fitted coherence times (a, b, c) and echo amplitudes (d, e, f) for simulated spin echo decay curves in the following cases: I = 3/2, Δν_{i} ≪ ν_{ij} (a, d); I = 3/2, Δν_{i} ≫ ν_{ij} (b, e); I = 9/2, Δν_{i} ≫ ν_{ij} (c, f).
The simulations with Δν_{i} ≫ ν_{ij} and realistic rf pulse durations (solid symbols) are in very good agreement with the experiments on QDs (Fig. 3). Namely, T_{2} increases under CPX (spin locking) and the echo amplitude reduces under CPY with growing n. The absolute simulated T_{2} values are only a factor of ~2 larger than in the experiment (which is due to the limited number of spins in the model, see Supplementary Fig. 5). Under CHASE10/20, there is a reduction in T_{2} with increasing n for I = 9/2 spins similar to the ^{115}In experiment (cf. Figs. 3c and 4c); there is a strong reduction of the echo amplitude for spin3/2 nuclei with Δν_{i} ≫ ν_{ij}, in agreement with the ^{75}As experiment (cf. Figs. 3b, e and 4b, e). Moreover, the pronounced increase of T_{2} for ^{71}Ga with its smaller inhomogeneous broadening matches well the simulation with Δν_{i} ≪ ν_{ij} (cf. Figs. 3a and 4a).
Good agreement with experiments means the simulations are valid and can be used to explore the experimentally inaccessible regimes of ideal (infinitely short t_{π} → 0) control pulses (open symbols in Fig. 4). We readily find that even with perfect spin rotations (t_{π} → 0), the T_{2} of an inhomogeneously broadened (Δν_{i} ≫ ν_{ij}) spin ensemble under CHASE10/20 shows a flat region for I = 3/2 (Fig. 4b) and even an initial decrease with growing n for I = 9/2 (Fig. 4c), which is contrary to lim_{n→∞}T_{2} = ∞, expected for the average Hamiltonian converging to zero. This unexpected discrepancy can be understood by noting that the Hahn echo (n = 1/2) coherence \(T_2^{{\mathrm{HE}}  {\mathrm{X}}/{\mathrm{Y}}}\) of spin3/2 nuclei is ~3 times longer for Δν_{i} ≫ ν_{ij} due to the freezing of the dipolar spin–spin flipflops by inhomogeneous broadening^{38}. When n is increased, the CPX/Y coherence time \(T_2^{{\mathrm{CP}}  {\mathrm{X}}/{\mathrm{Y}}}\) decreases (open squares and circles in Fig. 4b) and asymptotically approaches the \(T_2^{{\mathrm{HE}}  {\mathrm{X}}/{\mathrm{Y}}}\) obtained for Δν_{i} ≪ ν_{ij} where spin flipflops are allowed (Fig. 4a). This suggests that the fast spin rotations induced by the high repetition rate train of infinitely short πpulses effectively reduce the spin lifetime, broaden the homogeneous NMR linewidths and reenable the frozen dipolar flipflops in an inhomogeneous spin bath. Further simulations show that restoration of the flipflops at larger inhomogeneous broadening requires a proportionally larger pulse repetition rate, confirming our interpretation. The ‘heating’ of the spins by frequent control pulses is then also responsible for slow convergence of the average Hamiltonian under CHASE sequences: the growth in T_{2} can only be achieved when the flipflops are fully ‘thawed’ (at total gate times ≳200t_{π} in Fig. 4b, c). When projected to the I_{z} = ±1/2 states, the dipolar Hamiltonian has the form
so the role of the flipflops is inherently most pronounced for I = 9/2 nuclei explaining the particularly strong reduction of ^{115}In coherence under long control sequences. Flipflops in the I_{z} = ±1/2 subspace are possible only for pairs with opposite spins. Thus, the heating of spins by control pulses will depend on the actual spin state. In other words, under frequent spin rotations, the manybody Hamiltonian itself becomes statedependent, invalidating the AHT approximation. This mechanism is the main practical limitation in controlling the coherence of a disordered manybody spin system via global spin rotations.
Emergent thermodynamic behaviour in a spin bath
In addition to the measurable transverse spin magnetisation (coherence), numerical simulations permit probing quantities that are not directly observed in experiment. In particular, we now examine the evolution of the multipartite spin–spin quantum entanglement and demonstrate its fundamental role in the decoherence of an interacting spin bath. We quantify entanglement using the basisindependent intrinsic coherence measure C_{I} introduced in ref. ^{51}. Figure 5a shows the simulated time evolution of the transverse spin magnetisation 〈I_{x}(t)〉 of spin3/2 nuclei with large inhomogeneous broadening Δν_{i} ≫ ν_{ij}. The results are averaged over randomly chosen initial eigenstates, and calculations are done for a fixed free evolution time τ between pulses. Figure 5b shows the evolution of the entanglement measure 〈C_{I}(t)〉 for the same simulations, whereas Fig. 5c presents the same results and additional data for a homogeneous bath Δν_{i} ≪ ν_{ij} and spin9/2 nuclei as trajectories in the entanglementcoherence 〈C_{I}〉 − 〈I_{x}〉 phase space.
Under free induction decay (FID, crosses in Fig. 5), coherence 〈I_{x}(t)〉 is lost rapidly within ~30 μs, before the onset of entanglement 〈C_{I}〉, which happens within ~3 ms (with some dips arising from the quasiperiodic evolution of a small spin ensemble). Application of the CPY sequence with ideal pulses (t_{π} → 0, open circles) has no effect on entanglement rate but extends coherence decay to the same timescale as entanglement. Going from CPY to CHASE20 (open triangles), we observe a slowdown both in the 〈I_{x}〉 decay and 〈C_{I}〉 growth, with the actual timescales strongly dependent on τ. The systematic relation between coherence and entanglement is demonstrated in Fig. 5c. Firstly, we find that evolution of the spin ensemble is always accompanied by a nearly monotonic conversion of coherence into entanglement. Secondly, we observe that despite different decoherence and entanglement rates, CPY and CHASE20 with ideal pulses (t_{π} → 0) result in similar 〈C_{I}〉 − 〈I_{x}〉 trajectories—extensive simulations for CPX/Y and CHASEX/Y show that this similarity holds for a wide range of pulsetopulse delays τ and applies to the case of a small inhomogeneity Δν_{i} ≪ ν_{ij} as well as spin9/2 nuclei (a limited selection is shown in Fig. 5c to keep the graph readable). This gives an empirical evidence for the existence of a universal optimal trajectory, where 〈I_{x}〉 has maximal possible value at any given 〈C_{I}〉. Figure 5c reveals the factors leading to suboptimal trajectories: in the absence of pulsed spin control (FID), transverse coherence in an inhomogeneous ensemble (Δν_{i} ≫ ν_{ij}) is rapidly lost well before emergence of any significant spin–spin entanglement. Furthermore, in an inhomogeneous system under pulsed spin control, going from ideal (t_{π} → 0, open symbols) to finite (t_{π} = 10 μs, solid symbols) control pulses, the trajectories become nonideal with a shift towards the FID trajectory—this is very pronounced in the case of CPY (circles), while for CHASEY20 with the same τ≈200 μs, the trajectory is closer to ideal, demonstrating the robustness of the CHASE sequence.
The evolution of coherence and entanglement can be interpreted in the framework of thermodynamics, i.e. without the need to know the exact form of the internal interaction Hamiltonian. The simulation results presented above show that transverse magnetisation 〈I_{x}〉 (coherence) of the spin system is analogous to work, i.e. a useful resource—in the case of nuclear spins, this work can actually be extracted, e.g. in the form of a current induced in a coil. By contrast, the entanglement 〈C_{I}〉 is analogous to a wasteful heat, which cannot be used to generate coherence or extract work. The true decoherence of the spin ensemble is then associated with the irreversible generation of entanglement 〈C_{I}〉 via the spin–spin interactions^{52,53}. For example, the apparent quick loss of coherence in the FID of an inhomogeneous ensemble (Δν_{i} ≫ ν_{ij}) does not involve entanglement and in fact can be reversed by applying a πpulse train (CP sequence). By contrast, once the maximum level of entanglement is reached (〈C_{I}〉≈ 0.67 in Fig. 5c), it is impossible to recover coherence, regardless of the pulse sequence applied. It is important to note that there are no artificial sources of decoherence in the presented simulations, implying the evolution is unitary^{54} and thus demonstrating that the irreversible loss of coherence due to entanglement emerges naturally^{30} in a system of just a few dipolarcoupled spins. Entanglement due to the nonideal (finite) control pulses can be understood, since these pulses are equivalent to Hamiltonian quenches, which are generally known to induce irreversible evolution^{30}. The difficulty of improving I = 9/2 indium spin coherence with pulsed control (Figs. 3c and 4c) also finds its explanation within the quantum thermodynamics framework, since stronger spin–spin correlations, which scale as ∝ (I + 1/2)^{2}, generally result in faster entanglement growth^{53} and more irreversible evolution^{55}.
Discussion
Based on our results, we conclude with the following interpretation of the relation between disorder, coherence and entanglement in dipolarcoupled spin baths. Suppression of spin–spin entanglement growth is required to extend spin coherence. This can be achieved by applying CHASE pulse cycles which converge the spin–spin interaction Hamiltonian to zero with reducing cycle time t_{c} → 0. Realistic (finiteduration) control pulses impose a lower limit on t_{c}. Despite that, the Hamiltonian in a homogeneous spin bath does converge under CHASE, predicting at least a factor of ~100 growth in coherence time under realistic parameters (Fig. 4a). A robust performance under nonideal soft pulses is a unique feature of CHASE distinguishing it from the previously introduced timesuspension sequences^{28,56,57,58} (see additional simulations in Supplementary Note 5). In a strongly inhomogeneous (disordered) spin bath, the spin–spin interactions are difficult to eliminate, even if ideal control pulses were possible (Fig. 4b, c), as control pulses themselves make the spin–spin interactions statedependent, disrupting the convergence of the Hamiltonian. Finite control pulses impose further restrictions, limiting the maximum coherence time that can be realistically achieved in an inhomogeneous spin bath.
In the case of bare nuclear spin baths, CHASE can be used, for example, to preserve the quantum information transferred to the nuclei from the electron spin^{59}. Another potential application is for dynamical suppression of the nuclear spin bath decoherence and fluctuations, which limit the electron spin qubit coherence in quantum dots. In order for dynamical suppression to be efficient, a significant fraction of the bath nuclei must be controlled. In selfassembled (InGaAs) quantum dots, the highspin states I_{z} > 1/2 cannot be manipulated with coherent rf pulses, due to the extreme (fewmegahertz) resonance broadening. While this can be overcome by nuclear spin hyperpolarisation^{29} and a subsequent population transfer^{38,41} to the I_{z} = 1/2 states, the real fundamental limitation for the InGaAs dots arises from the strongly correlated nature of the indium spin9/2 subsystem. A more promising area of application are latticematched GaAs/AlGaAs quantum dots (both electrostatic^{15} and epitaxial^{29}) or II–VI quantum dots^{60}, where quadrupolar inhomogeneous broadening is small or absent, enabling significant extension of coherence times and the ability to operate in a wide range of magnetic fields ≳1–10 mT, requiring only that the nuclear Zeeman splitting is larger than the quadrupolar and dipolar nuclear spin interactions.
Interaction with a central (electron) spin complicates the coherent dynamics compared with the bare nuclear spin bath case. For example, the nuclear spin echo coherence in InGaAs electroncharged dots is few tens of microseconds^{41}, but is expected to be longer in GaAs dots, due to the larger number of nuclei and smaller hyperfine interaction per nucleus. Using few microsecondlong rf pulses, it should be possible to examine extension of nuclear spin coherence under CHASE sequences in the presence of an electron. This would be a basis for exploring CHASE control of the nuclei, synchronised with electron spin qubit control, with the aim of creating a deterministically evolving nuclear spin environment and leading to extended electron spin qubit coherence—this approach can compliment existing techniques, such as preparation of the nuclear spin bath in a narrowed state via electron–nuclear feedback^{61,62}.
More broadly, CHASE is promising for dopants and point defects in diamond, silicon and silicon carbide, where it can be applied both to the nuclear spin bath and directly to electron spin qubits to enhance their coherence beyond the limits of standard dynamical decoupling protocols by suppressing the instantaneous spin diffusion^{16,26} arising from electron–electron dipolar interactions. Further optimisation of spin bath decoherence freezing can be explored using techniques such as optimal control^{63}.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
The authors are grateful to Joseph Maguire and Ian Griffiths for assistance. This work was supported by the EPSRC Programme Grant EP/J007544/1, ITN S3NANO, UK Quantum Technology Hub for Quantum Communications Technologies and EPSRC grant EP/M013472/1. E.A.C. was supported by a University of Sheffield ViceChancellor’s Fellowship and a Royal Society University Research Fellowship. Computational resources were provided in part by the University of Sheffield HPC cluster Iceberg.
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M.H., P.S. and D.R. developed and grew the samples. A.M.W. and E.A.C. designed the pulse sequences. A.M.W., G.G. and G.R. conducted the experiments. E.A.C. conducted numerical simulations. A.M.W. and E.A.C. analysed the data. A.M.W., E.A.C. and M.S.S. wrote the paper with input from all authors. E.A.C. coordinated the project.
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Waeber, A.M., Gillard, G., Ragunathan, G. et al. Pulse control protocols for preserving coherence in dipolarcoupled nuclear spin baths. Nat Commun 10, 3157 (2019). https://doi.org/10.1038/s41467019111606
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