††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Hopping of the center-of-mass of single G centers in silicon-on-insulator

Alrik Durand  Laboratoire Charles Coulomb, Université de Montpellier and CNRS,
34095 Montpellier, France  Yoann Baron  Laboratoire Charles Coulomb,
Université de Montpellier and CNRS, 34095 Montpellier, France  Péter
Udvarhelyi  Wigner Research Centre for Physics, P.O. Box 49, H-1525 Budapest,
Hungary  Department of Atomic Physics, Institute of Physics, Budapest
University of Technology and Economics, Műegyetem rakpart 3., H-1111,
Budapest, Hungary  Félix Cache  Laboratoire Charles Coulomb, Université de
Montpellier and CNRS, 34095 Montpellier, France  Krithika V.R  Laboratoire
Charles Coulomb, Université de Montpellier and CNRS, 34095 Montpellier, France
Tobias Herzig  Division of Applied Quantum Systems, Felix-Bloch Institute for
Solid-State Physics, University Leipzig, Linnéestraße 5, 04103 Leipzig,
Germany  Mario Khoury  CNRS, Aix-Marseille Université, Centrale Marseille,
IM2NP, UMR 7334, Campus de St. Jérôme, 13397 Marseille, France  Sébastien
Pezzagna  Division of Applied Quantum Systems, Felix-Bloch Institute for
Solid-State Physics, University Leipzig, Linnéestraße 5, 04103 Leipzig,
Germany  Jan Meijer  Division of Applied Quantum Systems, Felix-Bloch
Institute for Solid-State Physics, University Leipzig, Linnéestraße 5, 04103
Leipzig, Germany  Jean-Michel Hartmann  Univ. Grenoble Alpes, CEA, LETI,
F-38000 Grenoble, France  Shay Reboh  Univ. Grenoble Alpes, CEA, LETI, F-38000
Grenoble, France  Marco Abbarchi  CNRS, Aix-Marseille Université, Centrale
Marseille, IM2NP, UMR 7334, Campus de St. Jérôme, 13397 Marseille, France
Solnil, 95 Rue de la République, 13002 Marseille, France  Isabelle Robert-
Philip  Laboratoire Charles Coulomb, Université de Montpellier and CNRS, 34095
Montpellier, France  Adam Gali  Wigner Research Centre for Physics, P.O. Box
49, H-1525 Budapest, Hungary  Department of Atomic Physics, Institute of
Physics, Budapest University of Technology and Economics, Műegyetem rakpart
3., H-1111, Budapest, Hungary  MTA-WFK Lendület “Momentum” Semiconductor
Nanostructures Research Group, Budapest, Hungary  Jean-Michel Gérard  Univ.
Grenoble Alpes, CEA, Grenoble INP, IRIG, PHELIQS, 38000 Grenoble, France
Vincent Jacques  Laboratoire Charles Coulomb, Université de Montpellier and
CNRS, 34095 Montpellier, France  Guillaume Cassabois  Laboratoire Charles
Coulomb, Université de Montpellier and CNRS, 34095 Montpellier, France
Institut Universitaire de France, 75231 Paris, France.  Anaïs Dréau
<EMAIL_ADDRESS>Laboratoire Charles Coulomb, Université de
Montpellier and CNRS, 34095 Montpellier, France<EMAIL_ADDRESS>
###### Abstract

Among the wealth of single fluorescent defects recently detected in silicon,
the G center catches interest for its telecom single-photon emission that
could be coupled to a metastable electron spin triplet. The G center is a
unique defect where the standard Born-Oppenheimer approximation breaks down as
one of its atoms can move between 6 lattice sites under optical excitation.
The impact of this atomic reconfiguration on the photoluminescence properties
of G centers is still largely unknown, especially in silicon-on-insulator
(SOI) samples. Here, we investigate the displacement of the center-of-mass of
the G center in silicon. We show that single G defects in SOI exhibit a
multipolar emission and zero-phonon line fine structures with splittings up to
$\sim 1$ meV, both indicating a motion of the defect central atom over time.
Combining polarization and spectral analysis at the single-photon level, we
evidence that the reconfiguration dynamics are drastically different from the
one of the unperturbed G center in bulk silicon. The SOI structure freezes the
delocalization of the G defect center-of-mass and as a result, enables to
isolate linearly polarized optical lines. Under above-bandgap optical
excitation, the central atom of G centers in SOI behaves as if it were in a
6-slot roulette wheel, randomly alternating between localized crystal sites at
each optical cycle. Comparative measurements in a bulk silicon sample and ab
initio calculations highlight that strain is likely the dominant perturbation
impacting the G center geometry. These results shed light on the importance of
the atomic reconfiguration dynamics to understand and control the
photoluminescence properties of the G center in silicon. More generally, these
findings emphasize the impact of strain fluctuations inherent to SOI wafers
for future quantum integrated photonics applications based on color centers in
silicon.

##  I Introduction

Color centers in silicon have recently received renewed attention as they
could offer a new building block for quantum technologies [1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26,
27, 28, 29, 30, 31]. Within a few years, single photon emission has been
demonstrated on a dozen of different types of fluorescent defects in silicon
[3, 5, 7, 29, 9, 16, 17, 26]. One of the next challenges is to provide single
spin qubits interfaced with telecom light and integrated into silicon-on-
insulator (SOI) photonic nanostructures [9]. Therefore, strong emphasis has
been placed on color centers with non-zero electron spin that could be
optically initialized and readout, such as the T center [8, 9, 10, 11, 12, 13,
14], erbium dopants [24, 25, 26] or the G center [32, 1, 2, 15, 16, 17, 18,
19, 20, 21, 22, 23]. This last defect is the brightest among the three and it
can be isolated at single-defect scale in unstructured SOI wafers [16].

The G center in silicon is a fluorescent carbon-based complex studied for
several decades [33, 34, 35], and commonly used as a non-destructive probe of
the carbon concentration [36]. Besides an optical emission in the telecom
O-band, the G center features a metastable electron spin triplet whose
magnetic resonances have been measured optically on ensembles [32, 37]. Single
isolated G centers can be created by co-implantation of carbon ions and
protons [16] or by Si-focused ion beam implantation in carbon-implanted
silicon with a spatial resolution of 50 nm [17]. Purcell-enhanced emission has
recently been demonstrated for ensembles of G centers [19] and single G
centers [21] in photonic cavity, as well as Hong-Ou-Mandel interferences using
an individual G defect integrated in a silicon waveguide [18].

Despite these advances, our knowledge on the photophysics of the G center in
silicon remains limited. Unlike usual solid-state systems, this defect cannot
be described under the standard Born-Oppenheimer approximation since its
microscopic structure allows its center-of-mass to move [15, 32, 35, 36, 38].
How this structural reconfiguration impacts the optical properties of the G
center in SOI is very little known, which hinders the development of potential
quantum applications.

Figure 1: The G center in silicon in the unperturbed case. (a) Microscopic
structure of the defect. Two substitutional carbon atoms C (black) connected
by an interstitial silicon atom Si(i)(purple) are aligned along the $\langle
111\rangle$ crystal axis. (b) In the $\langle 111\rangle$ direction, the G
center structure showing the 6 possible equivalent sites for the Si(i). (c) In
gray, sketch of the energy potential of the 6 identical coupled wells
experienced by the Si(i) along its reorientation path around the C-C axis.
This potential is similar in both the electronic ground and excited states of
the G center, albeit with different barrier heights [15]. In colors, density
distributions for the 6 rotational states of the defect center-of-mass, with
level degeneracy indicated on the right. The rotational energy splittings are
shifted for clarity and should not be compared to the potential barrier
energy. (d) Quartet energy level structure of the 6 delocalized states of the
center-of-mass of the G center. Energies are offset by $E_{0}$ (see main
text). (e) Theoretical zero-phonon line spectrum of the unperturbed G center
in both absorption and photoluminescence. $\Delta$ is the difference between
the quartet energy splittings in the ground and excited states. Each emission
line corresponds to optical transitions between rotational states where the
Si(i) is equally delocalized over the 6 sites.

In this work, we explore the motion of the center-of-mass of the G center in
silicon. We start by introducing the atomic configuration of the unperturbed
defect and the energy level fine structure resulting from the rotational
delocalization of its center-of-mass. Single G centers in SOI are then
investigated through low-temperature photoluminescence (PL) experiments. Their
single-photon polarization is first analyzed to reveal multipolar emission. We
then examine the zero-phonon line spectra of these individual defects and show
a fine structure with line splittings exceeding by two orders of magnitude the
ones of unperturbed G centers. By combining spectral and polarization
analysis, we demonstrate that the center-of-mass of these G centers is hopping
over time between localized positions, like a 6-slot roulette wheel. Finally,
performing ab initio calculations and spectral measurements in a bulk silicon
sample enables to evidence strain effects as the dominant perturbation
freezing the rotational delocalization of single G centers in SOI structures.
In conclusion, we discuss the impact of these findings on the control of the
quantum properties of single G centers in SOI platforms.

##  II Unperturbed G center in silicon

The microscopic structure of the luminescent form of the G center in silicon
was proposed in the 1980s [37, 35, 39] and recently confirmed by advanced ab
initio calculations [15]. This atomic configuration is displayed in Figure
1(a). It consists of 2 substitutional carbon atoms aligned along the $\langle
111\rangle$ crystal direction and connected by an interstitial silicon atom
Si(i). This defect configuration corresponds to a $C_{1h}$ symmetry, with a
plane of symmetry defined by these 3 atoms [15]. Unlike the tetravalent carbon
atoms, the Si(i) forms only two covalent bonds with its neighboring atoms and
is therefore less rigidly attached to the matrix. Indeed, recent density
functional theory (DFT) simulations have shown that this silicon atom does
rotate around the C-C axis [15]. More precisely, in both ground and excited
states, the interstitial silicon atom is delocalized by tunnel effect between
the 6 equivalent sites indicated in Figure 1(b) [15].

Due to the displacement of its center-of-mass, the standard Born-Oppenheimer
approximation breaks down for the G center in silicon. Nevertheless, this
rotational degree of freedom can be introduced in a simple tight-binding model
to extract the defect eigenstates and associated energies, as proposed in Ref.
[15]. Both in ground and excited states, the electronic wavefunction of the G
defect can be written as the product of its center-of-mass wavefunction
$\varphi(\bm{R}_{{}_{CM}})$ by the wavefunction of the electron cloud relative
to the center-of-mass $\phi(\bm{r}-\bm{R}_{{}_{CM}})$:

$\psi(\bm{r})=\varphi(\bm{R}_{{}_{CM}})\cdot\phi(\bm{r}-\bm{R}_{{}_{CM}}).$
(1)

To simplify notations, we omit the indices ${GS}$ and ${ES}$ referring to the
ground and excited states, respectively. The DFT calculated potential energy
surface (PES) along the reorientation path of the Si(i) can be approximated by
a 6-period cosine function in both ground and excited states, albeit with
different barrier amplitudes [15]. Due to the tunnel coupling between adjacent
sites, the silicon atom thus behaves like a quasiparticle in a potential of 6
identical coupled wells (Fig. 1(c)). Using the Bloch theorem, the 6 rotational
eigenstates of the defect center-of-mass can be expressed in Dirac notation
as:

$|\varphi^{(m)}\rangle=\frac{1}{\sqrt{6}}\sum_{n=0}^{5}e^{ik_{m}na}|n\rangle,$
(2)

where $m\in\llbracket 0,5\rrbracket$, $k_{m}$ is the wavevector of the Bloch
function, $a$ is the period of the potential and $|n\rangle$ the wavefunction
describing the Si(i) atom localized in the site $n$. From this equation, we
directly get that the Si(i) is equally delocalized on the 6 possible sites for
each eigenstate of the G center (Fig. 1(c)). The wavevector expression is
derived from the Born-von Karman boundary conditions ($e^{ik_{m}6a}=1$):

$k_{m}=\frac{m\pi}{3a}.$ (3)

The time-independent Schrödinger equation describing the stationary states of
the defect center-of-mass is
$\mathcal{H}|\varphi^{(m)}\rangle=E^{(m)}|\varphi^{(m)}\rangle$. Considering
$E_{0}=\langle n|\mathcal{H}|n\rangle$ the energy of each uncoupled site, and
$\delta_{0}=\langle n|\mathcal{H}|n\pm 1\rangle$ the tunneling coupling energy
between adjacent sites, the energy of the eigenstate $|\varphi^{(m)}\rangle$
writes as:

$E^{(m)}=E_{0}+2\delta_{0}\cos{(k_{m}a)}.$ (4)

Combining Eqs. (4) and (3), we finally obtain the formula for the 6 eigen-
energies of the defect center-of-mass:

$E^{(m)}=E_{0}+2\delta_{0}\cos{\left(m\frac{\pi}{3}\right)}.$ (5)

The rotational energy levels of the G center, in both ground and excited
states, are thus arranged in a quartet fine structure with degeneracy 1:2:2:1
and splittings $\delta_{0}$-$2\delta_{0}$-$\delta_{0}$, as represented in
Figure 1(d). However, due to a higher energy barrier for the Si(i) rotation,
the tunneling coupling energy is much smaller in the ground state [15]. As a
result, the quartet energy splitting in this level is negligible compared to
the excited one: $\delta_{0,_{GS}}\ll\delta_{0,_{ES}}$. We note that the same
quartet structure is found for the electronic states of the benzene molecule
[15].

In the Franck-Condon approximation, the selection rules for optical dipolar
transitions between these levels are given by:

$\langle\psi_{GS}|-\bm{d}\cdot\bm{E}|\psi_{ES}\rangle=\langle\varphi_{GS}^{(m)}|\varphi_{ES}^{(m^{\prime})}\rangle\cdot\langle\phi_{GS}|-\bm{d}\cdot\bm{E}|\phi_{ES}\rangle.$
(6)

Since
$\langle\varphi_{GS}^{(m)}|\varphi_{ES}^{(m^{\prime})}\rangle=\delta_{m,m^{\prime}}$
the Kronecker symbol, the rotational quantum number $m$ must be preserved
during the optical transition. Hence optical transitions are only allowed
between the same rotational eigenstates. Let $\Delta$ be the difference
between the tunneling coupling energy in the ground and excited states:
$\Delta=|\delta_{0,_{ES}}-\delta_{0,_{GS}}|$. From the optical selection
rules, it results that a quartet fine structure with relative line intensities
1:2:2:1 and splittings $\Delta$-$2\Delta$-$\Delta$ should be observed in
absorption and emission (Fig. 1(e)). Indeed, such quartet spectra have been
measured on an ensemble of G centers in isotopically purified bulk 28Si [2].
In both absorption and PL spectra, the energy splitting is
$\Delta^{(exp)}=2.5\pm 0.2\,\mu$eV$=0.60\pm 0.05$ GHz. However, neither this
quartet structure nor any other fine structure has been reported for G centers
at single-defect scale.

Figure 2: The G center in silicon. (a) Optical scan at 30 K of the SOI sample
co-implanted with carbon ions and protons [16]. Hotspots are isolated G
centers. (b) Low-resolution PL spectrum of a single G center showing the zero-
phonon line at 1279 nm and the E-line associated with the defect local-
vibrational mode. (c) Second-order autocorrelation function demonstrating
single-photon emission with an antibunching at zero delay $g^{(2)}(0)\simeq
0.17<1/2$. Figure 3: Polarization emission diagrams of single G centers in
silicon. (a, b) Typical polarization emission diagrams recorded on two single
G centers and corrected from background counts (see main text). The diagram
visibility $V$ and orientation angle $\phi$ are extracted from data fitting
(solid lines) with a function $V\cdot\cos^{2}(\theta-\phi)+1-V$. (c, d)
Histograms of the distribution of the angle $\phi$ and visibility $V$ measured
on 39 G centers, respectively. (e) Orientations of the emission dipoles of a
[111]-oriented G center depending on the position of the Si(i): dipoles along
$[1\bar{1}0]$, $[01\bar{1}]$ and $[\bar{1}01]$ for the Si(i) positions
$\\{0,3\\}$, $\\{1,4\\}$ and $\\{2,5\\}$, respectively. (f) Polarization
emission diagrams (dashed lines) for each of the 3 dipoles considering photon
collection from the (001)-top surface. Their amplitudes correspond to the
relative ratio of PL collected intensity (see main text). The indigo thick
line represents the total polarization diagram resulting from the contribution
of the three dipoles (Appendix A).

##  III Single G centers in SOI

The silicon sample used here is the SOI wafer from Ref. [16]. Its top silicon
layer is made of 54 nm of isotopically pure 28Si grown at 650∘, 20 Torr with
28SiH4 in a reduced pressure-chemical vapor deposition reactor, on top of 4 nm
of silicon with natural isotope abundance. To create G centers, it was locally
co-implanted with carbon ions and protons at 8keV and 6keV energies,
respectively, using several fluences $\geq 3\cdot 10^{10}$ cm-2 [16]. The
experimental setup is a home-made confocal microscope built in a He-closed
cycle cryostat, equipped with superconducting single-photon detectors (more
details in Ref. [3, 23]). We perform above-bandgap optical excitation using a
532-nm continuous laser and select sample areas in which well separated single
G centers can be optically addressed. For practical reasons, all measurements
have been performed at 30 K.

Optical raster scans of the SOI silicon sample with 28Si on top show well-
isolated light spots (Fig. 2(a)). Their PL spectra enable an unambiguous
identification of genuine G centers due to the presence of a zero-phonon line
(ZPL) at 1278 nm and the E-line at 1382 nm (Fig. 2(b)) [23]. The single photon
emission is confirmed by the observation of antibunching at zero delay in the
autocorrelation function $g^{(2)}(\tau)$ of the PL signal (Fig. 2(c)).

Figure 4:  Fine structure of single G centers in SOI. (a) High-resolution ZPL
spectra recorded on single G centers. (b)[(c)] From top to bottom, ZPL
spectrum measured on the defect G-0 [G-1] with a polariser selecting the
polarization perpendicular and parallel to the main axis of the defect
polarization diagram respectively, as indicated in the top left corners. Note
that the orientation of these 2 defects in the crystal is different:
$[\bar{1}11]$ or $[1\bar{1}1]$ for G-0, and $[111]$ or $[\bar{1}\bar{1}1]$ for
G-1. The spectra with no polarization filtering from (a) are reproduced in
dashed lines; the other PL spectra are normalized to its maximum. Colored
areas represent the 0.3-meV integration range used in Figure 5(a,b).

##  IV Analysis of the single-photon polarization

###  IV.1 Measurements

Analysis of the polarization of single photons from G centers reveals a
luminescence mechanism more complex than single dipole emission. Emission
polarization diagrams are acquired on single G defects by rotating the
polarizer axis in the detection path. PL photons are collected perpendicularly
to the (001) top surface of the 28SOI sample. To retain only the G center PL,
diagrams are corrected from unpolarized background contribution measured at a
single point close to defects. The first observation from Figure 3(a-b) is
that the polarization emission diagrams of single G centers are strongly
anisotropic. For some G defects, the PL signal is maximum at 90∘,
corresponding to the crystal axis $[1\bar{1}0]$ (Fig. 3(a)). In contrast for
others, a PL maximum is detected at 0∘, i.e. towards the $[110]$ axis (Fig.
3(b)). A statistical analysis of 39 G centers in 28SOI shows that only these
two directions are permitted, with 50/50 probability within statistical
fluctuations (Fig. 3(c)). Another information from these diagrams is that no
polarizer angle can cancel the PL signal from single G centers, as all
diagrams show a visibility $\leq 68\%$ (Fig. 3(d)). The average visibility
over the set is 58%, with a standard deviation of 7%. The polarization
diagrams of single G centers cannot be explained by the presence of a single
emission dipole. As a consequence, the G center luminescence originates from
multiple dipoles.

###  IV.2 Interpretation

To understand the multipolar emission of the G center, we need to return to
its microscopic structure. The interstitial silicon atom can take six
positions in the plane perpendicular to the $\langle 111\rangle$ defect main
axis, as shown in Figure 1(b). For a given static position of Si(i), symmetry
point group theory for $C_{1h}$ symmetry authorizes a single emission dipole
perpendicular to the symmetry plane [15]. For a $[111]$-oriented G center,
this leads to 3 pairs of possible dipoles aligned along the $[1\bar{1}0]$,
$[01\bar{1}]$ and $[\bar{1}01]$ axes for the $\\{0,3\\}$, $\\{1,4\\}$ and
$\\{2,5\\}$ Si(i) positions, respectively (Fig. 3(e)). When projected onto the
(001) sample top surface from which PL is collected, these dipoles produce
polarization diagrams oriented at angles of $90^{\circ}$, $45^{\circ}$ and
$135^{\circ}$ respectively, each with a visibility of 100%. None of these
dipoles taken alone can explain the single photon polarization observed from
individual G centers (see Fig. 3(a-d)). Numerical simulations with the finite-
difference time-domain (FDTD) method show that the PL collected for the
$[1\bar{1}0]$-dipole, that is parallel to the (001) top surface, is between
2.03 and 2.17 times more intense than that of the other two dipoles, depending
on the defect depth in the Si layer (Appendix A). Considering these three
emission dipoles as independent and equally probable, we obtain the
polarization diagram of Figure 3(f), pointing towards $[1\bar{1}0]$ and with a
visibility varying between 67.0% and 68.5% depending on depth. The same
polarization diagram would also be obtained for a $[\bar{1}\bar{1}1]$-oriented
G center, while G centers aligned along $[\bar{1}11]$ or $[1\bar{1}1]$ would
give diagrams turned by $90^{\circ}$. This model is in very good agreement
with the experimental data of Figures 3(a-d), allowing to conclude that the
interstitial silicon atom of these single G centers does move over time during
the measurements. Note that at this stage, it is not possible to deduce
whether or not these G defects are in the unperturbed geometry described in
Section II. Since the electron density is equally distributed over the 6 sites
for all energy levels, each of the 3 dipoles has the same probability to emit
and the polarization diagram should be identical to the one from Figure 3(f).
To gain more information about their center-of-mass rotation, we need to
examine the fine structure of these individual G centers.

##  V ZPL Fine structure

###  V.1 Measurements

High-resolution PL spectra recorded on individual G defects show a fine
structure radically different from the one presented in Figure 1(e). Figure
4(a) displays the ZPL spectra recorded on 7 individual G centers in 28SOI for
which antibunching with $g^{(2)}(0)<1/2$ has been measured with no spectral
filtering. Except for one defect (G-5), all G centers show at least two or
three ZPL lines. Since single defects are investigated here, these multiple
lines are evidence of a fine structure of the G center. This effect is to be
distinguished from a lifting of orientational degeneracy under stress between
non-equivalent defect orientations in ensemble of G centers [40, 33, 41].
These fine structures cannot be explained by a static configuration with no
Si(i) displacement, as the associated $C_{1h}$ monoclinic point group symmetry
would only allow a single transition between $A^{\prime}$ and
$A^{\prime\prime}$ singlet orbital levels [15]. The disparity in pattern and
line positions observed between defects indicates an extrinsic origin,
resulting from different local environments. Importantly, the quartet fine
structure reported on unperturbed G centers has a characteristic splitting of
10 $\mu$eV, more than two orders of magnitude smaller than in Figure 4(a) [2].
Due to the spectral resolution of our spectrometer ($\simeq 150\mu$eV), such
quartet would appear as a single emission line in the spectra. The fine
structures with meV-line splittings in Figure 4(a) rather indicate a
significant perturbation of single G centers induced by their local
environment.

Figure 5: Signature of the localization of Si(i) in single G centers. (a, b)
Evolution vs polarizer angle of the ZPL intensity for defects G-0 and G-1,
respectively (see Fig.4(b, c)). Solid lines are models considering the
emission lines result from the Si(i) sites indicated in the hexagons on the
right. (c) In gray, calculated potential energy surface (PES) along the
reorientation path of the interstitial silicon atom for a G center under 0.1%
$[1\bar{1}0]$-uniaxial strain. In colors, numerical solutions for the
resulting rotational states with level degeneracy indicated on the right. The
rotational energy splitting is increased for clarity and should not be
compared to the energy scale on the PES.

PL spectra of individual G centers in SOI strongly change with polarization
filtering. We focus on the single G-0 and G-1 defects that have both well-
resolved fine structures. The G-0 center shows a doublet fine structure with
an energy separation of $0.70\pm 0.02$ meV. As shown on Figure 4(b), when the
polarizer is perpendicular to the main axis of the defect emission diagram,
the highest-energy line $L_{0}$ is completely extinguished. When the polarizer
is parallel, the $L_{0}$ line is similar to the previous spectrum with no
polarization filtering. On the contrary, the second line $L_{1}$ has the same
intensity for both polarizer angles, corresponding to roughly one half
compared to the unfiltered spectrum. Similar behavior is observed on the
triplet fine structure of the G-1 defect whose splittings from $L_{0}$ to
$L_{2}$ are $1.00\pm 0.02$ meV and $0.86\pm 0.02$ meV (Fig. 4(c)). The
strongest line $L_{0}$ is globally unchanged for the polarizer parallel to the
orientation of the G-1 diagram and cancels out for the perpendicular position.
The intensity of the other two lines $L_{1}$ and $L_{2}$ is roughly decreased
by a factor of 2 compared to the unfiltered spectrum in both cases.

The intensity of the different ZPLs is measured as a function of the polarizer
angle for G-0 and G-1 centers in Figure 5(a,b). For both centers, we observe
that the $L_{0}$ line is linearly polarized as its intensity follows a
sinusoidal modulation with a visibility close to unity: $90\pm 10\%$ and
$94\pm 5\%$ for G-0 and G-1, respectively. In contrast, the other lines are
non-polarized because their intensities stay globally constant as a function
of polarizer angle. Furthermore, the intensity ratios when the $L_{0}$ line is
at maximum are roughly 2:1 and 4:1:1, for G-0 and G-1 respectively.

###  V.2 Interpretation

The ZPL pattern evolution with polarization shows that the G defect
environment lifts the quasi-degeneracy between the 3 emission dipoles of the
unperturbed case. The $L_{0}$ signal variation is not compatible with the
multipolar emission described above and results instead from a single dipole.
Since the $L_{0}$ intensity reaches a maximum at $0^{\circ}$ for G-0
[$90^{\circ}$ for G-1], we can associate it to the $[110]$-dipole
[$[1\bar{1}0]$-dipole for G-1]. According to Figure 3(f), such an emission can
only take place if the Si(i) is localized in one of the 2 interstitial sites
$0$ or $3$ (Fig. 5(a,b)). To produce the $L_{1}$ and $L_{2}$ lines, the only
possible explanation is an equal mixture of dipoles [101] and [011] for the
center G-0 [dipoles $[\bar{1}01]$ and $[01\bar{1}]$ for G-1]. For the G-0
doublet, this implies that the $L_{1}$ line corresponds to the 4 positions
$\\{1,2,4,5\\}$ of the interstitial silicon atom (Fig. 3(e), 5(a)). Assuming
that all Si(i) sites are equally occupied over the time of the acquisition,
the relative intensity of $L_{1}$ after the polarizer should be $\simeq 1/2.1$
of the maximum intensity of $L_{0}$ according to FDTD simulations (Appendix
A). Within error bars, these values well match the experimental data for G-0
displaying maximal intensity ratio of 2:1, as shown on Figure 5(a). Following
the same reasoning for defect G-1, possible Si(i) position configurations for
the lines $L_{2}$ and $L_{1}$ would be $\\{1,2\\}$ & $\\{4,5\\}$ or
$\\{1,5\\}$ & $\\{2,4\\}$ (Fig. 5(b)). Furthermore, their relative intensity
should be half that of the line $L_{1}$ of G-0 associated to twice more sites.
This interpretation is in good agreement with the experimental data for G-1
showing maximal intensity ratios of roughly 4:1:1 (Fig. 5(b)). We note that
the G-3 center that also has a triplet fine structure shows the same
polarization behavior as G-1 (see Appendix B). It is therefore very likely
that these 2 defects experience a similar perturbation at first order.

The ZPL evolution with polarization establishes that the defect center-of-mass
is not in delocalized rotational states, as expected for unperturbed G centers
(Sec. II). Instead, these G defects experience an external coupling distorting
their original symmetry and freezing the rotational dynamics of their center-
of-mass. Partial delocalization of Si(i) could still happen between adjacent
sites (i.e. for $\\{1,2\\}$ or $\\{4,5\\}$), but only if their energy
difference is on the order of or smaller than the tunneling coupling energy.
Given the energy scale at stake here compared to $\delta_{0,_{ES}}\simeq
2.5\mu$eV (see Sec. II), we can consider this tunneling effect to be unlikely
for the single G centers under study. Consequently, the center-of-mass
eigenstates in both ground and excited states are the wavefunctions localized
on each of the 6 Si(i) sites, and not the Bloch states derived previously in
Section II. The selection rules derived in the Franck-Condon approximation
naturally (Eq.6) imply that the localized site of the defect center-of-mass is
preserved during optical emission. As a result, the ZPL energy reflects the
energy difference of one given Si(i) site between the excited and ground
states. Since the G center has comparable geometric configurations in these
two states, the ZPL splitting pattern directly indicates the energy degeneracy
lifting between the different Si(i) sites.

Although the optical transitions preserve the configuration of the defect, our
observations show that the Si(i) of these single G centers does move over the
acquisition time. More precisely, it spends on average the same amount of time
in each of the 6 possible sites. This motion cannot be thermally-activated
hopping as $k_{B}T$ ($\simeq 2.6$ meV at 30K) is much smaller than the
potential barrier energies in both the ground and excited states (89 and 33
meV, respectively from Ref. [15]). Instead the motion of the center-of-mass of
G centers could result from the excess energy provided to the system during
the above-bandgap optical excitation. Furthermore, this off-resonant
excitation is not governed by the selection rules derived above for resonant
optical transitions (Eq. 6). At each optical excitation, the interstitial
silicon of the G center randomly hops between the different positions, like a
ball in a 6-slot casino roulette wheel. By repeating the optical cycles, each
Si(i) site is on average equally occupied, thus explaining our experimental
data from Sections IV,V. The next step is to determine which perturbation is
responsible for blocking the delocalization dynamics of these individual G
centers and generating the observed fine structures.

##  VI Perturbations from SOI structures

Figure 6: ZPL spectra recorded on isolated G center spots in a bulk 28Si
sample plotted (a) with and (b) without vertical offset.

Since single G centers are incorporated in a SOI structure, the first
perturbation that comes to mind is lattice strain. Indeed, the difference in
thermal expansion coefficients between Si and SiO2 leads to in-plane strain in
the top silicon layer [42]. Furthermore, even state-of-the-art commercial SOI
wafers are known to present lattice undulations at the $\mu$m-scale resulting
from the bonding process used during fabrication [43]. To test whether the ZPL
splitting patterns come from the SOI structure, we first investigate G defects
fabricated with a similar co-implantation method in a bulk silicon sample made
of 1.5 $\mu$m of 28Si on top of a natural Si substrate. G defects are created
using a fluence of $10^{10}$cm-2 for both carbon atoms and protons, and with
the same energies as for the SOI sample (Sec. III). Figure 6 shows the ZPL
spectra measured on randomly chosen isolated G center spots in the bulk
sample. For all investigated defects, a single ZPL line at 1278.6 nm was
observed and without any ZPL fine structure found. These results demonstrate
that the frozen delocalization of the G defect center-of-mass is indeed due to
its formation in the SOI structure.

To evaluate the impact of homogeneous uniaxial strain on the G center, we
apply DFT calculations and map the PES of the interstitial silicon atom of the
defect (see Appendix C for DFT methods). Since the G center has similar
geometric structures in both ground and excited states, we can focus only on
the external perturbation in the ground state. As optical transitions occur
between the same localized Si(i) sites, the splitting pattern in the ground
state rotational levels qualitatively explains the fine structure of the ZPL.

The ab initio PES calculated for a G center under $0.1\%$ $[1\bar{1}0]$
uniaxial strain is displayed as a gray curve in Figure 5(c). Unlike the
unperturbed case from Figure 1(c), the PES is not a sixfold symmetric periodic
potential anymore. In particular, the well minima of sites $0$ and $3$ are
shifted by $\simeq 1.9$ meV compared to the 4 other sites (Fig. 5(c)). Solving
the time-independent Schrödinger equation for this perturbed potential leads
to 2 separated subsets of rotational states, as shown on Figure 5(c). The
highest energy sublevel corresponds to the 2 degenerate states localized on
sites $0$ and $3$, whereas the lowest one is associated to the 4 degenerate
states localized on sites $\\{1,2,4,5\\}$. Consequently, such $[1\bar{1}0]$
uniaxial strain does not only freeze the delocalization of the Si(i) of the G
center but reproduces the doublet fine structure observed on G-0 defect (Fig.
5(a)). Our calculations indicate that the origin of the ZPL splitting pattern
is the energy difference of the center-of-mass potential minima between the
excited and ground states. To explain the G center triplet fine structure, an
extra perturbation needs to be added to the uniaxial strain to split the
4-fold sublevel (Fig. 5(b)). A possible candidate could be electric field,
possibly coming from localized charges at the silicon-to-oxide interface [44].
The impact of electric field on G centers is beyond the scope of the paper and
will be investigated in future works.

At last, to compare different uniaxial strain directions, the ZPL energies are
estimated using the Delta self-consistent-field ($\Delta$SCF) method for each
set of equivalent equilibrium frozen configurations of the G center [45]. For
the 4 simulated strain directions, namely $[1\bar{1}0]$, $[110]$, $[001]$ and
$[111]$, only the magnitude of the ZPL splitting and the overall energy shift
of the spectrum differ between different directions (see Appendix C).
Especially, hardly no splitting is observed for the [111]-strain because this
totally symmetric perturbation equally modifies the barrier energy for all
sites at first order. Assuming a linear strain regime, this calculation method
qualitatively reproduces the ZPL splittings observed on ensembles of G centers
under uniaxial stress along $(001)$, $(111)$ and $(110)$ [33] (see Appendix
C). This confirms its validity for estimating the behavior of the G center ZPL
doublet with strain direction.

##  VII Conclusion

In this work, we have investigated the center-of-mass motion of the G center
in silicon resulting from the motion of its intrinsic interstitial silicon
atom between 6 crystal sites. For the unperturbed G center, the distinctive
quartet fine structure arising from the Si(i) perfect delocalization by tunnel
effect and the subsequent selection rules for the optical transitions can be
derived from a simple tight-binding model. Single-photon polarization analysis
from individual defects shows that the G center emission emanates from 3
dipoles matching the 6 possible sites of the Si(i). Individual G centers in
28SOI exhibit ZPL fine structures that do not only strongly vary from one
center to another, but above all with $\sim 1$-meV energy splittings. Such ZPL
splittings are more than 2 orders of magnitude larger than the one observed on
an ensemble of unperturbed G centers in bulk 28Si [2]. The evolution of the
ZPL patterns with polarization evidences that the optical transitions of these
single G centers are not compatible with a center-of-mass delocalization and
instead occur while the Si(i) sits at a fixed position, chosen among the 6
potential minima. Ab initio simulations identify uniaxial strain as being the
dominant mechanism blocking the G center tunneling rotation by partially
lifting the energy degeneracy between the different Si(i) sites. The
correlation between the strain inherent to the SOI structure and the giant ZPL
fine structure measured on single G centers is corroborated by the lack of any
ZPL splitting for G centers fabricated with the same ion implantation method
but in a 28Si bulk sample. As above-bandgap excitation does not obey the
selection rules for direct optical transitions, we believe this off-resonant
excitation causes the Si(i) to hop randomly between the 6 sites, as if it were
shaken in a 6-slot roulette wheel.

The current results are a milestone on the road to controlling the quantum
properties of the G center in silicon. The sensitivity of its fine structure
to lattice distortion could be used as sub-micron scale optical strain sensor
in SOI based nanomechanical resonators [46]. Furthermore, strain tuning of the
defect optical emission wavelengths could be an alternative pathway to
standard electrical tuning based on Stark effect [47, 48]. As the motion of
the Si(i) is also allowed in the metastable electron spin triplet level [32,
37, 49, 15], investigating the rotational dynamics of the G center could also
be performed through its spin quantum degree of freedom.

Our results highlight the key role on the properties of color centers of the
inhomogeneous strain field in SOI films. The resulting ZPL fluctuations
between defects, as shown here for G centers, will have to be efficiently
compensated in view of developing quantum photonic chips relying on the
emission from multiple color centers. However far from being only a drawback,
strain perturbation looks as an attractive resource in the specific case of
the G center as it enables to isolate single-dipole optical transitions
following the freezing of its center-of-mass delocalization. This opens the
route towards an efficient coupling to single-mode SOI nanophotonic cavities
for cavity quantum electrodynamics experiments. For instance, in contrast,
Purcell enhanced PL will be reduced for the unpolarized optical transitions of
unperturbed G centers [50]. In this context, it is worth mentioning that
suspended silicon nanobeams fabricated by selectively etching the buried
silica are likely to remove the strain fluctuations resulting from the Si-SiO2
interface [43]. Combined with uniaxial strain from tensor arms [51, 52], these
nanostructures could provide a homogeneous strain environment for color
centers. As a result, the G centers will exhibit highly uniform optical
emission lines with no center-of-mass delocalization. Taking benefit of these
recent advances, one could use strain to engineer the PL properties of G
centers in silicon in a deterministic way.

## Acknowledgments

We acknowledge funding from the French National Research Agency (ANR) through
the projects OCTOPUS (No. ANR-18-CE47-0013-01) and QUASSIC (No.
ANR-18-ERC2-0005-01), the Plan France 2030 through the project OQuLuS
ANR-22-PETQ-0013, the Occitanie region through the SITEQ contract and the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (project SILEQS, Grant No. 101042075). A.
Durand acknowledges support from the French DGA. A. Gali acknowledges the
support by the Ministry of Culture and Innovation and the National Research,
Development and Innovation Office within the Quantum Information National
Laboratory of Hungary (Grant No. 2022-2.1.1-NL-2022-00004). A. Gali
additionally acknowledges the high-performance computational resources
provided by KIFÜ (Governmental Agency for IT Development) institute of Hungary
and the European Commission for project QuMicro (Grant No. 101046911). The
authors thank François Rieutord for enlightening discussions regarding lattice
strain in SOI wafers.

##  APPENDIX A DIPOLE COLLECTION EFFICIENCIES

Figure 7: Simulations of the dipole collected intensity as a function of the
defect depth. (a-c) Respectively, Purcell factor $F$, collection efficiency
$C_{\mathrm{eff}}$ and their product for the dipoles $[1\bar{1}0]$ (red) and
$[01\bar{1}]$ (blue). (d) Evolution of the relative PL contribution between
these two dipoles.

The relative PL contribution used in Figure 3(f) for the G center dipoles
inside the SOI sample is simulated using the fine-difference time-domain
method on the commercial software RSOFT from Synopsis (version 19.09). Since
the top surface of the SOI sample is (001)-oriented, both $[\bar{1}01]$ and
$[01\bar{1}]$ dipoles will lead to the same results (see Fig. 3(e)). The
dipole spontaneous emission enhancement factor $F$ resulting from Purcell
effect in the top silicon layer is computed as function of the defect depth
for $[1\bar{1}0]$ and $[01\bar{1}]$ dipoles (Fig. 7(a)). The value of $F$
indicates whether the dipole emission is enhanced ($F\\!>\\!1$) or reduced
($F\\!<\\!1$) compared to its value in bulk silicon. While the emission of the
$[1\bar{1}0]$ in-plane dipole is little affected, the one of the $[01\bar{1}]$
out-of-plane dipole is inhibited by a factor of $\simeq 2$ due to the SOI
structure. On the contrary, the dipole orientation has a small impact on its
collection efficiency $C_{\mathrm{eff}}$ through a 0.85-Numerical Aperture
microscope objective, as shown on Figure 7(b). However, it strongly varies
with the dipole depth, from $\simeq 2.5\%$ at the surface to $\simeq 4.5\%$ at
the oxide interface. The total collected PL is proportional to the product
$F\cdot C_{\mathrm{eff}}$ for each dipole (Fig. 7(c)). To obtain the relative
contribution between the in-plane and out-of-plane dipoles, we compute the
ratio $r=(F^{[1\bar{1}0]}\cdot
C_{\mathrm{eff}}^{[1\bar{1}0]})/(F^{[01\bar{1}]}\cdot
C_{\mathrm{eff}}^{[01\bar{1}]})$. As displayed in Figure 7(d), the defect
depth has little influence on this ratio that evolves in the [2.0-2.2] range
within the 60-nm thick silicon layer.

##  APPENDIX B TRIPLET FINE STRUCTURE ANALYSIS ON DEFECT G-3

Figure 8: Triplet ZPL fine structure of G-3 defect. (a) From left to right,
ZPL spectrum measured on the defect G-3 with a polariser selecting the
polarization perpendicular and parallel to the main axis of the defect
polarization diagram respectively, as indicated in the top left corners. The
spectra with no polarization filtering from Figure 4(a) are reproduced in
dashed lines; the y-axis is normalized to its maximum. Colored areas represent
the 0.2-meV integration range used in (b). (b) Evolution vs polarizer angle of
the intensity of the 3 ZPL (see Fig.4(b,c)). Solid lines are models
considering the emission lines result from the Si(i) sites indicated in figure
4(b).

The spectral analysis with polarization from Section V is performed on the
single G center G-3, that presents a triplet ZPL fine structure. The evolution
of its ZPL with polarization filtering reproduces the behavior observed on the
triplet fine structure of defect G-1 (Fig. 4(c), 5(b)). Especially, we measure
again that the highest optical line is fully cancelled for polarization
perpendicular to the defect emission diagram (Fig. 8(a)). The quantitative
analysis of the ZPL intensities as function of polarizer angle shows that this
line is indeed linearly polarized while the other 2 ZPL are unpolarized (Fig.
8(b)). Like the single G centers in SOI previously investigated, the PL from
the G-3 defect results from optical transitions occuring between localized
states of its center-of-mass. In particular, the ZPL fine structure of the G-3
center could be explained by the same localized Si(i) sites as for the G-1
center, such as the ones proposed in Figure 5(b).

##  APPENDIX C AB-INITIO CALCULATIONS FOR PERTURBED G CENTERS

###  C.1 DFT methods

We apply DFT calculations on a G center perturbed by uniaxial strain and map
the ground state PES along the rotational reorientation path of the central
silicon atom of the G center. Along the associated $Q$ vibrational coordinate,
we sample all the 6 minima and barrier points and one additional middle point
between these, using 24 single-point calculations in total for a single PES
calculation. These geometric configurations are relaxed by climbing image
nudged elastic band (NEB) method on the sampling images until a
$10^{-4}~{}\mathrm{eV/\AA}$ threshold is reached in the forces. This precision
is necessary for the accurate modeling of small perturbations resulting in
small relative energy differences. Consequently, we need to compromise the
accuracy in the total energy by using PBE functional. The calculations are
performed in the plane wave based Vienna Ab initio Simulation Package (VASP)
[53]. The defect is modeled in a large 512-atom supercell with single
$\Gamma$-point sampling in the reciprocal space. The core electrons are
treated in the PAW formalism [54, 55, 56, 57]. The atomic positions remain
fixed during this calculation. The rotational reorientation is modeled in
analogy to our previous work in Ref. [15].

Figure 9: Energy of the (a) ZPL doublet splitting and (b) average ZPL shift
for a [111]-oriented G center under various directions of uniaxial strain.

###  C.2 ZPL patterns for different uniaxial strains

Figure 10: Evolution of the energy shift of the 2 ZPL $L_{0}$ and $L_{1}$ of a
$[111]$-oriented G defect under different uniaxial strain directions to mimic
the behavior of an ensemble of G centers under $[001]$, $[111]$ and $[110]$
uniaxial strain, respectively.

We estimate the evolution of the ZPL patterns under different strain
directions using the $\Delta$SCF method in the non-equivalent local minima
positions as calculating the PES for each direction is very heavy on
computational resources. We can safely assume that the ZPL shift is dominated
by the relative energy shift in the local minima of the PES for the larger
strain regimes where frozen position approximation holds. We note that this
simulation method is not accurate to predict the correct ZPL energy ($\simeq
654$ meV for the unperturbed G center) but it enables to qualitatively extract
strain-induced ZPL splitting trends, as shown in the following. The ZPL energy
is calculated for each subset of equivalent equilibrium Si(i) positions for 4
uniaxial strain directions: $[001]$, $[110]$, $[1\bar{1}0]$ and $[111]$. As
visible on Figure 9(a), the ZPL splitting energies are proportional to the
strain amplitude in the $[-0.2\%,+0.2\%]$ investigated range. The largest
splitting is obtained for the $[110]$-strain, while hardly no splitting is
observed for the $[111]$-strain, as it corresponds to a totally symmetric
perturbation. All 4 simulated strain directions produce an overall shift of
the ZPL doublet fine structure (Fig. 9(b)).

As a safety check, we calculate the evolution of the ZPL patterns of an
ensemble of G centers subject to uniaxial strain along the crystal directions
used in a previous piezospectroscopic experiment [33] (Fig. 10). For a
$[001]$-strain, all 4 possible $\langle 111\rangle$ crystal axis of the G
center experience the same lattice distortion, hence the same doublet fine
structure. Under a $[111]$-strain, the $[111]$-oriented G centers will feel a
totally symmetric perturbation with no splitting while the 3 other G
orientations will undergo a perturbation equivalent to a
$[\bar{1}\bar{1}1]$-strain for [111]-oriented G center. Consequently, 3
optical lines are expected to split for a G-center ensemble under
[111]-strain. The $[110]$ strain will separate the orientations into 2
families: $[111]$\- and $[\bar{1}\bar{1}1]$-oriented centers on one hand, and
$[\bar{1}11]$ and $[1\bar{1}1]$-oriented defects on the other hand. The strain
experienced by these last 2 orientations will be equivalent to a
$[1\bar{1}0]$-strain acting on a [111]-oriented G center. As a result, 4
optical lines are expected to emerge for an ensemble of G defects under
$[110]$-uniaxial strain. Assuming the linear regime of Hooke’s law prevails,
the ZPL splittings calculated for these 3 uniaxial strains are in good
qualitative agreement with the uniaxial stress measurements performed on a
ensemble of G centers in [33].

## References

  * 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, Optical properties of an ensemble of G-centers in silicon, 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, Highly enriched ${}^{28}\mathrm{Si}$ reveals remarkable optical linewidths and fine structure for well-known damage centers, Physical Review B 98, 195201 (2018).
  * 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, Single artificial atoms in silicon emitting at telecom wavelengths, Nature Electronics 3, 738 (2020).
  * Simmons [2020] S. Simmons, A single silicon colour centre resolved, Nature Electronics 3, 734 (2020).
  * Hollenbach  _et al._ [2020] M. Hollenbach, Y. Berencén, U. Kentsch, M. Helm, and G. V. Astakhov, Engineering telecom single-photon emitters in silicon for scalable quantum photonics, Optics Express 28, 26111 (2020).
  * Zhang  _et al._ [2020] G. Zhang, Y. Cheng, J.-P. Chou, and A. Gali, Material platforms for defect qubits and single-photon emitters, Applied Physics Reviews 7, 031308 (2020).
  * Durand  _et al._ [2021] A. Durand, Y. Baron, W. Redjem, T. Herzig, A. Benali, S. Pezzagna, J. Meijer, A. Kuznetsov, J.-M. Gérard, I. Robert-Philip, M. Abbarchi, V. Jacques, G. Cassabois, and A. Dréau, Broad Diversity of Near-Infrared Single-Photon Emitters in Silicon, Physical Review Letters 126, 083602 (2021).
  * 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, Silicon-Integrated Telecommunications Photon-Spin Interface, PRX Quantum 1, 020301 (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, Optical observation of single spins in silicon, Nature 607, 266 (2022).
  * Higginbottom  _et al._ [2023] D. B. Higginbottom, F. K. Asadi, C. Chartrand, J.-W. Ji, L. Bergeron, M. L. Thewalt, C. Simon, and S. Simmons, Memory and Transduction Prospects for Silicon T Center Devices, PRX Quantum 4, 020308 (2023).
  * DeAbreu  _et al._ [2023] 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, Waveguide-integrated silicon T centres, Optics Express 31, 15045 (2023).
  * Johnston  _et al._ [2023] A. Johnston, U. Felix-Rendon, Y.-E. Wong, and S. Chen, Cavity-coupled telecom atomic source in silicon (2023), arXiv:2310.20014 [quant-ph].
  * Lee  _et al._ [2023] C.-M. Lee, F. Islam, S. Harper, M. A. Buyukkaya, D. Higginbottom, S. Simmons, and E. Waks, High-Efficiency Single Photon Emission from a Silicon T-Center in a Nanobeam, ACS Photonics 10, 3844 (2023).
  * Islam  _et al._ [2024] F. Islam, C.-M. Lee, S. Harper, M. H. Rahaman, Y. Zhao, N. K. Vij, and E. Waks, Cavity-Enhanced Emission from a Silicon T Center, Nano Letters 24, 319 (2024).
  * Udvarhelyi  _et al._ [2021] P. Udvarhelyi, B. Somogyi, G. Thiering, and A. Gali, Identification of a Telecom Wavelength Single Photon Emitter in Silicon, Physical Review Letters 127, 196402 (2021).
  * Baron  _et al._ [2022a] 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, Single G centers in silicon fabricated by co-implantation with carbon and proton, Applied Physics Letters 121, 084003 (2022a).
  * 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, Wafer-scale nanofabrication of telecom single-photon emitters in silicon, Nature Communications 13, 7683 (2022).
  * 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, Indistinguishable photons from an artificial atom in silicon photonics (2022), arXiv:2211.09305 [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, Cavity-enhanced zero-phonon emission from an ensemble of G centers in a silicon-on-insulator microring, Applied Physics Letters 122, 061109 (2023).
  * Prabhu  _et al._ [2023] M. Prabhu, C. Errando-Herranz, L. De Santis, I. Christen, C. Chen, C. Gerlach, and D. Englund, Individually addressable and spectrally programmable artificial atoms in silicon photonics, Nature Communications 14, 2380 (2023).
  * Saggio  _et al._ [2023] V. Saggio, C. Errando-Herranz, S. Gyger, C. Panuski, M. Prabhu, L. De Santis, I. Christen, D. Ornelas-Huerta, H. Raniwala, C. Gerlach, M. Colangelo, and D. Englund, Cavity-enhanced single artificial atoms in silicon (2023), arXiv:2302.10230 [quant-ph].
  * Day  _et al._ [2023] A. M. Day, M. Sutula, J. R. Dietz, A. Raun, D. D. Sukachev, M. K. Bhaskar, and E. L. Hu, Electrical Manipulation of Telecom Color Centers in Silicon (2023), arXiv:2311.08276 [quant-ph].
  * Durand  _et al._ [2024] A. Durand, Y. Baron, F. Cache, T. Herzig, M. Khoury, S. Pezzagna, J. Meijer, J.-M. Hartmann, S. Reboh, M. Abbarchi, I. Robert-Philip, J.-M. Gérard, V. Jacques, G. Cassabois, and A. Dréau, Genuine and faux single G centers in carbon-implanted silicon (2024), arXiv:2402.07705 [quant-ph].
  * Berkman  _et al._ [2023] I. R. Berkman, A. Lyasota, G. G. de Boo, J. G. Bartholomew, S. Q. Lim, B. C. Johnson, J. C. McCallum, B.-B. Xu, S. Xie, N. V. Abrosimov, H.-J. Pohl, R. L. Ahlefeldt, M. J. Sellars, C. Yin, and S. Rogge, Millisecond electron spin coherence time for erbium ions in silicon (2023), arXiv:2307.10021 [quant-ph].
  * Weiss  _et al._ [2021] L. Weiss, A. Gritsch, B. Merkel, and A. Reiserer, Erbium dopants in nanophotonic silicon waveguides, Optica 8, 40 (2021).
  * Gritsch  _et al._ [2023] A. Gritsch, A. Ulanowski, and A. Reiserer, Purcell enhancement of single-photon emitters in silicon, Optica 10, 783 (2023).
  * Redjem  _et al._ [2023] W. Redjem, Y. Zhiyenbayev, W. Qarony, V. Ivanov, C. Papapanos, W. Liu, K. Jhuria, Z. Y. Al Balushi, S. Dhuey, A. Schwartzberg, L. Z. Tan, T. Schenkel, and B. Kanté, All-silicon quantum light source by embedding an atomic emissive center in a nanophotonic cavity, Nature Communications 14, 3321 (2023).
  * Zhiyenbayev  _et al._ [2023] Y. Zhiyenbayev, W. Redjem, V. Ivanov, W. Qarony, C. Papapanos, J. Simoni, W. Liu, K. Jhuria, L. Z. Tan, T. Schenkel, and B. Kanté, Scalable manufacturing of quantum light emitters in silicon under rapid thermal annealing, Optics Express 31, 8352 (2023).
  * Baron  _et al._ [2022b] 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, Detection of Single W-Centers in Silicon, ACS Photonics 9, 2337 (2022b).
  * Lefaucher  _et al._ [2024] B. Lefaucher, J.-B. Jager, V. Calvo, F. Cache, A. Durand, V. Jacques, I. Robert-Philip, G. Cassabois, Y. Baron, F. Mazen, S. Kerdilès, S. Reboh, A. Dréau, and J.-M. Gérard, Purcell Enhancement of Silicon W Centers in Circular Bragg Grating Cavities, ACS Photonics 11, 24 (2024).
  * Udvarhelyi  _et al._ [2022] P. Udvarhelyi, A. Pershin, P. Deák, and A. Gali, An L-band emitter with quantum memory in silicon, npj Computational Materials 8, 1 (2022).
  * Lee  _et al._ [1982] K. M. Lee, K. P. O’Donnell, J. Weber, B. C. Cavenett, and G. D. Watkins, Optical Detection of Magnetic Resonance for a Deep-Level Defect in Silicon, Physical Review Letters 48, 37 (1982).
  * Foy  _et al._ [1981] C. P. Foy, M. C. d. Carmo, G. Davies, and E. C. Lightowlers, Uniaxial stress measurements on the 0.97 eV line in irradiated silicon, Journal of Physics C: Solid State Physics 14, L7 (1981).
  * Thonke  _et al._ [1981] K. Thonke, H. Klemisch, J. Weber, and R. Sauer, New model of the irradiation-induced 0.97-eV (${G}$) line in silicon: A $\mathrm{C_{S}}\\!-\\!\mathrm{Si}^{*}$ complex, Physical Review B 24, 5874 (1981).
  * Davies  _et al._ [1983] G. Davies, E. C. Lightowlers, and M. d. Carmo, Carbon-related vibronic bands in electron-irradiated silicon, Journal of Physics C: Solid State Physics 16, 5503 (1983).
  * Davies [1989] G. Davies, The optical properties of luminescence centres in silicon, Physics Reports 176, 83 (1989).
  * O’Donnell  _et al._ [1983] K. P. O’Donnell, K. M. Lee, and G. D. Watkins, Origin of the 0.97 eV luminescence in irradiated silicon, Physica B+C 116, 258 (1983).
  * Capaz  _et al._ [1998] R. B. Capaz, A. Dal Pino, and J. D. Joannopoulos, Theory of carbon-carbon pairs in silicon, Physical Review B 58, 9845 (1998).
  * Song  _et al._ [1990] L. W. Song, X. D. Zhan, B. W. Benson, and G. D. Watkins, Bistable interstitial-carbon–substitutional-carbon pair in silicon, Physical Review B 42, 5765 (1990).
  * Tkachev and Mudryi [1978] V. D. Tkachev and A. V. Mudryi, Piezospectroscopic effect on zero-phonon luminescence lines of silicon, Journal of Applied Spectroscopy 29, 1485 (1978).
  * Ristori  _et al._ [2024] A. Ristori, M. Khoury, M. Salvalaglio, A. Filippatos, M. Amato, T. Herzig, J. Meijer, S. Pezzagna, D. Hannani, M. Bollani, C. Barri, C. M. Ruiz, N. Granchi, F. Intonti, M. Abbarchi, and F. Biccari, Strain Engineering of the Electronic States of Silicon-Based Quantum Emitters, Advanced Optical Materials 12, 2301608 (2024).
  * Camassel  _et al._ [2000] J. Camassel, L. A. Falkovsky, and N. Planes, Strain effect in silicon-on-insulator materials: Investigation with optical phonons, Physical Review B 63, 035309 (2000).
  * Fukuda  _et al._ [2006] K. Fukuda, T. Yoshida, T. Shimura, K. Yasutake, M. Umeno, and S. Iida, White X-ray Topography of Lattice Undulation in Bonded Silicon-on-Insulator Wafers, Japanese Journal of Applied Physics 45, 6795 (2006).
  * Schroder [2005] D. K. Schroder,  _Semiconductor Material and Device Characterization_, 1st ed. (Wiley, 2005).
  * Gali  _et al._ [2009] A. Gali, E. Janzén, P. Deák, G. Kresse, and E. Kaxiras, Theory of Spin-Conserving Excitation of the N$-$V- Center in Diamond, Physical Review Letters 103, 186404 (2009).
  * Craighead [2000] H. G. Craighead, Nanoelectromechanical Systems, Science 290, 1532 (2000).
  * Bassett  _et al._ [2011] L. C. Bassett, F. J. Heremans, C. G. Yale, B. B. Buckley, and D. D. Awschalom, Electrical Tuning of Single Nitrogen-Vacancy Center Optical Transitions Enhanced by Photoinduced Fields, Physical Review Letters 107, 266403 (2011).
  * de las Casas  _et al._ [2017] C. F. de las Casas, D. J. Christle, J. Ul Hassan, T. Ohshima, N. T. Son, and D. D. Awschalom, Stark tuning and electrical charge state control of single divacancies in silicon carbide, Applied Physics Letters 111, 262403 (2017).
  * Vlasenko  _et al._ [1986] L. S. Vlasenko, M. P. Vlasenko, V. N. Lomasov, and V. A. Khramtsov, Spin-dependent effects due to triplet centers in irradiated silicon, Sov. Phys. JETP 64, 3 (1986).
  * Gérard [2003] J.-M. Gérard, Solid-State Cavity-Quantum Electrodynamics with Self-Assembled Quantum Dots, in  _Single Quantum Dots: Fundamentals, Applications, and New Concepts_, Topics in Applied Physics (Springer Berlin Heidelberg, Berlin, Heidelberg, 2003) pp. 269–314.
  * Chrétien  _et al._ [2019] J. Chrétien, N. Pauc, F. Armand Pilon, M. Bertrand, Q.-M. Thai, L. Casiez, N. Bernier, H. Dansas, P. Gergaud, E. Delamadeleine, R. Khazaka, H. Sigg, J. Faist, A. Chelnokov, V. Reboud, J.-M. Hartmann, and V. Calvo, GeSn Lasers Covering a Wide Wavelength Range Thanks to Uniaxial Tensile Strain, ACS Photonics 6, 2462 (2019).
  * Guilloy  _et al._ [2015] K. Guilloy, N. Pauc, A. Gassenq, P. Gentile, S. Tardif, F. Rieutord, and V. Calvo, Tensile Strained Germanium Nanowires Measured by Photocurrent Spectroscopy and X-ray Microdiffraction, Nano Letters 15, 2429 (2015).
  * Blöchl [1994] P. E. Blöchl, Projector augmented-wave method, Physical Review B 50, 17953 (1994).
  * Kresse and Hafner [1993] G. Kresse and J. Hafner, Ab initio molecular dynamics for liquid metals, Physical Review B 47, 558 (1993).
  * Kresse and Furthmüller [1996a] G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Physical Review B 54, 11169 (1996a).
  * Kresse and Furthmüller [1996b] G. Kresse and J. Furthmüller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Computational Materials Science 6, 15 (1996b).
  * Paier  _et al._ [2006] J. Paier, M. Marsman, K. Hummer, G. Kresse, I. C. Gerber, and J. G. Ángyán, Screened hybrid density functionals applied to solids, The Journal of Chemical Physics 124, 154709 (2006).