# Polyharmonic Green Functions and Nonlocal BMS Transformations of a Free
Scalar Field

Carles Batlle11footnotemark: 1<EMAIL_ADDRESS>Víctor
Campello22footnotemark: 2<EMAIL_ADDRESS>Joaquim Gomis33footnotemark: 3
<EMAIL_ADDRESS>Departament de Matemàtiques and IOC, Universitat
Politècnica de Catalunya
EPSEVG, Av. V. Balaguer 1, E-08800 Vilanova i la Geltrú, Spain  Departament de
Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via de les Corts
Catalanes 585, E-08007, Barcelona, Spain  Departament de Física Quàntica i
Astrofísica and Institut de Ciències del Cosmos (ICCUB), Universitat de
Barcelona, Martí i Franquès 1, E-08028 Barcelona, Spain

###### Abstract

We express the nonlocal BMS charges of a free massless Klein-Gordon scalar
field in $2+1$ in terms of the Green functions of the polyharmonic operators.
Using the properties of these Green functions, we are able to discuss the
asymptotic behaviour of the fields that ensures the existence of the charges,
and prove that one obtains a realization of the $2+1$ BMS algebra in canonical
phase space. We also discuss the transformations in configuration space, and
show that in this case the algebra closes only up to skew-symmetric
combinations of the equations of motion. The formulation of the charges, in
terms of Green functions, opens the way to the generalization of the formalism
to other dimensions and systems.

###### keywords:

BMS symmetry, Nonlocal transformations, Polyharmonic functions

##  1 Introduction

The BMS group of symmetry transformations [1, 2] has experienced a surge of
interest during the last decade. It has been used, for instance, to deduce
Weinberg’s soft graviton theorems [3] as Ward identities of BMS
supertranslations [4, 5, 6, 7]. A pedagogical overview of the role of BMS
symmetries in several problems in field theory and gravitation is presented in
[8]. The BMS algebra in the case of the $2+1$ space-time has been studied in
[9, 10], and some applications can be found in [11][12][13][14][15].

Following the ideas in [16], an explicit nonlocal realization of
supertranslations for scalar free fields in $2+1$ Minkowski space-time was
developed in [17]. The existence of the Noether charges that canonically
generate these transformations implies an asymptotic behaviour at spatial
infinity for the scalar field.444The asymptotic behaviour of the scalar fields
and the relation to soft theorems has been studied in [18]. The BMS asymptotic
symmetries at spatial infinity using the hamiltonian formalism has been
studied in [19, 20, 21].

For a massless field, it was shown in [17] that the functional variation of
the field $\phi$ and its canonical momentum $\pi$ under a supertranslation is
given by555The results for the massive case were also given in [17], as well
as the extension to superrotations; see also [22][23] for a discussion of the
nonrelativistic limit, and [24] for the addition of super-dilatations.

$\displaystyle\delta_{\ell}\phi(t,\vec{x})$
$\displaystyle=\int\text{d}^{2}y\,[f_{\ell}(\vec{x}-\vec{y})\phi(t,\vec{y})+g_{\ell}(\vec{x}-\vec{y})\pi(t,\vec{y})],$
(1.1) $\displaystyle\delta_{\ell}\pi(t,\vec{x})$
$\displaystyle=\int\text{d}^{2}y\,[h_{\ell}(\vec{x}-\vec{y})\phi(t,\vec{y})+f_{\ell}(\vec{x}-\vec{y})\pi(t,\vec{y})].$
(1.2)

with $\ell\in\mathbb{Z}$ and where integration is all over two-dimensional
space. The functions appearing in the above expressions are given by

$\displaystyle f_{\ell}(\vec{x})$ $\displaystyle=\quad 2\int
d\widetilde{k}\,\omega\,\omega_{\ell}(\vec{k})\sin(\vec{k}\cdot\vec{x}),$
(1.3) $\displaystyle g_{\ell}(\vec{x})$ $\displaystyle=\quad 2\int
d\widetilde{k}\,\omega_{\ell}(\vec{k})\cos(\vec{k}\cdot\vec{x}),$ (1.4)
$\displaystyle h_{\ell}(\vec{x})$ $\displaystyle=-2\int
d\widetilde{k}\,\omega^{2}\omega_{\ell}(\vec{k})\cos(\vec{k}\cdot\vec{x}),$
(1.5)

with $\omega=|\vec{k}|$ and
$\omega_{\ell}(\vec{k})=\omega^{1-\ell}(k_{1}+ik_{2})^{\ell}$, and where the
measure in momenta space is
$d\widetilde{k}\,=\text{d}^{2}k/((2\pi)^{2}2\omega)$. One can check that
$\omega_{-\ell}=\omega_{\ell}^{*}$ and thus one can work with $\ell\geq 0$ and
take the complex conjugate when negative indexes are needed. Furthermore,
since $\omega_{\ell}(-\vec{k})=(-1)^{\ell}\omega_{\ell}(\vec{k})$, one can see
from (1.3)(1.4)(1.5) that

$f_{2\ell}(\vec{x})=0,\quad g_{2\ell+1}(\vec{x})=0,\quad
h_{2\ell+1}(\vec{x})=0,\ \ \ell\in\mathbb{Z}.$ (1.6)

Notice that, in general, the transformations (1.1)(1.2) are nonlocal unless
the functions $f_{\ell}$, $g_{\ell}$, $h_{\ell}$ are proportional to a delta
function or a finite number of its derivatives. As we will see, this happens
only for $\ell=0,\pm 1$, which corresponds to ordinary space-time
translations.

Using standard equal-time Poisson brackets and the properties
$f_{\ell}(-\vec{x})=-f_{\ell}(\vec{x})$,
$g_{\ell}(-\vec{x})=g_{\ell}(\vec{x})$,
$h_{\ell}(\vec{x})=\nabla^{2}g_{\ell}(\vec{x})$, it can be seen that the field
transformations (1.1), (1.2) are generated by the supertranslation charges

$\displaystyle
Q_{\ell}(t)=\int\text{d}^{2}x\text{d}^{2}y\,(f_{\ell}(\vec{x}-\vec{y})\pi(t,\vec{x})\phi(t,\vec{y})+\dfrac{1}{2}g_{\ell}(\vec{x}-\vec{y})\pi(t,\vec{x})\pi(t,\vec{y})-\dfrac{1}{2}h_{\ell}(\vec{x}-\vec{y})\phi(t,\vec{x})\phi(t,\vec{y})).$
(1.7)

Once the asymptotic behaviour of $\phi(t,\vec{x}),\pi(t,\vec{x})$ at spatial
infinity is given, see A, and using standard equal-time Poisson brackets, it
can be shown (see B) that these charges have zero Poisson bracket with the
Hamiltonian of the massless scalar field

$H(t)=\int\text{d}^{2}x\left(\frac{1}{2}\pi^{2}(t,\vec{x})+\frac{1}{2}(\vec{\nabla}\phi(t,\vec{x}))^{2}\right),$
(1.8)

and are thus conserved. The proof relies solely on the symmetry properties of
the functions $f_{\ell}$ and $g_{\ell}$ and on the relation between $g_{\ell}$
and $h_{\ell}$. The main goal of the paper is to show that the functions
(1.3), (1.4) and (1.5) can be cast in terms of higher level objects, which
turn out to be Green functions, and to use their properties to discuss some
aspects of the transformations. The algebra of the transformations in terms
only of $\phi(t,\vec{x})$ closes only up to an antisymmetric combination of
the equations of motion.

The rest of the paper is organized as follows. Section 2 presents the
expression of the supercharges in terms of Green functions of appropriate
operators. Using these, the existence of the charges is discussed. Section 3
computes the Poisson brackets of the obtained charges, while Section 4
considers the BMS transformations in configuration space. Finally, in Section
5 we summarize our results and suggest that the formalism presented in this
paper can be generalized to other dimensions and systems. Detailed
calculations of all the results have been moved to the appendixes.

##  2 Nonlocal transformation of the fields in terms of polyharmonic functions

Using the explicit expression for $\omega_{\ell}$, one can write $g_{\ell}$ as

$\displaystyle g_{\ell}(\vec{x})$ $\displaystyle=\dfrac{1}{(2\pi)^{2}}\int
d^{2}k\,\omega^{-\ell}(k^{1}+ik^{2})^{\ell}\cos(\vec{k}\cdot\vec{x}).$ (2.1)

For $\ell$ odd, the integrand is antisymmetric in $\vec{k}$ and the integral
cancels out. For $\ell$ even and non-negative, one can write

$\displaystyle g_{2\ell}(\vec{x})$ $\displaystyle=\dfrac{1}{(2\pi)^{2}}\int
d^{2}k\,\omega^{-2\ell}(k^{1}+ik^{2})^{2\ell}\cos(\vec{k}\cdot\vec{x})$
$\displaystyle=(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}(-1)^{\ell}\dfrac{1}{(2\pi)^{2}}\int
d^{2}k\,\omega^{-2\ell}\cos(\vec{k}\cdot\vec{x}).$ (2.2)

Now, defining a distribution $G_{\ell}(\vec{x})$ such that

$G_{\ell}(\vec{x})=(-1)^{\ell}\dfrac{1}{(2\pi)^{2}}\int
d^{2}k\,\omega^{-2\ell}\cos(\vec{k}\cdot\vec{x}),$ (2.3)

one gets the equation for a polyharmonic Green function,

$(\nabla_{\vec{x}}^{2})^{\ell}G_{\ell}(\vec{x})=\dfrac{1}{(2\pi)^{2}}\int
d^{2}k\,\cos(\vec{k}\cdot\vec{x})=\delta(\vec{x}),\quad\ell\geq 0.$ (2.4)

One has that $G_{0}(\vec{x})=\delta(\vec{x})$ and, for $\ell\geq 1$, the Green
function is [25, 26]

$G_{\ell}(\vec{x},\vec{y})=G_{\ell}(\vec{x}-\vec{y})=\dfrac{|\vec{x}-\vec{y}|^{2(\ell-1)}}{[(\ell-1)!]^{2}2^{2\ell-1}\pi}(\log|\vec{x}-\vec{y}|-H_{\ell-1}),$
(2.5)

where $H_{\ell}=\sum_{i=1}^{\ell}\frac{1}{i}$ and $H_{0}=0$. It follows from
(2.4) that

$\nabla^{2}G_{\ell}(\vec{x}-\vec{y})=G_{\ell-1}(\vec{x}-\vec{y}),\quad\ell\geq
1.$ (2.6)

As shown in E, these functions satisfy also the convolution property

$\int\text{d}^{2}xG_{\ell}(\vec{y}-\vec{x})G_{m}(\vec{z}-\vec{x})=G_{\ell+m}(\vec{y}-\vec{z}).$
(2.7)

The expressions for $f_{\ell}$ and $h_{\ell}$ can then be directly obtained as
a function of $g_{\ell}$ by observing that
$f_{2\ell+1}=-(\partial_{x_{1}}+i\partial_{x_{2}})g_{2\ell}$ and
$h_{2\ell}=\nabla^{2}g_{2\ell}$, for $\ell\in\mathbb{N}$ and they are zero
otherwise. Thus, these functions can be written in terms of the polyharmonic
Green function $G_{\ell}(\vec{x})$ as

$\displaystyle g_{2\ell}(\vec{x})$
$\displaystyle=(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(\vec{x}),$
(2.8) $\displaystyle f_{2\ell+1}(\vec{x})$
$\displaystyle=-(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell+1}G_{\ell}(\vec{x}),$
(2.9) $\displaystyle h_{2\ell}(\vec{x})$
$\displaystyle=(\partial_{x_{1}}-i\partial_{x_{2}})(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell+1}G_{\ell}(\vec{x}),$
(2.10)

for $\ell\geq 0$. For $\ell=0$ one has $g_{0}(\vec{x})=\delta(\vec{x})$,
$f_{1}(\vec{x})=-(\partial_{x_{1}}+i\partial_{x_{2}})\delta(\vec{x})$ and
$h_{0}(\vec{x})=\nabla^{2}\delta(\vec{x})$, which yield the standard space-
time translations for the fields.

In terms of the $G_{\ell}$ the supertranslation charges (1.7) for $\ell\geq 0$
take the forms

$\displaystyle Q_{2\ell}(t)$
$\displaystyle=\int\text{d}^{2}x\,\text{d}^{2}y\,\left(\dfrac{1}{2}\pi(t,\vec{x})\pi(t,\vec{y})+\dfrac{1}{2}\vec{\nabla}\phi(t,\vec{x})\cdot\vec{\nabla}\phi(t,\vec{y})\right)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(\vec{x}-\vec{y}),$
(2.11) $\displaystyle Q_{2\ell+1}(t)$
$\displaystyle=\int\text{d}^{2}x\,\text{d}^{2}y\,(\partial_{x_{1}}+i\partial_{x_{2}})\pi(t,\vec{x})\phi(t,\vec{y})(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(\vec{x}-\vec{y})$
(2.12)
$\displaystyle=\int\text{d}^{2}x\,\text{d}^{2}y\,\phi(t,\vec{x})\pi(t,\vec{y})(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell+1}G_{\ell}(\vec{x}-\vec{y})$
(2.13)
$\displaystyle=-\int\text{d}^{2}x\,\text{d}^{2}y\,(\partial_{x_{1}}+i\partial_{x_{2}})\phi(t,\vec{x})\pi(t,\vec{y})(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(\vec{x}-\vec{y}).$
(2.14)

Using that $\omega_{\ell}^{*}=\omega_{-\ell}$ one has that for $\ell<0$ the
only difference is the appearance of $\partial_{x_{1}}-i\partial_{x_{2}}$
instead of $\partial_{x_{1}}+i\partial_{x_{2}}$, and thus

$Q_{-2\ell}(t)=Q_{2\ell}^{*}(t),\quad Q_{-(2\ell+1)}(t)=Q_{2\ell+1}^{*}(t).$
(2.15)

The supertranslation transformations in terms of the $G_{\ell}$ are given by

$\displaystyle\delta_{2\ell}\phi(t,\vec{x})$
$\displaystyle=\\{\phi(t,\vec{x}),Q_{2\ell}(t)\\}=\int\text{d}^{2}y\,\pi(t,\vec{y})(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(\vec{x}-\vec{y}),$
(2.16) $\displaystyle\delta_{2\ell+1}\phi(t,\vec{x})$
$\displaystyle=\\{\phi(t,\vec{x}),Q_{2\ell+1}(t)\\}=\int\text{d}^{2}z\,\phi(t,\vec{z})(\partial_{z_{1}}+i\partial_{z_{2}})^{2\ell+1}G_{\ell}(\vec{z}-\vec{x})$
$\displaystyle=-\int\text{d}^{2}y\,\phi(t,\vec{y})(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell+1}G_{\ell}(\vec{x}-\vec{y}),$
(2.17)

where we have used the Poisson bracket
$\\{\phi(t,\vec{x}),\pi(t,\vec{y})\\}=\delta(\vec{x}-\vec{y})$.

In particular, using that $G_{0}(\vec{x}-\vec{y})=\delta(\vec{x}-\vec{y})$,
one can see that $Q_{0}(t)=H(t)$ is the generator of time-translations and
that

$\displaystyle Q_{x_{1}}(t)$
$\displaystyle=\frac{Q_{1}(t)+Q_{-1}(t)}{2}=-\int\text{d}^{2}x\,\pi(t,\vec{x})\partial_{x_{1}}\phi(t,\vec{x}),$
(2.18) $\displaystyle Q_{x_{2}}(t)$
$\displaystyle=\frac{Q_{1}(t)-Q_{-1}(t)}{2i}=-\int\text{d}^{2}x\,\pi(t,\vec{x})\partial_{x_{2}}\phi(t,\vec{x}),$
(2.19)

generate the spatial translations.

We discuss next the asymptotic behaviour of the fields that guarantees the
existence of the supertranslation charges and of the symplectic form.

Let us first consider the kinetic term in the action, which eventually leads
to a well-defined Poisson bracket,

$\int\mbox{d}^{2}x\,\dot{\phi}(t,\vec{x})\pi(t,\vec{x}).$ (2.20)

If we assume asymptotic expansions

$\displaystyle\phi(t,\vec{x})$
$\displaystyle=\frac{\bar{\phi}_{1}}{|\vec{x}|}+\frac{\bar{\phi}_{2}}{|\vec{x}|^{2}}+\ldots,$
(2.21) $\displaystyle\pi(t,\vec{x})$
$\displaystyle=\frac{\bar{\pi}_{2}}{|\vec{x}|^{2}}+\frac{\bar{\pi}_{3}}{|\vec{x}|^{3}}+\ldots,$
(2.22)

where the $\bar{\phi}_{1}$, $\bar{\phi}_{2}$, $\bar{\pi}_{1}$,
$\bar{\pi}_{2}$, …, are functions depending on time and the angular variable
then

$\int\mbox{d}^{2}x\,\dot{\phi}(t,\vec{x})\pi(t,\vec{x})=\int\mbox{d}\theta\int
r\mbox{d}r\left(\dot{\bar{\phi}}_{1}\bar{\pi}_{2}\frac{1}{r^{3}}+O(r^{-4})\right)$
(2.23)

which makes the term well defined. It follows also from these conditions that
the field configuration has then a finite energy, and in fact the conditions
cannot be relaxed, by instance by assuming $\phi\sim\log r$ or $\pi\sim 1/r$,
if one wants to have a finite energy. Notice that under these conditions no
logarithmic divergence appears in (2.23), in contrast with the case in $3+1$
space-time discussed in [27].

The leading order behaviour of $G_{l}(\vec{x}-\vec{y})$ for large
$r=|\vec{x}-\vec{y}|$ is

$G_{\ell}(r)\sim r^{2(\ell-1)}\log r.$ (2.24)

As shown in A, the derivatives of order $2\ell$ which appear in (2.11) and
(2.14) behave as

$(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(r)\sim\frac{1}{r^{2}}\quad\forall\ell\geq
1.$ (2.25)

Taking this into account and comparing (1.8) with (2.11) and (2.14), it
follows that the supertranslation charges exist for field configurations
behaving as in (2.21), (2.22).

In order to have the BMS algebra we will need, besides the generators of
super-translations, those of the Lorentz symmetries, given by

$\displaystyle M^{12}(t)$
$\displaystyle=-\int\mbox{d}^{2}x\,\pi(t,\vec{x})\left(x_{1}\partial_{x_{2}}\phi(t,\vec{x})-x_{2}\partial_{x_{1}}\phi(t,\vec{x})\right)$
(2.26) $\displaystyle M^{0i}(t)$
$\displaystyle=-\int\mbox{d}^{2}x\,\left(t\pi(t,\vec{x})\partial_{x_{i}}\phi(t,\vec{x})+x_{i}{\cal
H}(t,\vec{x})\right),\quad i=1,2,$ (2.27)

with

${\cal
H}(t,\vec{x})=\frac{1}{2}(\pi^{2}(t,\vec{x})+(\vec{\nabla}\phi(t,\vec{x}))^{2})$
(2.28)

the energy density of the scalar field.

##  3 The BMS algebra in phase space

The abstract BMS algebra in $2+1$ is given by

$\displaystyle[L_{n},P_{m}]$ $\displaystyle=(n-m)P_{m+n},$ (3.1)
$\displaystyle[P_{m},P_{m^{\prime}}]$ $\displaystyle=0,$ (3.2)

with $n\in\\{-1,0,1\\}$ and $m,m^{\prime}\in\mathbb{Z}$, and with the $L_{n}$
satisfying $[L_{n},L_{n^{\prime}}]=(n-n^{\prime})L_{n+n^{\prime}}$ and
yielding the $2+1$ Lorentz algebra. The algebra can be extended to
$n,n^{\prime}\in\mathbb{Z}$ by introducing the superrotations $L_{n}$, $|n|>1$
[28][29].

We will show that the above supertranslation charges (2.11), (2.12), (2.15)
provide a realization of this algebra in terms of the standard equal-time
Poisson brackets and with the following combinations of Lorentz generators:

$\displaystyle L_{0}(t)$ $\displaystyle=\frac{1}{2i}M^{12}(t),$ (3.3)
$\displaystyle L_{1}(t)$ $\displaystyle=-M^{01}(t)-iM^{02}(t),$ (3.4)
$\displaystyle L_{-1}(t)$
$\displaystyle=M^{01}(t)-iM^{02}(t)=-(L_{1}(t))^{*}.$ (3.5)

The proof relies on the general symmetry properties of the Green functions
$G_{\ell}$ and their derivatives, as well as on a key identity that is proved
in D. The brackets between the supertranslation charges are discussed in B and
here we will discuss only those involving the Lorentz generators.

Let us consider first the Poisson bracket $\\{L_{0}(t),Q_{2\ell}(t)\\}$, for
$\ell\geq 0$. With the notation $\phi(x)=\phi(t,\vec{x})$ and so on, and
defining

${\cal
H}(x,y)=\frac{1}{2}\big{(}\pi(x)\pi(y)+\vec{\nabla}\phi(x)\cdot\vec{\nabla}\phi(y)\big{)},$
(3.6)

one has

$\displaystyle\\{L_{0}(t),Q_{2\ell}(t)\\}=$
$\displaystyle=-\frac{1}{2i}\int\text{d}^{2}x\text{d}^{2}y\text{d}^{2}z\left\\{\pi(x)(x_{1}\partial_{x_{2}}\phi(x)-x_{2}\partial_{x_{1}}\phi(x)),{\cal
H}(y,z)\right\\}(\partial_{y_{1}}+i\partial_{y_{2}})^{2\ell}G_{\ell}(\vec{y}-\vec{z}).$

This bracket is computed in C, and the result is (see (C.3))

$\displaystyle\\{L_{0}(t),Q_{2\ell}(t)\\}$ $\displaystyle=-2\ell
Q_{2\ell}(t),\quad\ell\geq 0.$ (3.7)

Using similar steps to those of C, one can also obtain

$\displaystyle\\{L_{0}(t),Q_{2\ell+1}(t)\\}$
$\displaystyle=-(2\ell+1)Q_{2\ell+1}(t).\quad\ell\geq 0.$ (3.8)

The above computations are only valid for $\ell\geq 0$. For $\ell<0$ one can
take the complex conjugate of (3.7) and (3.8), and use (2.15) and also
$L_{0}^{*}(t)=-L_{0}(t)$. In this way one obtains, for $\ell\geq 0$,

$\displaystyle\\{L_{0}(t),Q_{-2\ell}(t)\\}$ $\displaystyle=2\ell
Q_{-2\ell}(t)=-(-2\ell)Q_{-2\ell}(t),$ (3.9)
$\displaystyle\\{L_{0}(t),Q_{-(2\ell+1)}(t)\\}$
$\displaystyle=(2\ell+1)Q_{-(2\ell+1)}(t)=-(-(2\ell+1))Q_{-(2\ell+1)}(t),.$
(3.10)

Relations (3.7—3.10) give the complete set of BMS algebra relations involving
the rotation generator $L_{0}$.

Let us proceed now with the brackets involving the boost generators. Consider
first

$\displaystyle\\{L_{1}(t),Q_{2\ell}(t)\\}=$ $\displaystyle=$
$\displaystyle\int\text{d}^{2}x\text{d}^{2}y\text{d}^{2}z\Big{\\{}t\pi(x)(\partial_{x_{1}}+i\partial_{x_{2}})\phi(x)+(x_{1}+ix_{2}){\cal
H}(x),{\cal
H}(y,z)\Big{\\}}(\partial_{y_{1}}+i\partial_{y_{2}})^{2\ell}G_{\ell}(\vec{y}-\vec{z}).$

As shown in C, this can be seen to be (see equation (C.7))

$\displaystyle\\{L_{1}(t),Q_{2\ell}(t)\\}$
$\displaystyle=(1-2\ell)Q_{2\ell+1}(t),$ (3.11)

which is the correct action of $L_{1}$ on a supertranslation of order $2\ell$,
$\ell>0$, and can be extended to the trivial (Poincaré) case $\ell=0$.

Using similar computations, together with
$G_{\ell}=\vec{\nabla}^{2}G_{\ell+1}$ one can show that, for $\ell>0$,

$\displaystyle\\{L_{1}(t),Q_{2\ell+1}(t)\\}$
$\displaystyle=-2\ell\int\text{d}^{2}x\text{d}^{2}y{\cal
H}(x,y)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell+2}G_{\ell+1}(\vec{x}-\vec{y}).$
(3.12)

Changing now $\ell\to\ell-1$ one has

$\displaystyle\\{L_{1}(t),Q_{2\ell-1}(t)\\}$
$\displaystyle=-2(\ell-1)\int\text{d}^{2}x\text{d}^{2}y{\cal
H}(x,y)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(\vec{x}-\vec{y})$
$\displaystyle=-2(\ell-1)Q_{2\ell}(t)=\big{(}1-(2\ell-1)\big{)}Q_{2l}(t),\quad\ell>1,$
(3.13)

which is the correct relation, and which again can be extended to the Poincaré
$\ell=1$ case.

In order to get all the relations of the BMS algebra, an extra pair of
brackets must be computed. The final results are

$\displaystyle\\{L_{1}(t),Q_{-2\ell}(t)\\}$
$\displaystyle=(2\ell+1)Q_{-2\ell+1}(t)=(1-(-2\ell))Q_{-2\ell+1}(t),$ (3.14)
$\displaystyle\\{L_{1}(t),Q_{-(2\ell+1)}(t)\\}$
$\displaystyle=(2\ell+2)Q_{-2\ell}(t)=(1-(-(2\ell+1)))Q_{-2\ell}(t).$ (3.15)

In these cases the computations involve slightly different manipulations but
always using (D.5).

The brackets involving $L_{-1}$ can be computed from (3.11), (3.13), (3.14)
and (3.15) by complex conjugation and using $L_{1}(t)=-L_{-1}^{*}(t)$, and one
gets

$\displaystyle\\{L_{-1}(t),Q_{-2\ell}(t)\\}$
$\displaystyle=\big{(}-1+2\ell\big{)}Q_{-(2\ell+1)}(t)=\big{(}-1-(-2\ell)\big{)}Q_{-(2\ell+1)}(t),$
(3.16) $\displaystyle\\{L_{-1}(t),Q_{2\ell}(t)\\}$
$\displaystyle=-\big{(}2\ell+1\big{)}Q_{2\ell-1}(t)=\big{(}-1-2\ell)\big{)}Q_{2\ell-1}(t),$
(3.17) $\displaystyle\\{L_{-1}(t),Q_{2\ell+1}(t)\\}$
$\displaystyle=-\big{(}2\ell+2\big{)}Q_{2\ell}(t)=\big{(}-1-(2\ell+1)\big{)}Q_{2\ell}(t),$
(3.18) $\displaystyle\\{L_{-1}(t),Q_{-2\ell-1}(t)\\}$ $\displaystyle=2\ell
Q_{-2(\ell+1)}(t)=\big{(}-1-(-2\ell-1)\big{)}Q_{-2(\ell+1)}(t),.$ (3.19)

This completes the proof that the charges defined by (2.11), (2.12) and (2.15)
provide, together with the Lorentz generators, a realization of the $2+1$ BMS
algebra.

##  4 The BMS algebra in configuration space

In the previous section we have shown that the BMS algebra is obtained in
phase space using the expression of the supertranslation and Lorentz charges
in terms of the fields $\phi$, $\pi$. We will discuss now the transformations
in configuration space, using $\phi$ and $\dot{\phi}$ as independent fields.
The result is that one obtains a BMS algebra of transformations modulo trivial
symmetry transformations, given by skew-symmetric combinations of the
equations of motion of $\phi$.

In order to show this, let us consider the specific case of the Lorentz
transformation $\delta_{1}^{B}$ associated to $L_{1}$ and the transformation
$\delta_{2\ell}$ given by the supertranslation charge $Q_{2\ell}$, $\ell\geq
0$. In configuration space we must substitute $\dot{\phi}$ for $\pi$, and the
transformations are

$\displaystyle\delta^{B}_{1}\phi(x)$
$\displaystyle=t(\partial_{x_{1}}+i\partial_{x_{2}})\phi(x)+(x_{1}+ix_{2})\dot{\phi}(x),$
(4.1) $\displaystyle\delta_{{2\ell}}\phi(x)$
$\displaystyle=\int\text{d}^{2}y\,\dot{\phi}(y)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(\vec{x}-\vec{y}).$
(4.2)

Since these are functional variations, the transformation of $\dot{\phi}$ is
obtained by derivation, and one gets

$\displaystyle\delta^{B}_{1}\dot{\phi}(x)$
$\displaystyle=(\partial_{x_{1}}+i\partial_{x_{2}})\phi(x)+t(\partial_{x_{1}}+i\partial_{x_{2}})\dot{\phi}(x)+(x_{1}+ix_{2})\ddot{\phi}(x),$
(4.3) $\displaystyle\delta_{{2\ell}}\dot{\phi}(x)$
$\displaystyle=\int\text{d}^{2}y\,\ddot{\phi}(y)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(\vec{x}-\vec{y}).$
(4.4)

Now we can compute the compositions of transformations

$\displaystyle\delta_{{2\ell}}\delta^{B}_{1}\phi(x)$
$\displaystyle=t(\partial_{x_{1}}+i\partial_{x_{2}})\delta_{2\ell}\phi(x)+(x_{1}+ix_{2})\delta_{{2\ell}}\dot{\phi}(x)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!=t\int
d^{2}y\,\dot{\phi}(y)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell+1}G_{\ell}(\vec{x}-\vec{y})+(x_{1}+ix_{2})\int
d^{2}y\,\ddot{\phi}(y)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(\vec{x}-\vec{y}),$
(4.5)

and

$\displaystyle\delta^{B}_{1}\delta_{{2\ell}}\phi(x)$ $\displaystyle=\int
d^{2}y\,\delta^{B}_{1}\dot{\phi}(y)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(\vec{x}-\vec{y})$
$\displaystyle=\int
d^{2}y\,(\partial_{y_{1}}+i\partial_{y_{2}})\phi(y)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(\vec{x}-\vec{y})$
$\displaystyle+t\int
d^{2}y\,(\partial_{y_{1}}+i\partial_{y_{2}})\dot{\phi}(y)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(\vec{x}-\vec{y})$
$\displaystyle+\int
d^{2}y\,(y_{1}+iy_{2})\ddot{\phi}(y)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(\vec{x}-\vec{y}).$
(4.6)

The first term in (4.6),

$\int
d^{2}y\,(\partial_{y_{1}}+i\partial_{y_{2}})\phi(y)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(\vec{x}-\vec{y})=\int
d^{2}y\,\phi(y)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell+1}G_{\ell}(\vec{x}-\vec{y})$

is just $-\delta_{2\ell+1}\phi(x)$ and, assembling the remaining terms, the
commutator of the two transformations turns out to be

$\displaystyle[\delta^{B}_{1},\delta_{{2\ell}}]\phi(x)$
$\displaystyle=-\delta_{{2\ell+1}}\phi(x)-\int
d^{2}y\,\big{(}(x_{1}-y_{1})+i(x_{2}-y_{2})\big{)}\ddot{\phi}(y)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(\vec{x}-\vec{y}).$
(4.7)

Now we add and subtract $\vec{\nabla}^{2}\phi$ and obtain

$\displaystyle[\delta^{B}_{1},\delta_{{2\ell}}]\phi(x)=-\delta_{{2\ell+1}}\phi(x)-\int\text{d}^{2}y\,\big{(}(x_{1}-y_{1})+i(x_{2}-y_{2})\big{)}\vec{\nabla}^{2}_{y}\phi(y)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(\vec{x}-\vec{y})$
(4.8) $\displaystyle-$
$\displaystyle\int\text{d}^{2}y\,\big{(}(x_{1}-y_{1})+i(x_{2}-y_{2})\big{)}\big{(}\ddot{\phi}(y)-\vec{\nabla}^{2}_{y}\phi(y)\big{)}(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(\vec{x}-\vec{y})$
$\displaystyle=$
$\displaystyle-\delta_{{2\ell+1}}\phi(x)-\int\text{d}^{2}y\,\big{(}(x_{1}-y_{1})+i(x_{2}-y_{2})\phi(y)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}(\partial_{x_{1}}-i\partial_{x_{2}})G_{\ell}(\vec{x}-\vec{y})$
$\displaystyle-$ $\displaystyle
2\int\mbox{d}^{2}y\,\phi(y)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell+1}G_{\ell}(\vec{x}-\vec{y})-\int\text{d}^{2}y\,F_{\ell}(x,y)\big{(}\ddot{\phi}(y)-\vec{\nabla}^{2}_{y}\phi(y)\big{)},$

where we have integrated twice by parts the term $\vec{\nabla}^{2}\phi$ and
defined

$F_{\ell}(x,y)=\big{(}(x_{1}-y_{1})+i(x_{2}-y_{2})\big{)}(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(\vec{x}-\vec{y}).$
(4.9)

The third term in (4.8) is $2\delta_{2\ell+1}\phi(x)$, while the second term,
following the same steps that led to (C.6), becomes
$2(\ell-1)\delta_{2\ell+1}\phi(x)$. Putting everything together one has

$\displaystyle[\delta^{B}_{1},\delta_{{2\ell}}]\phi(x)$
$\displaystyle=(2\ell-1)\delta_{2\ell+1}\phi(x)-\int\text{d}^{2}y\,F_{\ell}(x,y)\big{(}\ddot{\phi}(y)-\vec{\nabla}^{2}_{y}\phi(y)\big{)}.$
(4.10)

Taking into account that, for any transformations generated by charges $A$,
$B$, one has that $[\delta_{A},\delta_{B}]\phi=-\delta_{\\{A,B\\}}\phi$, the
first term in (4.10) is the one expected from the BMS algebra and, because
$F_{\ell}(x,y)=-F_{\ell}(y,x)$, the extra term is a skew-symmetric linear
combination of the equations of motion,

$\delta_{\ell,\text{trivial}}\phi(x)=\int\text{d}^{2}y\,F_{\ell}(x,y)\big{(}\ddot{\phi}(y)-\vec{\nabla}^{2}_{y}\phi(y)\big{)},$
(4.11)

which is a trivial symmetry transformation of any system. Notice that, for
$\ell=0$, $F_{\ell}(x,y)=0$ and, as it must be, the extra term is not present
for the standard commutator of a time translation and a Lorentz boost.

Similar results are obtained for the other commutators of transformations, and
hence the algebra closes on-shell in a consistent way.

##  5 Discussion and outlook

We have obtained an explicit expression for the BMS supertranslation charges
of free massless scalar real scalar field in $2+1$ space-time, in terms of the
Green functions of the polyharmonic operator.

We work first in phase space, and discuss the asymptotic behaviour of the
fields that ensures the existence of the charges, as well as that of the
symplectic form associated to the Poisson brackets.

The conservation of the charges only depends on general symmetry properties of
the involved functions, but the commutative character of the algebra satisfied
by these charges relies on a convolution property of the Green functions.
Finally, the correct algebra with the Lorentz generators is obtained using
more specific properties of the polyharmonic Green functions.

We also discuss the closure of the transformations in configuration space, and
it turns out that the correct algebra is obtained modulo transformations given
by skew-symmetric combinations of the equations of motion, which are trivial
symmetry transformations of any system.

The form of the supertranslation charges presented in this paper, in terms of
Green functions of an appropriate operator, opens the way to the
generalization to other cases and/or dimensions. For instance, one could
consider the extension of the results to the $3+1$ space-time dimension case,
or to the case of the generators of superrotations which were also constructed
in [17].

Another subject worth of study is whether the supertranslation transformations
of the field $\phi$ can be interpreted as a base transformation of the
coordinates in Minkowski space, or if additional coordinates, each associated
to a supertranslation, must be introduced.

##  6 Acknowledgements

We acknowledge discussions with Marc Henneaux and Axel Kleinschmidt.

The work of CB is partially supported by Project PID2021-126001OB-C31 funded
by MCIN/ AEI/ 10.13039/ 501100011033/ ERDF, EU. JG has been supported in part
by MINECO FPA2016-76005-C2-1-P and PID2019- 105614GB-C21 and from the State
Agency for Research of the Spanish Ministry of Science and Innovation through
the Unit of Excellence Maria de Maeztu 2020-2023 award to the Institute of
Cosmos Sciences (CEX2019-000918-M).

##  Appendix A Asymptotic behaviour of the polyharmonic Green functions

The polyharmonic Green function $G_{\ell}$ has the form

$G_{\ell}(x)=A_{\ell}^{(0)}|\vec{x}|^{2\ell-2}\log|\vec{x}|+B_{\ell}^{(0)}|\vec{x}|^{2\ell-2},$
(A.1)

where the constants $A_{\ell}^{(0)}$ and $B_{\ell}^{(0)}$ can be read from
(2.5). Successive applications of $\partial_{x_{1}}+i\partial_{x_{2}}$ yield
expressions of the same form, with decreasing powers of $|\vec{x}|$, until one
reaches

$(\partial_{x_{1}}+i\partial_{x_{2}})^{\ell-1}G_{\ell}(\vec{x})=(x_{1}+ix_{2})^{\ell-1}\left(A_{\ell}^{(\ell-1)}\log|\vec{x}|+B_{\ell}^{(\ell-1)}\right).$
(A.2)

From this point the derivatives cease to content the $\log$ term and one can
see that the derivative of order $2\ell$ is of the form

$(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(\vec{x})=C^{(2\ell)}(x_{1}+ix_{2})^{2\ell}\frac{1}{|\vec{x}|^{2\ell+2}},$
(A.3)

with a constant $C^{(2\ell)}$. This is a rational function of $x_{1}$, $x_{2}$
with asymptotic behaviour

$(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(\vec{x})\sim\frac{1}{|\vec{x}|^{2}}\quad\forall\ell\geq
1\quad\text{for $|\vec{x}|\to\infty$},$ (A.4)

which is independent of $\ell\geq 1$. This allows us to study the conditions
that must be imposed on the fields so that the supertranslation charges are
finite. A general supertraslation charge $Q_{\ell}$, $\ell\geq 1$, has
integrals of the form

$Q_{\ell}(t)=\int\text{d}^{2}x\text{d}^{2}yF(t,\vec{x})G(t,\vec{y})(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(\vec{x}-\vec{y})$
(A.5)

where $F$ and $G$ are either $\pi$ or first order derivatives of $\phi$.
Performing a change of variables $\vec{x}=\vec{y}+\vec{r}$ and using the
asymptotic behaviour (A.4), the existence of the charge reduces to the
existence of the integral

$\int\mbox{d}^{2}r\mbox{d}^{2}yF(\vec{y}+\vec{r})G(\vec{y})\frac{1}{r^{2}}.$
(A.6)

Let us assume now that the fields $F$ and $G$ behave, for large argument, as

$F(\vec{y}+\vec{r})\sim\frac{\bar{F}}{|\vec{y}+\vec{r}|^{\alpha}},\quad
G(\vec{y})\sim\frac{\bar{G}}{|\vec{y}|^{\beta}},$ (A.7)

with $\bar{F}$, $\bar{G}$ depending on the angular variable and time. This
leads to the study of the integral

$\int\mbox{d}^{2}\Omega\,\bar{F}\bar{G}\int\,r\mbox{d}r\,y\mbox{d}y\,\frac{1}{|\vec{y}+\vec{r}|^{\alpha}|\vec{y}|^{\beta}r^{2}},$
(A.8)

where $\mbox{d}^{2}\Omega$ is the angular measure. Performing a change to
polar coordinates in $\mathbb{R}^{2}_{+}$ for the $\mbox{d}r\mbox{d}y$
measure, one finally gets

$\int\mbox{d}^{3}\Omega\,\bar{F}\bar{G}\int\,\rho\mbox{d}\rho\,\frac{1}{\rho^{\alpha+\beta}},$
(A.9)

with $\mbox{d}^{3}\Omega$ including the additional integration over the
angular coordinate of the polar change of variables. For this integral to
converge it is necessary that

$\alpha+\beta>2.$ (A.10)

Considering the forms of $F$ and $G$ for the different supertranslation
charges, one concludes that the asymptotic behaviour of the fields which
guarantees the existence of all the $Q_{\ell}$ is the one given in (2.21) and
(2.22).

##  Appendix B Brackets between the supertranslation charges

Consider two arbitrary supertranslation charges

$\displaystyle Q_{\ell}(t)$
$\displaystyle=\int\text{d}^{2}x\text{d}^{2}y\,(f_{\ell}(\vec{x}-\vec{y})\pi(x)\phi(y)+\dfrac{1}{2}g_{\ell}(\vec{x}-\vec{y})\pi(x)\pi(y)-\dfrac{1}{2}h_{\ell}(\vec{x}-\vec{y})\phi(x)\phi(y)),$
(B.1) $\displaystyle Q_{m}(t)$
$\displaystyle=\int\text{d}^{2}z\text{d}^{2}w\,(f_{m}(\vec{z}-\vec{w})\pi(z)\phi(w)+\dfrac{1}{2}g_{m}(\vec{z}-\vec{w})\pi(z)\pi(w)-\dfrac{1}{2}h_{m}(\vec{z}-\vec{w})\phi(z)\phi(w)).$
(B.2)

Using standard Poisson brackets, one gets

$\displaystyle\\{Q_{\ell}(t),Q_{m}(t)\\}$
$\displaystyle=\int\text{d}^{2}x\text{d}^{2}y\text{d}^{2}z\,\Big{(}f_{\ell}(\vec{x}-\vec{y})f_{m}(\vec{y}-\vec{z})\pi(x)\phi(z)-f_{\ell}(\vec{x}-\vec{y})f_{m}(\vec{z}-\vec{x})\phi(y)\pi(z)$
$\displaystyle+f_{\ell}(\vec{x}-\vec{y})g_{m}(\vec{y}-\vec{z})\pi(x)\pi(z)+f_{\ell}(\vec{x}-\vec{y})h_{m}(\vec{x}-\vec{z})\phi(y)\phi(z)$
$\displaystyle-
g_{\ell}(\vec{x}-\vec{y})f_{m}(\vec{z}-\vec{x})\pi(y)\pi(z)+g_{\ell}(\vec{x}-\vec{y})h_{m}(\vec{x}-\vec{z})\pi(y)\phi(z)$
$\displaystyle-
h_{\ell}(\vec{x}-\vec{y})f_{m}(\vec{x}-\vec{z})\phi(y)\phi(z)-h_{\ell}(\vec{x}-\vec{y})g_{m}(\vec{x}-\vec{z})\phi(y)\pi(z)\Big{)}.$
(B.3)

Let us consider first the case $m=0$, that is $Q_{m}(t)=Q_{0}(t)=H(t)$, for
which $f_{0}(\vec{x}-\vec{y})=0$,
$g_{0}(\vec{x}-\vec{y})=\delta(\vec{x}-\vec{y})$ and
$h_{0}(\vec{x}-\vec{y})=\vec{\nabla}^{2}_{x}\delta(\vec{x}-\vec{y})$. After
integration by parts one gets

$\displaystyle\dot{Q}_{\ell}(t)$
$\displaystyle=\\{Q_{\ell}(t),H(t)\\}=\int\text{d}^{2}x\text{d}^{2}y\,\Big{(}f_{\ell}(\vec{x}-\vec{y})\pi(x)\pi(y)+\vec{\nabla}^{2}_{x}f_{\ell}(\vec{x}-\vec{y})\phi(y)\phi(x)$
$\displaystyle+\vec{\nabla}^{2}_{x}g_{\ell}(\vec{x}-\vec{y})\pi(y)\phi(x)-h_{\ell}(\vec{x}-\vec{y})\phi(y)\pi(x)\Big{)}.$
(B.4)

The first two terms are zero, each by itself, due to the skew-symmetry of
$f_{\ell}$ and its even-order derivatives, while the two last terms cancel
each other after using
$\vec{\nabla}^{2}(x)g_{\ell}(\vec{x}-\vec{y})=h_{\ell}(\vec{x}-\vec{y})$ and
the symmetry of $h_{\ell}$. This shows that the supertraslation charges are
conserved by virtue of the symmetry properties of $f_{\ell}$, $g_{\ell}$ and
$h_{\ell}$, and the relation between $g_{\ell}$ and $h_{\ell}$, without using
the explicit form of these functions in terms of the polyharmonic Green
functions.

For general $\ell$ and $m$ one must consider the different cases separately.

  1. 1.

$\ell$ and $m$ odd. In this case $g_{\ell}=h_{\ell}=g_{m}=h_{m}=0$ and, after
renaming the variables of integration in the first non-zero contribution,

$\displaystyle\\{Q_{\ell}(t),Q_{m}(t)\\}$
$\displaystyle=\int\text{d}^{2}x\text{d}^{2}y\text{d}^{2}z\,\Big{(}f_{\ell}(\vec{z}-\vec{x})f_{m}(\vec{x}-\vec{y})-f_{\ell}(\vec{x}-\vec{y})f_{m}(\vec{z}-\vec{x})\Big{)}\phi(y)\pi(z).$
(B.5)

  2. 2.

$\ell$ even and $m$ odd. Now $f_{\ell}=0$ and $g_{m}=h_{m}=0$, and the result
can be written as

$\displaystyle\\{Q_{\ell}(t),Q_{m}(t)\\}$
$\displaystyle=-\int\text{d}^{2}x\text{d}^{2}y\text{d}^{2}z\,g_{\ell}(\vec{x}-\vec{y})f_{m}(\vec{z}-\vec{x})\pi(y)\pi(z)$
$\displaystyle-\int\text{d}^{2}x\text{d}^{2}y\text{d}^{2}z\,h_{\ell}(\vec{x}-\vec{y})f_{m}(\vec{x}-\vec{z})\phi(y)\phi(z).$
(B.6)

  3. 3.

$\ell$ and $m$ even. We have $f_{\ell}=f_{m}=0$ and (B.3) boils down to

$\displaystyle\\{Q_{\ell}(t),Q_{m}(t)\\}$
$\displaystyle=\int\text{d}^{2}x\text{d}^{2}y\text{d}^{2}z\,\Big{(}g_{\ell}(\vec{x}-\vec{z})h_{m}(\vec{x}-\vec{y})-h_{\ell}(\vec{x}-\vec{y})g_{m}(\vec{x}-\vec{z})\Big{)}\phi(y)\pi(z).$
(B.7)

The elementary symmetry properties used up to now are not enough to show that
the above expressions are actually zero. In order to do so, one must use the
fact that the functions $f_{\ell}$, $g_{\ell}$ and $h_{\ell}$ can be written
in terms of Green functions that obey the convolution property (2.7).

Let us prove, for instance, that the first term in (B.6) is zero. Changing
$\ell\to 2\ell$ and $m\to 2m+1$ one has, assuming $l>0$, $m>0$,

$\displaystyle-\int\text{d}^{2}x\text{d}^{2}y\text{d}^{2}z\,g_{\ell}(\vec{x}-\vec{y})f_{m}(\vec{z}-\vec{x})\pi(y)\pi(z)\rightarrow$
$\displaystyle-$
$\displaystyle\int\text{d}^{2}x\text{d}^{2}y\text{d}^{2}z\,(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(\vec{x}-\vec{y})(\partial_{x_{1}}+i\partial_{x_{2}})^{2m+1}G_{m}(\vec{x}-\vec{z})\pi(y)\pi(z)$
$\displaystyle=$
$\displaystyle-\int\text{d}^{2}x\text{d}^{2}y\text{d}^{2}z\,G_{\ell}(\vec{x}-\vec{y})G_{m}(\vec{x}-\vec{z})(\partial_{y_{1}}+i\partial_{y_{2}})^{2\ell}\pi(y)(\partial_{z_{1}}+i\partial_{z_{2}})^{2m+1}\pi(z)$
$\displaystyle\stackrel{{\scriptstyle(\ref{convGG})}}{{=}}$
$\displaystyle-\int\text{d}^{2}y\text{d}^{2}zG_{\ell+m}(\vec{y}-\vec{z})(\partial_{y_{1}}+i\partial_{y_{2}})^{2\ell}\pi(y)(\partial_{z_{1}}+i\partial_{z_{2}})^{2m+1}\pi(z)$
$\displaystyle=$
$\displaystyle-\int\text{d}^{2}y\text{d}^{2}z(\partial_{y_{1}}+i\partial_{y_{2}})^{2\ell+2m+1}G_{\ell+m}(\vec{y}-\vec{z})\pi(y)\pi(z)=0$

due to the skew-symmetry of
$(\partial_{y_{1}}+i\partial_{y_{2}})^{2\ell+2m+1}G_{\ell+m}(\vec{y}-\vec{z})$,
and the second term of (B.6) can also be shown to be zero using the same
manipulations. Notice that the same reasoning can be used for $\ell$ and/or
$m$ negative, since this amounts to change some
$\partial_{x_{1}}+i\partial_{x_{2}}$ to $\partial_{x_{1}}-i\partial_{x_{2}}$
and the result, which only depends on the number of derivatives, is the same.

Using the same techniques and the convolution property, the two terms which
appear in (B.5) or (B.7) can be shown to be the same and hence that the
corresponding brackets are zero. This completes the proof that the
supertraslation charges yield a commutative algebra under the Poisson
brackets.

##  Appendix C Detailed computation of some Poisson brackets

Consider first

$\displaystyle\\{L_{0}(t),Q_{2\ell}(t)\\}=$
$\displaystyle=-\frac{1}{2i}\int\text{d}^{2}x\text{d}^{2}y\text{d}^{2}z\left\\{\pi(x)(x_{1}\partial_{x_{2}}\phi(x)-x_{2}\partial_{x_{1}}\phi(x)),{\cal
H}(y,z)\right\\}(\partial_{y_{1}}+i\partial_{y_{2}})^{2\ell}G_{\ell}(\vec{y}-\vec{z}).$

Using that even-order derivatives of an even-symmetric function are even, and
the equal-time Poisson brackets
$\\{\phi(x),\pi(y)\\}=\delta(\vec{x}-\vec{y})$, the above boils down to

$\displaystyle\\{L_{0}(t),Q_{2\ell}(t)\\}$
$\displaystyle=-\frac{1}{2i}\int\text{d}^{2}x\text{d}^{2}y\text{d}^{2}z\Big{(}(x_{1}\partial_{x_{2}}\phi(x)-x_{2}\partial_{x_{1}}\phi(x))\vec{\nabla}\phi(z)\cdot\vec{\nabla}_{y}(-\delta(\vec{x}-\vec{y}))$
$\displaystyle+\pi(x)\pi(z)(x_{1}\partial_{x_{2}}-x-2\partial_{x_{1}})\delta(\vec{x}-\vec{y})\Big{)}(\partial_{y_{1}}+i\partial_{y_{2}})^{2\ell}G_{\ell}(\vec{y}-\vec{z})$
$\displaystyle=-\frac{1}{2i}\int\text{d}^{2}x\text{d}^{2}z\Big{(}{2\cal{H}}(x,z)\Big{)}(x_{1}\partial_{x_{2}}-x_{2}\partial_{x_{1}})(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(\vec{x}-\vec{z}),$

where several integrations by parts, assuming the appropriate asymptotic
behaviour for the fields, have been performed, and the relation
$\vec{\nabla}_{x}G_{\ell}(\vec{x}-\vec{y})=-\vec{\nabla}_{y}G_{\ell}(\vec{x}-\vec{y})$
has been used. Next, we use the commutator

$\big{[}x_{1}\partial_{x_{2}}-x_{2}\partial_{x_{1}},(\partial_{x_{1}}+i\partial_{x_{2}})^{n}\big{]}=in(\partial_{x_{1}}+i\partial_{x_{2}})^{n},\quad
n=0,1,\ldots$ (C.1)

to write the above as

$\displaystyle\\{L_{0}(t),Q_{2\ell}(t)\\}$
$\displaystyle=-\frac{1}{i}\int\text{d}^{2}x\text{d}^{2}z{\cal
H}(x,z)\Big{(}(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}(x_{1}\partial_{x_{2}}-x_{2}\partial_{x_{1}})$
$\displaystyle\quad+i\ 2\ell\
(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}\Big{)}G_{\ell}(\vec{x}-\vec{z}).$

One has that

$(x_{1}\partial_{x_{2}}-x_{2}\partial_{x_{1}})G_{\ell}(\vec{x}-\vec{z})=G^{\prime}_{\ell}(|\vec{x}-\vec{z}|)\frac{-x_{1}z_{2}+x_{2}z_{1}}{|\vec{x}-\vec{z}|}$
(C.2)

is and odd function under $\vec{x}\leftrightarrow\vec{z}$, and hence its
derivatives of even order are also odd. The product with ${\cal H}(x,z)$ is
also odd and the term vanishes under integration in $x$ and $z$. Thus one is
left only with the term from the commutator and

$\displaystyle\\{L_{0}(t),Q_{2\ell}(t)\\}$
$\displaystyle=-2\ell\int\text{d}^{2}x\text{d}^{2}z{\cal
H}(x,z)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}G_{\ell}(\vec{x}-\vec{z})$
$\displaystyle=-2\ell Q_{2\ell}(t),\quad\ell\geq 0.$ (C.3)

Let us compute now the Poisson bracket of the boost generator $L_{1}(t)$ with
an even-order supertranslation charge,

$\displaystyle\\{L_{1}(t),Q_{2\ell}(t)\\}=$ $\displaystyle=$
$\displaystyle\int\text{d}^{2}x\text{d}^{2}y\text{d}^{2}z\Big{\\{}t\pi(x)(\partial_{x_{1}}+i\partial_{x_{2}})\phi(x)+(x_{1}+ix_{2}){\cal
H}(x),{\cal
H}(y,z)\Big{\\}}(\partial_{y_{1}}+i\partial_{y_{2}})^{2\ell}G_{\ell}(\vec{y}-\vec{z}).$

After computing the Poisson brackets and using integration by parts and the
symmetry of $G_{\ell}$ and its derivatives, the terms containing $t$ can be
written as

$2t\int\text{d}^{2}x\text{d}^{2}z{\cal
H}(x,z)(\partial_{x_{1}}+i\partial_{x_{2}})G_{\ell}(\vec{x}-\vec{z}),$

which is zero due to the skew-symmetry of
$(\partial_{x_{1}}+i\partial_{x_{2}})G_{\ell}(\vec{x}-\vec{z})$ under
$\vec{x}\leftrightarrow\vec{z}$. Performing the same manipulations, the
remaining terms can be written as

$\displaystyle\\{L_{1}(t),Q_{2\ell}(t)\\}$
$\displaystyle=\int\text{d}^{2}x\text{d}^{2}z\Big{(}(x_{1}+ix_{2})\pi(x)\vec{\nabla}\phi(z)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}\cdot\vec{\nabla}_{x}G_{\ell}(\vec{x}-\vec{z})$
$\displaystyle\quad+(x_{1}+ix_{2})\vec{\nabla}\phi(x)\pi(z)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}\cdot\vec{\nabla}_{x}G_{\ell}(\vec{x}-\vec{z})\Big{)}.$
(C.4)

The $\vec{\nabla}\phi(z)$ in the first term can be integrated by parts,
yielding one term, while the integration by parts of $\vec{\nabla}\phi(x)$
yields two. After a change of variables and using
$(\partial_{z_{1}}+i\partial_{z_{2}})^{2\ell}\vec{\nabla}_{z}^{2}G_{l}(\vec{x}-\vec{z})=(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}\vec{\nabla}_{x}^{2}G_{l}(\vec{x}-\vec{z})$,
the two terms that are similar can be combined, and the result is

$\displaystyle\\{L_{1}(t),Q_{2\ell}(t)\\}$
$\displaystyle=\int\text{d}^{2}x\text{d}^{2}z\Big{(}((x_{1}-z_{1})+i(x_{2}-z_{2}))\pi(x)\phi(z)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}\cdot\vec{\nabla}_{x}^{2}G_{\ell}(\vec{x}-\vec{z})$
$\displaystyle\quad-\phi(x)\pi(z)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell}(\partial_{x_{1}}+i\partial_{x_{2}})G_{\ell}(\vec{x}-\vec{z})\Big{)}.$
(C.5)

The second term is just $-Q_{2\ell+1}(t)$ and the integrand in the first one
can be re-written, using that
$[x_{1}+ix_{2},\partial_{x_{1}}+i\partial_{x_{2}}]=0$ and
$\vec{\nabla}^{2}_{x}=(\partial_{x_{1}}+i\partial_{x_{2}})(\partial_{x_{1}}-i\partial_{x_{2}})$,
as

$\displaystyle\\{L_{1}(t),Q_{2\ell}(t)\\}=-Q_{2\ell+1}(t)$ (C.6)
$\displaystyle\\!+\int\text{d}^{2}x\text{d}^{2}z\Big{(}\pi(x)\phi(z)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell+1}((x_{1}-z_{1})+i(x_{2}-z_{2}))(\partial_{x_{1}}-i\partial_{x_{2}})G_{\ell}(\vec{x}-\vec{z})\Big{)}.$

Since we are considering $\ell\geq 1$ (the case $\ell=0$ correspond to the
standard Poincaré algebra) and $2\ell+1>2(\ell-1)$, we can use relation (D.5)
to rewrite (C.6) as

$\displaystyle\\{L_{1}(t),Q_{2\ell}(t)\\}$
$\displaystyle=-Q_{2\ell+1}(t)+2(\ell-1)\int\text{d}^{2}x\text{d}^{2}z\pi(x)\phi(z)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell+1}G_{\ell}(\vec{x}-\vec{z})$
$\displaystyle=-Q_{2\ell+1}(t)-2(\ell-1)\int\text{d}^{2}x\text{d}^{2}z\phi(x)\pi(z)(\partial_{x_{1}}+i\partial_{x_{2}})^{2\ell+1}G_{\ell}(\vec{x}-\vec{z})$
$\displaystyle=-Q_{2\ell+1}(t)-2(\ell-1)Q_{2\ell+1}(t)$
$\displaystyle=(1-2\ell)Q_{2\ell+1}(t).$ (C.7)

##  Appendix D Some identities satisfied by the polyharmonic Green functions

From (2.5) one has

$\displaystyle(\partial_{x_{1}}-i\partial_{x_{2}})G_{\ell}(\vec{x}-\vec{y})$
$\displaystyle=\frac{|\vec{x}-\vec{y}|^{2(\ell-2)}}{(\ell-1)[(\ell-2)!]^{2}2^{2\ell-2}\pi}((x_{1}-y_{1})-i(x_{2}-y_{2}))\big{(}\log|\vec{x}-\vec{y}|-H_{\ell-1}\big{)}$
$\displaystyle+\frac{|\vec{x}-\vec{y}|^{2(\ell-2)}}{[(\ell-1)!]^{2}2^{2\ell-1}\pi}((x_{1}-y_{1})-i(x_{2}-y_{2})).$
(D.1)

Assuming $\ell>1$ and using $H_{\ell-1}=H_{\ell-2}+1/(\ell-1)$ and re-
arranging terms, this can be rewritten as

$\displaystyle(\partial_{x_{1}}-i\partial_{x_{2}})G_{\ell}(\vec{x}-\vec{y})$
$\displaystyle=\frac{1}{2(\ell-1)}((x_{1}-y_{1})-i(x_{2}-y_{2}))G_{\ell-1}(\vec{x}-\vec{y})$
$\displaystyle-\frac{|\vec{x}-\vec{y}|^{2(\ell-2)}}{[(\ell-1)!]^{2}2^{2\ell-1}\pi}((x_{1}-y_{1})-i(x_{2}-y_{2})),$
(D.2)

which is a recurrence relation valid for $\ell>1$. Multiplying by
$(x_{1}-y_{1})+i(x_{2}-y_{2})$ one gets

$\displaystyle((x_{1}-y_{1})+i(x_{2}-y_{2}))(\partial_{x_{1}}-i\partial_{x_{2}})G_{\ell}(\vec{x}-\vec{y})$
$\displaystyle=\frac{1}{2(\ell-1)}|\vec{x}-\vec{y}|^{2}G_{\ell-1}(\vec{x}-\vec{y})$
$\displaystyle-\frac{|\vec{x}-\vec{y}|^{2(\ell-1)}}{[(\ell-1)!]^{2}2^{2\ell-1}\pi},$
(D.3)

which, except for polynomial terms, has the functional dependence of
$G_{\ell}$. Indeed, using again the relation between $H_{\ell-1}$ and
$H_{\ell-2}$, one obtains

$\displaystyle((x_{1}-y_{1})+i(x_{2}-y_{2}))(\partial_{x_{1}}-i\partial_{x_{2}})G_{\ell}(\vec{x}-\vec{y})=2(\ell-1)G_{\ell}(\vec{x}-\vec{y})+\frac{|\vec{x}-\vec{y}|^{2(\ell-1)}}{[(\ell-1)!]^{2}2^{2\ell-1}\pi}.$
(D.4)

Although (D.4) has been obtained under the assumption that $\ell>1$ it can be
checked by direct computation that it is also valid for $\ell=1$.

In the computations in Section 3 the left-hand side of this identity appears
with derivatives $\partial_{x_{1}}+i\partial_{x_{2}}$ acting on it. Since the
second term in (D.4) is a polynomial of order $2(\ell-1)$ in the components of
$\vec{x}$, it turns out that, for $\ell\geq 1$,

$(\partial_{x_{1}}+i\partial_{x_{2}})^{n}((x_{1}-y_{1})+i(x_{2}-y_{2}))(\partial_{x_{1}}-i\partial_{x_{2}})G_{\ell}(\vec{x}-\vec{y})=2(\ell-1)(\partial_{x_{1}}+i\partial_{x_{2}})^{n}G_{\ell}(\vec{x}-\vec{y})$
(D.5)

for $n>2(\ell-1)$.

##  Appendix E Convolution property of the polyharmonic Green functions

One has

$\displaystyle\vec{\nabla}_{y}^{2(\ell+m)}\int\text{d}^{2}x\,G_{\ell}(\vec{y}-\vec{x})G_{m}(\vec{z}-\vec{x})$
$\displaystyle=\int\text{d}^{2}x\,\vec{\nabla}_{y}^{2(\ell+m)}G_{\ell}{\vec{y}-\vec{x}}G_{m}(\vec{z}-\vec{x})=\int\text{d}^{2}x\,\vec{\nabla}_{y}^{2\ell}\vec{\nabla}_{x}^{2m}G_{\ell}{\vec{y}-\vec{x}}G_{m}(\vec{z}-\vec{x})$
$\displaystyle=\int\text{d}^{2}x\,\vec{\nabla}_{y}^{2\ell}G_{\ell}{\vec{y}-\vec{x}}\vec{\nabla}_{x}^{2m}G_{m}(\vec{z}-\vec{x})=\int\text{d}^{2}x\,\delta(\vec{y}-\vec{x})\delta(\vec{z}-\vec{x})$
$\displaystyle=\delta(\vec{y}-\vec{z}).$ (E.1)

Under standard regularity conditions, the homogeneous polyharmonic problem has
only the trivial solution [30], and the above computation proves (2.7).

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SMSM References

# Coherent spin dynamics of rare-earth doped crystals in the high-
cooperativity regime

Joseph Alexander<EMAIL_ADDRESS>London Centre for
Nanotechnology, University College London, London WC1H 0AH, United Kingdom
Gavin Dold  London Centre for Nanotechnology, University College London,
London WC1H 0AH, United Kingdom  National Physical Laboratory, Hampton Road,
Teddington TW11 0LW, United Kingdom  Oscar W. Kennedy  London Centre for
Nanotechnology, University College London, London WC1H 0AH, United Kingdom
Mantas Šimėnas  London Centre for Nanotechnology, University College London,
London WC1H 0AH, United Kingdom  James O’Sullivan  London Centre for
Nanotechnology, University College London, London WC1H 0AH, United Kingdom
Christoph W. Zollitsch  London Centre for Nanotechnology, University College
London, London WC1H 0AH, United Kingdom  Sacha Welinski  Université PSL,
Chimie ParisTech, CNRS, Institut de Recherche de Chimie Paris, 75005 Paris,
France  Thales Research and Technology, 1 Avenue Augustin Fresnel, 91767
Palaiseau, France  Alban Ferrier  Université PSL, Chimie ParisTech, CNRS,
Institut de Recherche de Chimie Paris, 75005 Paris, France  Faculté des
Sciences et Ingénierie, Sorbonne Université, 75005 Paris, France
Elo$̈\mathrm{i}$se Lafitte-Houssat  Université PSL, Chimie ParisTech, CNRS,
Institut de Recherche de Chimie Paris, 75005 Paris, France  Thales Research
and Technology, 1 Avenue Augustin Fresnel, 91767 Palaiseau, France  Tobias
Lindström  National Physical Laboratory, Hampton Road, Teddington TW11 0LW,
United Kingdom  Philippe Goldner  Université PSL, Chimie ParisTech, CNRS,
Institut de Recherche de Chimie Paris, 75005 Paris, France  John J. L. Morton
<EMAIL_ADDRESS>London Centre for Nanotechnology, University College
London, London WC1H 0AH, United Kingdom  Department of Electronic and
Electrical Engineering, UCL, London WC1E 7JE, United Kingdom

###### Abstract

Rare-earth doped crystals have long coherence times and the potential to
provide quantum interfaces between microwave and optical photons. Such
applications benefit from a high cooperativity between the spin ensemble and a
microwave cavity — this motivates an increase in the rare earth ion
concentration which in turn impacts the spin coherence lifetime. We measure
spin dynamics of two rare-earth spin species, ${}^{\text{\\!{145}}}$Nd and
${}^{\text{\\!{}}}$Yb doped into Y${}_{\text{2}}$SiO${}_{\text{5}}$, coupled
to a planar microwave resonator in the high cooperativity regime, in the
temperature range $1.2\text{\,}\mathrm{K}$ to $14\text{\,}\mathrm{mK}$. We
identify relevant decoherence mechanisms including instantaneous diffusion
arising from resonant spins and temperature-dependent spectral diffusion from
impurity electron and nuclear spins in the environment. We explore two methods
to mitigate the effects of spectral diffusion in the ${}^{\text{\\!{}}}$Yb
system in the low-temperature limit, first, using magnetic fields of up to 1 T
to suppress impurity spin dynamics and, second, using transitions with low
effective g-factors to reduce sensitivity to such dynamics. Finally, we
demonstrate how the ‘clock transition’ present in the ${}^{\text{\\!{171}}}$Yb
system at zero field can be used to increase coherence times up to
$T_{2}=$6(1)\text{\,}\mathrm{ms}$$.

##  I Introduction

Rare-earth ions (REIs) doped into yttrium orthosilicate
(Y${}_{\text{2}}$SiO${}_{\text{5}}$) are promising systems for use in solid-
state quantum technologies. They have optical [1] and microwave [2] spin
transitions with coherence times of milliseconds for electron spin resonance
transitions [3, 4, 5], and hours for nuclear spin transitions [6, 7]. These
properties have stimulated proposals for the use of such REIs in microwave [8,
9] or optical [10, 11, 12] multimode quantum memories and quantum microwave to
optical transducers [13, 14, 15]. High efficiency storage and retrieval
protocols based on spins coupled to microwave resonators require
cooperativity, $C\geq 1$, and consequently high spin density [16, 17, 18, 19].
However, such high spin densities introduce spin decoherence mechanisms such
as spectral [20] or instantaneous [21] diffusion which must be understood and
mitigated to achieve useful quantum memory lifetimes.

In this article we perform pulsed electron spin resonance (ESR) measurements
of two REI systems (${}^{\text{\\!{145}}}$Nd and ${}^{\text{\\!{nat}}}$Yb) in
YSO, using planar superconducting resonators [22] that are strongly coupled to
the spin ensemble ($C\sim$ 4-245). First we measure and identify instantaneous
diffusion as a limiting decoherence mechanism in highly doped spin systems. We
observe a strongly temperature-dependent coherence time, and show that this is
due to suppression of spectral diffusion arising from spin bath polarisation
at low temperatures. Nevertheless, even at a base temperature of 14 mK,
coherence times are limited by spectral diffusion from electron spin sub-
ensembles. Finally, we show how so-called ‘clock’ (or ZEFOZ - ZEro First Order
Zeeman [23]) transitions and isotopic purification of the REI impurities can
extend coherence times ($T_{2}$) at zero field to over
$6\text{\,}\mathrm{ms}$.

##  II Experimental and Spin System

Figure 1: a) The three rare earth ion doped YSO samples studied each had
superconducting NbN resonators patterned on them to enable ESR studies of the
spin systems. The ${}^{\text{\\!{145}}}$Nd sample (200 ppm) was measured using
a ‘thin-ring’ resonator [22] and then placed inside a 3D copper cavity. The
${}^{\text{\\!{}}}$Yb samples were measured using a spiral resonator and
placed on top of a PCB designed to maximise spin signal. b) Measurement of the
${}^{\text{\\!{145}}}$Nd $m_{\mathrm{I}}=+\tfrac{7}{2}$ transition around
$B=$326\text{\,}\mathrm{m}\mathrm{T}$$ along D${}_{\text{1}}$, showing the
avoided crossing in (a) continuous-wave measurement with a VNA, where the
background $\mathrm{S}_{\mathrm{21}}$ transmission of the 3D cavity has been
subtracted for clarity. c) Echo amplitude from a two-pulse Hahn echo sequence
with the echo-detected avoided crossing. The darker blue regions indicate
larger echo amplitude, these are points where there is more hybridisation
between the resonator and the spins. The operating point for subsequent pulsed
experiments on Nd is indicated by the red cross.

We study three Y${}_{\text{2}}$SiO${}_{\text{5}}$ samples: one doped with
(isotopically purified) ${}^{\text{\\!{145}}}$Nd${}^{\text{3+}}$ ions (nuclear
spin, $I=7/2$) at $200\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$
($4\text{\times}{10}^{18}\text{\,}{\mathrm{cm}}^{-3}$), one with natural
isotopic abundance ${}^{\text{\\!{}}}$Yb${}^{\text{3+}}$ at
$50\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$
($1\text{\times}{10}^{18}\text{\,}{\mathrm{cm}}^{-3}$), and one with
${}^{\text{\\!{171}}}$Yb3+ at $5\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$
($1\text{\times}{10}^{17}\text{\,}{\mathrm{cm}}^{-3}$). The crystals were cut
along their principal dielectric axes (D${}_{\text{1}}$, D${}_{\text{2}}$, b)
with faces perpendicular to b polished for resonator fabrication. Due to its
larger ionic radius, ${}^{\text{\\!{145}}}$Nd preferentially substitutes
${}^{\text{\\!{}}}$Y in one of the crystal sites in
Y${}_{\text{2}}$SiO${}_{\text{5}}$, whereas ${}^{\text{\\!{}}}$Yb equally
populates both sites. The ${}^{\text{\\!{}}}$Nd sample was studied using a
‘thin ring’ NbN resonator [22] of film thickness
$45\text{\,}\mathrm{n}\mathrm{m}$, resonant frequency
$f_{\mathrm{c}}=$8.07\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$, and loaded
quality factor $Q\approx$72\,000$$. For the ${}^{\text{\\!{}}}$Yb samples we
used several spiral NbN resonator with film thickness
$15\text{\,}\mathrm{n}\mathrm{m}$ to explore different regimes: a resonator
with $f_{\mathrm{c}}=$5.04\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$, and
$Q\approx$31\,000$$, as well as lower frequency resonators targeting the zero
field clock transitionwith
$f_{\mathrm{c}}=$2.372\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$, and
$Q\approx$26\,000$$. The spiral resonator design was better suited to the
lower target microwave frequencies for Yb, while maintaining a footprint of
about $500\text{\,}\mathrm{\SIUnitSymbolMicro m}$ $\times$
$800\text{\,}\mathrm{\SIUnitSymbolMicro m}$. All samples were measured using a
dilution refrigerator (base temperature $\sim$14 mK) with a (3-1-1) T vector
magnet. The Nd sample was mounted on a sapphire platform inside a copper box,
while the Yb samples were mounted over a copper strip line designed with
impedance steps for efficient collection of signal as outlined in the
Supplementary Information (SI). An overview of all three systems is shown in
Fig. 1a).

Spin transitions were located by monitoring $S_{21}(f)$ while sweeping applied
magnetic field and identifying avoided crossings between the resonator and
electron spin ensemble, as shown for Nd in Fig. 1b). A similar measurement is
possible using pulsed ESR by monitoring the two-pulse Hahn echo as a function
of both magnetic field and frequency, as shown in Fig. 1c). The maximum echo
intensity is measured at $B=$326\text{\,}\mathrm{m}\mathrm{T}$$ where the
resonator and spin line hybridise; we also note the weak background echo which
can be measured far from the spin line (see SI). Fitting this avoided crossing
and extracting ensemble coupling strength ($g_{ens}$), spin linewidth
($\gamma_{s}$) and resonator linewidth ($\kappa_{c}$) returns cooperativity
$C=g_{\mathrm{ens}}^{2}/\gamma_{\mathrm{s}}\kappa_{\mathrm{c}}=245$ with
$g_{\mathrm{ens}}>\kappa_{\mathrm{c}},\gamma_{\mathrm{s}}$, showing the system
to be in the strong coupling regime, well suited for high efficiency memories.

The Nd and Yb impurities studied here are well described by the spin
Hamiltonian:

$\mathcal{H}=\mathbf{S}\cdot\hat{A}\cdot\mathbf{I}+\mu_{B}\mathbf{B}\cdot\hat{g}\cdot\mathbf{S}-\mu_{n}\mathbf{B}\cdot
g_{n}\cdot\mathbf{I}$ (1)

where the electron spin ($\mathbf{S}$) interacts with the nuclear spin
($\mathbf{I}$) described by a hyperfine tensor, $\hat{A}$, and to an external
magnetic field ($\mathbf{B}$) via a g-tensor, $\hat{g}$. The nuclear spin also
interacts with the field via a nuclear g-factor ($g_{n}$), the values of which
are given in the SI. The large anisotropy of the electron g-tensor results in
a ESR spectrum which has a strong angular dependence and allows for regimes in
which dipole coupling between rare-earth spins can be suppressed [24].

##  III Spin Decoherence Mechanisms

###  III.1 Instantaneous Diffusion

Figure 2: Echo-detected field sweep of ${}^{\text{\\!{nat}}}$Yb at high field
(red curves). The ${}^{\text{\\!{I=0}}}$Yb a) and the ${}^{\text{\\!{171}}}$Yb
b) ESR lines can be separately studied at $905\text{\,}\mathrm{mT}$ and
$1075\text{\,}\mathrm{mT}$ respectively. The effect of ID can be observed as
$T_{2}$ (blue datapoints) is measured across the ESR line. For
${}^{\text{\\!{I=0}}}$Yb, the coherence time deceases significantly at the
centre of the spin line, where the density of resonant spins is maximal. In
both panels, the black curve is the calculated ID limit on $T_{2}$ using Eq. 2
with no fitting parameters (see main text).

A common decoherence mechanism in bulk doped spin systems — particularly
relevant in samples that are highly-doped to achieve strong coupling to
resonators — arises from the dipolar interaction between resonant spins [25].
Such interactions are not refocused in a two-pulse Hahn echo (because both
interacting spins are flipped), leading to a decoherence mechanism known as
_instantaneous diffusion_ (ID), whose rate is proportional to $n$, the density
of resonant spins, as [25, 26]:

$\displaystyle
T_{2}=\frac{9\sqrt{3}\hbar}{\pi\mu_{0}(g\mu_{\mathrm{B}})^{2}n}.$ (2)

Here, $g$ is the effective g-factor, while $h$, $\mu_{0}$ and
$\mu_{\mathrm{B}}$ are Planck’s constant, the vacuum permeability and Bohr
magneton, respectively.

We use two approaches to study the effect of resonant spin density on
coherence time: i) exploiting differences in the natural abundance of those
${}^{\text{\\!{}}}$Yb isotopes which can be spectrally separated due to their
nuclear spin; and ii) using the variation of spin density across the
inhomogenously broadened ESR transition. The resonant spin density can be
estimated based on the product of the impurity concentration of Yb, the
natural abundance of the relevant isotope(s), and the fraction of spins which
are addressed given a particular excitation/resonator bandwidth. For example,
here a bandwidth of about $66\text{\,}\mathrm{kHz}$ is set by the microwave
$\pi$-pulse length of $15\text{\,}\mathrm{\SIUnitSymbolMicro s}$, which is to
be compared against the inhomogenously broadened spin linewidth of
$\gamma_{s}/2\pi$ = $8.7\text{\,}\mathrm{MHz}$.

In Fig. 2 we show echo-detected magnetic field sweeps across two ESR
transitions: one arising from the set of Yb isotopes with nuclear spin $I=0$
(17.5 ppm) and the other from ${}^{\text{\\!{171}}}$Yb (3.5 ppm). The
resonator frequency shifts as the spin transition is approached (as
illustrated for Nd in Fig. 1(c), so the applied microwave frequency is
adjusted at each magnetic field value to follow the resonator. The drop in
echo amplitude seen in the centre of the line in Fig. 2(a) arises from the
strong resonator/spin hybridisation and not a result of a distortion of the
ESR lineshape, which we assume here to follow a simple Gaussian profile. We
measure $T_{2}$ across the inhomogenously broadened ESR lineshape and observe
a drop in the coherence time at the centre, where the resonant spin density is
greatest. Indeed, using the nominal
$50\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ concentration of
${}^{\text{\\!{}}}$Yb${}^{\text{3+}}$, natural isotopic abundance, expected
excitation bandwidth, and measured ESR linewidth with Eq. 2 we can accurately
reproduce the observed $T_{2}$ values with no free parameters. Towards the
tails of the ESR line, where the resonant spin density is lowest, the measured
$T_{2}$ begins to deviate from that limited by ID and saturates at 1–2 ms.
Performing the same measurement for the ${}^{\text{\\!{171}}}$Yb transition we
observe no change in $T_{2}$ over the ESR lineshape — this is expected because
we can use the results above to predict an ID-limited $T_{2}$ of
$11\text{\,}\mathrm{ms}$ (thanks to the lower spin concentration of
${}^{\text{\\!{171}}}$Yb), which is well above the measured values of
$6(2)\text{\,}\mathrm{ms}$. We therefore confirm that ID is not a limiting
decoherence mechanism in the ${}^{\text{\\!{171}}}$Yb system for this sample
and move on to explore alternative mechanisms.

###  III.2 Spectral diffusion

####  III.2.1 Temperature dependence

Cross-relaxation processes (or spin flip-flops) between spins cause the local
magnetic environment to fluctuate over time giving rise to _spectral
diffusion_ [25, 20, 27]. Reducing the temperature of the spin bath polarizes
it, reducing the rate of flip-flops and consequently, the effects of spectral
diffusion [28, 29]. Figure 3 shows such an increase in coherence time with
decreasing temperature, for both the ${}^{\text{\\!{145}}}$Nd and
${}^{\text{\\!{nat}}}$Yb samples: in Nd from
$24(1)\text{\,}\mathrm{\SIUnitSymbolMicro s}$ at $1.2\text{\,}\mathrm{K}$ to
$0.41(1)\text{\,}\mathrm{ms}$ at $14\text{\,}\mathrm{m}\mathrm{K}$ and in
${}^{\text{\\!{nat}}}$Yb increasing from
$33(7)\text{\,}\mathrm{\SIUnitSymbolMicro s}$ to $3.4(1)\text{\,}\mathrm{ms}$
over the same temperature range.

Figure 3: The decoherence rate ($1/T_{2}(T)$) measured from
$1.2\text{\,}\mathrm{K}$ to $14\text{\,}\mathrm{m}\mathrm{K}$ for (a) the
${}^{\text{\\!{145}}}$Nd sample and (b) the ${}^{\text{\\!{nat}}}$Yb sample.
In both samples, the decoherence rate drops with temperature due to
suppression of spectral diffusion below the Zeeman temperature of various spin
sub-ensembles present in the sample. The solid curve is a fit using Eqn. 7,
and dashed lines indicate the Zeeman temperatures of all the sub-ensembles
used in the fit, whose relative abundance is shown in pie charts. The inset on
a) and b) show the measurement of $T_{2}$ at the lowest temperature
$14\text{\,}\mathrm{m}\mathrm{K}$ using a two-pulse echo sequence. (c) and (d)
show stimulated (3-pulse) echo decays for the ${}^{\text{\\!{145}}}$Nd and
${}^{\text{\\!{nat}}}$Yb samples, respectively, aiming to determine the source
of decoherence at the base temperature of 14 mK. The values of $\tau$ in the
3-pulse echo sequence were chosen for each sample in proportion with $T_{2}$
at 14 mK. For ${}^{\text{\\!{nat}}}$Yb (${}^{\text{\\!{145}}}$Nd), the
magnetic field was 370 mT (326 mT) and microwave frequency 5.04 (8.07) GHz.

As in Ref. [9], we model the temperature dependence of $T_{2}$ assuming that
spectral diffusion is caused by one or more sub-ensembles of spins with
distinct resonant frequencies. In our 145Nd sample, the different projections
of the $I=7/2$ nuclear spin along the applied magnetic field form the dominant
contribution to such sub-ensembles, while in the ${}^{\text{\\!{nat}}}$Yb
sample, multiple isotopes with different nuclear spin and the projections of
nuclear spin for $I>0$ act as sub-ensembles.

We can write the decoherence rate $\Gamma=1/T_{2}$ of the ‘central spin’ being
studied as [20, 27]:

$\Gamma=\frac{R\Gamma_{\mathrm{SD}}}{2}\left(\sqrt{\Gamma_{0}^{2}+\frac{R\Gamma_{\mathrm{SD}}}{\pi}}-\Gamma_{0}\right)^{-1}\approx\frac{\sqrt{\pi
R\Gamma_{\rm SD}}}{2},$ (3)

where $R$ is the spin flip rate, $\Gamma_{\rm SD}$ is the spectral diffusion
linewidth, and $\Gamma_{0}$ is a residual decoherence rate in the absence of
spectral diffusion. When the dominant noise process is spectral diffusion
(i.e. $\Gamma_{0}\ll\sqrt{R\Gamma_{\rm SD}}$), we neglect the $\Gamma_{0}$
dependence of $T_{2}$. Assuming that spectral diffusion occurs from spin flips
due to spin-spin interactions between resonant sub-ensembles, we can write the
flip-flop rate within the $i$th sub-ensemble of spins as [24]:

$R_{i}=\beta_{\rm
ff}\left(\theta\right)\frac{n_{i}^{2}}{\Gamma_{i}}\operatorname{sech}^{2}\Big{(}\frac{T_{\mathrm{Z},i}}{T}\Big{)}$
(4)

where $n_{i}$ and $\Gamma_{i}$ are, respectively, the spin density and
linewidth of the $i$th sub-ensemble, and $\beta_{\rm ff}$ is a parameter which
encompasses the angular dependence of the coupling parameter and the g-tensor.
$T_{\mathrm{Z,i}}$ is the Zeeman temperature for each sub-ensemble -
$T_{z,i}=hf/k_{B}$. In the case of an isotropic medium $\beta_{\rm
ff}\propto\mu_{B}^{4}g^{4}$, however, this does not hold in the case of strong
anisotropy, as in YSO [30, 3, 31, 32]. This term is proportional to the rate
derived from Fermi’s golden rule, taking the dipole coupling as a perturbation
of the spin Hamiltonian. To account for this we rewrite
$\beta_{ff}=\xi\frac{\mu_{B}^{4}\mu_{0}^{2}}{h^{2}}M_{i}^{2}$ where $M_{i}$ is
the matrix element of each sub-ensemble transition, and $\xi$ is a coupling
rate fit parameter. We thus rewrite $R$ as:

$R_{i}=\xi\frac{\mu_{B}^{4}\mu_{0}^{2}}{h^{2}}M_{i}^{2}\frac{n_{i}^{2}}{\Gamma_{i}}\operatorname{sech}^{2}\Big{(}\frac{T_{\mathrm{Z},i}}{T}\Big{)}.$
(5)

The spectral diffusion linewidth of the $i$th sub-ensemble (the frequency
shift of the central spin by the spins in this sub-ensemble due to dipolar
interactions) is given by [20]:

$\Gamma_{\mathrm{SD},i}=\frac{\pi\mu_{0}\mu_{B}^{2}}{9\sqrt{3}h}n_{i}g_{i}g\operatorname{sech}^{2}\Big{(}\frac{T_{\mathrm{Z},i}}{T}\Big{)},$
(6)

where $g$ and $g_{i}$ are the effective g-factors of the central and $i$-th
spins, respectively. Combining Eqs. 3–6 gives an expression for the overall
temperature-dependent decoherence rate, where the spectral diffusion component
contains contributions from multiple sub-ensembles, similar in form to that
given in Ref. [9]:

$\Gamma(T)=\Gamma_{\mathrm{res}}+\sqrt{\frac{\pi\mu_{0}^{3}\mu_{B}^{6}}{9\sqrt{3}h^{3}\Gamma}g}\sum_{i}\frac{\sqrt{\xi}n_{i}^{3/2}M_{i}}{\Big{(}1+\mathrm{e}^{\tfrac{T_{\mathrm{Z}}^{i}}{T}}\Big{)}\Big{(}1+\mathrm{e}^{-\tfrac{T_{\mathrm{Z}}^{i}}{T}}\Big{)}}.$
(7)

$\Gamma_{\mathrm{res}}$, which we treat as a fit parameter, is a temperature-
independent decoherence rate due to, for example, spectral diffusion from
spins that remain unpolarised even at our base temperature of 14 mK. $\xi$ is
a single temperature-independent fit parameter reflecting the average
effective g-factor ($g_{i}$) of all sub-ensembles. $M_{i}$ was calculated
using EasySpin [33]. We assume the linewidth of each sub-ensembles is equal to
that of the central spin. The relative populations of the ground
$P_{\downarrow}$ and excited $P_{\uparrow}$ states involved in the flip-flop
processes for each sub-ensemble are
$P_{\downarrow}=[1+\exp(+T_{\mathrm{Z}}^{i}/T)]^{-1}$ and
$P_{\uparrow}=[1+\exp(-T_{\mathrm{Z}}^{i}/T)]^{-1}$ according to Boltzmann
statistics [28, 9].

Fits to Eqn. 7, where $\xi$ and $\Gamma_{\rm res}$ are the only free
parameters, are shown in Fig. 3. The resulting free parameter $\xi$ returns
values of $1.94(1)\text{\,}$ for ${}^{\text{\\!{}}}$Nd and $12(1)\text{\,}$
for ${}^{\text{\\!{}}}$Yb. As $\xi\propto g^{4}$, the difference in these
values is attributed to the differences in the g-tensor components, in
particular $g_{z}=4.17$ in ${}^{\text{\\!{}}}$Nd whereas $g_{z}=6.06$ in
${}^{\text{\\!{}}}$Yb site 2 [3, 31]. Notably, the Nd sample also has
substantially larger residual decoherence rate, $\Gamma_{\mathrm{res}}$, at
the base temperature of the dilution fridge than the Yb sample, however, we
believe this is partially due to poor thermalisation in the
${}^{\text{\\!{}}}$Nd sample, as outlined in the SI.

####  III.2.2 Low-temperature limit

Figure 4: Angular dependence of $T_{2}$ in the high field regime. a) The
measured coherence time of ${}^{\text{\\!{171}}}$Yb as the applied field is
rotated in the D1-D2 plane, the subsite degeneracy is lifted due to a slight
misalignment with the b axis. At $-131\text{\,}{}^{\circ}$, $T_{2}$ reaches a
maximum, which corresponds well with the modelled coherence time extracted
using $R\Gamma_{SD}$. b) As the angle of the applied field is rotated in the
D1-D2 crystal plane, the anisotropic g-tensor results in an angular dependence
of the spectral diffusion rate ($R$) and the ‘${}^{\text{\\!{171}}}$Yb’s
sensitivity to it ($\nabla f$). At angles of about $131\text{\,}{}^{\circ}$,
both $R$ and $\nabla f$ reach a minima resulting in an increase in the
coherence time.

To further investigate the residual decoherence mechanism present at low
temperatures, we used a three-pulse stimulated echo sequence
($\tfrac{\pi}{2}$–$\tau$–$\tfrac{\pi}{2}$–$T_{\mathrm{w}}$–$\tfrac{\pi}{2}$–$\tau$–echo)
[27] to directly measure the component due to spectral diffusion. The
inhomogeneous fields induced by planar resonators on bulk-doped crystals means
that the pulses are not true $\tfrac{\pi}{2}$ pulses. The stimulated echo
amplitude is given by [27]:

$\displaystyle A(\tau,T_{\mathrm{w}})$
$\displaystyle=A_{0}\exp\left[-\left(\frac{T_{\mathrm{w}}}{T_{1}}+2\pi\tau\Gamma_{\mathrm{eff}}\right)\right],$
(8)

where $A_{0}$ is a fitted amplitude, $\tau$ and $T_{\mathrm{w}}$ are delay
parameters used in the pulse sequence, and $T_{1}$ is the spin relaxation time
measured to be $696\text{\,}\mathrm{ms}$ (${}^{\text{\\!{}}}$Nd) and
$47\text{\,}\mathrm{ms}$ (${}^{\text{\\!{}}}$Yb) from data shown in the SI.
The effective decoherence rate, $\Gamma_{\mathrm{eff}}$, is:

$\displaystyle\Gamma_{\mathrm{eff}}$
$\displaystyle=\Gamma_{0}+\tfrac{1}{2}\Gamma_{\mathrm{SD}}(R\tau+1-\mathrm{e}^{-RT_{\mathrm{w}}}),$
(9)

where $\Gamma_{\mathrm{SD}}$ is the spectral diffusion linewidth, $R$ is the
total spin-flip rate, and $\Gamma_{0}$ is a residual decoherence rate. We
assume all decoherence occurs from spectral diffusion and set $\Gamma_{0}=0$.

As shown in the SI, fitting Eqn. 8 and 9 to the data does not uniquely
determine $R$ and $\Gamma_{\mathrm{SD}}$ due to their covariance, however, the
fit routine does reliably determine their product. In Nd these fits return
$R\Gamma_{\mathrm{SD}}=$3.5(4)\text{\times}{10}^{6}\text{\,}{\mathrm{Hz}}^{2}$$,
which allows us to extract a limit on the coherence time using Eqn. 3 of
$T_{2}=$0.60(3)\text{\,}\mathrm{ms}$$. Using equations 4 and 6 with the
stimulated echo fit, an estimated spin bath temperature of
$61\text{\,}\mathrm{mK}$ is obtained (significantly higher than the measured
base temperature). Further evidence of the poor thermalisation is seen in the
deviation from expected temperature dependence of the echo magnitude below
about $100\text{\,}\mathrm{mK}$ (see SI).

Following a similar analysis for the ${}^{\text{\\!{}}}$Yb sample, the fits
return $R\Gamma_{SD}=$1.3(1)\text{\times}{10}^{5}\text{\,}{\mathrm{Hz}}^{2}$$.
This results in a spectral diffusion-limited $T_{2}$ of
$3.1(7)\text{\,}\mathrm{ms}$, consistent with the 2-pulse echo value of
$3.38(9)\text{\,}\mathrm{ms}$, and an estimated spin bath temperature of
$38\text{\,}\mathrm{mK}$. We explore the source of spectral diffusion in the
${}^{\text{\\!{}}}$Yb sample and compute the effect of ${}^{\text{\\!{89}}}$Y
nuclear flip-flops. Following the methods presented in Ref. [20], described
further in the SI, we calculate a spectral diffusion rate and linewidth from
${}^{\text{\\!{89}}}$Y flip-flops;
$R_{Y}\Gamma_{SD,Y}=$$1.05\text{\times}{10}^{5}\text{\,}{\mathrm{Hz}}^{2}$
with a limiting $T_{2}=$ $3.48\text{\,}\mathrm{ms}$. This is in agreement with
our measured, $T_{2}$ and we therefore conclude that the low-temperature
coherence time is limited by ${}^{\text{\\!{89}}}$Y nuclear spin flip-flops.

##  IV Mitigating Spin decoherence

In Section III we found the coherence of ${}^{\text{\\!{171}}}$Yb electron
spins in ${}^{\text{\\!{nat}}}$Yb-doped YSO at 14 mK to be primarily limited
by spectral diffusion from ${}^{\text{\\!{89}}}$Y nuclear spin flip flops,
when measured at $5.04\text{\,}\mathrm{GHz}$ and 370 mT. We next explore two
strategies to further suppress spin decoherence: first, exploiting the
substantial anisotropy of the g-tensor and rotating the magnetic field to
orientations which give the lowest effective g-factor, and second, exploring
the ‘clock transition’ at zero magnetic field for which the first-order
sensitivity of the ESR transition frequency goes to zero with respect to
magnetic field.

###  IV.1 Increasing Magnetic Field

For a given ESR transition frequency, the magnetic field orientation at which
the effective g-factor is lowest is that which provides the ESR resonance at
the maximum magnetic field. We investigate this approach for 171Yb using a
lower resonator frequency ($2.43\text{\,}\mathrm{GHz}$) than that used above
so that the resonant fields remain within the range of our vector magnet. As
the magnetic field is rotated in the D1-D2 plane, the resonant field reaches a
peak of $1.2\text{\,}\mathrm{T}$, however, due to experimental limitations the
highest field we studied was $1.07\text{\,}\mathrm{T}$, at an angle of -131∘
from D1. Fig. 4a) shows how $T_{2}$ varies as the field is rotated in the
D1-D2 plane. Due to a small misalignment towards the b axis (estimated to be <
$0.8\text{\,}{}^{\circ}$), the degeneracy of the two crystal subsites is
lifted.

Considering first the coherence times measured at an angle of around
$-88^{\circ}$ from D1 (approximately along D2), we see that they fall
substantially below that predicted due to ${}^{\text{\\!{89}}}$Y nuclear spin
flip-flops. Here, the magnetic field was
$154\text{\,}$$-174\text{\,}\mathrm{mT}$, less than half that used in Section
III due to the lower resonator frequency. In this regime therefore, some
electron spin sub-ensembles appear to remain unpolarised at
$14\text{\,}\mathrm{mK}$ and contribute to spectral diffusion. As the magnetic
field orientation rotates further away from D1, the effective g-factor
decreases, demanding a larger magnetic field magnitude to satisfy the
resonance condition for 171Yb — this increased magnetic field in turn serves
to further polarise electron spin sub-ensembles in the bulk, as was achieved
in Section III by lowering temperature. Therefore, approaching the optimum
field orientation of $-133^{\circ}$ offers the twin benefits of i) reducing
magnetic field noise by polarising the bath at high field (minimising
$R\Gamma_{\rm SD}$), while also reducing the sensitivity, $\nabla_{B}$f, of
the central spin to that noise. Indeed, at $-131^{\circ}$, the coherence time
of subsite $a$ was measured to be $6(2)\text{\,}\mathrm{ms}$. Both
$\nabla_{B}\textit{f}$ and $R\Gamma_{\rm SD}$ are plotted in Fig. 4(b)), based
on the relationships in Eqs. 4 and 6 and the spin Hamiltonian (see SI for
further details). The resulting model for $T_{2}$ follows the observed data
well, including the different times seen for the two subsites. We can
therefore predict that if the $5.04\text{\,}\mathrm{GHz}$ resonator were used
in this experiment, we would expect a $T_{2}$ of $25.5\text{\,}\mathrm{ms}$
(similar to that measured in [5]) at $-132.7\text{\,}{}^{\circ}$, however this
would occur at $3.07\text{\,}\mathrm{T}$, beyond the capabilities of our
experimental setup.

###  IV.2 Around Zero Magnetic Field

We have shown above that using large magnetic fields (around
$1\text{\,}\mathrm{T}$) applied at specific orientations can extend the
${}^{\text{\\!{171}}}$Yb electron spin $T_{2}$ to several milliseconds.
However, such fields introduce practical challenges when embedding
${}^{\text{\\!{171}}}$Yb within more complex superconducting circuits, and
indeed to integrate with superconducting qubits it is desirable to operate
close to zero magnetic field. One of the attractive features of
${}^{\text{\\!{171}}}$Yb as a quantum memory is the ‘clock transition’ it
exhibits at zero field — here the sensitivity to magnetic field noise goes to
zero, to first order, which should reduce impact of spectral diffusion on the
spin coherence time.

Figure 5: Field dependence of the coherence time of ${}^{\text{\\!{171}}}$Yb
at low field for both the ${}^{\text{\\!{nat}}}$Yb doped (a) and isotopically
pure ${}^{\text{\\!{171}}}$Yb doped (b) systems. While the $T_{2}$ increases
substantially (to $1.77(6)\text{\,}\mathrm{ms}$) as the the field approaches
zero in the ${}^{\text{\\!{nat}}}$Yb system, the coherence time is not as long
as that measured at high field ($6(2)\text{\,}\mathrm{ms}$). The coherence
time deviates away from the trend in decreasing $\nabla_{B}f$ as shown by the
grey dashed line. This is due to the presence of an unpolarized spin bath of
other ${}^{\text{\\!{}}}$Yb isotopes at zero field, this is highlighted by the
increased coherence in the isotopically pure ${}^{\text{\\!{171}}}$Yb system
(b). Here both a longer maximum $T_{2}$ is measured
($6(2)\text{\,}\mathrm{ms}$) and a wider region of long coherence is observed.

The use of a microresonator to perform ESR at a clock transition at zero
magnetic field requires precise control of its resonance frequency to match
the ${}^{\text{\\!{171}}}$Yb zero-field splitting of
$2.370\text{\,}\mathrm{GHz}$. This challenge is made more acute by the
narrowing of the ESR linewidth which occurs around the clock transition. We
explore ESR in the low field regime of ${}^{\text{\\!{nat}}}$Yb doped YSO
using a resonator with frequency within 2 MHz of the clock transition
(comparable to the spin-resonator coupling strength).

To investigate the impact of the clock transition on spin coherence, we
perform magnetic field sweeps along a fixed direction of 49∘ in the D1-D2
plane — at this field orientation $\nabla_{B}f$ follows a local minimum,
leading to the longest expected coherence times (see SI). Fig. 5a) shows how
$T_{2}$ increases sharply as the field passes through zero, reaching a maximum
of $1.77(6)\text{\,}\mathrm{ms}$ at $0\text{\,}\mathrm{mT}$. While a strong
increase in coherence time (over an order of magnitude) is observed around
zero field compared to fields up to 40 mT at this orientation, the increase
remains well below that predicted by the decrease in sensitivity to magnetic
field noise, $\nabla_{B}f$. Indeed, the times measured here at the nominal
clock transition at zero field remain several times shorter than those
measured at high fields ($\sim 1$ T) described above where the effective
g-factor drops to 0.1. We attribute this behaviour to the impact of
environmental electron spins that are not only unpolarised, but also become
mutually resonant at zero magnetic field, leading to a large increase in the
effective magnetic field noise. In this ${}^{\text{\\!{nat}}}$Yb sample,
isotopes with zero nuclear spin (${}^{\text{\\!{I=0}}}$Yb) comprise
approximately 70% of the natural abundance of Yb impurities, providing a large
bath of unpolarized electron spins. Approaching zero magnetic field, we
conclude that the rate of spectral diffusion increases faster than
$\nabla_{B}f$ of ${}^{\text{\\!{171}}}$Yb decreases, limiting the enhancement
of $T_{2}$ (see SI for more details).

###  IV.3 Isotopic Purification of Yb

When seeking to exploit ${}^{\text{\\!{171}}}$Yb defect spins in YSO as
candidate quantum memory medium, any other isotopes of Yb present in the bulk
serve only to introduce additional sources of decoherence, particularly, at
the low magnetic fields used to access the clock transition. We next explore
the use of isotopically pure ${}^{\text{\\!{171}}}$Yb doped
Y${}_{\text{2}}$SiO${}_{\text{5}}$, using an identical resonator design to
that used in the ${}^{\text{\\!{nat}}}$Yb sample with resonant frequency
$2.368\text{\,}\mathrm{GHz}$. Fig. 5b) plots measured $T_{2}$ as a function of
magnetic field along the same 49∘ orientation as above (the lower resonator
frequency here limited the addressable field range). In the absence of other
${}^{\text{\\!{}}}$Yb isotopes the coherence times are now over an order of
magnitude longer in this field range of 0–10 mT, reaching a gentle maximum of
$6(1)\text{\,}\mathrm{m}\mathrm{s}$ at $2.5\text{\,}\mathrm{m}\mathrm{T}$ —
the longest ${}^{\text{\\!{171}}}$Yb $T_{2}$ measured in all the systems we
studied here. As the field magnitude reaches $<$ $1\text{\,}\mathrm{mT}$, the
measured $T_{2}$ shortens, which we attribute to the impact of
${}^{\text{\\!{89}}}$Y nuclear spins at low magnetic field. The coherence
decay curves themselves exhibit electron spin echo envelope modulation
(ESEEM), which occurs when the electron spin interacts with one or more
neighbouring nuclear spins and the coherent quantum state is transferred
between the two. As the frequency of these oscillations is proportional to the
Larmor frequency, the revival of electron spin echo goes to infinity as the
field goes to zero. This means that a coherence curve cannot be fitted to the
echo decay and so this appears as a decrease in the $T_{2}$, this is a problem
present in all systems with a strong hyperfine interaction [34]. Another
possibility for the dip in coherence at $0\text{\,}\mathrm{mT}$ could also be
due to the changing behaviour of the nuclear spin bath around the clock
transition [35], here the ${}^{\text{\\!{89}}}$Y nuclear spins are no longer
in the frozen core and so there is an increase in spectral diffusion. This
interaction between the electron spin and the ${}^{\text{\\!{89}}}$Y nuclear
spins is a general issue for operating quantum memory at zero field which
effects all rare-earths doped in Y${}_{\text{2}}$SiO${}_{\text{5}}$,
motivating studies in other host materials with lower nuclear spin
concentrations [36, 37].

##  V Conclusion

We used on-chip superconducting resonators to perform pulsed ESR measurements
on ${}^{\text{\\!{145}}}$Nd and ${}^{\text{\\!{171}}}$Yb in YSO in the high-
cooperativity regime $C=$ $4245$. Cooling the spin ensemble down to a nominal
base temperature of $14\text{\,}\mathrm{m}\mathrm{K}$ was shown to extend
coherence times by polarising the bath of electron spins in the environment,
leading to a $T_{2}=$0.41(1)\text{\,}\mathrm{ms}$$ for
${}^{\text{\\!{145}}}$Nd (likely limited by poor sample thermalisation) and
$T_{2}=$3.38(9)\text{\,}\mathrm{ms}$$ for ${}^{\text{\\!{171}}}$Yb, believed
to be limited by spectral diffusion from ${}^{\text{\\!{89}}}$Y nuclear spins
in the host. We explored two routes to suppressing spectral diffusion in
${}^{\text{\\!{171}}}$Yb. The first involved rotating the applied magnetic
field to access transitions with low effective g-factor to enable the
coherence time of $6(2)\text{\,}\mathrm{ms}$. The high magnetic field
$(>$1\text{\,}\mathrm{T}$)$ serves to polarise the spin bath, reducing the
spectral diffusion rate, while the low effective g-factor reduces the
sensitivity to spectral diffusion and so the coherence time is further
increased. Second, we studied coherence times around the clock transition,
showing that the strong reduction in sensitivity of magnetic field noise
around zero field is somewhat compensated by a large increase in magnetic
field noise from spin flip-flops from other Yb isotopes. Using an isotopically
pure ${}^{\text{\\!{171}}}$Yb:YSO sample we were able to increase the $T_{2}$
to a maximum of $6(1)\text{\,}\mathrm{ms}$.

Overall, based on the understanding we have obtained on decoherence mechanisms
in these system, we conclude that ${}^{\text{\\!{171}}}$Yb offers clear
advantages over ${}^{\text{\\!{145}}}$Nd for spin-based memories in YSO,
arising from its lower nuclear spin ($I=1/2$) and thus reduced concentration
of non-resonant electron spin in the environment. The longest coherence times
for ${}^{\text{\\!{171}}}$Yb are predicted to be achieved at higher magnetic
fields (e.g. 25 ms at fields of about 3 T), though these show limited
compatibility with superconducting circuits in general. ESEEM from
${}^{\text{\\!{89}}}$Y is likely to pose a limiting factor in the use of
${}^{\text{\\!{171}}}$Yb spins at zero magnetic field, however this could be
avoided by moving to other crystalline hosts, such as CaWO4 [5]. Quantum
memories schemes typically require high-fidelity control of the spins, which
is challenging for bulk-doped rare-earth ion spins such as those studied here
— spatially confining such spins is therefore likely to be advantageous, for
example through implantation or lithography.

###### Acknowledgements.

This work has received funding from the UK Engineering and Physical Sciences
Research Council (EPSRC) through the Centre for Doctoral Training in
Delivering Quantum Technologies (Grant No. EP/L015242/1), QUES2T
(EP/N015118/1) and the Skills Hub in Quantum Systems Engineering: Innovation
in Quantum Business, Applications, Technology and Engineering (InQuBATE),
Grant No. EP/P510270/1; as well as from the European Research Council (ERC)
via LOQOMOTIONS (H2020- EU.1.1., Grant No. 771493). We would like to
acknowledge support from the UK department for Business Energy and Industrial
Strategy (BEIS) through the UK national quantum technologies programme and the
ANR MIRESPIN project, Grant No. ANR-19-CE47-0011 of the French Agence
Nationale de la Recherche and DGA.

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