# VeHIF: An Accessible Vegetation High-Impedance Fault Data Set Format

Douglas P. S. Gomes, Cagil Ozansoy

###### Abstract

High-impedance faults are a challenging problem in power distribution systems.
They often do not trigger protection devices and can result in serious hazards
such as igniting fires when in contact with vegetation. The current research
field dedicated to studying these faults is extensive but suffers from a
constraining bottleneck of a lack of real experimental data. Many works set to
detect and localize such faults rely on high-impedance fault low-fidelity
models, and the lack of public data sets makes it impractical to have
objective performance benchmarks. This letter describes and proposes a format
for a data set of more than 900 vegetation high-impedance faults funded by the
Victorian Government in Australia recorded in high-sampling resolution. The
original data set is public, but it was made available through an obscure
format that limits its accessibility. The presented format in this letter uses
the standard hierarchical data format (HDF5), which makes it easily accessible
in many languages such as MATLAB, Python, C++, and more. The data set compiler
and visualizer script are also provided in the work repository
111https://github.com/dougpsg/hif_vegetation_data.

###### Index Terms:

accessible, data set, high-impedance fault, high-frequency, vegetation

##  I Introduction

Vegetation high-impedance faults (VHIFs) in power distribution systems are
characterized by the unintentional current flow (fault) from electrical
equipment and nearby vegetation where the resulting current is lower than
protection devices’ sensitivity. Despite not representing a threat to power
equipment, VHIFs are still dangerous disturbances with significant potential
to create safety risks and fire hazards. The research field studying high
impedance faults (HIFs) are relatively mature [1], but despite being
responsible for many developments, there are no consensus solution or
standards for the detection or addressing the problem. While the term HIF is
often used interchangeably with VHIFs, it is fair to say that the latter is a
sub-problem of the former, probably deserving separate treatment.

The damages created by the 2009 Black Saturday bushfires [2] — a series of
devastating bushfires with some ignited by electrical faults — highlighted the
importance of addressing these faults. The commission responsible for
investigating the fires explicitly recommended further research in the field,
resulting in a $750 million funding round. One of the projects, the
‘Vegetation Conduction Ignition Testing’, set to test around twenty plant
species were tested in hundreds of experiments on a real three-wire 22-kV
feeder to study fault behaviour, and foster protection technology development
[3]. The tests resulted in an extensive data set of vegetation HIF fault
recordings with high-resolution sampling and wideband signals, having a
significantly high potential for insights.

Having a publicly available data set of VHIFs is highly significant to its
related research field. It is fair to state that one of the bottlenecks for
the progress in the field of HIFs is the lack of shared benchmarks and data
sets. Most papers either make use of a private data set or rely on synthetic
data sets resulting from HIF model simulations (a large share of works) [1].
To the data set being discussed here (VeHIF), a few works have been published
using it [4, 5, 6], mainly by this letter’s authors. One impediment for
accessing the data is that, although public, it was disclosed in an obscure
format given by the sampling equipment used. Having an easily accessible
version of this data set where others can quickly access and interface in many
programming interfaces is therefore much valuable to the field.

Given the limited accessibility of the data set and to collaborate with future
research on the complex and evolving problem of VHIF detection, this letter
presents a proposed format for the data set, access to the re-organised data,
and example code on how to use it. The described data set format is compiled
from the data provided by the Victorian Government data website [7], but the
proposed format offers seamlessly access for experiments. As the data set
contains faulty and non-faulty example signals, this format suits
investigations that approach this as a classification problem. The data set is
publicly hosted on the data science platform Kaggle [8], and repository with
codes to the data set compiler and visualizer are also provided [9]. It is
important to note that the data set should be used in the context presented
and described by the official tests report [3]. There, one can find detailed
descriptions of the characteristics of the tests, vegetation species tested,
electrical configurations and specifications.

##  II The vegetation high-impedance fault experiments

The data was made public by the Victorian Government [7], and it contains
logs, video and signal recordings, graphs, and analysis of the tests. The data
set described here only includes tests (waveforms) recordings and some
metadata regarding the experiments. Such data should be enough for primary
analyses and using it in applications like signal classification and spectrum
analysis. The voltage and current signals were made available in a proprietary
format given by the hardware used in their recordings, thus not directly
accessible for manipulation.

Two sampling channels were used for each current and voltage signals. The
project team decided to have a channel dedicated to higher frequencies without
the influence of the main power frequency (50 Hz). All four channels were
connected to an analogue 6-pole Bessel low-pass filter with 10 MHz corner
frequency for anti-aliasing. A second low-pass anti-aliasing filter (Bessel
IIR digital) with proper data decimation was added to the low-frequency (LF)
and high-frequency (HF) channel; It had a cut-off frequency of 50 kHz in LF
channels and 1 MHz in HF channels. HF channels were also filtered by a 10 kHz
high-pass filter, characteristic of the capacitive voltage divider design. The
resulting effective band of LF channels was approximately 0 Hz to 50 kHz,
while the HF channels were effectively composed of frequencies from 10 kHz to
1 MHz.

The high-speed recorder continuously sampled all channels at 100 kSa/s, plus
the additional measurements in the HF channels. The latter was not sampled
continuously but by periodic intermittent short recordings (snapshots)
referred to as ‘sweeps’ with a duration of 20 ms. The sweeps were triggered by
a non-synchronous signal with 1 s period, meaning that HF recordings were
composed of one sweep per second (one power cycle per second). The many sweeps
per test were extracted and sequentially concatenated in a single vector in
the data set. The trigger signals are also included in the data, with their
square waves signalling the starting time of a sweep recording.

##  III VeHIF Data set format

###  III-A The HDF5 format

Hierarchical data format (HDF) has many advantages and characteristics that
fit the use for this data set. It is free, supported by many commercial and
non-commercial software platforms and programming languages (MATLAB, C, C++,
Fortran, Java), and has efficient partial I/O access to data on disk. Not
surprisingly, it has become of wide use in public benchmark data sets and
frameworks in fields like deep learning [10].

The file format is hierarchically represented main by two entities: Groups and
Data sets. Data sets, in this case, are the actual data, often organized and
accessed as arrays (slicing is also a feature). Groups are the hierarchical
representation (nodes) where sub-groups and data sets are mapped. A close
analogy is the one given by a file system in a computer where ‘groups’ are
directories and ‘data sets’ the files. In addition, groups and data sets can
have attributes, which can be seen as the metadata of a given node. Groups,
data sets, and attributes are all the concepts necessary to work with the
VeHIF data set.

###  III-B Data set structure

The root of the data set contains only two groups: ‘cal’ and ‘test’. The
latter are tests that were labelled are background noise recordings by the
program. As these are recordings where no vegetation was being tested, they
can be used as the negative class when using it as training data in a
classification problem. The former, ‘test’, represents all the tests available
on the data provided in the official data web page [7], which did not result
in an error when translating the data from the original format to the use
presented here. In total, there are 53 background and 983 fault tests, as 15
fault tests produced errors in the data set compilation. Both of these root
groups contain one attribute, which is a list of the original test number as
per the original report [3].

Every sub-group under the two root group represent a unique test recording.
They are labelled by their original number and contains at least four data
sets: ‘current_hf’, ‘current_lf’, ‘voltage_hf’, ‘voltage_lf’. The tests
subgroup contains one additional data set, ‘hf_trigger’, which represents the
waveform used to trigger the high-frequency sweeps recordings. The sub-groups
under ‘cals’ have two attributes: ‘filename’ and ‘cal_type’; the first refers
to the original filename and the second to the electrical configuration of the
test (phase-to-earth or phase-to-phase). The sub-groups under ‘test’ have four
attributes: ‘filename’, ‘test_type’, ‘max_current’, and ‘report_validity’.
Test type can be ‘bush’, ‘grass’, ‘phase-to-earth’, or ‘phase-to-phase’. The
maximum current and report validity are extracted from the program’s report.
Each test was set to end at a defined current value, and some tests were
labelled as invalid as per the test report [3].

Two examples are illustrated in the presented figures. Figure 1 shows the
recordings of voltage and current sampled from the LF channel. Differently
from the HF channel, this channel samples continuously, and thus it allows to
plot the waveform against time throughout the whole duration of the test.
Figure 2 illustrates two sweeps recordings from the HF channel. This channel
sampled the tests in sweeps of 20 ms with 2 MSa/s sampling rate, leading to
40k samples per sweep (axis in the figure). The HF sampling channel sampled
signals connected to the low-pass and high-pass filters, leading to LF and HF
pairs of the same sweeps. Figure 2 shows the pairs (LF and HF) of voltage
waveforms for the case where a fault is present (faulty) and in the absence of
fault (non-fault). Note how the small currents from such vegetation faults do
not introduce any visible effect on the voltage waveforms.

Figure 1: Voltage (a) and current (b) waveforms sampled from the LF channel.
One of the few tests with pre-fault voltage recordings. Figure 2: Two example
of sampled sweeps. LF recordings from a) Non-fault sweep and b) Faulty sweep.
HF recordings from c) Non-fault sweep and d) Faulty sweep.

###  III-C Accessibility

The data set is publicly hosted on the data science platform Kaggle [8]. In
addition to providing the data set in the accessible H5DF format, the data set
is accompanied by a repository with the code used to compile the data set and
a visualizer script in MATLAB [9]. As previously mentioned, such tests were
thoroughly described in the official report, and one should refer to it for
specific information on the tests and experimental configurations. Given the
public libraries such as the ‘h5py’ in Python and native H5DF integration to
MATLAB, access to the tests waveforms should be seamlessly and direct.

###  III-D Potential uses

While facilitating access to the data, such a data set format also allows for
quicker experimentation. The proposed categorisation in groups suits
investigations approaching the problem in a supervised learning fashion. The
group can represent classes, allowing for an objective comparison of methods
of studying them. The large number of waveforms can also be used in
investigations concerning signal features or patterns. As an example, using
machine learning, as in unsupervised learning, or dimensionality reduction
approaches to find relevant patterns in the signal. The VeHIF problem still
presents significant knowledge gaps, with the behaviour still to be clearly
defined. Researchers can benefit from such a format to enhance or objectively
compare distinguished investigations.

## References

  * [1] Douglas PS Gomes and Cagil Ozansoy,  “High-impedance faults in power distribution systems: A narrative of the field’s developments,”  ISA transactions, 2021.
  * [2] 2009 Victorian Bushfires Royal Commission,  “Final report,” July 2010.
  * [3] Tony Marxsen,  “Vegetation conduction ignition test report - final,”  Department of economic development jobs transport and resources, Marxsen Consulting Pty Ltd., 2015.
  * [4] D. P. S. Gomes, C. Ozansoy, and A. Ulhaq,  “High-sensitivity vegetation high-impedance fault detection based on signal’s high-frequency contents,”  IEEE Transactions on Power Delivery, vol. 33, no. 3, pp. 1398–1407, 2018.
  * [5] D. P. S. Gomes, C. Ozansoy, and A. Ulhaq,  “Vegetation high-impedance faults’ high-frequency signatures via sparse coding,”  IEEE Transactions on Instrumentation and Measurement, pp. 1–1, 2019\.
  * [6] Sevvandi Kandanaarachchi, Nandini Anantharama, and Mario A Munoz,  “Early detection of vegetation ignition due to powerline faults,”  IEEE Transactions on Power Delivery, vol. 36, no. 3, pp. 1324–1334, 2020.
  * [7] Department of Environment, Land, Water & Planning,  “Powerline bushfire safety program - vegetation conduction ignition test report and data,” https://discover.data.vic.gov.au/dataset/powerline-bushfire-safety-program-vegetation-conduction-ignition-test-report, 2016\.
  * [8] Douglas Gomes on Kaggle,  “High-impedance vegetation fault data set,” https://www.kaggle.com/dougpsg/highimpedance-vegetation-fault-data-set, 2021\.
  * [9] Douglas Gomes,  “High-impedance vegetation fault dataset,” https://github.com/dougpsg/hif_vegetation_data, 2021.
  * [10] Bart Van Merriënboer, Dzmitry Bahdanau, Vincent Dumoulin, Dmitriy Serdyuk, David Warde-Farley, Jan Chorowski, and Yoshua Bengio,  “Blocks and fuel: Frameworks for deep learning,”  arXiv preprint arXiv:1506.00619, 2015.


# Optical and Zeeman spectroscopy of individual Er ion pairs in silicon

Guangchong Hu  Centre for Quantum Computation and Communication Technology,
School of Physics, University of New South Wales, Australia  Rose L. Ahlefeldt
<EMAIL_ADDRESS>Centre for Quantum Computation and Communication
Technology, The Australian National University, Australia  Gabriele G. de Boo
Centre of Excellence for Quantum Computation and Communication Technology,
School of Physics, University of New South Wales, Australia  Alexey Lyasota
Centre for Quantum Computation and Communication Technology, School of
Physics, University of New South Wales, Australia  Brett C. Johnson  Centre
for Quantum Computation and Communication Technology, RMIT University,
Australia  Centre for Quantum Computation and Communication Technology, The
University of Melbourne, Australia  Jeffrey C. McCallum  Centre for Quantum
Computation and Communication Technology, The University of Melbourne,
Australia  Matthew J. Sellars  Centre for Quantum Computation and
Communication Technology, The Australian National University, Australia
Chunming Yin  Centre of Excellence for Quantum Computation and Communication
Technology, School of Physics, University of New South Wales, Australia  Hefei
National Laboratory for Physical Sciences at the Microscale, CAS Key
Laboratory of Microscale Magnetic Resonance and School of Physical Sciences,
University of Science and Technology of China, China  Sven Rogge  Centre of
Excellence for Quantum Computation and Communication Technology, School of
Physics, University of New South Wales, Australia

###### Abstract

We make the first study the optical energy level structure and interactions of
pairs of single rare earth ions using a hybrid electro-optical detection
method applied to Er-implanted silicon. Two examples of Er3+ pairs were
identified in the optical spectrum by their characteristic energy level
splitting patterns, and linear Zeeman spectra were used to characterise the
sites. One pair is positively identified as two identical Er3+ ions in sites
of at least C2 symmetry coupled via a large, 200 GHz Ising-like spin
interaction and 1.5 GHz resonant optical interaction. Small non-Ising
contributions to the spin interaction are attributed to distortion of the site
measurable because of the high resolution of the single-ion measurement. The
interactions are compared to previous measurements made using rare earth
ensemble systems, and the application of this type of strongly coupled ion
array to quantum computing is discussed.

##  I Introduction

Optically coupled single spins in solid state hosts are studied for a range of
quantum technologies, including quantum memories, repeaters, and computers due
to three main features: spin states with low decoherence, optical transitions
to interface the solid state qubit with flying qubits, and a physical platform
compatible with scalable device engineering [1]. For computing in particular,
the possibility of creating arrays of closely spaced, optically coupled spins
is attractive as these arrays can provide a multi-qubit system where the
short-range interactions between different qubits enable fast multi-qubit
gates. For instance, by using a set of 13C nuclear spins coupled to an
optically active diamond nitrogen vacancy center, entangling gate operations
have been achieved in a ten-qubit array [2].

While most work in this area has concentrated on bright optical color centers
such as the nitrogen vacancy in diamond, rare earth ions in various solids are
increasingly being studied. These materials have much weaker optical coupling
strengths, but several groups have recently shown the detection [3, 4, 5, 6,
7, 8] and coherent control [9, 10, 11, 12, 13] of single rare earth ions. Er3+
is particularly interesting as it offers an optical transition between its
4I15/2–4I13/2 levels that lies within the $1550\text{\,}\mathrm{nm}$
transition band of optical fibers, and crystals doped with Er3+ have shown
very long optical [14] and spin [15] coherence times in the absence of
magnetic noise. Single Er3+ ions have been observed using hybrid electrical-
optical [7, 16] and optical-cavity enhanced detection [17, 10, 9]. Six
individual Er3+ ions have been individually addressed in a single optical
cavity [9], but no arrays of Er ions have been studied to date.

Here, we report the first measurements of individual interacting Er ion pairs.
These measurements were made in silicon, which can be engineered into photonic
devices using CMOS fabrication techniques, and is a low-spin host offering a
quiet magnetic environment for the Er3+ ions. However, the solubility of Er3+
in silicon is low, and Er is typically included in the silicon lattice by ion
implantation or during molecular beam epitaxy growth, often co-doped with
oxygen or other light impurities to enhance the erbium-associated luminescence
[18, 19, 20]. Implanted Er3+ in Si has shown comparable spin coherence times
in low magnetic fields to Er substitutional dopants [21], but it can occupy a
large variety of possible sites in silicon, and the properties of these sites,
even for ensembles, are not well known. Sites have been studied with
photoluminescence [22, 23, 24], photoluminescence excitation [25, 26] and
electron paramagnetic resonance [27, 18, 28, 29]. However, only a small number
of sites have been positively identified, typically as Er-O complexes of low
symmetry, including multiple monoclinic [28] and trigonal [28] sites, and one
orthorhombic site [24]. Only a single bulk study [30] has reported a possible
Er3+ pair site, and no information on the interaction between the ions was
obtained.

Figure 1: Concept of studying Er pair sites using hybrid electro-optic
detection. Left: An Er ion pair sitting in the channel region underneath the
gate (G) of a fin-field effect transistor (finFET) is optically excited from
the (Kramers doublet) 4I15/2 ($\ket{0}$) state to the 4I13/2 ($\ket{1}$)
state, and the non-radiative decay via nearby trap sites produces a current
between source (S) and drain (D), forming the detected signal. Right:
representation of the energy level structure of an Er3+ ion pair, showing
transitions probed. The coupled pair has four electronic states, $\ket{00}$,
$\ket{10}$, $\ket{01}$, and $\ket{11}$, each composed of four electron spin
levels for the $I=0$ isotopes studied. We excite only transitions ”a” for the
non-identical site and ”a” and ”b” for the identical site, shown in orange,
although transitions ”c” and ”d” (gray) can in principle be excited via a two-
laser process.

In this work, we study the 4I15/2–4I13/2 transition of Er ion pairs using a
hybrid electrical-optical detection method [7] (see Fig 1). We present spectra
in zero and applied magnetic field for two different types of Er pairs sites:
one composed of Er3+ ions in non-identical sites, and a second of ions in
identical sites. We analyze these spectra using a simplified model for spin
and optical coupling between the two ions. This work is not only the first
study of individual rare earth ion pairs in the solid state, but the first
direct study of Er pair sites in silicon. The high resolution possible by
studying a single instance of the pair, rather than an ensemble of pairs,
allows precise determination of the properties of the individual Er ions as
well as of the interaction between them, improving our understanding of rare
earth interactions for future quantum computing applications [31].

##  II Interactions between rare earth ion pairs

While single rare earth ion pairs have not previously been studied, many
workers have investigated pair interactions in ensembles where the rare earth
ions are substitutional dopants. This work is reviewed by Cone and Meltzer
[32] (focused on optical energy transfer) and, more recently, Guillot-Noël et
al. [31] (focused on spin interactions). The possible interactions between two
ions in a pair can be separated into two classes: indirect and direct
interactions. The main indirect interaction is a strain shift arising from the
crystal distortion due to the incorporation of dopant ions. For an ion pair,
this strain means the presence of the partner ion shifts, in particular, the
optical transition frequency of each ion in the pair (either identically for
ions in the same site, or differently for ions in different sites). The shift
is unique to the pair site, which has allowed site-selective measurements of
pair interactions in ensemble systems. These strain shifts can vary from GHz
to THz (e.g. [33, 34, 35, 36]).

In a quantum computer, strain shifts would determine the optical frequency of
a pair of qubits, but would play no role in gate operations. Thus, here we are
concerned with direct interactions between the wavefunctions of the two ions
of the pair. There are four main mechanisms considered for these interactions:
exchange, magnetic multipole interactions, electric multipole interactions,
and virtual phonon exchange [37, 32, 38]. The relative contribution of these
terms is strongly dependent on the interacting ions, and in principle
accurately estimating it requires knowledge of the full electronic
wavefunction of the individual ions. However, the interaction can often be
split into an optical component, and a spin component, with the latter fitted
directly to experimental data. This is the approach taken here.

We will first briefly discuss the optical interaction. Modelling contributions
to this part does require an accurate crystal field model, which was not
available in our measurements as these sites were seen for the first time, and
we studied only a two crystal field levels (ground and excited state). The
off-diagonal part of the optical interaction leads to energy transfer, while
the diagonal part causes a frequency shift on one ion when the other is
excited. This latter “blockade” mechanism is utilized for multi-qubit gates in
rare earth quantum computing proposals, where the mechanism can be electric
dipole [39], magnetic dipole [40] or mixed [41]. In a single-laser experiment,
this blockade effect is difficult to distinguish from the strain shift, but it
can be measured in a two laser experiment, yielding values for nearest
neighbors from, for instance, 50 MHz in EuCl3.6H2O [42] to 5 GHz in Pr:LaF3
[43].

###  II.1 Spin interactions of ion pairs

The electron or nuclear spin splitting of an isolated electronic state for a
single rare earth ion can be accurately modelled using a reduced spin
Hamiltonian. Such a model includes only these degrees of freedom,
parametrizing any electronic contributions to the structure. The same model
can be used to study spin interactions between ion pairs [31]. Here, we
describe the spin Hamiltonian model appropriate to the Er3+ pairs we studied.

In low symmetry sites, the spin Hamiltonian of an individual Er ion is
anisotropic and different in every electronic state. Therefore, the most
general case we need to model for a pair of coupled ions is two optically
active Er ions in different low symmetry sites. Then, we need to develop
reduced Hamiltonians for three electronic levels: $\ket{00}$, with both ions
in the ground state; $\ket{01}$, with ion 1 in the ground state, and ion 2 in
the optical excited state; and $\ket{10}$, with ion 1 in the excited state,
and ion 2 in the ground state. We ignore $\ket{11}$ because we have not seen
signatures of this double-excitation in optical spectra. Assuming both Er ions
are isotopes with nuclear spin $I=0$ (the case for the pairs studied here),
the spin Hamiltonian for an electronic state can be written:

$H_{mn}=\bm{B\cdot M^{1m}\cdot\hat{S}_{1}}+\bm{B\cdot
M^{2n}\cdot\hat{S}_{2}}+\bm{\hat{S_{1}}\cdot J^{mn}\cdot\hat{S}_{2}}$ (1)

for $mn=\ket{00},\ket{10},\ket{01}$. In this equation, the first two terms
describe the Zeeman interactions of the two individual ions, and the third
term gives the coupling between the two spins. Specifically, $\bm{B}$ is the
external magnetic field, $\bm{M^{mi}}$ is the anisotropic Zeeman tensor
(g-tensor) of the ion $i$ ($i=1,2$) in state $m$, $\bm{\hat{S}_{i}}$ is the
spin operator for ion $i$, and $\bm{J^{mn}}$ is the spin-spin interaction
tensor for ion 1 in electronic state $\ket{m}$ and ion 2 in state $\ket{n}$.
The spin operators $\bm{\hat{S}_{i}}$ are normal spin-half vector operators
expressed in space of the coupled ions as:

$\displaystyle\bm{\hat{S_{1}}}$
$\displaystyle\implies\bm{\hat{I}\otimes\hat{S_{1}}}$ (2)
$\displaystyle\bm{\hat{S_{2}}}$
$\displaystyle\implies\bm{\hat{S_{2}}\otimes\hat{I}}$

for $\bm{\hat{I}}$ the identity operator. In the general case the interaction
between the two spins is completely anisotropic, with all nine terms of
$\bm{J}$ independent. It is common to write this spin interaction as a sum of
three terms, all of different shape:

$\displaystyle\bm{S_{1}\cdot J\cdot S_{2}}$ $\displaystyle=J_{0}\bm{S_{1}\cdot
S_{2}}+\sum_{\alpha,\beta}V_{\alpha\beta}\left(S_{1}^{\alpha}S_{2}^{\beta}+S_{1}^{\beta}S_{2}^{\alpha}\right)$
(3) $\displaystyle+\bm{D\cdot\left(S_{1}\times S_{2}\right)}$

These terms are, in order, the isotropic term, the symmetric traceless term,
and the antisymmetric term. This decomposition is particularly useful for
certain spins, such as $3d$ ions with quenched orbital angular momentum, where
these terms can be directly linked to different interaction mechanisms, and it
is generally true that $|J|\gg|V|\gg|D|$. The same is not true for rare earth
ions, where the large spin-orbit interaction means that even interactions that
are isotropic between the real electron spins on the two ions (such as
Heisenberg exchange) become anisotropic in this reduced pseudo-spin
Hamiltonian [37, 38]. For rare earth ions, all terms in the above equation can
be a similar size, although the values can be constrained by symmetry
arguments.

Of the four interaction mechanisms introduced above, only magnetic multipole
and exchange contribute to the interactions of Er3+ pairs to first order,
since the Kramers degeneracy of Er3+ implies a first order insensitivity to
electrically-mediated interactions. The magnetic multipole interaction is
dominated by the magnetic dipole-dipole term [31], and this can be accurately
determined once the Zeeman tensors of participating ions are known:

$\bm{J^{mn}_{dd}}=\frac{\mu_{0}h}{4\pi|\bm{r}|^{3}}\left[\bm{M^{1m}M^{2n}}-3\bm{(M^{1m}\hat{r})(M^{2n}\hat{r})^{T})}\right]$
(4)

where $\bm{r}$ is the position vector joining the two ions. For the general
case of anisotropic Zeeman tensors, $\bm{J^{mn}_{dd}}$ contributes to both the
isotropic and symmetric traceless parts of Eq. (3).

The direct exchange interaction is typically small and superexchange
(wavefunction overlap via intervening ions) dominates because the spatial
extent of $4f$ electronic wavefunctions is much smaller than typical inter-
atomic spacings. Since the environment of the sites studied here is not known,
superexchange contributions cannot be approximated, but isotropic and
anisotropic superexchange interactions of 1-100 GHz between rare earth ion
pairs have been seen in other materials [36, 44, 45], making this interaction
similar in size to the magnetic dipole term.

##  III Experimental method

The method for obtaining spectra of individual Er ions in silicon has been
described in detail elsewhere [7].Briefly, the devices used are finFETs
implanted with Er in the channel region, as shown in Fig. 1. A quantum dot is
formed by the confinement provided by the corner effect and residual barriers
in the access regions between the source/drain and the channel. This
configuration works as a single electron transistor, and is sensitive to any
nearby electrochemical potential fluctuation. If individual Er ions are
resonantly optically excited into the 4I13/2 state, they can decay non-
radiatively by ionizing a nearby trap, changing the charge environment. Then,
the tunneling current through the quantum dot shows a strong signal when the
laser frequency matches an Er 4I15/2-4I13/2 transition.

Two devices were studied here, labelled 1 and 2, with the same channel height
of $60\text{\,}\mathrm{nm}$. Both were implanted with 170Er3+ and oxygen with
a ratio of 1:6, to a simulated Er3+ concentration of $1.5\times 10^{17}$ cm-3,
and annealed at $700\text{\,}\mathrm{\SIUnitSymbolCelsius}$ in N2 for 10
minutes. Device 1 had dimensions of $890$$\times$$100\text{\,}\mathrm{nm}$
with 780 erbium ions estimated to be in the channel. Device 2 had a dimensions
of $100$$\times$$35\text{\,}\mathrm{nm}$, with 30 erbium ions estimated to be
in the channel.

The experiments were carried out at $4.2\text{\,}\mathrm{K}$ in an Oxford
helium bath cryostat with a 12 T superconducting magnet. The device was
mounted on a sample rod installed with a home-built vector magnet and a
single-mode fiber that was aligned with the channel region under a microscope.
The beam diameter was approximately 500 $\mu$m and both laser and magnetic
field were parallel to [001] and perpendicular to the plane of the device. The
incident polarization was unknown, although it could be manipulated from
outside the cryostat with fiber polarization paddles.

The device was biased near the Coulomb blockade region, so that discrete
Coulomb peaks were visible in a current – gate voltage trace. The gate voltage
was chosen to be on the rising edge of a discrete Coulomb peak where the
sensitivity to charge is the highest. Then, spectra were recorded by
monitoring the current while a wavelength-tunable laser was used to excite the
4I15/2-4I13/2 transition of single erbium ions.

##  IV Pair site spectra

In any one device studied as described above, multiple widely separated
spectral lines are observed over a range from 1529 to 1546 nm [16]. When the
laser is resonant with one of these lines, the electrical signal shows a two-
level behavior, confirming the signal arises from the excitation of a single
Er3+ ion [7]. The lines can typically be attributed to transitions of
individual Er3+ ions in the channel from the lowest crystal field level of
4I15/2 to the 4I13/2 multiplet. These single sites characteristically display
a single line in zero field that splits in an applied field into two or four
lines, the Zeeman splitting expected for sites of lower than cubic symmetry
(no cubic sites have been observed optically in Si). The only exception is
when the site is occupied by an Er ion with nuclear spin, in which case
multiple, GHz-spaced lines are seen in both zero and applied field, commonly
in groups of eight ($I=7/2$ for 167Er). Such sites are not expected in these
170Er-implanted samples.

In other cases we observe multiple lines in zero field separated by up to tens
of GHz. For these lines, it is important to distinguish structure due to
interacting pairs (or multi-sites) from two other cases: multiple, unrelated
single Er3+ ions with similar frequencies; and single Er3+ ions with
incidental near-degenerate crystal field ground or excited states. Pair sites
can be distinguished from unrelated single sites by: a) the existence of
relationships between the properties of individual lines in the spectrum (such
as similar polarization dependence or Zeeman shifts) and b) by the presence of
either bending of lines or anti-crossings between different lines at certain
magnetic field values. In particular, almost all spin interactions will lead
to some bending near zero field due to quenching of the magnetic moment of the
interacting states. Er3+ with a near-degenerate pair of crystal field levels
will also show bending or anticrossings, but only at one field value. The
number of related lines seen can help further distinguish a pair site from
this case, which can show up to eight lines only.

We have observed multiple different sites with signatures of pair
interactions, and discuss two examples below that represent the set. We
present linear Zeeman spectra for fields along the $z$ [001] axis, and for the
first site, along the $x$ [110] axis.

###  IV.1 Site A at 1538.306 nm

(a)

(b)  (c)

Figure 2:  Site A, which occurs at $194.884\,80\text{\,}\mathrm{THz}$
($1538.306\text{\,}\mathrm{nm}$). This spectrum consists of eight lines
appearing as four slightly split pairs separated by several GHz in zero field.
a) Peak locations (highest detected current) of these eight lines for fields
along the lab $z$ axis, a [001] direction. Color is used to distinguish the
lines of a pair. Asterisks denote lines that persist in high field, indicating
those lines originate in the lowest energy ground state. The black box denotes
the region for which raw spectra are shown in b), c). b) Example raw spectrum
for a magnetic field along [001]. Color axis is detected current. c) Example
raw spectrum for a magnetic field along [110].

Fig. 2(a), shows the lines associated with the first site, site A (seen in
device 1). As described below, we attribute this site to a pair of Er3+ ions
in non-identical sites. The spectrum displays eight lines grouped as four
barely split pairs for fields along $z$. The splittings vary from 0.5 to 0.8
GHz depending on the line pair, and are roughly constant for fields above 0.05
T. There is no Zeeman effect to first order at zero field, leading to the
characteristic bending expected of a coupled site as described above. The pair
line splittings approach zero in zero field, and the two inner pairs of lines
disappear (see Fig. 2(c)). When the field is rotated away from [001], the
splitting within the pair increases and anticrossings appear between different
pairs of lines, Fig. 2(c), indicating that the site is anisotropic. Along $x$
([110]), the anticrossing seen at 0.04 T in Fig. 2(b) becomes a near-crossing.
The behavior seen in the $x$ and $z$ directions suggest that [110] and [001]
represent symmetry axes of the site.

Given that the site is anisotropic, fitting a spin Hamiltonian of the form Eq.
1 would require spectra along several directions, which we did not collect.
However, much can be determined from the spectra in Fig. 2. Eight lines in
both low and high field suggest a pair site consisting of an Er3+ ion coupled
to a non-optically-active spin-half (or pseudo-spin-half) defect. Such a
system has four levels in each of the ground and excited state, and eight
lines are expected when the mixing of the states is such that the optical
field cannot drive flips of the non-optically active spin. Based on the large
zero field splitting, that defect must have an electronic (rather than
nuclear) moment.

Because four of the eight lines persist in high field, indicating that they
involve transitions from the lowest energy ground state, the energy level
structure must also consist of pairs of barely split levels for [001] fields.
Thus, the Zeeman splitting of the optically active site generates the large
splitting with $g_{z}=136\pm 4$ GHz/T in the ground state, and $g_{z}=48\pm 4$
GHz/T in the excited state. Meanwhile, the optically inactive site has
$g_{z}<4$ GHz/T with a much larger $g_{x}>50~{}$ GHz/T in the $x$ direction.
Since the optically inactive site is anisotropic with relatively large $g$
along certain directions, it can be identified as another Er ion in a
different crystallographic site whose optical transitions are at a different
frequency, although it may not be optically active at all.

Some conclusions can also be drawn about the interaction between spins. It is
certainly not primarily an isotropic exchange interaction, because that is
inconsistent with the zero field structure. Both symmetric and antisymmetric
interactions are possible, and specifically the magnitude of the zero field
splitting is consistent with a dipole-dipole interaction between two Er ions
separated by $\order{4}$ Å.

###  IV.2 Site B at 1534.64 nm

Figure 3: Composite optical spectrum for Site B at 1534.64 nm. This spectrum
is composed of 23 individual spectra, either low-resolution line-following
spectra or high-resolution grids around line crossings and anticrossings. The
color axis is detected current, and the scale is logarithmic. All spectra have
been background-subtracted. The dashed line marks zero magnetic field and the
red asterisks mark the two lines still visible in fields above 2 T. The four
pairs of zero field peaks are labelled P1-4. P1 and P3 split into two lines in
an applied field, and P2 and P4 into four.

Site B (device 2), shown in Fig. 3, has many more lines than site A. At zero
field, four groups of lines are present, labelled peaks 1 to 4. Of these,
peaks 2 and 4 are slightly split (around 150 MHz), and all peaks split near-
identically in fields along the $z$ axis ([001]) at a rate of 21.82 GHz/T. The
positive-frequency branches of peaks 1 and 3 persist in high fields,
indicating they arise from the lowest energy ground state. Peaks 2 and 4 anti-
cross near 0.2 T. The spectrum also displays many other avoided crossings
between the visible lines and other, invisible lines that only gain transition
intensity at the avoided crossings. The number and position of the avoided
crossings establishes that there are four straight lines crossing over this
spectrum from each side, with zero field offsets of the order of $\pm 100$ GHz
and slopes of the order of 200 GHz/T. In total, twenty lines can be directly
inferred from this spectrum.

Figure 4: Prototypical energy level structure for two Er ions coupled purely
via Ising spin interactions between spins in the ground and excited states,
the suggested system for site B. Optical states are labelled on the left side
of the energy structure, and spin eigenstates of the pair system on the right.
Spin zero field and Zeeman splittings are to scale for site B. Blue arrows and
text give the Zeeman splittings of the doublets in terms of ground and excited
state single ion g-values in the applied field direction, and red lines and
text give the optical splittings in terms of the Ising interaction $J_{zz}$.
Transitions $\ket{00}\rightarrow\ket{10}$ give rise to P1 and P2 in Fig. 3,
and transitions $\ket{00}\rightarrow\ket{01}$ to P3 and P4.

The spectrum satisfies both our criteria above for a pair site: lines with
similar properties, and the presence of anticrossings between lines. However,
in this case, we do not see the zero field bending described above. As
mentioned, this structure is almost always seen for a pair site. The only pair
interaction shape which can explain the lack of zero field bending is a near-
Ising interaction Hamiltonian, i.e. only one term of $\mathbf{J}$ in each
state is non-zero. Further, the fact that we can identify more than sixteen
lines means that both ions were being excited here, so they must have similar
optical frequencies and, thus, occupy identical sites. Overall, then, the
spectrum can be described by two optically active Er ions occupying near-
identical crystallographic sites coupled primarily by an Ising spin-
Hamiltonian $\mathbf{J}=J_{zz}$ in each electronic state. Additionally, $z$ is
a principal axis of the Zeeman tensor in all states and thus a symmetry axis
of the site. Below, we explain how this structure leads to the spectrum seen.

The energy level structure for two ions coupled only by Ising interactions is
shown in Fig. 4: it consists of two doublets in zero field for each of the
three relevant electronic states (the $\ket{00}$ ground state, and the
$\ket{10}$ and $\ket{01}$ singly-excited states), split by $J^{mn}_{zz}/2$.
The levels are labelled according to the spin eigenstates, and the Zeeman
splittings are shown as a function of the ground and excited state $g$ values
of the individual ions. The spectrum requires that optical transitions in this
system that preserve spin orientation (such as
$\ket{00}\ket{\uparrow\uparrow}\rightarrow\ket{10}\ket{\uparrow\uparrow}$) are
allowed, and these transitions have identical Zeeman splitting of
$\pm(g_{0}-g_{1})=21.82$ GHz for this site. These transitions explain the
visible lines in Fig. 4, with peaks 1 and 2 arising from spin-preserving
excitations to $\ket{10}$, and peaks 3 and 4 from excitations to $\ket{01}$.
The lines appearing only at anticrossings with these lines are due to non-
spin-preserving transitions.

This model of the energy level structure is sufficient to determine ground and
excited state $g$ values in the direction of the applied field ($z$) and
$J_{zz}$ for each state. We find near maximal $g$ value along the $z$
direction in both states, $g_{z}^{0}=232$ GHz/T and $g_{z}^{1}=188$ GHz/T, and
Ising interaction coefficients $J_{zz}^{00}=209$ GHz, $J_{zz}^{01}=220$ GHz,
$J_{zz}^{10}=233$ GHz. An additional optical offset between the $\ket{01}$ and
$\ket{10}$ transitions of 10.8 GHz is required to explain the offset of the
centroids of the peak pairs 1 and 2, and 3 and 4. As described in Section II,
this could be caused by several interactions that we cannot distinguish in
this single laser experiment: differing strain shifts due to the two ions
sitting in slightly different environments, the strain shift due to the other
ion of the pair, or a shift due to a diagonal optical interaction between the
pair.

The Ising model is a good first approximation to the overall structure of the
spectrum, but the details show other, smaller contributions must be included.
First, the spin anti-crossings visible between 0.45 and 0.65 T in Fig. 3
(corresponding to anti-crossings of the two doublets of a single electronic
state, see Fig. 4) are crossings in an Ising model (the anticrossing at 0.15 T
is an optical anticrossing, discussed later). Second, the small (0.15 GHz)
zero field splitting of the four lines contributing to peaks 2 and 4
(corresponding to transitions for peak 2, for example, of
$\ket{00}\frac{1}{\sqrt{2}}\quantity(\ket{\uparrow\downarrow}\pm\ket{\downarrow\uparrow})\rightarrow\ket{10}\ket{\uparrow\downarrow},\ket{10}\ket{\downarrow\uparrow}$)
is not reproduced. Both these features indicate that there is a small non-
Ising contribution to the interaction spin Hamiltonian. These contributions
can be isolated to a few terms in $\mathbf{J}$ in the three electronic states,
because each term in $\mathbf{J}$ has a different effect on the spectrum. The
lack of any zero-field anticrossings (bending) indicates that $J_{xx}\approx
J_{yy}\approx 0$ for all states. This also confirms $g_{x}\approx g_{y}\approx
0$ as any transverse $g$ gives rise to $J_{xx}$ and $J_{yy}$ terms due to the
magnetic dipole interaction. The spin anticrossings at non-zero fields are
caused by approximately 1 GHz $xz$ or $yz$ contributions in each state, while
the zero field splitting on 2 and 4 indicates an $xy$ contribution to $J^{00}$
only of approximately 0.15 GHz. Additionally, there is a slight ($<2$ GHz/T)
symmetric difference in the Zeeman splitting rates of the four transitions
contributing to peaks 2 and 4, and a very mild ($0.1$ GHz/T) asymmetry between
positive and negative Zeeman arms. These features suggest the two Er ions are
not exactly identical, with slightly different $g$ values or orientations of
the $g$-tensor. Consequently, the field-independent ground state doublet gains
a slight splitting in a magnetic field, with the eigenfunctions better
described with new eigenstates $\ket{\downarrow\uparrow}$ and
$\ket{\uparrow\downarrow}$. These differences likely indicate slightly
different crystalline environments for the two ions, although canting of the
moments due to the ion-ion interaction [46] could also explain small
differences in orientation.

Figure 5: Simulation of the optical spectrum of Fig. 3 using a near-Ising spin
Hamiltonian model for each of the three electronic states, as described in the
text. Color indicates the transition probability assuming a spin-preserving
selection rule. The color axis is logarithmic and matches Fig. 3,
$\log_{10}(P+0.03)$ where $P$ is the spin wavefunction overlap defined in-
text. Blue circles mark the position of optical anti-crossings seen in the
data.

In Fig. 5, we show that including these extra terms in the spin Ising model
reproduces the key elements of Fig. 3 quite well. In the figure, we simulate
the optical spectrum expected from near-Ising spin Hamiltonians for
$\ket{00}$, $\ket{10}$, and $\ket{01}$ with $g_{z}$ and $J_{zz}$ values as
given above, $g_{x}=g_{y}=0$ in all states, and with additional terms as
follows: for ion 2, $g_{z}$ decreased by 1.5 GHz in the ground state and 1 GHz
in the excited state (to account for the different Zeeman splittings of the
lines in P2 and P4); and the addition of an antisymmetric term (Eq. (3))
$\mathbf{D}^{00}=[2,0,0.2]$ GHz and $\mathbf{D}^{10}=\mathbf{D}^{01}=[1,0,0]$
GHz to account for the additional $J_{xy}$, $J_{xz}$ and Jyz terms as
described above. In the figure, energy differences were calculated between
ground and excited state spin Hamiltonians with optical frequency shifts
chosen to match the experimental spectrum. Finally, optical transition
intensities (color axis) were estimated assuming a spin-preserving selection
rule, by calculating the spin wavefunction overlap
$P=|\innerproduct{\Psi_{f}}{\Psi_{i}}|^{2}$ where $\ket{\Psi_{i}}$ and
$\ket{\Psi_{f}}$ are the spin wavefunctions of the states determined from the
appropriate ground and excited spin Hamiltonians. The model reproduces all
spin anticrossings seen in the spectrum, however, there are several optical
anticrossings that are not reproduced in this spin Hamiltonian simulation,
most notably the large anticrossing at 0.15 T. These indicate an additional
small optical interaction, discussed in Section IV.2.1 below.

Given the spin model described above can well explain the spectrum observed,
we now expand on the physical coupled site that could lead to the spin
parameters determined. The two constituent Er ions have identical g-tensors
with one principal axis along [001] in both sites, which requires a site
symmetry of at least C2. The g-tensors are near maximal and approximately
Ising-like ($g_{\perp}\approx 0$), implying the ground and excited states are
near-pure $\ket{M_{J}=\pm 15/2}$ and $\ket{M_{J}=\pm 13/2}$ doublets. This
shape commonly occurs in rare earth Kramers ions in crystals for certain site
symmetries or certain values of crystal field parameters (e.g. [47, 48]).
Indeed, an Er site with an Ising-like g-tensor with similar $g_{||}$ in both
ground and excited states has been previously observed in silicon [24],
suggested to be Er in either a substitutional or interstitial $T_{d}$ site (
although the latter is strongly preferred for Er [49]) distorted down to
$C_{2v}$.

That site had a different optical frequency (1538 nm) and was oriented
differently to our coupled site, with $g_{||}$ along [110]. However, it is
likely our coupled site is related, consisting of two Er ions sitting in
identical $T_{d}$ sites. These individual sites may be distorted to lower
symmetry by the incorporation of individual Er (possibly coordinated with
oxygen or another impurity), or the presence of the other Er may lower the
symmetry. For instance, placing Er ions in the two interstitial $T_{d}$ sites
separated along [001] by the unit cell spacing 5.4 Å would lower the symmetry
to $C_{2v}$ with the $C_{2}$ axis along [001], consistent with our measured
g-tensors. The resulting pair site would be symmetric about the intervening
substitutional Td site, whose symmetry would be lowered to D2d. It is likely
each Er ion would be displaced from the Td site by the strain caused by the
other Er ion (spacing different to 5.4 Å), but this displacement must be along
the [001] to maintain the D2d symmetry of the intervening site. Such a
displacement would lower the Er site symmetry to C2, which is still consistent
with the g-tensors presented here.

Such a description is also consistent with the optical selection rules. For
our geometry, with the light propagating along the C2 axis, only $\sigma$
polarization is accessible for both electric and magnetic dipole components of
the light. Both these have the same $\Delta M_{J}=\pm 1$ selection rule, which
allows only spin preserving transitions, such as
$\ket{00}\ket{\frac{-15}{2}\frac{-15}{2}}\rightarrow\ket{10}\ket{\frac{-13}{2}\frac{-15}{2}}$
(equivalent to
$\ket{00}\ket{\downarrow\downarrow}\rightarrow\ket{10}\ket{\downarrow\downarrow}$).

Finally, we consider the origin of the spin-spin interaction. For a site with
Ising g-tensors, an overall $C_{2}$ or higher symmetry of the individual
sites, and $D_{2d}$ symmetry of the whole site, both magnetic dipole-dipole
interactions and exchange interactions result in an Ising interaction
Hamiltonian, so these interactions cannot be separated on the basis of
interaction shape. This is unfortunate, because knowledge of the magnetic
dipole component would allow the distance between the ions to be determined,
since the magnetic dipole moments are known. We can, however, make some
estimates of the relative dipole and exchange components. The interaction is
not entirely dipole-dipole, because that would imply that $J_{zz}$ is
proportional to $g_{z}$ for each electronic state and thus a larger
interaction would be expected in the ground state, instead of in the excited
states as seen here. The smallest possible exchange contribution that can
explain the $J_{zz}$ values seen is 20% (for two ions separated by 3.25Å) but
a much larger exchange contribution is required for larger separations. For
instance, at the 5.4 Å lattice spacing of the tetrahedral sites, $>80\%$ of
$J_{zz}$ must come from exchange. Such a contribution is reasonable: in
Kramers rare earth ion pair sites, exchange interactions have been observed to
be similar in size to magnetic dipole-dipole out to separations of 9 Å[50, 51,
31].

####  IV.2.1 Optical anti-crossing

The large (1.5 GHz) anti-crossing at 0.15 T in Fig. 3, and several smaller
anticrossings at higher field are not reproduced in the spin Hamiltonian
simulation of Fig. 5, indicating that these are due to optical interactions:
they arise from anticrossings between spin states belonging to different
optical excited states $\ket{10}$ and $\ket{01}$. The large anticrossing at
0.15 T, in particular, has two interesting features: first, one of the two
anti-crossing states has near zero optical transition strength, indicating the
transition to this state has become forbidden; second, the position of this
dark state flips when the direction of the magnetic field is reversed, being
on the low frequency side of the anti-crossing for positive fields and the
high frequency side for negative fields. Here, we describe this behavior with
a simplified model.

From the spin model, we can determine that this large anti-crossing occurs
between the $\ket{10}\ket{\downarrow\uparrow}$ and
$\ket{01}\ket{\downarrow\uparrow}$ states. The behavior can be understood by
considering a new reduced Hamiltonian acting on the four electronic states
with spin component $\ket{\downarrow\uparrow}$, which we will write as
$\ket{0^{\prime}0^{\prime}},\ket{1^{\prime}0^{\prime}},\ket{0^{\prime}1^{\prime}},\ket{1^{\prime}1^{\prime}}$.
Then, the Hamiltonian of the system can be generically represented as:

$H=H_{1}+H_{2}+H_{12}$ (5)

where $H_{1},H_{2}$ represent the Hamiltonians of the individual ions, and
$H_{12}$ describes the interaction. When the two ions are resonant at the
anticrossing, the wavefunctions are the three symmetric states
$\ket{0^{\prime}0^{\prime}},\ket{1^{\prime}1^{\prime}}$ and
$\frac{1}{\sqrt{2}}\left[\ket{0^{\prime}1^{\prime}}+\ket{1^{\prime}0^{\prime}}\right]$
and the antisymmetric state
$\frac{1}{\sqrt{2}}\left[\ket{0^{\prime}1^{\prime}}-\ket{1^{\prime}0^{\prime}}\right]$.
The latter two states are split by
$2\bra{0^{\prime}1^{\prime}}H_{12}\ket{1^{\prime}0^{\prime}}=1.6$ GHz.
Transitions between the symmetric states retain the same selection rules for
the individual sites, so are allowed in our $\sigma$ polarization. However,
transitions from $\ket{0^{\prime}0^{\prime}}$ to the antisymmetric state are
forbidden: the transition operators (electric dipole and magnetic dipole) are
identical for the two single-ion transitions for $\sigma$ light due to the
overall D2d symmetry of the coupled site, and so the contributions from the
two component states to the antisymmetric wavefunction cancel. Thus, one
branch of the anti-crossing is dark. Note that the transition can be allowed
for $\pi$ light. An example is seen in the ensemble coupled site in Pr:LaF3
studied by Ref. [43]).

The fact that the dark state flips across the anticrossing when the field is
reversed implies that the interaction is purely or mostly odd under time
reversal symmetry: for negative fields, the relevant anticrossing states have
spin projection $\ket{\uparrow\downarrow}$, the time-reversed copy of the
positive magnetic field states $\ket{\downarrow\uparrow}$. Interaction
mechanisms such as exchange are even under time reversal; the odd symmetry
means the interaction must contain a magnetic term. A possible interaction is
a photon-mediated (Forster) interaction with a magnetic dipole–electric dipole
character. The optical transition of Er3+ is frequently of mixed
electric/magnetic dipole character, making this interaction feasible. However,
since this electrical detection method does not provide information on the
optical transition dipole moments, it is difficult to estimate the size of
this odd interaction in Si.

##  V Discussion

Over several Er implanted devices with similar Er concentrations to those
studied here, we have observed about 200 unique resonances. Of these,
approximately 10% show signatures of pairing according to our definition in
Section IV. This high proportion is, perhaps, not unexpected as Er is well
known to cluster in silicon [49]. Studying these pair sites will give more
information about the sorts of clusters formed. Here, we chose two
representative pair sites with relatively simple optical structure to
investigate in more detail.

Site B showed both optical and spin interactions. We saw a 1.5 GHz off-
diagonal optical interaction with an unusual odd time-reversal symmetry, and
cannot rule out a diagonal contribution to the 10.8 GHz optical frequency
detuning of the two ions of the pair. These interactions are similar to those
seen in several other rare earth crystals, including Pr:LaF3 [43], GdCl3 and
Gd(OH)3 [52], although much larger optical interactions are possible [53].

Both sites showed strong spin-spin coupling, 10-200 GHz. In each case, the
isotropic exchange contribution was unmeasurable, $<0.1$ GHz, and both
magnetic dipole and exchange were likely to contribute to the coupling. In
particular, for site B, we showed that at least 25%, 50 GHz, of the spin-spin
interaction must be an anisotropic exchange term, with the remainder arising
from the magnetic dipole-dipole contribution. The exchange contribution
differed significantly between the ground and excited states, which is
consistent with previous measurements of spin interactions in photoexcited
ensembles, where superexchange contributions varied by up to an order of
magnitude between levels [45, 54].

The spin interactions seen here can be compared with previous measurements of
isotropic exchange, anisotropic exchange, and magnetic dipole spin
interactions for different rare earth pairs in ensembles, reviewed by Guillot-
Noël et al. [31]. Our magnetic dipole interactions are much larger than most
other materials, because the two ions have nearly the maximum $g_{z}$ possible
for a rare earth ion, and interact along the direction of strongest coupling.
The large anisotropic exchange interaction is more surprising. It is nearly 2
orders of magnitude greater than most of the materials listed by Guillot-Noël
et al., even those with small pair spacings for rare earth ions of 3-4 Å.
However, it is exactly pairs with strong interactions that are difficult to
measure using the techniques employed in most of those studies. Most
measurements used EPR, with which it is challenging to even observe lines
spaced by $\approx 100$ GHz in zero field. The situation is slightly improved
when using optical methods, since they have a much larger bandwidth. However,
for large interactions comparable to the strain shifts of the pair sites, it
would be difficult to distinguish these two contributions to the satellite
structure and associate lines with particular pair sites. Thus, the very
strong exchange interactions seen here may be more common in rare earth pairs
than previous ensemble pair site measurements suggest. Similar contributions
from anisotropic exchange have been seen, for instance, using optical
measurements in NdCl3 (ion separation of 4.2 Å)[54], and studying magnetic
ordering in GdVO4 (ion separation of 3.9Å) [55].

We showed that site B was best explained by an Ising spin interaction, but
that this was perturbed slightly with very small non-Ising terms, 0.1-1%.
These contributions are likely due to strain in the site, causing a slight
deviation from the overall proposed D2d symmetry. We were able to measure
these very small contributions remarkably accurately because we studied only a
single instance of the site; in an ensemble, this type of strain variation
leads to inhomogeneous broadening, and so only average effects of strain can
be characterized. Individually studying multiple instances of the same site
would be an excellent way to better understand the microscopic nature of
strain effects in rare earth crystals. Among the 20 pair sites we have seen so
far, site B has not reoccurred, although we have seen a related site with an
Ising coupling and similar g values, so a wider survey of sites would be
required. We also have not seen evidence of an Er3+ single site with the C2v
symmetry and Ising $g$-tensors of the two constituent sites of this pair site.
This does not necessarily imply that the site cannot be occupied by only one
Er3+ ion. There is no requirement for the single site to be spectrally close
to the pair since strain shifts of pair sites can be well in excess of 500 GHz
(4 nm) [34], and we have not yet characterized the symmetry of all 200-odd
single Er3+ sites observed in these devices.

Proposals for quantum computing with rare earth ion arrays often consider
using the optical excitation blockade interaction to perform two qubit gates
[39, 41], which is the frequency shift of one ion on the excitation of
another. In our coupled model Fig. 1, this corresponds to the frequency
difference between, for instance, the spin-preserving transitions ”a”
($\ket{00}\rightarrow\ket{10}$) and ”c” ($\ket{10}\rightarrow\ket{11}$). Here,
a blockade interaction of approximately 30 GHz arises from the spin
interactions alone. This is three orders of magnitude greater than the
electric dipole blockade interaction seen for nearest-neighbor Eu3+ pairs
[42], and six orders of magnitude larger than the interaction used for the
only rare earth two qubit gate to date [56]. This type of strong magnetic
dipole interaction offers the prospect of fast gates times for blockade [39]
and direct interaction gates [57, 40].

We do note that a first step to using these interactions for quantum computing
in silicon, specifically, is to demonstrate initialization and spin-state
selective readout of individual ions, which has not been possible so far with
this electro-optical detection method. However, recent measurements have shown
that implanted Er3+ in Si does have good spin coherence properties [21], and
all-optical methods for detection of single Er3+ ions are being developed [26,
25].

##  VI Conclusion

We presented optical measurements of two different examples of individual Er3+
pair sites in silicon: pair in identical crystallographic sites and a second
pair in non-identical sites. We showed that the identical pair displayed
Ising-like $g$-tensors and a very strong, 200 GHz, Ising spin interaction in
addition to a 1.5 GHz optical interaction between the ions of the pair. We
were able to resolve a small distortion to the Ising symmetry of 0.1%
attributed to strain in the site. To our knowledge, this is the first study of
a rare earth pair site seen either in ensemble or single instance coupled
simultaneously by spin and optical interactions, and the rich structure and
strong interactions seen here suggest such coupled sites could be a useful
resource for quantum computing.

##  VII Acknowledgements

We thank John Bartholomew for insightful discussions, and acknowledge funding
from the Australian Research Council Centre of Excellence for Quantum
Computation and Communication Technology (Grant No. CE170100012). We
acknowledge the AFAiiR node of the NCRIS Heavy Ion Capability for access to
ion-implantation facilities.

GH and RLA contributed equally to this work.

## References

  * Awschalom  _et al._ [2018] D. D. Awschalom, R. Hanson, J. Wrachtrup, and B. B. Zhou, Nature Photonics 12, 516 (2018).
  * Bradley  _et al._ [2019] C. E. Bradley, J. Randall, M. H. Abobeih, R. C. Berrevoets, M. J. Degen, M. A. Bakker, M. Markham, D. J. Twitchen, and T. H. Taminiau, Physical Review X 9, 031045 (2019).
  * Kolesov  _et al._ [2012] R. Kolesov, K. Xia, R. Reuter, R. Stöhr, A. Zappe, J. Meijer, P. R. Hemmer, and J. Wrachtrup, Nature Communications 3, 1029 (2012).
  * Utikal  _et al._ [2014] T. Utikal, E. Eichhammer, L. Petersen, A. Renn, S. Götzinger, and V. Sandoghdar, Nature Communications 5, 3627 (2014).
  * Nakamura  _et al._ [2014] I. Nakamura, T. Yoshihiro, H. Inagawa, S. Fujiyoshi, and M. Matsushita, Scientific Reports 4, 7364 (2014).
  * Kolesov  _et al._ [2013] R. Kolesov, K. Xia, R. Reuter, M. Jamali, R. Stöhr, T. Inal, P. Siyushev, and J. Wrachtrup, Physical Review Letters 111, 120502 (2013).
  * Yin  _et al._ [2013] C. Yin, M. Rancic, G. G. de Boo, N. Stavrias, J. C. McCallum, M. J. Sellars, and S. Rogge, Nature 497, 91 (2013).
  * Groot-Berning  _et al._ [2019] K. Groot-Berning, T. Kornher, G. Jacob, F. Stopp, S. T. Dawkins, R. Kolesov, J. Wrachtrup, K. Singer, and F. Schmidt-Kaler, Physical Review Letters 123, 106802 (2019).
  * Chen  _et al._ [2020] S. Chen, M. Raha, C. M. Phenicie, S. Ourari, and J. D. Thompson, Science 370, 592 (2020).
  * Raha  _et al._ [2020] M. Raha, S. Chen, C. M. Phenicie, S. Ourari, A. M. Dibos, and J. D. Thompson, Nature Communications 11, 1605 (2020).
  * Siyushev  _et al._ [2014] P. Siyushev, K. Xia, R. Reuter, M. Jamali, N. Zhao, N. Yang, C. Duan, N. Kukharchyk, A. D. Wieck, R. Kolesov, and J. Wrachtrup, Nature Communications 5, 3895 (2014).
  * Kindem  _et al._ [2020] J. M. Kindem, A. Ruskuc, J. G. Bartholomew, J. Rochman, Y. Q. Huan, and A. Faraon, Nature 580, 201 (2020).
  * Kornher  _et al._ [2020] T. Kornher, D.-W. Xiao, K. Xia, F. Sardi, N. Zhao, R. Kolesov, and J. Wrachtrup, Physical Review Letters 124, 170402 (2020).
  * Böttger  _et al._ [2009] T. Böttger, C. W. Thiel, R. L. Cone, and Y. Sun, Physical Review B 79, 115104 (2009).
  * Rančić  _et al._ [2018] M. Rančić, M. P. Hedges, R. L. Ahlefeldt, and M. J. Sellars, Nature Physics 14, 50 (2018).
  * de Boo  _et al._ [2020] G. G. de Boo, C. Yin, M. Rančić, B. C. Johnson, J. C. McCallum, M. J. Sellars, and S. Rogge, Physical Review B 102, 155309 (2020).
  * Dibos  _et al._ [2018] A. M. Dibos, M. Raha, C. M. Phenicie, and J. D. Thompson, Physical Review Letters 120, 243601 (2018).
  * Carey  _et al._ [1998] J. D. Carey, R. C. Barklie, J. F. Donegan, F. Priolo, G. Franzò, and S. Coffa, Journal of Luminescence 80, 297 (1998).
  * Priolo  _et al._ [1995] F. Priolo, G. Franzò, S. Coffa, A. Polman, S. Libertino, R. Barklie, and D. Carey, Journal of Applied Physics 78, 3874 (1995).
  * Liu  _et al._ [1995] P. Liu, J. P. Zhang, R. J. Wilson, G. Curello, S. S. Rao, and P. L. F. Hemment, Applied Physics Letters 66, 3158 (1995).
  * Hughes  _et al._ [2021] M. A. Hughes, N. A. Panjwani, M. Urdampilleta, K. P. Homewood, B. Murdin, and J. D. Carey, Applied Physics Letters 118, 194001 (2021).
  * Przybylinska  _et al._ [1996] H. Przybylinska, W. Jantsch, Y. Suprun-Belevitch, M. Stepikhova, L. Palmetshofer, G. Hendorfer, A. Kozanecki, R. J. Wilson, and B. J. Sealy, Physical Review B 54, 2532 (1996).
  * Tang  _et al._ [1989] Y. S. Tang, K. C. Heasman, W. P. Gillin, and B. J. Sealy, Applied Physics Letters 55, 432 (1989).
  * Vinh  _et al._ [2003] N. Q. Vinh, H. Przybylińska, Z. F. Krasil’nik, and T. Gregorkiewicz, Physical Review Letters 90, 066401 (2003).
  * Weiss  _et al._ [2021] L. Weiss, A. Gritsch, B. Merkel, and A. Reiserer, Optica 8, 40 (2021).
  * Berkman  _et al._ [tion] I. Berkman, G. de Boo, A. Lyasota, J. G. Bartholomew, B. C. Johnson, J. McCallum, B.-B. Xu, S. Xie, R. Ahlefeldt, M. Sellars, C. Yin, and S. Rogge (in preparation).
  * Carey  _et al._ [1996] J. D. Carey, J. F. Donegan, R. C. Barklie, F. Priolo, G. Franzò, and S. Coffa, Applied Physics Letters 69, 3854 (1996).
  * Carey  _et al._ [1999] J. D. Carey, R. C. Barklie, J. F. Donegan, F. Priolo, G. Franzò, and S. Coffa, Physical Review B 59, 2773 (1999).
  * Carey [2002] J. D. Carey, Journal of Physics: Condensed Matter 14, 8537 (2002).
  * Hughes  _et al._ [2019] M. A. Hughes, H. Li, N. Theodoropoulou, and J. D. Carey, Scientific Reports 9, 19031 (2019).
  * Guillot-Noël  _et al._ [2004] O. Guillot-Noël, P. Goldner, P. Higel, and D. Gourier, Journal of Physics: Condensed Matter 16, R1 (2004).
  * Cone and Meltzer [1987] R. L. Cone and R. S. Meltzer, in  _Spectroscopy of Solids Containing Rare Earth Ions_ , edited by A. A. Kaplyanskii and R. M. Macfarlane (North Holland, 1987) p. 481.
  * Fricke [1979] W. Fricke, Zeitschrift für Physik B Condensed Matter 33, 261 (1979).
  * Yamaguchi  _et al._ [1998] M. Yamaguchi, K. Koyama, T. Suemoto, and M. Mitsunaga, Journal of Luminescence 76-77, 681 (1998).
  * Ahlefeldt  _et al._ [2013a] R. L. Ahlefeldt, W. D. Hutchison, N. B. Manson, and M. J. Sellars, Physical Review B 88, 184424 (2013a).
  * Laplane  _et al._ [2016] C. Laplane, E. Zambrini Cruzeiro, F. Fröwis, P. Goldner, and M. Afzelius, Physical Review Letters 117, 037203 (2016).
  * Wolf [1971] W. P. Wolf, Le Journal de Physique Colloques 32, C1 (1971).
  * Baker [1971] J. M. Baker, Reports on Progress in Physics 34, 109 (1971).
  * Ohlsson  _et al._ [2002] N. Ohlsson, R. Krishna Mohan, and S. Kröll, Optics Communications 201, 71 (2002).
  * Grimm  _et al._ [2021] M. Grimm, A. Beckert, G. Aeppli, and M. Müller, PRX Quantum 2, 010312 (2021).
  * Ahlefeldt  _et al._ [2020] R. L. Ahlefeldt, M. J. Pearce, M. R. Hush, and M. J. Sellars, Physical Review A 101, 012309 (2020).
  * Ahlefeldt  _et al._ [2013b] R. L. Ahlefeldt, D. L. McAuslan, J. J. Longdell, N. B. Manson, and M. J. Sellars, Physical Review Letters 111, 240501 (2013b).
  * Lukac and Hahn [1989] M. Lukac and E. Hahn, Optics Communications 70, 195 (1989).
  * Guillot-Noël  _et al._ [2000] O. Guillot-Noël, V. Mehta, B. Viana, D. Gourier, M. Boukhris, and S. Jandl, Physical Review B 61, 15338 (2000).
  * Clemens and Hutchison [1983] J. M. Clemens and C. A. Hutchison, Physical Review B 28, 50 (1983).
  * Erdös [1966] P. Erdös, Journal of Physics and Chemistry of Solids 27, 1705 (1966).
  * Carey [2009] J. D. Carey, Journal of Physics: Condensed Matter 21, 175601 (2009).
  * Wolf [2000] W. P. Wolf, Brazilian Journal of Physics 30, 794 (2000).
  * Kenyon [2005] A. J. Kenyon, Semiconductor Science and Technology 20, R65 (2005).
  * Birgeneau [1968] R. J. Birgeneau, Applied Physics Letters 13, 193 (1968).
  * Brower  _et al._ [1966] K. L. Brower, H. J. Stapleton, and E. O. Brower, Physical Review 146, 233 (1966).
  * Cone and Meltzer [1973] R. L. Cone and R. S. Meltzer, Physical Review Letters 30, 859 (1973).
  * Cone and Meltzer [1975] R. L. Cone and R. S. Meltzer, The Journal of Chemical Physics 62, 3573 (1975).
  * Prinz [1966] G. A. Prinz, Physical Review 152, 474 (1966).
  * Abraham  _et al._ [1992] M. M. Abraham, J. M. Baker, B. Bleaney, J. Z. Pfeffer, and M. R. Wells, Journal of Physics: Condensed Matter 4, 5443 (1992).
  * Longdell  _et al._ [2004] J. J. Longdell, M. J. Sellars, and N. B. Manson, Physical Review Letters 93, 130503 (2004).
  * Kinos  _et al._ [2021] A. Kinos, D. Hunger, R. Kolesov, K. Mølmer, H. de Riedmatten, P. Goldner, A. Tallaire, L. Morvan, P. Berger, S. Welinski, K. Karrai, L. Rippe, S. Kröll, and A. Walther, arXiv:2103.15743 [quant-ph] (2021), arXiv:2103.15743 [quant-ph] .