# Cavity-enhanced single artificial atoms in silicon

Valeria Saggio1,∗, Carlos Errando-Herranz1,2,∗, Samuel Gyger1,3, Christopher
Panuski1, Mihika Prabhu1, Lorenzo De Santis1,4, Ian Christen1, Dalia Ornelas-
Huerta1, Hamza Raniwala1, Connor Gerlach1, Marco Colangelo1, and Dirk
Englund1<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>1Massachusetts
Institute of Technology, Cambridge, USA
2University of Münster, Münster, Germany
3KTH Royal Institute of Technology, Stockholm, Sweden
4QuTech, Delft University of Technology, Delft, Netherlands

###### Abstract

Artificial atoms in solids are leading candidates for quantum networks Bhaskar
_et al._ (2020); Pompili  _et al._ (2021), scalable quantum computing
Childress and Hanson (2013); Choi  _et al._ (2019), and sensing Degen  _et
al._ (2017), as they combine long-lived spins with mobile and robust photonic
qubits. The central requirements for the spin-photon interface at the heart of
these systems are long spin coherence times and efficient spin-photon coupling
at telecommunication wavelengths. Artificial atoms in silicon Redjem  _et al._
(2020); Higginbottom  _et al._ (2022) have a unique potential to combine the
long coherence times of spins in silicon Saeedi  _et al._ (2013) with
telecommunication wavelength photons in the world’s most advanced
microelectronics and photonics platform. However, a current bottleneck is the
naturally weak emission rate of artificial atoms. An open challenge is to
enhance this interaction via coupling to an optical cavity. Here, we
demonstrate cavity-enhanced single artificial atoms at telecommunication
wavelengths in silicon. We optimize photonic crystal cavities via inverse
design and show controllable cavity-coupling of single G-centers in the
telecommunications O-band. Our results illustrate the potential to achieve a
deterministic spin-photon interface in silicon at telecommunication
wavelengths, paving the way for scalable quantum information processing.

##  I Introduction

Artificial atoms employ controlled coherent spin-photon interfaces to transfer
quantum states between long-lived stationary spins and flying photons. Three
central requirements for a scalable spin-photon interface are a long spin
coherence time, efficient spin-photon coupling, and operation at
telecommunication wavelengths Ruf  _et al._ (2021). However, current material
platforms fail to meet these requirements at once Zaporski  _et al._ (2023);
Pompili  _et al._ (2021); Bhaskar  _et al._ (2020).

A central challenge is to enhance the naturally weak coherent radiative
emission rate while suppressing other excited-state decoherence processes. In
particular, the modified local density of optical states in a cavity can
increase the radiative emission fraction $\beta$ into a desired mode while
suppressing emission into other modes:

$\beta=\frac{(1+F_{\text{P}})\gamma_{\text{R}}}{(1+F_{\text{P}})\gamma_{\text{R}}+\gamma_{0}},$
(1)

where $\gamma_{R}$ is the radiative rate for the transition of interest and
$\gamma_{0}$ encompasses the rates for other radiative and non-radiative
transitions. $F_{\text{P}}$ is the cavity Purcell factor, which in the case of
perfect cavity-atom coupling is defined as Purcell (1995)

$F_{\text{P}}=\frac{3\lambda^{3}}{4\pi^{2}}\frac{Q}{V},$ (2)

with $\lambda$ the wavelength in the material, $Q$ the quality factor, and $V$
the effective mode volume of the cavity. To enable efficient collection of the
light emitted from the cavity, it is required that the cavity far-field
emission is matched to the optical mode of interest — such as the mode of an
optical fiber. This light collection is quantified using the coupling
efficiency $\eta$. The net collection efficiency $\beta\eta$ defines the
performance of a quantum network built with such devices. Accommodating both
high $Q/V$ and high $\eta$ in a single device is a nontrivial design challenge
that depends largely on the materials, the fabrication, and the operation
wavelengths of the artificial atom of choice. These challenges have so far
resulted in weak and small-scale spin-photon coupling for current leading
artificial atom platforms Ruf  _et al._ (2021).

Figure 1: Description of the system. a) Illustration of the system under
study, consisting of optimized 2D photonic crystal cavities coupled to single
G-centers in silicon. b) The G-center consists of two substitutional carbon
atoms (blue spheres) and a silicon interstitial atom (gray spheres). c) The
O-band radiative transition occurs between singlet states 1 and 2 and can be
excited via above-band excitation. The system features an additional triplet
state (3). d) Spectrum showing the photoluminescence (PL) of a single cavity-
coupled G-center with a zero phonon line (ZPL) around 1279 nm.

Recently, there has been a resurgence of interest in silicon as a host
material for single artificial atoms operating in the telecommunication bands
Bergeron  _et al._ (2020); Baron  _et al._ (2021); Durand  _et al._ (2021);
Higginbottom  _et al._ (2022). Initial reports have focused on the G-center
Hollenbach  _et al._ (2020); Redjem  _et al._ (2020), the T-center
Higginbottom  _et al._ (2022); DeAbreu  _et al._ (2022), and the W-center
Baron  _et al._ (2021), and include the first optical observation of an
isolated spin in silicon Higginbottom  _et al._ (2022), and the first
isolation of single artificial atoms in silicon waveguides and their spectral
programming Prabhu  _et al._ (2022). These demonstrations, combined with the
experimentally reported 30-min-long spin coherence times in ionized donors in
28Si Saeedi  _et al._ (2013) and the success of silicon microelectronics and
photonics Vivien and Pavesi (2016), make this technology compelling for large-
scale quantum information processing. However, although high $Q/V$ and $\eta$
optical cavities have recently been shown in silicon at a large scale Panuski
_et al._ (2022), their coupling to single artificial atoms remains a
challenge.

Here, we report on the inverse design of high $Q/V$, $\eta$-optimized photonic
crystal cavities, and demonstrate cavity-enhanced interaction of light with
single artificial atoms at telecommunication wavelengths in silicon.

##  II Results

Our device, illustrated in Fig. 1a, consists of single G-centers coupled to
inverse-designed 2D photonic crystal cavities. The G-center is a quantum
emitter formed by two substitutional carbon atoms and a silicon interstitial
(Fig. 1b), and features a zero phonon line (ZPL) transition at 970 meV (1279
nm) in the telecommunications O-band along with a spin triplet metastable
state Udvarhelyi  _et al._ (2021) (Figs. 1c, d).

Our cavities were designed following our previous work Panuski  _et al._
(2022) to simultaneously achieve a target $Q/V$ while optimizing for vertical
coupling $\eta$ by matching the emission to a narrow numerical aperture in the
O-band. Fig. 2 shows one of our cavity designs, including its near-field
cavity mode (Fig. 2a) and its far-field scattering profile (Fig. 2b), with
more than 70% of the emitted power simulated to radiate into an objective NA
of 0.55. More information on the optimization can be found in Methods, and the
results of the cavity optimization in SI Section .3 and Fig. S1. The
fabrication of our device follows our previous work Prabhu  _et al._ (2022)
with the addition of an underetch step, and is described in Methods.

Figure 2: Optimized cavities. Simulated a) near-field electric field amplitude
and b) scattered far-field power for the 2D photonic crystal cavity design
used in this work. c) Scanning electron micrograph showing one of our
optimized photonic crystal cavities. d) Confocal PL 2D scan showing the
reflectivity map (blue scale) overlaid with a PL map (red and green) of the
same area, highlighting the PL emission from the cavity center.

The measurements on our device were performed with a setup consisting of a
home-built cryogenic confocal microscope featuring temperature and CO2 gas
control, and optimized for visible light excitation and infrared collection
into a single-mode fiber (details in SI Section .4). A scanning electron
micrograph of a representative cavity in our chip is shown in Fig. 2c. Its
reflectivity was characterized in cross-polarization Altug and Vucković
(2005); Panuski  _et al._ (2022); DeAbreu  _et al._ (2022) (details in SI
Section .5). Fig. 2d shows a photoluminescence (PL) 2D scan of one of our
systems, where the cavity-coupled artificial atom is evidenced by the color-
labeled IR emission in the cavity center upon excitation with green light. The
spectral signature of the PL, shown in Fig. 1d, features a ZPL centered at
around 1279 nm and thus aligns with the previously reported G-center ZPLs
Hollenbach  _et al._ (2020); Baron  _et al._ (2021); Prabhu  _et al._ (2022).

For the cavities of interest, we measure quality factors of $\sim 3700$ and
2100, and center wavelengths around 1279 nm, near the G-center ZPL.

Fig. 3 shows the experimental results confirming the presence of single
artificial atoms in our photonic crystal cavities. The coupling between the
atom and the cavity results in an enhancement of the atom single-photon
emission. We observe linearly polarized PL emission (Fig. 3a) from our system,
indicative of coupling through the expected transverse electric cavity mode.
Measuring the PL saturation under increasing excitation power yields that of a
two-level emitter model (see Fig. 3b and SI Section .6). We further confirmed
the addressing of a single artificial atom by demonstrating single-photon
emission via a Hanbury-Brown-Twiss (HBT) experiment. Our second-order
autocorrelation results (Fig. 3c) show excellent antibunching with a fitted
$g^{(2)}(0)$ value of $0.03\begin{subarray}{c}+0.07\\\ -0.03\end{subarray}$
without background correction (details in SI Section .7). This value is nearly
an order of magnitude lower than the rest of the literature (see SI Table 1),
and indicates high-purity single-photon emission. The bunching near $\pm 10$
ns delay conforms with the presence of a third dark state, which has been
attributed to a metastable triplet state Udvarhelyi  _et al._ (2021). These
measurements demonstrate the presence of a single G-center in our cavity.

To confirm the cavity enhancement of our single artificial atom, we detuned
the cavity resonance wavelength from the G-center ZPL using two different
methods, i.e. thermal and gas tuning. In the thermal tuning experiments,
starting with a cryostat temperature of 4 K, we brought the temperature up to
24 K and consequently shifted the cavity away from our G-center ZPL. This
effect is visible in Fig. 3d, where the pink (orange) curves show the cavity
and ZPL profiles before (after) the temperature increase. Different detunings
$\delta_{\mathrm{t}}$ and $\delta_{\mathrm{t}}^{\prime}$, defined as the
difference between the cavity and ZPL wavelengths, are therefore achieved at 4
K and 24 K, respectively. While a significant cavity wavelength shift occurs,
we also observe a much smaller ZPL shift, not shown in the figure. Therefore,
this plot shall be indicative only of the relative shift between the cavity
and ZPL wavelength. A more detailed discussion about the figure can be found
in SI Section .8a. We note that temperatures below 30 K have been reported to
not affect G-center ensembles Beaufils  _et al._ (2018); Chartrand  _et al._
(2018). We observe an intensity enhancement of $6.078\pm 0.218$ with a cavity
$Q$ of $\sim 3700$ and a mode volume of $V<1(\lambda/n)^{3}$.

To validate that the intensity enhancement does not originate from the
temperature change induced by thermal cavity tuning, we performed additional
experiments using gas tuning. We injected CO2 gas into the cryogenic sample
chamber to coat the cavity with solid CO2, followed by selective gas
sublimation using a $532$ nm continuos wave (CW) laser. A further description
of the process can be found in SI Section .8b. Analogously to the thermal
tuning case, Fig. 3e shows the cavity reflectivity and G-center ZPL now under
gas tuning for two different detunings $\delta_{\mathrm{g}}$ and
$\delta_{\mathrm{g}}^{\prime}$. We calculate an enhancement in the PL emission
of $3.837\pm 0.074$ with a cavity $Q$ of $\sim 2100$ and a mode volume of
$V<1(\lambda/n)^{3}$. Also in this case, we observe a ZPL shift (not shown in
the figure). More details are to be found in SI Section .8b.

Ultimately, we measured the excited state lifetime of our artificial atoms
under both gas and thermal tuning using a 0.5 ns pulsed laser at 532 nm for
all cavity detunings (see SI Section .6 for details about these measurements).
Fig. 3f shows our measured lifetimes and the emission rates for all of our
experiments. We do not observe a statistically significant lifetime
modification even under a clear cavity enhancement of the emission rates above
6x.

Figure 3: Cavity-enhanced single-photon emission. a) Polarization plot of the
PL emission from our system, which matches that of an electric dipole. b) PL
counts at increasing CW excitation powers. Saturation matching that of a two-
level system is observed. c) The second-order autocorrelation function, which
yields $g^{(2)}(0)\approx 0$, demonstrating high-purity single-photon
emission. d) Cavity-atom coupling is demonstrated by changing the sample
temperature from 4 K (pink) to 24 K (orange) to spectrally tune a cavity with
respect to the G-center ZPL, which results in a reduction of the PL magnitude.
e) This effect is confirmed with a second cavity-atom system on the same chip
before (green) and after (blue) gas detuning. f) PL counts and lifetimes for
both systems under the measured cavity-atom detuning magnitudes.

The quantum efficiency (QE) of any silicon color center is one of the central
unanswered questions in the field. Recent reports have estimated the QE of a
single G-center to be above 1% from waveguide-coupled counts Prabhu  _et al._
(2022); Komza  _et al._ (2022), and $<$ 10% for ensembles coupled to separate
cavities Lefaucher  _et al._ (2023). To gain a valid estimate of the QE of the
G-center, an experiment varying the coupling rate between the same single
artificial atom and a cavity was required. Our measurements allow us to
extract such a value. Using the derivation described in SI Section .9
Lefaucher  _et al._ (2023), the literature value of the Debye-Waller factor
$F_{\text{DW}}=0.15$ Beaufils  _et al._ (2018), and our measured values of
off-resonance lifetime $\tau_{\text{off}}=6.09\pm 0.25$ ns and count rate
enhancement for thermal tuning, we obtain a quantum efficiency bounded as
$\text{QE}\leavevmode\nobreak\ <\leavevmode\nobreak\ 18\%$.

##  III Discussion

We show, for the first time, strong enhancement of the quantum emission of
individual artificial atoms coupled to silicon nanocavities. A central
requirement for the scalability of our system is localized spatial and
spectral alignment of both many cavities and many atoms to a common global
frequency. The spatial alignment of the cavity and atom can be achieved by
making use of the recently reported localized implantation of single G- and
W-centers Hollenbach  _et al._ (2022). Silicon artificial atoms can be
spectrally aligned using the recently reported non-volatile optical tuning for
G-centers Prabhu  _et al._ (2022) or methods used in other artificial atom
systems such as tuning via electric fields Anderson  _et al._ (2019), or
mechanical strain Wan  _et al._ (2020). Cavity tuning via local thermal
oxidation of silicon has been achieved on a large scale Panuski  _et al._
(2022), and a similar method could be used to align large arrays of cavity-
atom systems at room or cryogenic temperatures. Our approach directly applies
to other silicon artificial atoms, such as the T-center, which would enable
direct access to a spin outside of a metastable state Higginbottom  _et al._
(2022).

A hypothesis was recently raised regarding the possibility of two different
physical systems being reported as G-centers Baron  _et al._ (2022). We
believe that our work provides conclusive evidence for this claim. Table 1 in
SI Section .10 compares the reported experimental results for single G-center
labeled artificial atoms in silicon, and shows two clear clusters. The first
group comprises the single emitter reports in Refs. Hollenbach  _et al._
(2020); Baron  _et al._ (2022); Prabhu  _et al._ (2022); Komza  _et al._
(2022), and aligns with G-center ensemble work Lefaucher  _et al._ (2023).
These studies show a ZPL centered around 1279 nm and a narrow inhomogeneous
linewidth < 1.1 nm, a QE between 1 and 18%, and a short excited state lifetime
< 10 ns that does not change significantly under Purcell enhancement,
confirming the QE magnitude. The second group comprises Refs. Redjem  _et al._
(2020, 2023), and features a shorter ZPL centered around 1270 nm and a larger
inhomogeneous linewidth of 9.1 nm, a QE$\sim$50%, and a longer excited state
lifetime > 30 ns, which changes significantly under Purcell enhancement and
thus qualitatively aligns with the estimated QE. Our measurements align with
the first system, i.e. the originally reported G-centers, and provide the
first upper bound for the QE of single G-centers, previously estimated to be
between 1% Prabhu  _et al._ (2022); Komza  _et al._ (2022) and 10% for
ensembles Lefaucher  _et al._ (2023). More information on this comparison can
be found in SI Section .10. We conclude that our work highlights the need for
further theoretical and experimental investigation regarding the creation
process and the photophysics of G-center-like artificial atoms in silicon
platforms.

##  IV Conclusion

We showed cavity-enhanced single artificial atoms in silicon by integrating
single G-centers into inverse-designed photonic crystal nanocavities. We
demonstrated an intensity enhancement of $\sim$6x with a cavity featuring a
$Q/V>3700$, which yielded the highest purity single-photon emission for
silicon color centers in the literature, and the first bound to the QE of
single G-centers of $<18$%. Our demonstration lays the groundwork for
efficient spin-photon interfaces at telecommunication wavelengths for large-
scale quantum information processing in silicon.

## Methods

###  IV.1 Sample fabrication

The fabrication process follows Redjem  _et al._ (2020), starting from a
commercial SOI wafer with $220$ nm silicon on $2$ µm silicon dioxide. Cleaved
chips from this wafer were implanted with 12C with a dose of $5\times 10^{13}$
ions/cm2 and $36$ keV energy, and subsequently annealed at
$1000\leavevmode\nobreak\ ^{\circ}$C for $20$ s to form G-centers in the
silicon layer. The samples were then processed by a foundry (Applied
Nanotools) for electron beam patterning and etching, resulting in through-
etched silicon cavities with SiO2 bottom cladding and air as top cladding. The
silicon etching was performed using inductively coupled plasma reactive ion
etching with SF6-C4F8 mixed-gas. As a final step, the samples were under-
etched in a $49\%$ solution of hydrofluoric acid for $2$ min and dried using a
critical point dryer.

###  IV.2 Cavity far-field optimization

Traditional photonic crystal cavity optimization aims to cancel radiative loss
to enhance quality factor $Q$, which also reduces collection efficiency. To
avoid this, we incorporate the far-field collection efficiency $\eta$ to the
optimization objective function alongside maximizing $Q$ and minimizing mode
volume $V$ Panuski  _et al._ (2022) (Fig. S1). This process is implemented
using the open-source package Legume Minkov  _et al._ (2020) which maps the
problem of cavity design onto efficient and auto-differentiable guided mode
expansion (GME) for gradient-based optimization.

In practice, we observe quality factors much lower than the
$Q\sim\mathcal{O}(10^{6})$ result expected from both simulation (Fig. S1) and
previous statistical studies on thousands of photonic crystals designed for
$\sim$1550 nm operation under the same optimization method Panuski  _et al._
(2022). We attribute this disparity to the high carbon doping density used to
produce cavity-coupled G-centers with sufficient probability. Reducing the
doping density or applying localized doping Hollenbach  _et al._ (2020) could
play a role in recovering performance closer to intrinsic silicon. Applying
large-scale characterization techniques Sutula  _et al._ (2022) to locate
ideal emitters and fabricate cavities around these positions could enhance the
yield of coupled emitters in the case of reduced doping density.

## Notes

During the preparation of this manuscript, we became aware of a manuscript
reporting on a cavity-coupled single silicon color center Redjem  _et al._
(2023) and a second manuscript reporting on single rare-earth ions in silicon
cavities Gritsch  _et al._ (2023).

## Acknowledgements

The authors acknowledge Kevin C. Chen, Chao Li, Hugo Larocque and Mohamed
ElKabbash for helpful discussions. C.E-H. and L.D. acknowledge funding from
the European Union’s Horizon 2020 research and innovation program under the
Marie Sklodowska-Curie grant agreements No.896401 and 840393. M.P.
acknowledges funding from the National Science Foundation (NSF) Convergence
Accelerator Program under grant No.OIA-2040695 and Harvard MURI under grant
No.W911NF-15-1-0548. I.C. acknowledges funding from the National Defense
Science and Engineering Graduate (NDSEG) Fellowship Program and NSF award
DMR-1747426. M.C. acknowledges support from MIT Claude E. Shannon award. D.E.
acknowledges support from the NSF RAISE TAQS program. This material is based
on research sponsored by the Air Force Research Laboratory (AFRL), under
agreement number FA8750-20-2-1007. The U.S. Government is authorized to
reproduce and distribute reprints for Governmental purposes notwithstanding
any copyright notation thereon. The views and conclusions contained herein are
those of the authors and should not be interpreted as necessarily representing
the official policies or endorsements, either expressed or implied, of the Air
Force Research Laboratory (AFRL), or the U.S. Government.

## References

  * Bhaskar  _et al._ (2020) M. K. Bhaskar, R. Riedinger, B. Machielse, D. S. Levonian, C. T. Nguyen, E. N. Knall, H. Park, D. Englund, M. Lončar, D. D. Sukachev, and M. D. Lukin, Nature 580, 60 (2020).
  * Pompili  _et al._ (2021) M. Pompili, S. L. N. Hermans, S. Baier, H. K. C. Beukers, P. C. Humphreys, R. N. Schouten, R. F. L. Vermeulen, M. J. Tiggelman, L. dos Santos Martins, B. Dirkse, S. Wehner, and R. Hanson, Science 372, 259 (2021).
  * Childress and Hanson (2013) L. Childress and R. Hanson, MRS Bulletin 38, 134 (2013).
  * Choi  _et al._ (2019) H. Choi, M. Pant, S. Guha, and D. Englund, npj Quantum Information 5, 1 (2019).
  * Degen  _et al._ (2017) C. L. Degen, F. Reinhard, and P. Cappellaro, Reviews of Modern Physics 89, 035002 (2017).
  * Redjem  _et al._ (2020) W. Redjem, A. Durand, T. Herzig, A. Benali, S. Pezzagna, J. Meijer, A. Y. Kuznetsov, H. S. Nguyen, S. Cueff, J.-M. Gérard, I. Robert-Philip, B. Gil, D. Caliste, P. Pochet, M. Abbarchi, V. Jacques, A. Dréau, and G. Cassabois, Nature Electronics 3, 738 (2020).
  * Higginbottom  _et al._ (2022) D. B. Higginbottom, A. T. K. Kurkjian, C. Chartrand, M. Kazemi, N. A. Brunelle, E. R. MacQuarrie, J. R. Klein, N. R. Lee-Hone, J. Stacho, M. Ruether, C. Bowness, L. Bergeron, A. DeAbreu, S. R. Harrigan, J. Kanaganayagam, D. W. Marsden, T. S. Richards, L. A. Stott, S. Roorda, K. J. Morse, M. L. W. Thewalt, and S. Simmons, Nature 607, 266 (2022).
  * Saeedi  _et al._ (2013) K. Saeedi, S. Simmons, J. Z. Salvail, P. Dluhy, H. Riemann, N. V. Abrosimov, P. Becker, H.-J. Pohl, J. J. L. Morton, and M. L. W. Thewalt, Science 342, 830 (2013).
  * Ruf  _et al._ (2021) M. Ruf, N. H. Wan, H. Choi, D. Englund, and R. Hanson, Journal of Applied Physics 130, 070901 (2021).
  * Zaporski  _et al._ (2023) L. Zaporski, N. Shofer, J. H. Bodey, S. Manna, G. Gillard, M. H. Appel, C. Schimpf, S. F. Covre da Silva, J. Jarman, G. Delamare, G. Park, U. Haeusler, E. A. Chekhovich, A. Rastelli, D. A. Gangloff, M. Atatüre, and C. Le Gall, Nature Nanotechnology , 1 (2023).
  * Purcell (1995) E. M. Purcell, in  _Confined Electrons and Photons: New Physics and Applications_, NATO ASI Series, edited by E. Burstein and C. Weisbuch (Boston, MA, 1995) pp. 839–839.
  * Bergeron  _et al._ (2020) L. Bergeron, C. Chartrand, A. T. K. Kurkjian, K. J. Morse, H. Riemann, N. V. Abrosimov, P. Becker, H.-J. Pohl, M. L. W. Thewalt, and S. Simmons, PRX Quantum 1, 020301 (2020).
  * Baron  _et al._ (2021) Y. Baron, A. Durand, P. Udvarhelyi, T. Herzig, M. Khoury, S. Pezzagna, J. Meijer, I. Robert-Philip, M. Abbarchi, J.-M. Hartmann, V. Mazzocchi, J.-M. Gérard, A. Gali, V. Jacques, G. Cassabois, and A. Dréau, (2021), arXiv:2108.04283 [physics, physics:quant-ph] .
  * Durand  _et al._ (2021) A. Durand, Y. Baron, W. Redjem, T. Herzig, A. Benali, S. Pezzagna, J. Meijer, A. Y. Kuznetsov, J.-M. Gérard, I. Robert-Philip, M. Abbarchi, V. Jacques, G. Cassabois, and A. Dréau, Physical Review Letters 126, 083602 (2021).
  * Hollenbach  _et al._ (2020) M. Hollenbach, Y. Berencén, U. Kentsch, M. Helm, and G. V. Astakhov, Optics Express 28, 26111 (2020).
  * DeAbreu  _et al._ (2022) A. DeAbreu, C. Bowness, A. Alizadeh, C. Chartrand, N. A. Brunelle, E. R. MacQuarrie, N. R. Lee-Hone, M. Ruether, M. Kazemi, A. T. K. Kurkjian, S. Roorda, N. V. Abrosimov, H.-J. Pohl, M. L. W. Thewalt, D. B. Higginbottom, and S. Simmons, (2022), arXiv:arXiv.2209.14260 .
  * Prabhu  _et al._ (2022) M. Prabhu, C. Errando-Herranz, L. De Santis, I. Christen, C. Chen, and D. R. Englund, (2022), arXiv:2202.02342 [physics, physics:quant-ph] .
  * Vivien and Pavesi (2016) L. Vivien and L. Pavesi,  _Handbook of Silicon Photonics_ (CRC Press, Boca Raton, FL, USA, 2016).
  * Panuski  _et al._ (2022) C. L. Panuski, I. R. Christen, M. Minkov, C. J. Brabec, S. Trajtenberg-Mills, A. D. Griffiths, J. J. D. McKendry, G. L. Leake, D. J. Coleman, C. Tran, J. S. Louis, J. Mucci, C. Horvath, J. N. Westwood-Bachman, S. F. Preble, M. D. Dawson, M. J. Strain, M. L. Fanto, and D. R. Englund, Nature Photonics 16, 834 (2022).
  * Udvarhelyi  _et al._ (2021) P. Udvarhelyi, B. Somogyi, G. Thiering, and A. Gali, Physical Review Letters 127, 196402 (2021).
  * Altug and Vucković (2005) H. Altug and J. Vucković, Optics Letters 30, 982 (2005).
  * Beaufils  _et al._ (2018) C. Beaufils, W. Redjem, E. Rousseau, V. Jacques, A. Y. Kuznetsov, C. Raynaud, C. Voisin, A. Benali, T. Herzig, S. Pezzagna, J. Meijer, M. Abbarchi, and G. Cassabois, Physical Review B 97, 035303 (2018).
  * Chartrand  _et al._ (2018) C. Chartrand, L. Bergeron, K. J. Morse, H. Riemann, N. V. Abrosimov, P. Becker, H.-J. Pohl, S. Simmons, and M. L. W. Thewalt, Physical Review B 98, 195201 (2018).
  * Komza  _et al._ (2022) L. Komza, P. Samutpraphoot, M. Odeh, Y.-L. Tang, M. Mathew, J. Chang, H. Song, M.-K. Kim, Y. Xiong, G. Hautier, and A. Sipahigil,  (2022), arXiv:2211.09305 [cond-mat, physics:physics, physics:quant-ph] .
  * Lefaucher  _et al._ (2023) B. Lefaucher, J.-B. Jager, V. Calvo, A. Durand, Y. Baron, F. Cache, V. Jacques, I. Robert-Philip, G. Cassabois, T. Herzig, J. Meijer, S. Pezzagna, M. Khoury, M. Abbarchi, A. Dréau, and J.-M. Gérard, Applied Physics Letters 122, 061109 (2023).
  * Hollenbach  _et al._ (2022) M. Hollenbach, N. Klingner, N. S. Jagtap, L. Bischoff, C. Fowley, U. Kentsch, G. Hlawacek, A. Erbe, N. V. Abrosimov, M. Helm, Y. Berencén, and G. V. Astakhov, Nature Communications 13, 7683 (2022).
  * Anderson  _et al._ (2019) C. P. Anderson, A. Bourassa, K. C. Miao, G. Wolfowicz, P. J. Mintun, A. L. Crook, H. Abe, J. Ul Hassan, N. T. Son, T. Ohshima, and D. D. Awschalom, Science 366, 1225 (2019).
  * Wan  _et al._ (2020) N. H. Wan, T.-J. Lu, K. C. Chen, M. P. Walsh, M. E. Trusheim, L. De Santis, E. A. Bersin, I. B. Harris, S. L. Mouradian, I. R. Christen, E. S. Bielejec, and D. Englund, Nature 583, 226 (2020).
  * Baron  _et al._ (2022) Y. Baron, A. Durand, T. Herzig, M. Khoury, S. Pezzagna, J. Meijer, I. Robert-Philip, M. Abbarchi, J.-M. Hartmann, S. Reboh, J.-M. Gérard, V. Jacques, G. Cassabois, and A. Dréau, Applied Physics Letters 121, 084003 (2022).
  * Redjem  _et al._ (2023) W. Redjem, Y. Zhiyenbayev, W. Qarony, V. Ivanov, C. Papapanos, W. Liu, K. Jhuria, Z. A. Balushi, S. Dhuey, A. Schwartzberg, L. Tan, T. Schenkel, and B. Kanté,  (2023), arXiv:2301.06654 [quant-ph] .
  * Minkov  _et al._ (2020) M. Minkov, I. A. D. Williamson, L. C. Andreani, D. Gerace, B. Lou, A. Y. Song, T. W. Hughes, and S. Fan, ACS Photonics 7, 1729 (2020).
  * Sutula  _et al._ (2022) M. Sutula, I. Christen, E. Bersin, M. P. Walsh, K. C. Chen, J. Mallek, A. Melville, M. Titze, E. S. Bielejec, S. Hamilton, D. Braje, P. B. Dixon, and D. R. Englund, (2022), arXiv:2210.13643 .
  * Gritsch  _et al._ (2023) A. Gritsch, A. Ulanowski, and A. Reiserer,  (2023), arXiv:2301.07753 [quant-ph] .
  * Lin  _et al._ (2021) Z. Lin, L. Schweickert, S. Gyger, K. D. Jöns, and V. Zwiller, Journal of Instrumentation 16, T08016 (2021).
  * Warren (1986) S. G. Warren, Applied Optics 25, 2650 (1986).
  * Ismail  _et al._ (2016) N. Ismail, C. C. Kores, D. Geskus, and M. Pollnau, Optics Express 24, 16366 (2016).

## Supplementary Information

###  .3 Cavity optimization

Figure S1 shows the course of optimization for the two cavities we employed in
our experiments, with the only difference being the hole sizes. Sweeps with
several varying parameters (e.g. lattice constant, cavity and hole size) were
fabricated in order to target systems with the desired performance.

Figure S1: 2D photonic crystal cavity optimization path. The course of
optimization for one of our designs over 222 iterations, plotting the
simulated quality factor $Q$, a result proportional to the mode volume $V$,
the scattering efficiency $\eta$ into the 0th Brillouin zone, and the
composite figure of merit FoM targeted by the optimization.

###  .4 Measurement setup

Fig. S2 shows a schematic of our measurement setup, which consists of 3 main
optical paths: 1) excitation, 2) cryogenic 4F, and 3) collection.

The excitation path combines IR and visible collimated lasers with a dichroic,
and routes them via a linear polarizer into a polarizing beam splitter (PBS).
For our characterization, we use two visible lasers and two infrared lasers.
Our visible lasers consist of a continuous wave Coherent Verdi G5 at 532 nm,
and a pulsed laser from NKT Photonics SuperK laser with a maximum repetition
rate of 78 MHz and filtered by a bandpass filter centered at 532 nm with a
bandwidth of 0.2 nm. Our infrared lasers are a tunable CW O-band TSL570C from
Santec, and a superluminescent diode S5FC1018S from Thorlabs with a broadband
emission centered at 1310 nm. A set of scanning mirrors is placed in the
excitation path for precise beam positioning and PL mapping of our cavities.

The transmitted polarization component is imaged into our cryostat by using a
4F lens system consisting of scanning piezoelectric mirrors, two lenses, and
an objective, with preceding IR quarter and half wave plates for polarization
rotation. Our microscope objective is a collar corrected objective LCPLN50XIR
from Olympus with a NA of 0.65, and is external to the cryostat. Our cryostat
is a Montana Instruments system. The sample is mounted on a XYZ cryogenic
piezoelectric stage (Attocube).

The PL and reflection from our sample are collected via the PBS reflection
through an IR polarizer and a filtering station into a fiber switch, which
routes the collected light into superconducting nanowire single-photon
detectors (SNSPDs) or an IR spectrometer. Our filtering setup consists of a
longpass filter (cutoff wavelength at 1250 nm) and a shortpass filter (cutoff
wavelength at 1300 nm). Additionally, in several experiments we used a tunable
fiber filter from WLPhotonics with FWHM transmission bandwidth of $0.10$ nm.
Our two SNSPDs (NIST) feature detection efficiencies of $21\%$ and $24\%$, and
are readout with a Swabian Instruments Timetagger 20. Our IR spectrometer
consists of a PyLon IR CCD from Princeton Instruments and switchable gratings,
one with a density of 300 gr/mm and a 1.2 µm blaze and another with a density
of 900 gr/mm and a 1.3 µm blaze, leading to pixel-defined resolutions of 155
pm and 40 pm respectively. For the second-order autocorrelation measurements
we used a fiber beam splitter (Thorlabs TW1300R5F1) after the filtering
station. In addition, we image our sample using a flip mirror before the fiber
switch and an InGaAs cooled CCD camera (Allied Vision Goldeye), preceded by a
flip lens that allows us to switch between near-field and far-field imaging.

The setup comprises also a gas line equipped with a needle valve and several
ball valves to enable controllable gas injection into our cryostat.

Figure S2: Measurement setup schematic. Our setup consists of a cryogenic
confocal microscope. Laser light at visible (532 nm) and infrared (1280 nm)
shines through a polarizer and a polarizing beam splitter (PBS), a set of
galvanometers (scanning mirrors), a 4F system, and polarization rotation
components, into an objective and into the cryostat. The IR PL or cavity
reflectivity from the sample reflects off the PBS through a polarizer and a
filtering stage into the fiber switch, which leads to SNSPDs or an IR
spectrometer. A flip mirror and a lens between the filters and the fiber
switch into an IR camera enable the visualization of the image plane, and
flipping away the lens provides us with a back focal plane image.

###  .5 Cross-polarization cavity characterization

We characterized our cavities via a cross-polarization measurement, as
previously reported in Ref. Altug and Vucković (2005). The measurement
protocol consists of preparing the polarization of the input and output IR
beams to be orthogonal and $45^{\circ}$ rotated with respect to the cavity
axis. This was achieved by setting the input and output polarizers
perpendicular to each other, and by rotating the common half-wave plate to
align the fields to the cavity (see schematic in Fig. S2). In our setup, we
used a PBS for increased polarization extinction.

###  .6 Measurements under pulsed excitation

All measurements under pulsed excitation were performed with our SuperK laser
using a repetition rate of 39 MHz, and varying the power from a minimum of 0.2
$\upmu$W to a maximum of 2 $\upmu$W.

As already discussed in the main text, our artificial atom emission fits well
to the characteristic two-level emitter saturation model, which is given by

$\displaystyle I(P)=I_{\infty}\frac{P}{P+P_{\text{sat}}}.$ (3)

Here, $I$ and $P$ are intensity and power, respectively, and $I_{\infty}$ and
$P_{\text{sat}}$ are the corresponding saturation values. Fitting our
experimental data to this theoretical model, we find $I_{\infty}=(12\pm 2$)
kcounts/min and $P_{\text{sat}}=(7.4\pm 1.2)$ $\upmu$W. The data and
corresponding fit are shown in Fig. S3a. The same theoretical model is used to
fit the data taken under CW excitation power reported in the main text. In
that case (Fig. 3b in the main text), we obtain
$I_{\infty}^{\mathrm{CW}}=(93\pm 5$) kcounts/min and
$P_{\text{sat}}^{\mathrm{CW}}=(28\pm 3)$ $\upmu$W.

Fig. S3b shows an example of our lifetime data and fits. The plot displays one
data set acquired at detuning $\delta_{\mathrm{g}}$ (refer to Fig. 3e in the
main text) with a pulsed excitation power of 0.3 $\upmu$W fitted to a mono-
exponential model

$\displaystyle f(t)=x\ e^{-\frac{t-y}{\tau}}+z,$ (4)

with $x$ being a fitting constant for the amplitude, and $y$ and $z$ offsets
for the decay time $t$ and $f(t)$, respectively. The fit returns a lifetime of
$\tau=5.55\pm 0.06$ ns. All lifetime data sets are fitted using the same
model, which returns the lifetime values plotted in Fig. S3c for all the
detuning values. In both thermal and gas tuning cases, the lifetime does not
differ significantly for different detunings. This figure displays the values
that are used to estimate the bound on the QE. The single value shown in Fig.
3f in the main text for detuning $\delta_{\mathrm{g}}$ is the weighted average
between the two values reported in Fig. S3c. However, for the sake of
completeness, we also report additional lifetimes acquired at detuning
$\delta_{\mathrm{g}}$ under likely different laser conditions, originating
from an error related to our laser control electronics board. The values are
shown in Fig. S3d. In this case, we measure lifetimes all below $4.75$ ns. As
these measurements were taken under possibly different experimental
conditions, we decided not to include them in our theoretical analysis.
However, they confirm that the lifetime of our emitter remains essentially
unchanged with increasing excitation power.

Figure S3: Measurements under pulsed excitation. a) Saturation curve obtained
acquiring PL counts while varying the pulsed excitation power. b) Example of a
lifetime fit, where a mono-exponential is used to obtain the lifetime from the
experimental data. c) Lifetimes displayed both for the gas and thermal tuning
cases. In each case, we measured the lifetimes at similar pulsed excitation
powers. d) Set of lifetime measurements collected at detuning
$\delta_{\mathrm{g}}$ under possibly different laser conditions.

###  .7 Second-order autocorrelation measurements

The second-order autocorrelation measurements were performed using a HBT
interferometer. We excited each of our emitters with a $532$ nm CW pump and
sent the generated photons to a fiber beam splitter whose outputs were
connected to two SNSPDs, and we then analyzed the coincidence counts at
different time delays between the two outputs. Fitting our second-order
autocorrelation data with a three-level system equation

$g^{(2)}(t)=a\left[1-(1-b)\left((1+c)e^{-\frac{\absolutevalue{t-t_{\mathrm{shift}}}}{\tau_{1}}}-c\
e^{-\frac{\absolutevalue{t-t_{\mathrm{shift}}}}{\tau_{2}}}\right)\right],$ (5)

with $a$, $b$, and $c$ fitting parameters, $t_{\text{shift}}$ the offset for
the time delay $t$, and $\tau_{1}$ and $\tau_{2}$ the lifetimes, we obtain the
$g^{(2)}(0)$ value after data normalization as $g^{(2)}(0)=b$ at
$t=t_{\mathrm{shift}}=0$. HBT measurement data and fits are shown in Figs. S4a
and b for the thermal tuning and gas tuning cases, respectively. In both cases
we obtain a $g^{(2)}(0)$ close to 0, thus confirming genuine single-photon
emission. In the gas tuning case, we find
$g^{(2)}(0)^{\mathrm{g}}=0.03\begin{subarray}{c}+0.07\\\ -0.03\end{subarray}$,
$\tau_{1}^{\mathrm{g}}=(3.16\pm 0.5)$ ns and $\tau_{2}^{\mathrm{g}}=(9.05\pm
1.3)$ ns. The data was collected filtering the region around the ZPL, and thus
reducing the noise contribution coming from elsewhere. In the thermal tuning
case, we used the longpass and shortpass filters to isolate a $\sim 50$ nm-
wide region comprising the ZPL, while we improved the gas-tuning measurement
by filtering a much narrower region with the only help of the electrically
tunable bandpass fiber filter described in SI Section .4. We used a CW
excitation power of 6 $\upmu$W (10 $\upmu$W) in the thermal (gas) tuning case.
In both cases the data was acquired as raw time tags and is not background-
corrected. The coincidence counts were evaluated with the software ETA Lin
_et al._ (2021) using a time binning of $200$ ps ($512$ ps) in the thermal
(gas) tuning case.

The G-center is known to feature three energy levels: a ground and excited
singlet state and a metastable triplet state Udvarhelyi  _et al._ (2021). The
metastable state introduces a bunching effect in the second-order correlation,
and has been previously observed for G-centers Redjem  _et al._ (2020);
Hollenbach  _et al._ (2020). This bunching effect is clearly visible in Fig.
S4b, while it is less evident in Fig. S4a. We attribute this difference to the
fact that different emitters have different electron trapping rates and other
mesoscopic properties that dictate the bunching.

Figure S4: Second-order autocorrelation measurements. a) HBT measurement
result from the thermally tuned cavity-atom system, and b) zoomed out version
of the gas tuning cavity-atom system shown in Fig. 3d. Both measurements show
clear $g^{(2)}(0)\approx 0$, demonstrating single-photon emission.

###  .8 Tuning of cavity resonances

Here we expand on the methods used to decouple the G-centers from our
cavities. As mentioned in the main text, we performed both thermal and gas
tuning to achieve decoupling.

####  .8.1 Thermal tuning

Starting with a cryostat chamber temperature of $\sim 4$ K, we define the
detuning $\delta_{\mathrm{t}}$ as
$\delta_{\mathrm{t}}=\lambda_{\mathrm{cav}}^{\mathrm{t}}-\lambda_{\mathrm{ZPL}}^{\mathrm{t}}$,
where $\lambda_{\mathrm{ZPL}}^{\mathrm{t}}$ is the G-center ZPL wavelength and
$\lambda_{\mathrm{cav}}^{\mathrm{t}}$ the cavity resonance wavelength. Fitting
both the cavity and ZPL profiles to a Lorentzian function, we find the
wavelengths to be $\lambda_{\mathrm{cav}}^{\mathrm{t}}=(1279.747\pm 0.002)$ nm
and $\lambda_{\mathrm{ZPL}}^{\mathrm{t}}=(1279.850\pm 0.004)$ nm, and thus
calculate $\delta_{\mathrm{t}}=(-0.103\pm 0.004)$ nm. The cavity and ZPL
profiles at $\sim 4$ K are displayed in pink in Fig. S5. The extracted $Q$
factor of our cavity is $3725\pm 50$.

In order to decouple the cavity-atom system, we warmed up the chamber to $\sim
24$ K and observed the subsequent effect. As visible in orange in Fig. S5, the
cavity shifted away spectrally from the ZPL, resulting in a reduction of the
ZPL intensity. The cavity resonance is now found at
$\lambda_{\mathrm{cav}}^{\mathrm{t}^{\prime}}=(1279.057\pm 0.001)$ nm, and a
less significant spectral shift is also observed in the ZPL wavelength, now at
$\lambda_{\mathrm{ZPL}}^{\mathrm{t}^{\prime}}=(1279.781\pm 0.001)$ nm. The
latter effect is in line with what we recently reported in Ref. Prabhu  _et
al._ (2022), where a non-volatile spectral shift in the ZPL of our single
G-centers was observed. After the temperature variation, the detuning is found
to be ${\delta^{\prime}_{\mathrm{t}}}=(-0.724\pm 0.001)$ nm. Fig. 3d in the
main text condenses this information by combining both detuning cases
${\delta_{\mathrm{t}}}$ and ${\delta^{\prime}_{\mathrm{t}}}$ in one plot. To
highlight these detunings between the cavity and ZPL, we shifted the x-axis of
the bottom plot in Fig. S5 by ($\lambda_{\mathrm{ZPL}}^{\mathrm{t}}$ \-
$\lambda_{\mathrm{ZPL}}^{\mathrm{t}^{\prime}}$ ). In this way, the detunings
between the cavity and ZPL are more clearly visible. However, because of this
shift, it should be noted that the plot in Fig. 3d does not reflect the real
difference between the cavity wavelengths before and after tuning. This
information is preserved in Fig. S5.

All spectra were acquired with a grating density of $300$ gr/mm. The cavity
reflectivity measurements were performed with our tunable CW narrowband IR
laser while sweeping its wavelength and recording the cavity spectrum at each
step. All spectra were then merged to obtain the complete cavity profile.

Figure S5: Thermal tuning of a cavity. Decoupling of the cavity-atom system is
achieved by changing the temperature in the cryostat chamber from $\sim 4$ K
to $\sim 24$ K, leading to a detuning of $\delta_{\mathrm{t}}$ (top) and
$\delta_{\mathrm{t}}^{\prime}$ (bottom), respectively. A significant cavity
wavelength shift — from $\lambda_{\mathrm{cav}}^{\mathrm{t}}$ to
${\lambda_{\mathrm{cav}}^{\mathrm{t}^{\prime}}}$ — as well as a much less
evident emitter ZPL shift — from $\lambda_{\mathrm{ZPL}}^{\mathrm{t}}$ to
${\lambda_{\mathrm{ZPL}}^{\mathrm{t}^{\prime}}}$ — are observed.

####  .8.2 Gas tuning

To validate what we observed in the thermal tuning case, we spectrally shifted
the cavity of a second cavity-atom system via a different tuning mechanism
based on gas deposition and subsequent sublimation. In this case, the chamber
temperature remained constant (at $\sim 8$ K) throughout our measurements.
First, we coated our sample with a thin layer of gas (CO2) by injecting the
gas into our cryostat chamber via a dedicated gas line. The gas layer alters
the mode index of the cavity and thus redshifts its wavelength. We assume
first-order perturbation theory and follow the derivation in the SI of Ref.
Panuski  _et al._ (2022). Using a refractive index for solid CO2 of 1.4 Warren
(1986), and a representative mode for the photonic crystal cavity, we estimate
a resonance shift of the order of $10$ nm for an infinite thickness of solid
CO2.

Figure S6: In-situ gas tuning of a cavity. a) Starting from its nominal
wavelength (center of purple curve), the cavity resonance is redshifted by
introducing CO2, and controllably blueshifted by sequentially burning gas
using a $532$ nm CW pump at different powers (indicated in the legend). b)
Decoupling of the emitter from the cavity using this gas tuning method. The
figure shows the ZPL and cavity profiles under different detunings
$\delta_{\mathrm{g}}$ (before gas tuning, top) and
$\delta_{\mathrm{g}}^{\prime}$ (after gas tuning, bottom). A cavity wavelength
shift — from $\lambda_{\mathrm{cav}}^{\mathrm{g}}$ to
${\lambda_{\mathrm{cav}}^{\mathrm{g}^{\prime}}}$ — as well as a ZPL shift —
from $\lambda_{\mathrm{ZPL}}^{\mathrm{g}}$ to
${\lambda_{\mathrm{ZPL}}^{\mathrm{g}^{\prime}}}$ — are observed.

We then performed gradual gas sublimation to achieve a controllable cavity
resonance blueshift. Fig. S6a shows a representative experimental gas shifting
of cavity spectra. Our maximum tuning, in the order of $1$ nm, aligns
reasonably well with our theoretical estimate. Starting from the nominal
cavity resonance wavelength (purple curve), we injected CO2 and thus
redshifted the cavity mode (black curve). We then illuminated the cavity with
a $532$ nm CW pump for $\sim 0.5$ s at increasingly higher powers (displayed
in the figure’s legend), and thus achieved a controllable cavity resonance
shift. We note that our cavity blueshifts further than its initial central
wavelength, indicating that there is likely a significant leakage of CO2 or
other gases during the cooling, which we can sublimate using our optical
pumping technique.

After depositing CO2 on our sample, we acquired the ZPL and cavity spectra. We
find their center wavelengths to be
$\lambda_{\mathrm{ZPL}}^{\mathrm{g}}=(1279.277\pm 0.001)$ nm and
$\lambda_{\mathrm{cav}}^{\mathrm{g}}=(1279.354\pm 0.002)$ nm, leading to a
very small detuning of
$\delta_{\mathrm{g}}=\lambda_{\mathrm{cav}}^{\mathrm{g}}-\lambda_{\mathrm{ZPL}}^{\mathrm{g}}=(0.077\pm
0.002)$ nm as shown in green in Fig. S6b. Unlike the previous case, the cavity
reflectivity measurements shown in Fig. S6b were performed with our
superluminescent diode by simply illuminating the cavity and recording its
reflectivity spectrum. Moreover, these spectra were acquired with a higher
resolution compared to the previous experiments (here we used a grating
density of $900$ gr/mm) which resulted in a slight wavelength offset likely
due to calibration errors. This offset was taken into account when plotting
the data, in order to enable a fair wavelength comparison between the thermal
and gas tuning cases.

While the value of $\lambda_{\mathrm{ZPL}}^{\mathrm{g}}$ was still obtained by
fitting the ZPL profile to a Lorentzian function, the cavity spectrum required
a different analysis due to the presence of parasitic oscillations in its
profile, as visible in Fig. S6b. This behaviour was not observed in the
thermal tuning case because of the lower resolution arising from a smaller
grating density. It is known that these parasitic cavities are Fabry-Pérot
(FP) cavities arising from reflections in the silicon substrate Panuski  _et
al._ (2022). A FP cavity is described by an airy distribution Ismail  _et al._
(2016) as

$f_{\text{FP}}=\frac{A(1-R_{1})^{2}R_{2}}{(1-\sqrt{R_{1}R_{2}})^{2}+4\sqrt{R_{1}R_{2}}\sin^{2}{\phi}},$
(6)

with $A$ an amplitude fitting constant, $\phi=\pi(v-v_{0})/v_{\text{FSR}}$,
$v-v_{0}$ the optical frequency detuning, $v_{\text{FSR}}=c/2L$ the cavity
free spectral range with $c$ the speed of light and $L$ the cavity length, and
$R_{1}$ and $R_{2}$ the mirror reflectivities.

The photonic crystal cavity resonance is described by a Lorentzian as follows

$f_{\text{L}}=\frac{B}{(\lambda-\lambda_{0})^{2}+(\Gamma/2)^{2}}+C,$ (7)

with $B$ and $C$ being fitting constants for the amplitude and offset,
$\lambda$ and $\lambda_{0}$ the wavelength and wavelength offset, and $\Gamma$
the full width at half maximum (FWHM). We estimate the effect of the coupled
system by using a product function

$f_{\text{tot}}=f_{\text{FP}}f_{\text{L}}.$ (8)

We extract a cavity Q factor of $2136\pm 30$.

Figure S7: Cavity resonance fit. The combined Airy-Lorentzian (red) fits well
our cavity reflectivity measurements (blue stars), and can be decomposed in a
Fabry-Pérot cavity (orange dashed) and a Lorentzian (green dashed).

To decouple our cavity-atom system, we illuminated a region in the vicinity of
our cavity with $532$ nm CW laser light with powers up to 500 $\upmu$W. This
resulted in a shift of our cavity resonance wavelength from
$\lambda_{\mathrm{cav}}^{\mathrm{g}}$ to
$\lambda_{\mathrm{cav}}^{\mathrm{g}^{\prime}}=(1278.976\pm 0.001)$ nm. We note
that also in this case the optical pumping required for gas tuning introduces
a non-volatile spectral shift in the ZPL of our single emitters, as reported
in Ref. Prabhu  _et al._ (2022). The ZPL wavelength reads now
$\lambda_{\mathrm{ZPL}}^{\mathrm{g}^{\prime}}=(1279.4587\pm 0.0003)$ nm. A
detuning of ${\delta^{\prime}_{\mathrm{g}}}=(-0.483\pm 0.001)$ nm is therefore
achieved. These results are shown in blue in Fig. S6b. Considerations
analogous to how the plot in Fig. 3d is realized (see previous subsection)
hold for Fig. 3e as well.

###  .9 Derivation of quantum efficiency

In the following, we derive the relevant rates from first principles and use
them to estimate the G-center quantum efficiency of our system. We assume that
1) the cavity detuning does not significantly affect the Purcell factor of
non-ZPL radiative phenomena (part of $\gamma_{0}$), and that 2) there is a
negligible contribution of other radiative modes in our measurements due to
our $0.1$ nm narrow filtering of the ZPL. Under these assumptions, we write
the total emission rate $1/\tau$ and the collected photon flux $\phi$ for both
the on- and off-resonance cases as

$\displaystyle\quad\frac{1}{\tau_{\text{on}}}=\gamma_{\text{R}}+F_{\text{P}}\gamma_{\text{R}}+\gamma_{\text{0}},$
(9)
$\displaystyle\quad\frac{1}{\tau_{\text{off}}}=\gamma_{\text{R}}+\gamma_{\text{0}},$
(10)
$\displaystyle\quad\phi_{\text{on}}=\eta(\gamma_{\text{R}}+F_{\text{P}}\gamma_{\text{R}}),$
(11) $\displaystyle\quad\phi_{\text{off}}=\eta\gamma_{\text{R}},$ (12)

with $F_{\text{P}}$ the Purcell factor and $\eta=0.7$ the coupling efficiency
into the wanted mode extracted from our simulations in Fig. 2. The quantum
efficiency QE is defined as

$\text{QE}=\frac{\gamma_{\text{R}}}{\gamma_{\text{R}}+\gamma_{\text{0}}}\frac{1}{F_{\text{DW}}},$
(13)

where $F_{\text{DW}}$ is the Debye-Waller factor, i.e. the fraction of the PL
intensity emitted in the ZPL, and is extracted from the literature. Extracting
$\gamma_{0}$ from the equation above and substituting it in the ratio between
Eqs. 9 and 10, that is $\tau_{\text{off}}/\tau_{\text{on}}$, we find an
expression for $\tau_{\text{off}}/\tau_{\text{on}}$ as a function of QE:

$\frac{\tau_{\text{off}}}{\tau_{\text{on}}}=1+F_{\text{P}}F_{\text{DW}}\text{QE}.$
(14)

Following the procedure in Ref. Lefaucher  _et al._ (2023), given that in our
measurements the lifetimes remain constant within the error, we can define a
bound on our QE by setting a threshold for the detection of the longest
lifetime allowed by our off-resonance standard deviation $\tau_{\text{off-
th}}=\tau_{\text{off}}-3\sigma_{\text{off}}$. As the on- and off-resonance
lifetimes do not differ significantly, it holds that
$\tau_{\text{on}}>\tau_{\text{off-th}}$. Therefore, we can derive an upper
bound for the QE:

$\text{QE}<\frac{\frac{\tau_{\text{off}}}{\tau_{\text{off-
th}}}-1}{F_{\text{P}}F_{\text{DW}}}.$ (15)

To calculate QE from Eq. 15, an estimate of the Purcell factor $F_{\text{P}}$
is needed. We can derive $F_{\text{P}}$ from the ratio between the photon
fluxes $\phi_{\text{on}}$ and $\phi_{\text{off}}$ in Eqs. 11 and 12:

$\frac{\phi_{\text{on}}}{\phi_{\text{off}}}=(1+F_{\text{P}}),$ (16)

and thus find

$F_{\text{P}}=\frac{\phi_{\text{on}}}{\phi_{\text{off}}}-1.$ (17)

Deriving the Purcell factor $F_{\text{P}}$ from Eq. 17 using our measured
photon fluxes and substituting it in 15, we can estimate the QE to be bounded
by $\sim 18\%$ for a measured off-resonance lifetime value of
$\tau_{\text{off}}=(6.09\pm 0.25)$ ns.

###  .10 Comparison to the G-center literature

Table 1 summarizes the reported literature on single emitters in silicon
identified as G-centers in terms of their ZPL, estimated QE, $g^{(2)}(0)$, and
excited state lifetimes. We include a recent report on cavity-coupled
ensembles Lefaucher  _et al._ (2023) and the simultaneous report on cavity-
enhancement of singles Redjem  _et al._ (2023), and use these to add cavity-
enhanced excited state lifetime and $Q/V$ to the comparison. What reported in
Table 1 aligns with the hypothesis, originally brought forward by Ref. Baron
_et al._ (2022), that the community is likely studying two different defects.
The first one, which we label $G_{a}$ Hollenbach  _et al._ (2020); Baron  _et
al._ (2022); Hollenbach  _et al._ (2022); Prabhu  _et al._ (2022); Komza  _et
al._ (2022); Lefaucher  _et al._ (2023), aligns with the original G-center
ensembles, and features a ZPL at around $1278$ nm with a narrow inhomogeneous
linewidth, a lower QE, and a short excited state lifetime. The second one,
$G_{b}$ Redjem  _et al._ (2020, 2023), features a blue-shifted ZPL at $1270$
nm with a high QE and a longer excited-state lifetime.

Although our results suggest the possibility of two different artificial atom
systems, the reported differences may still be due to a different host
material, measurement setups, or fabrication protocols.

Reference | ZPL | QE | $g^{(2)}(0)$ | $\tau$ | $\tau^{\mathrm{cav}}$ | $Q/V_{\mathrm{cav}}$ | Notes
---|---|---|---|---|---|---|---
| (nm) | ($\%$) |  | (ns) | (ns) | $(\lambda/n)^{-3}$ |
Redjem et al. Redjem  _et al._ (2020) | $1270.0\pm 9.1$ |  $\sim 50$* | 0.3 | $35.8\pm 0.2$ |  |  | 29 emitters
Hollenbach et al. Hollenbach  _et al._ (2020) | $1278.3\pm 0.1$ |  | $0.07\pm 0.04^{\dagger}$ | $3.8$ |  |  | 12 µm SOI, 12 emitters
Baron et al. Baron  _et al._ (2022) | $1279$ |  | $\sim 0.3$ | $4.5$ |  |  | 1 emitter measured
Hollenbach et al. Hollenbach  _et al._ (2022) | $1278$ |  | $0.36\pm 0.06$ | $10.0$ |  |  | 1 emitter measured
Prabhu et al. Prabhu  _et al._ (2022) | $1278.7\pm 1.1$ | >1* | $0.38\pm 0.08$ | $8.3\pm 0.7$ |  |  | Waveguide, 37 emitters
Komza et al. Komza  _et al._ (2022) | $1278$ | >2* | $0.15\pm 0.02$ | $4.6$ |  |  | Waveguide, 1 emitter
Redjem et al. Redjem  _et al._ (2023) | $1275$ |  | $0.30\pm 0.07$ | $33.3$ | $6.7$ | 4862 | L3 cavity, 1 emitter
Lefaucher et al. Lefaucher  _et al._ (2023) | $1279$ | <10** |  | $5.6\pm 0.1$ | $5.6\pm 0.1$ | 417 | Ensembles, static rings
This work | $1279$ | <18 | $0.03\pm\begin{subarray}{c}0.07\\\ 0.03\end{subarray}$ | $6.1\pm 0.3$ | $5.50\pm 0.05$ | $>3725$ | Opt. L3 cavity, 1 emitter
Table 1: Comparison of measured properties for the reported G-centers in the
literature. All the listed references other than Lefaucher et al. Lefaucher
_et al._ (2023) report single-photon emitters. If several emitters are listed
in Notes, the reported error is the statistical standard deviation. Otherwise
it is the measurement/fit error. *Estimated from photon counts and setup loss.
**Extracted from ensembles and static cavities. $\dagger$ Background
corrected.

*