1. Introduction

Impurities in solid-state crystals with optical access have received significant attention as potential spin-qubit candidates for quantum memories [1,2], transduction [3,4], and networks [5,6], with applications in quantum computing [7–9] and quantum communication [10,11]. Advantageous properties include their potential for scalable device integration [12], strong radiative oscillator strength [2,5], spin-dependent optical transitions [13], and long spin coherence times [14,15]. In this work, we characterize the optical linewidth of the donor-bound exciton transition for ensembles of three shallow donors in ZnO: Al, Ga, and In.

The neutral shallow donor (D⁰) in ZnO, a direct bandgap semiconductor, is a spin-1/2 electron qubit system [16] with the potential for further coupling to the donor nuclear spin. In the presence of a magnetic field, the D⁰-bound electron states demonstrate spin-relaxation times up 0.5 s [17] and ensemble coherence times of 50 µs in natural ZnO [18]. The potential for longer coherence times through isotope and chemical purification [18,19] make D⁰ in ZnO an attractive spin-qubit candidate for photon-based quantum technologies.

The donor is optically coupled to the donor-bound exciton (D⁰X). Two figures of merit for optically active quantum defects are the oscillator strength and the ratio of the optical transition linewidth to the Fourier-transform-limited linewidth. High oscillator strengths are desirable since they are proportional to the photon emission rate of single-photon sources and high optical depth is beneficial for optical quantum memories. In ZnO, the D⁰X is bright with a radiative lifetime of 0.86, 1.06, and 1.35 ns for Al, Ga, and In D⁰X respectively [20]. A linewidth broader than the lifetime transform limit impacts photon indistinguishability and the strength of the photon–spin interaction, for both ensemble [2] and single-defect applications [21].

Our goal is to elucidate the various sources—both homogeneous and inhomogeneous—that contribute to the D⁰ $\leftrightarrow$ D⁰X transition linewidth beyond this radiative limit. After describing the experimental platform in Section 2, in Section 3.1, we show that the inhomogeneous ensemble optical linewidth of D⁰ $\leftrightarrow$ D⁰X at 1.8 K can be as low as 7 GHz, compared to the $\mathcal {O}(100\;{\text{MHz}})$ Fourier-transformed lifetime linewidth. Transmission measurements also show a very high estimated optical depth of 25 to 300 for the Ga and Al ensembles, approaching that of cold atoms [22,23]. By measuring the linewidth as a function of temperature in Section 3.2, we find that the dominant phonon contribution to the linewidth is via population relaxation between the D⁰X excited states. This homogeneous broadening mechanism is negligible at 2 K. In Section 3.3, we calculate an intrinsic inhomogeneous broadening due to isotopic variation of the order of a few GHz. Finally, in Section 3.4, we probe the homogeneous linewidth via spectral anti-hole burning. We find surprisingly that the spectral anti-hole linewidth, which is measured on microsecond time scales, is the dominant contributor to the ensemble linewidth. We conclude with a discussion on how these properties may impact donor qubit operation and how they may be further improved.

2. Experimental Setup

The two samples studied in this work are 300-µm-thick Tokyo Denpa single-crystal substrates from the same parent crystal. Sample A is untreated, and Sample B has undergone indium implantation and annealing to form In donors at the surface ($\sim$200 nm deep) [24]. Three donor species are studied: Al, Ga, and In substituting for Zn. The donor concentrations in the first two microns of sample B’s surface were determined by secondary ion mass spectroscopy (SIMS) measurements as 1.2$\times 10^{15}$ cm$^{-3}$ for Al and 9.2$\times 10^{15}$ cm$^{-3}$ for Ga [24]. The bulk doping concentration for In was below the SIMS detection limit. We note that the PL intensity of the individual lines can vary by an order of magnitude across the sample. One possible cause would be non-uniform incorporation during growth.

The samples are mounted in a helium immersion cryostat with a superconducting magnet. The magnetic field $\vec {B}$ direction is fixed. The [0001] sample surface is perpendicular to crystal axis $\hat {c}$. We access different magnetic field geometries by rotating the sample to either the $\vec {B}\parallel \hat {c}$ window (Faraday geometry) or $\vec {B}\perp \hat {c}$ window (Voigt geometry). The optical axis $\hat {k}$ is parallel to the crystal axis $\hat {c}$ in all measurements. The optical path for each experiment in this manuscript and equipment part numbers used for these experiments are detailed in Supplement 1, Section S1.

The spin-1/2 neutral shallow donor electron (D⁰) is optically coupled to the donor-bound exciton (D⁰X) state, where an electron–hole pair is bound to the neutral donor. In an applied magnetic field, the spin-1/2 electron of D⁰ splits to $\vert {\downarrow }\rangle$ and $\vert {\uparrow}\rangle$. The electron $g$-factor of the donor is nearly isotropic with $g_e=1.97$ [18]. The D⁰X also splits in applied field due to the bound hole spin $\vert{\Uparrow }\rangle$ and $\vert{\Downarrow}\rangle$. The hole $g$-factor is highly anisotropic [25] ranging from $-$1.2 in Faraday to 0.3 in Voigt [17,18]. We can optically access and manipulate the two D⁰ spins via two $\Lambda$-systems [Fig. 1(a)]. In the Voigt geometry, the relative strength of the transitions is nearly identical. In the Faraday geometry, the branching ratio is 99:1 between the $\sigma ^\pm$ and $z$-polarized transitions [26,27].

 

Fig. 1. (a) Energy diagram and selection rules in Voigt (left) and Faraday (right) geometries. Here, $V_x$ and $H_x$ denote vertically and horizontally polarized light with $x$ denoting the D⁰ ground electron-spin state. Circular polarizations are denoted as $\sigma ^+$ and $\sigma ^+$, while $\hat {z}$ corresponds to linear polarization parallel to the optical axis. (b) PL spectrum of Sample B. Al, Ga, and In (Al$^*$, Ga$^*$, and In$^*$) label the corresponding D⁰ $\leftrightarrow$ D⁰X (D⁰ $\leftrightarrow$ D⁰X^*) transitions. Excitation energy is 3.44 eV, power is 30 nW and diameter is 600 nm. (c) PLE of the three donor species at 1.7 K for Sample A. Fits are to a Voigt profile and take into account incident power oscillations due to a beam splitter (Section S2). Excitation power is 200 nW for Al, 100 nW for Ga, and 1.15 µW for In. The beam diameter is 600 nm. (d) Optical density for the three donor species at 1.7 K in Sample A. Fits are to a Voigt profile. For Al and Ga, we constrain the FWHM to the FWHM found in the PLE measurements in part (c). The calculation of OD from transmission is given in Section S3. Excitation power is 15 nW with a diameter of 380 µm. PL spectra in this experimental configuration are given in Section S4.

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3. Results

3.1 Ensemble Optical Linewidth

A photoluminescence (PL) spectrum showing emission from all three donors in Sample B is shown in Fig. 1(b). We observe six peaks between 3.355 eV and 3.363 eV. The transitions labeled Al, Ga, and In correspond to the transition between the 1s D⁰ and the lowest energy D⁰X state of the respective donor species (D⁰ $\leftrightarrow$ D⁰X) [20]. The transitions Al$^*$, Ga$^*$, and In$^*$ correspond to transitions from an excited D⁰X state, which we denote as D⁰X^*. The D⁰X^* intensity becomes brighter with increasing temperature. The energy differences between the D⁰X^* and D⁰X for Al, Ga, and In are 1.26 meV, 1.46 meV, and 2.05 meV, respectively. As discussed in Section 3.2, the presence of these excited states contribute to a phonon-induced broadening of the D⁰X transitions at elevated temperatures.

The linewidths in Fig. 1(b) are spectrometer resolution-limited (55 GHz). We thus utilize micro-photoluminescence excitation (PLE) spectroscopy to determine the ensemble linewidths. In PLE measurements, we scan a continuous-wave laser near the D⁰ $\leftrightarrow$ D⁰X resonance and sum over the collected sideband PL. The excitation laser is measured by a wavemeter with a resolution of 0.5 GHz. The sideband PL consists of the two electron satellite (TES) D⁰ (2s or 2p) $\leftrightarrow$ D⁰X, first and second phonon replicas (1LO and 2LO), and first phonon replica of the TES (1LO-TES), as described in prior work [17,24]. The measured PLE linewidths of the Al, Ga, and In D⁰X transitions are $7.1\pm 0.1$ GHz, $11.1\pm 0.3$ GHz, and $7\pm 0.3$ GHz, respectively.

The lifetime-limited linewidth $\Gamma$ is given by $\Gamma =1/(2\pi \tau )$, where $\tau$ is the lifetime measured in the literature [20]. The Al and Ga lifetimes contain a non-radiative component which is attributed to a non-radiative surface recombination mechanism [28], but could also be due to exciton dissociation. Using the faster component as the lifetime value, the lifetime-limited linewidths are 0.5 GHz for Al, 0.4 GHz for Ga, and 0.1 GHz for In. The PLE linewidth is almost two orders of magnitude larger than the expected lifetime limit. The observed broadening could be due to homogeneous factors that would affect a single center on the time scale of our measurements including phonon-broadening, spectral diffusion, and hyperfine interactions. It could also be due to inhomogeneous factors such as static microscopic electric and strain fields, and isotope disorder. While significantly broader than the lifetime limit, the PLE ensemble linewidth is still remarkably narrow and is less than 100 times the lifetime limit. In comparison, the best ratio of inhomogeneous:radiative linewidths for in situ doped nitrogen vacancy (NV) centers is 1000 [29]; this ratio is even larger for rare-earth-doped ions (REIs) [30].

This narrow linewidth persists in transmission measurements. Figure 1(d) shows the optical depth (OD) $\alpha d$ through the $d=300$ µm substrate, where $\alpha$ is the frequency-dependent absorption constant. The In D⁰X absorption linewidth through the 300-µm-thick sample is only 10% broader than the micro-PLE linewidth, suggesting that high optical homogeneity persists over large volumes. The Al and Ga D⁰X transmission measurements saturate at 11.5 OD. This saturation occurs when the resonant PL intensity from the sample exceeds the transmitted laser power. Assuming that the linewidth does not significantly vary between micro-PLE and transmission measurements, a fit to the wings of the transmission spectra give a peak OD of 25 for Ga and more than 300 for Al. The area under each OD peak is proportional to the number of donors in the probed ensemble. We estimate the average donor density for each donor species; $N_{\rm Al} = 7.5\times 10^{15}\,\textrm{cm}^{-3}$, $N_{\rm Ga} = 9.9\times 10^{14}\,\textrm{cm}^{-3}$, and $N_{\rm In} = 7.4\times 10^{13}\,\textrm{cm}^{-3}$ (Section S5). These values, measured for Sample A, are within an order of magnitude of the SIMS values measured for Sample B. The relative concentrations are consistent with the PL spectra for this sample (Section S4).

An optical depth $\alpha d \gg 1$ is a critical requirement for the development of efficient optical quantum memories [2]. Warm atomic vapors [31] and cold atoms [32] can achieve optical depths from the tens up to a thousand in cm-scale devices. These high optical depths are more challenging to achieve in solid-state systems which typically have much lower oscillator strengths [30,33] and large inhomogeneous broadening. In comparison, the ZnO donor-bound exciton system combines high oscillator strength, high homogeneity, and optical depth at residual donor densities.

3.2 Temperature-Dependent Phonon Broadening

Defect–phonon interactions can be a dominant homogeneous optical dephasing mechanism, hence we need to confirm that we are performing our linewidth study below the thermal broadening limit. Figure 2(a) depicts the PLE linewidth dependence of the D⁰ $\leftrightarrow$ D⁰X transition for temperatures varying from 1.5 to 18 K for all three in situ doped donors in Sample A. Implanted In (Sample B) at low implantation doses follows a similar dependence; however, a stronger temperature dependence is observed at higher doses (Section S6).

 

Fig. 2. (a) PLE linewidth as a function of temperature for Al, Ga, and In. Each dataset is shown with the 0 K linewidth subtracted. All datasets are fit to Eq. (1), with $\Delta E$ determined from Fig. 1(b). Here, $\Delta \nu _0 =7.4 \pm 0.4$ GHz, $11.8 \pm 0.8$ GHz, and $6.5 \pm 0.5$ GHz and $a = 110 \pm 5$ GHz, $99 \pm 6$ GHz, and $59 \pm 4$ GHz, for Al, Ga, and In, respectively. (b) Magneto-PL of Al donor in Sample A. (left) Voigt (T=7.4 K), (right) Faraday (T = 5.5 K). Excitation at 3.44 eV. In Voigt, the splitting of the (unresolved) doublets corresponds to $g_e = 1.95$, while the D⁰X^* splitting corresponds to effective $g$-factors of 1.87 between observed transitions. In Faraday, the D⁰X splitting corresponds to an exciton $g$-factor $g_{e-h} = 0.83$. The D⁰X^* splitting corresponds to an effective $g$-factor of 3.39.

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The onset and rate of the temperature-dependent linewidth broadening is different for the different donor species. The observed temperature dependence can be modeled by a single-phonon absorption process with rate $\Gamma (T)$ from the lowest-energy D⁰X to an excited D⁰X^* state. In ZnO, the primary interaction is likely the piezo–phonon interaction, as has been observed in longitudinal spin relaxation [17]. The precise phonon absorption rate is unclear due to the lack of a satisfactory model for D⁰X^*; however, the rate of phonon absorption is proportional to the phonon number $N_{ph}$ at energy $\Delta E$, the energy splitting between the D⁰X and D⁰X^* states. The $N_{ph}$ will follow a Bose–Einstein distribution. This allows us to express the total linewidth as

A fit to Eq. (1) using the experimentally measured $\Delta E$ [Fig. 1(a)] shows good agreement to the model [Fig. 2(a)] at least up to 20 K. The presence of the excited state places a fundamental limit on the maximum temperature at which indistinguishable photons can be obtained from an emitter. In the absence of Purcell enhancement, the temperature at which the temperature component of the linewidth becomes equal to the radiative limit, i.e., $\Delta \nu _\text {\,T} = \Delta \nu _{\text {rad}}$ is $T = 2.7, \,3.1, \,3.8$ K for Al, Ga, and In, respectively. We note that at the lowest temperatures in this study, T = 1.7 K, the phonon contribution to the linewidth is negligible, $\Delta \nu _{\left (\text {T}\,=\,1.7\,\text {K}\right )} = 20, \,4.6, \,0.05$ MHz for Al, Ga, and In, respectively. A difference in the scaling factor $a$ is observed between donors. This difference may be due to the difference in the polarizability of the donors [17,35], which inversely depends on state localization [36]. Polarizability will increase from In to Ga to Al, which is consistent with the observed increase in $a$.

To gain insight into the nature of the D⁰X^* state, we collected PL spectra as a function of magnetic field. Figure 2(b) depicts the Al D⁰X and D⁰X^* field dependence. Section S7 includes magneto-PL for all donors. In both geometries, the D⁰X observed splittings are consistent with the reported $g$-factors for the electron and hole [17,18]. However we observe three transitions for D⁰X^* in Voigt and a very large exciton splitting ($g_{\text {exciton}} = 3.39$) in Faraday. Our $g$-factor measurements differ from Ref. [37] in which it was reported that the $g$-factor of D⁰X^* is the same as that of D⁰X. We attribute this discrepancy to the high density of overlapping lines around the Ga and Al D⁰X^* energies in Ref. [37] which could result in state misidentification. The origin of the three transitions (versus 2 or 4 observed for D⁰X) and large $g$-factor is currently unknown. Interpretation of the excited states of the bound exciton is discussed further in Section S9.

3.3 Inhomogeneous Broadening Due to Isotopic Composition

After thermal homogeneous broadening, inhomogeneous broadening is often another dominant broadening source. Inhomogeneous broadening of the D⁰ $\leftrightarrow$ D⁰X transition energy can be caused by extrinsic factors such as local variations in strain and electric fields due to point and extended defects. Intrinsic contributions to inhomogeneous broadening can occur due to nuclear spin or isotopic mass composition of the emitter’s environment. The latter is a dominant inhomogeneous broadening mechanism in high quality natural silicon [38–40]. Here we estimate the effect of isotopic mass composition on the ZnO D⁰X optical linewidth. The effect of the nuclear spin environment on the linewidth was also considered; however, with the exception of a large In donor hyperfine splitting [24], this was not found to have a significant effect (Section S10).

The local isotopic mass environment can effect the D⁰ $\leftrightarrow$ D⁰X transition by local variation in the zinc and oxygen isotopes in the defect’s environment, by or variation of the isotope of the impurity atom. The D⁰ $\leftrightarrow$ D⁰X transition closely follows the local bandgap, which is determined in part by the zero-point electron–phonon renormalization [41,42]. This zero-point renormalization energy depends on the average mass of atoms present in the local environment [42]. To assess the effect of isotopic substitution on a specific state or set of states, the bandgap variation must be decomposed into conduction band and valence band shifts [43,44]. These have different temperature dependencies, and thus different zero-point renormalization energies [42]. In ZnO, the valence band will shift by approximately 80${\% }$ of the total bandgap shift, while the conduction band exhibits a shift in the opposite direction of 20${\% }$ [45].

To quantify the resulting shifts, we follow the method presented for silicon in Ref. [39], modified for ZnO. Here the carrier wavefunction is discretized on atomic sites $\vec r_i$. The model for the effective mass envelope functions of the D⁰ electron, and D⁰X electrons and hole are given in Section S8. In this model, the energy shift for a given carrier state due to perturbation of its isotopic environment is given by $\langle\Phi _{\text {S,c}}(\vec {r}_i) | H_i^{\text {iso}} | \Phi _{\text {S,c}}(\vec {r}_{i'}) \rangle = \delta _{i,i'}W_{i, c}$, where $\vert{\Phi _\text {S,c}\left (\vec {r_i}\right )\rangle}$ refers to the Bloch function for state S (D⁰ or D⁰X), carrier c (e = electron or h = hole), the lattice site $i$ at a distance $\vec {r_i}$ from the impurity, $H_i^{iso}$ is the perturbation term, and $W_{i, c}$ is the energy shift due to the isotopic variation. The energy shift results from a shift in the top of the valence band or bottom of the conduction band, depending on if the carrier is a hole or an electron, respectively. Shifts are relative to the lowest mass isotope. The values for these shifts are listed in Table S4 in Section S12.

The total shift $\Delta E^{iso}_{S,c}$ on each state $S$ and carrier $c$ is

 

Fig. 3. Isotopic broadening for each shallow donor type. Different local environments will shift the energies of the D⁰ and D⁰X by different but correlated amounts, with the D⁰X shifting more due to the mobility of the valence band in ZnO. We calculate 1.9 GHz broadening for Al, 2.0 GHz for Ga, and 2.2 GHz for In.

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The isotopic mass environment broadening is determined numerically using 2000 simulated environments for each D⁰ and D⁰X. As shown in Fig. 3(b), the estimated total contribution of the isotopic mass environment to the inhomogeneous broadening is 1.9 GHz, 2.0 GHz, and 2.2 GHz for Al, Ga, and In, respectively. Less localized states (e.g., Al) are affected by a larger number of environmental lattice sites, leading to smaller deviations in the local isotopic mass environment. Using this model for the phosphorus D⁰ $\leftrightarrow$ D⁰X transition in natural silicon, which is known to be isotopically broadened, yields an inhomogeneous linewidth that is in good agreement with observation (Section S12) [40]. Thus, in the current samples, while not negligible, the isotopic environment is not the dominant broadening mechanism in the ensemble linewidth.

3.4 Homogeneous Spectral Anti-Hole Linewidth

We thus probe the homogeneous linewidth of the Al donor with spectral anti-hole burning measurements to elucidate the nature of the dominant broadening mechanism. The anti-hole linewidth will not be affected by the static varying isotopic environment [40]. We perform these two-laser experiments in a pump–probe configuration to probe time-dependent mechanisms such as spectral diffusion. We have also performed continuous-wave (cw) measurements which yield similar linewidths (Section S13).

Setting a single probe laser resonant to the $\sigma ^-$ transition, as shown in Fig. 4(a), we optically pump (OP) the sub-ensemble population from the $\vert{\uparrow}\rangle$ to the $\vert{\downarrow}\rangle$ spin state. Hence, when collecting the sideband emission, we observe a decrease in signal that is proportional to the population depletion of the $\vert{\downarrow}\rangle$ spin state [Fig. 4(b)]. In the absence of a pump beam, the probe signal is small due to the small thermal population in the $\vert{\uparrow}\rangle$ state.

 

Fig. 4. (a) An energy diagram for spectral hole burning experiment. (b) Single laser optical pumping (OP) curve with 230 nW probe laser power. (c) OP curves in pump–probe experiment with the 230-nW probe laser, 440-nW pump power, and 100-µs pump pulse length. Three excitation frequencies are shown (on resonance, +2.2 GHz, and +3.3 GHz detuned). Both panels (b) and (c) contain a schematic of the pulse sequence. Between cycles, the wait time is $6\times$ the 1.5 ms longitudinal spin relaxation time ($T_1$). (d) PLE curves for transient experiments. Fit is to a Voigt profile. For clarity, each spectrum has been offset vertically, and the steady state and single laser initial state curves are scaled $\times 10$. Faraday geometry at 7 T and 1.8 K.

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When we perform optical pumping as a function of probe laser wavelength, intensity at the start of the optical pumping curve varies as the laser is scanned over the resonance. If we integrate this signal in the first 2 microseconds and plot the intensity as a function of probe frequency, we observe a linewidth of $4.2\pm 2.8$ GHz. This is comparable to the 7 GHz 0-field cw PLE linewidth. If instead we integrate the signal at the end of the optical pumping curve, i.e., when the system is in the optically pumped state, a linewidth of $16.0\pm 0.9$ GHz is observed [Fig. 4(d)]. The broader linewidth in the optically pumped state is expected as efficient optical pumping on-resonance decreases the peak intensity. The similarity between the zero-field cw measurements and the time-dependent PLE in which the start of the optical-pumping curve is integrated is expected as optical pumping should not occur at zero-field.

We next introduce a second excitation laser resonant to the $\sigma ^+$ transition [“Pump” laser in Fig. 4(a)]. The pump laser initializes the sub-ensemble with which it is resonant from the $\vert{\downarrow}\rangle$ to the $\vert{\uparrow}\rangle$ spin state. After a wait time $\tau _w$, the probe laser optically pumps the sub-ensemble $\vert{\uparrow}\rangle$ to the $\vert{\downarrow}\rangle$ spin state. In Fig. 4(c), an enhanced optical pumping signal is observed when the probe laser is applied after the pump. The amplitude of the optical pumping signal, plotted as a function of probe frequency, determines the linewidth of the resonantly pumped sub-ensemble and thus, the homogeneous linewidth of the D⁰ $\leftrightarrow$ D⁰X transition. As shown in Fig. 4(d), we observe a linewidth of $3.9\pm 0.5$ GHz. Varying the wait time does not affect initial amplitude of the OP curve (Section S14). Thus, the homogeneous linewidth appears to be the dominant component to the spectral anti-hole linewidth.

The observed spectral anti-hole linewidth can be attributed to the D⁰X state; two laser coherent population trapping experiments, which probe the D⁰ coherence report a two-photon spin linewidth in the tens of MHz for Al [24]. The time dynamics of the optical pumping curve [Fig. 4(c)] suggests a laser-induced diffusion process is occurring. In contrast to single-laser experiments, in which the optical pumping curve can be characterized by a single exponential [18], complex temporal dynamics of the curve are observed following the pump pulse. As both the intensity and frequency of the probe pulse are fixed, this temporal dynamics suggests a time-dependent change in the transition frequency/line shape of the probed donors. We expect spectral diffusion is also occurring during the pump pulse, i.e., as D⁰X population is depleted at the resonant pump frequency, near-in-frequency D⁰X population diffuse into this depletion region. Future studies are required to elucidate the origin of the spectral diffusion process. Potential sources include changes in the microscopic charge environment due to nearby impurities and defects, and instantaneous spectral diffusion (ISD) [46], in which excited D⁰X within the ensemble interact electromagnetically. This is a well-documented phenomenon in REIs where the interaction is dipolar. In the D⁰ case, the effect may be more severe. The D⁰ and D⁰X effective mass states are significantly extended as compared to REI’s with a significant difference between the single electron D⁰ and three-carrier D⁰X wavefunctions (Section S8).

4. Discussion and Outlook

Further studies are required to confirm the origin of the homogeneous linewidth. If it is due to ISD, there are two immediate impacts on quantum information technologies. First, for single quantum defect applications, lower donor densities are required. In REIs, this can be achieved by burning a large spectral hole from which a narrow, low-density anti-hole can be established [47]. This strategy is only feasible when the homogeneous linewidth is much narrower than the inhomogeneous linewidth, a condition not satisfied in our samples. Thus, lower donor dopant densities will be required. On the flip side, a large ISD linewidth indicates a large D⁰X–D⁰X interaction which long-term could provide a mechanism for D⁰–D⁰ gates [48].

The total line shape will be a convolution of the inhomogeneous and homogeneous line shapes. In the simple model in which we take the homogeneous line shape to be Lorentzian with FWHM $\Delta \nu _\mathrm {L}$ and the inhomogeneous line shape to be Gaussian with FWHM $\Delta \nu _\mathrm {G}$, the ensemble linewidth can be approximated by $\Delta \nu _\mathrm {L}/2 + \sqrt {\Delta \nu _\mathrm {L}^2/4+\Delta \nu _\mathrm {G}^2}$ [49]. For Al, the narrowest measured homogeneous linewidth is 3.9 GHz with an ensemble linewidth ranging from 4.2 to 7 GHz, resulting in an inhomogeneous linewidth ranging from 1.1 to 4.6 GHz. This range is consistent with the 1.9 GHz inhomogeneous broadening estimated due to isotope disorder and suggests isotope broadening is the dominant inhomogeneous broadening mechanism.

Both chemical and isotope purification will thus be key to the development of optical quantum technologies with ZnO. Isotope purification, already required to improve the D⁰ spin coherence by removing non-zero spin nuclear isotopes [18], will be further advantageous to reduce broadening from mass disorder. Chemical purity will improve the homogeneous linewidth, either by eliminating impurities/defects that contribute to spectral diffusion and/or by lowering the inter-donor spacing to reduce instantaneous spectral diffusion. Even without these material improvements, the current linewidth properties are sufficient for several applications if donors can be integrated into photonic devices. A Purcell enhancement of less than 100 can enable the generation of indistinguishable photons. Detuned Raman excitation schemes can also mitigate against excited-state dephasing [50,51]. Finally, if single donors can be isolated, theoretical schemes to entangle donors with trapped ions [52] should be possible with the current optical properties.