# Variational principles for nonlinear pde systems via duality

Amit Acharya Department of Civil & Environmental Engineering, and Center for
Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, PA 15213, email:
<EMAIL_ADDRESS>
###### Abstract

A formal methodology for developing variational principles corresponding to a
given nonlinear pde system is discussed. The scheme is demonstrated in the
context of the incompressible Navier-Stokes equations, systems of first-order
conservation laws, and systems of Hamilton-Jacobi equations.

##  1 Introduction

This work answers a question raised in [Ach22], namely, identifying the basic
ingredients necessary for developing a variational principle all, or some, of
whose Euler-Lagrange (E-L) equations are a given system of pde; the functional
to be developed is required to have space-time derivatives of its fundamental
fields in ‘more-than-linear’ combinations. Such a question arose from the
purely practical issue of developing a basis for application of Effective
Field Theory techniques in Physics (cf. [Kle89, BNW+17, BNWZ17]) to the system
of nonlinear dislocation dynamics [Ach22] in continuum mechanics and materials
science. Despite the success in formulating an appropriate action functional,
that effort also exposed a certain flexibility in the adopted scheme whose
details remained to be understood. Here, we are able to understand those
details and abstract out the essence of the technique. The idea is then
demonstrated on a wider setting of important classes of physical systems of
nonlinear pde. We note here that the question of finding a variational
principle(s) (some of) whose E-L equations are a given system of pde is
different from the one adopted in the ‘Least-Squares Method,’ (cf. [AKM19, Ch.
10]), as explained in [SW68, Sec. 1] and [Vis14]; the E-L equations of the
Least Squares functional are not necessarily the pde system from which the
Least-Squares functional is developed. With the null minimizer requirement,
minimizers of the Least Squares functional are solutions of the pde system
involved. In the approach adopted herein, a family of functionals is developed
which satisfy the stated requirement. Mathematically rigorous considerations
of the Least-Squares approach rely strongly on convex duality - the present
approach relies on elements of convex duality even at a formal level.

This paper is organized as follows: in the following paragraph of this section
we motivate the main idea of this work through an easy computation. In Secs.
2-5 we still use the ‘ad-hoc’ procedure adopted in [Ach22] to demonstrate our
ideas in the context of the incompressible Navier-Stokes (N-S) equations for a
homogeneous fluid. Related work can be found in [Ker99, Liu07] which review
earlier work in the physics and mechanics literature, including that of
mathematicians C. Doering and P. Constantin; a sampling of the mathematical
literature on the matter can be found in [GM09, OSS18] and references provided
therein (an important disclaimer here is that I am in no way an expert in the
theory or practice of the N-S equations). In Sec. 6, the scheme is generalized
to understand the fundamental restrictions, at least at a formal level,
required to make it work, exploiting its full level of flexibility; this is
demonstrated in the context of a first-order system of conservation laws in
divergence form and a system of Hamilton-Jacobi equations. Some closing
observations are recorded in Sec. 7. A word on notation: the Einstein
summation convention for repeated indices is always invoked; $\Omega$ will
represent a fixed domain in ambient, 3-d Euclidean space, and $[0,T]$ an
interval of time.

Before moving on to the physical models we wish to discuss, we discuss the
‘toy’ model of the heat equation in one space dimension and time to give the
pattern of the typical computations that are involved in each case that
follows, only nonlinear and in more space dimensions. Consider the heat
equation

$\partial_{t}\theta=\partial_{x}\left(k\partial_{x}\theta\right)$

and the functional

$\displaystyle\widehat{S}[\theta,\lambda]$
$\displaystyle:=\int_{I\times[0,T]}dtdx\
\lambda(\partial_{t}\theta-\partial_{x}(k\partial_{x}\theta))+H(\theta)$
$\displaystyle\ =\int_{I\times[0,T]}dtdx\
\theta(-\partial_{t}\lambda-\partial_{x}(k\partial_{x}\lambda))+H(\theta),$

for any convex function $H$ so that $p=H^{\prime}(\theta)$ is uniquely
solvable for $\theta(p)$, and assuming that $\lambda$ vanishes on the boundary
of the interval $I\subset\mathbb{R}$ (representing the spatial domain). The
goal is to propose an ‘action’ functional whose E-L equation is the original
(system of) pde in question, here the heat equation. Here, we have imposed the
pde (system) with a Lagrange multiplier field (to generate a scalar), exposed
linear terms in the basic primal field(s) (here $\theta$) and added a convex
term in the basic field, the latter two actions in anticipation of performing
a Legendre transform. We note that the Lagrange multiplier field necessarily
enters in a ‘linear’ manner in the functional, even when the pde is nonlinear
(when the pde contains nonlinear terms - as in the models considered later -
all such nonlinear terms are combined additively with the function $H$ to
define a function $M$). Motivated by the structure of $\widehat{S}$, now
define

$\displaystyle M^{*}(p)$ $\displaystyle:=p\,\theta(p)-H(\theta(p))$
$\displaystyle p$
$\displaystyle:=-(-\partial_{t}\lambda-\partial_{x}(k\partial_{x}\lambda)).$

With the above definitions, define the functional on a reduced state space

$S[\lambda]=\int_{I\times[0,T]}dtdx\ -M^{*}(p)$

whose first variation is given by

$\delta S=\int_{I\times[0,T]}dtdx\ -\partial_{p}M^{*}(p)\delta p.$

Noting that $\partial_{p}M^{*}(p)=\theta(p)$,

$\delta S=\int_{I\times[0,T]}dtdx\
\theta(p)\left(-\partial_{t}\delta\lambda-\partial_{x}\left(k\partial_{x}\delta\lambda\right)\right)$

yielding the E-L equation

$\partial_{t}(\theta(p))-\partial_{x}\left(k\partial_{x}(\theta(p))\right)=0.$

With $H(\theta)=\frac{1}{2}\theta^{2}$, $M^{*}(p)=\frac{1}{2}p^{2}$ and
$\theta(p)=p$, and the functional $S[\lambda]$ is bounded above. The Euler-
Lagrange equation in terms of the function $\lambda$ is
$\partial^{2}_{t}\lambda-k^{2}\partial^{4}_{x}\lambda=0$ which has the wrong
sign when viewed as an initial value problem in time $t$ with dispersion
relationship $\omega^{2}=-k^{2}m^{4}$ and eigenmode with wave number $m$
associated with growth factors $e^{\pm|k|m^{2}t}$. However, the functional can
be maximized or its critical points approached by a gradient ascent in a ‘fake
time’ variable, say $s$, given by $\partial_{s}\lambda=\frac{\delta
S}{\delta\lambda}$, and it can be expected that the decaying solution is
automatically picked up by such methods. For $k>0$, it can be checked that
initial and boundary conditions on $\theta$ can be translated, non-uniquely,
to constraints on the field $\lambda$ at the domain space-time boundaries. One
also would seem to have the formal guarantee that any solution in the dual
variable $\lambda$ that respects the boundary and initial constraints obtained
from the primal problem must generate the unique solution to the primal
problem through the mapping
$\theta(x,t)=p(x,t)=\partial_{t}\lambda(x,t)+k\,\partial^{2}_{x}\lambda(x,t)$
(the mapping varies, of course, depending on the choice of the convex function
$H$, but this conclusion remains unaltered).

We note that it is not our intent with the remarks above to suggest that the
heat equation be solved in this manner, but only to motivate how the
variational principles we develop henceforth may be ‘solved’ or approximated.

After completing this work, we became aware of the work of Brenier[Bre18].
There are strong connections of the ideas presented in this paper with those
in [Bre18] \- for instance, the functional (1) with $\hat{\nu}=0$ and
$G(x)=\frac{1}{2}dx^{2},d\to 0$ so that $G^{*}(\xi)=\frac{1}{2d}\xi^{2}$
formally recovers Brenier’s variational principle for the Euler equation (up
to accounting for the initial condition). Brenier does rigorous analysis to
show existence of generalized solutions to his maximization principle which
may be considered as encouraging for the ideas presented herein (the
mathematical analysis in [Bre18] is beyond the scope of this paper).

##  2 A ‘dual’ action for the incompressible Navier-Stokes equation

For the fields

$\displaystyle\gamma$ $\displaystyle:\Omega\times[0,T]\to\mathbb{R}$
$\displaystyle\lambda$ $\displaystyle:\Omega\times[0,T]\to\mathbb{R}^{3}$

consider the functional

$\displaystyle
S_{d}[\gamma,\lambda]=\int_{\Omega\times[0,T]}dtd^{3}x-\frac{1}{2}p_{i}\mathbb{K}_{ij}p_{j}-G^{*}(\xi),$
(1)

where

$\displaystyle p_{k}$
$\displaystyle:=-[\hat{\nu}(\partial_{j}\partial_{j}\lambda_{k}+\partial_{k}\partial_{i}\lambda_{i})-\partial_{k}\gamma+\partial_{t}\lambda_{k}]$
(2) $\displaystyle\xi$ $\displaystyle:=-\partial_{i}\lambda_{i}$
$\displaystyle\mathbb{L}_{ij}(\nabla\lambda)$
$\displaystyle:=c\,\delta_{ij}+\partial_{j}\lambda_{i}+\partial_{i}\lambda_{j};\qquad\qquad\mathbb{K}_{ij}(\nabla\lambda):=(\mathbb{L}(\nabla\lambda))^{-1}_{ij};\qquad\qquad\mathbb{K}_{ij}=\mathbb{K}_{ji}$
$\displaystyle\omega(\xi)$ $\displaystyle:=G^{\prime}{}^{-1}(\xi)$
$\displaystyle G^{*}(\xi)$ $\displaystyle:=\omega(\xi)\xi-G(\omega(\xi))$

for $c>0$ an arbitrary, non-dimensional constant, $G:\mathbb{R}\to\mathbb{R}$
an arbitrary convex (smooth) function, and
$G^{\prime}:\mathbb{R}\to\mathbb{R}$ refers to the derivative of the function
$G$ whose inverse function exists. We make the assumption here that
$\mathbb{L}$ is an invertible matrix for all possible values of its argument
(presumably this can be arranged by taking a sufficiently large value of the
constant $c$). For $\nu>0$ the shear viscosity and $\rho_{0}>0$ the constant
density of the homogeneous, incompressible fluid,
$\hat{\nu}:=\frac{\nu}{\rho_{0}}$.

In what follows, we will not always explicitly write the arguments of
$\mathbb{L},\mathbb{K},\omega$, and $G^{*}$.

The first variation of $S_{d}$, assuming all variations vanish on the boundary
of $\Omega\times[0,T]$111Here, we are interested in interior field equations;
natural ‘boundary’ conditions can be inferred by not assuming the variations
to vanish on the boundary of the space-time domain and adding applied initial
and natural spatial boundary conditions in standard fashion. An added term,
canceling the contribution at time $T$ arising from integration by parts, in
time would also be required. is given by

$\displaystyle\delta S_{d}=\int_{\Omega\times[0,T]}dtd^{3}x$ $\displaystyle-
p_{i}\mathbb{K}_{ik}[-(\hat{\nu}(\partial_{j}\partial_{j}\delta\lambda_{k}+\partial_{k}\partial_{i}\delta\lambda_{i})-\partial_{k}\delta\gamma+\partial_{t}\delta\lambda_{k})]$
(3) $\displaystyle-G^{*}{}^{\prime}[-\partial_{i}\delta\lambda_{i}]$
$\displaystyle-p_{i}p_{k}\,\delta\mathbb{K}_{ik},$

and noting that

$\delta\mathbb{K}_{ik}=-\mathbb{K}_{ij}\delta\mathbb{L}_{jm}\mathbb{K}_{mk}=-\mathbb{K}_{ij}\mathbb{K}_{mk}[\partial_{j}\delta\lambda_{m}+\partial_{m}\delta\lambda_{j}]$

along with the definition

$v_{k}(p,\nabla\lambda):=\mathbb{K}_{ik}(\nabla\lambda)p_{i}$ (4)

we have

$\displaystyle\delta S_{d}=\int_{\Omega\times[0,T]}dtd^{3}x$
$\displaystyle\hat{\nu}(\delta\lambda_{k}\partial_{j}\partial_{j}v_{k}+\delta\lambda_{i}\partial_{i}\partial_{j}v_{j})$
$\displaystyle+(\partial_{k}v_{k})\delta\gamma-(\partial_{t}v_{k})\delta\lambda_{k}$
$\displaystyle-\frac{1}{2}[\partial_{j}(v_{j}v_{k})+\partial_{m}(v_{k}v_{m})]\delta\lambda_{k}$
$\displaystyle-\partial_{i}(G^{*}{}^{\prime})\delta\lambda_{i}.$

Noting that

$G^{*}(\xi)=\omega(\xi)\xi-G(\omega(\xi))\ \ \mbox{and}\ \
G^{\prime}(\omega(\xi))=\xi\Longrightarrow G^{*}{}^{\prime}(\xi)=\omega(\xi)$

and defining

$\mathcal{P}:=\rho_{0}\,\omega$ (5)

the following Euler-Lagrange equations are obtained:

$\displaystyle\delta\lambda_{k}:$
$\displaystyle-\partial_{t}v_{k}-\partial_{j}(v_{k}v_{j})+\partial_{j}\left(\hat{\nu}(\partial_{j}v_{k}+\partial_{k}v_{j})-\frac{\mathcal{P}}{\rho_{0}}\delta_{kj}\right)=0$
(6) $\displaystyle\delta\gamma:$ $\displaystyle\partial_{k}v_{k}=0.$

These are the Navier-Stokes system for an incompressible, homogeneous fluid if
the fields $v,\mathcal{P}$ (that are defined in terms of the fields $\lambda$
and $\gamma$) are interpreted as the velocity and the pressure fields,
respectively.

##  3 The primal action for (1) and its reduced state space

We motivate how the functional (1) was arrived at. With all definitions and
notation of the previous section enforced and in terms of the fields

$\displaystyle v:\Omega\times[0,T]$ $\displaystyle\to\mathbb{R}^{3}$
$\displaystyle\gamma:\Omega\times[0,T]$ $\displaystyle\to\mathbb{R}$
$\displaystyle\lambda:\Omega\times[0,T]$ $\displaystyle\to\mathbb{R}^{3}$
$\displaystyle\omega:\Omega\times[0,T]$ $\displaystyle\to\mathbb{R}$

consider the functional

$\displaystyle\widehat{S}_{d}[v,\gamma,\lambda,\omega]=\int_{\Omega\times[0,T]}dtd^{3}x$
$\displaystyle c\frac{1}{2}v_{i}v_{i}+G(\omega)$
$\displaystyle+\gamma(\partial_{i}v_{i})$
$\displaystyle+\lambda_{i}(\partial_{j}(-\omega\delta_{ij}+\hat{\nu}(\partial_{j}v_{i}+\partial_{i}v_{j}))-\partial_{t}v_{i}-\partial_{j}(v_{i}v_{j})),$

where the incompressible Navier-Stokes equations have been enforced via
Lagrange multipliers.

Each term in the action density has physical dimensions of
$\frac{Length^{2}}{Time^{2}}$; multiplying through by $\rho_{0}$ gives units
of energy per unit volume.

The main idea is to invoke a Legendre transform based change of variables and
then consider the variational principle in a reduced state space. Assuming for
the moment that the Lagrange multiplier fields vanish on the boundary, we have

$\displaystyle\widehat{S}_{d}[v,\gamma,\lambda,\omega]=\int_{\Omega\times[0,T]}dtd^{3}x$
$\displaystyle
c\frac{1}{2}v_{i}v_{i}+v_{i}v_{j}\partial_{j}\lambda_{i}+G(\omega)$
$\displaystyle+(\hat{\nu}(\partial_{j}\partial_{j}\lambda_{i}+\partial_{i}\partial_{k}\lambda_{k})-\partial_{i}\gamma+\partial_{t}\lambda_{i})v_{i}$
$\displaystyle+\omega\partial_{i}\lambda_{i}.$

Working with the definitions (2) and (4), we affect a reduction in the state
space of $\widehat{S}$ to define

$\displaystyle S_{d}[\gamma,\lambda]=\int_{\Omega\times[0,T]}dtd^{3}x$
$\displaystyle\frac{1}{2}v_{i}(p,\nabla\lambda)\mathbb{L}_{ij}(\nabla\lambda)v_{j}(p,\nabla\lambda)-p_{i}v_{i}(p,\nabla\lambda)+G(\omega(\xi))-\omega(\xi)\xi.$
(7)

Using the definitions (2) and (4) once again and noting that

$v_{i}p_{i}-\frac{1}{2}v_{i}\mathbb{L}_{ij}v_{j}=\frac{1}{2}p_{i}\mathbb{K}_{ik}p_{k},$
(8)

we note that (7) is the functional (1).

##  4 A ‘mixed’ action for the incompressible Navier-Stokes equation

For the fields

$\displaystyle A$ $\displaystyle:\Omega\times[0,T]\to\mathbb{R}^{3\times
3}_{sym}$ $\displaystyle\gamma$ $\displaystyle:\Omega\times[0,T]\to\mathbb{R}$
$\displaystyle\lambda$ $\displaystyle:\Omega\times[0,T]\to\mathbb{R}^{3}$
$\displaystyle\omega$ $\displaystyle:\Omega\times[0,T]\to\mathbb{R}$

consider the functional

$\displaystyle
S_{m}[A,\gamma,\lambda,\omega]=\int_{\Omega\times[0,T]}dtd^{3}x-\frac{1}{2}p_{i}\mathbb{K}_{ij}p_{j}\mp
R^{*}(\tau)+\omega\partial_{i}\lambda_{i},$ (9)

where

$\displaystyle p_{k}$
$\displaystyle:=-[-\partial_{j}A_{kj}-\partial_{k}\gamma+\partial_{t}\lambda_{k}]$
(10) $\displaystyle\tau_{kl}$
$\displaystyle:=\mp[-A_{kl}-\hat{\nu}(\partial_{l}\lambda_{k}+\partial_{k}\lambda_{l})]$
$\displaystyle\mathbb{L}_{ij}(\nabla\lambda)$
$\displaystyle:=c\,\delta_{ij}+\partial_{j}\lambda_{i}+\partial_{i}\lambda_{j};\qquad\qquad\mathbb{K}_{ij}(\nabla\lambda):=(\mathbb{L}(\nabla\lambda))^{-1}_{ij};\qquad\qquad\mathbb{K}_{ij}=\mathbb{K}_{ji}$
$\displaystyle D_{ij}(\tau)$
$\displaystyle:=\left(R^{\prime-1}\right)_{ij}(\tau)$ $\displaystyle
R^{*}(\tau)$ $\displaystyle:=D_{ij}(\tau)\tau_{ij}-R(D(\tau))$

for $R:\mathbb{R}^{3\times 3}_{sym}\to\mathbb{R}$ an arbitrary convex function
on the space of symmetric tensors, and $R^{\prime}_{ij}:\mathbb{R}^{3\times
3}_{sym}\to\mathbb{R}^{3\times 3}_{sym}$ refers to the function
$\partial_{D_{ij}}R$ and $R^{\prime-1}_{ij}$ to its inverse function.

The matrix field $\mathbb{L}$ and the constant $\hat{\nu}$ are defined exactly
as in Sec. 2.

The first variation of $S_{m}$, assuming all variations vanish on the boundary
of $\Omega\times[0,T]$ (with the same understanding for what needs to be done
to include natural boundary and initial conditions) is given by

$\displaystyle\delta S_{m}=\int_{\Omega\times[0,T]}dtd^{3}x$ $\displaystyle-
p_{i}\mathbb{K}_{ik}(\nabla\lambda)[-(-\partial_{j}\delta
A_{kj}-\partial_{k}\delta\gamma+\partial_{t}\delta\lambda_{k})]$ (11)
$\displaystyle\mp R^{*}{}^{\prime}_{kl}[\mp(-\delta
A_{kl}-2\hat{\nu}\partial_{l}\delta\lambda_{k})]$
$\displaystyle+\delta\omega\partial_{i}\lambda_{i}-(\partial_{i}\omega)\delta\lambda_{i}-\frac{1}{2}p_{i}p_{k}\delta\mathbb{K}_{ik},$

and noting that

$\delta\mathbb{K}_{ik}=-\mathbb{K}_{ij}\delta\mathbb{L}_{jm}\mathbb{K}_{mk}$

we have

$\displaystyle\delta S_{m}=\int_{\Omega\times[0,T]}dtd^{3}x$
$\displaystyle\partial_{j}(p_{i}\mathbb{K}_{ik}(\nabla\lambda))\delta
A_{kj}+\partial_{k}(p_{i}\mathbb{K}_{ik}(\nabla\lambda))\delta\gamma-\partial_{t}(p_{i}\mathbb{K}_{ik}(\nabla\lambda))\delta\lambda_{k}$
$\displaystyle-R^{*}_{kl}\delta
A_{kl}+\partial_{l}(2\hat{\nu}R^{*}{}^{\prime}_{kl})\delta\lambda_{k}+(\partial_{i}\lambda_{i})\delta\omega$
$\displaystyle-\frac{1}{2}[\delta\lambda_{m}\partial_{j}(\mathbb{K}_{ij}(\nabla\lambda)p_{i}\mathbb{K}_{mk}(\nabla\lambda)p_{k})+\delta\lambda_{j}\partial_{m}(\mathbb{K}_{ij}(\nabla\lambda)p_{i}\mathbb{K}_{mk}(\nabla\lambda)p_{k})],$

leading to the Euler-Lagrange equations

$\displaystyle\delta A_{ik}:$
$\displaystyle\qquad\frac{1}{2}[\partial_{j}(p_{i}\mathbb{K}_{ik}(\nabla\lambda))+\partial_{k}(p_{i}\mathbb{K}_{ij}(\nabla\lambda))]-R^{*}{}^{\prime}_{kj}(\tau)=0$
(12) $\displaystyle\delta\gamma:$
$\displaystyle\qquad\partial_{k}(p_{i}\mathbb{K}_{ik}(\nabla\lambda))=0$
$\displaystyle\delta\lambda_{k}:$
$\displaystyle\qquad-\partial_{t}(p_{i}\mathbb{K}_{ik}(\nabla\lambda))+\partial_{l}(2\hat{\nu}R^{*}{}^{\prime}_{kl}(\tau))-\partial_{k}\omega$
$\displaystyle\qquad-\frac{1}{2}[\partial_{j}(\mathbb{K}_{ij}(\nabla\lambda)p_{i}\mathbb{K}_{km}(\nabla\lambda)p_{m})+\partial_{m}(\mathbb{K}_{ik}(\nabla\lambda)p_{i}\mathbb{K}_{mr}(\nabla\lambda)p_{r})]=0$
$\displaystyle\delta\omega:$ $\displaystyle\qquad\partial_{i}\lambda_{i}=0.$

Defining a velocity and a pressure field as

$v_{k}:=\mathbb{K}_{ik}p_{i};\qquad\qquad\mathcal{P}:=\rho_{0}\,\omega$ (13)

we note that the first three equations of (12) imply the equations

$\displaystyle\partial_{k}v_{k}$ $\displaystyle=0$ (14)
$\displaystyle-\partial_{t}v_{k}-\partial_{j}(v_{k}v_{j})+\partial_{j}\left[\frac{\nu}{\rho_{0}}(\partial_{j}v_{k}+\partial_{k}v_{j})-\frac{\mathcal{P}}{\rho_{0}}\delta_{kj}\right]$
$\displaystyle=0,$

which is the Navier-Stokes system for a homogeneous, incompressible fluid.

##  5 The primal action for (9) and its reduced state space

We motivate how the functional (9) was arrived at. With all definitions and
notation of previous sections enforced and in terms of the fields

$\displaystyle v:\Omega\times[0,T]$ $\displaystyle\to\mathbb{R}^{3}$
$\displaystyle D:\Omega\times[0,T]$ $\displaystyle\to\mathbb{R}^{3\times
3}_{sym}$

consider the functional

$\displaystyle\widehat{S}_{m}[v,D,\omega,A,\gamma,\lambda]=\int_{\Omega\times[0,T]}dtd^{3}x$
$\displaystyle c\frac{1}{2}v_{i}v_{i}\pm R(D)$
$\displaystyle+A_{ij}(\partial_{j}v_{i}-D_{ij})$
$\displaystyle+\gamma(\partial_{i}v_{i})$
$\displaystyle+\lambda_{i}(\partial_{j}(-\omega\delta_{ij}+2\hat{\nu}D_{ij})-\partial_{t}v_{i}-\partial_{j}(v_{i}v_{j})),$

where the incompressible Navier-Stokes equations have been enforced via
Lagrange multipliers.

As before, we invoke a Legendre transform based change of variables and then
consider the variational principle in a reduced state space. Assuming for the
moment that the Lagrange multiplier fields vanish on the boundary, we have

$\displaystyle\widehat{S}_{m}[v,D,\omega,A,\gamma,\lambda]=\int_{\Omega\times[0,T]}dtd^{3}x$
$\displaystyle c\frac{1}{2}v_{i}v_{i}\pm R(D)$
$\displaystyle+(-\partial_{j}A_{ij}-\partial_{i}\gamma+\partial_{t}\lambda_{i})v_{i}$
$\displaystyle+(-A_{kl}-\hat{\nu}(\partial_{l}\lambda_{k}+\partial_{k}\lambda_{l}))D_{kl}$
$\displaystyle+\omega\partial_{i}\lambda_{i}+v_{i}v_{j}\partial_{j}\lambda_{i}.$

Working with the definitions (10) and (13), we affect a reduction in the state
space of $\widehat{S}_{m}$ to define

$\displaystyle
S_{m}[A,\gamma,\lambda,\omega]=\int_{\Omega\times[0,T]}dtd^{3}x$
$\displaystyle\frac{1}{2}v_{i}(p,\nabla\lambda)\mathbb{L}_{ij}(\nabla\lambda)v_{j}(p,\nabla\lambda)-p_{i}v_{i}(p,\nabla\lambda)$
(15) $\displaystyle\pm
R(D(\tau))\mp\tau_{kl}D_{kl}(\tau)+\omega\partial_{i}\lambda_{i}.$

Using the definitions (10) and (13) once again and noting (8), we note that
(15) is the functional (9).

##  6 Generalizations

In the previous sections and in [Ach22] the kinetic energy was chosen as the
added potential associated with the velocity field in the primal actions. It
is however clear that apart from a solvability condition, this potential
should be amenable to a (more or less) arbitrary choice. In this section we
demonstrate this feature of the proposed scheme through a discussion of
important classes of pde.

In the following subsections, we will repeatedly make use of an assumption and
its consequence, which we write out in detail before proceeding. It is
essentially related to a Legendre transform in the presence of a parameter.
For $M:\mathbb{R}^{n}\times\mathbb{R}^{m}\to\mathbb{R}$,
$H:\mathbb{R}^{n}\to\mathbb{R}$, $L\in\mathbb{R}^{m}$, and
$F:\mathbb{R}^{n}\to\mathbb{R}^{m}$ satisfying

$M(U,L)=H(U)-L\cdot F(U),$ (16)

we assume that for given $P$ and $L$, there exists a unique function $U(P,L)$
which satisfies the relation

$P=\partial_{U}M(U(P,L),L).$ (17)

for all likely values of $L$ and $P$ to be encountered, i.e., the algebraic
system of equations $P=\partial_{U}M(U,L)$ is uniquely solvable for $U$ in
terms of $P,L$. If $M$ were to be convex in $U$ for all $L$, then such a
condition would certainly hold. In the following, the function $F$ will be
specified from the physical pde system to be solved and the function $H$ will
be free to choose, so this condition is essentially a constraint on the choice
of the class of functions to which $H$ belongs.

Assuming the above, consider

$\displaystyle M^{*}(P,L)$ $\displaystyle:=U(P,L)\cdot P-M(U(P,L),L)$ (18)
$\displaystyle\Longrightarrow\partial_{L}M^{*}(P,L)$
$\displaystyle=\partial_{L}U_{i}(P,L)P_{i}-\partial_{U_{i}}M(U(P,L),L)\partial_{L}U_{i}(P,L)-\partial_{L}M(U(P,L),L)$
$\displaystyle=-\partial_{L}M(U(P,L),L),$

due to (17). Given the form of (16), we have

$\partial_{L}M^{*}(P,L)=F(U(P,L)).$ (19)

We also have that

$\partial_{P}M^{*}(P,L)=\partial_{P}U_{i}(P,L)P_{i}+U(P,L)-\partial_{U_{i}}M(U(P,L),L)\partial_{P}U_{i}(P,L)=U(P,L)$
(20)

by (17).

###  6.1 Dual variational principles for first-order systems of conservation
laws in divergence form

The pde system of interest is of the form

$\partial_{t}u_{I}+\partial_{i}f_{Ii}(u)=0,$ (21)

where $I=1,\ldots,n$, and $i=1,\ldots,d$, $d$ being the number of space
dimensions.

In terms of the field

$\displaystyle\lambda$ $\displaystyle:\Omega\times[0,T]\to\mathbb{R}^{n}$

consider the functional

$S_{cl}[p,\lambda]=\int_{\Omega\times[0,T]}dtd^{3}x\,-M^{*}(p,\nabla\lambda),$
(22)

where $(\nabla\lambda)_{Ij}=\partial_{j}\lambda_{I}$, along with the
identifications $u=U$, $p=P$, $\nabla\lambda=L$, $m=nd$, $F=f$ in (16)-(17)
and the definitions

$\displaystyle p_{I}$ $\displaystyle:=\partial_{t}\lambda_{I}$ (23)
$\displaystyle M(u(p,\nabla\lambda),\nabla\lambda)$
$\displaystyle:=H(u(p,\nabla\lambda))-\nabla\lambda\cdot
f(u(p,\nabla\lambda))$ $\displaystyle M^{*}(p,\nabla\lambda)$
$\displaystyle:=u(p,\nabla\lambda)\cdot p-M(u(p,\nabla\lambda),\nabla\lambda)$

for any choice of the function $H$ that allows (17) to hold. Then the first
variation is given by

$\delta
S_{cl}=\int_{\Omega\times[0,T]}dtd^{3}x\,-\left(\partial_{p}M^{*}\cdot\partial_{t}\delta\lambda+\partial_{\,\nabla\lambda}M^{*}\cdot\nabla\delta\lambda\right)$

which leads to the E-L equations

$\partial_{t}u_{I}(p,\nabla\lambda)+\partial_{i}f_{Ii}(u(p,\nabla\lambda))=0$

by (19)-(20). Thus, every solution of the E-L equations of (22) defines a
solution of (21) through the definition of the function $U(P,L)$ related to
(17).

We also note that for given $f$, each member of the entire class of functions
$M^{*}$, defined through a choice of an admissible state function $H$,
satisfies the conservation law (21) in the sense

$\partial_{t}\left(\partial_{p}M^{*}\right)_{I}+\partial_{i}\left(\partial_{\,\nabla\lambda}M^{*}\right)_{Ii}=0.$
(24)

###  6.2 Dual variational principle for a second order system of Hamilton-
Jacobi equations

Consider the system of Hamilton-Jacobi equations

$\displaystyle\partial_{t}u_{I}$ $\displaystyle=f_{I}\left(u,B,C\right)$ (25)
$\displaystyle\partial_{i}u_{I}$ $\displaystyle=B_{Ii}$
$\displaystyle\partial_{i}\partial_{j}u_{I}$ $\displaystyle=C_{Iji}$

where $f_{I}$ is a smooth function of its arguments.

In terms of the fields

$\displaystyle\lambda$ $\displaystyle:\Omega\times[0,T]\to\mathbb{R}^{n}$
$\displaystyle\gamma$ $\displaystyle:\Omega\times[0,T]\to\mathbb{R}^{n\times
d}$ $\displaystyle\rho$ $\displaystyle:\Omega\times[0,T]\to\mathbb{R}^{n\times
d\times d}$

and the definitions

$\displaystyle P$
$\displaystyle:=\left(\partial_{t}\lambda+\nabla\cdot\gamma-\nabla^{2}:\rho,\gamma,\rho\right)$
(26) $\displaystyle L$ $\displaystyle:=\lambda$ $\displaystyle F$
$\displaystyle:=f,$

where $(\nabla\cdot\gamma)_{I}=\partial_{i}\gamma_{Ii}$ and
$(\nabla^{2}:\rho)_{I}=\partial_{j}\partial_{i}\rho_{Iij}$, we consider
functions $H:\mathbb{R}^{n}\times\mathbb{R}^{n\times
d}\times\mathbb{R}^{n\times d\times d}\to\mathbb{R}$ such that, for the
generic element in its domain referred to as

$U:=(u,B,C),$

the function $M$ in (16) is defined with the solvability property (17), and in
terms of it, the function $M^{*}$ in (18).

Consider the functional

$S_{HJ}[\lambda,\gamma,\rho]=\int_{\Omega\times[0,T]}dtd^{3}x-M^{*}(P,L),$
(27)

whose E-L equations are (assuming $(\lambda,\gamma,\rho)$ have compact support
on $\Omega$)

$\displaystyle-\partial_{t}u_{I}(P,L)+f_{I}(U(P,L))$ $\displaystyle=0$
$\displaystyle-\partial_{i}u_{I}(P,L)+B_{Ii}(P,L)=0$
$\displaystyle-\partial_{i}\partial_{j}u_{I}(P,L)+C_{Iji}(P,L)=0.$

We note that the static system $\partial_{i}f_{Ii}(B)=0$,
$B_{Ii}=\partial_{i}u_{I}$ can be dealt with as an Hamilton-Jacobi system as
well as by taking account of its conservation structure by the proposed
technique, with $L=\nabla\lambda$ in the latter case.

##  7 Concluding remarks

The proposed scheme for generating variational principles for nonlinear pde
systems may be abstracted and summarized as follows : We first pose the given
system of pde as a first-order system (introducing extra fields representing
(higher-order) space and time derivatives of the fields of the given system);
as before let us denote this collection of fields by $U$. ‘Multiplying’ by
dual Lagrange multiplier fields, denoted by $D$, adding a function $H(U)$,
solely in the variables $U$ (the purpose of which, and associated
requirements, will be clear shortly), and integrating by parts over the space-
time domain, we form a ‘mixed’ functional in the primal and dual fields given
by

$\widehat{S}_{H}[U,D]=\int_{[0,T]\times\Omega}dtd^{3}x\
\mathcal{L}_{H}(\mathcal{D},U)$

where $\mathcal{D}$ is a collection of local objects in $D$ and at most its
first order derivatives. We then require that the family of allowable
functions $H$ be such that it allows the definition of a function
$U_{H}(\mathcal{D})$ such that

$\frac{\partial\mathcal{L}_{H}}{\partial U}(\mathcal{D},U_{H}(\mathcal{D}))=0$

so that the dual functional, defined solely on the space of the dual fields
$D$, given by

$S_{H}[D]=\int_{[0,T]\times\Omega}dtd^{3}x\
\mathcal{L}_{H}(\mathcal{D},U_{H}(\mathcal{D}))$

has the first variation

$\delta S_{H}=\int_{[0,T]\times\Omega}dtd^{3}x\
\frac{\partial\mathcal{L}_{H}}{\partial\mathcal{D}}\delta\mathcal{D}.$

By the process of formation of the functional $\widehat{S}_{H}$, it can then
be seen that the (formal) E-L equations arising from $\delta S_{H}$ have to be
the original first-order primal system, with $U$ substituted by
$U_{H}(\mathcal{D})$, regardless of the $H$ employed.

Furthermore, assume the Lagrangian $\mathcal{L}_{H}$ can be expressed in the
form

$\mathcal{L}_{H}(\mathcal{D},U):=-P(\mathcal{D})\cdot U+f(U,D)+H(U)$

for some function $P$ defined by the structure of the primal first-order
system ((linear terms in) first derivatives of $U$ after multiplication by the
dual fields and integration by parts always produce such terms), and for some
function $f$ which, when non-zero, does not contain any linear dependence in
$U$. Our scheme requires the existence of a function $U_{H}$ defined from
‘solving $\frac{\partial\mathcal{L}}{\partial U}(\mathcal{D},U)=0$ for $U$,’
i.e. $\exists\ U_{H}(P(\mathcal{D}),\mathcal{D})$ s.t. the equation

$-P(\mathcal{D})+\frac{\partial f}{\partial
U}(U_{H}(P(\mathcal{D}),\mathcal{D}),\mathcal{D})+\frac{\partial H}{\partial
U}\left(U_{H}(P(\mathcal{D}),\mathcal{D})\right)=0$

is satisfied. This requirement may be understood as follows: define

$f(U,\mathcal{D})+H(U)=:M(U,\mathcal{D})$

and assume that it is possible, through the choice of $H$, to make the
function $\frac{\partial M}{\partial U}(U,\mathcal{D})$ monotone in $U$ so
that a function $U_{H}(P,\mathcal{D})$ can be defined that satisfies

$\frac{\partial M}{\partial
U}(U_{H}(P,\mathcal{D}),\mathcal{D})=P,\quad\forall P.$

Then the Lagrangian is

$\mathcal{L}(\mathcal{D},U_{H}(P(\mathcal{D}),\mathcal{D}))=-P(\mathcal{D})\cdot
U_{H}(P(\mathcal{D}),\mathcal{D})+M(U_{H}(P(\mathcal{D}),\mathcal{D}),\mathcal{D})=:-M^{*}(P(\mathcal{D}),\mathcal{D})$

where $M^{*}(P,\mathcal{D})$ is the Legendre transform of the function $M$
w.r.t $U$, with $\mathcal{D}$ considered as a parameter.

We conclude with a few observations and directions for future work:

  1. 1.

Setting $\hat{\nu}=0$ in the functionals (1) and (9) yields stationary
principles for the incompressible Euler equations.

  2. 2.

The mixed variational principle for the incompressible N-S equations in Secs.
4-5 affords the addition of added viscosity through the function $R$ in the
solution of the dual problem - it is an interesting question whether this
feature of the problem can help in the analysis and solution of problems of
turbulence.

  3. 3.

The pointwise invertibility of the matrix field $\mathbb{L}$ in Secs. 2-5
appears to be a key issue in the formalism. It would be interesting to
understand the effect of the condition $\det(\mathbb{L})=0$ on the proposed
scheme and whether there can be a relation between the Reynolds number and the
value of $c$ for optimal performance of the scheme.

In the general setting, this condition translates to the validity of the
assumption (17).

  4. 4.

Constrained by appropriate boundary and initial conditions, when the pde
system has unique solutions, it is clear that any choice of the function $H$
within the admissible class, coupled with the mapping from the dual fields to
the primal fields, must lead to the unique solution. In the absence of
uniqueness, it is an interesting question whether specific choices of $H$
(along with the dual-to-primal mapping) act as a selection mechanism for
picking up particular solutions. For instance, an appropriate quadratic choice
of $H$ for the ‘simple’ equation $\partial_{x}f(B)=0,B=\partial_{x}u$ where
$f$ is nonconvex results in an essentially semilinear second-order dual
problem that is definitely simpler than the quasilinear primal problem. The
structure of the dual problem in this case does not make the expectation of a
smooth solution an absurd one. If this is indeed borne out in reality, then
the solution to the primal problem that is defined through the dual-to-primal
mapping may also be expected to be smooth.

  5. 5.

A detailed study of how boundary and initial conditions of the primal problem
can be transferred to the same for the dual problem is warranted, most
importantly for practical purposes of generating numerical approximations to
the proposed formal mathematical scheme.

  6. 6.

In the context of first-order systems of conservation laws, what connection,
if any, might exist between the large class of functions $M^{*}$ and entropies
[Daf16] of conservation laws is an important question to resolve.

  7. 7.

There does not exist well-established computational approximation schemes for
systems of H-J equations. This work provides a variational structure for such
systems which naturally lends itself to, say a finite element, discretization.
Whether such an idea has any practical merit would be interesting to explore.
If so, this can be very useful for certain systems related to the mechanics of
fracture and plastic deformation of solids, see [MA21, ZAWB15, AA19].

  8. 8.

For static conservation laws of the form $\partial_{i}f_{Ii}(\nabla
u)=0,(\nabla u)_{Ij}:=\partial_{j}u_{I}$, the scheme produces variational
principles even when there does not exist a potential $\psi(\nabla u)$ such
that $f_{Ii}=\partial_{(\nabla u)_{Ii}}\psi$.

  9. 9.

The considerations herein show that even a given variational principle can be
associated with a vastly different dual variational principle through the
choice of the function $H$ by associating the former’s E-L equations with the
latter following the proposed scheme. This seems to open up fascinating points
of convergence between apparently different classes of physical models
described in terms of pde and/or variational principles. A particularly
intriguing question is whether the present considerations have any connection
to the correspondences like AdS-CFT or AdS-CMT [ZLSS15, HLS18] in the string
theory-high energy-condensed matter-gravitational physics nexus.

  10. 10.

The proposed duality scheme seems to suggest that for a given pde system,
elements of the class of admissible potentials $H$ form symmetry operations
for the system. The algebraic structure of this class of potentials, starting
from whether they form a group, is of independent interest, as well as whether
knowledge of such structure can help in the understanding of solutions to the
pde system.

## Acknowledgments

This work was supported by the grant NSF OIA-DMR #2021019. It is a pleasure to
acknowledge discussions with Marshall Slemrod and Vladimir Sverak (VS) and to
thank them for taking the time to take a look at the paper. I also thank VS
for pointing me to [AKM19] and for encouragement.

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# Coherent optical-microwave interface for manipulation of low-field
electronic clock transitions in 171Yb3+:Y2SiO5

L. Nicolas<EMAIL_ADDRESS>Department of Applied Physics, University
of Geneva, CH-1211 Geneva 4, Switzerland  M. Businger  Department of Applied
Physics, University of Geneva, CH-1211 Geneva 4, Switzerland  T. Sanchez Mejia
Department of Applied Physics, University of Geneva, CH-1211 Geneva 4,
Switzerland  A. Tiranov  Center for Hybrid Quantum Networks (Hy-Q), The Niels
Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark  T.
Chanelière  Université Grenoble Alpes, CNRS, Grenoble INP, Institut Néel,
38000 Grenoble, France  E. Lafitte-Houssat  Chimie ParisTech, PSL University,
CNRS, Institut de Recherche de Chimie Paris, 75005 Paris, France  Thales
Research and Technology, 91767 Palaiseau, France  A. Ferrier  Chimie
ParisTech, PSL University, CNRS, Institut de Recherche de Chimie Paris, 75005
Paris, France  Faculté des Sciences et Ingénierie, Sorbonne Université, UFR
933, 75005 Paris, France  P. Goldner  Chimie ParisTech, PSL University, CNRS,
Institut de Recherche de Chimie Paris, 75005 Paris, France  Mikael Afzelius
Department of Applied Physics, University of Geneva, CH-1211 Geneva 4,
Switzerland

###### Abstract

The coherent interaction of solid-state spins with both optical and microwave
fields provides a platform for a range of quantum technologies, such as
quantum sensing, microwave-to-optical quantum transduction and optical quantum
memories. Rare-earth ions with electronic spins are interesting in this
context, but it is challenging to simultaneously and efficiently drive both
optical and microwave transitions over a long crystal. In this work, we use a
loop-gap microwave resonator to coherently drive optical and microwave clock
transitions in 171Yb3+:Y2SiO5, at close to zero external magnetic field. The
low magnetic field regime is particularly interesting for interfacing these
spin transitions with superconducting circuits. We achieve a Rabi frequency of
0.56 MHz at 2.497 GHz, over a 1-cm long crystal. Furthermore, we provide new
insights into the spin dephasing mechanism at very low fields, showing that
superhyperfine-induced collapse of the Hahn echo signal plays an important
role at low fields. Our calculations and measurements reveal that the
effective magnetic moment can be manipulated in 171Yb3+:Y2SiO5, allowing to
suppress the superhyperfine interaction at the clock transition. At a doping
concentration of 2 ppm and a temperature of $3.4$ K, we achieve the longest
spin coherence time of $10.0\pm 0.4~{}\text{ms}$ reported in 171Yb3+:Y2SiO5.

##  I Introduction

Solid-state electronic spins coupled to microwave (MW) cavities have been
extensively studied for realizing various coherent interfaces for quantum
information applications. This is due to the possibility for fast manipulation
and their high degree of tunability Morton and Bertet (2018), yet achieving
long coherence times with electronic spins remains challenging due to their
magnetic sensitivity. Efficient and coherent spin-MW interfaces have been
developped for various solid-state electronic spin systems, such as nitrogen-
vacancy centers in diamond Kubo  _et al._ (2010); Zhu  _et al._ (2011); Amsüss
_et al._ (2011); Ranjan  _et al._ (2013); Putz  _et al._ (2014), rare-earth
ion doped crystals Bushev  _et al._ (2011); Probst  _et al._ (2013); Chen  _et
al._ (2016); Fariña  _et al._ (2021); King  _et al._ (2021), phosphorus donors
in silicon Zollitsch  _et al._ (2015); Weichselbaumer  _et al._ (2020) and
ferri- and ferromagnetic magnons Huebl  _et al._ (2013); Tabuchi  _et al._
(2015); Abdurakhimov  _et al._ (2015); Everts  _et al._ (2020). Furthermore,
electronic spin systems that simultaneously couple coherently to light allow
reversible coupling between optical and MW modes, which is a key feature of
optical quantum memories Heshami  _et al._ (2016); Rakonjac  _et al._ (2021);
Ortu  _et al._ (2022); Businger  _et al._ (2020); Jin  _et al._ (2022) and MW-
optical quantum transducers Williamson  _et al._ (2014); Blum  _et al._
(2015); Fernandez-Gonzalvo  _et al._ (2015); Hisatomi  _et al._ (2016);
Bartholomew  _et al._ (2020).

In this work, we explore coherent optical and MW coupling in the context of
optical quantum memories, based on rare-earth (RE) ion doped crystals. An open
challenge on this platform is to realize a broadband ($\geq$ 100 MHz) memory
while storing the optical photon as a long-lived spin excitation. Broadband
operation requires large hyperfine splittings, which can be achieved with
electronic spin systems Businger  _et al._ (2020), i.e. with Kramers RE ions.
These, however, are prone to magnetic-induced dephasing due to their large
moments. Yet, long electronic spin coherence times can be reached by
combinations of moderate to large magnetic fields, sub-Kelvin temperatures
and/or ppb-level RE concentrations Kindem  _et al._ (2018); Rančić  _et al._
(2018); Dantec  _et al._ (2021), reaching up to 23 ms in a Er3+:CaWO4 crystal
at 0.7 ppb concentration at 10 mK Dantec  _et al._ (2021). Another approach is
to exploit zero first-order Zeeman (ZEFOZ) transitions (i.e. clock
transitions) that appear at zero magnetic field in Kramers ions having
anisotropic hyperfine interactions Ortu  _et al._ (2018); Rakonjac  _et al._
(2020); Kindem  _et al._ (2020); Chiossi  _et al._ (2022), which can be
reached without superconducting coils and with conventional cryo coolers
working at 2.5-4 K. Zero-field ZEFOZ transitions have yielded Hahn-echo spin
coherence times of up to 2.5 ms on the 655 MHz transition in 171Yb3+:Y2SiO5
Welinski  _et al._ (2020) at 10 ppm doping concentration and 3 K. In the MW
regime, a Hahn-echo coherence time of 35 $\mu$s was obtained on the 3.369 GHz
transition in the optically excited state of 171Yb3+:YVO4 at zero field
Bartholomew  _et al._ (2020). An important step is to simultaneously
demonstrate long spin coherence times in the MW regime $>$1 GHz at the ZEFOZ
condition of the ground state, efficient MW manipulation of a spin ensemble,
and optical coupling to the same ensemble. In addition, in order to preserve
the optical depth for efficient optical quantum memory operation, the coupling
must be homogeneous over the length of the crystal.

Figure 1: a) Energy level diagram for the optical
2F${}_{7/2}(0)\leftrightarrow^{2}$F${}_{5/2}(0)$ transition of site II in
171Yb3+:Y2SiO5, with relevant optical and MW transition used in the
experiments. b) Picture of the loop-gap resonator and the crystal illustrating
the common volume excited by the laser $\vec{E}$ and microwave
$\vec{B}_{\text{MW}}$ fields, aligned along the D2 and $b$ crystal axes,
respectively. c) MW transmission spectra recorded through the resonator at 5 K
and 300 K. d) Rabi oscillation time sequence, see text for details, with an
example of laser power transmission showing the Rabi oscillations. e) MW Hahn
echo time sequence, see text for details. The inset shows an example of the
optical RHS echo signal, detected through balanced heterodyne detection at a
beat frequency of 3 MHz.

Here, we employ a lumped-element resonator in order to achieve simultaneous
optical-MW coupling over a 1-cm long Y2SiO5 crystal. The resonator was tuned
to a zero-field ZEFOZ ground-state transition of the 171Yb3+ ion at 2.497 GHz.
The enhanced MW field gives a 0.56 MHz Rabi frequency and Hahn-echo
measurements resulted in an electronic spin coherence time of 10 ms, with a
171Yb3+ concentration of 2 ppm and a temperature of 3.4 K. With small magnetic
fields ($<$1 mT), our results provide evidence of strong superhyperfine
coupling to neighbouring yttrium ions (mostly 89Y3+ ions which have a 100%
abundance) Car  _et al._ (2018a), suggesting that superhyperfine-induced
collapse of the spin echo signal plays an important role in the dephasing, as
recently observed with photon echoes in Er-doped Y2SiO5 Car  _et al._ (2020a).
This effect can, however, be suppressed at certain angles of the magnetic
field and at the ZEFOZ point. Our results shows the possibility of operating
an optical quantum memory using any of the MW transitions available in
171Yb3+:Y2SiO5, over a sample volume compatible with crystals employed for
quantum memories.

## Results

###  I.1 Cavity-enhanced spin-microwave interaction

In this work, we study 171Yb3+ ions occupying site II of the Y2SiO5 crystal,
with the relevant optical transition
2F${}_{7/2}(0)\leftrightarrow^{2}$F${}_{5/2}(0)$ at a wavelength of 978.854 nm
Welinski  _et al._ (2016). The 171Yb3+ doping concentration is 2 ppm, with an
isotopic purity of $\approx 95\%$, and the crystal is cut along the D1, D2 and
$b$ axes Li  _et al._ (1992). The two electronic states have four
nondegenerate hyperfine states at zero magnetic field, see Fig. 1(a), due to
the highly anistropic hyperfine interaction in Y2SiO5 Tiranov  _et al._
(2018). Previous zero-field spin coherence measurements focused on the low-
frequency transitions at 528 and 655 MHz transitions Ortu  _et al._ (2018);
Welinski  _et al._ (2020), which can be rather efficiently excited by a
solenoid coiled around the crystal Ortu  _et al._ (2018); Businger  _et al._
(2020). It is interesting, however, to be able to address any MW transition,
as other so-called $\Lambda$-systems involving the high-frequency MW
transitions could have interesting features for optical quantum memories and
transducers. For instance, the electronic spin component of the hyperfine
states results in a polarization dependence of the relative transition
strengths of the optical-hyperfine transitions (see Supplementary
Information), which can be exploited to maximize the coupling strengths of
both optical transitions in a $\Lambda$-system. This is in contrast to non-
Kramers ions possessing only nuclear hyperfine states, such as Eu3+ Lauritzen
_et al._ (2012) and Pr3+, where the relative strengths are typically
polarization-independent, which severely restricts the number of useful
$\Lambda$-systems. Here, we specifically focus on the
$\ket{2_{g}}$-$\ket{4_{g}}$ transition at $\omega_{\mathrm{MW}}=2496.55$ MHz,
because calculations show a particularly low magnetic sensitivity for small
bias fields (see Supplementary Information).

To drive the high-frequency MW transition, we designed a loop-gap resonator
based on the work by Angerer et al. Angerer  _et al._ (2016), which enables
efficient and homogeneous driving of the spins over the entire 1-cm length of
the Y2SiO5 crystal. In short, the resonator consists of bow-tie type elements,
see Fig. 1(b), resulting in a lumped element LC-type electrical circuit where
the resonance frequency can be tuned by adjusting the distance between the bow
ties and the lid. The design of Ref. Angerer  _et al._ (2016) was modified by
adding optical access through two small holes, allowing optical and MW
excitation of a common mode, as shown in Fig. 1(b). The crystal was placed in
between the bow ties, with the crystal $b$-axis along the optical beam. The
crystal downshifted the resonance frequency by about 60-80 MHz, due to the
anisotropic dielectric permittivity of the crystal Carvalho  _et al._ (2015).

In Fig. 1(c), we show transmission spectra acquired with a vector network
analyzer (VNA), at a temperature of 300 K and 5 K. Cooling down the resonator
typically increased the frequency by $\approx$10 MHz, which was consistent
enough between cool downs such that tuning at room temperature was possible.
The transmission linewidth is 4.5 MHz, corresponding to a quality factor of
$Q=555$. The resonator was made out of standard copper, without mirror
polishing, lowering the $Q$-factor in comparison to Ref. Angerer  _et al._
(2016). However, in this context, the strong coupling regime is not the goal.
We use the resonator for a better impedance matching at high MW frequencies
and for producing homogeneous oscillating magnetic field $B_{\text{MW}}$ in
the crystal volume. For the following measurements, the crystal and resonator
were cooled down to about 3.4 K.

The coherent MW driving was measured through optical detection of MW Rabi
oscillations. A continuous probe laser tuned to the $\omega_{41}$ transition
pumps spins into the $\ket{2_{g}}$ state, increasing the population difference
between the $\ket{4_{g}}$ and $\ket{2_{g}}$ states. The transmitted power of
the probe laser is recorded on a photodiode, effectively measuring the
population in $\ket{4_{g}}$. The strong $\omega_{41}$ transition allows a high
contrast measurement of the population. An amplified MW pulse at 2496.55 MHz
was sent to the resonantor, reaching a power of 43 dBm at the MW input of the
cryostat. As shown in Fig. 1(d), the MW field induces clear Rabi oscillations
between the $\ket{2_{g}}$ and the $\ket{4_{g}}$ states, reaching a frequency
of $2\pi\times 560$ kHz. Note that the Rabi frequency is comparable to the
inhomogeneous spin broadening, which is about 680 kHz.

The spin coherence time can be measured using a Hahn echo sequence on the MW
transition, which can be optically detected using Raman heterodyne scattering
(RHS) Mlynek  _et al._ (1983); Ortu  _et al._ (2018); Everts  _et al._ (2020),
see Fig. 1(e). A pulsed probe laser tuned to the $\omega_{21}$ transition
first polarizes spins into the $\ket{4_{g}}$ state. Then the MW Hahn echo
sequence is applied, consisting of a 0.42-$\mu$s long $\pi/2$-pulse and a
0.84-$\mu$s long $\pi$-pulse, separated by a time $\tau$. The MW echo is
detected by applying another probe pulse at the moment of the echo, which
produces a RHS signal on the strong $\omega_{41}$ transition. For an increased
sensitivity, the RHS signal field amplitude is detected through a balanced
heterodyne detection with a local oscillator (LO) detuned by 3 MHz from the
RHS signal, see Fig. 1(e). The spin echo results will be discussed in the
following section.

###  I.2 Low-field spin properties of 171Yb3+:Y2SiO5

The $C_{1}$ point symmetry of the doping sites in Y2SiO5 results in an
effective electronic spin $S=1/2$ of the Yb3+ ion, and the isotope 171Yb3+ has
a nuclear spin $I=1/2$. The effective spin Hamiltonian can then be decomposed
into a hyperfine and an electronic Zeeman part Tiranov  _et al._ (2018)
(neglecting the weak nuclear Zeeman effect), for the ground ($g$) and
electronic level ($e$), respectively,

$\bm{\mathcal{H}}=\mathbf{S}\cdot\mathbf{A}_{g,e}\cdot\mathbf{I}+\mathbf{B}\cdot\bm{\mu}_{g,e}.$
(1)

The electronic spin $\mathbf{S}$ and the nuclear spin $\mathbf{I}$ components
are coupled through the hyperfine tensor $\mathbf{A}_{g,e}$. The magnetic
dipole moment is
$\bm{\mu}_{g,e}=\mu_{\text{B}}\mathbf{g}_{g,e}\cdot\mathbf{S}$, where
$\mathbf{g}_{g,e}$ is the Zeeman tensor and $\mu_{\text{B}}$ the Bohr
magneton. In a $C_{1}$ point symmetry, $A_{x}\neq A_{y}\neq A_{z}$, which
completely lifts degeneracy at $B=0$ and results in four hyperfine eigenstates
$\ket{k_{i}}$ ($k=1$ to $4$) in each electronic level ($i=g,e$), see Fig.
1(a). At $B=0$, the hyperfine states are completely hybridized in their
electronic and nuclear components, such that
$\expectationvalue{\mathbf{S}}=\expectationvalue{\mathbf{I}}=0$. As a
consequence, the expectation value of the dipole moment is zero
$\expectationvalue{\bm{\mu}_{g,e}}=0$, and to first order there is no linear
Zeeman effect in either electronic level. The zero first-order Zeeman effect
(ZEFOZ) effect at $B=0$ leads to a strong increase in both optical and spin
coherence times Ortu  _et al._ (2018). The zero effective magnetic dipole
moment also quenches superhyperfine interactions to first order (see
Supplementary Information), thereby quenching the electron spin echo envelope
modulation (ESEEM) Rowan  _et al._ (1965); Car  _et al._ (2018b); Lovrić  _et
al._ (2011), as first observed experimentally in Ref. Ortu  _et al._ (2018).

Figure 2: a) Map of the relative Hahn echo amplitude normalized to the maximum
amplitude for a range of magnetic fields in the D1-D2 plane, with the pulse
separation fixed at $\tau=5$ ms. b) Numerically computed $S_{1}$ gradient for
the 2.497 GHz transition in the D1-D2 plane, as a function of the angle
$\varphi$ to the D1 axis. The solid lines in a) and c) show the expected
minimum at $\varphi=55.9^{\circ}$, while the dashed lines show the estimated
angle of maximum echo amplitude, at an angle of $\varphi=48^{\circ}$. They
agree within the expected error of the field direction and of the cuts of the
crystal surfaces. c) Similar to the plot in a), but showing data taken at a
higher resolution in the field step. The color-coded crosses and circles show
the field values at which the echo decays shown in e) and Fig. 3(b)-(c) were
recorded, respectively. d) Map of the echo decays as a function of time delay
$\tau$, for a range of fields along $B_{\text{D}_{1}}$ and for fixed
$B_{\text{D}_{2}}=-155~{}\mu$T. The dashed line shows the yttrium Larmor
period. Examples of decay curves from the map are shown in e), from top to
bottom $B_{\text{D}_{1}}=0,-66,-106,-202,-278~{}\mu$T, indicated with crosses
in panel c).

When a weak magnetic field is applied, the states $\ket{k}$ are mixed trough
the Zeeman interaction (note that in the following we have omitted the index
$i=g,e$ indicating the electronic level). The weakly perturbed states
$\ket{l}$ result in a non-zero expectation value $\bra{l}\mathbf{S}\ket{l}\neq
0$, such that each state acquires a weak magnetic dipole moment
$\bra{l}\bm{\mu}\ket{l}\neq 0$. If the $\mathbf{A}$ and $\mathbf{g}$ tensors
are aligned, a good approximation here Tiranov  _et al._ (2018), then a first-
order perturbation calculation shows (see Supplementary Information) that the
energy $E_{l}$ of the perturbed state $\ket{l}$ has a quadratic Zeeman effect,
with an associated field-gradient
$dE_{l}/dB_{m}=\mu_{\text{B}}g_{m}\bra{l}S_{m}\ket{l}$, where $m=x,y,z$ is the
direction of the weak magnetic field. It can further be shown that the
effective dipole moment component along $m$, for state $\ket{l}$, is given by
$\bra{l}S_{m}\ket{l}=(1/2)\mu_{\text{B}}B_{m}g_{m}/(E_{l}-E_{l^{\prime}})$,
where $\ket{l^{\prime}}$ is the only other hyperfine state such that
$\bra{l^{\prime}}S_{m}\ket{l}\neq 0$. Note that the zero-field energies
$E_{l}$ can easily be calculated from the $\mathbf{A}$ tensor elements Tiranov
_et al._ (2018). The direction-averaged, first-order, transition frequency
gradient $S_{1}$ is calculated by taking the difference of the energy
gradients of the connected states.

The calculations emphasize the direct connection between the first-order
dipole moment and the energy gradient when a weak field is applied, as
expected for a classical dipole. Both are zero to first order in the ZEFOZ
point, but can also be strongly quenched when applying a weak field, by tuning
the field direction to minimize the nominator term $g_{m}^{2}$ and to maximize
the denominator term $E_{l}-E_{l^{\prime}}$, as shown experimentally in Ref.
Ortu  _et al._ (2018). In the following, we will see how both effects change
the observed spin dephasing time, and the strength of superhyperfine
interaction and the ESEEM effect.

###  I.3 Spin echo measurements

The Hahn echo signal amplitude was measured as a function of magnetic field in
the D1-D2 plane, i.e. with $B_{\text{D}_{1}}$ and $B_{\text{D}_{2}}$
components, for a fixed pulse separation of $\tau=5$ ms, see Fig. 2(a). The
echo signal is strongest in the vicinity of zero applied field, as expected
for the ZEFOZ point, but also remains strong along a particular line. The
intense line corresponds well to the minimum $S_{1}$ gradient in the D1-D2
plane, expected at an angle of $\varphi=55.9^{\circ}$ to the D1 axis (see Fig.
2(b) and Supplementary Information). In addition, one can observe a distinct
ESEEM oscillation pattern, which is a sign of dipole-dipole coupling to
neighbouring 89Y3+ ions (i.e. superhyperfine interaction).

Figure 3: Map of the echo decays as a function of time delay $\tau$, for a
range of fields along $B_{\text{D}_{1}}$ and for fixed
$B_{\text{D}_{2}}=-220~{}\mu$T. The white dashed lines show multiples of the
yttrium Larmor period, $nT_{\text{Y}}$, where $n=1,2,3$. b) The echo decay
obtained at $B_{\text{D}_{1}}=-248~{}\mu$T shows clear ESEEM peaks at the
multiples of $T_{\text{Y}}$ (vertical dashed line). c) The echo decay at
$B_{\text{D}_{1}}=-51~{}\mu$T shows no discernible modulation. The magnetic
fields for the two decays in b) and c) are indicated as horizontal dashed
lines in a), and with circles in Fig. 2(c).

A map with finer resolution in the field was also recorded around the point of
highest echo amplitude, for $\tau=5$ ms, see Fig. 2(c). A symmetric pattern
appears around a field offset of around $B_{\text{D}_{2}}=-155~{}\mu$T along
the D2 axis. Such offsets have been observed in all our past echo measurements
in 171Yb3+:Y2SiO5 Ortu  _et al._ (2018); Businger  _et al._ (2020), and it
varies depending on the cryostat and electromagnetic environment, showing that
it is produced externally from the crystal. In general, these slowly varying
background fields can make it challenging to experimentally find and maintain
the ZEFOZ point. In the following, we assume that the true zero field, i.e.
the ZEFOZ point, is located close to $B_{\text{D}_{1}}=0$ and
$B_{\text{D}_{2}}=-155~{}\mu$T.

The strong ESEEM pattern seen in Fig. 2(c) indicates that the superhyperfine
coupling contributes significantly to the effective spin dephasing mechanism
away from the ZEFOZ point. To investigate this further, we recorded Hahn echo
decay curves as a function of $\tau$, for a varying field $B_{\text{D}_{1}}$
along the D1 axis. A constant field of $B_{\text{D}_{2}}=-155~{}\mu$T was
applied in the D2 axis, to compensate for the observed lab bias field. The
data is presented as a 2D map in Fig. 2(d), and a few examples of decay curves
are shown in Fig. 2(e). For the highest fields along D1, there is a distinct
ESEEM modulation appearing, with an almost complete collapse of the echo
signal before the first ESEEM revival. This shows that the superhyperfine
coupling plays a major role in the observed spin dephasing, as was also
recently observed with photon echoes in Er-doped Y2SiO5 Car  _et al._ (2020a).
Our measurements also suggests that coupling to 89Y3+ ions is dominating, as
the time of the first ESEEM revival closely follows the Lamor period of the
yttrium spin $T_{\text{Y}}=1/(B\gamma_{\text{Y}})$, where
$\gamma_{\text{Y}}=2.095$ MHz/T, see Fig. 2(d). Note that in the model, the
field amplitude $B$ is taken to be the field applied along D1, i.e.
$B=B_{\text{D}_{1}}$, as the D2 field only compensates the lab bias field.

The ESEEM oscillation seen in Figs 2(d)-(e) increases in period and decreases
in amplitude, as the $B$ field approaches the ZEFOZ condition at the estimated
zero-field point. The increased period is clearly due to the yttrium Larmor
frequency going to zero, while we attribute the decrease in amplitude to the
quenching of the effective 171Yb3+ dipole moment in the ZEFOZ point. Note that
this is a very effective method for finding the true $B=0$ point
experimentally, by pushing the oscillation to longer periods until its
amplitude vanishes. However, as both the Larmor frequency and the first-order
effective dipole moment goes to zero, it is not possible to discriminate the
two effects while approaching $B=0$.

A clearer way of showing how the ESEEM oscillation can be quenched by
minimizing the effective dipole moment is to approach the line of minimum
$S_{1}$ gradient, while applying a constant offset field along D2 such that
the yttrium Larmor frequency is strictly non-zero for all measurements. In
Fig. 3(a), we show a 2D map of Hahn echo decays, as a function of fields
applied along the D1 axis. The D2 field was held constant at
$B_{\text{D}_{2}}=-220~{}\mu$T, meaning there was a constant offset of
$-65~{}\mu$T along D2 from the estimated true $B=0$ point. The data shows up
to three clear ESEEM oscillations for the highest fields along D1, see the
example in Fig. 3(b). However, as the D1 field is tuned towards the line of
minimum dipole moment, the oscillations completely vanish, although the Larmor
frequency is non-zero at all times. For the field of
$B_{\text{D}_{1}}=-51~{}\mu$T, which is exactly on the line of minimum
gradient (cf. Fig. 2(c)), the echo amplitude displays an exponential decay
without oscillations, see Fig. 3(c). The dependence of the period of
oscillation on the field magnitude $B$ follows closely the Larmor frequency,
shown by the theoretical lines based on the model field amplitude
$B=\sqrt{(65~{}\mu\text{T})^{2}+B_{\text{D}_{1}}^{2}}$, see Fig. 3(a). The
quenching of the ESEEM modulation at a non-zero field magnitude unambiguously
demonstrates that the effective dipole moment is field dependent in 171Yb3+
and that it can be minimized at specific orientations. This could provide an
interesting tool for studying superhyperfine interactions in detail.

Figure 4: a) Hahn echo decays for fields $B_{b}=18,42,85,230~{}\mu$T along the
$b$-axis, with a constant offset field $B_{\text{D}_{2}}=-155~{}\mu\text{T}$
to compensate for the lab bias field along D2. Solid lines show fits to a
stretched exponential. b) Spin coherence times measured along the $b$ axis.
The solid line shows a simple empirical model (see text for details). c) The
longest Hahn echo decay was obtained by carefully adjusting the bias magnetic
field around the estimated ZEFOZ point, while optimizing the echo amplitude at
a long delay, resulting in a spin coherence time of $T_{2}=(10.0\pm
0.4)~{}\text{ms}$.

The Hahn echo decay was also measured while applying a varying field in the
$b$ axis, starting from the estimated ZEFOZ point (i.e. with
$B_{\text{D}_{2}}=-155~{}\mu\text{T}$ constant). Along this axis, no
oscillations appear for up to $B_{b}=300~{}\mu\text{T}$, as shown in Fig.
4(a). However, the timescale of the decay is similar as measured along the D1
axis, indicating that the effective spin dephasing time is also governed by
superhyperfine coupling along the $b$-axis. The spin decay curves were fitted
to a stretched exponential $E(\tau)=E(0)\exp(-(2\tau/T_{2})^{m})$, where $m$
is the Mims factor, see Fig. 4(a).

The spin coherence time decreases as the $B_{b}$ is increased, as shown in
Fig. 4(b), following a $1/B_{b}$ dependence at higher fields, as already
observed in Ref. Ortu  _et al._ (2018). The coherence time data fits well to a
simple empirical formula
$T_{2}(B)=1/(1/T_{2}(0)+\pi\kappa\absolutevalue{B-B_{0}})$, where $T_{2}(0)$
is the estimated zero-field coherence time, $\kappa$ the field dependent
homogeneous linewidth, and $B_{0}$ the offset bias field along the b-axis. The
fit to data yields $T_{2}(0)=(10.3\pm 0.2)~{}\text{ms}$, $\kappa=(1.48\pm
0.04)\text{MHz/T}$ and $B_{0}=(14.1\pm 0.4)~{}\mu\text{T}$. Similar values
were found when fitting only the first collapses before the first ESEEM
revival of the D1 scan shown in Fig. 2(d) and (e) (see Supplementary
Information). Two aspects stand out in this simple analysis, which is the
universal $1/B$ dependence (also observed in Ref. Ortu  _et al._ (2018)) and a
field-dependent homogeneous linewidth $\kappa$ that is rather close to the
yttrium gyromagnetic ratio $\gamma_{\text{Y}}$. As it stands, we do not have a
detailed model to explain these observations, but we observe that the screened
superhyperfine decay model in Ref. Car  _et al._ (2020a) cannot be applied
here since the dipole moment is proportional to $B$ (see Supplementary
Information). In any case, it appears clearly that a model of the spin
dephasing times close to zero field must include superhyperfine interactions
with yttrium ions.

Finally, we briefly discuss the longest spin coherence times obtained in this
study. As observed in our dataset, magnetic bias fields of the order of a few
$\mu\text{T}$ affect the Hahn echo decay. Exponential functions were fitted to
the longest decays obtained for the D1 and $b$ axis scans, see Figs 2(e) and
Fig. 4(a), from which we obtained $T_{2}=(9.6\pm 0.8)~{}\text{ms}$ and
$T_{2}=(8.5\pm 0.6)~{}\text{ms}$, respectively. In a separate measurement,
where we finely adjusted the bias field around the expected ZEFOZ point, we
obtained $T_{2}=(10.0\pm 0.4)~{}\text{ms}$ as shown in Fig. 4(c). This can be
compared to the $T_{2}=(6\pm 1)~{}\text{ms}$ obtained in a parallel study by
Alexander et al. Alexander  _et al._ (2022) in a Y2SiO5 crystal doped with 5
ppm of 171Yb3+. In this context, it is also worth noting that we measured the
optical coherence time in both crystals, and found a similar relative
difference of $(0.610\pm 0.050)~{}\text{ms}$ and $(1.05\pm 0.130)~{}\text{ms}$
for the 5 and 2 ppm doping, respectively (see Supplementary Information).

##  II Conclusions and Discussion

The lumped-element resonator provides a method for coherent driving of both
optical and MW transitions over a large sample volume, giving us access to all
the higher MW spin transitions for quantum memory schemes Businger  _et al._
(2020). The polarization-dependent optical transition strengths between the
hyperfine levels opens up many possible $\Lambda$-schemes, particularly
involving the high-frequency spin transitions such as the 1.841 GHz and 3.025
GHz transitions (see Supplementary Information), allowing one to optimize the
memory scheme in terms of other parameters, such as the maximum memory
bandwidth. Integrating the lumped-element resonator in a spin-wave quantum
memory experiment, cf. Ref. Businger  _et al._ (2020), is an important next
step.

The experiments presented in this work show that superhyperfine-induced ESEEM
oscillations of the Hahn echo signal have a strong influence on the dephasing
dynamics at low fields. We have also shown that the effective magnetic dipole
moment is field-orientation dependent at low fields and can be quenched at the
zero-field ZEFOZ point, or when the $S_{1}$ gradient is very low. These
experimental observations are also well-supported by our calculations. There
remains many open and interesting questions, however, such as the influence of
a stochastic time-dependent spectral diffusion process de Lange  _et al._
(2010), which has been successfully applied to europium-doped Y2SiO5 Holzäpfel
_et al._ (2020); Ortu  _et al._ (2022). The simple empirical model to explain
the $B$-field dependence of the coherence time includes a zero-field
$T_{2}(0)$, which could be a sign that the observed zero-field coherence time
could be dominated by spectral diffusion. In addition, one would expect to see
more distinct ESEEM oscillations at longer delays, as seen in Ref. Car  _et
al._ (2020b), which would also suggest another mechanism at long time scales.
Since the effective magnetic dipole moment can be tuned by the strength and
orientation of the magnetic field, we believe that 171Yb3+:Y2SiO5 provides a
particularly interesting system for further studies of these dephasing
mechanisms.

##  III Methods

The magnetic fields along the D1 and D2 axes were generated by two pairs of
coils, placed outside the cryostat. A superconducting coil placed inside of
the cryostat generated the field along the $b$-axis. The magnetic fields were
calibrated by measuring the Zeeman frequency shifts they induced, and
comparing these to the shifts calculated with the model presented in Tiranov
_et al._ (2018).

To increase the signal of the RHS echo, the spin echo sequence is preceded by
a population preparation step, in order to increase the population difference
between states $\ket{4_{g}}$ and $\ket{2_{g}}$. To do so, the laser is first
tuned to the $\ket{2_{g}}$-$\ket{1_{e}}$ transition to decrease the population
in the $\ket{2_{g}}$ state. Then, the frequency of the laser is scanned to
address optical transitions involving the $\ket{1_{g}}$, $\ket{2_{g}}$ and
$\ket{3_{g}}$ states, thereby pumping ions into the state $\ket{4_{g}}$.

To drive the $\ket{2_{g}}$-$\ket{4_{g}}$ MW-transition with high power, a
standard wifi amplifier was used (Sunhans SH24Gi20W).

## DATA AVAILABILITY

The datasets shown in the figures of this article can be accessed from the
Zenodo data repository 10.5281/zenodo.7064041.

## ACKNOWLEDGEMENTS

We acknowledge funding from the Swiss National Science Foundation (SNSF)
through project 197168, the French Agence Nationale de la Recherche through
project MIRESPIN (Grant No. ANR-19-CE47- 0011) and support from DGA.

## AUTHOR CONTRIBUTIONS

A.T. and T.S.M. designed and made initial tests of the loop-gap resonator. The
crystal growth and initial optical spectroscopy was done by E.L.H., A.F. and
P.G.. All the optically detected spin resonance measurements were carried out
by L.N., with support from M.B. and T.S.M.. L.N. did all the data reduction
and analysis, with support from M.A. The theoretical superhyperfin
calculations were done by M.A. and T.C.. The manuscript was mainly written by
N.L. and M.A., with contributions from all the authors. M.A. provided overall
oversight of the project.

## COMPETING INTERESTS

The authors declare no competing interests.

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