# Transition metal ion ensembles in crystals as solid-state coherent spin-
photon interfaces: The case of nickel in magnesium oxide

E. Poem<EMAIL_ADDRESS>Department of Physics of Complex Systems,
Weizmann Institute of Science, Rehovot 7610001, Israel  S. Gupta  Prizker
School of Molecular Engineering, The University of Chicago, Chicago, IL 60637,
USA  I. Morris  K. Klink  Department of Physics and Astronomy, Michigan State
University, 567 Wilson Rd., East Lansing, MI 48824, USA  L. Singh  Department
of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001,
Israel  T. Zhong  Prizker School of Molecular Engineering, The University of
Chicago, Chicago, IL 60637, USA  J. N. Becker  Department of Physics and
Astronomy, Michigan State University, 567 Wilson Rd., East Lansing, MI 48824,
USA  Coatings and Diamond Technologies Division, Center Midwest (CMW),
Fraunhofer USA Inc., 1449 Engineering Research Ct., East Lansing, MI 48824,
USA  O. Firstenberg  Department of Physics of Complex Systems, Weizmann
Institute of Science, Rehovot 7610001, Israel

###### Abstract

We present general guidelines for finding solid-state systems that could serve
as coherent spin-photon interfaces and show that transition metal ions in
various crystals could comply with these guidelines. As an illustrative
example, we focus on divalent nickel ions in magnesium oxide. We perform
electron spin resonance spectroscopy and polarization-sensitive magneto-
optical fluorescence spectroscopy of a dense ensemble of these ions, and find
that (i) the ground-state electron spin stays coherent at liquid-helium
temperatures for several microseconds, and (ii) there exists energetically
well-isolated excited states which can couple to two ground state spin
sublevels via optical transitions of orthogonal polarizations. The latter
implies that fast, coherent optical spin control is possible. We then propose
schemes for optical initialization and control of the ground-state spin using
polarized optical pulses, as well as two schemes for implementing a noise-
free, broadband quantum-optical memory at near-telecom wavelengths in this
material system.

††preprint: APS/123-QED

##  I Introduction

Quantum communication and networks require coherent coupling between
travelling qubits, carrying the quantum information for long distances, and
stationary qubits, which can act as memories and processors, storing and
manipulating the quantum information [1]. Photons at wavelengths in the
telecommunication bands are arguably the best travelling qubits, as they have
very low loss probability and can maintain their coherence over long distances
in standard telecom fibers. Solid-state defect spins are promising stationary
qubits, as they can have coherence times as long as milliseconds and are
embedded in a miniaturizable platform [2, 3]. One of the outstanding
challenges towards the realization of optical quantum networks is the coherent
coupling of telecom photons and solid-state qubits. For defect spins, this
would mean the coherent coupling of light and spin.

As light (at optical wavelengths, in the far field) directly affects only the
orbit of the electron but not its spin [4], any spin-photon coupling has to be
mediated by additional internal interactions within the quantum system, such
as the relativistic effect of spin-orbit (SO) coupling. This leads to the two
following requirements.

First, to coherently couple light at a certain frequency to a spin at a given
temperature, there should exist an excited state at that frequency (with
respect to the ground state) in which the SO coupling rate is much faster than
the decoherence rate at that temperature. For ensembles of spin defects with
typical inhomogeneous broadenings of the optical transitions on the order of a
few to tens of GHz, even at low temperatures [2, 5], this requires _THz-scale
SO coupling_.

Second, the ground-state spin should couple to the electron orbit as little as
possible, as such coupling would expose it to decoherence due to the
interactions of the orbital degree of freedom with phonons. One way to achieve
this in the presence of strong SO interaction is to use an electronic system
with a _ground state orbital singlet_ , and therefore zero effective orbital
angular momentum.

The tension between these two requirements can be illustrated with a few
example cases.

The first case is that of ensembles of NV- centers in diamond [6]. While these
ensembles comply with the second requirement (3A2 ground state manifold), they
do not comply with the first ($\sim$3 GHz SO coupling within the 3E excited
state manifold). Therefore, they have excellent spin properties [7] but only
incoherent spin-photon coupling (coherent spin-photon coupling was achieved
only for single NV- centers at low temperatures [8, 9, 10, 11, 12, 13]).

The second case includes both negatively-charged group-IV-vacancy centers in
diamond [14] and commonly-used rare-earth ion ensembles [15, 2], like Pr3+,
Eu3+, or Er3+. These systems feature orthogonal challenges to the NV- center,
as they have strong SO in the excited state, but an orbital multiplet ground
state manifold. Therefore, while they may support high-temperature coherent
spin-photon coupling, their ground state spin quickly decoheres above liquid
helium temperatures.

The third example case is that of the three known zero orbital angular
momentum (S-state) rare-earth ions, Gd3+ [16, 17], Eu2+ [18, 19], and Tb4+
[18, 20], and the S-state actinide ion Cm3+ [21]. These ions comply with both
requirements and indeed exhibit very narrow spin distributions and optical
line widths, some of them even up to room temperature. Unfortunately, the
first three have optical transitions only in the ultra-violet, and the last
one, while having transitions in the visible, is radioactive. This makes them
less suitable for optical communications purposes.

In contrast to the examples above, transition metal ions, even in high
symmetry lattice sites of cubic crystals [22], can comply with both
requirements while having optical transitions at infrared or even telecom
wavelengths. First, due to their relatively large atomic number, the SO
interaction is usually on the order of a few THz (similarly to the case of
rare earth ions). Second, due to the effect of the crystal field, which can be
much stronger than in rare earth ions, there are multiple cases with ground
state orbital singlets (similar to the NV-). This happens whenever the highest
set of degenerate single-electron orbitals is half-filled. For weak crystal
fields, forming high-spin configurations, the only relevant configuration is
$d^{5}$ (in analogy to the $f^{7}$ configuration of the S-state rare earth
ions). For stronger crystal fields, however, where low-spin configurations
form, there are more options. For cubic sites, these include $d^{8}$ and
$d^{3}$ ions in octahedral sites, and $d^{2}$ and $d^{7}$ ions in tetrahedral
sites. Of these, the configurations with an even number of electrons have an
$S=1$ ground state, while those with an odd number have an
$S=\nicefrac{{3}}{{2}}$ ground state. While some transition metal ion systems
were investigated in the context of quantum information processing [23, 24,
25, 26], the only such system having an orbital singlet ground state that has
been investigated in this context, to the best of our knowledge, is Cr4+ in
GaN and SiC [27, 28], having a d2 configuration in a tetrahedral site. It
indeed displays infrared emission (around 1090 nm) and a coherent ground state
spin up to at least 15 K. This spin could potentially be coherently controlled
optically, though, to the best of our knowledge, this has not been
demonstrated yet.

Here we focus on another such example: divalent nickel (Ni2+) substituting for
a magnesium ion in magnesium oxide (MgO). It has a d8 configuration, and due
to the octahedral geometry of its site, its ground state is an orbital singlet
and spin triplet. This system was very thoroughly studied in the past few
decades, both for fundamental characterization [29, 30, 31, 32, 33, 34, 35,
36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50] and for
applications, mostly as a potential gain medium for tunable and pulsed
infrared lasers [51, 52, 53]. The spin triplet nature of the ground state was
confirmed by electron spin resonance (ESR) studies already over 60 years ago
[29, 30, 31]. The spin-lattice relaxation time (T1) was measured to be as long
as 1 ms even at higher-than liquid helium temperatures [35, 36]. Optical
studies revealed rich emission and absorption spectra [33, 37, 39, 53, 47,
48]. In particular, the lowest emission energy zero-phonon lines (ZPLs) are
very close to the telecom O-band, at 1220 nm and 1250 nm. Remarkably, these
lines remain well separated up to temperatures as high as 150 K. The optical
lifetime of these lines is very long, about 3.6 ms [52, 53]. While this
indicates a weak transition dipole moment, predominantly magnetic, due to the
perfect solubility of NiO in MgO [49], optically dense ion ensembles could be
readily made, compensating for the weak optical response of individual ions
and enabling a strong collective response. Furthermore, as only 5% of the
atoms in MgO made with natural isotope abundances have non-zero nuclear spin
(due to 25Mg), the spin dephasing rates due to nuclear spin-bath noise should
be low and were theoretically predicted to be below 1 kHz [54, 55].
Importantly, like diamond [7], this material can in principle be made nuclear-
spin-free by using isotopically purified precursors.

These compelling features lead us to re-examine this material system for use
as a coherent spin-photon interface at above-liquid helium temperatures. We
experimentally investigate both the ground-state spin decoherence mechanisms
and the spin structure of the excited state, and show that they are compatible
with THz bandwidth coherent optical spin control even at temperatures
exceeding that of liquid helium.

The rest of the manuscript is organized as follows. In Sect. II, we describe
the level structure of Ni2+ in MgO, both of the ground state and of the
excited states, where for the latter we focus on the difference between the
case of a weakly-perturbed SO coupling and that of quenched SO coupling due to
a strong dynamic Jahn-Teller (DJT) distortion of the excited state orbitals
[56]. In Sect. III, using ESR and temperature-dependent spin-echo (SE)
measurements, we show that for a high-density ensemble, the main decoherence
mechanism is dipolar interaction between the ensemble spins. For a few GHz of
ground state splitting (induced by an external magnetic field), this
interaction is already saturated at liquid-helium temperatures, leading to a
decoherence time of 3 $\mu$s. In Sect. IV, we use polarization-sensitive
magneto-optical fluorescence spectroscopy on the two ZPLs and show that the
excited-state spin structure is indeed determined mostly by the SO interaction
since the DJT distortion of the excited-state orbitals is weak. Following
these findings, in Sect. V, we propose protocols for optical spin-state
preparation, measurement, and manipulation, as well as for noise-free optical
quantum memory. Finally in Sect. VI, we summarize our results and outline
possible directions for future research.

##  II level structure

###  II.1 Ground state

The ground state of a Ni2+ ion in an octahedral site of a cubic lattice (Oh
symmetry group) contains two electrons (or two electron-holes) occupying two
degenerate $e_{g}$ single-electron orbitals (formed by the $d_{x^{2}-y^{2}}$
and $d_{z^{2}}$ $d$-orbitals), as shown in Fig. 1(a). As this is a half-filled
shell, there is only one many-electron orbital. The total spin of the two
electrons can be either 0 or 1, but the spin-1 states have lower energy. The
ground state is therefore an orbital-singlet spin-triplet, 3A2g, the spin-
orbit representation of which is $T_{2u}$. The effective Hamiltonian for this
manifold is

$H^{\mathrm{g}}=\mu_{\mathrm{B}}g^{\mathrm{g}}_{s}\mathbf{B\cdot
S}+\mathbf{S\cdot q(\epsilon)\cdot S}+\mu_{\mathrm{B}}\mathbf{B\cdot\delta
g_{S}(\epsilon)\cdot S},$ (1)

where $\mathbf{S}$ is the vector of spin-1 operators, $\mathbf{B}$ is the
magnetic field vector, $\mathbf{q}(\epsilon)$ is the strain-induced magnetic
quadrupole moment, $\mathbf{\delta g_{S}}(\epsilon)$ is the strain-induced
g-tensor [57], and $\epsilon$ is the strain tensor. Here $\mu_{\mathrm{B}}$ is
the Bohr magneton, and $g^{\mathrm{g}}_{s}$ is the ground state g-factor. The
three spin states are degenerate at zero fields and strains. A constant
magnetic field along the z-direction splits the states via a Zeeman
interaction [first term in Eq. (1)]. Due to the second term in Eq. (1), local
random strains (of a cubic lattice) can shift the $T_{2u,0}$ state with
respect to the $T_{2u,\pm 1}$ states to first order in the ratios of the
strain energies and the Zeeman energy. This inhomogeneously broadens the
$T_{2u,1}\leftrightarrow T_{2u,0}$ and the $T_{2u,0}\leftrightarrow T_{2u,-1}$
spin transitions. The same term can also split the $T_{2u,\pm 1}$ states,
however only to second order in strain-to-Zeeman energy ratio [57]. This is
because the relevant strain terms are off-diagonal in the magnetic-field
Hamiltonian, such that their effect is quenched as the magnetic field becomes
large, leading to a reduced broadening of the $T_{2u,1}\leftrightarrow
T_{2u,-1}$ spin transition. The third term in Eq. (1) cannot shift the
$T_{2u,0}$ state with respect to the $T_{2u,\pm 1}$ states, but can split the
$T_{2u,\pm 1}$ states to first order. However, in MgO, for an applied magnetic
field on the order of 100 mT (few GHz $T_{2u,1}\leftrightarrow T_{2u,0}$
Zeeman splitting), this splitting is about two orders of magnitude smaller
than the first-order shifts induced by the second term [31, 57, 58, 59, 60],
keeping the broadening of the $T_{2u,1}\leftrightarrow T_{2u,-1}$ spin
transition smaller than that of the $T_{2u,0}\leftrightarrow$$T_{2u,\pm 1}$
transitions. The level splitting and broadenings are schematically presented
in Fig. 1(d). In Sect. III below, we present ESR measurements of these
inhomogeneous broadenings, as well as SE measurements of the homogeneous
decoherence time at different temperatures.

Figure 1:  Energy level structure of Ni2+:MgO. (a) Ground-state electronic
configuration. The arrows represent electrons, and the ellipses represent
holes. (b) Electronic configuration of the first excited state. (c) [(e)]
Ground and excited state level structures resulting from SO interaction in the
presence of a weak [strong] DJT. (d) [(f)] Splitting of the ground and excited
state spin sublevels in a magnetic field for weak [strong] DJT. The grey
shaded regions in the weak DJT ground state represent random-strain-induced
inhomogeneous broadening. A polarized $\Lambda$-system forms for weak DJT.
Other optical transitions neglected for simplicity. (g) [(h)] Splitting of the
optical emission lines under a magnetic field (field increases to the right)
in Faraday configuration, for the case of weak [strong] DJT. Purple (orange)
lines represent $\sigma^{+}$ ($\sigma^{-}$) polarized transitions. Faded-
colored (black) lines represent partially-polarized (unpolarized) transitions.
$\pi$-polarized transitions are not shown.

###  II.2 Excited state

The first excited state of Ni2+ in MgO is composed of one electron-hole in one
of two $e_{g}$ single-electron orbitals, and one hole in one of three $t_{2g}$
single-electron orbitals (formed by the $d_{yz}$, $d_{xz}$, and $d_{xy}$
$d$-orbitals), as shown in Fig. 1(b). Therefore, there are six possible two-
electron orbitals. The cubic symmetry splits these six states into two orbital
triplets, T1g and T2g, where the latter has a lower energy [33]. The spin
state of the two electrons in the lowest excited-state manifold is again a
spin-1 triplet. Thus, the lowest excited-state manifold is 3T2g, which
contains 9 states in total.

These states are coupled and split by the spin-orbit interaction. Its
magnitude depends on the strength of the dynamic Jahn-Teller (DJT) coupling
between the electronic orbitals and lattice vibrations, as the latter may
affect the shapes of the orbitals and thus their effective angular momentum
[56]. The effective Hamiltonian for the 3T2g manifold, in the presence of a
magnetic field, is

$H^{\mathrm{e}}(\kappa)=H_{\mathrm{so}}^{\mathrm{e}}(\kappa)+H_{B}^{\mathrm{e}}(\kappa),$
(2)

where $\kappa=3E_{\mathrm{JT}}/\hbar\omega_{\mathrm{ph}}$ is the relative
strength of the DJT coupling. It is proportional to the ratio between the
electron-phonon coupling energy $E_{\mathrm{JT}}$ and the energy of the lowest
phonon mode $\hbar\omega_{\mathrm{ph}}$.

The zero-field Hamiltonian is [56]

$H_{\mathrm{so}}^{\mathrm{e}}(\kappa)=e^{-\kappa/2}\zeta\mathbf{L\cdot S}+[\mu
e^{-\kappa/2}+K_{1}(\kappa)](\mathbf{L\cdot
S})^{2}+[\rho+\mu(1-e^{-\kappa/2})+K_{2}(\kappa)]A,$ (3)

where $\mathbf{L}$ ($\mathbf{S}$) is the orbital (spin) angular momentum
vector operator, and
$A=L_{x}^{2}S_{x}^{2}+L_{y}^{2}S_{y}^{2}+L_{z}^{2}S_{z}^{2}$ is a second-order
cubic-symmetry spin-orbit term. Here, both $\mathbf{L}$ and $\mathbf{S}$ are
spin-1 operators. The quadratic SO terms (the last two terms) arise from
second-order perturbation theory applied to the full Hamiltonian [56]. The
energies $K_{1}$ and $K_{2}$ are given by,
$K_{1}=(g_{L}^{2}\zeta^{2}/\hbar\omega_{\mathrm{ph}})e^{-\kappa}G(\kappa/2)$,
and
$K_{2}=(g_{L}^{2}\zeta^{2}/\hbar\omega_{\mathrm{ph}})e^{-\kappa}[G(\kappa)-G(\kappa/2)]$,
where $G(x)=\int_{0}^{x}dt(e^{t}-1)/t$ [56, 61]. The magnetic-field dependent
Hamiltonian is

$H_{B}^{\mathrm{e}}=\mu_{\mathrm{B}}(g_{L}e^{-\kappa/2}\mathbf{L}+g^{\mathrm{e}}_{s}\mathbf{S})\cdot\mathbf{B},$
(4)

where $g_{s}^{\mathrm{e}}$ is the excited state g-factor, and $g_{L}$ its
orbital gyromagnetic ratio. Here we neglected the static strain shifts.

For weak DJT distortion, that is, for $\kappa<1$, the “static lattice” SO
structure survives, and the nine-fold degenerate state-space splits into four
distinct energy levels: a doublet ($E_{u}$), two triplets ($T_{1u}$ and
$T_{2u}$), and a singlet ($A_{2u}$), as shown in Fig. 1(c). Out of these, the
$T_{1u}$ triplet contains one state (marked in the figure) that mixes the
$m_{s}=\pm 1$ spin states, each coupled to a different orbital. This state,
together with the $T_{2u,\pm 1}$ ground states, manifests a polarized
$\Lambda$ system [Fig. 1(d)]. Such a level system enables the control of the
$T_{2u,\pm 1}$ ground-state two-level system using polarized light [62, 63,
10, 64].

For the opposite case of a strong DJT distortion, where $\kappa\gg 1$, only
the $A$ term in the Hamiltonian of Eq. (3) survives, and there are only two
energy levels: a triplet (composed of $E_{u}$ and $A_{2u}$) and a sextuplet
(composed of $T_{1u}$ and $T_{2u}$), as shown in Fig. 1(e). In this case, due
to destructive interference between different states of the same level, no
$\Lambda$-system can form [65], and coherent optical spin manipulation is
prohibited [Fig. 1(f)].

It is therefore crucial to distinguish between the weak and strong DJT
regimes. However, the absorption and emission spectra are ambiguous at first
glance and more detailed studies are required to elucidate the strength of the
DJT in this system. If the absorption spectrum would feature four distinct
narrow lines, as is the case, for example, for Ni2+ in forsterite [66], the
favorable weak DJT case would be clearly identified. However, only two of the
observed absorption lines are narrow, while the rest are broad [33]. This
could be for one of two reasons: (i) the broad lines include the two remaining
“static lattice” lines, but mixed with high-energy vibrations (which are not
included in the above model), or (ii) the broad lines are purely vibrational
lines, and the two narrow lines are the result of a strong DJT distortion of
the electronic levels.

In the literature, the common interpretation is that of weak DJT, and some
studies assign energies to the upper two electronic transitions [33, 53].
However, so far, this interpretation has not been validated other than via a
theoretical analysis of the possible vibrational modes of Ni2+ in MgO [41,
40], and recent theoretical studies have questioned it [50]. In Appendix A, we
show that both the strong and weak DJT cases can fit the observed spectra.
Furthermore, cases of other transition metal spin-1 systems where only two of
the four expected narrow absorption lines were observed, namely V3+ in GaAs,
GaP, InP [67, 68, 69, 70, 71, 72, 73, 74, 75, 76], and ZnO [77], have been
reported, and the strong DJT distortion case was shown to be valid for these
systems. This was achieved by analyzing the magnetic-field dependence of the
absorption spectrum [70, 77]. For the case of Ni2+ in MgO, while magnetic
circular dichroism has been probed in the past [42, 43], no conclusion
regarding the excited state spin structure was drawn.

Figures 1(g) and 1(h) present the magnetic splitting of the optical
transitions for the cases of weak and strong DJT, respectively (see Appendix B
for the derivation). It is clearly seen that both the number of spectral
components and their polarizations differ between the two cases, allowing for
a clear distinction between them. In Sect. IV below, we present polarized
optical magneto-fluorescence spectroscopy measurements for the two lowest-
energy ZPLs of Ni2+ in MgO, which unambiguously support the case of a weak DJT
distortion.

##  III Ground state spin coherence

For the experiment, we use a 5$\times$5$\times$1 mm single-crystal MgO sample,
cut along the (001) planes and optically-polished on the two large facets,
grown by Goodfellow Inc. It was intentionally doped with 450 ppm (2.4$\times
10^{19}$ cm-3) of nickel (nominally) and had a nominal concentration of 10 ppm
of unintentional dopants. The sample was investigated as grown, with no
further processing.

Figure 2:  Electron spin resonance (ESR) spectrum of the ground-state spin of
Ni2+:MgO. The magnetic field was calibrated using the Cr3+ line at $g=1.98$.
The inset magnifies the spectrum around the $\Delta m_{s}=2$ transition.

For ESR and SE measurement, the sample was placed in a 5.006 GHz aluminum
microwave loop-gap cavity with a linewidth of 1 MHz. The cavity was mounted on
the mixing chamber of a Bluefors LD250 dilution refrigerator reaching a base
temperature of 8 mK. A three-axis vector magnet (AMI Model 430) was used to
apply a magnetic field. A cryogenic amplifier (Low noise factory LNF-
LNC0.3_14a) was used to pre-amplify the spin echo signal, limiting the maximum
microwave power to $<1$ mW and the maximum temperature to 4 K.

The ESR spectrum measured at 4 K is presented in Fig. 2 with the magnetic
field approximately aligned along the [100] axis. The relatively narrow
feature around 180 mT ($g=1.98$) is a well-known transition of Cr3+ [78]. We
verified the presence of Cr3+ in our sample also using fluorescence
spectroscopy (see Sect. IV below). We used the known g-factor of the Cr3+
transition to calibrate the magnetic field. The dominant broad feature around
160 mT ($g=2.24$) is related to Ni2+. The measured g-factor matches the known
value of 2.214 [30] to the precision of our magnetic field calibration. Its
width (peak-to-peak) is 13 mT, corresponding to 400 MHz line width. This large
broadening is probably due to random strain introduced by the high
concentration of dopants, limiting the inhomogeneous coherence time of the
spin ensemble to $\sim 1$ ns. Similar widths have been measured previously,
and it is also known that high-temperature annealing reduces the width by
about a factor of two [38]. We did not see the ‘inverse line’ previously
observed in the center of the Ni2+ line [30, 38]. As this line was attributed
to a homogeneous, resonant cross-relaxation process [38], it could be that
this process was quenched due to the low temperature in our experiment. We
also could not observe the ‘double-quantum’ line, due to a two-photon
transition between the $T_{2u,-1}$ and $T_{2u,1}$ states [30, 35, 38], most
probably due to our microwave-power limitation.

In addition to the strong Ni2+ and Cr3+ lines, the measured spectrum exhibits
a small feature very close to half the magnetic field of the main Ni2+
resonance, as shown in the inset of Fig. 2. We attribute this feature, which
has the form of a Fano resonance (‘bound state in a continuum’ [4]), to the
forbidden, $\Delta m_{s}=2$ single-photon transition between the $T_{2u,-1}$
and $T_{2u,1}$ states of the Ni2+ ground state. The transition becomes
partially allowed due to strain [57]. The Fano shape, also seen in previous
works [35], is most probably due to interference with the wide background
coming from the $\Delta m_{s}=1$ transition. The width of this line (FWHM) is
1.5 mT, corresponding to about 90 MHz and $\sim 5$ ns inhomogeneous coherence
time. This narrow width (relative to the main transition) results from the
lower strain sensitivity of the energy gap between the $T_{2u,\pm 1}$ levels
[57]. A similar ratio was also measured for the double-quantum transition [30,
35, 38].

Next, we set the magnetic field to 141 mT, at the edge of the distribution,
and measure the spin echo following excitation with two 500 ns-long pulses.
The measured echo amplitude versus the time between the pulses, taken at 9 mK,
is presented in Fig. 3(a). Most strikingly, we observe pronounced
oscillations. The Fourier transform of this pattern is presented in Fig. 3(b).
The main frequency component is at 375$\pm$10 kHz, fitting the predicted 366
kHz of nuclear Zeeman splitting of 25Mg at the applied field rather well.
Additional components at the second and third harmonics of this frequency are
also visible. The oscillations can therefore be explained as an electron spin-
echo envelope modulation (ESEEM) [79, 80] caused by the coupling of the Ni2+
electronic spin to the nuclear spins of neighboring 25Mg atoms. The modulation
frequencies exactly match multiples of the nuclear Zeeman splitting [marked by
vertical dashed lines in Fig. 3(b)] and are not affected by any hyperfine
coupling terms despite the strong modulation depth, in principle necessitating
strong hyperfine coupling. This can be explained by the zero spin component of
the excited electronic state, limiting hyperfine coupling to the ground state
electron spin, which is fully occupied at the experiment temperature. However,
as the nuclear spin-state is still fully mixed at the experiment temperature,
only transitions with the same ground-state nuclear spin and different excited
state nuclear spin would interfere, and thus hyperfine coupling does not show
up in the modulation frequency. Using the model presented in Refs. [79, 80],
adapted to the case of initial (final) electron spin component of $-1$ (0) and
a nuclear spin of 5/2 (see Appendix C), we calculate the expected modulation
frequencies and their amplitudes for the applied pulse and cavity bandwidths.
These are presented as yellow vertical bars in Fig. 3(b).

Figure 3:  Spin echo at 9 mK and 141 mT. (a) Amplitude versus the time between
the two driving pulses. The orange line is a fit of the mean of the
oscillation envelopes to a biexponential decay curve. (b) The frequency
content of the measured echo decay. A wide background peaked at zero frequency
was subtracted. The yellow bars are the relative oscillation amplitudes
predicted by the ESEEM model (see text and Appendix C) multiplied by the
instrumental spectral response (gray line).

The decay of the mean envelope of the oscillations can be fitted to a bi-
exponential function [orange line in fig. 3(a)], with a short decay time of
4.50$\pm 0.03$ $\mu$s, and a long decay time of 109$\pm 2$ $\mu$s.

To explain the shape of the decay curve and its temperature dependence (see
below), we consider three main dephasing mechanisms [81]: direct flip-flop of
neighboring spins within the sub-ensemble probed by the cavity; instantaneous
diffusion dephasing due to the flipping of neighboring spins by the
$\pi$-pulse; and stochastic energy shifts (‘spectral diffusion’) of the probed
spins due to flip-flops of the entire ensemble [82, 83]. We neglect dephasing
due to nuclear spins (25Mg and 61Ni), as the product of their density and
magnetic moments is much lower than that of the Ni2+ electron spin ensemble.

At low temperatures, the first two processes usually dominate, as they involve
resonant dipole-dipole interaction between close-by spins. However, in an
inhomogeneous ensemble, some of the probed spins will have fewer probed-spin
neighbors than others, leading to a distribution of decay times and to a bi-
exponential decay curve [84]. In general, in the sub-ensemble of probed spins
for which the immediate environment is of low density, the spectral diffusion
dephasing would have a more significant contribution to the total dephasing
rate. However, at low temperatures the first two processes only weakly depend
on temperature [81], while the spectral diffusion dephasing, which depends on
the number of spin pairs that can perform flip-flop, vanishes at low
temperatures [82]. Thus, at very low temperatures, the short decay time is
caused by the sub-ensemble of probed spins with strong instantaneous diffusion
and direct flip-flop, and the long decay time originates from the sub-ensemble
in which these interactions are weak.

The low-temperature short-time coherence therefore strongly depends on the
average density of the probed sub-ensemble [83]. This can be observed by
scanning the field across the inhomogeneous broadening of the spin ensemble
and probing the spin-echo amplitude for a fixed, short time difference (here
5.6 $\mu$s), as shown in Fig. 4(a). It is clearly seen that the amplitude
drops near the center of the distribution, where the density of probed spins
is the highest, and thus most of the probed population would experience strong
direct dephasing. The strong coherence peak at 180 mT is due to Cr3+ ions, the
density of which is much lower than that of the Ni2+ ions.

As the temperature increases, the spectral diffusion rate increases, first
affecting only the lower density sub-ensemble, until at a certain temperature
it will dominate even over the direct processes in the denser sub-ensemble, at
which point the coherence decay will become mono-exponential. Figure 4(b)
presents the extracted long decay times versus temperature. The line is a two-
parameter fit to a model including both a temperature-independent, relatively
weak component, due to instantaneous diffusion and direct flip-flop in the
low-density sub-ensemble, and the temperature-dependent, spectral diffusion
dephasing rate [82, 83], the latter adapted to a spin-1 bath (see Appendix D).

Figure 4:  Dependence of spin coherence on density and temperature. (a) Spin
echo amplitude at $\tau=5.6$ $\mu$s and a temperature of 9 mK versus the
applied magnetic field. The shaded gray area is the integrated ESR signal,
proportional to the defect density probed at each magnetic-field value. The
dash-dotted gray lines mark the peaks of the Ni2+ and Cr3+ distributions. The
black dashed line marks the field at which all time-dependent echo
measurements were performed. (b) Time dependence of the echo signal vs.
temperature. The dots are the echo-envelope decay times, extracted by
exponential fits to the data. The error bars are the 68% confidence level of
the fits. The solid line is a two-parameter fit of the measured times to a
model including the temperature-dependent, spectral-diffusion dephasing rate,
and an additional, temperature-independent rate accounting for the direct
flip-flop and the instantaneous diffusion processes (see Appendix D). The
shaded purple area marks the 68% confidence interval of this fit. The bottom-
left (top-right) inset presents the measured echo signals versus the time
delay (dots), for the three lowest (highest) temperatures, together with the
fitted exponential functions (solid lines). The colors correspond to the
temperature, as marked in the main plot by the solid circles.

The spin decoherence rate saturates around liquid-helium temperature, yielding
a coherence time of about 3 $\mu$s. At these temperatures, the coherence decay
is indeed mono-exponential [see top right inset to Fig. 4(b)]. As the rates of
all the three dephasing processes we consider depend at-least linearly on the
ensemble density [82], there is a prospect of considerably prolonging the
coherence time by using less dense ensembles. This, in combination with
thermal annealing, could also considerably decrease the inhomogeneous
broadening, bringing that of the $\delta m_{s}=2$ transition to the level of a
few MHz, which would enable dynamical decoupling of the entire ensemble using
nanosecond microwave or optical pulses. Note that this requirement could be
considerably alleviated if picosecond or even femtosecond optical pulses could
be used (see Sect. V below).

##  IV Optical spectroscopy

###  IV.1 Emission spectrum

For optical fluorescence spectroscopy, we placed the sample in a closed cycle,
low-vibration helium flow cryostat (ARS CS204-DMX-20-OM). A diode laser at 660
nm (Thorlabs LP660-SF50) was used for excitation (into a vibrational sideband
of the 3T1g multiplet). The fluorescence was collected using an infrared-
optimized, NA=0.8 objective lens (Shibuya M ePLAN NIR 100A) and analyzed by a
0.75 m spectrometer (Teledyne-Princeton Instruments SpectraPro HRS-750),
equipped with a 300 g/mm grating (resolution limit of 30 GHz around 1250 nm),
and a liquid-nitrogen-cooled InGaAs CCD array detector (Teledyne-Princeton-
Instruments PyLoN IR). Figure 5(a) presents the measured emission spectra at
different temperatures. Two distinct lines, at 1220 nm and 1250 nm dominate
the spectrum, corresponding to the optical transitions from the first two
excited states to the ground state. The (inhomogeneous) width of the lines up
to temperatures of about 60 K is $\sim$100 GHz, much narrower than the
splitting between them, 5.28 THz. While the lines further broaden at higher
temperatures (mostly homogeneously), it is clearly seen that they remain well-
separated up to temperatures as high as 150 K. The inset presents the emission
spectrum around 698 nm, detected by the same spectrometer using a 1200 g/mm
grating and a silicon CCD array detector (Teledyne-Princeton-Instruments Blaze
HR). The narrow emission line (18 GHz, close to the resolution limit of 15
GHz) of Cr3+ [85] is clearly seen, supporting the identification of the ESR
line at 180 mT (Fig. 2). By modulating the laser current and gating the CCD
camera accordingly, we measured the time dependence of the fluorescence
following the laser pulse, and confirmed that the 1250 nm fluorescence decay
time in our sample is indeed $\sim$3.6 ms, as previously reported for Ni2+ in
MgO [52]. Figure 5(b) presents the measured lifetime of the excited level
versus the temperature. The inset presents an exemplary measurement. Very
little change ($\sim$2%) of the decay time is observed even up to 100 K [52].

Figure 5:  Optical spectroscopy of Ni2+:MgO. (a) Temperature dependence of the
emission spectrum. The inset shows the Cr3+ line around 698 nm. (b)
Fluorescence decay-time versus temperature. The inset presents a
characteristic fluorescence decay curve (semi-logarithmic scale).

These features suggest that, if one of the two excited electronic levels
leading to the observed emission contains a SO-coupled state, optical spin
manipulation with high fidelity could be performed using short pulses even at
high temperatures. We use magneto-optical spectroscopy measurements to verify
that such a state indeed exists.

###  IV.2 Magneto-optical spectroscopy

For performing polarized magneto-fluorescence spectroscopy, we placed the
sample in a closed-cycle helium cryostat (attocube attoDRY 2100), equipped
with a 9-T superconducting magnet. The sample temperature could be varied from
1.7 K up to room temperature, independently of the magnet temperature, which
was kept low and constant. A Ti:Sapphire laser (Sirah Matisse CS) was used for
optical excitation at 690 nm. Two sets of measurements were performed, one at
1.7 K base temperature, and the other one at 60 K. In each set, the magnetic
field was varied from 0 to 9 T and the emission spectrum was measured in two
orthogonal circular polarizations. The emission was dispersed using the
HRS-750 spectrometer, equipped with a 600 g/mm grating (resolution limit of 15
GHz around 1250 nm), and recorded using an electrically-cooled CCD array
camera (Teledyne-Princeton Instruments NIRvana HS). The first set focuses on
the 1250 nm line (emission from the $E_{u}$ level) and the second on the 1220
nm line (emission from the $T_{1u}$ level). The 1220 nm emission was measured
at an elevated temperature as it is extremely weak at lower temperatures, due
to thermalization to the lowest excited state [see also Fig. 5(a)]. Figure 6
presents the polarized spectra for the two transitions.

Figure 6:  Polarized magneto-fluorescence spectroscopy for determining spin-
orbit coupling in the excited levels of Ni2+:MgO. (a) [(b)] Measured spectra
around 1250 nm for $\sigma^{+}$ [$\sigma^{-}$] circular polarization versus
the applied magnetic field. (c) The line centers (dots) extracted from (a) and
(b), together with linear regression fits (lines). The error bars present the
errors in the line centers. The data points to the left of the vertical gray
line were excluded from the linear regression. (d)-(f) The same as (a)-(c),
for the 1220 nm emission. The inset in (c) presents the mean difference
between the frequencies of oppositely-polarized transitions of the same sign
of the Zeeman shift versus the magnetic field $B$. At high fields, a
significant difference from zero is seen. The solid line is a fit of
$\beta_{\mathrm{meas}}B^{2}$ excluding the points to the left of the gray
vertical line.

At a high enough magnetic field, the 1220 nm emission splits into four, fully
polarized lines, and the 1250 nm emission splits into two, partially polarized
lines, with a polarization degree of about 50%. These patterns exactly match
the prediction for the weak DJT case. In that case, the theory also predicts
the Zeeman shifts of all emission lines (Appendix B).

In order to compare our measurements to the predicted Zeeman shifts, we fitted
each of the measured fluorescence spectra to a double hyperbolic-secant
function and extracted the energies of the two peaks for every value of the
applied magnetic field. These energies are presented in Figs. 6(c) and 6(f).
We then fitted the magnetic field dependence of the peak energies to straight
lines with a common origin. The best-fitted lines are also presented in the
figures.

For the 1250 nm emission, the difference between $\sigma^{+}$ and $\sigma^{-}$
polarized lines of opposite-sign slopes is predicted to be equal to the
splitting of the ground state. That is, by dividing the slope difference by
$2\mu_{\mathrm{B}}$, one should obtain $g_{s}^{\mathrm{g}}$. In this way, we
obtain a value of $g_{s}^{\mathrm{g}}=2.242\pm 0.003$. This value indeed
closely matches the value directly measured using ESR. The theory predicts
that the two $E_{u}$ states should not split in a magnetic field, to first
order. However, due to the (very small) magnetic-field-induced mixing of the
$E_{u,\epsilon}$ state with the $T_{1u,0}$ state, there should be a negative
quadratic shift of its energy with the magnetic field [70]. Using second-order
perturbation theory, the coefficient of this quadratic shift is given by
$\beta=-(g_{s}^{\mathrm{e}}+\tilde{g}_{L})^{2}\mu_{\mathrm{B}}^{2}/\Delta_{21}$,
where $\tilde{g}_{L}=g_{L}e^{-\kappa/2}$, and $\Delta_{21}$ is the zero-field
splitting between the $E_{u}$ and $T_{1u}$ states. Using the measured values
of $\Delta_{21}=5.28\pm 0.02$ THz and
$g_{s}^{\mathrm{e}}+\tilde{g}_{L}=1.33\pm 0.02$ (obtained from the Zeeman
splitting of the $T_{1u}$ states, as shown below), we obtain
$\beta_{\mathrm{calc}}=-66\pm 2$ MHz/T2. While this effect is very small, we
could directly measure it by looking at the difference between the energies of
$\sigma^{+}$ and $\sigma^{-}$ polarized lines having same-sign Zeeman shifts.
At high enough magnetic fields we could measure a significant ($>3\sigma$)
deviation from zero, as presented in the inset of Fig 6(c). It is also clear
that the dependence of this deviation on the magnetic field is non-linear. The
solid line is a quadratic fit to the points for which $B\geq 3.5$ T. The
dashed lines present the 68% confidence level of this fit (mostly due to the
uncertainty in the measured energy differences). From the fit, we obtain a
value of $\beta_{\mathrm{meas}}=-90\pm 35$ MHz/T2, which agrees with the
calculated value to within the measurement error. A similar effect, though
much larger, was previously observed for V3+ ions in GaAs [70].

For the 1220 nm emission, the slope difference between the two outer lines is
again only due to the ground state and is again predicted to be
$2\mu_{\mathrm{B}}g_{s}^{\mathrm{g}}$. We indeed extract a value of
$g_{s}^{\mathrm{g}}=2.22\pm 0.025$, which is consistent with the value
extracted from the $E_{u}$ lines and with the ESR value. In addition to the
linear slope, there should also be a small quadratic shift, common to both
lines, due to the small magnetic coupling between the $T_{1u,0}$ and the
$E_{u,\epsilon}$ states. The magnitude of this effect should be the same as
for the $E_{u,\epsilon}$ state, but its sign should be opposite, that is, we
expect a positive quadratic shift. However, as the signal here was much weaker
than in the 1250 nm line, the accuracy of the assigned energies was lower, and
we could not resolve this effect.

The slope difference between the two inner lines is related only to the
splitting of the excited $T_{1u}$ states and is predicted to be
$\alpha=\mu_{\mathrm{B}}(\tilde{g}_{L}+g_{s}^{\mathrm{e}})$. Assuming that the
deviation of the electron g-factor from the vacuum electron g-factor,
$g_{0}=2.0023$, is due to SO mixing alone (that is, neglecting the crystal
field contribution), and taking into account only the closest $T_{1u}$ state
(which is that arising from the 3T1 manifold), one obtains
$g_{s,\mathrm{th}}^{\mathrm{e}}=1.84$ (Appendix E). Together with the
theoretical value of $\tilde{g}_{L,\mathrm{th}}=-0.47$ (see Appendix A and
Appendix F), we obtain $\alpha_{\mathrm{th}}=1.37\mu_{\mathrm{B}}$. This
predicted value is in a good agreement with the measured value of
$\alpha_{\mathrm{meas}}=(1.33\pm 0.02)\mu_{\mathrm{B}}$. An even better
agreement may be obtained if the effects of the crystal field on the g-factor
would be taken into account [86].

##  V Proposed protocols

Having established that the 3T2 excited states of Ni2+ in MgO involve
unquenched, THz level SO coupling, and can thus mediate fast spin-photon
coupling, in this section we present several protocols for optical control of
the ground-state spin.

###  V.1 Spin state preparation and measurement

Figure 7(a) presents the polarization selection rules between the ground-state
manifold and the two lowest excited-state manifolds, where the definitions of
the polarizations with respect to the magnetic field and crystal axes are
presented in Fig. 7(b) (see also Table B1 in Appendix B). These selection
rules allow for polarization-based spin state preparation. Figure 7(c)
presents the basic principle: excitation with a defined polarization leaves
one of the three ground state spin sublevels uncoupled to the excitation
field. If the lifetime of this state is longer than the decay time from the
excited state, most of the population will eventually accumulate in this
state.

Figure 7:  Optical spin preparation, measurement, and control schemes. (a) The
polarization selection rules for optical transitions between the ground state
manifold, $T_{2u}$, and the first two excited state manifolds, $E_{u}$ and
$T_{1u}$. Purple, orange, and grey arrows represent transitions in
$\sigma^{+}$, $\sigma^{-}$, and $\pi$ polarizations, respectively. The widths
of the arrows represent the strength of the transitions (see also Table B1 in
Appendix B). (b) The geometry of exciting optical beams with respect to the
crystal axes and the applied magnetic field. (c) Polarized optical pumping to
the $T_{2u,-1}$ state. Dashed purple, orange, and grey (black) lines indicate
spontaneous (non)radiative decay. Similar schemes with different pump
polarizations allow pumping to the other two ground states. (d) Pulsed state
preparation. Here, the pumping is done using $\pi$-pulses (of
$\sigma^{+}$-polarized light, for example) that transfer all undesired state
population to the upper excited state. From there, this population quickly
decays to the lower excited state. It can either be shelved there for the
lifetime of the lower excited state or could be forced down to a phonon-
sideband, which then quickly decays to the ground state. In the latter case,
to achieve full spin-state polarization, the process has to be repeated a few
times. (e) Fluorescence decay curves without (purple) and with (orange) the
introduction of laser light at the phonon side-band. A clear increase in the
decay rate is observed. (f) [(g)] Coherent optical arbitrary spin rotation on
the $\\{T_{2u,1},T_{2u,-1}\\}$ qubit space using a polarized pulse around the
$T_{1u}$ [$E_{u}$] transition. (h) Coherent optical arbitrary spin rotation on
the $\\{T_{2u,1},T_{2u,0}\\}$ qubit space.

While the nonradiative decay from the $T_{1u}$ to the $E_{u}$ excited states
is fast even at low temperature [as evident from the thermalization of the
excited state population, Fig. 5(a)], the decay from the $E_{u}$ excited state
to the $T_{2u}$ ground states takes a few milliseconds, even at high
temperatures [Fig. 5(b)]. As the ground-state spin lifetime has been measured
to be on the orders of milliseconds only at temperatures of a few Kelvin [36],
the standard optical pumping, Fig. 7(c), may not work at higher temperatures.

There are at least two solutions to this problem. One is to use the fact that
the $E_{u}$ state is long-lived and use it as a shelving state that stores the
unwanted spin population while coherent operations are performed on the
population that remains in the ground state, which is only of the desired spin
state. For this, the polarized optical field should transfer all the unwanted
population to the excited state before it decays. This can be achieved using
an ultrafast, optical $\pi$-pulse, resonant with the 1220 nm transition
($T_{2u}\rightarrow T_{1u}$). In order to transfer the entire population, the
excitation has to be coherent. This means that the pulse bandwidth has to be
much larger than the optical linewidth ($\sim 100$ GHz). However, in order to
not involve the $E_{u}$ levels, which will spoil the polarization selection
rules, the pulse spectrum has to be narrower than the energy difference
between the $E_{u}$ and $T_{1u}$ spectral lines (5.28 THz). A $\sim 1$ THz
wide pulse, that is, of a few hundred fs duration, would fit this range. The
downside of this solution is that one decreases the optical density (the
effective defect density) by a factor of three (as only a third of the defects
are left in the ground state).

The second solution may overcome this problem. As shown in Fig. 7(d), a second
pulse, at a frequency matching the transition from the shelving $E_{u}$ state
to a phonon sideband of the ground state, would stimulate the transition of
the shelved population to the phonon sideband. From there, the population
would quickly decay back to the ground state. Repeating the shelving and
stimulating pulses a few times would result in most of the population being
pumped into the ground state decoupled from the shelving pulse, in a similar
manner to standard optical pumping.

To test the feasibility of this ‘stimulated optical pumping’ concept, we
introduced about 0.5 W of laser light at 1319 nm (Sanctity Laser
SSL-1319-1500-10TM-D-LED) during the decay of the population after its
excitation by the 660-nm pulse. Figure 7(e) presents the fluorescence decay
with and without the addition of the 1319 nm laser. A clear decrease in the
fluorescence decay time is observed, indicating the stimulation of population
decay from the shelving state. Using a CW laser, however, is inefficient, as
only a small part of the phonon side-band is used, and the added power heats
up the sample. Using an ultra-short pulse for the stimulated de-excitation
should solve these issues.

For optically measuring the spin state, one can turn on again the polarized
pumping light at 1220 nm and monitor the resulting fluorescence at 1250 nm. As
for each of the three polarizations ($\sigma^{+}$, $\sigma^{-}$, and $\pi$),
one of the three spin states is uncoupled from the polarized pumping field,
the fluorescence intensity will be inversely proportional to the population of
that state. The combined information from all three measurements would thus
yield the populations of all three states.

###  V.2 Coherent spin control

The polarized $\Lambda$-systems present in the level structure allow for
polarization-based, full coherent control of the spin states using short
optical pulses. Figures 7(f) and 7(g) present the relevant transitions for
coherent control of the $\\{T_{2u,1},T_{2u,-1}\\}$ qubit manifold, through the
$T_{1u,0}$ state or the $E_{u}$ states, respectively. In the latter case, the
coupling paths through the two $E_{u}$ states destructively interfere only
partially, still enabling control. In both cases, the control pulse can be
off-resonance or near-resonance, whereas in the latter case the pulse spectrum
should be much wider than the linewidth of the relevant transition. The axis
of rotation in the Bloch sphere is determined by the pulse polarization. The
angle of rotation about this axis is determined either by the intensity of the
pulse (off-resonant pulse) [87] or by the detuning of the pulse (near-resonant
pulse) [88]; in the latter case, the intensity is set to create a full $2\pi$
rotation, ending back at the ground state [62]. Figure 7(h) shows the
transitions employed to control the $\\{T_{2u,1},T_{2u,0}\\}$ qubit manifold.
As they involve both $\sigma$ and $\pi$ polarizations, they have to be applied
from orthogonal directions [Fig. 7(b)]. A similar arrangement with the
opposite $\sigma$ polarization would drive the $\\{T_{2u,0},T_{2u,-1}\\}$
qubit manifold. Here too, both near- and off-resonance control pulses may be
applied.

###  V.3 Noise-free quantum memory

The ability to optically prepare and coherently control their spin state,
combined with the near-telecom optical transitions and the possibility of
creating high optical-density ensembles, naturally suggests the application of
divalent nickel ion ensembles in MgO as quantum-optical memories [89, 90]. The
quantum memory scheme most suitable to a medium with a large broadening of the
excited state is the far-detuned Raman scheme [91]. The optical cooperativity
$\mathcal{C}$ of the system, which governs the memory efficiency [65], can be
estimated using the transition dipole moment of the $T_{2u}\leftrightarrow
T_{1u}$ transition, $3.4\times 10^{-32}$ Cm, as derived from the oscillator
strength of $4\times 10^{-7}$ [33]. We assume a density of $2.4\times 10^{18}$
cm-3 (10 times lower than the density of the current sample), a detuning of
200 GHz, a waveguide of 5 mm in length, and a cross-section of 5$\times 5$
$\mu$m2. For control pulses of 1 $\mu$J, which are readily produced by
standard optical parametric amplifiers, we obtain $\mathcal{C}\approx 2$,
indicating a total storage and retrieval efficiency of
$\mathcal{C}^{2}/(1+\mathcal{C})^{2}\approx 45\%$ [91]. Thus, it seems that
efficient Raman storage should be possible using realistic parameters.
Furthermore, much higher cooperativity, $\mathcal{F}\times\mathcal{C}$, and
hence higher efficiency, can be obtained by adding an optical cavity with
moderate finesse $\mathcal{F}$ [92, 93, 94], e.g., by using a ring-resonator
structure or imprinting Bragg mirrors onto the waveguide.

One prevalent source of noise in a Raman memory scheme is due to four-wave
mixing [95]. This is the case when the control field couples to the full
ground-level manifold, and not only to the empty ground state, a situation
aggravated at detunings larger than the ground-state splitting. However, in a
Raman memory based on a polarized $\Lambda$ system, a polarized control field
couples only to a single ground state, and thus four-wave-mixing noise is
suppressed [65, 64].

A second source of noise is the leakage of control light into the signal
channel. When the signal and control are oppositely polarized, they can be
separated by their polarizations. Usually, however, this is not enough, and a
second separation mechanism, such as spectral filtering is invoked [96, 95,
94, 97, 98]. This would be possible here only if the spectrum of the control
pulse is narrower than the ground-state splitting, but that would limit the
bandwidth of the memory. Furthermore, tight spectral filtering usually lowers
the efficiency of the memory. Here we propose to replace spectral filtering
with spatial filtering, by introducing an angle between the signal and control
modes. This is possible as, in contrast to warm atomic vapors, here the
emitters do not move during storage and cannot create any motional dephasing
due to signal and control wave-vector mismatch [99, 100]. Figure. 8 presents
two polarized Raman-memory schemes with an additional spatial-mode mismatch,
applicable in the Ni2+:MgO system.

Figure 8:  Noise-free memory schemes. (a)-(c) A single-mode, optionally
rephased memory scheme. (a) Spatial configuration (top) and timing (bottom) of
the optical pulses. (b) Storage of a $\sigma^{+}$ signal by creating a
$T_{2u,1}-T_{2u,-1}$ coherence in a medium initially prepared in the
$T_{2u,1}$ ground state, by a $\sigma^{-}$ control field. A second application
of the control before the inhomogeneous dephasing time of the
$T_{2u,1}-T_{2u,-1}$ coherence would read the signal out. (c) A pair of
rephasing pulses of vertical polarization ($V$) can be applied between storage
and retrieval to cancel out the inhomogeneous dephasing, thus prolonging the
memory time. (d)-(f) A polarization-conserving, optionally rephased memory
scheme. (d) Spatial configuration (top) and timing (bottom) of the optical
pulses. (b) Storage of an arbitrarily-polarized signal on the pair of
coherences, $T_{2u,0}-T_{2u,1}$ and $T_{2u,0}-T_{2u,-1}$, using a
$\pi$-polarized pulse on a medium initially prepared in the $T_{2u,0}$ ground
state. A second application of the control before these coherences decay would
read the signal out. (f) Here too, a pair of rephasing pulses of $V$
polarization can be applied.

The first scheme, described in the top three panels of Fig. 8, uses the
$\sigma^{+}$-polarized transition $T_{2u,1}\leftrightarrow T_{1u,0}$ for the
signal, and the $\sigma^{-}$-polarized transition $T_{2u,-1}\leftrightarrow
T_{1u,0}$ for the control, where the ensemble is first prepared in the
$T_{2u,1}$ state. As shown in Fig. 8(a) (top), while the control field
propagates along the magnetic field (setting the quantization axis), the
signal field is at a small angle with respect to it. As shown in Fig. 8(b),
this scheme is based on the $\\{T_{2u,1},T_{2u,-1}\\}$ qubit system. As shown
in Fig. 2, while the relevant spin transition is narrower than that of the
$\\{T_{2u,1},T_{2u,0}\\}$ qubit system, it is still considerably broadened.
However, most of this broadening is due to inhomogeneous strain distribution,
which could be mitigated by using spin-echo sequence (and, generally, would be
narrower in annealed samples [38]). Figure 8(c) presents a possible way to
introduce the echo pulses optically. This could be done using a vertically-
polarized pulse near the $E_{u}$ resonance, flipping between the $T_{2u,1}$
and $T_{2u,-1}$ states. As shown in Fig. 8(a) (top), this pulse could
potentially be applied from the side of the sample, perpendicular to the
control and the signal, to minimize scattering into the signal mode, but it
could also be applied along their direction, as it could rather easily be
spectrally filtered-out.

Figure 8(a) (bottom) presents a possible pulse scheme including two rephasing
pulses between storage and retrieval, where the time between the memory
control pulses and the rephasing pulses is $\tau$, and the time between the
two rephasing pulses is $2\tau$. This scheme would return the qubit state to
its initial state in time for retrieval. This is similar to the revival of
silenced echo (ROSE) quantum memory scheme [101], as there is no microwave
cavity, and the spin echo between the two rephasing pulses is very weak.

Figure 8(d) presents a second noise-free memory scheme. This scheme, in
contrast to the first one, stores both polarization modes of the signal. As
shown in Fig. 8(e), Starting from the system initialized in the $T_{2u,0}$
state, a $\pi$-polarized control pulse stores an arbitrarily
$\sigma^{\pm}$-polarized signal in the $T_{2u,0}-T_{2u,1}$ and the
$T_{2u,0}-T_{2u,-1}$ coherences. The $\pi$-polarized control pulse is
introduced from the side so, despite being in the same frequency as the
signal, can be separated by both polarization and spatial filtering. As the
storage coherences include the broad $T_{2u,0}$ state, here rephasing is
critical. Fortunately, the same two-pulse rephasing scheme described above can
be used here as well. As the $T_{2u,1}$ and $T_{2u,-1}$ states are switched,
the phase dispersion of both $T_{2u,0}-T_{2u,1}$ and $T_{2u,0}-T_{2u,-1}$
coherences is reversed [Fig. 8(f)]. One downside of this scheme is that, as
the control comes from the side, the area it has to illuminate is much larger,
necessitating much larger pulse energy.

For the two memory schemes proposed above, using two rephasing pulses limits
the applicability of such memories to cases where the required storage time is
known in advance. This is the case for the synchronization of random events to
fixed, pre-determined time bins. Nevertheless, the memory schemes could be
adapted to cases where the release time-bin is not pre-determined. This can be
done by setting the total rephasing time, $4\tau$, to the minimum cycle time
of the synchronized system, repeating the two-pulse rephasing sequence until
retrieval is required, and then introducing the retrieving control at the end
of the last rephasing cycle. This would amount to applying a Carr-Purcell-
Meiboom-Gill (CPMG) sequence of variable length between storage and retrieval,
and would thus have the added value of protecting the stored coherence from
dynamical external noise (up to a frequency of $1/4\tau$). Moreover, the
applied pulse sequence must not be limited to CPMG. It could be any other
periodic pulse sequence that can be stopped after an arbitrary number of
periods. For example, one could apply a combination of the CPMG sequence and
the Waugh-Huber-Haeberlen (WAHUHA) sequence [102], which protects both against
external dynamic noise and against noise created by dipolar interactions
within the ensemble, thus prolonging the memory time beyond the limit set by
dipolar dephasing. This would alleviate the limit on the density of the
ensemble, potentially increasing the memory efficiency.

##  VI Conclusion

We have introduced general guidelines for identifying spin defects in solid-
state systems that could coherently couple to light in the presence of
significant optical decoherence, may it be caused by inhomogeneous broadening
in an ensemble, high temperature, or both. The two main requirements are an
excited-state SO interaction faster than the optical decoherence and an
orbital-singlet ground state. Following these guidelines, we propose to study
transition-metal ion ensembles in various crystals, and, as an example, we
present the case of Ni2+ in MgO.

First, we perform ESR measurements of Ni2+:MgO and extract both the
inhomogeneous and homogeneous coherence times. In our sample, these are on the
order of a few ns and a few $\mu$s, respectively, at liquid-helium
temperature. Using temperature-dependant spin-echo measurement, we find that
the main homogeneous broadening mechanism at this temperature is spectral
diffusion due to dipolar interactions within the ensemble and that it is fully
saturated. As the dephasing rate due to this mechanism depends on the density
of the ensemble, the dephasing time could be prolonged by working with lower
densities. Our particular sample is dense, containing $\sim 450$ ppm of Ni,
and indeed lower densities would still be high enough to allow for efficient
interaction with light. Additionally, one could use designated dynamical-
decoupling sequences to protect against dipolar dephasing.

We could also detect ESEEM, which we attributed to 25Mg nuclei in the host MgO
crystal. While we have not measured the nuclear spin coherence time, we can
tell that it is longer than the electronic coherence time. This opens up the
possibility of using the nuclear spin ensemble as a long-term quantum memory
[103, 104, 105].

Second, we perform magneto-optical spectroscopy measurements and verify that
the excited states are indeed SO-coupled states, despite a weak DJT distortion
of the electronic orbitals. This, combined with the observations of
spectrally-separated emission lines and an almost fixed fluorescence decay
time up to higher than liquid nitrogen temperatures, suggests that this system
may be used as a coherent spin-photon interface at relatively high
temperatures.

Third, we propose detailed protocols for optical spin-state initialization and
measurement and optical coherent spin control. Specifically, even when the
spin lifetime is shorter than the excited-state lifetime, spin initialization
is still possible using either shelving or pulsed stimulated decay. Based on
these basic protocols, we then propose two noise-free, high-bandwidth quantum
memory protocols, possibly combining dynamical decoupling. While the first
protocol is limited to single-mode storage, the second is for a two-mode,
polarization-preserving memory. Much more work is required in order to
implement these protocols and to fully explore the potential of Ni2+ in MgO as
a solid-state light-spin interface.

Moving forward, other transition-metal dopant systems could be explored. Some
interesting candidates are Co2+ [106] and Ni3+ [107] (d7 ions), or Fe6+ [108],
Nb3+ [109] and Ta3+ [110] (d2 ions), in tetrahedral sites. All of these are
known to have optical transitions between 1550 and 1700 nm, with weak or non-
existing DJT quenching. In addition, many other transition-metal-ion doped
crystals that match our general guidelines have not been studied at all.

We thus believe that this work can open the way to further investigations of
transition-metal ions in crystals as a new family of materials with the
potential of serving as the long-sought-after high-temperature, coherent
solid-state spin-photon interfaces, with major applications in quantum
networks.

###### Acknowledgements.

EP would like to thank H. Bernien and D. D. Awschalom for facilitating his
visit to the University of Chicago and for fruitful discussions. EP was
supported in part by a grant from Fran Morris Rosman and Richard Rosman. EP,
LS, and OF acknowledge support from the US-Israel Binational Science
Foundation (BSF) and US National Science Foundation (NSF), and the Estate of
Louise Yasgour. SG and TZ acknowledge support by the National Science
Foundation (NSF) Faculty Early Career Development Program (CAREER) Grant (No.
1944715), partial support by the University of Chicago Materials Research
Science and Engineering Center, which is funded by the NSF under award number
DMR- 2011854, and the Army Research Office grant no. W911NF2010296. IM was
supported by an Alfred J. and Ruth Zeits Research Fellowship. JNB acknowledges
support by the Cowen Family Endowment at MSU.

##  Appendix A DJT model and the absorption spectra

The model in Eqs. (2)-(4) has eight independent parameters: the electron-
phonon coupling energy $E_{\mathrm{JT}}$, the energy of the lowest phonon mode
$\hbar\omega_{\mathrm{ph}}$, the pure-electronic spin-orbit coupling
parameters $\zeta$, $\mu$, and $\rho$, the spin gyromagnetic ratios
$g_{s}^{\mathrm{g}}$ and $g_{s}^{\mathrm{e}}$, and the orbital gyromagnetic
ratio $g_{L}$. Below we show how the values of these parameters can be found
from existing spectral measurements under two different interpretations, the
weak and strong DJT distortion.

First, we find the relevant model parameters for the case of weak DJT
distortion. Using the energy level assignments from the literature [33, 40,
53], combined with our value for the ground-state spin g-factor
$g_{s}^{\mathrm{g}}$, one could find values for the first six parameters of
the model [61], as detailed in Table A1. The value of $g_{s}^{\mathrm{e}}$ can
be found from the value of $\zeta$ as shown in Appendix E, and
$g_{L}=-\nicefrac{{1}}{{2}}$ can be calculated from the orbital structure, as
shown in Appendix F.

Table A1: List of SO and DJT parameter values and their derivations from independently measured values assuming a _weak_ DJT distortion. Parameter | Value | Relation to measured values | Measured values | Refs.
---|---|---|---|---
$g_{s}^{\mathrm{g}}$ | 2.242 | EPR and magneto-spectroscopy | 2.242 | This work
$\hbar\omega_{\mathrm{ph}}$ | 6.15 THz | Optical spectroscopy | 6.15 THz | [40]
$\zeta$ | -7.73 THz | $\zeta=(g_{0}-g_{s}^{\mathrm{g}})E_{ge}/4g_{0}$ |  $E_{ge}=258$ THz111‘Center of mass’ of the optical absorption line | [31, 33]
$E_{\mathrm{JT}}$ | 0.26 THz | $E_{\mathrm{JT}}=-(2/3)\hbar\omega_{\mathrm{ph}}\ln{(-\Delta_{32}/\zeta)}$ |  $\Delta_{32}=7.26$ THz222Splitting between the $T_{2u}$ and $T_{1u}$ lines | [53]
$\mu$ | 4.92 THz | $\mu=(\Delta_{43}+\Delta_{21}+e^{-\kappa/2}\zeta/2-3K_{1})/3e^{-\kappa/2}$ |  $\Delta_{43}=12.6$ THz333Splitting between the $A_{2u}$ and $T_{2u}$ lines | [53]
$\rho$ | -5.74 THz | $\rho=-\Delta_{21}-(1-e^{-\kappa/2})\mu-K_{2}$ |  $\Delta_{21}=5.28$ THz444Splitting between the $T_{1u}$ the $E_{u}$ lines | [33]

The obtained values of $E_{\mathrm{JT}}$, $\hbar\omega_{\mathrm{ph}}$,
$\zeta$, and $g_{L}$ then yield $\kappa=0.13$, $K_{1}=0.14$ THz, and
$K_{2}=0.15$ THz, consistent with a weak DJT distortion.

For the second case, for which $\kappa\gg 1$, the only relevant spectroscopic
data is the splitting between the two narrow, low-energy lines, $\Delta_{21}$.
In this limit, $K_{1},K_{2}\rightarrow 0$, and thus $\Delta_{32}\approx-2\zeta
e^{-\kappa/2}$. That is, it is becomes very small and may be below the
resolution limit. Similarly, $\Delta_{43}\approx-\Delta_{21}-\zeta
e^{-\kappa/2}$ and becomes indiscernible from $-\Delta_{21}$. With these
approximated values one can see that $\mu$ has to be 0, and $\rho$ has to be
$-\Delta_{21}$. All the other parameters are independent of the optical
spectrum and can thus take the same values as in Table A1.

We thus see that the same general model, though with different $\mu$ and
$\rho$, can fit the available spectroscopic data also if we assume that the
DJT distortion is strong.

It is therefore impossible to discern between the case of weak DJT distortion
and that of strong DJT distortion given only this data, and additional
measurements are required.

##  Appendix B Polarization selection rules

The three ground states can be described as spin-orbit product states, between
the $|$A${}_{2g}\rangle$ orbital and the three spin states $|1\rangle_{s}$,
$|0\rangle_{s}$, and $|-1\rangle_{s}$. The excited states can be described as
superpositions of spin-orbit product states between the three T2g orbitals
$|$T${}_{2g,1}\rangle$, $|$T${}_{2g,0}\rangle$, and $|$T${}_{2g,-1}\rangle$,
and the three spin states.

The optical transition probabilities between any excited state $|j\rangle_{e}$
and ground state $|i\rangle_{g}$ are given by

$P_{ij}^{k}=|_{e}\langle j|D_{M}^{k}|i\rangle_{g}|^{2},$ (B1)

where $D_{M}^{k}$ is the magnetic-dipole transition matrix in polarization
$k$. In the spin-orbit product basis used here, these matrices are given by

$D_{M}^{k}\propto|k\rangle_{l}\langle\mathrm{A}_{2g}|\otimes I_{s}.$ (B2)

The state $|k\rangle_{l}$ is the T2g orbital state corresponding to the
polarization $k$, according to [111]

$\\{\sigma^{+},\pi,\sigma^{-}\\}\Leftrightarrow\\{|\mathrm{T}_{2g,-1}\rangle,|\mathrm{T}_{2g,0}\rangle,|\mathrm{T}_{2g,1}\rangle\\}.$
(B3)

The matrix $I_{s}$ is a unit matrix in the spin-space, and “$\otimes$” is the
Kronecker product. The proportionality constant is not important for the
calculation of relative rates.

The ground and excited-state wavefunctions, the relative optical transition
rates between them (in all polarizations), and the Zeeman shift coefficient
for these transitions are given in Table B1 and Table B2 for weak and strong
DJT distortion, respectively. The tables neglect second-order magnetic
coupling effects.

Table B1: Transition Zeeman shift in units of $\mu_{\mathrm{B}}B$ (left) and relative transition probabilities in the three polarizations, [$\sigma^{+}$,$\pi$,$\sigma^{-}$] (right), for the case of a _weak_ DJT distortion. The orbital $|$T${}_{2g,k}\rangle$ is abbreviated by $|k\rangle_{l}$. |  |  $|$A${}_{2g}\rangle|\mathrm{-}1\rangle_{s}$ |  $|$A${}_{2g}\rangle|0\rangle_{s}$ |  $|$A${}_{2g}\rangle|1\rangle_{s}$
---|---|---|---|---
$E_{u,\epsilon}$ | $\tfrac{1}{\sqrt{2}}(|1\rangle_{l}|1\rangle_{s}+|\mathrm{-}1\rangle_{l}|\mathrm{-}1\rangle_{s})$ |  $g_{s}^{\mathrm{g}}$ [$\tfrac{1}{2}$ 0 0] | 0 [0 0 0] |  $-g_{s}^{\mathrm{g}}$ [0 0 $\tfrac{1}{2}$]
$E_{u,\theta}$ | $\tfrac{1}{\sqrt{6}}(|1\rangle_{l}|\mathrm{-}1\rangle_{s}+|\mathrm{-}1\rangle_{l}|1\rangle_{s})+\tfrac{\sqrt{2}}{\sqrt{3}}|0\rangle_{l}|0\rangle_{s}$ |  $g_{s}^{\mathrm{g}}$ [0 0 $\tfrac{1}{6}$] | 0 [0 $\tfrac{2}{3}$ 0] |  $-g_{s}^{\mathrm{g}}$ [$\tfrac{1}{6}$ 0 0]
$T_{1u,-1}$ | $-\tfrac{1}{\sqrt{2}}(|1\rangle_{l}|0\rangle_{s}+|0\rangle_{l}|1\rangle_{s})$ |  $\tfrac{1}{2}(2g_{s}^{\mathrm{g}}+g_{s}^{\mathrm{e}}+\tilde{g}_{L})$ [0 0 0] |  $\tfrac{1}{2}(g_{s}^{\mathrm{e}}+\tilde{g}_{L})$ [0 0 $\tfrac{1}{2}$] |  $-\tfrac{1}{2}(2g_{s}^{\mathrm{g}}-g_{s}^{\mathrm{e}}-\tilde{g}_{L}$) [0 $\tfrac{1}{2}$ 0]
$T_{1u,0}$ | $\tfrac{1}{\sqrt{2}}(|1\rangle_{l}|1\rangle_{s}-|\mathrm{-}1\rangle_{l}|\mathrm{-}1\rangle_{s})$ |  $g_{s}^{\mathrm{g}}$ [$\tfrac{1}{2}$ 0 0] | 0 [0 0 0] |  $-g_{s}^{\mathrm{g}}$ [0 0 $\tfrac{1}{2}$]
$T_{1u,1}$ | $\tfrac{1}{\sqrt{2}}(|\mathrm{-}1\rangle_{l}|0\rangle_{s}+|0\rangle_{l}|\mathrm{-}1\rangle_{s})$ |  $-\tfrac{1}{2}(2g_{s}^{\mathrm{g}}+g_{s}^{\mathrm{e}}+\tilde{g}_{L})$ [0 $\tfrac{1}{2}$ 0] |  $-\tfrac{1}{2}(g_{s}^{\mathrm{e}}+\tilde{g}_{L})$ [$\tfrac{1}{2}$ 0 0] |  $\tfrac{1}{2}(2g_{s}^{\mathrm{g}}-g_{s}^{\mathrm{e}}-\tilde{g}_{L}$) [0 0 0]
Table B2: Transition Zeeman shift in units of $\mu_{\mathrm{B}}B$ (right) and relative transition probabilities in the three polarizations, [$\sigma^{+}$,$\pi$,$\sigma^{-}$] (left), for the case of _strong_ DJT distortion. The orbital $|$T${}_{2g,k}\rangle$ is abbreviated by $|k\rangle_{l}$. |  |  $|$A${}_{2g}\rangle|\mathrm{-}1\rangle_{s}$ |  $|$A${}_{2g}\rangle|0\rangle_{s}$ |  $|$A${}_{2g}\rangle|1\rangle_{s}$
---|---|---|---|---
$E_{u,\epsilon}$ | $\tfrac{1}{\sqrt{2}}(|1\rangle_{l}|1\rangle_{s}+|\mathrm{-}1\rangle_{l}|\mathrm{-}1\rangle_{s})$ |  $g_{s}^{\mathrm{g}}$ [$\tfrac{1}{2}$ 0 0] | 0 [0 0 0] |  $-g_{s}^{\mathrm{g}}$ [0 0 $\tfrac{1}{2}$]
$E_{u,\theta}$ | $\tfrac{1}{\sqrt{6}}(|1\rangle_{l}|\mathrm{-}1\rangle_{s}+|\mathrm{-}1\rangle_{l}|1\rangle_{s})+\tfrac{\sqrt{2}}{\sqrt{3}}|0\rangle_{l}|0\rangle_{s}$ |  $g_{s}^{\mathrm{g}}$ [0 0 $\tfrac{1}{6}$] | 0 [0 $\tfrac{2}{3}$ 0] |  $-g_{s}^{\mathrm{g}}$ [$\tfrac{1}{6}$ 0 0]
$A_{2u}$ | $\tfrac{1}{\sqrt{3}}(|1\rangle_{l}|\mathrm{-}1\rangle_{s}+|\mathrm{-}1\rangle_{l}|1\rangle_{s})-|0\rangle_{l}|0\rangle_{s})$ |  $g_{s}^{\mathrm{g}}$ [0 0 $\tfrac{1}{3}$] | 0 [0 $\tfrac{1}{3}$ 0] |  $-g_{s}^{\mathrm{g}}$ [$\tfrac{1}{3}$ 0 0]
$\tfrac{1}{\sqrt{2}}(T_{2u,1}-T_{1u,-1})$ | $|0\rangle_{l}|1\rangle_{s}$ |  $g_{s}^{\mathrm{g}}-g_{s}^{\mathrm{e}}$ [0 1 0] |  $-g_{s}^{\mathrm{e}}$ [0 0 0] |  $-g_{s}^{\mathrm{g}}-g_{s}^{\mathrm{e}}$ [0 0 0]
$T_{1u,0}$ | $\tfrac{1}{\sqrt{2}}(|1\rangle_{l}|1\rangle_{s}-|\mathrm{-}1\rangle_{l}|\mathrm{-}1\rangle_{s})$ |  $g_{s}^{\mathrm{g}}$ [$\tfrac{1}{2}$ 0 0] | 0 [0 0 0] |  $-g_{s}^{\mathrm{g}}$ [0 0 $\tfrac{1}{2}$]
$\tfrac{1}{\sqrt{2}}(T_{1u,1}-T_{2u,-1})$ | $|0\rangle_{l}|\mathrm{-}1\rangle_{s}$ |  $g_{s}^{\mathrm{g}}+g_{s}^{\mathrm{e}}$ [0 0 0] |  $g_{s}^{\mathrm{e}}$ [0 0 0] |  $-g_{s}^{\mathrm{g}}+g_{s}^{\mathrm{e}}$ [0 1 0]
$-\tfrac{1}{\sqrt{2}}(T_{1u,-1}+T_{2u,1})$ | $|1\rangle_{l}|0\rangle_{s}$ |  $g_{s}^{\mathrm{g}}$ [0 0 0] | 0 [1 0 0] |  $-g_{s}^{\mathrm{g}}$ [0 0 0]
$T_{2u,0}$ | $\tfrac{1}{\sqrt{2}}(|1\rangle_{l}|\mathrm{-}1\rangle_{s}-|\mathrm{-}1\rangle_{l}|1\rangle_{s})$ |  $g_{s}^{\mathrm{g}}$ [$\tfrac{1}{2}$ 0 0] | 0 [0 0 0] |  $-g_{s}^{\mathrm{g}}$ [0 0 $\tfrac{1}{2}$]
$\tfrac{1}{\sqrt{2}}(T_{1u,1}+T_{2u,-1})$ | $|\mathrm{-}1\rangle_{l}|0\rangle_{s}$ |  $g_{s}^{\mathrm{g}}$ [0 0 0] | 0 [0 0 1] |  $-g_{s}^{\mathrm{g}}$ [0 0 0]

##  Appendix C ESEEM for $S=1$ and $I=\nicefrac{{5}}{{2}}$

The Hamiltonian for electron-spin–nuclear-spin interaction involving two
electron spin levels split in a magnetic field along the $z$-direction can be
written as [80]

$H=$ | $\hbar\omega_{S}S_{z}+\left(-\hbar\omega_{I}+\frac{m_{s}^{\alpha}+m_{s}^{\beta}}{2}A_{zz}\right)I_{z}$
---|---
| $+\frac{m_{s}^{\alpha}+m_{s}^{\beta}}{2}A_{zx}I_{x}$
| $+(m_{s}^{\alpha}-m_{s}^{\beta})(A_{zz}S_{z}I_{z}+A_{zx}S_{z}I_{x}),$
(C1)

where $m_{s}^{\alpha(\beta)}$ is the magnetic quantum number of lower (higher)
electron spin level, $\omega_{S(I)}$ is the Zeeman splitting between adjacent
electron (nuclear) spin levels, $S_{x},\ S_{z}$ are effective
spin-$\nicefrac{{1}}{{2}}$ operators for the electron spin, $I_{x},\ I_{z}$
are nuclear spin operators (here taken as $I=\nicefrac{{5}}{{2}}$ operators),
and $A_{zz}$ and $A_{zx}$ are electron-nuclear spin coupling constants.

For the case of electron spin levels of magnetic quantum numbers
$m_{s}^{\alpha}=-1$ and $m_{s}^{\beta}=0$ (which is the relevant case for low
temperatures, where the population of the $m_{s}=1$ level can be neglected),
Eq. (C1) is reduced to

$H=\hbar\omega_{S}S_{z}-\hbar\omega_{I}I_{z}+A_{zz}\left(S_{z}-\tfrac{1}{2}\right)I_{z}+A_{zx}\left(S_{z}-\tfrac{1}{2}\right)I_{x}.$
(C2)

It can be seen that, as all the electronic spin operators are diagonal, this
Hamiltonian separates into two blocks, one for each electronic spin state.
These blocks operate only in the nuclear spin state space. For the lower
electronic state ($m_{s}=-1$), we obtain

$H_{1}=-\hbar\omega_{s}/2-(\hbar\omega_{I}+A_{zz})I_{z}-A_{zx}I_{x},$ (C3)

while for the higher electronic state ($m_{s}=0$) we obtain

$H_{0}=\hbar\omega_{s}/2-\hbar\omega_{I}I_{z}.$ (C4)

It is clear that, while in the lower electronic spin state manifold there is
mixing between the nuclear spin state, no such mixing is present in the higher
electronic spin state manifold, and the separation between the states in that
manifold is determined only by the nuclear Zeeman frequency $\omega_{I}$, even
if $A_{zz}$ and $A_{zx}$ are non-zero. This is due to the fact that here
$|m_{s}^{\alpha}+m_{s}^{\beta}|=|m_{s}^{\alpha}-m_{s}^{\beta}|$.

At the temperature of the experiment, while most of the population is in the
ground electronic spin state, the nuclear spin state is still fully mixed.
Therefore, transitions starting in different ground states would not
interfere. Hence, the frequencies of interference fringes are determined only
by the frequency differences in the excited state, and will therefore be
harmonics of the nuclear Zeeman splitting. This is in striking contrast to the
more commonly-studied case of $S=\nicefrac{{1}}{{2}}$ electronic spin states
[79, 80].

For a localized electronic spin with an isotropic g-factor
$g_{s}^{\mathrm{g}}$ interacting with neighboring nuclear spins, the contact
interaction is negligible, and the components of the electron-spin–nuclear-
spin interaction tensor are given by the dipole-dipole interaction [80]

$A_{ij}=\frac{3\mu_{0}}{4\pi|r|^{5}}\mu_{\mathrm{B}}\mu_{N}g_{s}^{\mathrm{g}}g_{n}(|r|^{2}\delta_{ij}-3r_{i}r_{j}),$
(C5)

where $\mu_{0}$ is the permiability of vacuum, $\mu_{N}$ and $g_{n}$ are the
nuclear magneton and the nuclear g-factor, respectively, and $\vec{r}$ is the
position vector of the nuclear spin with respect to the electron spin. For
MgO, the nearest Mg neighbors to the Ni substitutional site are located in all
12 permutations and sign combinations of
$\vec{r}=a(\nicefrac{{1}}{{2}},\nicefrac{{1}}{{2}},0)$, where $a=0.42$ nm is
the lattice constant of MgO. The length of all these vectors is
$|r|=a/\sqrt{2}$. Out of the 12 possibilities, four do not contain the
$z$-component, and for them $A_{zz}=A_{zx}=0$. For the other eight,

$A_{zz}=$ | $-\frac{3\sqrt{2}\mu_{0}}{2\pi a^{3}}\mu_{\mathrm{B}}\mu_{N}g_{s}^{\mathrm{g}}g_{n},$
---|---
$A_{zx}=$ | $\frac{3}{2}A_{zz}.$
(C6)

Note that, due to symmetry, replacing $x$ with $y$ does not change the
coupling coefficient, and thus all 8 sites have the same coupling strength.
Using $g_{n}=-0.34$ for 25Mg, we obtain $A_{zz}/h=165$ kHz and $A_{zx}/h=248$
kHz, where $h$ is Planck’s constant. These values are comparable in absolute
value to the nuclear Zeeman splitting at $B=141$ mT, $\omega_{I}/2\pi=-366$
kHz (the minus sign is due to the negative nuclear g-factor), leading to a
significant modulation depth. For the simple case of $I=\nicefrac{{1}}{{2}}$,
the visibility (which is only due to the nuclear spin state mixing in the
$m_{s}=-1$ state), is given by [80]

$V=P_{25}\frac{|A_{xz}|/2}{\sqrt{(\hbar\omega_{I}+A_{zz})^{2}+A_{zx}^{2}/4}},$
(C7)

where $P_{25}=1-(1-p_{25})^{N_{nn}}$ is the probability of a nearest-neighbour
Mg atom to be 25Mg. Here $p_{25}=0.1$ is the natural aboundance of 25Mg and
$N_{nn}=8$ is the number of relevant nearest neighbours. This expression
yields $V=0.3$. This value is of the same order of magnitude as the measured
visibility of $\sim 0.5$.

As the nuclear spin of 25Mg is $\nicefrac{{5}}{{2}}$ and not
$\nicefrac{{1}}{{2}}$, additional modulation frequencies, harmonics of
$\omega_{I}$ up to the fifth harmonic, are possible. Indeed, this is seen in
the measurement [Fig. 3(b)]. In order to calculate the ratios between the
magnitude of the oscillations in the different frequencies, we numerically
diagonalize the full $I=\nicefrac{{5}}{{2}}$ Hamiltonian, Eq. (C3), using the
same parameter values of Eq. (C7), and calculate the relative transition
amplitudes using $S_{x}$ as the transition operator [79, 80]. The result,
including bandwidth limitations due to the pulse duration and cavity width,
are presented by the yellow bars in Fig. 3(b).

##  Appendix D Spectral diffusion dephasing for $S=1$

Spectral diffusion of the probed spins can be caused by flip-flop processes
within the surrounding spin bath, which stochastically change the magnetic
field environment of the probed spins. The resulting dephasing rate is
proportional to the number of spin pairs that can flip-flop. As the flip-flop
process conserves energy, one should count only the spin pairs having the same
energy splitting. This number depends on the spin populations and hence
depends on temperature, making the spectral diffusion dephasing rate
temperature-dependent.

In order to find the temperature dependence, we first calculate the number of
spin pairs at a given temperature, for a certain strain detuning of the
$m_{s}=0$ level, $\delta$. We note that in a S=1 system there are three
possible flip-flop processes:

$|1,0\rangle$ | $\leftrightarrow$ | $|0,1\rangle,$
---|---|---
$|-1,0\rangle$ | $\leftrightarrow$ | $|0,-1\rangle,$
$|1,-1\rangle$ | $\leftrightarrow$ | $|0,0\rangle,$
(D1)

where $|m_{s,1},m_{s,2}\rangle$ represents a state of a spin pair in levels
$m_{s,1}$ and $m_{s,2}$. The numbers of spin-pairs that conserve energy for
these processes are, respectively,

$n_{1}(\delta)n_{0}(\delta),$
---
$n_{-1}(\delta)n_{0}(\delta),$
$\tfrac{1}{2}[n_{-1}(\delta)n_{1}(-\delta)+n_{0}(\delta)n_{0}(-\delta)],$
(D2)

where $n_{m_{s}}(\delta)$ is the number of spins (per unit detuning)
populating the level $m_{s}$ for a system with $m_{s}=0$ detuning of $\delta$.
At a given temperature, the populations of the three levels are

$n_{-1}(\delta)$ | $\propto$ | $1/Z(\delta),$
---|---|---
$n_{0}(\delta)$ | $\propto$ | $e^{-\beta(\varepsilon_{B}-\delta)}/Z(\delta),$
$n_{1}(\delta)$ | $\propto$ | $e^{-2\beta\varepsilon_{B}}/Z(\delta),$
(D3)

where here $\beta=1/k_{B}T$ is the inverse temperature in units of energy
($k_{B}$ is the Bolzmann constant) and
$\varepsilon_{B}=g_{s}^{\mathrm{g}}\mu_{\mathrm{B}}B$ is the Zeeman energy
splitting.
$Z(\delta)=1+e^{-\beta(\varepsilon_{B}-\delta)}+e^{-2\beta\varepsilon_{B}}$ is
the partition function. Note that here we neglected any strain shift of the
$m_{s}=1$ level.

The total spectral diffusion dephasing rate is then given by summing the
number of pairs for the three processes for a given $\delta$, multiplying by
the detuning distribution function
$P(\delta)=\tfrac{1}{\sqrt{2\pi}\sigma}e^{-\delta^{2}/2\sigma^{2}}$ ($\sigma$
being the width of the distribution), and integrating over $\delta$. Adding
also a temperature independent dephasing rate, representing the direct flilp-
flop and instanteneous diffusion processes, which are effectively constant at
the (low) temperatures where they are significant, this yields

$\frac{1}{T_{2}}=\frac{1}{T_{2,LT}}+\frac{3}{T_{2,SD,\mathrm{sat}}}\int_{-\infty}^{\infty}d\delta
P(\delta)\left(\frac{e^{-\beta(\varepsilon_{B}-\delta)}+e^{-\beta(3\varepsilon_{b}-\delta)}}{Z^{2}(\delta)}+\frac{e^{-2\beta\varepsilon_{B}}}{Z(\delta)Z(-\delta)}\right),$
(D4)

where $T_{2,LT}$ is the low-temperature dephasing time and
$T_{2,SD,\mathrm{sat}}$ is the spectral diffusion dephasing time at saturation
(note that for low temperatures the integral tends to 0, while for high
temperatures it tends to $\nicefrac{{1}}{{3}}$), and we used the fact that
$P(\delta)$ is even.

For the fit used in Fig. 4, we used the measured values of $\sigma=170$ MHz
and $\varepsilon_{B}=4386$ MHz ($B=141$ mT), and fitted the values of
$T_{2,LT}$ and $T_{2,SD,\mathrm{sat}}$ to the data. We obtain $T_{2,LT}=85\pm
10$ $\mu$s and $T_{2,SD,\mathrm{sat}}=2.65\pm 0.2$ $\mu$s.

##  Appendix E Spin g-factor in the 3T2g$(T_{1u})$ states

The SO interaction may alter the value of the spin g-factor of a state by
mixing it with other states of the same SO representation [86]. For the
$T_{1u}$ SO states of the 3T2g manifold, the closest such states are the
$T_{1u}$ SO states of the 3T1g manifold. We thus assume a wavefnction of the
form,

$|\psi_{j}\rangle=N_{j}\left(|T_{1u,j}\rangle_{{}^{3}\mathrm{T}_{2g}}+\sum_{k=-1}^{1}\gamma_{k,j}|T_{1u,k}\rangle_{{}^{3}\mathrm{T}_{1g}}\right),$
(E1)

where $\gamma_{k,j}$ is the ‘mixing fraction’ of the state
$|T_{1u,k}\rangle_{{}^{3}\mathrm{T}_{1g}}$ in $\psi_{j}$, and
$N_{j}=(1+\sum_{k=-1}^{1}\gamma_{k,j}^{2})^{-1/2}$ it a normalization
constant. Using second-order perturbation theory, the mixing fractions are
given by,

$\gamma_{k,j}=\frac{{}_{{}^{3}\mathrm{T}_{2g}}\langle
T_{1u,k}|H_{SO}|T_{1u,j}\rangle_{{}^{3}\mathrm{T}_{1g}}}{E_{{}^{3}\mathrm{T}_{1g}}^{T_{1u}}-E_{{}^{3}\mathrm{T}_{2g}}^{T_{1u}}},$
(E2)

where $H_{SO}$ is the SO Hamiltonian, and $E_{{}^{3}\mathrm{T}_{1g}}^{T_{1u}}$
($E_{{}^{3}\mathrm{T}_{2g}}^{T_{1u}}$) is the energy of the $T_{1u}$ states in
the ${}^{3}\mathrm{T}_{1g}$ (${}^{3}\mathrm{T}_{2g}$) manifold. The effective
g-factor is then given by

$g_{T_{1u}}=\tfrac{1}{\hbar}\langle\psi_{1}|L_{z}+g_{0}S_{z}|\psi_{1}\rangle.$
(E3)

Using the known structure of the $T_{1u}$ wavefunctions [111] and assuming
$\gamma_{k,j}\ll 1$, one obtains,

$g_{T_{1u}}\approx\tfrac{1}{2}\left[g_{L}+g_{0}(1-2\gamma_{1,-1})\right],$
(E4)

where $g_{L}=\langle\mathrm{T}_{2g,1}|L|\mathrm{T}_{2g,1}\rangle$ is
calculated in Appendix F below. Note that this calculation does not include
DJT distortion. Its inclusion amounts to replacing $g_{L}$ with
$\tilde{g}_{L}$. By table B1 we identify

$g_{s}^{\mathrm{e}}=g_{0}(1-2\gamma_{1,-1}).$ (E5)

Using
$E_{{}^{3}\mathrm{T}_{1g}}^{T_{1u}}-E_{{}^{3}\mathrm{T}_{2g}}^{T_{1u}}=160$
THz [53], ${}_{{}^{3}\mathrm{T}_{2g}}\langle
T_{1u,k}|H_{SO}|T_{1u,j}\rangle_{{}^{3}\mathrm{T}_{1g}}=-\frac{\sqrt{3}}{2}\zeta$
[112, 111], and the value of $\zeta$ extracted from the g-factor of the
ground-state (Appendix A), $\zeta=-7.73$ THz, we obtain $\gamma_{1,-1}=0.04$,
yielding $g_{s}^{\mathrm{e}}=1.84$.

##  Appendix F Orbital g-factor in the 3T2g manifold

In the $d^{8}$ configuration, it is much easier to use the hole notation, as
then one has to consider only the two empty electron orbitals instead of the
eight full ones. We therefore first find the orbital angular momentum of the
two-electron orbitals from which electrons are missing, and then take the
negative of the result, as we are interested in the orbital angular momentum
of a full shell (which is 0) _minus_ those two electrons.

All the relevant two-electron orbitals are composed mostly of products of two
$d$ ($l=2$) single-electron states. Therefore, they are mostly composed of
doubly-quadratic functions of the Cartesian coordinates.

For spin-triplet states, which are exchange-symmetric, the orbitals also have
to be exchange-anti-symmetric, as the total electronic wavefunction must be
exchange-anti-symmetric.

Out of all the possible T2g two-electron orbitals, we focus here on those in
which one electron is of the $e_{g}$ single-electron orbital and the other is
of the $t_{2g}$ single-electron orbital [see Fig. 1(b)].

Therefore, we have to use a set of spanning functions of the T2g
representation of the $O_{h}$ point group, which are doubly-quadratic in $x$,
$y$, and $z$, are composed of products of quadratic basis functions of the Eg
and T2g representations, and are exchange-anti-symmetric.

These constraints leave only one possible choice (up to internal unitary
transformations). Choosing the main axis to be $z$ and all the functions to be
eigenfunctions of the $z$ component of the angular momentum, we are left with
[111]

$|$T${}_{2g,1}\rangle_{\mathrm{e}}$ | $=$ | $-\tfrac{a}{\sqrt{2}}\\{x_{1}z_{1}(3y_{2}^{2}-r_{2}^{2})+iy_{1}z_{1}(3x_{2}^{2}-r_{2}^{2})\\}_{\mathrm{a.s.}},$
---|---|---
$|$T${}_{2g,0}\rangle_{\mathrm{e}}$ | $=$ | $ia\\{x_{1}y_{1}(3z_{2}^{2}-r_{2}^{2})\\}_{\mathrm{a.s.}},$
$|$T${}_{2g,-1}\rangle_{\mathrm{e}}$ | $=$ | $\tfrac{a}{\sqrt{2}}\\{x_{1}z_{1}(3y_{2}^{2}-r_{2}^{2})-iy_{1}z_{1}(3x_{2}^{2}-r_{2}^{2})\\}_{\mathrm{a.s.}},$
(F1)

where the subscript $\mathrm{e}$ indicates that these are electronic orbitals,
$\\{\ \\}_{\mathrm{a.s.}}$ stands for exchange-anti-symmetrization,
$r_{1(2)}^{2}=x_{1(2)}^{2}+y_{1(2)}^{2}+z_{1(2)}^{2}$, and
$a=5/(4\pi\sqrt{2})$ is a normalization constant.

It can be shown [111] that these functions can be represented as the following
combinations of products of single-electron $d$ orbitals,

$|$T${}_{2g,1}\rangle_{\mathrm{e}}$ | $=$ | $\\{\tfrac{\sqrt{3}}{\sqrt{8}}(|1,2\rangle+|1,\mathrm{-}2\rangle)-\tfrac{1}{2}|\mathrm{-}1,0\rangle\\}_{\mathrm{a.s.}},$
---|---|---
$|$T${}_{2g,0}\rangle_{\mathrm{e}}$ | $=$ | $\tfrac{1}{\sqrt{2}}\\{|2,0\rangle-|\mathrm{-}2,0\rangle\\}_{\mathrm{a.s.}},$
$|$T${}_{2g,-1}\rangle_{\mathrm{e}}$ | $=$ | $-\\{\tfrac{\sqrt{3}}{\sqrt{8}}(|\mathrm{-}1,2\rangle+|\mathrm{-}1,\mathrm{-}2\rangle)-\tfrac{1}{2}|1,0\rangle\\}_{\mathrm{a.s.}},$
(F2)

where the two numbers in the kets stand for the eigenvalues of the $z$
component of the angular momentum of the single-electron $d$ orbitals of the
two electrons.

With the wavefunctions cast in this form, it is straightforward to calculate
their two-electron angular momentum matrix elements. One obtains,

$\displaystyle\langle
L_{x}\rangle=\frac{\hbar}{2\sqrt{2}}\left(\begin{array}[]{ccc}0&1&0\\\
1&0&1\\\ 0&1&0\end{array}\right),$ (F6) $\displaystyle\langle
L_{y}\rangle=\frac{\hbar}{2\sqrt{2}}\left(\begin{array}[]{ccc}0&-i&0\\\
i&0&-i\\\ 0&i&0\end{array}\right),$ (F10) $\displaystyle\langle
L_{z}\rangle=\frac{\hbar}{2}\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&0\\\
0&0&-1\end{array}\right).$ (F14)

It is clear that these are spin-1 matrices, as expected, just multiplied by a
common factor of $\nicefrac{{1}}{{2}}$. Thus, one can treat the T2g two-
electron orbitals as an effective $l=1$ system, with an effective orbital
g-factor of $\nicefrac{{1}}{{2}}$.

Now, as we are interested in the orbital angular momentum of a full $d$ shell
_missing_ two electrons in the above states, we should take the _negative_ of
the above result. That is,

$g_{L}=-\nicefrac{{1}}{{2}}.$ (F15)

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