&\to\int_{Y}\left(\int_{X_{y}}f_{y}\cdot|F_{B_{o}}|^{2}\operatorname{vol}_{X_{y}}\right)\operatorname{vol}_{Y}+\sum_{x\in\boldsymbol{x}}\int_{Y}f\operatorname{vol}_{Y}\\\
&=\int_{M}f\cdot|F_{A_{o}}|^{2}\operatorname{vol}_{\varphi}+\sum_{x\in\boldsymbol{x}}\int_{Y}f\operatorname{vol}_{Y}.\end{split}$
(5.31)

On the other hand, given any compact subset $K\subseteq M\backslash
Y\times\boldsymbol{x}$, we can choose a compact subset $K_{X}\subseteq
X\backslash\boldsymbol{x}$ so that $K\subseteq Y\times K_{X}$. After applying
gauge transformations, we know that $B_{n}$ converges to $B_{o}$ in
$C^{\infty}(K_{X})$. It follows from the estimate (5.12) that $A_{n}$
converges to $A_{o}$ in $C^{\infty}(Y\times K_{X})$ up to gauge
transformations. Thus, we have proven that the closure of
$M_{\varphi}(X,P_{\rho})$ inside $\overline{M}_{\varphi}(X,P_{\rho})$ is
compact. However, by the gluing results of Taubes, any ideal instanton can
occur as a limit. This completes the proof. ∎

##  6\. Hyperkähler Fibrations in Dimension 8

Our discussion of $\operatorname{Spin}(7)$-instantons will follow a similar
pattern as that of $G_{2}$-instantons. The major difference comes from the
self-duality of the calibration $4$-form over
$\operatorname{Spin}(7)$-manifolds. So for fibrations, both the fibre and base
can be chosen as hyperkähler $4$-manifolds. We shall prove that these
fibrations have to arise as a finite quotient of a product manifold. Then we
explain over what kinds of bundles one can implement the techniques of section
3 to deduce corresponding results for the moduli space of
$\operatorname{Spin}(7)$-instantons.

###  6.1. Cayley Fibrations

In this subsection, we focus on $\operatorname{Spin}(7)$-manifolds fibred by
Cayley submanifolds. This is a type of calibrated submanifold introduced in
[HL82], whose local deformation theory was studied by [McL98]. We shall write
$(M,\Phi)$ for a complete, torsion-free $\operatorname{Spin}(7)$-manifold. In
particular, $M$ carries an induced metric $g_{\Phi}$ and a volume form
$\operatorname{vol}_{\Phi}$.

######  Definition .

An oriented $4$-dimensional submanifold $X\subseteq M$ is called a Cayley
submanifold if $\Phi|_{X}=\operatorname{vol}_{X}$.

Given a Cayley submanifold $X\subseteq M$, we identify the normal bundle $NX$
with the orthogonal complement $(TX)^{\perp_{\Phi}}\subseteq TM|_{X}$ with
respect to the metric $g_{\Phi}$. We then have a canonical identification

$\begin{split}\Lambda^{+}N^{*}X\longrightarrow\Lambda^{+}T^{*}X.\\\
\omega\longmapsto(\omega\lrcorner\;\Phi)|_{X}\end{split}$ (6.1)

Now suppose $(M,\Phi)$ admits a fibration structure $\pi:M\to Y$ with smooth
Cayley fibres. The metric $g_{\Phi}$ induces an Ehresmann connection
$H_{\Phi}\subseteq TM$. Writing $V=\ker(\pi_{*}:TM\to TY)$ for the vertical
subbundle, we get a bigraded orthogonal decomposition of differential forms as
before

$\Omega^{r}(M)=\bigoplus_{s+t=r}\Omega^{s,t}_{\Phi}=:\bigoplus_{s+t=r}C^{\infty}(M,\Lambda^{s}H^{*}_{\Phi}\otimes\Lambda^{t}V^{*}).$
(6.2)

The exterior derivative decomposes accordingly as

$d=d_{H_{\Phi}}+d_{f}+F_{H_{\Phi}}:\Omega^{s,t}_{\Phi}\longrightarrow\Omega^{s+1,t}_{\Phi}\oplus\Omega^{s,t+1}_{\Phi}\oplus\Omega^{s+2,t-1}_{\Phi}.$
(6.3)

Although the filtration (5.1) will give us an intrinsic type decomposition,
there is no canonical decomposition of $\Phi$ analogous to that of the
$G_{2}$-calibration form $\varphi$.

######  Lemma .

The $\operatorname{Spin}(7)$-structure $\Phi$ induces a canonical conformal
class $c_{\Phi}$ on $Y$.

###### Proof.

Given $y\in Y$, we choose a lift $x_{o}\in X_{y}$. Then
$\pi_{*}:H_{\Phi,x_{o}}\to T_{y}Y$ is an isomorphism. We claim that the image
$\operatorname{im}(\pi_{*}:\Lambda^{+}H_{\Phi,x_{o}}\to\Lambda^{2}T^{*}Y)$ is
independent of the choice of the lift $x_{o}$. By declaring
$\Lambda^{+}T^{*}Y$ as the image, we get a conformal structure on $Y$.

To prove the claim, we choose $\omega_{o}\in\Lambda^{+}H^{*}_{\Phi,x_{o}}$,
which yields a section

$\omega_{y}\in C^{\infty}(X_{y},\Lambda^{2}H^{*}_{\Phi}|_{X_{y}})$

by lifting $\pi_{*}(\omega_{o})\in\Lambda^{2}T^{*}Y$. This gives us a smooth
function $a(x)$ on $X_{y}$ specified by

$\omega_{y}\wedge\omega_{y}=a(x)\operatorname{vol}_{X_{y}}.$

Since $X_{y}$ is Cayler, (6.1) tells us that $a(x)$ is positive. This proves
that the images of the space of self-dual $2$-forms on $H_{\Phi}|_{X_{y}}$ are
identified in $\Lambda^{2}T^{*}Y$. ∎

Let us write $\underline{\omega}_{\Phi}=-\Phi^{2,2}\in\Omega^{2,2}_{\Phi}$ for
the $(2,2)$-component of the calibration form $\Phi$. Choosing a local frame
$(\bar{\tau}_{1},\bar{\tau}_{2},\bar{\tau}_{3})$ of $\Lambda^{+}T^{*}Y$ with
respect to the conformal structure $c_{\Phi}$, we can write

$\underline{\omega}_{\Phi}=\sum_{i=1}^{3}\tau_{i}\wedge\omega_{i},$

where $\tau_{i}\in\Omega^{2,0}_{\Phi}$ is the $H_{\Phi}$-lift of
$\bar{\tau}_{i}$, and $\omega_{i}\in\Omega^{0,2}_{\Phi}$.

From the standard $\operatorname{Spin}(7)$-form $\Phi_{0}$ in (2.31), we can
decompose $\Phi$ into

$\Phi=\underline{\nu}_{\Phi}-\underline{\omega}_{\Phi}+\underline{\mu}_{\Phi}\in\Omega^{4,0}_{\Phi}\oplus\Omega^{2,2}_{\Phi}\oplus\Omega^{0,4}_{\Phi}.$
(6.4)

The self-duality of $\Phi$ implies that
$\star_{\Phi}\underline{\nu}_{\Phi}=\underline{\mu}_{\Phi}$ and
$\star_{\Phi}\underline{\omega}_{\Phi}=\underline{\omega}_{\Phi}$. The
torsion-free condition gives us the structure equations

$d_{H_{\Phi}}\underline{\omega}_{\Phi}=0\qquad
d_{H_{\Phi}}\underline{\mu}_{\Phi}=0\qquad
d_{f}\underline{\omega}_{\Phi}=F_{H_{\Phi}}\underline{\mu}_{\Phi}\qquad
d_{f}\underline{\nu}_{\Phi}=F_{H_{\Phi}}\underline{\omega}_{\Phi}$ (6.5)

######  Definition .

We say the Cayley fibration $\pi:M\to Y$ is hyperkähler if near each $y\in Y$
one can choose a local frame $(\bar{\tau}_{1},\bar{\tau}_{2},\bar{\tau}_{3})$
of $\Lambda^{+}T^{*}Y$ so that in the expression
$\underline{\omega}_{\Phi}=\sum_{i=1}^{3}\tau_{i}\wedge\omega_{i}$ we have

$\omega_{i}\wedge\omega_{j}|_{X_{y}}=2\delta_{ij}\cdot\operatorname{vol}_{X_{y}}.$

We prove the analogue of subsection 5.1 for Cayley fibrbations.

######  Lemma .

Let $\pi:M\to Y$ be a Cayley fibration of a complete
$\operatorname{Spin}(7)$-manifold. Then the following are equivalent.

  1. (a)

$\pi:M\to Y$ is hyperkähler.

  2. (b)

$\pi:M\to Y$ is flat, i.e. $F_{H_{\Phi}}=0$.

  3. (c)

There exists a hyperkähler metric $g_{Y}$ on $Y$ so that
$\pi:(M,g_{\varphi})\to(Y,g_{Y})$ is a Riemannian submersion.

###### Proof.

$(a)\Longleftrightarrow(b)$. It follows from (6.5) that (b) is equivalent to
$d_{f}\underline{\omega}_{\Phi}=0$. With respect to a local frame of
$\Lambda^{+}T^{*}Y$, we can write
$\omega_{i}\wedge\omega_{j}|_{X_{y}}=2a_{ij}(y)\cdot\operatorname{vol}_{X_{y}}$,
where $(a_{ij}(y)):X_{y}\to\mathbb{R}^{3\times 3}$ is a smooth function taking
values in the space of positive definite symmetric matrices. Then
$d_{f}\underline{\omega}_{\Phi}=0$ is equivalent to $da_{ij}(y)=0$, which
means we can choose a local frame to satisfy the hyperkähler condition.

$(a)\Longrightarrow(c)$. Suppose $\pi:M\to Y$ is hyperkähler. Locally, we have
a frame $\bar{\tau}_{1},\bar{\tau}_{2},\bar{\tau}_{3}$ of $\Lambda^{+}T^{*}Y$
and $\underline{\omega}_{\Phi}=\sum_{i}\tau_{i}\wedge\omega_{i}$ satisfying
$\omega_{i}\wedge\omega_{j}=2\delta_{ij}\cdot\operatorname{vol}_{X_{y}}$. Then
we get a metric $g_{Y}$ on $Y$ characterized by

$\bar{\tau}_{i}\wedge\bar{\tau}_{j}=2\delta_{ij}\cdot\operatorname{vol}_{g_{Y}}.$
(6.6)

Since any two such frames differ by an action of elements in $SO(3)$, the
metric is well-defined. It follows from (6.1) that
$\tau_{i}\wedge\tau_{j}=2\delta_{ij}\cdot\underline{\nu}_{\Phi}$. Thus, the
projection $\pi_{*}:H_{\Phi,x}\to T_{y}Y$ is an isometry for each lift $x\in
X_{y}$, which means $\pi$ is a Riemannian submersion. We note that the
holonomy group $\operatorname{Hol}(g_{Y})\subseteq\operatorname{Spin}(7)\cap
SO(4)=SU(2)$. Thus the induced metric $g_{Y}$ is hyperkähler.

$(c)\Longrightarrow(a)$. Let $\bar{\tau}_{1},\bar{\tau}_{2},\bar{\tau}_{3}$ be
hyperkähler $2$-forms on $(Y,g_{Y})$ satisfying (6.6). Denote by $\tau_{i}$
the $H_{\Phi}$-lift of $\bar{\tau}_{i}$, $i=1,2,3$. Then we can write
$\underline{\omega}_{\Phi}=\sum_{i}\tau_{i}\wedge\omega_{i}$. Then (2.32)
implies that
$\omega_{i}\wedge\omega_{j}=2\delta_{ij}\cdot\operatorname{vol}_{X_{y}}$. ∎

The same argument as that of subsection 5.1 gives us the corresponding result
for Cayley fibrations.

######  Proposition .

Let $\pi:M\to Y$ be hyperkähler Cayley fibration of a complete torsion-free
$\operatorname{Spin}(7)$-manifold $(M,\Phi)$. Then we get a hyperkähler metric
$g_{Y}$ and a hyperkähler $4$-manifold $(X,g_{X})$ so that $(M,g_{\Phi})$ is
isometric to $\tilde{Y}\times_{\pi_{1}(Y)}X$, where $\tilde{Y}$ is the
universal cover of $Y$ and $\pi_{1}(Y)$ acts on $X$ via diffeomorphisms
preserving its hyperkähler structure.

Due to subsection 6.1 we shall focus on $\operatorname{Spin}(7)$-manifolds
given by the product of two hyperkähler $4$-manifolds from now on.

###  6.2. $\operatorname{Spin}(7)$-Instantons on Cayley Fibrations

Let $(Y,\tau_{1},\tau_{2},\tau_{3})$ and
$(X,\omega_{1},\omega_{2},\omega_{3})$ be two compact hyperkähler
$4$-manifolds. Their product $M=Y\times X$ is equipped with a torsion-free
$\operatorname{Spin}(7)$-structure

$\Phi:=\pi_{1}^{*}\operatorname{vol}_{Y}+\pi_{2}^{*}\operatorname{vol}_{X}-\sum_{i=1}^{3}\pi_{1}^{*}\tau_{i}\wedge\pi_{2}^{*}\omega_{i}.$
(6.7)

Then the projection $\pi_{1}:M\to Y$ is a hyperkähler Cayley fibration. Let us
write
$\underline{\eta}=\pi_{2}^{*}\operatorname{vol}_{X}-\sum_{i=1}^{3}\pi_{1}^{*}\tau_{i}\wedge\pi_{2}^{*}\omega_{i}$.
Then Theorem 4.1 applies in this case by setting $\alpha=\underline{\eta}$.

Appealing to Theorem 1.1, we see that the moduli space
$\mathcal{M}_{\Phi}(M,P)$ of $\operatorname{Spin}(7)$-instantons over an
$\underline{\eta}$-null bundle is non-empty if and only if
$\mathcal{N}(X,P_{X})$ is non-empty, and
$\underline{P}=\tilde{Y}\times_{\rho}P_{X}$ for some representation
$\rho:\pi_{1}(Y,y_{0})\to\mathscr{G}_{P_{X}}$ which stabilizes some ASD
connection on $P_{X}$.

Now we can give an explicit description of the moduli space through the
twisting map (5.10) constructed for the $G_{2}$-instanton case above. Again,
we start with the irreducible locus. Let
$\rho:\pi_{1}(Y,y_{0})\to\mathbb{Z}/r$ be a representation which gives us the
$\underline{\eta}$-null bundle $\pi:P_{\rho}\to M$. Then we define the
twisting map $\Theta^{*}_{\rho}$ the same way as in (5.10). We can rephrase
Theorem 1.3 as follows in the current set-up.

######  Theorem 6.1.

Let $M=Y\times X$ be the product of two compact hyperkähler $4$-manifolds
equipped with the torsion-free $\operatorname{Spin}(7)$-structure $\Phi$ and
$\pi_{X}:P_{X}\to X$ a principal $SU(r)$-bundle satisfying
$\mathcal{N}^{*}(X,P_{X})\neq\varnothing$. Then for any representation
$\rho:\pi_{1}(Y,y_{0})\to\mathbb{Z}/r$, the map

$\Theta^{*}_{\rho}:H^{1}(Y;\mathbb{Z}/r)\times\mathcal{N}^{*}(X,P_{X})\longrightarrow\mathcal{M}^{*}_{\Phi}(M,P_{\rho})$

is an isomorphism between moduli spaces.

###### Proof.

The reason that $\Theta^{*}_{\rho}$ defines a homeomorphism follows from the
argument of Theorem 5.1 verbatim. It remains to argue that the differential of
$\Theta^{*}_{\rho}$ identifies the Zariski tangent spaces on both sides.
Recall that the deformation complex of $\mathcal{M}_{\Phi}(M,P_{\rho})$ at a
$\operatorname{Spin}(7)$-instanton $[A]$ is given by

$\Omega^{0}(M,\mathfrak{g}_{P_{\rho}})\xrightarrow{-d_{A}}\Omega^{1}(M,\mathfrak{g}_{P_{\rho}})\xrightarrow{\pi_{7}\circ
d_{A}}\Omega^{2}_{7}(M,\mathfrak{g}_{P_{\rho}})$ ($E_{\Phi,A}(M)$)

To repeat the argument of subsection 5.2, it only remains to prove a version
of subsection 5.2 in this $\operatorname{Spin}(7)$ case, which we do next. ∎

######  Lemma .

Let $(M,\Phi)$ be a product of two compact hyperkähler surfaces as above and
$\pi:P\to M$ an $\underline{\eta}$-null $SU(r)$-bundle. Suppose $A$ is a
$\operatorname{Spin}(7)$-instanton on $M$ and
$a\in\Omega^{1}(M,\mathfrak{g}_{P})$ satisfies $\pi_{7}(d_{A}a)=0$. Then

$(d_{A}a)^{2,0}=0\qquad(d_{A}a)^{1,1}=0\qquad d^{+}_{B_{y}}b_{y}=0,\;\forall
y\in Y,$

where $B_{y}=A|_{P_{y}}$, and $b_{y}=a|_{X_{y}}$.

###### Proof.

The proof is based on a similar argument to that of subsection 5.2. Appealing
to (2.34), we get

$\|F_{A+a}\|^{2}_{L^{2}(M)}=4\|\pi_{7}(F_{A+a})\|^{2}_{L^{2}(M)}+16r\pi^{2}\langle
c_{2}(P_{X}),[X]\rangle\operatorname{Vol}(Y).$ (6.8)

Since $\operatorname{tr}((a\wedge a)\wedge(a\wedge a))=0$, and
$\pi_{7}(F_{A}+d_{A}a)=0$, (2.34) implies

$\|F_{A+a}\|^{2}_{L^{2}(M)}=\|a\wedge a\|^{2}_{L^{2}(M)}+16r\pi^{2}\langle
c_{2}(P_{X}),[X]\rangle\operatorname{Vol}(Y).$ (6.9)

The computation of $\|F^{0,2}_{A+a}\|^{2}_{L^{2}(M)}$ results in the same
formula (5.19), which leads to (5.21) as well. The conclusion then follows
from the same rescaling argument as in the proof of subsection 5.2. ∎

When the structure group of $P$ is $SU(2)$, we can deal with the abelian
strata of the moduli spaces in the same way as subsection 5.2, which gives
Theorem 1.4 for the product of two hyperkähler surfaces. We only need to
supply a lemma corresponding to subsection 5.2. We write
$E^{\mathbb{R}}_{\Phi,A}(M)$ for the deformation complex equipped with
$\mathbb{R}$-coefficient.

######  Lemma .

With the set-up above, we have

$H^{1}(E^{\mathbb{R}}_{\Phi,A}(M))\simeq H^{1}(M;\mathbb{R})$

###### Proof.

Given $a\in\Omega^{1}(M,\mathbb{R})$ satisfying $\pi_{7}(da)=0$, (2.34)
implies

$d(a\wedge da\wedge\Phi)=da\wedge
da\wedge\Phi=\left(3|\pi_{7}(da)|^{2}-|\pi_{21}(da)|^{2}\right)\operatorname{vol}_{\Phi}.$

Thus, we conclude that
$\|\pi_{21}(da)\|^{2}_{L^{2}(M)}=3\|\pi_{3}(da)\|^{2}_{L^{2}(M)}=0$. In
particular, we get $da=0$, which is all we need for the identification. ∎

The compactification of moduli spaces of $\operatorname{Spin}(7)$-instantons
defined on an $\underline{\eta}$-null $SU(2)$-bundle $P_{\rho}$ is exactly the
same as in the case of $G_{2}$-instantons on $\underline{\omega}$-null
bundles. We write $\langle c_{2}(P_{X},[X]\rangle=\kappa\in\mathbb{N}$. Then
for each integer $0\leq s\leq\kappa$, we get the bundles $P_{X}(s)\to X$ and
$P_{\rho}(s)\to M$ as before. The space of ideal
$\operatorname{Spin}(7)$-instantons over $P_{\rho}$ is defined as before,
where the convergence is given the same way as in subsection 1.2. The argument
of Theorem 5.3 then gives us the analogous compactification result.

######  Theorem 6.2.

The moduli space

$\overline{\mathcal{M}}_{\Phi}(M,P_{\rho})=\bigcup_{s=1}^{\kappa}\mathcal{M}_{\Phi}(M,P_{\rho}(s))\times\operatorname{Sym}^{s}(X)\cup\mathcal{M}_{\Phi}(X,P_{\rho})$
(6.10)

is compact.

##  7\. Codimension-1 Reduction

In this section, we discuss the codimension-$1$ case of dimension reduction,
which recovers and extends the results of Wang [Wan20] in a more explicit
form.

###  7.1. $G_{2}$-Instantons over $S^{1}\times Z$

We start by recalling the notion of half-closed $SU(3)$-structure on
$6$-manifolds following Wang [Wan20].

######  Definition .

Let $Z$ be a smooth $6$-manifold. A half-closed $SU(3)$-structure on $Z$ is a
quadruple $(J,g,\omega,\Omega)$ consisting of

  1. (1)

an almost complex structure $J$,

  2. (2)

an Hermitian metric $g$ with respect to $J$,

  3. (3)

a positive real $(1,1)$-form $\omega$ given by $\omega(-,-):=g(J-,-)$,

  4. (4)

a nowhere vanishing $(3,0)$-form $\Omega$,

which satisfies the conditions

$\frac{\omega^{3}}{6}=\frac{1}{4}\operatorname{Re}\Omega\wedge\operatorname{Im}\Omega\qquad\text{
and }\qquad d(\operatorname{Re}\Omega)=0.$

To simplify notation, we shall write $(\omega,\Omega)$ for an
$SU(3)$-structure and omit $J$ and $g$. Let $Z$ be a closed smooth
$6$-manifold equipped with a half-closed $SU(3)$-structure $(\omega,\Omega)$.
Then we get a closed $7$-manifold $M=S^{1}\times Z$ equipped with a closed
$G_{2}$-structure

$\varphi=\pi_{1}^{*}dt\wedge\pi_{2}^{*}\omega+\pi_{2}^{*}\operatorname{Re}\Omega,$
(7.1)

where $t$ is the arc-length parameter on the circle $S^{1}$ of length $2\pi$.
For simplicity, we shall omit writing out the pull-back maps $\pi_{1}^{*}$ and
$\pi_{2}^{*}$ in the notation for differential forms later when there is no
ambiguity.

Consider a principal $U(r)$-bundle $\pi:P\to M$. We fix a reference point
$t_{0}\in S^{1}$ and identify $P|_{Z_{t_{0}}}\simeq P_{Z}$ with a
$U(r)$-bundle over $Z$, where $Z_{t}:=\\{t\\}\times Z$ denotes the fibre over
$t\in S^{1}$. Recall that the degree of $P_{Z}$ is defined to be $\deg
P_{Z}=\langle c_{1}(P_{Z})\smile[\omega]^{2},[Z]\rangle$. We start with a
simple observation that is covered in [Wan20, Theorem 1.17] when
$P=\pi_{2}^{*}P_{Z}$ is the pull-back bundle.

######  Lemma .

Let $M=S^{1}\times Z$ be the product $G_{2}$-manifold as above. Suppose the
moduli space of $G_{2}$-instantons $\mathcal{M}_{\varphi}(M,P)$ is non-empty.
Then $\deg P_{Z}=0$.

###### Proof.

Let $A$ be a $G_{2}$-instanton on $P$. Then (2.27) gives us

$\star_{\varphi}(F_{A}\wedge\star_{\varphi}\varphi)=F_{A}\;\lrcorner\;\varphi=0.$
(7.2)

We pick $I_{\epsilon}=(t_{0}-\epsilon,t_{0}+\epsilon)\subseteq S^{1}$ a small
neighborhood of $t_{0}$ in $S^{1}$, and choose an identification
$P|_{I_{\epsilon}\times Z}\simeq I_{\epsilon}\times P_{Z}$. Then over
$I_{\epsilon}\times Z$, we can write $A=B_{t}+\sigma dt$, where $B_{t}$ is a
family of connection on $P_{Z}$, and
$\sigma(t,-)\in\Omega^{0}(Z,\mathfrak{g}_{P_{Z}})$. The curvature of $F_{A}$
then takes the form

$F_{A}=F_{B_{t}}+dt\wedge\left(d_{B_{t}}\sigma-\dot{B}_{t}\right).$ (7.3)

Since $\varphi=dt\wedge\omega+\operatorname{Re}\Omega$, the $(1,0)$-component
of $F_{A}\;\lrcorner\;\varphi$ is $dt\wedge(F_{B_{t}}\;\lrcorner\;\omega)$,
which must vanish for each $t\in I_{\epsilon}$. Thus, we conclude that

$0=\int_{Z_{t_{0}}}\operatorname{tr}(F_{B_{t_{0}}})\;\lrcorner\;\omega\operatorname{vol}_{Z_{t_{0}}}=\frac{1}{2}\int_{Z_{t}}\operatorname{tr}(F_{B_{t_{0}}})\wedge\omega^{2}=-\pi
i\langle c_{1}(P_{Z})\smile[\omega]^{2},[Z]\rangle,$ (7.4)

which finishes the proof. ∎

Note that no extra assumptions are imposed on the bundle $P$ for the
conclusion of subsection 7.1 to hold. From now on, we shall assume $\deg
P_{Z}=0$. Then the absolute minimizers of the Yang–Mills functional are
$G_{2}$-instantons over $S^{1}\times Z$ and HYM connections over $Z$, which
satisfy the set-up (4.1) and (4.2) in section 4. Thus the conclusion of
Theorem 4.1 continues to hold in this case, which we state as the following.

######  Theorem 7.1.

Let $M=S^{1}\times Z$ be the product $G_{2}$-manifold of a circle and a closed
$6$-manifold $Z$ equipped with a half-closed $SU(3)$-structure
$(\omega,\Omega)$. Suppose $\pi:P\to M$ is a $U(r)$-bundle satisfying $\deg
P|_{Z_{t_{0}}}=0$ for a fixed basepoint $t_{0}\in S^{1}$.

  1. (a)

If $P$ is $\operatorname{Re}\Omega$-negative, then
$\mathcal{M}_{\varphi}(M,P)=\varnothing$.

  2. (b)

If $P$ is $\operatorname{Re}\Omega$-null, then each $G_{2}$-instanton $A$ on
$P$ satisfies

$F_{A}^{2,0}=0\quad\text{ and }\quad F_{A}^{1,1}=0.$

Moreover, the moduli space $\mathcal{M}_{\varphi}(M,P)$ is non-empty if and
only if

  1. (b.1)

The moduli space $\mathcal{M}_{\omega}(Z,P_{Z})$ of HYM connections on $P_{Z}$
is non-empty.

  2. (b.2)

The family bundle is $\underline{P}=\mathbb{R}\times_{\rho}P_{Z}$ for some
representation $\rho:\pi_{1}(S^{1},t_{0})\to\mathscr{G}_{P_{Z}}$ satisfying
$\operatorname{im}\rho\subseteq\operatorname{Stab}(B)$ for some HYM connection
$B$ on $P_{Z}$.

The pull-back bundle $\pi_{2}^{*}P_{Z}\to M$ considered in [Wan20] satisfies
the $\operatorname{Re}\Omega$-null assumption automatically, which corresponds
to the trivial representation
$\rho:\pi_{1}(S^{1},t_{0})\to\mathscr{G}_{P_{Z}}$. When the structure group is
$U(r)$, irreducible connections are defined to be those having $U(1)$ as
stabilizers. Since the Picard torus $\operatorname{Pic}(S^{1})$ can be
identified with the circle $S^{1}$ itself, the twisting map (5.10) is defined
on $\operatorname{Pic}(S^{1})\times\mathcal{B}^{*}_{P_{Z}}$ and restricts as a
homeomorphism between $S^{1}\times\mathcal{M}_{\omega}(Z,P_{Z})$ and
$\mathcal{M}_{\varphi}(M,P)$, following the argument of Theorem 5.1. Thus, the
main result of Wang [Wan20, Theorem 1.17 $\mathbb{I}$] can be recovered from
Theorem 7.1.

To get an elliptic deformation theory for $G_{2}$-instantons, the calibration
form $\varphi$ is required to be coclosed. To this end, we consider Calabi–Yau
fibres. Note that a half-closed $SU(3)$-structure $(J,g,\omega,\Omega)$
defines a Calabi–Yau structure if we further require

$(a)\;J\text{ is integrable}\quad(b)\;\omega\text{ is
closed}\quad(c)\;\Omega\text{ is holomorphic}$ (7.5)

In particular, the $G_{2}$-structure $\varphi$ in (7.1) is torsion-free when
$Z$ is a Calabi–Yau $3$-fold. Although it is not necessary to replace the
structure group $U(r)$ by $SU(r)$, we switch to the latter case to be
consistent with Theorem 1.3. Note that the degree of any $SU(r)$-bundle is
automatically zero. To complete the proof of Theorem 1.3 and Theorem 1.4 in
this case, it remains to identify the Zariski tangent spaces.

Let $M=S^{1}\times Z$ be the product of a circle with a Calabi–Yau $3$-fold,
and $\pi:P\to M$ a $\operatorname{Re}\Omega$-null $SU(r)$-bundle. Let $A$ be a
$G_{2}$-instanton on $P$. Then Theorem 4.1 implies that $B_{t}=A|_{P_{t}}$ is
a HYM connection for each $t\in S^{1}$. In this case, HYM connections are
characterized by (cf. [DS11])

$F_{B}\wedge\operatorname{Im}\Omega=0\text{ and }F_{B}\wedge\omega^{2}=0.$
(7.6)

Appealing to the argument of subsection 5.2, the identification of Zariski
tangent spaces is complete once we prove the following result.

######  Lemma .

Let $A$ be a $G_{2}$-instanton on $P$ as above. Given
$a\in\Omega^{1}(M,\mathfrak{g}_{P})$, we write $b_{t}=a|_{Z_{t}}$. Then

$d_{A}a\wedge\star_{\varphi}\varphi=0\;\;\Longrightarrow\;\;d_{B_{t}}b_{t}\wedge\operatorname{Im}\Omega=0,\text{
and }d_{B_{t}}b_{t}\wedge\omega^{2}=0,\;\forall t\in S^{1}.$

###### Proof.

In constrast to the proof of subsection 5.2, we give an alternative argument
for this case. Over $S^{1}\times Z$, we have
$\star_{\varphi}\varphi=dt\wedge\operatorname{Im}\Omega+\omega^{2}$. We write
$a=a^{1,0}+a^{0,1}$. Then the (0, 6)-component of the equation
$d_{A}a\wedge\star_{\varphi}\varphi=0$ is

$d_{f,A}a^{0,1}\wedge\omega^{2}=0,$ (7.7)

from which we conclude that $d_{B_{t}}b_{t}\wedge\omega^{2}=0$.

The $(1,5)$-component of $d_{A}a\wedge\star_{\varphi}\varphi=0$ is

$d_{f,A}a^{0,1}\wedge
dt\wedge\operatorname{Im}\Omega+(d_{A}a)^{1,1}\wedge\omega^{2}=0.$ (7.8)

Let us write $\mathfrak{b}_{t}=(d_{A}a)^{1,1}|_{Z_{t}}$, which decomposes as
$\mathfrak{b}_{t}=\mathfrak{b}^{(1,0)}_{t}\oplus\mathfrak{b}^{(0,1)}_{t}\in\Omega^{(1,0)}_{Z_{t}}\oplus\Omega^{(0,1)}_{Z_{t}}$.
Since $d_{B_{t}}b_{t}\wedge\operatorname{Im}\Omega\in\Omega^{(3,2)}_{Z_{t}}$,
we see that
$\mathfrak{b}^{(0,1)}_{t}\wedge\omega^{2}=0\in\Omega^{(2,3)}_{Z_{t}}$. Since
both $\mathfrak{b}_{t}$ and $\omega^{2}$ are real, we see that

$\mathfrak{b}^{(1,0)}_{t}\wedge\omega^{2}=\left(\mathfrak{b}^{(0,1)}_{t}\wedge\omega^{2}\right)^{conj}=0,$

where the superscript $conj$ means to take complex conjugation. It follows
that $\mathfrak{b}_{t}\wedge\omega^{2}=0$, hence
$d_{B_{t}}b_{t}\wedge\operatorname{Im}\Omega=0$ by (7.8), as desired. ∎

As one can see, the argument in the proof of subsection 7.1 is local, thus
does not use the $\operatorname{Re}\Omega$-null condition on $P$. We think the
main reason is that $\dim S^{1}=1$ implies that the family bundle
$\underline{P}$ is necessarily flat, and $(d_{A}a)^{2,0}=0$ automatically.
Then one can make use of the complex structure on the fibres $Z_{t}$ to deal
with the equation mixing $(d_{A}a)^{1,1}$ and $(d_{A}a)^{0,2}$.

To identify the abelian strata in the case of $SU(2)$-bundles, we already know
that $H^{1}(E^{\mathbb{R}}_{\varphi,A}(M))\simeq H^{1}(M;\mathbb{R})$ from
subsection 5.2. So it remains to identify the first cohomology on fibres,
which we prove in a more general setting.

######  Lemma .

Let $(Z,\omega)$ be a closed Kähler manifold of complex dimension $m$. Suppose
$a\in\Omega^{1}(Z,\mathbb{R})$ satisfies $(da)^{(2,0)}=(da)^{(0,2)}=0$ and
$da\wedge\omega^{m-1}=0$. Then $da=0$.

###### Proof.

It suffices to prove that $(da)^{(1,1)}=0$ under the given assumption. For a
$(1,1)$-form $\eta$, the identity (2.17) reads

$\frac{1}{(m-2)!}\eta\wedge\eta\wedge\omega^{m-2}=\left(|\eta|^{2}-\frac{1}{m}|\widehat{\eta}|^{2}\right)\operatorname{vol},$

where $\widehat{\eta}=\eta\;\lrcorner\;\omega$. Replacing $\eta$ by $da$, we
get

$\frac{1}{(m-2)!}d(a\wedge da\wedge\omega^{m-2})=|da|^{2}\operatorname{vol},$

from which we conclude that $da=0$ as desired. ∎

###  7.2. $\operatorname{Spin}(7)$-Instantons over $S^{1}\times X$

Let $(X,\varphi)$ be a closed, torsion-free $G_{2}$-manifold. The product
manifold $M=S^{1}\times X$ is equipped with a
$\operatorname{Spin}(7)$-structure

$\Phi=dt\wedge\varphi+\psi,$ (7.9)

where we have omitted writing out the pull-back maps $\pi_{1}^{*}$ and
$\pi_{2}^{*}$. Let $\pi:P\to M$ be an $U(r)$-bundle and $\pi_{X}:P_{X}\to X$
the corresponding restriction to $\\{t_{0}\\}\times X$ for a fixed basepoint
$t_{0}\in S^{1}$. Comparing the minima of the Yang–Mills functional on
$\mathscr{A}_{P}$ and $\mathscr{A}_{P_{X}}$ from section 2, we see that
Theorem 4.1 holds in this case, which covers the second part of Wang’s main
result [Wan20, Theorem 1.17 $\mathbb{II}$] for the same reason explained in
the paragraph following Theorem 7.1.

Now we let $\pi:P\to M$ be a $\psi$-null $SU(r)$-bundle. To prove Theorem 1.3
and Theorem 1.4 in this case, it remains to identify the corresponding Zariski
tangent spaces. By the same argument of subsection 5.2, all we need is to
prove the following result, which is analogous to subsection 5.2.

######  Lemma .

Let $A$ be a $\operatorname{Spin}(7)$-instanton on the $\psi$-null
$SU(r)$-bundle $P$ as above. Suppose $a\in\Omega^{1}(M,\mathfrak{g}_{P_{t}})$
satisfies $\star_{\Phi}(d_{A}a\wedge\Phi)=-d_{A}a$. Then

$(d_{A}a)^{1,1}=0\qquad\star_{\varphi}(d_{B_{t}}b_{t}\wedge\varphi)=-d_{B_{t}}b_{t},\;\forall
t\in S^{1},$

where $B_{t}=A|_{P_{t}}$, and $b_{t}=a|_{X_{t}}$.

###### Proof.

Since $\pi_{7}(d_{A}a)=0$, we can copy the formula (6.9) to this case:

$\|F_{A+a}\|^{2}_{L^{2}(M)}=\|a\wedge a\|^{2}_{L^{2}(M)}+16r\pi^{2}\langle
c_{2}(P_{X}),[X]\rangle\operatorname{Vol}(S^{1}).$ (7.10)

The computation of $\|F^{0,2}_{A+a}\|^{2}_{L^{2}(M)}$ can be derived in a
similar way:

$\|F^{0,2}_{A+a}\|^{2}_{L^{2}(M)}=3\int_{S^{1}}\|\pi_{7}(F_{B_{y}+b_{y}})\|^{2}_{L^{2}(X_{t})}dt+16r\pi^{2}\langle
c_{2}(P_{X}),[X]\rangle\operatorname{Vol}(S^{1}).$ (7.11)

Since $B_{y}$ is a $G_{2}$-instanton, we see that
$\pi_{7}(F_{B_{y}+b_{y}})=\pi_{7}(d_{B_{y}}b_{y})+\pi_{7}(b_{y}\wedge b_{y})$.
Appealing to the fact that $F^{1,1}_{A}=0$, we get

$\|a\wedge a\|^{2}_{L^{2}(M)}=\|(d_{A}a+a\wedge
a)^{1,1}\|^{2}_{L^{2}(M)}+3\int_{S^{1}}\|\pi_{7}(d_{B_{y}}b_{y}+b_{y}\wedge
b_{y})\|^{2}_{L^{2}(X_{t})}dt$ (7.12)

Replacing $a$ by $\epsilon a$ using a small positive constant $\epsilon$ in
(7.12), we get

$\epsilon^{2}\left(\|(d_{A}a)^{1,1}\|^{2}_{L^{2}(M)}+3\int_{S^{1}}\|\pi_{7}(d_{B_{y}}b_{y})\|^{2}_{L^{2}(X_{t})}dt\right)+\mathcal{O}(\epsilon^{3})=0.$
(7.13)

The claimed relations now follow by dividing through by $\epsilon^{2}$ and
letting $\epsilon\to 0$. ∎

One might hope to carry out a proof of subsection 7.2 in a similar fashion of
subsection 7.1. It turns out the equations corresponding to the
$(1,1)$-component and $(0,2)$-component of the given assumption
$\star_{\Phi}(d_{A}a\wedge\Phi)=-d_{A}a$ are equivalent, so one obtains

$(d_{A}a)^{0,2}\wedge\psi=-\star_{\Phi}(d_{A}a)^{1,1}.$ (7.14)

It follows that the two equations in the conclusion of subsection 7.2 are
equivalent. However, to get the vanishing condition on either side of (7.14),
it is necessary to make use of the $\psi$-null condition on $P$, which is a
global one.

##  8\. Product Kähler Manifolds

In this section, we investigate the last case in Table 1. Let $(Y,\omega_{1})$
and $(X,\omega_{2})$ be compact Kähler manifolds of complex dimension $m_{1}$
and $m_{2}$, respectively. Let $M=Y\times X$ be the product Kähler manifold
equipped with the Kähler form
$\omega=\pi_{1}^{*}\omega_{1}+\pi_{2}^{*}\omega^{2}$. Let $\pi:P\to M$ be a
principal $SU(r)$-bundle whose restriction on the fibres can be identified
with $\pi_{X}:P_{X}\to X$.

To comply with the set-up of section 4, we let

$\theta=\omega^{m-2}=\operatorname{vol}_{Y}\wedge\frac{(m-2)!}{(m_{2}-2)!}\omega_{2}^{m_{2}-2}+\alpha=\operatorname{vol}_{Y}\wedge\beta+\alpha,$
(8.1)

where $m=m_{1}+m_{2}$. It follows from (2.18) that the absolute minimum of the
Yang–Mills functional on $\mathscr{A}_{P}$ is

$\frac{16r\pi^{2}}{(m-2)!}\langle\kappa(P)\smile[\omega]^{m-2},[M]\rangle=\frac{16r\pi^{2}}{(m-2)!}\langle\kappa(P)\smile[\theta],[M]\rangle,$
(8.2)

while the absolute minimum of the Yang–Mills functional on
$\mathscr{A}_{P_{X}}$ is

$\frac{16r\pi^{2}}{(m_{2}-2)!}\langle\kappa(P_{X})\smile[\omega_{2}]^{m_{2}-2},[X]\rangle=\frac{16r\pi^{2}}{(m-2)!}\langle\kappa(P_{X})\smile[\beta],[X]\rangle.$
(8.3)

Since the coefficients of (8.2) and (8.3) match up, the conclusion of Theorem
4.1 applies in this case.

Let $\pi:P\to M$ be an $\alpha$-null bundle. Theorem 1.3 and Theorem 1.4 can
be proved similarly as the other cases except we need the corresponding result
of subsection 5.2 to identify the Zariski tangent spaces. To this end, we
prove the following result.

######  Lemma .

Let $A$ be a HYM connection on an $\alpha$-bundle $P$ as above. Suppose
$a\in\Omega^{1}(M,\mathfrak{g}_{P})$ satisfies
$(d_{A}a)^{(2,0)}=(d_{A}a)^{(0,2)}=0$ and $d_{A}a\wedge\omega^{m-1}=0$. Then

$(d_{B_{y}}b_{y})^{(2,0)}=(d_{B_{y}}b_{y})^{(0,2)}=0,\text{ and
}d_{B_{y}}b_{y}\wedge\omega_{2}^{m_{2}-1}=0,\;\forall y\in Y,$

where $B_{y}=A|_{P_{y}}$, and $b_{y}=a|_{X_{y}}$.

###### Proof.

We adopt the same argument as in subsection 5.2. (2.18) tells us that

$\begin{split}\|F_{A+a}\|^{2}_{L^{2}(M)}&=4\|F^{(2,0)}_{A+a}\|^{2}_{L^{2}(M)}+\frac{1}{m}\|\widehat{F}_{A+a}\|^{2}_{L^{2}(M)}\\\
&+\frac{16r\pi^{2}}{(m-2)!}\langle
c_{2}(P_{X})\smile[\beta],[X]\rangle\operatorname{Vol}(Y).\end{split}$ (8.4)

Since $F_{A}^{(2,0)}=(d_{A}a)^{(2,0)}=0$ and
$F_{A}\;\lrcorner\;\omega=d_{A}a\lrcorner\omega=0$, replacing $F_{B}$ by
$a\wedge a$ in (2.17) tells us that

$\|F_{A+a}\|^{2}_{L^{2}(M)}=\|a\wedge
a\|^{2}_{L^{2}(M)}+\frac{16r\pi^{2}}{(m-2)!}\langle
c_{2}(P_{X})\smile[\beta],[X]\rangle\operatorname{Vol}(Y).$ (8.5)

On the other hand, applying the argument to $F_{B_{y}+b_{y}}$ on the fibres
$X_{y}$ gives us

$\begin{split}\|F^{0,2}_{A+a}\|^{2}_{L^{2}(M)}&=4\int_{Y}\|(d_{B_{y}}b_{y}+b_{y}\wedge
b_{y})^{(2,0)}\|^{2}_{L^{2}(X_{y})}\operatorname{vol}_{Y}\\\
&+\frac{1}{m_{2}}\int_{Y}\|(d_{B_{y}}b_{y}+b_{y}\wedge
b_{y})\lrcorner\omega_{2}\|^{2}_{L^{2}(X_{y})}\operatorname{vol}_{Y}\\\
&+\frac{16r\pi^{2}}{(m-2)!}\langle
c_{2}(P_{X})\smile[\beta],[X]\rangle\operatorname{Vol}(Y).\end{split}$ (8.6)

Combining with the fact that $F_{A}^{2,0}=F_{A}^{1,1}=0$, we get

$\begin{split}\|a\wedge a\|^{2}_{L^{2}(M)}&=\|(d_{A}a+a\wedge
a)^{2,0}\|^{2}_{L^{2}(M)}+\|(d_{A}a+a\wedge a)^{1,1}\|^{2}_{L^{2}(M)}\\\
&+4\int_{Y}\|(d_{B_{y}}b_{y}+b_{y}\wedge
b_{y})^{(2,0)}\|^{2}_{L^{2}(X_{y})}\operatorname{vol}_{Y}\\\
&+\frac{1}{m_{2}}\int_{Y}\|(d_{B_{y}}b_{y}+b_{y}\wedge
b_{y})\lrcorner\omega_{2}\|^{2}_{L^{2}(X_{y})}\operatorname{vol}_{Y}.\end{split}$
(8.7)

Replacing $a$ by $\epsilon a$, dividing through by $\epsilon^{2}$, and letting
$\epsilon\to 0$, we see that all the linear integrands in (8.7) have to
vanish. In particular

$(d_{B_{y}}b_{y})^{(2,0)}=0\text{ and }d_{B_{y}}b_{y}\lrcorner\omega_{2}=0.$

This completes the proof. ∎

When $m_{2}=2$, HYM connections on $X$ are exactly the ASD connections. We
then have the corresponding compactification Theorem 1.5 for such cases.

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# Two-dimensional optomechanical crystal resonator in gallium arsenide

Rhys G. Povey<EMAIL_ADDRESS>Department of Physics, University of
Chicago, Chicago IL 60637, USA  Pritzker School of Molecular Engineering,
University of Chicago, Chicago IL 60637, USA  Ming-Han Chou  Department of
Physics, University of Chicago, Chicago IL 60637, USA  Pritzker School of
Molecular Engineering, University of Chicago, Chicago IL 60637, USA  Gustav
Andersson  Pritzker School of Molecular Engineering, University of Chicago,
Chicago IL 60637, USA  Christopher R. Conner  Pritzker School of Molecular
Engineering, University of Chicago, Chicago IL 60637, USA  Joel Grebel
Pritzker School of Molecular Engineering, University of Chicago, Chicago IL
60637, USA  Yash J. Joshi  Pritzker School of Molecular Engineering,
University of Chicago, Chicago IL 60637, USA  Jacob M. Miller  Department of
Physics, University of Chicago, Chicago IL 60637, USA  Pritzker School of
Molecular Engineering, University of Chicago, Chicago IL 60637, USA  Hong Qiao
Pritzker School of Molecular Engineering, University of Chicago, Chicago IL
60637, USA  Xuntao Wu  Pritzker School of Molecular Engineering, University of
Chicago, Chicago IL 60637, USA  Haoxiong Yan  Pritzker School of Molecular
Engineering, University of Chicago, Chicago IL 60637, USA  Andrew N. Cleland
<EMAIL_ADDRESS>Pritzker School of Molecular Engineering, University of
Chicago, Chicago IL 60637, USA  Center for Molecular Engineering and Materials
Science Division, Argonne National Laboratory, Lemont IL 60439, USA

###### Abstract

In the field of quantum computation and communication there is a compelling
need for quantum-coherent frequency conversion between microwave electronics
and infra-red optics. A promising platform for this is an optomechanical
crystal resonator that uses simultaneous photonic and phononic crystals to
create a co-localized cavity coupling an electromagnetic mode to an acoustic
mode, which then via electromechanical interactions can undergo direct
transduction to electronics. The majority of work in this area has been on
one-dimensional nanobeam resonators which provide strong optomechanical
couplings but, due to their geometry, suffer from an inability to dissipate
heat produced by the laser pumping required for operation. Recently, a quasi-
two-dimensional optomechanical crystal cavity was developed in silicon
exhibiting similarly strong coupling with better thermalization, but at a
mechanical frequency above optimal qubit operating frequencies. Here we adapt
this design to gallium arsenide, a natural thin-film single-crystal
piezoelectric that can incorporate electromechanical interactions, obtaining a
mechanical resonant mode at $f_{\mathrm{m}}\approx 4.5\,\mathrm{GHz}$ ideal
for superconducting qubits, and demonstrating optomechanical coupling
$g_{\mathrm{om}}/(2\,\pi)\approx 650\,\mathrm{kHz}$.

##  I Introduction

Spurred on by the promise of quantum computing [1, 2], superconducting qubits
[3, 4, 5, 6, 7, 8, 9] have become a ubiquitous topic of research and
development. These devices, typically operating with microwave electronic
frequencies of $4\text{--}8\,\mathrm{GHz}$ and housed in dilution
refrigerators at $\sim 10\,\mathrm{mK}$, are excellent candidates for quantum
information processing but suffer from an inability to engage in long-range
communication due to the lossy environment and large amount of microwave
thermal noise at room temperature. To facilitate such connections outside of
cryogenic temperatures, a quantum transducer that is able to coherently
convert a microwave-electronic quantum state to an infra-red fiber optic
signal, ideally in the telecom band of $1530\text{--}1565\,\mathrm{nm}$, is
desired [10, 11, 12, 13]. Of the many approaches being explored for this
challenging task, a propitious route is through an optomechanical crystal
cavity [14, 15] that emerges by placing a defect in a simultaneous photonic
and phononic crystal. This defect site is able to support both a microwave-
frequency acoustic resonant mode and an infra-red electromagnetic resonant
mode, allowing them to couple to each other. Acoustic modes can then be
directly transduced to electrical signals through a piezoelectric material.

The optomechanical Hamiltonian for an optical resonant mode at
$\omega_{\mathrm{o}}$, annihilation operator $\hat{a}$, and a mechanical
resonant mode at $\omega_{\mathrm{m}}$, annihilation operator $\hat{b}$, is
given by

$\hat{H}=\hbar\,\omega_{\mathrm{o}}\,\hat{a}^{\dagger}\,\hat{a}+\hbar\,\omega_{\mathrm{m}}\,\hat{b}^{\dagger}\,\hat{b}+\hbar\,g_{\mathrm{om}}\,\hat{a}^{\dagger}\,\hat{a}\,\bigl{(}\hat{b}^{\dagger}+\hat{b}\bigr{)}\;,$
(1)

where the interaction term, with optomechanical coupling $g_{\mathrm{om}}$,
can be derived by considering an optical Fabry-Pérot mirror attached to a
mechanical spring undergoing small oscillations [16, 17]. To obtain an
exchange interaction we need to include a strong laser pump at frequency
$\omega_{\mathrm{L}}$, adding the term
$\hbar\,L\,\hat{a}^{\dagger}\,\mathrm{e}^{-\mathrm{i}\,\omega_{\mathrm{L}}\,t}$,
where $L$ is measure of the pump strength. For a strong pump we can expand
$\hat{a}(t)=\hat{a}^{\prime}(t)+\bar{a}\,\mathrm{e}^{-\mathrm{i}\,\omega_{\mathrm{L}}\,t}$,
with $\bar{a}\propto L$, and linearize for $\bar{a}\gg\hat{a}^{\prime}$. With
the laser red-detuned from the optical resonance,
$\omega_{\mathrm{L}}=\omega_{\mathrm{o}}-\omega_{\mathrm{m}}$, under the
rotating wave approximation our interaction term then becomes

$\hat{H}_{\mathrm{int}}\approx\hbar\,g_{\mathrm{om}}\,\bigl{(}\bar{a}^{\ast}\,\hat{a}^{\prime}\,\hat{b}^{\dagger}+\bar{a}\,\hat{a}^{\prime\,\dagger}\,\hat{b}\bigr{)}\;,$
(2)

providing a ‘beam-splitter’ swapping interaction that allows infra-red optical
excitations to be exchanged with microwave-frequency mechanical excitations.
As can be seen in Eq.(2), the optomechanical swapping interaction is enhanced
by the average photon number, $\lvert\bar{a}\rvert^{2}$, which is proportional
to the laser pump power. Thus to maximize the effective optomechanical
coupling, the laser pump should be as strong as possible before detrimental
effects, such as heating, set in.

Previous work [18, 19, 20] and the majority of attention in optomechanical
crystals for transduction has been on one-dimensional nanobeam designs [21,
22, 23, 24, 25, 26]. These structures exhibit strong optomechanical coupling,
$g_{\mathrm{om}}/(2\,\pi)\sim 1\,\mathrm{MHz}$, and high mechanical and
optical quality factors but are plagued by poor heat dissipation due to their
one-dimensional nature limiting thermal conductivity [27]. Recently, a quasi-
two-dimensional optomechanical cavity design consisting of C-shape holes
within the waveguide of a snowflake hole crystal lattice was demonstrated in
silicon [28]. This device showed improved heat management whilst maintaining
good optomechanical coupling and quality factors, however, targeting
$194\,\mathrm{THz}$ optics, operates at a mechanical frequency near
$10\,\mathrm{GHz}$ — somewhat above the optimal frequency for popular
superconducting qubit archetypes. Here, we adapt the design to gallium
arsenide (GaAs), having half the acoustic wave speed of silicon, to produce a
two-dimensional optomechanical resonator at a superconducting qubit compatible
frequency $\sim 5\,\mathrm{GHz}$. Furthermore, the inherent piezoelectricity
in gallium arsenide allows for future direct coupling to an electrical
circuit.

##  II Design

Figure 1:  (a) Layout and geometric parameterization of the optomechanical
crystal resonator. The crystal (mirror) unit cell is highlighted. Parameter
values in the gradient section between the defect and crystal unit cells are
scaled with a $\mathrm{SmoothStep}_{1}$ function plotted by the purple line
below. (b) Acoustic dispersion relations from finite element model simulations
of the mirror unit cell. Band gaps for $\overset{+}{y}\,\overset{+}{z}$
symmetric modes are highlighted by yellow bands. (c) Electromagnetic
dispersion relations from finite element model simulations of the mirror unit
cell. Band gaps for $\overset{+}{z}$ symmetric modes are highlighted in yellow
and cover our infra-red frequency of interest, $194\,\mathrm{THz}$, marked
with a dashed red line. (d) Simulation of the displacement field, $Q_{y}$, for
the paddle breathing mode at $4.63\,\mathrm{GHz}$. Displacements in paddles
neighbouring the defect are too small to make out but are plotted above on a
log scale. (e) Simulations of the electric field for an electromagnetic
resonance at $197\,\mathrm{THz}$. Specific dimensions used for simulations are
given in the supplementary materials [29].

To describe the symmetries present in the optomechanical crystal design we use
the notation $\overset{\pm}{x}$ to denote a mirror ($+$) or anti-mirror ($-$)
symmetry through the $x=0$ plane, such that $P_{\vec{x}}\cdot
F[P_{\vec{x}}\cdot\vec{r}]=\pm F[\vec{r}]$ for some tensor field $F$, where
$P$ is the Householder reflection transform. The main two-dimensional
optomechanical crystal is made from snowflake shaped holes in a periodic
lattice that exhibits $\pi/3$ rotational symmetry and mirror symmetries
(wallpaper group p6m). This provides a complete acoustic band gap and $z$
mirror symmetric ($\overset{+}{z}$) electromagnetic band gap [29]. Removing a
row of snowflakes, and with additional stretching, creates a waveguide though
the two-dimensional optomechanical crystal. C-shape holes along this waveguide
in a ‘vertebrae’ pattern form a one-dimensional optomechanical crystal that
can be gradually transitioned to create a defect cavity. The layout and
paramterization of these holes to form the resonator is given in Fig. 1(a).

C-shape hole dimensions in the gradient region between the mirror and defect
are scaled according to the differentiable $\mathrm{SmoothStep}_{1}$ function,
plotted in Fig. 1(a), with definition
$\mathrm{SmoothStep}_{n}:[0,1]\rightarrow[0,1]$ by $x\mapsto
x^{n+1}\,\sum_{k=0}^{n}\,(\begin{smallmatrix}n+k\\\
k\end{smallmatrix})\,(\begin{smallmatrix}2\,n+1\\\
n-k\end{smallmatrix})\,(-x)^{k}$, where $(\begin{smallmatrix}n\\\
k\end{smallmatrix})$ are binomial coefficients. This is the lowest order
polynomial between constant zero and one that is $n$-differentiable.

Here, we focus on $n_{\text{defect}}=1$ single paddle resonances. Although
$n_{\text{defect}}>1$ resonators provide better optomechanical couplings in
simulation, they are prone to fabrication imperfections, splitting the
resonance into separate modes. Simulations showed that a $+1\,\mathrm{nm}$
change in paddle length ($\mathsf{p}$) shifted the individual resonance by
$-17\,\mathrm{MHz}$.

Starting with C-shape hole parameters used in silicon [28, 30], finite element
model simulations [31] were carried out in gallium arsenide and iterated upon
to find a set of dimensions that produced the desired band gaps. Dispersion
relations for such parameters over a vertebrae unit cell are given in Fig.
1(b,c). The acoustic resonance displacement field must be mirror-symmetric
about every geometric mirror plane in order to have non-vanishing
optomechanical coupling [32]. For the vertebrae resonator design, this is the
paddle breathing mode with mirror symmetries about each cardinal plane through
the center ($\overset{+}{x}\,\overset{+}{y}\,\overset{+}{z}$) as depicted in
Fig. 1(d). By changing the paddle length ($\mathsf{p}$) between the mirror
cell and defect cell, we can move the breathing mode from outside the band gap
to within.

Eigenvalue simulations of the full vertebrae resonator are shown in Fig.
1(d,e). In order to accommodate future acoustic piezoelectric wave
propagation, the waveguide is aligned along the $[110]$ GaAs crystal axis
[32]. The optomechanical coupling, calculated from displacement and electric
fields, due to the moving boundary and photo-elastic effect, is
$g_{\mathrm{om}}/(2\,\pi)=599\,\mathrm{kHz}$.

##  III Fabrication

Figure 2:  (a) Scanning electron microscope image, taken at $45^{\circ}$, of a
device similar to the one measured here. Two large suspended grating coupler
pads deflect light in and out of plane to couple to overhead optical fibers.
The optomechanical crystal plate is suspended in the middle with large
serpentine tethers to relieve tension during the release process. (b)
Magnified image of the optomechanical plate from (a), taken top down, showing
the vertebrae resonator. Proximity effect correction during electron beam
lithography is essential for uniform snowflakes. (c) Simplified diagram of the
optical measurement setup showing only elements discussed in the text. The
phase modulator is driven by a microwave frequency signal generator and
provides a calibration of power measured by the spectrum analyzer. The
V-groove assembly holds fiber optics above the device. A portion of the output
signal is used in a dither locking system, utilizing a lock-in amplifier and
proportional-integral-derivative (PID) controller, that adjusts the laser
output wavelength. The detection chain consists of an erbium doped fiber
amplifier (EDFA) followed by a filter to cut out amplified spontaneous
emission, and then a microwave-frequency bandwidth photodetector.

Devices are fabricated on heterostructure wafers consisting of
$250\,\mathrm{nm}$ $\mathrm{Ga}\mathrm{As}$, on a $1\,\upmu\mathrm{m}$
$\mathrm{Al}_{0.9}\mathrm{Ga}_{0.1}\mathrm{As}$ sacrificial layer, on
$635\,\upmu\mathrm{m}$ bulk $(100)$ $\mathrm{Ga}\mathrm{As}$. The relatively
low suspension height provided by the sacrificial layer
($\mathsf{t}_{\text{gap}}<\lambda_{\mathrm{o}}$) leads to optical fields
leaking into the bulk, limiting optical quality factors and grating coupler
performance. Patterning is performed with electron beam lithography using
$14\%$ by weight hydrogen silsesquioxane in methyl isobutyl ketone resist and
a chlorine, argon inductively coupled plasma etch. Proximity effect correction
is vital when patterning nanometer scale features over hundreds of square
microns on a heavy substrate with significant electron back scatter, details
of our point spread function used are provided in the supplementary materials
[29]. To account for uncertainty in precise slab thickness, produced hole
sizes, and simulations, devices are fabricated in series of varying global
scale factor, typically in steps of $1\%$. Parameter dimensions for the
pattern used to write the device presented here are listed in the
supplementary materials [29].

To couple light to the device, two suspended grating coupler pads are used to
deflect light in and out of plane to optical fibers positioned above the
device. The resonator is then interposed along a connecting suspended
waveguide as depicted in Fig. 2(a,b). Although less efficient than tapered-
beam edge coupling, this approach allowed up to 100 devices to be fabricated
and tested on a single $1\,\mathrm{cm}^{2}$ chip.

##  IV Measurement

Figure 3:  (a) Optical resonance transmission (orange) and reflection (cyan)
with colour-coded probing regions covered by the dither lock. (b) Thermal
mechanical spectrum imprinted on the optical signal when probed at the optical
resonance and inflection points (colour coded accordingly). Each trace is the
average of 100 scans with a 100 point moving average, statistical uncertainty
of $\pm 1$ standard deviation is depicted in light shading about the trace.
(c) Scans of the thermal mechanical spectrum with different input powers,
referenced to the V-groove assembly. Inset shows fit spectrum amplitude
$\mathcal{A}$ as a function of input power over our photodetector’s linear
operating regime, with $P^{2}$ fit curves.

Optical measurements are carried out at room temperature using an infra-red
fiber optic setup and tunable laser. A simplified diagram including only major
elements is shown in Fig. 2(c). To compensate for slow thermal drifts of the
optical resonance frequency and overall transmission, a second-harmonic dither
locking system is used to track the resonance inflection points. A diagram of
the complete setup, and details on the dither locking, are included in the
supplementary material [29].

Transmission through the optical resonance is given by

$\lvert
S_{2\,1}(\omega)\rvert^{2}=\frac{\left(\gamma_{\mathrm{o}}^{\mathrm{wg}}\right)^{2}}{\left(\omega-\omega_{\mathrm{o}}\right)^{2}+\left(\frac{\gamma_{\mathrm{o}}}{2}\right)^{2}}\;,$
(3)

with
$\gamma_{\mathrm{o}}=\gamma_{\mathrm{o}}^{0}+2\,\gamma_{\mathrm{o}}^{\mathrm{wg}}$,
where $\gamma_{\mathrm{o}}^{0}$ is intrinsic power loss rate and
$\gamma_{\mathrm{o}}^{\mathrm{wg}}$ is power loss rate to each waveguide port.
The loaded quality factor is defined as
$Q_{\mathrm{o}}^{\mathrm{L}}=\omega_{\mathrm{o}}/\gamma_{\mathrm{o}}$, and the
internal quality factor is
$Q_{\mathrm{o}}^{0}=\omega_{\mathrm{o}}/\gamma_{\mathrm{o}}^{0}$. A captured
transmission profile is given in Fig. 3(a).

At room temperature, the mechanical resonance has a thermal population
$\bar{n}_{\mathrm{m}}\approx\frac{k_{\mathrm{B}}\,T}{\hbar\,\omega_{\mathrm{m}}}=1371\pm
14$, which through the optomechanical interaction will modulate the optical
resonance at the mechanical resonance frequency. Probing the side of the
optical resonance, we are sensitive to the resulting power fluctuation using a
high bandwidth photodetector, and measuring with a spectrum analyzer can
observe the mechanical thermal spectrum.

Conveniently, the transmission factor of this optomechanical spectrum is
identical to the transmission factor of a weak phase-modulated signal [33],
allowing us to make a calibrated measurement of the optomechanical coupling.
At laser frequency $\omega_{\mathrm{L}}$, the phase modulated input is
$\exp\bigl{[}-\mathrm{i}\,\omega_{\mathrm{L}}\,t+\mathrm{i}\,A_{\Phi}\,\sin[\omega_{\Phi}\,t]\bigr{]}$.
For hot optomechanics, $\bar{n}_{\mathrm{m}}\gg 1$, and weak phase modulation,
$A_{\Phi}\ll 1$, our power spectral density is, to leading order in
$A_{\Phi}$, [32]

$S_{XX}(\omega>0)\approx\mathcal{T}(\omega_{\mathrm{L}};\omega)\,\Bigl{(}{g_{\mathrm{om}}}^{2}\,\frac{\bar{n}_{\mathrm{m}}\,\gamma_{\mathrm{m}}}{\left(\omega-\omega_{\mathrm{m}}\right)^{2}+\left(\frac{\gamma_{\mathrm{m}}}{2}\right)^{2}}+\frac{{A_{\Phi}}^{2}\,{\omega_{\Phi}}^{2}}{4}\,2\,\pi\,\delta(\omega-\omega_{\Phi})\Bigr{)}\;,$
(4)

where $\gamma_{\mathrm{m}}=\omega_{\mathrm{m}}/Q_{\mathrm{m}}$ is the
mechanical mode power loss rate, and $\mathcal{T}(\omega_{\mathrm{L}};\omega)$
is a complicated function of the laser frequency and spectrum frequency that
includes transmission through the optical cavity and all efficiencies. Using a
spectrum analyzer with window function $w(f_{\mathrm{SA}},\mathrm{RBW};f)$,
where $f_{\mathrm{SA}}$ is the measurement frequency and $\mathrm{RBW}$ is the
resolution bandwidth, with step size $\Delta
f_{\mathrm{SA}}\ll\mathrm{RBW}\ll\gamma_{\mathrm{m}}/(2\,\pi)$, the measured
power is

$P(\omega_{\mathrm{SA}})=\mathcal{T}(\omega_{\mathrm{L}};\omega_{\mathrm{SA}})\,\Bigl{(}\mathrm{RBW}\,2\,{g_{\mathrm{om}}}^{2}\,\frac{\bar{n}_{\mathrm{m}}\,\gamma_{\mathrm{m}}}{\left(\omega_{\mathrm{SA}}-\omega_{\mathrm{m}}\right)^{2}+\left(\frac{\gamma_{\mathrm{m}}}{2}\right)^{2}}+\frac{{A_{\Phi}}^{2}\,{\omega_{\Phi}}^{2}}{2}\,w(\omega_{\mathrm{SA}},\mathrm{RBW};\omega_{\Phi})+\text{bg}\Bigr{)}\;.$
(5)

A broad background, denoted by $\mathrm{bg}$, can be removed with a low-order
polynomial fit. If $\omega_{\Phi}\approx\omega_{\mathrm{m}}$, then
$\mathcal{T}(\omega_{L};\omega_{\mathrm{SA}})$ is approximately constant.
Fitting the window function over the phase modulated signal (with known
amplitude $A_{\Phi}$) allows us to determine $\mathcal{T}$, then fitting the
thermal spectrum gives us $g_{\mathrm{om}}$. For fast scans with no averaging,
there is a dither imprint that can be included in the fit or ignored.

In practice, a fit over the thermal spectrum gives us $\omega_{\mathrm{m}}$,
$\gamma_{\mathrm{m}}$, and the parameter
$\mathcal{A}=2\,\mathcal{T}\,{g_{\mathrm{om}}}^{2}\,\bar{n}_{\mathrm{m}}\,\gamma_{\mathrm{m}}$,
whilst a window fit over the phase modulation signal gives us
$P_{f_{\Phi}}=\mathcal{T}\,{A_{\Phi}}^{2}\,{\omega_{\Phi}}^{2}/2$, such that

$g_{\mathrm{om}}=\frac{\omega_{\Phi}\,A_{\Phi}}{\sqrt{k_{\mathrm{B}}\,T/\hbar}}\,\sqrt{\frac{\omega_{\mathrm{m}}\,\mathcal{A}}{4\,\gamma_{\mathrm{m}}\,P_{f_{\Phi}}}}\;.$
(6)

Spectrum scans are fit individually with each producing
$\sqrt{\frac{\omega_{\mathrm{m}}\,\mathcal{A}}{4\,\gamma_{\mathrm{m}}\,P_{f_{\Phi}}}}$,
and these are combined together to give a factor that contains all the
statistical uncertainty. Temperature, $T$, and phase modulation strength,
$A_{\Phi}$, are included with systematic uncertainties.

The inset of Fig. 3(a) shows the optical resonance transmission and
reflection, fits give $f_{\mathrm{o}}=194.5\,\mathrm{THz}$ with loaded quality
factor $Q_{\mathrm{o}}^{\mathrm{L}}\approx 4300$ and internal quality factor
$Q_{\mathrm{o}}^{\mathrm{0}}\approx 8600$. Using a first-harmonic dither lock
to the resonance peak (shown in green) there is no optomechanical spectrum
visible, as expected. With a second-harmonic dither lock to either of the
resonance inflection points (blue and red), the thermal optomechanical
spectrum can be seen at $f_{\mathrm{m}}=4.488\,\mathrm{GHz}$ with quality
factor $Q_{\mathrm{m}}\approx 600$. Fig. 3(b) shows the optomechanical
spectrum averaged over 100 scans.

Optomechanical couplings are calculated for each individual scan by fitting
the thermal spectrum and phase modulate signal. The weighted mean from 100
scans at both the blue and red inflection points yielded
$g_{\mathrm{om}}/(2\,\pi)=(649\pm 8)\,\mathrm{kHz}$.

Scans made at varied laser probe powers are shown in Fig. 3(c). The current
output of our photodetector is proportional to the laser power received,
giving the power of the measured signal proportional to the square of laser
power. This scaling of the thermal power spectrum amplitude, $\mathcal{A}$, is
shown in the inset.

##  V Discussion

We have demonstrated a quasi-two-dimensional optomechanical resonator in
gallium arsenide, using the vertebrae design, with an optical resonance in the
infra-red C band and mechanical resonance at $\sim 4.5\,\mathrm{GHz}$
compatible with transmon superconducting qubits. The optomechanical coupling
of $\sim 650\,\mathrm{kHz}$ is slightly less than the best demonstrated
couplings of $\sim 1\,\mathrm{MHz}$ [21, 28], however there is significant
room for improvement in design parameter optimization, fabrication, and
cryogenic operation. Switching to a $3\,\upmu\mathrm{m}$
$\mathrm{Al}_{0.7}\mathrm{Ga}_{0.3}\mathrm{As}$ sacrificial layer should
provide easily obtainable gains [21]. Transitioning to edge coupling in
reflection will open up the other side of the vertebrae resonator to
electromechanical coupling.

##  VI Acknowledgements

The authors would like to thank Peter Duda for helpful discussions. This work
was supported by the LPS/ARO award W911NF-23-1-0077, NSF QLCI for HQAN (award
2016136), by the Air Force Office of Scientific Research, and in part based on
work supported by the U.S. Department of Energy Office of Science National
Quantum Information Science Research Centers, and by UChicago’s MRSEC (NSF
award DMR-2011854). This work made use of the Pritzker Nanofabrication
Facility, part of the Pritzker School of Molecular Engineering at the
University of Chicago, which receives support from Soft and Hybrid
Nanotechnology Experimental (SHyNE) Resource (NSF ECCS-2025633), a node of the
National Science Foundation’s National Nanotechnology Coordinated
Infrastructure.

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##  Appendix A Snowflake crystal dispersion

Figure 4: Snowflake in GaAs, (left) acoustic and (right) electromagnetic
dispersion plots from finite element simulations.

Plots of the two-dimensional snowflake optomechanical crystal dispersion
relations are given in Fig. 4. Snowflake parameters are defined in Fig. 1.

##  Appendix B Parameter specifications

Figure 5: Fig 1(a) showing the vertebrae resonator dimension specifications.  Parameter | Simulation | Fabrication
---|---|---
$\mathsf{t}$ | $250\,\mathrm{nm}$ | $\approx 250\,\mathrm{nm}$
$\mathsf{a}$ | $550\,\mathrm{nm}$ | $572\,\mathrm{nm}$
$\mathsf{r}$ | $245\,\mathrm{nm}$ | $245\,\mathrm{nm}$
$\mathsf{w}$ | $87\,\mathrm{nm}$ | $62\,\mathrm{nm}$
$\mathsf{q}_{\text{mirror}}$ | $320\,\mathrm{nm}$ | $299\,\mathrm{nm}$
$\mathsf{v}_{\text{mirror}}$ | $480\,\mathrm{nm}$ | $472\,\mathrm{nm}$
$\mathsf{p}_{\text{mirror}}$ | $175\,\mathrm{nm}$ | $182\,\mathrm{nm}$
$\mathsf{u}_{\text{mirror}}$ | $210\,\mathrm{nm}$ | $252\,\mathrm{nm}$
$\mathsf{q}_{\text{defect}}$ | $310\,\mathrm{nm}$ | $285\,\mathrm{nm}$
$\mathsf{v}_{\text{defect}}$ | $470\,\mathrm{nm}$ | $453\,\mathrm{nm}$
$\mathsf{p}_{\text{defect}}$ | $220\,\mathrm{nm}$ | $228\,\mathrm{nm}$
$\mathsf{u}_{\text{defect}}$ | $210\,\mathrm{nm}$ | $249\,\mathrm{nm}$
$\mathsf{d}$ | $80\,\mathrm{nm}$ | $115\,\mathrm{nm}$
$\mathsf{s}$ | $1503\,\mathrm{nm}$ | $1563\,\mathrm{nm}$
$n_{\text{defect}}$ | $1$ | $1$
$n_{\text{gradient}}$ | $3$ | $3$
$n_{\text{mirror}}$ | $6$ | $3$
$n_{\text{taper}}$ | $0$ | $2$
Table 1: Parameters used for finite element model simulations and what was
eventually used in the electron beam lithography pattern.

Dimensional parameters, defined in Fig. 1(a) (reproduced in Fig. 5) with the
addition of $\mathsf{t}$ for slab thickness, used for given simulations (Fig.
1) and the electron beam lithography pattern of the measured device (Fig. 3)
are listed in Tab. 1. Snowflake and C-shape holes in simulations include a
chamfer of $\mathsf{c}=20\,\mathrm{nm}$ to account for imperfect corners in
fabrication.

GaAs is aligned such that $[110]\parallel\vec{x}$ along the waveguide
direction. This is chosen to accommodate future electromechanical elements —
symmetries in the displacement and voltage fields restrict piezoelectric wave
propagation in GaAs to along $[110]$.

Figure 6: Diagram of grating coupler dimensions.  Parameter | Fabrication
---|---
$\mathsf{t}$ | $\approx 250\,\mathrm{nm}$
$\mathsf{h}$ | $\approx 150\,\mathrm{nm}$
$n_{\mathrm{grates}}$ | $40$
$\mathsf{\Lambda}$ | $632\,\mathrm{nm}$
$\mathsf{f}$ | $0.58$
$b$ | $-0.05$
$\mathsf{w}$ | $1\,\upmu\mathrm{m}$
Table 2: Grating coupler parameters used in photolithography.

The grating couplers, with dimensions defined in Fig. 6, have parameters given
in Tab. 2, with curves following

$a=b\,x+\sqrt{x^{2}+y^{2}}\;,$ (7)

about focus $(0,0)$, where $a$ is the $y$-intercept distance.

Tethers in a zigzag pattern to hold up the suspended beam waveguide are kept
thin ($125\,\mathrm{nm}$) such as not to support optical modes. Larger
serpentine tethers hold up the main optomechanical crystal slab and provide
some give during the release process. Windows around the open area mitigate
possible damage from large cracks that form during the release. A gentler
release (with lower aluminium content sacrificial layer) could allow for a
more direct connection.

Parameter | Simulation
---|---
$\rho$ | $5317\,\mathrm{kg}\,\mathrm{m}^{-3}$
$\tilde{c}^{1\,1}$ | $118.41\,\mathrm{GPa}$
$\tilde{c}^{1\,2}$ | $53.78\,\mathrm{GPa}$
$\tilde{c}^{4\,4}$ | $59.12\,\mathrm{GPa}$
$\tilde{e}^{1\,4}$ | $-0.16\,\mathrm{C}\,\mathrm{m}^{-2}$
$\tilde{p}^{1\,1}$ | $-0.165$
$\tilde{p}^{1\,2}$ | $-0.140$
$\tilde{p}^{4\,4}$ | $-0.072$
$\varepsilon_{\mathrm{r}}$ | $11.361$
Table 3: GaAs material properties used in finite element modelling.

Gallium arsenide material properties are listed in Tab. 3. $\rho$ is density,
$c$ is stiffness, $p$ is photo-elastic, and $\varepsilon_{\mathrm{r}}$ is
relative permittivity.

##  Appendix C Optomechanical coupling calculation

The optomechanical coupling was calculated from simulations according to

$g_{\mathrm{om}}=\left\lvert g_{\begin{subarray}{c}\mathrm{om}\\\
\mathrm{MB}\end{subarray}}+g_{\begin{subarray}{c}\mathrm{om}\\\
\mathrm{PE}\end{subarray}}\right\rvert\;.$ (8)

The moving boundary contribution is

$g_{\begin{subarray}{c}\mathrm{om}\\\
\mathrm{MB}\end{subarray}}=-\frac{\omega_{\mathrm{o}}}{2}\sqrt{\frac{\hbar}{2\,\omega_{\mathrm{m}}}}\\\
\frac{\int_{\partial
S}Q_{k}(\vec{r})\,n_{k}(\vec{r})\,\left(E^{\parallel}_{i}(\vec{r})^{\ast}\,\left((\varepsilon_{S})_{i\,j}-(\varepsilon_{0})_{i\,j}\right)\,E^{\parallel}_{j}(\vec{r})-D^{\perp}_{i}(\vec{r})^{\ast}\,\Bigl{(}{(\varepsilon_{S})^{-1}}_{i\,j}-{(\varepsilon_{0})^{-1}}_{i\,j}\Bigr{)}\,D^{\perp}_{j}(\vec{r})\right)\,\mathrm{d}^{2}\vec{r}}{\sqrt{\int_{S}Q_{i}(\vec{r})^{\ast}\,\rho(\vec{r})\,Q_{i}(\vec{r})\,\mathrm{d}^{3}\vec{r}}\quad\int_{V}E_{i}(\vec{r})^{\ast}\,\varepsilon_{i\,j}(\vec{r})\,E_{j}(\vec{r})\,\mathrm{d}^{3}\vec{r}}\;,$
(9)

where $\vec{Q}$ is the complex harmonic displacement field, $\vec{E}$ and
$\vec{D}$ are the complex harmonic electric and electric displacement fields,
$\rho$ is density, $\varepsilon$ is permittivity, $V$ is all space, and $S$ is
the solid with boundary $\partial S$ and normal $\vec{n}$. The photoelectric
contribution is

$g_{\begin{subarray}{c}\mathrm{om}\\\
\mathrm{PE}\end{subarray}}=\frac{\omega_{\mathrm{o}}}{2}\sqrt{\frac{\hbar}{2\,\omega_{\mathrm{m}}}}\frac{\frac{1}{\varepsilon_{0}}\,\int_{S}E_{i}(\vec{r})^{\ast}\,(\varepsilon_{S})_{i\,a}\,p_{a\,b\,c\,d}\,\epsilon_{c\,d}(\vec{r})\,(\varepsilon_{S})_{b\,j}\,E_{j}(\vec{r})\,\mathrm{d}^{3}\vec{r}}{\sqrt{\int_{S}Q_{i}(\vec{r})^{\ast}\,\rho(\vec{r})\,Q_{i}(\vec{r})\,\mathrm{d}^{3}\vec{r}}\quad\int_{V}E_{i}(\vec{r})^{\ast}\,\varepsilon_{i\,j}(\vec{r})\,E_{j}(\vec{r})\,\mathrm{d}^{3}\vec{r}}\;,$
(10)

where $\epsilon$ is strain, and $p$ is the photo-elastic tensor noting that in
Voigt notation it transforms stress-like.

##  Appendix D Proximity effect correction

Due to the greater atomic weight of gallium and arsenide (compared to silicon,
for example), electron back scatter during the pattern writing process leads
to significant proximity dependent over-dosing. This is particularly
problematic for fashioning uniform nanoscale features over large areas, as is
needed for a two-dimensional optomechanical crystal. To combat this we employ
proximity effect corrections using a point spread function developed over many
fabrication cycles.

Our point spread function is modelled using Gaussian-Pearson VII product terms
of the form

$f_{\mathrm{GP}}[\sigma,\gamma,\nu;r]=\frac{\exp\bigl{[}-\gamma^{2}/\sigma^{2}\bigr{]}}{\pi\,\gamma^{2}\,\mathrm{E}_{\nu}\bigl{[}\gamma^{2}/\sigma^{2}\bigr{]}}\,\exp\bigl{[}-r^{2}/\sigma^{2}\bigr{]}\,\left(\frac{r^{2}}{\gamma^{2}}+1\right)^{-\nu}\;,$
(11)

where $E$ is the exponential integral function. This form is able to include a
basic Gaussian scattering radius, $\sigma$, and diffusion-like process with
range $\gamma$ and exponent $\nu$.

For fabricating snowflake crystal patterns, the normalized point spread
function was

$\displaystyle\mathrm{PSF}[r]=$
$\displaystyle\;0.143885\,f_{\mathrm{GP}}[685\,\mathrm{nm},1\,\mathrm{nm},1;r]$
$\displaystyle\;+0.172662\,f_{\mathrm{GP}}[5\,\mathrm{nm},0,0;r]$
$\displaystyle\;+0.683453\,f_{\mathrm{GP}}[13\,\upmu\mathrm{m},0,0;r]\;,$

implemented with a short range cut-off of $0.1\,\upmu\mathrm{m}$ .

##  Appendix E Optical measurement setup

Figure 7: Full diagram of the optical measurement setup. Thick vertical lines
indicate flat physical contact (PC) connections, whilst thick angled lines
indicate angled physical connections (APC). Fiber optic colours mimic their
real-life colour, lengths are not to scale.

The measurement setup is a C band infra-red fiber optic based system with free
space coupling to a chip in transmission and a high bandwidth photodetector
that can resolve microwave frequencies. A complete diagram of the system is
given in Fig. 7. From the laser, the optic signal first goes through optional
phase modulation, and then a variable attenuator for power control, and a
polarizer. It is directed onto the chip by a V-groove clamp that holds the
fiber in place above the chip at an angle of $8^{\circ}$ from normal. This
angle was optimized for silicon on insulator grating couplers of a previous
experiment. A second fiber is clamped $127\,\upmu\mathrm{m}$ away to pick up
the transmitted signal. A splitter takes a small amount of the signal for
frequency locking whilst the rest goes through an amplifier and filter before
hitting the fast photodetector.

Simple side-locking to a transmission value is not adequate for tracking the
optical resonance edge due to the prevalence of global efficiency drifts
thought to be due to temperature fluctuations. To deal with both frequency and
transmission drift, a dither lock feedback system is implemented. The laser
wavelength is modulated at $f_{\text{dither}}=100\,\mathrm{Hz}$ and a lock-in
amplifier operating at the $n$’th harmonic measures the $n$’th derivative of
the output transmission profile. A proportional-integral-derivative
controller, with a set point of 0 for the aforementioned signal, shifts the
laser center wavelength accordingly. Tracking the inflection point ($2$nd
derivative equal to zero) is done using the second harmonic.