# Spin emitters beyond the point dipole approximation in nanomagnonic cavities

Derek S. Wang<EMAIL_ADDRESS>Harvard John A. Paulson School of
Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA
Tomáš Neuman  Harvard John A. Paulson School of Engineering and Applied
Sciences, Harvard University, Cambridge, MA 02138, USA  Prineha Narang
<EMAIL_ADDRESS>Harvard John A. Paulson School of Engineering and
Applied Sciences, Harvard University, Cambridge, MA 02138, USA

###### Abstract

Control over transition rates between spin states of emitters is crucial in a
wide variety of fields ranging from quantum information science to the
nanochemistry of free radicals. We present an approach to drive a both
electric and magnetic dipole-forbidden transition of a spin emitter by placing
it in a nanomagnonic cavity, requiring a description of both the spin emitter
beyond the point dipole approximation and the vacuum magnetic fields of the
nanomagnonic cavity with a large spatial gradient over the volume of the spin
emitter. We specifically study the SiV- defect in diamond, whose Zeeman-split
ground states comprise a logical qubit for solid-state quantum information
processing, coupled to a magnetic nanoparticle serving as a model nanomagnonic
cavity capable of concentrating microwave magnetic fields into deeply
subwavelength volumes. Through first principles modeling of the SiV- spin
orbitals, we calculate the spin transition densities of magnetic dipole-
allowed and -forbidden transitions and calculate their coupling rates to
various multipolar modes of the nanomagnonic cavity. We envision using such a
framework for quantum state transduction and state preparation of spin qubits
at GHz frequency scales.

††preprint: APS/123-QED

##  I Introduction

Precise control over transition rates between states of an emitter and, thus,
absorption and emission of phonons, photons, and magnons is crucial for a
variety of applications ranging from quantum information processing to energy
harvesting in artificial and biological structures [1, 2, 3, 4, 5]. These
transition rates are governed by selection rules that generally assume that
these external fields are plane waves with negligible spatial gradient and
that the emitter can be represented as a point dipole [6, 7, 8]. Violating
either or both of these approximations, however, can lead to exotic optical
and chemical phenomena, such as multipolar transitions that are faster than
dipolar ones, multiquanta emission, efficient spin-flip processes, and
spatially-dependent near-field emission [9, 10, 11, 5, 12]. Further
experimental development of such processes to tailor transition rates may lead
to facile generation of highly entangled multipartite states, for instance
[13, 14, 15].

Although previous research in breaking selection rules has been largely
focused on the interaction between plasmonic systems with emitters in the
optical frequency range, there have been comparatively fewer studies on
breaking selection rules in spin emitters [16, 17, 18, 19, 20, 21, 22, 23, 24]
with transition frequencies in the microwave range, where magnetic dipolar
transitions are inherently orders of magnitude slower than their electric
dipolar counterparts [25]. Spin emitters are of large technological interest
given, for instance, the proposed usage of defect spin emitters as spin-photon
interfaces in quantum information processing systems [26, 27, 28, 29, 30] and
radical pair-based mechanisms as the source of biological magneto-reception
[31, 32].

In this Article, we present an approach to couple isolated spin emitters to
fields of magnon modes with length scales on the order of single molecular or
defect emitters. The magnon modes are realized by ferromagnetic or
ferrimagnetic nanoparticles as nanomagnonic cavities that concentrate
microwave magnetic fields into deeply subwavelength volumes. This
configuration is illustrated in Fig. 1, where the spin emitter is positioned
close to the nanoparticle surface and efficiently coupled via fields of
dipolar or higher order magnon modes. The large spatial gradients of coupled
magnon modes requires description of the emitters beyond the point-dipole
approximation and enables selection rule-breaking of orbital-spin transitions,
This phenomenon is analogous to those realized in studies that have
demonstrated breaking of electric dipole-based selection rules in plasmonic
systems [9, 10, 11, 5, 12]. While Ref. [6, 7] consider how magnetic
nanostructures can generate magnetic fields with large gradients and catalyze
intersystem crossing in molecular radical pairs, neither the full spectral
profile of the magnetic nanostructure nor the spatial variation of the spin
emitters was considered. To demonstrate selection rule breaking, we show a
non-zero transition rate between the logical $|0\rangle$ and $|1\rangle$
states in the ground state manifold of the negative silicon vacancy (SiV-) in
diamond, a leading materials candidate for spin-photon coupling in solid-state
qubits for quantum technologies [22, 23, 24, 33, 34, 35], coupled to a
spherical nanoparticle of yttrium iron garnet (YIG) that serves as the
nanomagnonic cavity. This transition is typically forbidden under electric and
magnetic dipole selection rules. We show that this magnon-spin coupling rate
can reach kHz frequency scales with the potential to enter the strong magnon-
spin coupling regime upon the development of lower-loss magnetic materials. We
envision leveraging these phenomena to more flexibly manipulate the quantum
states of spin qubits or to mediate qubit-qubit interactions necessary for
quantum technologies.

Figure 1: The spatially-varying vacuum magnetic field of a nanomagnonic cavity
realized with a magnetic nanoparticle with radius $R$ drives the spin- and
strain-forbidden transition between the $|e^{+},\uparrow\rangle$ and
$|e^{-},\downarrow\rangle$ states of a model spin emitter, the SiV- defect
center in diamond, placed $a$ away from the surface of the nanoparticle.

##  II SiV- as a model spin emitter

Figure 2: (a) Energy-level diagram of the ground state of the SiV- defect
center in diamond. The magnetic dipole-allowed and forbidden transitions are
labelled in green and red, respectively. The real and imaginary parts of the
components of the spin transition densities (b)-(c) $S_{x}(\mathbf{r})$ and
(d)-(e) $S_{y}(\mathbf{r})$. $S_{z}(\mathbf{r})$ is 0.

As an example of a spin emitter with a transition forbidden by magnetic diple-
based selection rules, we study the SiV- defect in diamond, a leading spin-
photon interface candidate for various quantum information processing
technologies [22, 23, 24]. The SiV- is a split-vacancy defect that consists of
the silicon atom located between two adjacent vacant sites in diamond,
resulting in $D_{3d}$ symmetry [36, 37]. The degeneracy of the ground state
manifold is broken by spin-orbit coupling that has been experimentally
observed at $\sim$50 GHz. This value can be closely reproduced with _ab
initio_ density functional theory calculations with a correction due to the
Jahn-Teller effect, although this effect is not necessary to predict the
orbital and spin character [37]. The resulting ground state manifold—with
lower energy degenerate states $\psi_{e^{+}}(\mathbf{r})|\downarrow\rangle$
and $\psi_{e^{-}}(\mathbf{r})|\uparrow\rangle$ and higher energy degenerate
$\psi_{e^{+}}(\mathbf{r})|\uparrow\rangle$ and
$\psi_{e^{-}}(\mathbf{r})|\downarrow\rangle$—can be coupled together via
microwave drive. The degeneracies of the lower and upper levels of the ground
state manifold can be broken in the presence of longitudinal magnetic field
along the [111] direction. The lowest two states
$\psi_{e^{+}}(\mathbf{r})|\downarrow\rangle$ and
$\psi_{e^{-}}(\mathbf{r})|\uparrow\rangle$ are of special interest, as they
comprise a spin qubit $|0\rangle$ and $|1\rangle$, respectively, used
successfully in Refs. [38, 39]. Notably, transitions within this spin qubit
are forbidden via homogeneous magnetic field due to the opposite parities of
the spatial orbitals $\psi_{e^{+}}(\mathbf{r})$ and $\psi_{e^{+}}(\mathbf{r})$
and via a homogeneous electric field due to their opposite spins [40]. This
transition, however, can be allowed with a combination of transverse magnetic
field, such that the eigenstates are mixtures of
$\psi_{e^{+}}(\mathbf{r})|\downarrow\rangle$ and
$\psi_{e^{-}}(\mathbf{r})|\uparrow\rangle$, and/or strain [41, 42, 40, 43]. We
show here that this transition can be driven solely with the vacuum magnetic
field with a large spatial gradient of a nanomagnonic cavity, physically
realized here as a ferrimagnetic YIG nanoparticle.

##  III Magnonic modes of a magnetic nanosphere

Figure 3: (a) Real and imaginary components of the surface potential $\phi(R)$
looking along the $y$-axis of the equatorial multipolar modes of the
nanomagnonic cavity, specifically the dipolar mode where $n=1$ and $m=-1$,
quadrupolar mode where $n=2$ and $m=-2$, and octupolar mode where $n=3$ and
$m=-3$. (b) Spectral density $J(\omega)$ of the nanomagnonic cavity coupled to
a magnetic point dipole including multipolar terms up to $n=25$ for $R=30$ nm,
$a=5$ nm. The relevant dipolar, quadrupolar, and octupolar modes are
highlighted in red, pink, and blue, respectively.

We briefly present the magnonic modes of a homogeneously magnetized nanosphere
with an external magnetic field aligned along the magnetization direction $z$
following the treatment of Walker [44, 45] and Fletcher [46] and discussed
more extensively in Ref. [47]. In particular we seek the solutions for a
magnon mode:

$\displaystyle(1+\chi)\left(\frac{\partial^{2}\phi}{\partial
x^{2}}+\frac{\partial^{2}\phi}{\partial
y^{2}}\right)+\frac{\partial^{2}\phi}{\partial z^{2}}=0,$ (1)

where the magnetic scalar potential $\phi$ is linked with the quasi-static
magnetic field by ${\bf H}(\omega)=-\nabla\phi(\omega)$ and $\omega$ is the
magnon frequency. $\chi\equiv\chi_{xx}=\chi_{yy}$ and
$\mathrm{i}\kappa\equiv\chi_{xy}=\chi_{yx}^{*}$ are the frequency-dependent
components of the susceptibility tensor $\bm{\chi}(\omega)$ defined by the
magnetic response to external fields in the Landau-Lifshitz-Gilbert equation
[47]. The inputs required to calculate the susceptibility tensor
$\bm{\chi}(\omega)$ are the gyromagnetic ratio $\gamma\approx 176$
$\mathrm{G}\mathrm{rad}\cdot\mathrm{T}^{-1}$, the phenomenological damping
parameter $\Gamma=1$ Mrad/s, external magnetic field $H_{e}=0.5$ T, and
saturation magnetization $\mu_{0}M_{\rm s}=0.178$ $\mathrm{T}$ [47], which
closely correspond to the values observed experimentally in YIG [48].

For the outer region of the sphere we define the field in the standard
spherical coordinates as shown in Figure 1:

$\displaystyle x$ $\displaystyle=r\sin(\theta)\cos(\varphi),$ (2)
$\displaystyle y$ $\displaystyle=r\sin(\theta)\sin(\varphi),$ (3)
$\displaystyle z$ $\displaystyle=r\cos(\theta),$ (4)

where the magnetic scalar potential outside the sphere $\phi_{\mathrm{out}}$
is

$\displaystyle\phi_{\rm
out}=B_{nm}r^{-n-1}P_{n}^{m}(\cos\theta)e^{im\varphi},$ (5)

where $P_{n}^{m}(\cos\theta)$ is the associated Legendre polynomial and each
magnon mode is defined by a pair of indices $n$ and $m$. The magnetic scalar
potential inside the sphere $\phi_{\mathrm{in}}$ can be expressed in
ellipsoidal coordinates:

$\displaystyle x$
$\displaystyle=R\sqrt{-\chi}\sqrt{1-\xi^{2}}\sin(\eta)\cos(\varphi),$ (6)
$\displaystyle y$
$\displaystyle=R\sqrt{-\chi}\sqrt{1-\xi^{2}}\sin(\eta)\sin(\varphi),$ (7)
$\displaystyle z$ $\displaystyle=R\sqrt{\frac{\chi}{1+\chi}}\xi\cos(\eta),$
(8)

resulting in the sought solution:

$\displaystyle\phi_{\rm
in}=A_{nm}P_{n}^{m}(\xi)P_{n}^{m}(\cos\eta)e^{im\varphi}.$ (9)

The solutions must fulfill the boundary conditions [46]:

$\displaystyle\left(\phi_{\rm in}\right)_{R}=\left(\phi_{\rm out}\right)_{R},$
(10)

and

$\displaystyle\left(\frac{\partial\phi_{\rm out}}{\partial r}\right)_{R}$
$\displaystyle=(1+\chi\sin^{2}\theta)\left(\frac{\partial\phi_{\rm
in}}{\partial r}\right)_{R}$
$\displaystyle+\frac{\chi\sin\theta\cos\theta}{R}\left(\frac{\partial\phi_{\rm
in}}{\partial\theta}\right)_{R}$
$\displaystyle+\frac{i\kappa}{R}\left(\frac{\partial\phi_{\rm
in}}{\partial\varphi}\right)_{R}.$ (11)

Note that the sign convention $e^{-i\omega t}$ has been used. After inserting
$\phi_{\rm in}$ and $\phi_{\rm out}$ into the boundary conditions and noticing
that on the particle surface $\xi_{0}=[(1+\chi)/\chi]^{1/2}$ and
$\eta=\theta$, we obtain:

$\displaystyle
B_{mn}=-A_{nm}\left[\frac{\xi_{0}P_{n}^{m}{}^{\prime}(\xi_{0})-m\kappa
P_{n}^{m}(\xi_{0})}{n+1}\right]R^{n+1}$ (12)

and

$\displaystyle B_{nm}=A_{nm}R^{n+1}P_{n}^{m}(\xi_{0}).$ (13)

Equations (12) and (13) must be fulfilled simultaneously, therefore posing a
condition on the acceptable value of frequency $\omega$ appearing in
$\xi_{0}(\omega)$ [i.e. $\chi(\omega)$] and $\kappa(\omega)$. The acceptable
frequencies (resonance frequencies) follow from the secular equation:

$\displaystyle(n+1-m\kappa)P_{n}^{m}(\xi_{0})+\xi_{0}P_{n}^{m}{}^{\prime}(\xi_{0})=0.$
(14)

The secular equation [Eq. (14)] can be solved for low values of $n$ and $m$
analytically as shown in Ref. [46]. The magnonic modes to which we couple the
spin emitter are plotted in Fig. 3(a), where $n=1$, 2, and 3 and $m=-n$
corresponding to the equatorial dipolar, quadrupolar, and octupolar modes,
respectively.

##  IV Magnon-emitter coupling

To determine the magnon-emitter coupling rate $g$, we first obtain the spin
transition density ${\mathbf{S}}({\bm{r}})$ for the forbidden transition,
where the $\alpha\in\\{x,y,z\\}$ component can be expressed as

$\displaystyle{S}_{\alpha}(\mathbf{r})=\psi_{e^{+}}(\mathbf{r})^{*}\psi_{e^{-}}(\mathbf{r})\langle\downarrow|\sigma_{\alpha}|\uparrow\rangle,$
(15)

where the Pauli vector ${\mathbf{\sigma}}=(\sigma_{x},\sigma_{y},\sigma_{z})$.
We plot each component of the spin transition densities in Fig. 2, where the
spin orbitals are calculated as described in Appendix A.

The spectral density $J(\omega)$ for a spatially-delocalized spin emitter can
then be written as

$\displaystyle J(\omega)$ $\displaystyle=\frac{\mu_{0}\mu_{\rm
B}^{2}}{\hbar\pi}$ $\displaystyle{\rm Im}\left\\{\iint{\bf S}^{\ast}({\bf
r}^{\prime})\cdot k_{0}^{2}{\bf G}_{\rm m}({\bf r}^{\prime},{\bf
r},\omega)\cdot{\bf S}({\bf r}){\rm d}^{3}{\bf r}{\rm d}^{3}{\bf
r}^{\prime}\right\\},$ (16)

where $\mu_{\rm B}$ is the Bohr magneton and ${\bf G}_{\rm m}({\bf
r}^{\prime},{\bf r},\omega)$ is the magnetostatic Green’s tensor. This
expression is generalized from the expression for the coupling between a
quantized magnon mode and a point-like spin emitter [47]. To evaluate the
Green’s tensor ${\bf G}_{\rm m}({\bf r}^{\prime},{\bf r},\omega)$ we solve the
Poisson’s equation in spherical geometry as described in Ref. [47] using the
expansion of the magnetic field in the magnon modes. The spectral function
$J(\omega)$ determines the dynamics of the spin excitation, with $c_{\rm e}$
being the excited-state amplitude, as

$\displaystyle\dot{\tilde{c}}_{\rm
e}=-\int_{0}^{t}f(t-t^{\prime}){\tilde{c}}_{\rm e}(t^{\prime})\,{\rm
d}t^{\prime},$ (17)

where ${\tilde{c}}_{\rm e}(t^{\prime})e^{-i\omega_{0}t^{\prime}}={{c}}_{\rm
e}(t^{\prime})$ and

$\displaystyle
f(t-t^{\prime})=\int_{-\infty}^{\infty}J(\omega)e^{i(\omega_{0}-\omega)(t-t^{\prime})}{\rm
d}\omega.$ (18)

Assuming that the spectral density $J(\omega)$ within $\Gamma$ around the
frequency $\omega_{n,m}$ of the magnon mode defined by multipole index $n$ and
angular momentum index $m$ to which the spin transition frequency is tuned
corresponds only to this single magnon mode, valid in the limit of small
damping $\Gamma=1$ Mrad/s for YIG, we can determine the coupling rate
$g_{n,m}$ by fitting

$\displaystyle\tilde{J}(\omega)=\frac{1}{2\pi}\frac{\Gamma
g_{n,m}^{2}}{(\Gamma/2)^{2}+(\omega-\omega_{n,m})^{2}}$ (19)

to the full magnetostatic solution for $J(\omega)$. To calculate the coupling
rate for the magnetic dipole-allowed transition, we can repeat this procedure
after first writing the spin transition density as

$\displaystyle{S}_{\alpha}(\mathbf{r})=\psi_{e^{+}}(\mathbf{r})^{*}\psi_{e^{+}}(\mathbf{r})\langle\downarrow|\sigma_{\alpha}|\uparrow\rangle.$
(20)

In Fig. 3(b), for a nanosphere with radius $R=30$ nm and magnetic point dipole
position $a=5$ nm away, we plot the spectral density $J(\omega)$ under the
dipolar approximation, where
$\psi_{e^{+}}(\mathbf{r})^{*}\psi_{e^{+}}(\mathbf{r})$ in Eq. (20) can be
approximated by a delta function and Eq. (16) reduces to the form in Ref.
[47]. Note that the three modes plotted in Fig. 3(a) have relatively large
values of $J(\omega)$ and their peaks are are well-separated from other magnon
frequencies within their linewidths $\sim\Gamma$, in contrast to the near-
continuum of higher-order magnon modes above $\sim$16.3 GHz.

Figure 4:  Coupling rate $g_{n,m}$ between the dipolar, quadrupolar, and
octupolar modes in red, purple, and blue, respectively, for the (a) allowed
$\psi_{e^{+}}(\mathbf{r})|\uparrow\rangle\rightarrow\psi_{e^{+}}(\mathbf{r})|\downarrow\rangle$
and (b) forbidden transition
$\psi_{e^{+}}(\mathbf{r})|\uparrow\rangle\rightarrow\psi_{e^{-}}(\mathbf{r})|\downarrow\rangle$
of SiV- for varying distance $a$ from the surface of the magnetic nanoparticle
at $\theta=\pi/2$ and $\phi=0$. Note that the y-axes are in different units.

We couple the SiV- spin emitter to equatorial multipolar modes $n=-m$ of the
magnonic cavity realized by a nanomagnetic sphere to demonstrate non-zero,
distance-dependent coupling $g$ between the forbidden spin transition and
magnon modes in Fig. 4(b). As a comparison, we also calculate the coupling
rate $g$ to the magnetic dipole-allowed transition between
$\psi_{e^{+}}(\mathbf{r})|\uparrow\rangle$ and
$\psi_{e^{+}}(\mathbf{r})|\downarrow\rangle$ in Fig. 4(a). For the allowed
transitions coupled to the dipolar, quadrupolar, and octupolar modes in Fig.
2(a), we find $g\sim 1$ MHz, where the ordering of coupling rates with respect
to indices $n$ and $m$ mirror the trends in $J(\omega)$ in Fig. 3(b). Given
that the coupling rate $g$ is much larger than the damping $\Gamma$, this
transition lies within the strong-coupling regime when the magnon mode is
resonantly tuned to the transition, resulting in, for instance, Jaynes-
Cummings-like coherent energy transfer between the singly-excited magnon mode
and spin emitter in accordance with calculations in Ref. [47]. In Fig. 4(b)
plotting $g$ as a function of sphere-emitter distance $a$ for the forbidden
transition, the ordering of coupling rates with respect to indices $n$ and $m$
identically mirror the trends in Fig. 4(a). However, the magnitudes of $g\sim
10$ kHz for the forbidden transition are two orders of magnitude lower than
$g$ of the allowed transition and four orders of magnitude lower than the
damping rate $\Gamma$, placing this transition within the weak-coupling regime
and enabling incoherent, accelerated energy transfer from the singly excited
magnon mode to the spin emitter’s $|1\rangle$ from the $|0\rangle$ state, or
vice versa. As the cavity-enhanced decay rate follows $g^{2}/\Gamma$, we
expect decay rates $\sim$1 kHz for the forbidden transition, as compared to no
decay whatsoever in a vacuum according to conventional selection rules. Lower-
loss magnetic materials, such as V[TCNE]x with magnon damping rate two orders
of magnitude lower than YIG’s [49], could be used to boost the nanomagnonic
cavity-enhanced decay rate of the spin emitter and realize the strong-coupling
regime where $g\gg\Gamma$. By comparison, in spin emitters realized by defects
in solid-state materials [50], phonon fields with frequencies also in the
microwave range can be used to drive spin transitions [51] with Rabi
frequencies on the order of 1-50 MHz [42, 43], but these interactions can
require careful control and engineering of phononic cavities.

We justify the assumption that the allowed and forbidden spin transition
frequencies can be tuned in resonance with the magnon modes of interest. The
spin transition energy $E_{\rm s}$ in the presence of a total magnetic field
$B_{z}$ along the direction of the spin eigenbasis is [36, 52]

$\displaystyle E_{\mathrm{s}}=(f\gamma_{L}+\gamma_{S})H_{z}/\mu_{0},$ (21)

where the orbital gyromagnetic ratio $\gamma_{L}=\mu_{B}$, the spin
gyromagnetic ratio $\gamma_{S}=2\mu_{B}$, and the coefficient of the orbital
Zeeman quenching effect $f\approx-0.1$. The total magnetic field
$H_{z}=|\bm{H}_{\mathrm{e}}|+|\bm{H}_{\mathrm{sphere}}|=|\bm{H}_{\mathrm{e}}^{\mathrm{em}}|$,
where all magnetic fields point along the $z$-direction, the external magnetic
field applied to the spin emitter SiV-
$\mu_{0}\bm{H}_{\mathrm{e}}^{\mathrm{em}}$ can be chosen such that the spin
transition frequency is resonant with the magnon frequencies plotted in Fig.
3, and $\mu_{0}\bm{H}_{\mathrm{sphere}}$ is equivalent to the magnetic field
of a magnetic point dipole at the center of the sphere with magnetic dipole
moment $\bm{M}_{s}V$. Due to the magnetic field of the sphere acting on the
spin emitter in addition to the externally applied magnetic field, there is an
frequency gap between the magnon eigenfrequencies and the frequency of the
forbidden transition of the spin emitter of $\sim 1$ GHz. Therefore, for this
particular sphere-emitter setup, we require a difference of
$\mu_{0}\bm{H}_{\mathrm{e}}^{\mathrm{em}}\sim 35$ mT in externally applied
magnetic field between the sphere and the spin emitter separated by 10 nm,
feasible in recent demonstrations of microcoils fabricated on diamond chips
with electrically tuneable magnetic field gradients of up to $\sim 10$ mT/nm
[53, 54, 55].

##  V Conclusion and outlook

The present study provides the theoretical basis for breaking selection rules
in spin emitters with magnetic nanoparticles. Specifically, we show that
magnetic nanoparticles serving as nanomagnonic cavities to concentrate
microwave magnetic fields generate fields with spatial gradients with length
scales on the order of the spatial delocalization of spin emitters. As an
example, we couple magnon modes of a YIG nanosphere to the SiV- defect center
and show coupling on the order of tens of kHz between several equatorial,
multipolar modes of the nanosphere and the forbidden transition between the
canonical spin-based qubit states. Natural extensions of the present work
include exploration of the coupling between planar interfaces and spatially
delocalized emitters, which can support larger magnetic field gradients,
analogous to the large electric near-fields supported by plasmonic surfaces.

## Acknowledgements

We acknowledge fruitful discussions with Christopher J. Ciccarino, Isaac
Harris, and Michael Haas. This material is based upon work supported by the
U.S. Department of Energy, Office of Science, National Quantum Information
Science Research Centers. This work was also partially supported by the
Department of Energy ‘Photonics at Thermodynamic Limits’ Energy Frontier
Research Center under grant DE-SC0019140. D.S.W. is supported by the Army
Research Office MURI (Ab-Initio Solid-State Quantum Materials) grant number
W911NF-18-1-0431 and the National Science Foundation Graduate Research
Fellowship. P.N. is a Moore Inventor Fellow through Grant GBMF8048 from the
Gordon and Betty Moore Foundation. This research used resources of the
National Energy Research Scientific Computing Center, a DOE Office of Science
User Facility supported by the Office of Science of the U.S. Department of
Energy under Contract No.DE-AC02-05CH11231.

##  Appendix A SiV- orbitals

To extract spin-orbit-coupled orbitals of SiV-, we use the periodic density
functional theory (DFT) code Quantum ESPRESSO [56, 57, 58] with a geometry-
optimized 511-atom supercell of diamond with a SiV- defect, the Perdew-Burke-
Ernzerhof (PBE) exchange-correlation functional [59], non-collinear spin-orbit
coupling, kinetic energy cutoff of 80 Rydberg, and energy and force
convergence thresholds of $10^{-8}$ a.u. and $10^{-5}$ a.u., respectively. The
resulting four Kohn-Sham spin orbitals within the band
gap–$\Psi_{A}(\mathbf{r}),\Psi_{B}(\mathbf{r}),\Psi_{C}(\mathbf{r}),\Psi_{D}(\mathbf{r})$–are
split by spin-orbit coupling into two degenerate pairs, such that
$E_{A}=E_{B}<E_{C}=E_{D}$. We can extract the spatial wavefunctions
$\psi_{e^{+}}(\mathbf{r}),\psi_{e^{-}}(\mathbf{r})$ from either pair, but for
specificity, we focus on the two Kohn-Sham orbitals with lower energies and
note that we can equivalently apply this approach to the pair with higher
energies. The lower energy pair is by default output as

$\displaystyle\Psi_{A}(\mathbf{r})=a_{z}(\mathbf{r})|\uparrow_{z}\rangle+b_{z}(\mathbf{r})|\downarrow_{z}\rangle,$
(22)
$\displaystyle\Psi_{B}(\mathbf{r})=c_{z}(\mathbf{r})|\uparrow_{z}\rangle+d_{z}(\mathbf{r})|\downarrow_{z}\rangle,$
(23)

where $a_{z}(\mathbf{r})$ to $d_{z}(\mathbf{r})$ are spatially-varying
coefficients for the spin basis along the $z$-direction $|\uparrow_{z}\rangle$
and $|\downarrow_{z}\rangle$. However, we seek orbitals spin-polarized along
the [111] direction, so we rotate the spinors by applying the spinor rotation
matrix

$\bm{R}=\exp\left(\frac{-{\rm i}\bm{\sigma}\cdot\bm{n}\Delta\phi}{2}\right),$
(24)

where $\bm{\sigma}$ is the Pauli vector, $\bm{n}$ is the axis of rotation, and
$\Delta\phi$ is the angle of rotation. We can therefore re-write
$\Psi_{A}(\mathbf{r})$ and $\Psi_{B}(\mathbf{r})$ as

$\displaystyle\Psi_{A}(\mathbf{r})=a(\mathbf{r})|\uparrow\rangle+b(\mathbf{r})|\downarrow\rangle,$
(25)
$\displaystyle\Psi_{B}(\mathbf{r})=c(\mathbf{r})|\uparrow\rangle+d(\mathbf{r})|\downarrow\rangle,$
(26)

where $a(\mathbf{r})$ to $d(\mathbf{r})$ are spatially-varying coefficients
for the spin basis along the $[111]$ direction $|\uparrow\rangle$ and
$|\downarrow\rangle$. Due to spin-orbit coupling, the spatial coefficients $a$
and $c$ (or $b$ and $d$) corresponding to the $|\uparrow\rangle$
($|\downarrow\rangle$) basis can be separated into the desired spatial
wavefunctions $\psi_{e^{+}}(\mathbf{r})$ ($\psi_{e^{-}}(\mathbf{r})$) defined
to a phase:

$\displaystyle\Psi_{A}(\mathbf{r})=\alpha\psi_{e^{+}}(\mathbf{r})|\uparrow\rangle+\beta\psi_{e^{-}}(\mathbf{r})|\downarrow\rangle,$
(27)
$\displaystyle\Psi_{B}(\mathbf{r})=\gamma\psi_{e^{+}}(\mathbf{r})|\uparrow\rangle+\delta\psi_{e^{-}}(\mathbf{r})|\downarrow\rangle,$
(28)

where $\alpha$ to $\delta$ are complex numbers fulfilling
$\alpha\gamma^{*}+\beta\delta^{*}=0$ and
$|\alpha|^{2}+|\beta|^{2}=|\gamma|^{2}+|\delta|^{2}=1$.

## References

  * Quidant  _et al._ [2010] R. Quidant, N. F. V. Hulst, A. G. Curto, G. Volpe, T. H. Taminiau, and M. P. Kreuzer, Unidirectional Emission of a Quantum Dot Coupled to a Nanoantenna, Science 329, 930 (2010).
  * Song  _et al._ [2012] P. Song, S. Meng, P. Nordlander, and S. Gao, Quantum plasmonics: Symmetry-dependent plasmon-molecule coupling and quantized photoconductances, Phys. Rev. B 86, 121410 (2012).
  * Tame  _et al._ [2013] M. S. Tame, K. R. McEnery, S. K. Özdemir, J. Lee, S. A. Maier, and M. S. Kim, Quantum plasmonics, Nat. Phys. 9, 329 (2013).
  * Hayat  _et al._ [2011] A. Hayat, A. Nevet, P. Ginzburg, and M. Orenstein, Applications of two-photon processes in semiconductor photonic devices: Invited review, Semicond. Sci. Technol. 26, 083001 (2011).
  * Rivera  _et al._ [2016] N. Rivera, I. Kaminer, B. Zhen, J. D. Joannopoulous, and M. Soljačić, Shrinking light to allow forbidden transitions on the atomic scale, Science 353, 263 (2016).
  * Cohen [2009] A. E. Cohen, Nanomagnetic control of intersystem crossing, J. Phys. Chem. A 113, 11084 (2009).
  * Yang and Cohen [2011] N. Yang and A. E. Cohen, Local geometry of electromagnetic fields and its role in molecular multipole transitions, J. Phys. Chem. B 115, 5304 (2011).
  * Lee  _et al._ [2011] H. Lee, N. Yang, and A. E. Cohen, Mapping nanomagnetic fields using a radical pair reaction, Nano Lett. 11, 5367 (2011).
  * Andersen  _et al._ [2011] M. L. Andersen, S. Stobbe, A. S. Sørensen, and P. Lodahl, Strongly modified plasmon–matter interaction with mesoscopic quantum emitters, Nat. Phys. 7, 215 (2011).
  * Takase  _et al._ [2013] M. Takase, H. Ajiki, Y. Mizumoto, K. Komeda, M. Nara, H. Nabika, S. Yasuda, H. Ishihara, and K. Murakoshi, Selection-rule breakdown in plasmon-induced electronic excitation of an isolated single-walled carbon nanotube, Nat. Photonics 7, 550 (2013).
  * Cotrufo and Fiore [2015] M. Cotrufo and A. Fiore, Spontaneous emission from dipole-forbidden transitions in semiconductor quantum dots, Phys. Rev. B 92, 125302 (2015).
  * Neuman  _et al._ [2018] T. Neuman, R. Esteban, D. Casanova, F. J. García-Vidal, and J. Aizpurua, Coupling of Molecular Emitters and Plasmonic Cavities beyond the Point-Dipole Approximation, Nano Lett. 18, 2358 (2018).
  * Huber  _et al._ [2018] D. Huber, M. Reindl, J. Aberl, A. Rastelli, and R. Trotta, Semiconductor quantum dots as an ideal source of polarization-entangled photon pairs on-demand: A review, J. Opt. 20, 073002 (2018).
  * Zeeshan  _et al._ [2019] M. Zeeshan, N. Sherlekar, A. Ahmadi, R. L. Williams, and M. E. Reimer, Proposed Scheme to Generate Bright Entangled Photon Pairs by Application of a Quadrupole Field to a Single Quantum Dot, Phys. Rev. Lett. 122, 227401 (2019).
  * Wang  _et al._ [2020] D. S. Wang, T. Neuman, and P. Narang, Dipole-coupled emitters as deterministic entangled photon-pair sources, Phys. Rev. Res. 2, 043328 (2020).
  * Imamo·∏°lu  _et al._ [2003] A. Imamoǧlu, E. Knill, L. Tian, and P. Zoller, Optical Pumping of Quantum-Dot Nuclear Spins, Phys. Rev. Lett. 91, 017402 (2003).
  * Cerletti  _et al._ [2005] V. Cerletti, W. A. Coish, O. Gywat, and D. Loss, Recipes for spin-based quantum computing, Nanotechnology 16, R27 (2005).
  * Lehmann  _et al._ [2007] J. Lehmann, A. Gaita-Arino, E. Coronado, and D. Loss, Spin qubits with electrically gated polyoxometalate molecules, Nat. Nanotechnol. 2, 312 (2007).
  * Trauzettel  _et al._ [2007] B. Trauzettel, D. V. Bulaev, D. Loss, and G. Burkard, Spin qubits in graphene quantum dots, Nat. Phys. 3, 192 (2007).
  * Müller  _et al._ [2008] M. Müller, L. Liang, I. Lesanovsky, and P. Zoller, Trapped Rydberg ions: From spin chains to fast quantum gates, New J. Phys. 10, 093009 (2008).
  * Kloeffel and Loss [2013] C. Kloeffel and D. Loss, Prospects for spin-based quantum computing in quantum dots, Annu. Rev. Condens. Matter Phys. 4, 51 (2013).
  * Sukachev  _et al._ [2017] D. D. Sukachev, A. Sipahigil, C. T. Nguyen, M. K. Bhaskar, R. E. Evans, F. Jelezko, and M. D. Lukin, Silicon-vacancy spin qubit in diamond: A quantum memory exceeding 10 ms with single-shot state readout, Phys. Rev. Lett. 119, 223602 (2017).
  * Pingault  _et al._ [2017] B. Pingault, D. D. Jarausch, C. Hepp, L. Klintberg, J. N. Becker, M. Markham, C. Becher, and M. Atatüre, Coherent control of the silicon-vacancy spin in diamond, Nat. Commun. 8, 15579 (2017).
  * Becker  _et al._ [2018] J. N. Becker, B. Pingault, D. Groß, M. Gündoğan, N. Kukharchyk, M. Markham, A. Edmonds, M. Atatüre, P. Bushev, and C. Becher, All-optical control of the silicon-vacancy spin in diamond at millikelvin temperatures, Phys. Rev. Lett. 120, 053603 (2018).
  * Cohen-Tannoudji  _et al._ [1977] C. Cohen-Tannoudji, B. Diu, and F. Laloe,  _Quantum mechanics_ (Wiley, New York, NY, 1977).
  * Kurizki  _et al._ [2015] G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, Quantum technologies with hybrid systems, Proc. Natl. Acad. Sci. U.S.A. 112, 3866 (2015).
  * Li  _et al._ [2015] L. Li, T. Schröder, E. H. Chen, M. Walsh, I. Bayn, J. Goldstein, O. Gaathon, M. E. Trusheim, M. Lu, J. Mower, M. Cotlet, M. L. Markham, D. J. Twitchen, and D. Englund, Coherent spin control of a nanocavity-enhanced qubit in diamond, Nat. Commun. 6, 6173 (2015).
  * Dréau  _et al._ [2018] A. Dréau, A. Tcheborateva, A. E. Mahdaoui, C. Bonato, and R. Hanson, Quantum Frequency Conversion of Single Photons from a Nitrogen-Vacancy Center in Diamond to Telecommunication Wavelengths, Phys. Rev. Appl. 9, 064031 (2018).
  * Awschalom  _et al._ [2018] D. D. Awschalom, R. Hanson, J. Wrachtrup, and B. B. Zhou, Quantum technologies with optically interfaced solid-state spins, Nat. Photonics 12, 516 (2018).
  * Christle  _et al._ [2015] D. J. Christle, A. L. Falk, P. Andrich, P. V. Klimov, J. U. Hassan, N. T. Son, E. Janzén, T. Ohshima, and D. D. Awschalom, Isolated electron spins in silicon carbide with millisecond coherence times, Nat. Mater. 14, 160 (2015).
  * Grissom [1995] C. B. Grissom, Magnetic Field Effects in Biology: A Survey of Possible Mechanisms with Emphasis on Radical-Pair Recombination, Chem. Rev. 95, 3 (1995).
  * Rodgers and Hore [2009] C. T. Rodgers and P. J. Hore, Chemical magnetoreception in birds: The radical pair mechanism, Proc. Natl. Acad. Sci. U.S.A. 106, 353 (2009).
  * Harris  _et al._ [2019] I. Harris, C. J. Ciccarino, J. Flick, D. R. Englund, and P. Narang, Group iii quantum defects in diamond are stable spin-1 color centers, Phys. Rev. B 102, 195206 (2019).
  * Ciccarino  _et al._ [2020] C. J. Ciccarino, J. Flick, I. B. Harris, M. E. Trusheim, D. R. Englund, and P. Narang, Strong spin-orbit quenching via the product jahn-teller effect in neutral group iv artificial atom qubits in diamond, npj Quantum Mater, 5, 75 (2020).
  * Neuman  _et al._ [2020a] T. Neuman, M. Trusheim, and P. Narang, Selective acoustic control of photon-mediated qubit-qubit interactions, Phys. Rev. A 101, 052342 (2020a), 1912.06160 .
  * Hepp  _et al._ [2014] C. Hepp, T. Müller, V. Waselowski, J. N. Becker, B. Pingault, H. Sternschulte, D. Steinmüller-Nethl, A. Gali, J. R. Maze, M. Atatüre, and C. Becher, Electronic structure of the silicon vacancy color center in diamond, Phys. Rev. Lett. 112, 1 (2014).
  * Thiering and Gali [2018] G. Thiering and A. Gali, Ab-Initio Magneto-Optical Spectrum of Group-IV Vacancy Color Centers in Diamond, Phys. Rev. X 8, 021063 (2018).
  * Bhaskar  _et al._ [2019] M. K. Bhaskar, R. Riedinger, B. Machielse, D. S. Levonian, C. T. Nguyen, E. N. Knall, H. Park, D. Englund, M. Lončar, D. D. Sukachev, and M. D. Lukin, Experimental demonstration of memory-enhanced quantum communication, Nature 580, 60 (2019).
  * Nguyen  _et al._ [2019] C. T. Nguyen, D. D. Sukachev, M. K. Bhaskar, B. Machielse, D. S. Levonian, E. N. Knall, P. Stroganov, R. Riedinger, H. Park, M. Lončar, and M. D. Lukin, Quantum network nodes based on diamond qubits with an efficient nanophotonic interface, Phys. Rev. Lett. 123, 183602 (2019).
  * Neuman  _et al._ [2020b] T. Neuman, M. Eichenfield, M. Trusheim, L. Hackett, P. Narang, and D. Englund, A Phononic Bus for Coherent Interfaces Between a Superconducting Quantum Processor, Spin Memory, and Photonic Quantum Networks, arXiv:2003.08383 (2020b).
  * Udvarhelyi  _et al._ [2018] P. Udvarhelyi, V. O. Shkolnikov, A. Gali, G. Burkard, and A. Pályi, Spin-strain interaction in nitrogen-vacancy centers in diamond, Phys. Rev. B 98, 075201 (2018).
  * Meesala  _et al._ [2018] S. Meesala, Y.-i. Sohn, B. Pingault, L. Shao, H. A. Atikian, J. Holzgrafe, M. Gündo, C. Stavrakas, A. Sipahigil, C. Chia, R. Evans, M. J. Burek, M. Zhang, L. Wu, J. L. Pacheco, J. Abraham, E. Bielejec, M. D. Lukin, M. Atatüre, and M. Lon, Strain engineering of the silicon-vacancy center in diamond, Phys. Rev. B 97, 205444 (2018).
  * Maity  _et al._ [2020] S. Maity, L. Shao, S. Bogdanovi, S. Meesala, Y.-i. Sohn, N. Sinclair, B. Pingault, M. Chalupnik, C. Chia, L. Zheng, K. Lai, and M. Lon, Coherent acoustic control of a single silicon vacancy spin in diamond, Nat. Commun. 11, 193 (2020).
  * Walker [1957] L. R. Walker, Magnetostatic modes in ferromagnetic resonance, Phys. Rev. 105, 390 (1957).
  * Walker [1958] L. R. Walker, Resonant modes of ferromagnetic spheroids, J. Appl. Phys. 29, 318 (1958).
  * Fletcher and Bell [1959] P. C. Fletcher and R. O. Bell, Ferrimagnetic resonance modes in spheres, J. Appl. Phys. 30, 687 (1959).
  * Neuman  _et al._ [2020c] T. Neuman, D. S. Wang, and P. Narang, Nanomagnonic cavities for strong spin-magnon coupling, arXiv:2007.11595 (2020c).
  * Wu [2010] M. Wu, Nonlinear spin waves in magnetic film feedback rings (Academic Press, 2010) pp. 163 – 224.
  * Candido  _et al._ [2020] D. R. Candido, G. D. Fuchs, E. Johnston-Halperin, and M. Flatté, Predicted strong coupling of solid-state spins via a single magnon mode, Mat. Quantum Technol. 10.1088/2633-4356/ab9a55 (2020), arXiv:2003.04341 .
  * Narang  _et al._ [2019] P. Narang, C. J. Ciccarino, J. Flick, and D. Englund, Quantum Materials with Atomic Precision: Artificial Atoms in Solids: Ab Initio Design, Control, and Integration of Single Photon Emitters in Artificial Quantum Materials, Adv. Funct. Mater. 29, 1904557 (2019).
  * Bennett  _et al._ [2013] S. D. Bennett, N. Y. Yao, J. Otterbach, P. Zoller, P. Rabl, and M. D. Lukin, Phonon-induced spin-spin interactions in diamond nanostructures: Application to spin squeezing, Phys. Rev. Lett. 110, 156402 (2013).
  * Lemonde  _et al._ [2018] M.-A. Lemonde, S. Meesala, A. Sipahigil, M. J. A. Schuetz, M. D. Lukin, M. Lončar, and P. Rabl, Phonon networks with silicon-vacancy centers in diamond waveguides, Phys. Rev. Lett. 120, 213603 (2018).
  * Zhang  _et al._ [2017] H. Zhang, K. Arai, C. Belthangady, J.-C. Jaskula, and R. L. Walsworth, Selective addressing of solid-state spins at the nanoscale via magnetic resonance frequency encoding, npj Quantum Inf. 3, 31 (2017).
  * Jakobi  _et al._ [2017] I. Jakobi, P. Neumann, Y. Wang, D. B. R. Dasari, F. El Hallak, M. A. Bashir, M. Markham, A. Edmonds, D. Twitchen, and J. Wrachtrup, Measuring broadband magnetic fields on the nanoscale using a hybrid quantum register, Nat. Nanotechnol. 12, 67 (2017).
  * Bodenstedt  _et al._ [2018] S. Bodenstedt, I. Jakobi, J. Michl, I. Gerhardt, P. Neumann, and J. Wrachtrup, Nanoscale Spin Manipulation with Pulsed Magnetic Gradient Fields from a Hard Disc Drive Writer, Nano Lett. 18, 5389 (2018).
  * Giannozzi  _et al._ [2009] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. De Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov, P. Umari, and R. M. Wentzcovitch, QUANTUM ESPRESSO: A modular and open-source software project for quantum simulations of materials, J. Phys.: Condens. Matter 21, 395502 (2009).
  * Giannozzi  _et al._ [2017] P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. B. Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo, A. D. Corso, S. de Gironcoli, P. Delugas, R. A. DiStasio, A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, R. Gebauer, U. Gerstmann, F. Giustino, T. Gorni, J. Jia, M. Kawamura, H.-Y. Ko, A. Kokalj, E. Kucukbenli, M. Lazzeri, M. Marsili, N. Marzari, F. Mauri, N. L. Nguyen, H.-V. Nguyen, A. O. de-la Roza, L. Paulatto, S. Poncé, D. Rocca, R. Sabatini, B. Santra, M. Schlipf, A. P. Seitsonen, A. Smogunov, I. Timrov, T. Thonhauser, P. Umari, N. Vast, X. Wu, and S. Baroni, Advanced capabilities for materials modelling with Quantum ESPRESSO, J. Phys.: Condens. Matter 27, 465901 (2017).
  * Giannozzi  _et al._ [2020] P. Giannozzi, A. Ferretti, N. Marzari, P. Bonfà, I. Timrov, D. Brunato, R. Car, P. Delugas, S. D. Gironcoli, F. F. Ruffino, A. Urru, S. Baroni, P. Bonfà, and I. Carnimeo, Quantum ESPRESSO toward the exascale, J. Chem. Phys. 152, 154105 (2020).
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).