# Characterization of single shallow silicon-vacancy centers in $4H$-SiC

Harpreet Singh1,2, Mario Alex Hollberg1, Misagh Ghezellou3, Jawad Ul-Hassan3,
Florian Kaiser4 and Dieter Suter1
1Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany.
2Department of Chemistry, University of California, Berkeley, Berkeley, CA
94720, USA. 3Department of Physics, Chemistry and Biology, Linköping
University, Linköping, Sweden. 43rd Institute of Physics and Research Center
SCOPE, University of Stuttgart, Stuttgart, Germany

###### Abstract

Shallow negatively charged silicon-vacancy centers have applications in
magnetic quantum sensing and other quantum applications. Vacancy centers near
the surface (within 100 nm) have different spin relaxation rates and optical
spin polarization, affecting the optically detected magnetic resonance (ODMR)
signal. This makes it essential to characterize these centers. Here we present
the relevant spin properties of such centers. ODMR with a contrast of up to 6
%, which is better than the state of the art, allowed us to determine the zero
field splitting, which is relevant for most sensing applications. We also
present intensity-correlation data to verify that the signal originates from a
single center and to extract transition rates between different electronic
states.

##  I Introduction

Silicon vacancies in silicon carbide (SiC) are excellent candidates for
quantum sensing and information applications [1, 2, 3, 4]. Negatively charged
silicon vacancies ($V_{Si}^{-}$) have spin 3/2 [5, 6]. The spin of
$V_{Si}^{-}$ can be initialized into a particular spin groundstate by optical
pumping [5, 4, 7] and it has sufficiently long coherence and relaxation times
for many interesting applications [3, 8, 9, 2, 10, 11, 12]. Debye-Waller
factors of about 6% were found for $V_{1}$ and $V_{2}$ type of $V_{Si}^{-}$
[13, 14], which is vital for the quantum application that rely on
inditinguishable photon emitters. The electronically excited state zero-field
splitting of the $V_{2}$ centers in $4H$ -SiC exhibits a significant thermal
shift, which makes them useful for thermometry applications [15]. $V_{Si}^{-}$
in SiC provides a low-cost and simple approach to quantum sensing of magnetic
fields and a sensitivity of 50 nT/$\sqrt{\text{Hz}}$, which was achieved
without complex photonic engineering, control protocols, or applying
excitation (optical and radio-frequency) powers greater than a watt [16].
Absolute dc magnetometry, which is immune to thermal noise and strain
inhomogeneity has been demonstrated using all four ground-state spin levels of
$V_{Si}^{-}$ [17]. Similar to NV in diamond, $V_{Si}^{-}$ based magnetometers
can also be applied to unique magnetic sensing challenges, such as resolving
an individual nuclear spin in a molecule [18]. Since the magnetic field of a
dipole decreases with distance as 1/$r^{3}$, it is essential that the sensor
can be brought close to the source [18], which is only possible if the
$V_{Si}^{-}$ is close to the surface.

Silicon vacancies can be created, e.g., by irradiating the sample with
neutrons or electrons [19, 20, 21]. The charge on the silicon-vacancy depends
on the Fermi-level: it is at least 1.24 eV above the valence band, the $k$
lattice site becomes negatively charged and at 1.5 eV also the $h$ lattice
site [22]. Irradiation by proton or other charged ions allow one to create a
plane of vacancies at a specific depth that depends upon the energy and type
of the ions [23, 24, 25]. Here we focus on single silicon vacancy centers
created with 6 keV He+ ion irradiation. Using a lithographic polymethyl
methacrylate (PMMA) mask with 100 nm holes, an array of silicon vacancies was
created [23] at a depth of 30-40 nm below the surface. Additional details of
the sample preparation are given in Appendix A.

In this work, we focus on the $V_{Si}^{-}$ in the $4H$-SiC polytype,
specifically $V_{Si}^{-}$ centers at hexagonal sites $h$ and cubic sites $k$,
which are commonly known as $V_{1}$ and $V_{2}$ [26]. We characterize the
properties of several individual centers and obtain useful values for the
optical transitions rates as well as for the radiation-less population
dynamics. We demonstrate that the optical pumping results in spin polarization
that is sufficiently high to observe optically detected magnetic resonance
(ODMR) spectra of single silicon vacancy centers at room temperature with high
contrast. From a series of ODMR spectra with different RF powers, we could
extract the transverse dephasing time $T_{2}^{*}$.

This paper is organized as follows. Section II introduces the optical and
magnetic properties of a single spin system. Section III provides details of
correlation measurement experiments and analysis to estimate the rates
involved in the population dynamics of the center during optical excitation.
Section IV shows continuous-wave optically detected magnetic resonance (cw-
ODMR) experiments at different RF powers. Finally, Sec.V contains a brief
discussion and some concluding remarks.

##  II System and Setup

Figure 1: (a) Experimental confocal setup used for PL and intensity
autocorrelation measurements.

The $V_{2}$ center has $C_{3v}$ symmetry and an electron spin $S=$ 3/2. In the
absence of a magnetic field, the spin Hamiltonian is

${\cal H}=D(S_{z}^{2}-\frac{S(S+1)}{3}),$ (1)

where the zero field splitting (ZFS) in the electronic ground state is $2D$=
70 MHz in frequency units ($h=1$) [3, 27]. $\vec{S}$ = {$S_{x},$ $S_{y}$,
$S_{z}$} is the vector of spin operators and we have chosen the $z$-axis
parallel to the $c$-axis of the crystal ($C_{3}$ symmetry axis).

Figure 1 shows the experimental confocal setup for measuring photoluminescence
(PL) and the second-order intensity autocorrelation. A 785 nm laser beam was
focused on the sample with an objective lens of numerical aperture (NA) 1.3.
With the help of a dichroic mirror, the PL emitted by the sample was filtered
and allowed to pass through a $\lambda/2$ waveplate, a set of optical filters
and a polarizing beam splitter (PBS). Two single-photon detectors (SPD) were
used for the two possible paths. With the $\lambda/2$ waveplate, we controlled
the ratio of photons detected by these two detectors. From the arrival time of
the photons, we calculated the intensity autocorrelation function $g^{(2)}$.
For taking PL scans, the counts of both SPDs were added and plotted against
the XY position. For simplicity, some of the mirrors and lenses are not shown
in Fig. 8, details are summarised in appendix A.

Figure 2 shows a confocal PL scan of a part of the sample. The plot shows that
most $V_{Si}^{-}$ centers lead to a photon detection rate of $\sim$8000 cps
(counts/s) at room temperature. Higher count rates are also observed in some
locations, indicating multiple close centers. The counts due to impurities and
dirt are higher than the counts of the vacancies and usually bleach away after
a few minutes of laser illumination.

Figure 2: Confocal scan of the defect center array in the XY plane. Large
count rates are due to dirt and impurities. The count rate is color-coded, and
its value can be taken from the bar on the right.

##  III Population Dynamics

We measured the second-order correlation function of the PL light emitted by a
$V_{Si}^{-}$. This allowed us to verify that the light is emitted by a single
center as well as to study the population dynamics. The second order
correlation function

$\displaystyle g^{(2)}(\tau)$ $\displaystyle=$ $\displaystyle\frac{\langle
n_{1}(t)n_{2}(t+\tau)\rangle}{\langle n_{1}(t)\rangle\langle
n_{2}(t+\tau)\rangle}$

is the conditional probability of measuring two photons with a delay $\tau$.
Here, $n_{i}(t)$ are the number of photons detected at time $t$ [28]. For an
ideal single photon emitter the conditional probability of detecting two
photons simultaneously drops to zero, $g^{(2)}(\tau=0)$=0 [28]. Experimentally
$g^{(2)}(0)<0.5$ is evidence for a single photon emitter. Figures 3 and 9 show
the experimental correlation data. The minimum in the curve is shifted from
$\tau=0$ by $\tau_{0}=1.0\pm 0.2$ ns, which originates in unequal delays of
the optical and electrical detection paths of the two SPDs.

Figure 3: Intensity autocorrelation measurements performed with 2.6 kW/cm2 and
378 kW/cm2 laser intensity in the focus. In the data recorded with intensity
$I$ = 2.6 kW/cm2, the background count rate was negligible, but it was
significant for $I$=378 kW/cm2. We therefore performed a background
correction. Each solid line follows the three-level correlation function from
Eq. (17).

We performed correlation measurements with different laser intensities. Figure
3 , shows the intensity autocorrelation vs. the delay $\tau$ for laser
intensities 2.6 kW/cm2 and 378 kW/ cm2 with orange diamonds and green stars,
respectively. The bin width, count rates, and background details used for
normalizing the plots are given in appendix B along with data for additional
laser intensities. Experimental correlation functions with $I$<135 kW/cm2,
reveal an antibunching dip, indicating that the selected PL source is a single
photon emitter $g^{(2)}(\tau=0)$ = 0.17 for laser intensity $I$ = 2.6 kW/cm2.
As shown in Fig. 10, the signal increases linearly with the laser intensity
$I$ for low power but saturates for $I>$130 kW/cm2. The background increases
linearly with $I$, which results in a decreasing signal-to-background ratio at
higher powers $I$ and a reduced antibunching dip ($g^{(2)}$ $(\tau=0)$ > 0.5),
for data taken at $I$ >135 kW/cm2(plotted in Fig. 9 in appendix B) [29]. A
bunching behaviour is observed for $|\tau|$ > 9 ns for the laser intensity of
378 kW/cm² in Fig. 3).

These data can be modeled by considering at least a three-level system, as
shown in Fig. 4. Absorption of laser light brings the system from the ground
state $|G\rangle$ to the excited state $|E\rangle$ with a rate $k_{ge}$, which
is proportional to the laser intensity $I$. Due to spontaneous emission from
the excited state $|E\rangle$, most of the population returns to the ground
state $|G\rangle$ with a rate $k_{eg}$ and emits PL. Some of the population
goes to the shelving state $|S_{1}\rangle$ due to the intersystem crossing
(ISC) with a rate $k_{es}$. From the shelving state, the population returns to
the ground state with a rate $k_{sg}$. From the shelving state population
preferentially populates the $|\pm 1/2\rangle$ spin substates. The resulting
population difference of up to 80% [5] makes ODMR measurements possible.
Writing $n_{g}$, $n_{e}$, and $n_{s}$ for the populations of the electronic
ground state $|G\rangle$, excited state $|E\rangle$, and shelving state
$|S_{1}\rangle$ respectively, the population dynamics can be written as

$\displaystyle\frac{d}{dt}\vec{n}=\frac{d}{dt}\left(\begin{array}[]{c}n_{g}\\\
n_{e}\\\ n_{s}\end{array}\right)=$
$\displaystyle\left(\begin{array}[]{ccc}-k_{ge}&k_{eg}&k_{sg}\\\
k_{ge}&-k_{eg}-k_{es}\\\ &k_{es}&-k_{sg}\end{array}\right)$
$\displaystyle\vec{n}.$ (8)

The eigenvectors $\vec{v_{i}}$ are given in appendix D and the eigenvalues for
Eq.(8) are

$\displaystyle\vec{\lambda}$ $\displaystyle=\frac{1}{2}$
$\displaystyle\left(\begin{array}[]{c}0\\\ A+\sqrt{A^{2}-4B}\\\
A-\sqrt{A^{2}-4B}\end{array}\right),$ (12)

where $A$ = $k_{ge}+k_{eg}+k_{es}+k_{sg}$ and $B$ =
$k_{sg}(k_{eg}+k_{es}+k_{ge})+k_{es}k_{ge}$. The resulting stationary state is

$\displaystyle\vec{n}^{st}$ $\displaystyle=$
$\displaystyle\frac{1}{B}\left(\begin{array}[]{c}k_{sg}(k_{eg}+k_{es})\\\
k_{ge}k_{sg}\\\ k_{es}k_{ge}\end{array}\right).$ (16)

To calculate the theoretical correlation function $g^{2}(\tau)$, we need the
excited state population at time $\tau$ when the system is initially in the
ground state ($n_{g}(0)=1$ and $n_{e}(0)=n_{s}(0)=0$):

$\displaystyle\frac{n_{e}(t)}{n_{e}^{st}}$
$\displaystyle\overset{t=\tau}{=}g^{(2)}(\tau)=$ $\displaystyle
1-(1+c)e^{-\tau/\tau_{1}}+c\;e^{-\tau/\tau_{2}},$ (17)

with the inverse of the eigenvalues

$\tau_{1,2}=2/(A\pm\sqrt{A^{2}-4B})$ (18)

and the bunching amplitude

$\displaystyle c$ $\displaystyle=$
$\displaystyle\frac{1-\tau_{2}k_{sg}}{k_{sg}(\tau_{2}-\tau_{1})}.$ (19)

Figure 4: Electronic energy levels scheme for $V_{Si}^{-}$ on the cubic site
($V_{2}$). The optical excitation with a 785 nm laser, from the ground state
to the excited state is indicated by the thick red arrow. Direct radiative
(PL) relaxation from the excited state $|E\rangle$ to the ground state
$|G\rangle$ is represented by a thin dark-red arrow. From the excited state
the center can undergo inter-system crossing (ISC) to the shelving state
$|S_{1}\rangle$, as indicated by black arrows. The ground state spin-sublevels
are separated by the ZFS. $k_{ge}$, $k_{eg}$, $k_{es}$ and $k_{sg}$ are the
transition rates from the ground to the excited state, excited state to ground
state, excited state to shelving state and shelving state to ground state,
respectively. The deshelving process, corresponds to a transfer of population
from $|S_{1}\rangle$ to a higher energy level, $|S_{2}\rangle$, followed by
another ISC.

In Fig. 3, the solid dark red lines are the fitted curves obtained using Eq.
(17). In Fig. 5, we plot the extracted fit parameters $\tau_{1}$, $\tau_{2}$,
and $c$ vs. the laser intensity. The experimental data points are compared to
theoretical curves that can be obtained from the intensity dependence of the
effective rate coefficients

$\displaystyle k_{sg}$ $\displaystyle=$
$\displaystyle\frac{1}{(1+c^{\infty})\tau_{2}^{\infty}},$ $\displaystyle
k_{es}$ $\displaystyle=$ $\displaystyle k_{sg}c^{\infty},$ $\displaystyle
k_{eg}$ $\displaystyle=$
$\displaystyle\frac{1}{\tau_{1}^{0}}-k_{23}\,\,(k_{eg}+k_{es})>k_{sg},$ (20)

where the values with superscript $0$ refer to the low-intensity limit and
those with superscript $\infty$ to the high intensity limit [30, 19]. The
absorption cross section $\sigma$ was obtained from the transition rate
$k_{ge}$ from the ground state to the excited state:

$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\frac{h\nu}{I}(\frac{k_{ge}}{n_{g}})$

with the population $n_{g}$ of the ground state and $I/h\nu$ the number of
incident photons per second per unit area. At saturation intensity $I_{0}$,
the population $n_{g}$ drops to 1/2:

$\frac{k_{ge}}{n_{g}(\infty)}=\frac{k_{sg}\left(k_{eg}+k_{es}\right)}{k_{sg}+k_{eg}}.$
(21)

The absorption cross section can be written as

$\sigma=\frac{k_{ge}}{n_{g}}\frac{h\text{{\hbox{\nu}}}}{I_{0}}=\frac{k_{sg}\left(k_{eg}+k_{es}\right)}{k_{sg}+k_{es}}\frac{h\text{{\hbox{\nu}}}}{I_{0}},$

Fitting simultaneously the obtained values of $\tau_{1}^{0}$ = (7.5$\pm$ 1.1)
ns, $\tau_{2}^{\infty}$ =(17.2 $\pm$1.4) ns and $c^{\infty}$ = 6.0 $\pm$ 0.5
from the data plotted in Fig. 5 after simultaneously fitting in Eqs. (12) and
(19). As in previous works [19, 30], the three-level model describes well the
antibunching decay time constant $\tau_{1}$ (the black dashed line in Fig.
5(a)) and the bunching amplitude $c\,(I)$ (the black dashed line in Fig. 5
(c)). Table 1 shows the calculated rates and absorption cross section obtained
from Eqs. (20). For low intensity ($I$ < 104 kW/ cm2) the bunching time
constant reaches several hundred ns, much longer than the window considered in
Fig. 3. At higher intensities, the bunching time gets shorter. In Ref. [30,
19], this is explained as a de-shelving process that can be described as shown
in Fig. 4: The laser re-excites the system from the shelving state
$|S_{1}\rangle$ to a higher-lying state $|S_{2}\rangle$ from where it can fall
back to $|E\rangle$. The following are the rates for the 4-level model

$\displaystyle k_{sg}$ $\displaystyle=$
$\displaystyle\frac{d\,I}{I+I_{0}}+k_{sg}^{0},$ $\displaystyle k_{sg}^{0}$
$\displaystyle=$ $\displaystyle\frac{1}{\tau_{2}^{0}},$ $\displaystyle
k_{sg}^{\infty}$ $\displaystyle=$ $\displaystyle\frac{1}{\tau_{2}^{\infty}},$
$\displaystyle k_{es}$ $\displaystyle=$
$\displaystyle\frac{1}{\tau_{2}^{\infty}}-k_{sg}^{0}-d$ $\displaystyle k_{eg}$
$\displaystyle=$ $\displaystyle\frac{1}{\tau_{1}^{0}}-k_{es},$ (22)

where $d=\frac{1/\tau_{2}^{\infty}-(1+c^{\infty})/\tau_{2}^{0}}{1+c^{\infty}}$
[30, 19]. Inclusion of the deshelving process results in the intensity-
dependent rate from shelving state to ground state $k_{sg}$ [30]. Fitting
simultaneously the obtained values of $\tau_{1}^{0}$ = (6.7 $\pm$ 1.4) ns,
$\tau_{2}^{0}$ =(204.4 $\pm$ 166.3) ns, $\tau_{2}^{\infty}$ =(14.9 $\pm$ 3.4)
ns and $c^{\infty}$ = 6.3 $\pm$ 0.9 from the data plotted in Fig. (5) after
simultaneously fitting in Eqs. (12) and (19), using 4-level rates given in
Eqs. (22). The blue curves in Fig. 5 correspond to the 4-level model, which
matches the experimental data better than the black dashed curves from the
3-level model. We now compare our results to previous work [19], in which
silicon vacancies were created through high-energy neutron irradiation without
post-annealing. The extracted transition times and cross-sections from the
4-level model were $1/k_{eg}$=7.6 ns, 1/$k_{es}$=16.8 ns, 1/$k_{sg}^{0}$=150
ns, 1/$k_{sg}^{\infty}$=123 ns, and $\sigma$= 1.6 $\times$ 10-16cm2 [19]. The
transition times $1/k_{eg}$, 1/$k_{es}$ and $1/k_{sg}^{0}$ measured by us for
the shallow silicon vacancy center are compatible, within the experimental
uncertainties, with those of Fuchs _et al_. [19]. Further the cross-section
$\sigma$ error bar’s lower end is close to the cross-section measured by Fuchs
et al. [19]. The main difference between the two samples is the defect
generation method and the depth of the center. We tentatively assign the
defect generation method to the $\approx 5\times$ increase that we observed in
our rate $k_{sg}^{\infty}$. In our sample, emitters are created with low-
energy Helium ion implantation, followed by 600°C annealing to remove
parasitic lattice damage. As has been shown in previous work [23], high damage
defect generation can significantly alter radiative and non-radiative decay
processes, which may modify relaxation rates [31]. Since our emitters are
located within $\approx$40 nm at the surface, we additionally expect coupling
to surface charge states and traps, which may lead to a symmetry reduction
through which additional decay channels can be allowed. We believe that a more
thorough analysis on the modification of the rates as a function of the defect
generation method would be useful, however, this would require low-temperature
experiments [32], which go beyond the scope of this paper.

Over the duration of the intensity autocorrelation measurement, the count rate
of the selected single center dropped significantly. We therefore used a
different center (we call it center 2) for the ODMR experiment. The intensity
autocorrelation measurement data for that center confirm that it is also a
single center.

Figure 5: Time constants $\tau_{1}$, $\tau_{2}$ and bunching amplitude $c$ of the second order correlation function for various laser power intensities. The experimental data are fitted with the functions of Eqs. (12) and (19). All three datasets are fitted simultaneously which results in the following parameters $\tau_{1}^{0}$ = (7.5$\pm$ 1.1) ns, $\tau_{2}^{\infty}$ =(17.2 $\pm$1.4) ns and $c^{\infty}$ = 6.0 $\pm$ 0.5 for the 3-level model (black dashed curves); $\tau_{1}^{0}$ = (6.7 $\pm$ 1.4) ns, $\tau_{2}^{0}$ =(204.4 $\pm$ 166.3) ns, $\tau_{2}^{\infty}$ =(14.9 $\pm$ 3.4) ns and $c^{\infty}$ = 6.3 $\pm$ 0.9 for the 4-level model (blue curves). Inverse rate coefficients in ns | Cross section
---|---
Model | $1/k_{eg}$ | $1/k_{es}$ | $1/k_{sg}^{0}$ | $1/k_{sg}^{\infty}$ |  $\sigma$ ($10^{-16}$cm2)
3 levels | 12 $\pm$ 3 | 20 $\pm$ 2 | - | 120 $\pm$ 13 | 1.1$\pm$.2
4 levels | 11 $\pm$ 4 | 17 $\pm$ 8 | 204 $\pm$ 166 | 15 $\pm$ 3 | 4.9 $\pm$ 1.2
Table 1: Inverse rate coefficients and absorption cross section of the
$V_{Si}^{-}$ center.

##  IV Optically Detected Magnetic Resonance

In conventional electron spin resonance (ESR), spins are measured by inductive
detection, which limits the sensitivity of the technique. Using optical
polarization of the spin system and optical detection, it is possible to
sufficiently increase the sensitivity for experiments on single spins [33,
34]. In this optically detected magnetic resonance (ODMR), the electron spin
is excited with laser light to its optical excited state, and the PL emitted
during re-emission is detected while a radio-frequency (RF) or microwave (MW)
field drives the spin system. The different spin states contribute differently
to the PL rate, so that a change in the spin polarization leads to a change of
the PL rate, which can be used to measure spin polarization. We used this
technique for measuring the ODMR of a single silicon vacancy in $4H$-SiC.

For recording the single center’s continuous wave (CW) ODMR, the center was
continuously illuminated with an optimized laser intensity of 68 kW/cm2 where
the signal to background ratio is maximal; more details are given in Appendix
F. The ODMR was recorded in “lock-in mode”: while stepping the RF over the
relevant frequency range, PL counts were recorded with and without RF. The
difference between the two vlaues ΔPL = PLRF−PLoff , is less sensitive to
fluctuations of the laser intensity.

Figure 6: (a) ODMR signal vs. frequency recorded for center 2 with different
RF powers in absence of applied magnetic fields. The horizontal axis is the
frequency in MHz, and the vertical axis is the relative change of the PL. (b)
ODMR signal vs. RF field strength and (c) linewidth vs. RF field strength.

Figure 6 (a) shows the recorded ODMR signal at different RF powers (0.1 W, 0.5
W, 1.0 W and 1.8 W). The ODMR spectrum recorded with 0.1 W fitted in double
Lorentzian function, and the obtained resonance frequencies are (72.8 $\pm$
0.3) MHz for the main peak and (36.4 $\pm$ 0.2) MHz for the smaller peak.The
frequency of the larger peak matches the literature value of the zero-field
splitting of the $V_{2}$ type $V_{Si}^{-}$ in $4H$-SiC [27]. At high RF power
the Rabi frequency of the system is comparable to the resonance frequency of
the system, the rotating wave approximation is no longer valid and causes
nonlinear processes, such as multi-RF photon absorption [35]. In our previous
work [35], we studied the multi-RF process in detail. In those experiment, the
large signal-to-noise ratio of the $V_{3}$ spin ensemble in allowed detailed
studies of the 1-, 2-, 3-, and 4-RF photon resonances. Here, the smaller peak
is at half the frequency of the main peak, so we attribute it to the 2-RF
photon peak. In the simulated spectra shown in Appendix G, Fig 13 (a), the
multi-photon peaks are more clearly visible. In the experiment, the signal due
to higher multi-photon absorption are not resolved due to limited signal-to-
noise ratio. To take these unresolved signal contributions into account, we
add a third Lorentzian component to the fitting function of the ODMR spectra
when the applied RF power is higher than 0.5 W. Figures 6 (b) and (c) show the
RF power dependence of the amplitudes and the linewidths for 1- and 2-photons
peaks. The dependence of the amplitude is fitted with the function

$S(P)=S_{max}[(P^{c/2}/(\varLambda_{0}+P^{c/2})],$ (23)

where S(P) is the ODMR signal amplitude and $P$ is the RF power. The exponent
$c$ is 1 (2) for the 1- (2)-photon transitions [35]. The values for $S_{max}$
and $\varLambda_{0}$ for the peaks are given in Table 2.

The linewidth $LW$ vs. the square root of the RF power $\sqrt{P}$ is fitted to
the function

$LW(P)=LW_{0}+a\,\sqrt{P}.$ (24)

The obtained fitting parameters, intercept $LW_{0}$ and slope $a$ are given in
Table 2. The low-power limit $LW_{0}$ corresponds to the inverse dephasing
time $1/(\pi LW_{0})$ [36]. The low-power limit of the linewidth corresponds
to a dephasing time of $32\pm 7$ ns. One contribution to the linewidth are
stray magnetic fields of about (0.5 G), which cause extra 2.8 MHz peak
broadening. Apart from that, the contrast and the linewidth depend on the RF
and laser intensity [37]. For the same kind of centers, $T_{2}^{*}$ = 34 $\pm$
4 $\mu$s was measured at 10 K using Ramsey interferometry in an external
magnetic field of $B_{0}$ = 36 G oriented along the crystal’s $c$-axis [23],
and similar values were obtained in deep-bulk defects in similar SiC crystals
[38, 4]. Due to differences in experimental conditions between the present
work and the study by Babin et al. [23], a quantitative comparison of the
dephasing rates estimated here from the low-power limit, $LW_{0}$, and the
$T_{2}^{*}$ measured in their study is not meaningful. However, it is worth
considering some conditions contributing to the differences. Babin et al.
performed their measurements at a low temperature of 10 K, while the
measurements presented here were obtained at room temperature. Additionally,
Babin et al. used the Ramsey method to measure free precession, whereas the
present measurements were obtained using the CW method. Furthermore, in Babin
et al.’s study, selective low-power resonant excitation ensured the system was
predominantly in the ground state. In contrast, in the present experiment, CW
excitation led to the system being mostly in the metastable state, which
contributes to the dephasing and line broadening. Lastly, Babin et al. applied
a magnetic field to lift the degeneracy of the electron spin states, which
reduces line-broadening from magnetic field noise and changes the dynamics of
the nuclear spin bath and its contribution to the line broadening [39]. Fig.
13 (b) of Appendix H shows the linewidth of the simulated spectra vs. the RF
coupling strength ($\Omega_{1}=g\mu_{B}B_{1})$, which looks similar to the
experimental data in Fig. 6 (c). As shown in Fig 13 (b) the width of the
2-photon resonance for low power tends to zero (which corresponds to
$T_{2}^{*}$ $\longrightarrow$ $\infty$ for 2-photon resonance) and the slope
$a_{1}$ of the 2-photon resonance is smaller than that of the 1-photon
resonance.

Peak | $c$ |  $S_{max}$(%) |  $\varLambda_{0}$($W^{c/2}$) |  $LW_{0}$ (MHz) |  $a$ (MHz/$\sqrt{W}$)
---|---|---|---|---|---
$1$-photon | 1 | 3.51$\pm$0.25 | 0.59$\pm$0.10 | 10.0 $\pm$ 2.3 | 35.9$\pm$3.3
$2$-photon | 2 | 1.2$\pm$0.6 | 0.6$\pm$0.6 | 3.3${}_{+4.0}^{-3.3}$ | 5$\pm$5
Table 2: Obtained fitting parameters from Eq. (23) for the ODMR-signal
amplitude and Eq. (24) for the linewidth vs RF field strength. Figure 7: ODMR
contrast for different centers.

We recorded the ODMR of ten more centers with 9 W RF power and 185 $k$W/cm2
laser intensity. We observed the contrast range and Fig. 7 shows the plot of
number of centers vs. ODMR contrast. On average, we observed a $\approx$ 3 %
contrast per center, with a minimum contrast of 1% and a maximum contrast of
6%. We also recorded ODMR spectra of one of the high contrast center for the
different RF and laser Powers; more details are given in appendix G.

##  V Discussion and Conclusion

Vacancy centers in silicon carbide are useful tools for sensing magnetic
fields and for other quantum technologies. With existing fabrication
technology, $V_{Si}^{-}$ can be produced on-demand with a high probability,
and reasonable resolution. Previously, the spin properties of a $V_{2}$ center
deep inside a $4H$-SiC sample have been studied using a solid immersion lens
to enhance the PL of the $V_{2}$ center [3]. Here we characterized shallow
single silicon-vacancy centers, which are very useful for quantum sensing
applications. The single center is located approximately 40 nm below the
surface, can be optically spin-polarized, and the measured optical transition
rates are slightly different from the previously measured deep silicon-vacancy
centers [3] and fitted well in 4-level model [19, 30].

The $T_{2}^{*}$ of transition $\pm$3/2 $\longleftrightarrow$$\pm$1/2 was 32
ns. Longer dephasing times are expected for the +1/2 $\longleftrightarrow$
-1/2 transition, since it is not affected by the zero-field splitting [17, 7];
this may be interesting for sensing applications. Observing this transition is
possible in an external magnetic field, which lifts the degeneracy of these
levels and can result in a population imbalance between them. The maximum ODMR
contrast of $\approx$6 % was obtained with 9 W RF power and 185 $k$W/cm2 laser
intensity.

To understand the exact cause of the difference between these transition rates
and the ODMR contrast of the shallow and deep centers in our future work, we
plan to perform experiments where we change the depth of the center, e.g. by
etching techique. Illumination for more than 24 hours with an intensity of 378
kW/ cm² reduces the count rates. The reduced count rate may be due to charge
state conversion [40]. Here we used a 785 nm laser, i.e. non-resonant pumping.
It may be possible to avoid the reduction of the count rate by using a
different laser wavelength close to zero phonon line. We will try this
approach in our future work.

In conclusion, we reported room temperature ODMR data from single shallow
$V_{Si}^{-}$ centers in $4H$-SiC. We observed higher ODMR contrast from in the
centers that were created with He+ implantation than in previous studies where
centers had been created by neutron irradiation [3, 41, 19]. Due to the higher
contrast, we are able to record ODMR of a center with PL $\approx$ 8$k$cps. An
ensemble of $V_{si}^{-}$ in SiC has a single dipole orientation, in contrast
to an ensemble of NV centers in diamond with four possible dipole
orientations. For some applications, this increases the signal by a factor of
four. So, this increased ODMR contrastdue to shallowness makes the silicon-
vacancy center a good candidate as NV center in diamonds for room temperatures
sensing applications such as NMR of a small sample volume and MRI of single
cells [42, 18, 43, 44].

## Appendix A: Sample

Arrays of $V_{Si}^{-}$ centres via implantation of He+ ions. He+ ions were
sent through a polymethyl methacryate (PMMA) mask with 100 nm diameter holes,
lithographed on the a face of an epitaxially grown SiC sample with a low
nitrogen concentration in the [N] = 4 $\times$ 1013 cm-3 range. The low He+
ion energy of 6 keV was chosen to minimize the crystal damage. To remove
residual lattice damage, the sample was annealed in argon atmosphere at 600° C
for 30 min [23]. Figure 2 shows the room-temperature PL map of an implanted
array using off-resonant excitation at 785 nm.

## Appendix B: ODMR setup

The excitation path (Fig. 8, grey background) begins with a 785 nm 250 mW
iBEAM-SMART-785-S-HP Laser (LA1) from Toptica which induces electronic
transitions of the $4H$-SiC-system. The 520 nm Lasertack 100 mW single mode
diode laser (LA2) is used for alignment. Each laser beam is directed by two
mirrors (M1 and M2 / M3 and M4) and coupled by the Thorlabs fiberports (C1/C2)
into a Thorlabs 532 nm/ 785 nm NG71F1 y-fiber which combines both beams. To
perform pulse experiments, an Acousto-Optic-Modulator (AOM) is placed in front
of the red laser (LA1). The zero order beam is detected by a S2386-44K6K
photodiode from Hamamatsu which monitors the stability of the laser output.
The 1st order diffracted beam is coupled into the y-fiber. Finally the
combined laser beam is coupled out of the y-fiber by fiberport (C3) and
directed by mirror (M5) to the DMLP805 long-pass dichroic mirror (DM) from
Thorlabs, which transmits wavelengths greater than 805 nm. Hence the combined
light beam is reflected and directed by mirror (M6) into an Olympus
UPLFLN100XO2 objective. It is mounted on a PI P-733.3CD XYZ-nanopositioner
which covers a range of 100 $\mu$m in X-/ Y- and 10 $\mu$m in Z-direction.
This XYZ-nanopositioner is operated by a PI E-727 Digital Multi-channel Piezo
Controller. The combined laser beam is focused onto the sample which is
mounted onto a circuit board, where RF-fields are applied through a 50 $\mu$m
copper wire. These RF-fields are generated by an Analog Devices AD9915 2.5
GSPS Direct Digital Synthesiser (DDS). The RF-power of range 20 MHz − 512 MHz
can be increased for several watts by a 50 W Mini Circuit LZY-1 amplifier
(AMP). It is important to connect a 50 $\Omega$ resistor at the other end of
the circuit board to prevent damage of the electronic devices due to RF-
reflections. Since the resistor is limited to 1 W, the RF-power is increased
by an attenuator (AT) beforehand. The circuit board itself is screwed tight
onto the sample stage that allows movements in Z-direction by a motorised
Standa 8MT173 translation stage which is operated by the Standa 8SMC5-USB-B8-1
motor controller. In Fig. 8, the circuit board is shown in yellow whereas the
objective is illustrated as a circle below it. All components of the detection
path (Fig. 8, green background) are selected for the near-infrared range of
805 nm-1000 nm. PL emitted from the sample is collected by the microscope
objective and directed by the mirror (M6) through the dichroic mirror (DM)
(bright red line in Fig. 8). It is then directed by the mirrors (M7) and (M8)
and focused by the lens (L1) with focal length of 100 mm, passes a 50 μm
pinhole (PH) and is then collimated by the lens (L2) with a focal length of
150 mm. The following waveplate ($\lambda$/2) allows to adjust the ratio
between transmitted and reflected PL at the polarized beam splitter (PBS) in
the box. Before entering the box via a tube, the PL is filtered by a 850 nm
long-pass-filter (F1). Depending on the type of measurement, a second filter
(F2) like a (950 $\pm$ 25) nm band-pass-filter or a 850 nm long-pass-filter is
used. The PL is splitted by the PBS and directed to the lens (L3/ L4), each
with focal length of 100 mm, which focuses the PL onto the Laser Components
Count-100N single photon detectors (SPD1 and SPD2, with photon detection
efficiency at 810 nm is 68 % and 59%). When performing an ODMR-measurement,
the registered events are counted by a Measurement Computing 1808X-USB DAQ-
card (DAQ), which has a 50 $\Omega$ resistor at each input to prevent
reflections. For intensity autocorrelation measurements, we used time tagger
and counters of quPSI from quτools.

Figure 8: Experimental confocal scan setup for the CW-ODMR (zfODMR), intensity
autocorrelation and pulse measurements. The detectors are isolated in a box
(in reddish-brown) to prevent noise from external light sources. PL can enter
the box via a tube (between F1 and F2). Excitation path (gray background): the
center is optically excited by a 785 nm laser. RF transitions are induced by a
50 $\mu$m copper wire over the sample. Detection path (green background): the
PL is detected by two single-photon detectors (SPD1 and SPD2), which are
connected to the DAQ-card (DAQ) for ODMR measurements. When performing a
$g^{2}$-measurement, the detectors are connected to the quPSI TCSPC device
(indicated by dashed lines).

## Appendix C: Details of intensity autocorrelation measurements

All but one $g^{2}$-measurements shown in Fig. 9 were taken with a bin width
$W_{b}$ of 486 ps, whereby the histogram data for $I$ = 378 kW/cm2 was
obtained with W of 162 ps. Eventually, all unnormalized histogram datasets
were normalized by $N_{norm}$ = $T$ $W_{b}$ $R_{1}$$R_{2}$ , with total time
$T$, bin width $W_{b}$ and count rates $R_{i}$ of the detectors $D_{i}$ and
afterwards smoothed to a bin width of 972 ps. The total count rates $R$ =
$R_{1}$ \+ $R_{2}$ are obtained by dividing the total number of counts by the
total time for the $g^{2}$-measurement. Finally, the individual count rates
were then obtained by $R_{1}$ = 0.6 $R$ and $R_{2}$ = 0.4 $R$. At $I$ = 2.6
kW/cm2, only the dark count rates of $R_{1}$, D = 251 counts/s and $R_{2}$,
$D$ = 129 counts/s at each detector contribute to the background due to
slightly different models. But a linear increase in background for increasing
excitation power is observed in Fig. 10, which can be considered by applying a
background-correction. Since the background-correction cannot be applied for a
background which only consists of the detectors dark counts, it was applied on
all histogram datasets except for the one at lowest excitation powers of 2.6
kW/cm2. Only the obtained normalized histogram data at smallest (orange
diamond) and at greatest (green star) optical excitation power are shown in
Fig. 3. All histogram datasets are shown in Fig. 9.

Figure 9: Complete series of intensity autocorrelation measurements performed
from 2.6 kW/cm2 to 378 kW/cm2. The histogram data is first normalized by
$N$norm = T $W_{b}$ $R_{1}$ $R_{2}$ and then smoothed to a bin width of 972
ps. Besides for the lowest $I$ = 2.6 kW/cm2, all histogram dataset are
corrected for their background. Each solid line follows the three-level
correlation function from Eq. (17). Figure 10: PL-signal at different
intensities $I$ of the center on which $g^{2}$-measurements were performed.
The PL-signal is fitted by: $S=S_{max}I/(I_{0}+I)$ with (7.8 $\pm$ 0.3)
kcounts/s and $I_{0}$= (44 $\pm$ 3) kW/cm2, whereas the background increases
linearly with the slope m = (15.9 $\pm$ 0.6) counts cm2/(kW s) and intercept b
= (309 $\pm$ 20) counts/s.

## Appendix D: Eigenvectors

Eigenvectors of the three level model

$\vec{v}_{1}=\left(\begin{array}[]{c}1\\\ \frac{k_{sg}}{k_{es}}\\\
\frac{k_{sg}(k_{eg}+k_{es})}{k_{es}k_{ge}}\end{array}\right)$ ,
$\vec{v}_{2}=\left(\begin{array}[]{c}-1\\\ G\\\ -F\end{array}\right)$,
$\vec{v}_{3}=\left(\begin{array}[]{c}-1\\\ -G\\\ F\end{array}\right)$,

where

$F=\frac{H+k_{eg}-k_{es}+k_{ge}-k_{sg}}{2k_{es}}$;

$G=\frac{H+k_{eg}+k_{es}+k_{ge}-k_{sg}}{2k_{es}}$

$H=\sqrt{k_{eg}^{2}+2k_{eg}(k_{es}+k_{ge}-k_{sg})-J}$

$J=2\,k_{sg}(k_{es}+k_{ge})+(k_{es}-k_{ge})^{2}+k_{sg}^{2}$

## Appendix F: Optimal laser power for ODMR

Figure 11: PL-signal at different intensities $I$. The PL-signal is fitted by:
$S=S_{max}I/(I_{0}+I)$ with (6.3 $\pm$ 0.1) $k$cps and $I_{0}$= (53$\pm$ 2)
kW/cm2, whereas the background increases linearly with the slope m = (4.9
$\pm$ 0.1) counts cm2/(kW s) and intercept b = (130 $\pm$ 8) cps.

A suitable electronical excitation Intensity $I$ can be obtained from the
saturation behaviour of the PL-signal. In Fig. 11, a maximum PL- signal of
(6.3 $\pm$ 0.1) $k$cps can be obtained, while the background increases
linearly with increasing $I$. Therefore, $I_{0}$ = (53 $\pm$ 2) kW/cm2 is
chosen for ODMR-measurements, since the PL-signal is almost saturated while
the background is relatively small. Even if a high background is mostly
compensated by the references detection ΔPL, it has the advantage that the
autofocus works more reliably at low background.

## Appendix G: ODMR simulations

Figure 12: Experimental and simulated ODMR plots for silicon vacancy with the
RF Hamiltonian of Eq. 25. Figure 13: (a) Simulated ODMR spectra at different
RF strengths. (b) Linewidth of 1-photon and 2-photon resonances vs. RF
coupling strength.

For the numerical simulation, we used the Lindblad master equation

$\displaystyle\frac{\partial\rho}{\partial t}$ $\displaystyle=$
$\displaystyle-2\pi i\>[{\cal
H}_{t}(t),\rho]+\sum_{\alpha,\beta,\delta_{1,..,5}}L_{i}^{\dagger}.\rho.L_{i}-\frac{1}{2}\left\\{L_{i}^{\dagger}L_{i},\rho\right\\}$
(25)

here ${\cal H}_{t}(t)={\cal H}+{\cal H}_{RF}(t)$,
$L_{\alpha}=\sqrt{2}\left(\begin{array}[]{cccc}0&\sqrt{\gamma}&0&0\\\
\sqrt{\gamma}&0&\sqrt{\alpha}&0\\\ 0&\sqrt{\alpha}&0&\sqrt{\gamma}\\\
0&0&\sqrt{\gamma}&0\end{array}\right)\thickapprox\sqrt{2\alpha}\>S_{x}$ drives
the spin-lattice relaxation process, $\gamma=3\alpha/4$,
$L_{\beta}=\sqrt{2\beta}\>S_{z}$ is the Lindblad operator for the dephasing
process, and $L_{\delta_{1,..,5}}$ are the Lindblad operators for the optical
pumping [35]

$L_{\delta_{1}}=\sqrt{\delta}\left(\begin{array}[]{cccc}0&0&0&0\\\ 1&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\end{array}\right)$,
$L_{\delta_{2}}=\sqrt{\delta}\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&1\\\
0&0&0&0\\\ 0&0&0&0\end{array}\right)$,

$L_{\delta_{3}}=\sqrt{\delta}\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\
1&0&0&0\\\ 0&0&0&0\end{array}\right)$,
$L_{\delta_{4}}=\sqrt{\delta}\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\
0&0&0&1\\\ 0&0&0&0\end{array}\right)$,

$L_{\delta_{5}}=\sqrt{\delta}\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&1&0\\\
0&1&0&0\\\ 0&0&0&0\end{array}\right)$, and $\delta$ is the optical pumping
rate [7].

The interaction Hamiltonian between the RF field and the spins is

${\cal H}_{RF}(t)=\Omega_{1}\text{cos}(2\pi\omega t)(S_{x}+S_{z}),$

where $\Omega_{1}$ = $g\mu_{B}B_{1}$ represents the strength of the coupling
to the RF field $B_{1}$ in frequency units and $\omega$ is the oscillation
frequency of the field. Figure 12 shows the experimental and simulated ODMR
spectra at 8 MHz RF coupling strengths with $\alpha$ = 7 ms-1, $\beta$ = 10
$\mu$s-1, and $\delta=$ 185 ms-1; the simulated spectrum matches well with the
experimentally recorded ODMR at 1 W RF power. Since we are using simple
optical pumping model [7], the amplitude of the simulated spectrum is
normalized to the maximal ODMR contrast for the plot. For the simulated
spectrum, we integrated Eq. (25) for 1.5 $\mu$s for every frequency.
Additional details can be found in our previous work [35]. Figure 13 (a) shows
the simulated ODMR spectra at different RF coupling strengths with the same
parameters used for simulating ODMR spectrum in Fig 12 except for the value of
$\beta$, which we reduced here to 2.5 $\mu s^{-1}$ to better resolve the
multi-photon peaks.

Peak |  $LW_{0}$(MHz) | $a$
---|---|---
1 | 0.34 $\pm$0.21 | 6.80 $\pm$0.05
2 |  $\approx$0 | 0.63 $\pm$0.05
Table 3: Obtained fitting parameters from Eq. (26) for the linewidth vs RF
field strength.

In Fig 13, with an RF coupling strength of 2 MHz, we can see 1-photon, and
2-photon peaks at 72 MHz and 36 MHz and a small signal of 3-photon peak at 24
MHz. At 8 MHz, the 5-, 4- and 3-photon peaks are also visible and close to
each other, and therefore difficult to resolve. The linewidth of the 1- and
2-photon peaks are extracted by fitting the simulated ODMR to a sum of
Lorentzians. Fig 13 (b) shows the resulting dependence of the linewidth $LW$
vs. the square root of the RF power $\varLambda$ itogether with a fir to the
function

$LW(P)=LW_{0}+a_{1}\,\Omega_{1}.$ (26)

The obtained fitting parameters, intercept $LW_{0}$ and the slope $a$ are
given in Table (3).

## Appendix H: ODMR with different rf and laser powers

Figure 14: ODMR spectra recorded for one of the high contrast centers (out of
ten measured centers), (a) with different applied RF power and 184 kW/ cm²
laser intensity; (b) with different laser intensities and 2.6 W RF power. The
horizontal axis is the frequency in MHz, and the vertical axis is the relative
change of the PL%.

Figure 14 (a) shows the ODMR signal recorded for one of the high-contrast
centers with different RF power, keeping the laser intensity at 184 kW/ cm²
and (b) for different Laser intensity at 2 W RF power. At 9 W RF power, the
ODMR signal contrast is 5.3 %, and the laser intensity does not affect the
ODMR contrast significantly. In the course of recording these data, the PL
count of the center fell from 8 kcps to 4 kcps which impacts the signal-to-
noise ratio of the later recorded variable laser intensity ODMR spectra.

###### Acknowledgements.

This work was supported by the Deutsche Forschungsgemeinschaft in the frame of
the ICRC TRR 160 (Project No. C7). J.U.H. acknowledge support from Swedish
Research Council under VR grant No.2020-05444 and Knut and Alice Wallenberg
Foundation (grant no. KAW 2018.0071). J.U.H. and F.K. acknowledge support from
the European Union through the QuantERA grant InQuRe. F.K. acknowledges
support from the German ministry of education and research (BMBF, grant no.
16KIS1639K, 16KISQ013 and 13N16219).

## References

  * Koehl  _et al._ [2011] W. F. Koehl, B. B. Buckley, F. J. Heremans, G. Calusine, and D. D. Awschalom, Room temperature coherent control of defect spin qubits in silicon carbide, Nature 479, 84 (2011).
  * Kraus  _et al._ [2013] H. Kraus, V. A. Soltamov, D. Riedel, S. Väth, F. Fuchs, A. Sperlich, P. G. Baranov, V. Dyakonov, and G. V. Astakhov, Room-temperature quantum microwave emitters based on spin defects in silicon carbide, Nature Physics 10, 157 (2013).
  * Widmann  _et al._ [2015] M. Widmann, S.-Y. Lee, T. Rendler, N. T. Son, H. Fedder, S. Paik, L.-P. Yang, N. Zhao, S. Yang, I. Booker, _et al._ , Coherent control of single spins in silicon carbide at room temperature, Nature materials 14, 164 (2015).
  * Nagy  _et al._ [2019] R. Nagy, M. Niethammer, M. Widmann, Y.-C. Chen, P. Udvarhelyi, C. Bonato, J. U. Hassan, R. Karhu, I. G. Ivanov, N. T. Son, _et al._ , High-fidelity spin and optical control of single silicon-vacancy centres in silicon carbide, Nature communications 10, 1954 (2019).
  * Soltamov  _et al._ [2012] V. A. Soltamov, A. A. Soltamova, P. G. Baranov, and I. I. Proskuryakov, Room temperature coherent spin alignment of silicon vacancies in $4h$\- and $6h$-sic, Phys. Rev. Lett. 108, 226402 (2012).
  * Riedel  _et al._ [2012] D. Riedel, F. Fuchs, H. Kraus, S. Väth, A. Sperlich, V. Dyakonov, A. A. Soltamova, P. G. Baranov, V. A. Ilyin, and G. V. Astakhov, Resonant addressing and manipulation of silicon vacancy qubits in silicon carbide, Physical review letters 109, 226402 (2012).
  * Singh  _et al._ [2021] H. Singh, A. N. Anisimov, I. D. Breev, P. G. Baranov, and D. Suter, Optical spin initialization of spin-$\frac{3}{2}$ silicon vacancy centers in $6h\text{$-$}\mathrm{SiC}$ at room temperature, Phys. Rev. B 103, 104103 (2021).
  * Carter  _et al._ [2015] S. G. Carter, O. O. Soykal, P. Dev, S. E. Economou, and E. R. Glaser, Spin coherence and echo modulation of the silicon vacancy in $4h-\mathrm{SiC}$ at room temperature, Phys. Rev. B 92, 161202(R) (2015).
  * Simin  _et al._ [2017] D. Simin, H. Kraus, A. Sperlich, T. Ohshima, G. V. Astakhov, and V. Dyakonov, Locking of electron spin coherence above 20 ms in natural silicon carbide, Phys. Rev. B 95, 161201(R) (2017).
  * Christle  _et al._ [2015] D. J. Christle, A. L. Falk, P. Andrich, P. V. Klimov, J. U. Hassan, N. T. Son, E. Janzén, T. Ohshima, and D. D. Awschalom, Isolated electron spins in silicon carbide with millisecond coherence times, Nature materials 14, 160 (2015).
  * Singh  _et al._ [2020] H. Singh, A. N. Anisimov, S. S. Nagalyuk, E. N. Mokhov, P. G. Baranov, and D. Suter, Experimental characterization of spin-3/2 silicon vacancy centers in $6h$-sic, Phys. Rev. B 101, 134110 (2020).
  * Lekavicius  _et al._ [2022] I. Lekavicius, R.L. Myers-Ward, D.J. Pennachio, J.R. Hajzus, D.K. Gaskill, A.P. Purdy, A.L. Yeats, P.G. Brereton, E.R. Glaser, T.L. Reinecke, and S.G. Carter, Orders of magnitude improvement in coherence of silicon-vacancy ensembles in isotopically purified $4\mathrm{H}$-$\mathrm{SiC}$, PRX Quantum 3, 010343 (2022).
  * Udvarhelyi  _et al._ [2020] P. Udvarhelyi, G. m. H. Thiering, N. Morioka, C. Babin, F. Kaiser, D. Lukin, T. Ohshima, J. Ul-Hassan, N. T. Son, J. Vučković, J. Wrachtrup, and A. Gali, Vibronic states and their effect on the temperature and strain dependence of silicon-vacancy qubits in $4h$-$\mathrm{Si}\mathrm{C}$, Phys. Rev. Applied 13, 054017 (2020).
  * Shang  _et al._ [2020] Z. Shang, A. Hashemi, Y. Berencén, H.-P. Komsa, P. Erhart, S. Zhou, M. Helm, A. V. Krasheninnikov, and G. V. Astakhov, Local vibrational modes of si vacancy spin qubits in sic, Phys. Rev. B 101, 144109 (2020).
  * Anisimov  _et al._ [2016] A. Anisimov, D. Simin, V. A. Soltamov, S. P. Lebedev, P. G. Baranov, G. V. Astakhov, and V. Dyakonov, Optical thermometry based on level anticrossing in silicon carbide, Scientific reports 6, 33301 (2016).
  * Abraham  _et al._ [2021] J. B. S. Abraham, C. Gutgsell, D. Todorovski, S. Sperling, J. E. Epstein, B. S. Tien-Street, T. M. Sweeney, J. J. Wathen, E. A. Pogue, P. G. Brereton, T. M. McQueen, W. Frey, B. D. Clader, and R. Osiander, Nanotesla magnetometry with the silicon vacancy in silicon carbide, Phys. Rev. Appl. 15, 064022 (2021).
  * Soltamov  _et al._ [2019] V. A. Soltamov, C. Kasper, A. V. Poshakinskiy, A. N. Anisimov, E. N. Mokhov, A. Sperlich, S. A. Tarasenko, P. G. Baranov, G. V. Astakhov, and V. Dyakonov, Excitation and coherent control of spin qudit modes in silicon carbide at room temperature, Nature Communications 10, 1678 (2019).
  * Taylor  _et al._ [2008] J. M. Taylor, P. Cappellaro, L. Childress, L. Jiang, D. Budker, P. Hemmer, A. Yacoby, R. Walsworth, and M. Lukin, High-sensitivity diamond magnetometer with nanoscale resolution, Nature Physics 4, 810 (2008).
  * Fuchs  _et al._ [2015] F. Fuchs, B. Stender, M. Trupke, D. Simin, J. Pflaum, V. Dyakonov, and G. Astakhov, Engineering near-infrared single-photon emitters with optically active spins in ultrapure silicon carbide, Nature communications 6, 1 (2015).
  * Hain  _et al._ [2014] T. C. Hain, F. Fuchs, V. A. Soltamov, P. G. Baranov, G. V. Astakhov, T. Hertel, and V. Dyakonov, Excitation and recombination dynamics of vacancy-related spin centers in silicon carbide, Journal of Applied Physics 115, 133508 (2014), https://doi.org/10.1063/1.4870456 .
  * Kasper  _et al._ [2020] C. Kasper, D. Klenkert, Z. Shang, D. Simin, A. Gottscholl, A. Sperlich, H. Kraus, C. Schneider, S. Zhou, M. Trupke, W. Kada, T. Ohshima, V. Dyakonov, and G. V. Astakhov, Influence of irradiation on defect spin coherence in silicon carbide, Phys. Rev. Appl. 13, 044054 (2020).
  * Hornos  _et al._ [2011] T. Hornos, A. Gali, and B. G. Svensson, Large-scale electronic structure calculations of vacancies in 4h-sic using the heyd-scuseria-ernzerhof screened hybrid density functional, in  _Materials Science Forum_ , Vol. 679 (Trans Tech Publ, 2011) pp. 261–264.
  * Babin  _et al._ [2022] C. Babin, R. Stöhr, N. Morioka, T. Linkewitz, T. Steidl, R. Wörnle, D. Liu, E. Hesselmeier, V. Vorobyov, A. Denisenko, _et al._ , Fabrication and nanophotonic waveguide integration of silicon carbide colour centres with preserved spin-optical coherence, Nature materials 21, 67 (2022).
  * Wang  _et al._ [2017] J. Wang, X. Zhang, Y. Zhou, K. Li, Z. Wang, P. Peddibhotla, F. Liu, S. Bauerdick, A. Rudzinski, Z. Liu, _et al._ , Scalable fabrication of single silicon vacancy defect arrays in silicon carbide using focused ion beam, Acs Photonics 4, 1054 (2017).
  * Pavunny  _et al._ [2021] S. P. Pavunny, A. L. Yeats, H. B. Banks, E. Bielejec, R. L. Myers-Ward, M. T. DeJarld, A. S. Bracker, D. K. Gaskill, and S. G. Carter, Arrays of si vacancies in 4h-sic produced by focused li ion beam implantation, Scientific reports 11, 1 (2021).
  * Ivády  _et al._ [2017] V. Ivády, J. Davidsson, N. T. Son, T. Ohshima, I. A. Abrikosov, and A. Gali, Identification of si-vacancy related room-temperature qubits in $4h$ silicon carbide, Phys. Rev. B 96, 161114(R) (2017).
  * Baranov  _et al._ [2011] P. G. Baranov, A. P. Bundakova, A. A. Soltamova, S. B. Orlinskii, I. V. Borovykh, R. Zondervan, R. Verberk, and J. Schmidt, Silicon vacancy in sic as a promising quantum system for single-defect and single-photon spectroscopy, Phys. Rev. B 83, 125203 (2011).
  * Fox  _et al._ [2006] A. M. Fox, M. Fox, _et al._ ,  _Quantum optics: an introduction_ , Vol. 15 (Oxford university press, 2006).
  * Basché  _et al._ [1992] T. Basché, W. E. Moerner, M. Orrit, and H. Talon, Photon antibunching in the fluorescence of a single dye molecule trapped in a solid, Phys. Rev. Lett. 69, 1516 (1992).
  * Neu  _et al._ [2011] E. Neu, D. Steinmetz, J. Riedrich-Möller, S. Gsell, M. Fischer, M. Schreck, and C. Becher, Single photon emission from silicon-vacancy colour centres in chemical vapour deposition nano-diamonds on iridium, New Journal of Physics 13, 025012 (2011).
  * Kasper [2021] C. A. Kasper,  _Engineering of Highly Coherent Silicon Vacancy Defects in Silicon Carbide_ , Ph.D. thesis, Universität Würzburg (2021).
  * Morioka  _et al._ [2022] N. Morioka, D. Liu, O. O. Soykal, I. Gediz, C. Babin, R. Stöhr, T. Ohshima, N. T. Son, J. Ul-Hassan, F. Kaiser, and J. Wrachtrup, Spin-optical dynamics and quantum efficiency of a single v1 center in silicon carbide, Phys. Rev. Appl. 17, 054005 (2022).
  * Suter [2020] D. Suter, Optical detection of magnetic resonance, Magnetic Resonance 1, 115 (2020).
  * Suter and Jelezko [2017] D. Suter and F. Jelezko, Single-spin magnetic resonance in the nitrogen-vacancy center of diamond, Progress in Nuclear Magnetic Resonance Spectroscopy 98-99, 50 (2017).
  * Singh  _et al._ [2022] H. Singh, M. A. Hollberg, A. N. Anisimov, P. G. Baranov, and D. Suter, Multi-photon multi-quantum transitions in the spin-$\frac{3}{2}$ silicon-vacancy centers of sic, Phys. Rev. Research 4, 023022 (2022).
  * Wertz [2012] J. Wertz,  _Electron spin resonance: elementary theory and practical applications_ (Springer Science & Business Media, 2012).
  * Jensen  _et al._ [2013] K. Jensen, V. M. Acosta, A. Jarmola, and D. Budker, Light narrowing of magnetic resonances in ensembles of nitrogen-vacancy centers in diamond, Phys. Rev. B 87, 014115 (2013).
  * Bourassa  _et al._ [2020] A. Bourassa, C. P. Anderson, K. C. Miao, M. Onizhuk, H. Ma, A. L. Crook, H. Abe, J. Ul-Hassan, T. Ohshima, N. T. Son, _et al._ , Entanglement and control of single nuclear spins in isotopically engineered silicon carbide, Nature Materials 19, 1319 (2020).
  * Bulancea-Lindvall  _et al._ [2021] O. Bulancea-Lindvall, N. T. Son, I. A. Abrikosov, and V. Ivády, Dipolar spin relaxation of divacancy qubits in silicon carbide, npj Computational Materials 7, 213 (2021).
  * Wolfowicz  _et al._ [2017] G. Wolfowicz, C. P. Anderson, A. L. Yeats, S. J. Whiteley, J. Niklas, O. G. Poluektov, F. J. Heremans, and D. D. Awschalom, Optical charge state control of spin defects in 4h-sic, Nature communications 8, 1 (2017).
  * Wang  _et al._ [2019] J.-F. Wang, Q. Li, F.-F. Yan, H. Liu, G.-P. Guo, W.-P. Zhang, X. Zhou, L.-P. Guo, Z.-H. Lin, J.-M. Cui, _et al._ , On-demand generation of single silicon vacancy defects in silicon carbide, Acs Photonics 6, 1736 (2019).
  * Glenn  _et al._ [2015] D. R. Glenn, K. Lee, H. Park, R. Weissleder, A. Yacoby, M. D. Lukin, H. Lee, R. L. Walsworth, and C. B. Connolly, Single-cell magnetic imaging using a quantum diamond microscope, Nature methods 12, 736 (2015).
  * Glenn  _et al._ [2018] D. R. Glenn, D. B. Bucher, J. Lee, M. D. Lukin, H. Park, and R. L. Walsworth, High-resolution magnetic resonance spectroscopy using a solid-state spin sensor, Nature 555, 351 (2018).
  * Bucher  _et al._ [2020] D. B. Bucher, D. R. Glenn, H. Park, M. D. Lukin, and R. L. Walsworth, Hyperpolarization-enhanced nmr spectroscopy with femtomole sensitivity using quantum defects in diamond, Phys. Rev. X 10, 021053 (2020).


* [14] T. Scaffidi, D. E. Parker, and R. Vasseur, “Gapless symmetry-protected topological order,” Physical Review X 7 no. 4, (2017) 041048\.
  * [15] R. Verresen, R. Thorngren, N. G. Jones, and F. Pollmann, “Gapless topological phases and symmetry-enriched quantum criticality,” arXiv e-prints (May, 2019) arXiv:1905.06969, arXiv:1905.06969 [cond-mat.str-el].
  * [16] R. Thorngren, A. Vishwanath, and R. Verresen, “Intrinsically Gapless Topological Phases,” arXiv:2008.06638 [cond-mat.str-el].
  * [17] S. C. Furuya and M. Oshikawa, “Symmetry Protection of Critical Phases and a Global Anomaly in $1+1$ Dimensions,” Phys. Rev. Lett. 118 no. 2, (2017) 021601, arXiv:1503.07292 [cond-mat.stat-mech].
  * [18] Y. Yao, C.-T. Hsieh, and M. Oshikawa, “Anomaly matching and symmetry-protected critical phases in $SU(N)$ spin systems in 1+1 dimensions,” Phys. Rev. Lett. 123 no. 18, (2019) 180201, arXiv:1805.06885 [cond-mat.str-el].
  * [19] R. Verresen, J. Bibo, and F. Pollmann, “Quotient symmetry protected topological phenomena,” arXiv preprint arXiv:2102.08967 (2021) .
  * [20] A. Keselman and E. Berg, “Gapless symmetry-protected topological phase of fermions in one dimension,” Physical Review B 91 no. 23, (2015) 235309.
  * [21] T. Grover and A. Vishwanath, “Quantum Criticality in Topological Insulators and Superconductors: Emergence of Strongly Coupled Majoranas and Supersymmetry,” arXiv:1206.1332 [cond-mat.str-el].
  * [22] W. Ji, S.-H. Shao, and X.-G. Wen, “Topological Transition on the Conformal Manifold,” Phys. Rev. Res. 2 no. 3, (2020) 033317, arXiv:1909.01425 [cond-mat.str-el].
  * [23] U. Borla, R. Verresen, J. Shah, and S. Moroz, “Gauging the Kitaev chain,” SciPost Phys. 10 (2021) 148. https://scipost.org/10.21468/SciPostPhys.10.6.148.
  * [24] X. Wang, L. Li, and J. Wu, “Stability and fine structure of symmetry-enriched quantum criticality in a spin ladder triangular model,” arXiv preprint arXiv:2306.11446 (2023) .
  * [25] Y. Hidaka, S. C. Furuya, A. Ueda, and Y. Tada, “Gapless symmetry-protected topological phase of quantum antiferromagnets on anisotropic triangular strip,” Phys. Rev. B 106 no. 14, (2022) 144436, arXiv:2205.15525 [cond-mat.str-el].
  * [26] X.-J. Yu, R.-Z. Huang, H.-H. Song, L. Xu, C. Ding, and L. Zhang, “Conformal boundary conditions of symmetry-enriched quantum critical spin chains,” Phys. Rev. Lett. 129 (Nov, 2022) 210601. https://link.aps.org/doi/10.1103/PhysRevLett.129.210601.
  * [27] D. E. Parker, T. Scaffidi, and R. Vasseur, “Topological Luttinger liquids from decorated domain walls,” Phys. Rev. B 97 no. 16, (Apr., 2018) 165114, arXiv:1711.09106 [cond-mat.str-el].
  * [28] D. E. Parker, R. Vasseur, and T. Scaffidi, “Topologically Protected Long Edge Coherence Times in Symmetry-Broken Phases,” Phys. Rev. Lett. 122 no. 24, (June, 2019) 240605, arXiv:1808.07485 [cond-mat.str-el].
  * [29] J. Zhao, Z. Yan, M. Cheng, and Z. Y. Meng, “Higher-form symmetry breaking at Ising transitions,” Phys. Rev. Res. 3 no. 3, (2021) 033024, arXiv:2011.12543 [cond-mat.str-el].
  * [30] H. Yang, L. Li, K. Okunishi, and H. Katsura, “Duality, criticality, anomaly, and topology in quantum spin-1 chains,” arXiv preprint arXiv:2203.15791 (2022) .
  * [31] R. Ma, L. Zou, and C. Wang, “Edge physics at the deconfined transition between a quantum spin Hall insulator and a superconductor,” SciPost Phys. 12 (2022) 196. https://scipost.org/10.21468/SciPostPhys.12.6.196.
  * [32] R. Wen and A. C. Potter, “Bulk-boundary correspondence for intrinsically-gapless SPTs from group cohomology,” arXiv:2208.09001 [cond-mat.str-el].
  * [33] L. Li, M. Oshikawa, and Y. Zheng, “Intrinsically/Purely Gapless-SPT from Non-Invertible Duality Transformations,” arXiv:2307.04788 [cond-mat.str-el].
  * [34] J. Wang, X.-G. Wen, and E. Witten, “Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions,” Phys. Rev. X 8 no. 3, (2018) 031048, arXiv:1705.06728 [cond-mat.str-el].
  * [35] L. Li, M. Oshikawa, and Y. Zheng, “Non-Invertible Duality Transformation Between SPT and SSB Phases,” arXiv:2301.07899 [cond-mat.str-el].
  * [36] X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, “Symmetry protected topological orders and the group cohomology of their symmetry group,” Phys. Rev. B 87 no. 15, (2013) 155114, arXiv:1106.4772 [cond-mat.str-el].
  * [37] Q.-R. Wang, Y. Qi, and Z.-C. Gu, “Anomalous Symmetry Protected Topological States in Interacting Fermion Systems,” Phys. Rev. Lett. 123 no. 20, (Nov., 2019) 207003, arXiv:1810.12899 [cond-mat.str-el].
  * [38] E. Witten, “The ”Parity” Anomaly On An Unorientable Manifold,” Phys. Rev. B 94 no. 19, (2016) 195150, arXiv:1605.02391 [hep-th].
  * [39] X. Chen, Z.-C. Gu, and X.-G. Wen, “Classification of gapped symmetric phases in one-dimensional spin systems,” Physical Review B 83 no. 3, (Jan, 2011) . http://dx.doi.org/10.1103/PhysRevB.83.035107.
  * [40] F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa, “Symmetry protection of topological phases in one-dimensional quantum spin systems,” Physical Review B 85 no. 7, (Feb, 2012) . http://dx.doi.org/10.1103/PhysRevB.85.075125.
  * [41] J. C. Wang, Z.-C. Gu, and X.-G. Wen, “Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond,” Phys. Rev. Lett. 114 no. 3, (2015) 031601, arXiv:1405.7689 [cond-mat.str-el].
  * [42] M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases,” Physical Review B 86 no. 11, (2012) 115109\.
  * [43] Y. Yao and M. Oshikawa, “Twisted boundary condition and Lieb-Schultz-Mattis ingappability for discrete symmetries,” Phys. Rev. Lett. 126 no. 21, (2021) 217201, arXiv:2010.09244 [cond-mat.str-el].
  * [44] U. Borla, R. Verresen, J. Shah, and S. Moroz, “Gauging the Kitaev chain,” SciPost Phys. 10 no. 6, (2021) 148, arXiv:2010.00607 [cond-mat.str-el].
  * [45] M. Suzuki, “Relationship among Exactly Soluble Models of Critical Phenomena. I*): 2D Ising Model, Dimer Problem and the Generalized XY-Model,” Progress of Theoretical Physics 46 no. 5, (11, 1971) 1337–1359, https://academic.oup.com/ptp/article-pdf/46/5/1337/5268367/46-5-1337.pdf. https://doi.org/10.1143/PTP.46.1337.
  * [46] W. Son, L. Amico, R. Fazio, A. Hamma, S. Pascazio, and V. Vedral, “Quantum phase transition between cluster and antiferromagnetic states,” EPL (Europhysics Letters) 95 no. 5, (Aug, 2011) 50001. https://doi.org/10.1209/0295-5075/95/50001.
  * [47] B. Zeng, X. Chen, D.-L. Zhou, and X.-G. Wen, “Quantum information meets quantum matter – from quantum entanglement to topological phase in many-body systems,” 2018.
  * [48] T. Kennedy and H. Tasaki, “Hidden ${\mathrm{z}}_{2}$×${\mathrm{z}}_{2}$ symmetry breaking in haldane-gap antiferromagnets,” Phys. Rev. B 45 (Jan, 1992) 304–307. https://link.aps.org/doi/10.1103/PhysRevB.45.304.
  * [49] T. Kennedy and H. Tasaki, “Hidden symmetry breaking and the haldane phase ins=1 quantum spin chains,” Communications in Mathematical Physics 147 no. 3, (1992) 431–484. https://doi.org/10.1007/BF02097239.
  * [50] M. Oshikawa, “Hidden z2* z2 symmetry in quantum spin chains with arbitrary integer spin,” Journal of Physics: Condensed Matter 4 no. 36, (1992) 7469.
  * [51] X.-G. Wen, “Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions,” Phys. Rev. B 89 no. 3, (2014) 035147, arXiv:1301.7675 [cond-mat.str-el].
  * [52] L. H. Santos and J. Wang, “Symmetry-protected many-body Aharonov-Bohm effect,” Phys. Rev. B 89 no. 19, (2014) 195122, arXiv:1310.8291 [quant-ph].
  * [53] J. Wang and X.-G. Wen, “Non-Abelian string and particle braiding in topological order: Modular SL(3,$\mathbb{Z}$) representation and (3+1) -dimensional twisted gauge theory,” Phys. Rev. B 91 no. 3, (2015) 035134, arXiv:1404.7854 [cond-mat.str-el].
  * [54] C. Wang and M. Levin, “Braiding Statistics of Loop Excitations in Three Dimensions,” Phys. Rev. Lett. 113 no. 8, (Aug., 2014) 080403, arXiv:1403.7437 [cond-mat.str-el].
  * [55] Z.-C. Gu, J. C. Wang, and X.-G. Wen, “Multi-kink topological terms and charge-binding domain-wall condensation induced symmetry-protected topological states: Beyond Chern-Simons/BF theory,” Phys. Rev. B 93 no. 11, (2016) 115136, arXiv:1503.01768 [cond-mat.str-el].
  * [56] M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases,” Phys. Rev. B 86 no. 11, (Sept., 2012) 115109, arXiv:1202.3120 [cond-mat.str-el].
  * [57] C. Wang and M. Levin, “Topological invariants for gauge theories and symmetry-protected topological phases,” Phys. Rev. B 91 no. 16, (Apr., 2015) 165119, arXiv:1412.1781 [cond-mat.str-el].
  * [58] E. Lieb, T. Schultz, and D. Mattis, “Two soluble models of an antiferromagnetic chain,” Annals of Physics 16 no. 3, (1961) 407–466.
  * [59] A. Seidel and D.-H. Lee, “The luther-emery liquid: Spin gap and anomalous flux period,” Phys. Rev. B 71 (Jan, 2005) 045113. https://link.aps.org/doi/10.1103/PhysRevB.71.045113.
  * [60] N. Seiberg, “Thoughts About Quantum Field Theory,” vision talk in Strings 2019 .
  * [61] H. Yang, L. Li, K. Okunishi, and H. Katsura, “Duality, criticality, anomaly, and topology in quantum spin-1 chains,” 2022.  https://arxiv.org/abs/2203.15791.
  * [62] I. Affleck and E. H. Lieb, “A proof of part of haldane’s conjecture on spin chains,” in Condensed Matter Physics and Exactly Soluble Models, pp. 235–247.  Springer, 1986.
  * [63] M. Oshikawa, M. Yamanaka, and I. Affleck, “Magnetization plateaus in spin chains: “haldane gap” for half-integer spins,” Phys. Rev. Lett. 78 (Mar, 1997) 1984–1987. https://link.aps.org/doi/10.1103/PhysRevLett.78.1984.
  * [64] M. Cheng, M. Zaletel, M. Barkeshli, A. Vishwanath, and P. Bonderson, “Translational symmetry and microscopic constraints on symmetry-enriched topological phases: A view from the surface,” Phys. Rev. X 6 (Dec, 2016) 041068. https://link.aps.org/doi/10.1103/PhysRevX.6.041068.
  * [65] X. Chen, Y.-M. Lu, and A. Vishwanath, “Symmetry-protected topological phases from decorated domain walls.,” Nature communications 5 (2014) 3507.
  * [66] E. Brezin and J. Zinnjustin, Fields, strings and critical phenomena.  Fields, strings and critical phenomena, 1990.
  * [67] Y. Fuji, “Effective field theory for one-dimensional valence-bond-solid phases and their symmetry protection,” Phys. Rev. B 93 (Mar, 2016) 104425. https://link.aps.org/doi/10.1103/PhysRevB.93.104425.