#  A note on the homogenization of incommensurate thin films

Irene Anello, Andrea Braides and Fabrizio Caragiulo
SISSA, via Bonomea 265, Trieste, Italy

###### Abstract

Dimension-reduction homogenization results for thin films have been obtained
under hypotheses of periodicity or almost-periodicity of the energies in the
directions of the mid-plane of the film. In this note we consider thin films,
obtained as sections of a periodic medium with a mid-plane that may be
incommensurate; that is, not containing periods other than $0$. A geometric
almost-periodicity argument similar to the cut-and-project argument used for
quasicrystals allows to prove a general homogenization result.

††Preprint SISSA 22/2022/MATE

##  1 Introduction

The energy of heterogeneous thin films of a hyperelastic material in
$\mathbb{R}^{d+1}$ can be described by integral functionals of the form

$\frac{1}{2\varepsilon}\int_{\omega\times(-\varepsilon,\varepsilon)}f_{\varepsilon}(x,\nabla
u)\,\mathrm{d}x,$ (1.1)

where $f_{\varepsilon}$ are hyperelastic energy densities satisfying suitable
growth conditions, $\omega\subset\mathbb{R}^{d}$ is the middle section of the
thin film, $2\varepsilon$ is its thickness, and
$u:\omega\times(-\varepsilon,\varepsilon)\to\mathbb{R}^{m}$ is the
displacement (in the physical case, $d=2$ and $m=3$). We use $\mathrm{d}x$ to
denote integration with respect to the Lebesgue measure in $\mathbb{R}^{k}$
both for $k=d$ and $k={d+1}$, which one being clear from the context. In the
seminal paper by Le Dret and Raoult [9], in the case $f_{\varepsilon}=f$
independent of $\varepsilon$ and homogeneous (i.e., $x$-independent), the
behaviour of such energies as $\varepsilon\to 0$ has been proven to be
described by a dimensionally reduced hyperelastic energy of the form

$\int_{\omega}f^{\rm LDR}(\nabla v)\,\mathrm{d}y,$ (1.2)

where now $v:\omega\to\mathbb{R}^{m}$ has a domain identified as the mid-
section of the film, and $f^{\rm LDR}$ is a quasiconvex function explicitly
described from $f$. For general energies (1.1) a compactness theorem [5]
ensures that, up to subsequences, their behaviour can be described by a
possibly heterogeneous dimensionally reduced energy

$\int_{\omega}f_{0}(y,\nabla v)\,\mathrm{d}y,$ (1.3)

with $f_{0}$ possibly depending on the subsequence. In the case of a periodic
integrand we can suppose, as this is the relevant case, that the oscillation
be at the same scale of the thickness, so that we assume
$f_{\varepsilon}(x,A)=f\bigl{(}{x\over\varepsilon},A\bigr{)}$ with
$f(\cdot,A)$ periodic in the coordinate directions. In this case we have
homogenization [5]; that is, the limit $f_{0}$ is homogeneous and independent
of the subsequence of $\varepsilon$, and described by a suitable asymptotic
formula. In order to prove this result, a key argument is the invariance
property of the energies by translations of the form $\varepsilon e_{j}$ for
$j\in\\{1,\ldots,d\\}$, where $\\{e_{i}:i\in\\{1,\ldots,d+1\\}\\}$ is the
canonical orthonormal basis of $\mathbb{R}^{d+1}$, entailing the translation
invariance of $f_{0}$. This argument is made possible by the assumption that
the middle plane of the thin film contains a $d$-dimensional lattice of
periods for $f$; that is, it is ‘commensurate’ with $\mathbb{Z}^{d+1}$.

In this paper we consider the general case of a periodic energy density $f$,
when the middle plane of the thin film contains a $n$-dimensional lattice of
periods for $f$ with $n\leq d$. ‘Incommensurate’ thin films are those with
$n=0$; that is, when the middle plane of the thin film does not contain any
period for $f$ except $0$.

In order to treat general, possibly incommensurate, thin films, we introduce
some notation, slightly different from the one used for commensurate thin
films, due to the necessity to distinguish between the periods of the energy
density and the directions of the thin film. We choose to maintain
$\mathbb{Z}^{d+1}$ as set of periods, considering a Carathéodory function
$\widetilde{f}:\mathbb{R}^{d+1}\times\mathbb{M}^{m\times{(d+1)}}\to\left[0,+\infty\right)$
satisfying the _standard p-growth conditions_

$\alpha|A|^{p}\leq\widetilde{f}(x,A)\leq\beta(1+|A|^{p})$ (1.4)

for all $x\in\mathbb{R}^{d+1}$ and $A\in\mathbb{M}^{m\times{d+1}}$ and some
$\alpha,\beta>0$, which is $1$-periodic in all coordinate directions; i.e.,

$\widetilde{f}(x+{e}_{i},A)=\widetilde{f}(x,A)$ (1.5)

for all $x\in\mathbb{R}^{d+1}$, all matrices $A\in\mathbb{M}^{m\times{(d+1)}}$
and all vectors ${e}_{i}$ of the standard orthonormal basis of
$\mathbb{R}^{d+1}$. We use the notation $\mathbb{M}^{m\times k}$ for the space
of ${m\times k}$ matrices with real entries.

We fix a unit vector $\nu$ and consider the hyperplane $\Pi=\\{x:\langle
x,\nu\rangle=0\\}$. The interesting case, to keep in mind as the most relevant
one, is when $\Pi$ is an _irrational_ hyperplane in $\mathbb{R}^{d+1}$; that
is, such that

$\Pi\cap\mathbb{Z}^{d+1}=\\{0\\}.$ (1.6)

We fix $\widetilde{\omega}$ a bounded subset of $\Pi$, open in the relative
topology, $h>0$, and for each $\varepsilon>0$ consider the functional

$\widetilde{I}_{\varepsilon}(u)=\frac{1}{2h\varepsilon}\int_{\widetilde{\omega}+(-h\varepsilon,h\varepsilon)\nu}\widetilde{f}\Big{(}\frac{x}{\varepsilon},\nabla{u}\Big{)}\,\mathrm{d}x,$
(1.7)

with domain
$\mathrm{W}^{1,p}(\widetilde{\omega}+(-h\varepsilon,h\varepsilon)\nu,\mathbb{R}^{m})$,
which ideally represents the elastic free energy of a thin film of size
$2h\varepsilon>0$ around $\widetilde{\omega}$. The introduction of a constant
$h>0$ amounts to assuming that thickness and periods are comparable, so that
$h$ represents their ratio, which we highlight for possible future reference.
We will prove that there exists a function
$\widetilde{f}_{\text{hom}}\colon\mathbb{M}^{m\times d}\to[0,+\infty)$
satisfying an asymptotic formula, such that

$\Gamma{\text{-}}\mkern-1.0mu\lim_{\varepsilon\to
0}\widetilde{I}_{\varepsilon}(u)=\int_{\widetilde{\omega}}\widetilde{f}_{\text{hom}}\big{(}\nabla
u(y)\big{)}\,\mathrm{d}y.$

The $\Gamma$-limit is performed in a dimension-reduction fashion, using a
convergence of
$u_{\varepsilon}\in\mathrm{W}^{1,p}(\widetilde{\omega}+(-h\varepsilon,h\varepsilon)\nu,\mathbb{R}^{m})$
to $u\in\mathrm{W}^{1,p}(\widetilde{\omega},\mathbb{R}^{m})$, where
$\widetilde{\omega}$ is identified with a subset of $\mathbb{R}^{d}$ in order
for the integration to be well defined.

The result is obtained by first resorting to the theory of [5]. To that end we
rewrite the functionals in the form (1.1), with
$f_{\varepsilon}(x,A)=f\smash{\bigl{(}{x\over\varepsilon},A\bigr{)}}$, and $f$
obtained from $\smash{\widetilde{f}}$ by a linear change of variables. Using
the dimension-reduction convergence of functions as in [9], we can then apply
the compactness theorem in [5] to obtain a limit of the form (1.3). Since $f$
may not be periodic in the coordinate directions we cannot immediately
conclude that homogenization takes place, since an invariance-by-translation
argument does not apply. However, we can use a sort of geometric almost
periodicity property: the set of periods for $\widetilde{f}$ which are close
to $\Pi$ (closeness suitably quantified by a small parameter) is projected to
a uniformly dense set in $\Pi$. We remark that the former set corresponds also
to the set of periods for $f$ which are close to the hyperplane identified
with $\mathbb{R}^{d}$, This argument reminds the cut-and-project arguments
typical of quasicrystalline structures (see [7, 2, 8, 10, 6, 3]). The
existence of such geometric quasi-periods is not sufficient to prove the
necessary translation-invariance properties for $f_{0}$ and the homogenization
asymptotic formula. To that end it is necessary to construct test functions by
using translation arguments, which are not directly at hand since translation
by a quasi-period may exit the thin film domain. The key technical point of
the paper is a novel lemma, which ensures that in the constructions of test
functions it is sufficient to modify functions defined in a smaller thin film,
which is then mapped inside the original thin film by any of the above-
mentioned translations (scaled by $\varepsilon$).

###  1.1 Statement of the results

We now formalize what we anticipated in the Introduction, rewriting the
energies $\widetilde{I}_{\varepsilon}$ in (1.7) in order to apply the results
in [5] with more ease. To that end, we make a change to coordinates more
suitable to the problem: we let $\phi$ be the linear isometry in
$\mathbb{R}^{d+1}$ sending ${e}_{1},\ldots,{e}_{d}$ to an orthonormal basis
${\pi}_{1},\ldots,{\pi}_{d}$ of $\Pi$ and ${e}_{d+1}$ to $\nu$. Let $\omega$
be the open set in $\mathbb{R}^{d}$ such that
$\phi^{-1}(\widetilde{\omega})=\omega\times\\{0\\}$, so that

$\phi^{-1}\big{(}\widetilde{\omega}+(-h\varepsilon,h\varepsilon)\nu\big{)}=\omega\times(-h\varepsilon,h\varepsilon)\eqqcolon\Omega_{\varepsilon}\subset\mathbb{R}^{d}\times\mathbb{R}$

After this change of variables, setting

${I}_{\varepsilon}(u)=\frac{1}{2h\varepsilon}\int_{\Omega_{\varepsilon}}f\Big{(}\frac{x}{\varepsilon},\nabla{u(x)}\Big{)}\,\mathrm{d}x,$
(1.8)

where $u\in\mathrm{W}^{1,p}(\Omega_{\varepsilon},\mathbb{R}^{m})$ and, denoted
by $R$ the constant matrix equal to $\nabla\phi$, having set

$f(x,A)=\widetilde{f}(\phi(x),AR),$ (1.9)

we have

${I}_{\varepsilon}(u)=\widetilde{I}_{\varepsilon}(\widetilde{u}),\quad\hbox{
where }\quad\widetilde{u}(\widetilde{x})=u(\phi^{-1}(\widetilde{x})).$ (1.10)

Hence, the functionals ${I}_{\varepsilon}$ and $\widetilde{I}_{\varepsilon}$
are equivalent, up to a linear change of variables. In order to simplify the
statement of the convergence, as is customary we scale all functionals to a
common domain, obtaining

$\begin{split}F_{\varepsilon}(u)&\coloneqq\frac{1}{2h}\int_{\omega\times(-h,h)}f\Big{(}{x\over\varepsilon},y,\Big{(}\nabla_{x}u(x,y)\Big{|}{1\over\varepsilon}{\partial_{y}u(x,y)}\Big{)}\Big{)}\,\mathrm{d}x\,\mathrm{d}y\end{split}$
(1.11)

for $u\in W^{1,p}(\omega\times(-h,h),\mathbb{R}^{m})$, so that
$F_{\varepsilon}(\overline{u})=I_{\varepsilon}(u)$, where
$\overline{u}(x,y)=u(x,\varepsilon y)$. Here, we have rewritten, with a little
abuse of notation, a variable in $\mathbb{R}^{d+1}$ as a pair $(x,y)$ with
$x\in\mathbb{R}^{d}$, and identified a matrix $A\in\mathbb{M}^{m\times{d+1}}$
with a pair which we denote $(A^{\prime}|\xi)$, where
$A^{\prime}\in\mathbb{M}^{m\times d}$ and $\xi\in\mathbb{R}^{m}$ is a (column)
vector. The notation $\nabla_{x}u$ denotes the gradient with respect to the
coordinates of $x$. We will keep the standard notation $\nabla u$ when the
gradient is performed with respect to all the variables in the domain, be it
in $\mathbb{R}^{d}$ or $\mathbb{R}^{d+1}$.

We will use the following notion of convergence.

######  Definition 1.1 (convergence to dimensionally reduced parameters).

A sequence of functions $u_{\varepsilon}$ with
$u_{\varepsilon}\in{W}^{1,p}(\Omega_{\varepsilon},\mathbb{R}^{m})$ converges
to $u\in{W}^{1,p}(\omega,\mathbb{R}^{m})$ if the corresponding functions
$\overline{u}_{\varepsilon}$ defined by
$\overline{u}_{\varepsilon}(x,y)=u_{\varepsilon}(x,\varepsilon y)$ converge to
some $\overline{u}$ weakly in $W^{1,p}(\omega\times(-h,h),\mathbb{R}^{m})$ and
$\overline{u}(x,y)=u(x)$.

We recall that $I_{\varepsilon}$ are equicoercive with respect to this
convergence, in the sense that if $I_{\varepsilon}(u_{\varepsilon})$ is
equibounded and $\overline{u}_{\varepsilon}$ are bounded in
$L^{p}(\Omega\times(-h,h),\mathbb{R}^{m})$ then $u_{\varepsilon}$ converge to
some $u$, up to subsequences [5].

######  Definition 1.2 (convergence to dimensionally reduced energies).

We say that $I_{\varepsilon}$ $\Gamma$-converges to $F_{0}$ with respect to
the convergence of $u_{\varepsilon}$ above if the corresponding
$F_{\varepsilon}$ $\Gamma$-converge to the functional $\overline{F}_{0}$ on
functions independent of $y$, defined by
$F_{0}(\overline{u})=\overline{F}_{0}(u)$ if $\overline{u}(x,y)=u(x)$, with
respect to the weak convergence in $W^{1,p}$.

We can then state the homogenization result as follows.

######  Theorem 1.3 (homogenization theorem).

Let $\nu$ be a unit vector and $\Pi=\\{x:\langle x,\nu\rangle=0\\}$ be a
hyperplane in $\mathbb{R}^{d+1}$. Let
$\smash{\widetilde{f}\colon\mathbb{R}^{d+1}\times\mathbb{M}^{m\times(d+1)}\to\mathbb{R}}$
be a Carathéodory function satisfying the $p$-growth conditions (1.4) for
$p>1$ and periodic in the coordinate direction as in (1.5). Let $\phi$ be the
linear isometry defined above and let
$f\colon\mathbb{R}^{d+1}\times\mathbb{M}^{m\times(d+1)}\to\mathbb{R}$ be the
Carathéodory integrand defined by (1.9). Let $h>0$, let $\omega$ be an open
and bounded subset of $\mathbb{R}^{d}$ and let $I_{\varepsilon}$ be defined by
(1.8), where $\Omega_{\varepsilon}=\omega\times(-h\varepsilon,h\varepsilon)$.
Then $I_{\varepsilon}$ $\Gamma$-converges to $F_{\rm hom}$ with respect to the
convergence in Definition 1.1 as $\varepsilon\to 0$, where

$F_{\rm hom}(u)=\int_{\omega}f_{\rm hom}(\nabla u)\,\mathrm{d}x$ (1.12)

for $u\in W^{1,p}(\omega,\mathbb{R}^{m})$, and $f_{\rm
hom}\colon\mathbb{M}^{m\times d}\to\mathbb{R}$ is a quasiconvex function
satisfying a $p$-growth conditions and the asymptotic homogenization formula

$\begin{split}f_{\mathrm{hom}}(A)=\lim_{T\to+\infty}\frac{1}{T^{d}}\inf_{u\in\mathcal{W}_{T}}\bigg{\\{}\frac{1}{2h}\int_{(0,T)^{d}\times(-h,h)}&f\big{(}x,y,\big{(}A+\nabla_{x}u\big{|}{\partial_{y}u}\big{)}\big{)}\,\mathrm{d}x\,\mathrm{d}y\Bigg{\\}}\end{split}$
(1.13)

where

$\mathcal{W}_{T}\coloneqq\Big{\\{}u\in\mathrm{W}^{1,p}\big{(}(0,T)^{d}\times(-h,h),\,\mathbb{R}^{m}\big{)}\colon
u=0\text{ on }\partial\big{(}(0,T)^{d}\big{)}\times(-h,h)\Big{\\}}.$

######  Remark 1.4 (homogenization on $\Pi$).

By using (1.10) we can interpret the result as a homogenization theorem
directly on $\Pi$, with the related homogenized energy

$\widetilde{F}_{\rm hom}(u)=\int_{\widetilde{\omega}}f_{\rm hom}(\nabla
u\,\widetilde{R}^{-1})\,\mathrm{d}{\mathcal{H}}^{d}(x),$

where $\widetilde{R}$ is the matrix related to the restriction of the linear
isometry to the subspace of $\mathbb{R}^{d+1}$ parameterized as
$\mathbb{R}^{d}$, and the Sobolev space
$W^{1,p}(\widetilde{\omega},\mathbb{R}^{m})$ (with underlying measure the
$d$-dimensional Hausdorff measure restricted to $\widetilde{\omega}$) is
suitably defined.

Figure 1: An irrational $\mathbb{Z}^{2}$-periodic thin film in
$\mathbb{R}^{2}$ with an $\eta$-neighbourhood of its middle line.

######  Remark 1.5 (connections with almost periodicity).

The theorem above has been proved in [5] if $f$ is periodic in the first
variable, which is the case if the lattice $\Pi\cap\mathbb{Z}^{d+1}$ has
dimension $d$ (or, equivalently, if it spans $\Pi$). In the case at hand we
will use a sort of geometric quasi-periodicity. Namely, we will use the fact
that, for fixed $\eta>0$ the set $\mathcal{T}_{\eta}$ of $\tau\in\Pi$ such
that dist$(\tau,\mathbb{Z}^{d+1})<\eta$ is uniformly dense in $\Pi$; that is,
there exists an inclusion length $L_{\eta}>0$ such that
$\mathcal{T}_{\eta}+[0,L_{\eta}]^{d+1}\supset\Pi$. This is an immediate
consequence of the periodicity of the function $\psi(x)={\rm
dist}(x,\mathbb{Z}^{d+1})$ on $\mathbb{R}^{d+1}$ and the consequent quasi-
periodicity of its restriction to $\Pi$. This property in turn implies its
uniform almost periodicity (see [1] Definition 1.7), which exactly states that
for all $\eta>0$ there exist a uniformly dense set of $\eta$-almost periods;
i.e., $\tau$ such that $|\psi(x+\tau)-\psi(x)|<\eta$ for all $x\in\Pi$. In
particular, taking $x=0$, we have that $\psi(\tau)\leq\eta$, which proves the
claim. Note that this property is most relevant if $\Pi$ is irrational, and is
trivial if $\Pi\cap\mathbb{Z}^{d+1}$ has dimension $d+1$. In Fig. 1, in a two-
dimensional setting, we picture an element of ${\mathcal{T}}_{\eta}$ on $\Pi$
(black dot) and its closest element in $\mathbb{Z}^{2}$ (grey dot).

This quasi-periodicity argument also shows that if $\widetilde{f}$ is
continuous in the first variable uniformly with respect to the second one,
then $f$ is almost periodic in the $x$-directions uniformly with respect to
the second variable, and we can apply the results of [4] Chapter 24. This
observation suggests that we can then generalize Theorem 1.3 by supposing that
for all $\eta>0$ there exists a uniformly dense set $\mathcal{T}_{\eta}$ in
$\Pi$ such that for all $\tau\in\mathcal{T}_{\eta}$ there exists
$z_{\tau}\in\mathbb{R}^{d+1}$ such that $\|\tau-z_{\tau}\|<\eta$ and
$|\widetilde{f}(x+z_{\tau},A)-\widetilde{f}(x,A)|\leq\eta(1+|A|^{p})$ for all
$x\in\mathbb{R}^{d+1}$ and $A$. This trivially holds if $\widetilde{f}$ is
periodic as above taking $z_{\tau}\in\mathbb{Z}^{d+1}$.

The geometric quasi-periodicity property highlighted above must be
complemented by a lemma, which will be used to cope with the fact that periods
may not belong to $\Pi$, so that translation arguments within the thin film
cannot be directly applied.

######  Lemma 1.6.

Let $h>0$ and let $g:[0,h]\to[0,+\infty)$ be an integrable function and define

$C\coloneqq\int_{0}^{h}g(y)\,\mathrm{d}y<+\infty.$

Then, for every $\delta>\eta>0$, the set

$E_{\eta}\coloneqq\bigg{\\{}y\in[h-\delta,h]\,:\,(h+\eta-y)g(y)\leq\frac{C}{\log\big{(}\delta/\eta\big{)}}\bigg{\\}}$
(1.14)

has positive measure.

###### Proof.

Suppose, by contradiction, that for some $\delta$ there exists a $\eta$ such
that $E_{\eta}$ has null measure, then

$(h+\eta-y)g(y)>\frac{C}{\log\big{(}\delta/\eta\big{)}},\quad\text{ for almost
all }y\in[h-\delta,h].$

Thus the strict inequality would persist under integration, and

$\displaystyle C$ $\displaystyle\geq$
$\displaystyle\int_{[h-\delta,h]}g(y)\,\mathrm{d}y>\frac{C}{\log\big{(}\delta/\eta\big{)}}\int_{[h-\delta,h]}\frac{1}{h+\eta-y}\,\mathrm{d}y$
$\displaystyle=$
$\displaystyle\frac{C}{\log\big{(}\delta/\eta\big{)}}\big{(}\log(\delta+\eta)-\log(\eta)\big{)}=C\frac{\log{\big{(}\delta/\eta+1\big{)}}}{\log{\big{(}\delta/\eta\big{)}}}>C$

providing a contradiction. ∎

Figure 2: Translation of a rectangle $R$ by an $\eta$-almost period.

The key geometric argument in the proof will be the use of Lemma 1.6 to define
test functions on translated copied of sets by an almost period. In Fig. 2 we
have drawn a cartoon in two dimensions of the argument in the original
irrational thin-film geometrical setting (that is, before changing variables
with $\phi$), supposing $\tilde{f}$ periodic: we have a function $u_{R}$
defined on a rectangle $R$ inside the thin film and need to define a function
in the translation of $R$ by some $\eta$-almost period $\tau\in\Pi$ with an
energy that differs very little from that of the original function. We cannot
directly use a translation by the period corresponding to $\tau$, which in
this setting is of the form $\tau+z_{\tau}\nu\in\mathbb{Z}^{2}$, since to do
this we would need that the function be defined on the (slightly) translated
rectangle $z_{\tau}\nu+R$. We then restrict the original function $u_{R}$
slightly inside $R$ is such a way that we control the energy on the upper and
lower boundary (dotted lines) and extend this restriction to a function
$\tilde{u}_{R}$ defined on the whole strip orthogonal to the thin film, with
controlled energy. This can be done by a construction that uses Lemma 1.6. It
is then possible to simply use a translation of this extended function by
$\tau+z_{\tau}\nu$, and restrict it to $\tau+R$.

##  2 Proof of the homogenization theorem

We directly prove the theorem under the more general assumptions in Remark
1.5, which can be stated as follows for the function $f$: for all $\eta>0$
there exists $L_{\eta}>0$ and $\mathcal{T}_{\eta}$ in $\mathbb{R}^{d}$ such
that

$\mathcal{T}_{\eta}+[0,L_{\eta}]^{d}=\mathbb{R}^{d},$

and for all $\tau\in\mathcal{T}_{\eta}$ there exists $z_{\tau}\in\mathbb{R}$
such that $|z_{\tau}|<\eta$ and

$|f(x+(\tau,z_{\tau}),A)-f(x,A)|\leq\eta(1+|A|^{p})$ (2.1)

for all $x\in\mathbb{R}^{d+1}$ and $A$.

We first note that, by the compactness theorem for thin structures ([5] and
[4] Theorem 24.20) for every sequence $\\{\varepsilon_{k}\\}_{k}$ of positive
real numbers, $\varepsilon_{k}\to 0$, there exist a subsequence (still denoted
by $\varepsilon_{k}$) and a Carathéodory function
$f_{0}:\mathbb{R}^{d}\times\mathbb{M}^{m\times d}\to[0,+\infty)$ satisfying

$0\leq f_{0}(x,A)\leq\beta(1+|A|^{p}),$

for all $x\in\omega$ and $A\in\mathbb{M}^{m\times d}$ such that

$\Gamma\hbox{-}\mkern-10.0mu\lim_{k\to+\infty}I_{\varepsilon_{k}}(u)=\int_{\omega}f_{0}(x,\nabla
u)\,\mathrm{d}x$ (2.2)

for all $u\in W^{1,p}(\omega,\mathbb{R}^{m})$ with respect to the convergence
in Definition 1.1. We then have to prove that $f_{0}$ does not depend on the
space variable and is given by formula (1.13), which also implies that it does
not depend on the subsequence $\varepsilon_{k}$. To that end, we will use
localization arguments and results ensuring the possibility of fixing boundary
values taken from [5] and Lemma 1.6.

In the following result we prove that $f_{0}$ depends only on the gradient,
$f_{0}(x,A)=f_{0}(A)$.

######  Proposition 2.1.

For any sequence $\\{\varepsilon_{k}\\}_{k}$ such that the $\Gamma$-limit in
(2.2) exists, $f_{0}$ depends only on the gradient; that is,
$f_{0}(x,A)=f_{0}(A)$.

###### Proof.

For any open subset $U\subset\omega$ we consider the _localized_ functionals

$\begin{split}F_{\varepsilon_{k}}(u,U)&\coloneqq\frac{1}{2h}\int_{U\times(-h,h)}f\big{(}{\varepsilon_{k}^{-1}}x,y,\big{(}\nabla_{x}u\big{|}{\varepsilon_{k}^{-1}}{\partial_{y}u}\big{)}\big{)}\mathrm{d}x\,\mathrm{d}y\\\
F_{0}(u,U)&\coloneqq{1\over
2h}\int_{U\times(-h,h)}f_{0}(x,\nabla_{x}u)\,\mathrm{d}x\,\mathrm{d}y.\end{split}$

Let $B^{d}_{\rho}(x)$ denote the ball of radius $\rho$ and center $x$ in
$\mathbb{R}^{d}$. It suffices to prove that given any
$x^{\prime},x^{\prime\prime}\in\omega$ and $\rho>0$ such that
$B^{d}_{\rho}(x^{\prime})\Subset\omega$ and
$B^{d}_{\rho}(x^{\prime\prime})\Subset\omega$, and any
$A\in\mathbb{R}^{m\times d}$, we have

$F_{0}\big{(}Ax,B^{d}_{\rho}(x^{\prime})\big{)}=F_{0}\big{(}Ax,B^{d}_{\rho}(x^{\prime\prime})\big{)}.$
(2.3)

Up to fixing lateral boundary values, we find a sequence
$\\{u_{k}\\}_{k}\in\mathrm{W}^{1,p}\big{(}B^{d}_{\rho}(x^{\prime})\times(-h,h),\mathbb{R}^{m}\big{)}$
with $u_{k}=0\text{ on }\big{(}\partial
B^{d}_{\rho}(x^{\prime})\big{)}\times(-h,h)$ and such that $u_{k}\to 0$ in
$\mathrm{L}^{p}(\Omega_{1},\mathbb{R}^{m})$ and

$\lim_{k\to+\infty}F_{\varepsilon_{k}}\big{(}Ax+u_{k},B^{d}_{\rho}(x^{\prime})\big{)}=F_{0}\big{(}Ax,B^{d}_{\rho}(x^{\prime})\big{)}$
(2.4)

(see [5]).

Let $\delta>0$ be fixed, and let $\eta\in(0,\delta)$. From now on we use the
shorthand $\partial_{y}$ for the partial derivative with respect to $y$.
Applying Lemma 1.6 to the integrable functions

$g(t)=\int_{B^{d}_{\rho}(x^{\prime})}\big{|}\big{(}A+\nabla_{x}u_{k}(x,t)\big{|}\varepsilon_{k}^{-1}\,\partial_{y}u_{k}(x,t)\big{)}\big{|}^{p}\,\mathrm{d}x$

for $t\in[0,h]$, we get the existence of $y_{\eta}^{+}\in(h-\delta,h)$ such
that

$\displaystyle{\int_{y_{\eta}^{+}}^{h+\eta}}\int_{B^{d}_{\rho}(x^{\prime})}\big{|}\big{(}A+\nabla_{x}u_{k}(x,y_{\eta}^{+})\big{|}\,\varepsilon_{k}^{-1}\partial_{y}u_{k}(x,y_{\eta}^{+})\big{)}\big{|}^{p}\,\mathrm{d}x\,{\mathrm{d}y}$
$\displaystyle\leq$
$\displaystyle\frac{1}{|\log(\delta/\eta)|}\int_{B^{d}_{\rho}(x^{\prime})\times(0,h)}\big{|}\big{(}A+\nabla_{x}u_{k}(x,y_{\eta}^{+})\big{|}\,\varepsilon_{k}^{-1}\partial_{y}u_{k}(x,y_{\eta}^{+})\big{)}\big{|}^{p}\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle\leq$ $\displaystyle\frac{C_{k}}{\alpha|\log(\delta/\eta)|},$

where $C_{k}\coloneqq F_{\varepsilon_{k}}(Ax+u_{k},B^{d}_{\rho}(x^{\prime}))$.
Hence, we obtain

$\displaystyle\hskip-28.45274pt{\int_{y_{\eta}^{+}}^{h+\eta}}\int_{B^{d}_{\rho}(x^{\prime})}f\Big{(}\varepsilon_{k}^{-1}x,y,\big{(}A+\nabla_{x}u_{k}(x,y_{\eta}^{+})\big{|}\,\varepsilon_{k}^{-1}\partial_{y}u_{k}(x,y_{\eta}^{+})\big{)}\Big{)}\,\mathrm{d}x\,{\mathrm{d}y}$
(2.5) $\displaystyle\leq$
$\displaystyle\beta\left(|B^{d}_{\rho}(x^{\prime})|{(\delta+\eta)}+\frac{C_{k}}{\alpha|\log(\delta/\eta)|}\right)\,,$
$\displaystyle=$
$\displaystyle\beta\left(c\rho^{d}{(\delta+\eta)}+\frac{C_{k}}{\alpha|\log(\delta/\eta)|}\right)\,,$
(2.6)

being $c\coloneqq|B^{1}|$. Similarly, we obtain the existence of
$y_{\eta}^{-}\in(-h,-h+\delta)$ such that

$\displaystyle\hskip-28.45274pt{\int_{-h-\eta}^{y_{\eta}^{-}}}\int_{B^{d}_{\rho}(x^{\prime})}f\Big{(}\varepsilon_{k}^{-1}x,y,\big{(}A+\nabla_{x}u_{k}(x,y_{\eta}^{-})\big{|}\,\varepsilon_{k}^{-1}\partial_{y}u_{k}(x,y_{\eta}^{-})\big{)}\Big{)}\,\mathrm{d}x\,{\mathrm{d}y}$
(2.7) $\displaystyle\leq$
$\displaystyle\beta\left(c\rho^{d}{(\delta+\eta)}+\frac{C_{k}}{\alpha|\log(\delta/\eta)|}\right).$

Define
$\widetilde{u}_{k}:B^{d}_{\rho}(x^{\prime})\times\mathbb{R}\to\mathbb{R}^{m}$
as

$\widetilde{u}_{k}(x,y)\coloneqq\begin{cases}u_{k}(x,y_{\eta}^{+})&\text{ if
}y\geq y^{+}_{\eta}\\\ u_{k}(x,y)&\text{ if }y_{\eta}^{-}\leq y\leq
y_{\eta}^{+}\\\ u_{k}(x,y_{\eta}^{-})&\text{ if }y\leq
y^{-}_{\eta}.\end{cases}$

Note that $\widetilde{u}_{k}(x,y)=0$ if $x\in\partial
B^{d}_{\rho}(x^{\prime})$.

Let $\\{\tau_{k}\\}_{k}$ be a sequence in $\varepsilon_{k}\mathcal{T}_{\eta}$
(the set of scaled almost-periods) such that $\tau_{k}\to
x^{\prime\prime}-x^{\prime}$. Such a sequence exists by the uniform density of
$\mathcal{T}_{\eta}$. Let $z_{k}=z_{\tau_{k}}$ be such that (2.1) holds, and
define $v_{k}(x,y)=\widetilde{u}_{k}(x-\tau_{k},y-z_{k})$. Noting that $v_{k}$
can be extended by $0$ outside
$\big{(}\tau_{k}+B^{d}_{\rho}(x^{\prime})\big{)}\times(-h,h)$ we get from
almost-periodicity, (2.5) and (2.7):

$\displaystyle
F_{\varepsilon_{k}}\big{(}Ax+v_{k},\,\tau_{k}+B^{d}_{\rho}(x^{\prime})\big{)}$
$\displaystyle=$
$\displaystyle\frac{1}{2h}\int_{\big{(}\tau_{k}+B^{d}_{\rho}(x^{\prime})\big{)}\times(-h,h)}f\bigg{(}\frac{x}{\varepsilon_{k}},y,\Big{(}A+\nabla_{x}v_{k}\Big{|}\frac{1}{\varepsilon_{k}}\partial_{y}v_{k}\Big{)}\bigg{)}\mathrm{d}x\,\mathrm{d}y$
$\displaystyle=$
$\displaystyle\frac{1}{2h}\int_{B^{d}_{\rho}(x^{\prime})\times(-h-z_{k},h+z_{k})}f\bigg{(}\frac{x}{\varepsilon_{k}}+\frac{\tau_{k}}{\varepsilon_{k}},y+z_{k},\Big{(}A+\nabla_{x}\widetilde{u}_{k}\Big{|}\frac{1}{\varepsilon_{k}}\partial_{y}\widetilde{u}_{k}\Big{)}\bigg{)}\mathrm{d}x\,\mathrm{d}y$
$\displaystyle\leq$
$\displaystyle\frac{1}{2h}\int_{B^{d}_{\rho}(x^{\prime})\times(-h-\eta,h+\eta)}f\bigg{(}\frac{x}{\varepsilon_{k}}+\frac{\tau_{k}}{\varepsilon_{k}},y+z_{k},\Big{(}A+\nabla_{x}\widetilde{u}_{k}\Big{|}\frac{1}{\varepsilon_{k}}\partial_{y}\widetilde{u}_{k}\Big{)}\bigg{)}\mathrm{d}x\,\mathrm{d}y$
$\displaystyle\leq$ $\displaystyle\frac{\beta c\rho^{d}(\eta+\delta)}{h}$
$\displaystyle\mkern
50.0mu+\frac{1}{2h}\Big{(}1+\frac{2\beta}{\alpha|\log(\delta/\eta)|}\Big{)}\int_{B^{d}_{\rho}(x^{\prime})\times(-h,h)}f\bigg{(}\frac{x}{\varepsilon_{k}}+\frac{\tau_{k}}{\varepsilon_{k}},y+z_{k},\Big{(}A+\nabla_{x}{u}_{k}\Big{|}\frac{1}{\varepsilon_{k}}\partial_{y}{u}_{k}\Big{)}\bigg{)}\mathrm{d}x\,\mathrm{d}y$
$\displaystyle\leq$ $\displaystyle\frac{\beta
c\rho^{d}}{h}+\frac{1}{2h}\Big{(}1+\frac{2\beta}{\alpha|\log(\delta/\eta)|}\Big{)}\int_{B^{d}_{\rho}(x^{\prime})\times(-h,h)}f\bigg{(}\frac{x}{\varepsilon_{k}},y,\Big{(}A+\nabla_{x}u_{k}\Big{|}\frac{1}{\varepsilon_{k}}\partial_{y}u_{k}\Big{)}\bigg{)}\mathrm{d}x\,\mathrm{d}y$
$\displaystyle\mkern
50.0mu+\frac{\eta}{2h}\Big{(}1+\frac{2\beta}{\alpha|\log(\delta/\eta)|}\Big{)}\int_{B^{d}_{\rho}(x^{\prime})\times(-h,h)}\Bigl{(}1+\Big{|}\Big{(}A+\nabla_{x}u_{k}\Big{|}\frac{1}{\varepsilon_{k}}\partial_{y}u_{k}\Big{)}\Big{|}^{p}\Bigr{)}\mathrm{d}x)\,\mathrm{d}y$
$\displaystyle\leq$ $\displaystyle\frac{\beta
c\rho^{d}(\eta+\delta)}{h}+\Big{(}1+\frac{2\beta}{\alpha|\log(\delta/\eta)|}\Big{)}F_{\varepsilon_{k}}\big{(}Ax+u_{k},\,B^{d}_{\rho}(x^{\prime})\big{)}$
$\displaystyle\mkern
50.0mu+\frac{\eta}{2h}\Big{(}1+\frac{2\beta}{\alpha|\log(\delta/\eta)|}\Big{)}\Big{(}2{h}c\rho^{d}+\frac{1}{\alpha}F_{\varepsilon_{k}}\big{(}Ax+u_{k},\,B^{d}_{\rho}(x^{\prime})\big{)}\Big{)}.$

Fix now $r>1$, and note that for $k$ large enough we have
$\tau_{k}+B^{d}_{\rho}(x^{\prime})\subset
B^{d}_{r\rho}(x^{\prime\prime})\Subset\omega$. Since $v_{k}\to 0$ in
$L^{p}\big{(}B^{d}_{r\rho}(x^{\prime})\times(-h,h),\mathbb{R}^{m}\big{)}$:

$\displaystyle F_{0}\big{(}Ax,B^{d}_{\rho}(x^{\prime\prime})\big{)}$
$\displaystyle\leq$
$\displaystyle\,F_{0}\big{(}Ax,B^{d}_{r\rho}(x^{\prime\prime})\big{)}$
$\displaystyle\leq$
$\displaystyle\,\liminf_{k\to+\infty}F_{\varepsilon_{k}}\big{(}Ax+v_{k},B^{d}_{r\rho}(x^{\prime\prime})\big{)}$
$\displaystyle\leq$
$\displaystyle\,\Big{(}1+\frac{\eta}{\alpha}\Big{)}\Big{(}1+\frac{2\beta}{\alpha|\log(\delta/\eta)|}\Big{)}\liminf_{k\to+\infty}F_{\varepsilon_{k}}\big{(}Ax+u_{k},B^{d}_{\rho}(x^{\prime})\big{)}$
$\displaystyle\mkern
40.0mu+\frac{c\rho^{d}\eta}{h}\Big{(}1+\frac{2\beta}{\alpha|\log(\delta/\eta)|}\Big{)}+\frac{\beta
c\rho^{d}(\eta+\delta)}{h}$ $\displaystyle\mkern
40.0mu+\frac{\beta}{h}\big{(}1+|A|^{p}\big{)}\big{|}B^{d}_{r\rho}(x^{\prime\prime})\setminus
B^{d}_{\rho}(x^{\prime\prime})\big{|}$ $\displaystyle\leq$
$\displaystyle\,\Big{(}1+\frac{\eta}{\alpha}\Big{)}\Big{(}1+\frac{2\beta}{\alpha|\log(\delta/\eta)|}\Big{)}F_{0}\big{(}u_{k},B^{d}_{\rho}(x^{\prime})\big{)}$
$\displaystyle\mkern
40.0mu+\frac{c\rho^{d}\eta}{h}\Big{(}1+\frac{2\beta}{\alpha|\log(\delta/\eta)|}\Big{)}+\frac{\beta
c\rho^{d}}{h}(\eta+\delta)$ $\displaystyle\mkern
40.0mu+\frac{\beta}{h}\big{(}1+|A|^{p}\big{)}\big{|}B^{d}_{r\rho}(x^{\prime\prime})\setminus
B^{d}_{\rho}(x^{\prime\prime})\big{|}.$

Letting first $r\to 1$, then $\eta\to 0$ and eventually $\delta\to 0$, we
finally obtain the inequality

$F_{0}\big{(}Ax,B^{d}_{\rho}(x^{\prime\prime})\big{)}\leq
F_{0}\big{(}Ax,B^{d}_{\rho}(x^{\prime})\big{)}.$

By symmetry we then obtain also equality and the claim. ∎

We can then proceed in the proof of Theorem 1.3. Take $\omega=(0,1)^{d}$ and
let $\\{\varepsilon_{k}\\}$ be a subsequence given by (2.2). The
$\Gamma$-limit of the family $F_{\varepsilon_{k}}$ exists and, for Proposition
2.1, is equal to

$F_{0}(u)=\int_{(0,1)^{d}}f_{0}(\nabla u)\,\mathrm{d}x.$

This functional is sequentially weakly lower semicontinuous on
$\mathrm{W}^{1,p}(\Omega_{1},\mathbb{R}^{m})$ so the function $f_{0}$ is
$\mathrm{W}^{1,p}$-quasiconvex. This means we can write

$f_{0}(A)=\min\biggl{\\{}\int_{(0,1)^{d}}f_{0}(A+\nabla
u)\,\mathrm{d}x:u=0\hbox{ on }\partial(0,1)^{d}\biggr{\\}}$ (2.8)

for all $A\in\mathbb{M}^{m\times d}$. From the property of convergence of
minima for $\Gamma$-convergence and a change of variable in the right-hand
side we obtain

$\displaystyle f_{0}(A)$ $\displaystyle=$
$\displaystyle\min\bigg{\\{}\int_{(0,1)^{d}}f_{0}(A+\nabla
u)\,\mathrm{d}x:u=0\hbox{ on }\partial(0,1)^{d}\bigg{\\}}$ (2.9)
$\displaystyle=$
$\displaystyle\min\bigg{\\{}\frac{1}{2h}\int_{(0,1)^{d}\times(-h,h)}f_{0}(A+\nabla_{x}u)\,\mathrm{d}x:u=0\hbox{
on }\partial\big{(}(0,1)^{d}\big{)}\times(-h,h)\bigg{\\}}$ $\displaystyle=$
$\displaystyle\lim_{k\to+\infty}\inf\biggl{\\{}\frac{1}{2h}\int_{(0,1)^{d}\times(-h,h)}f\left(\frac{x}{\varepsilon_{k}},y,\Big{(}A+\nabla_{x}u\Big{|}\frac{1}{\varepsilon_{k}}\partial_{y}u\Big{)}\right)\,\mathrm{d}x\,\mathrm{d}y:{u\in\mathcal{W}_{1}}\biggr{\\}}$
$\displaystyle=$
$\displaystyle\lim_{k\to+\infty}\inf\biggl{\\{}\frac{1}{2hT_{k}^{d}}\int_{(0,T_{k})^{d}\times(-h,h)}f\big{(}x,y,(A+\nabla_{x}u|\partial_{y}u)\big{)}\,\mathrm{d}x\,\mathrm{d}y:{u\in\mathcal{W}_{T_{k}}}\biggr{\\}},$

where $T_{k}=1/\varepsilon_{k}$.

The following proposition states the existence of the limit in (1.13), giving
in particular the independence of $f_{0}$ from the subsequence.

######  Proposition 2.2.

For every $A\in\mathbb{M}^{m\times d}$ the following limit exists

$f_{\hom}(A)=\lim_{T\to+\infty}\inf_{u\in\mathcal{W}_{T}}\biggl{\\{}\frac{1}{2hT^{d}}\int_{(0,T)^{d}\times(-h,h)}f\big{(}x,y,(A+\nabla_{x}u|\partial_{y}u)\big{)}\,\mathrm{d}x\,\mathrm{d}y\biggr{\\}}.$
(2.10)

###### Proof.

For every $T>0$ let

$g_{A}(T)\coloneqq\inf_{u\in\mathcal{W}_{T}}\biggl{\\{}\frac{1}{2hT^{d}}\int_{(0,T)^{d}\times(-h,h)}f\big{(}x,y,(A+\nabla_{x}u|\partial_{y}u)\big{)}\,\mathrm{d}x\,\mathrm{d}y\biggr{\\}}.$

Fix $\delta>0$ and let $\eta$ be such that $\delta>\eta>0$. Now fix $T>0$ and
let $u_{T}\in\mathcal{W}_{T}$ be such that

$\frac{1}{2hT^{d}}\underbrace{\int_{(0,T)^{d}\times(-h,h)}f\big{(}x,y,(A+\nabla_{x}u_{T}|\partial_{y}u_{T})\big{)}\,\mathrm{d}x\,\mathrm{d}y}_{\eqqcolon
C_{T}}\leq g_{A}(T)+\frac{1}{T}.$

We want to estimate $g_{A}(S)$ in terms of $g_{A}(T)$ when $S\gg T$. For this
purpose, we construct test functions $u_{S}\in\mathcal{W}_{S}$ by a patchwork
procedure, and we then exploit the almost-periodicity of the energy density
$f$, extending $u_{T}$ to a function which has an energy contribution of
$\mathrm{o}(\frac{1}{\eta})C_{T}$. More precisely, we apply Lemma 1.6 to the
integrable functions defined for $t\in[0,h]$ by

$g(t)\coloneqq\int_{(0,T)^{d}}|A+\nabla_{x}u_{T}(x,\pm{t}),\partial_{y}u_{T}(x,\pm{t})|^{p}\,\mathrm{d}x.$

We then obtain two values $y_{\eta}^{+}\in(h-\delta,h)$ and
$y_{\eta}^{-}\in(h,-h+\delta)$ such that

$\displaystyle\int_{[0,T]^{d}\times{(y_{\eta}^{+},h+\eta)}}f\big{(}x,y,\big{(}A+\nabla_{x}u_{T}(x,y_{\eta}^{+})\big{|}\,\partial_{y}u_{T}(x,y_{\eta}^{+})\big{)}\big{)}\,\mathrm{d}x\,\mathrm{d}y$
(2.11) $\displaystyle\leq$
$\displaystyle\beta\int_{[0,T]^{d}\times{(y_{\eta}^{+},h+\eta)}}\Bigl{(}1+\big{|}\big{(}A+\nabla_{x}u_{T}(x,y_{\eta}^{+})\big{|}\,\partial_{y}u_{T}(x,y_{\eta}^{+})\big{)}\big{|}^{p}\Bigr{)}\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle\leq$ $\displaystyle\beta T^{d}{|h+\eta-
y_{\eta}^{+}|}+\frac{\beta}{|\log(\delta/\eta)|}\int_{[0,T]^{d}\times(-h,h)}\big{|}\big{(}A+\nabla_{x}u_{T}(x,y_{\eta}^{+})\big{|}\,\partial_{y}u_{T}(x,y_{\eta}^{+})\big{)}\big{|}^{p}\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle\leq$
$\displaystyle\beta\left(T^{d}(\eta+\delta)+\frac{C_{T}}{\alpha|\log(\delta/\eta)|}\right)$

and

$\displaystyle\hskip-56.9055pt\int_{[0,T]^{d}\times{(-h-\eta,y_{\eta}^{-})}}f\Big{(}x,y,\big{(}A+\nabla_{x}u_{T}(x,y_{\eta}^{+})\big{|}\,\partial_{y}u_{T}(x,y_{\eta}^{+})\big{)}\Big{)}\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle\leq\beta\left(T^{d}(\eta+\delta)+\frac{C_{T}}{\alpha|\log(\delta/\eta)|}\right).$
(2.12)

Thus, we can modify $u_{T}$ setting

$\widetilde{u}_{T}(x,y)\coloneqq\begin{cases}u_{T}(x,y_{\eta}^{+})&\text{ if
}y\geq y^{+}_{\eta}\\\ u_{T}(x,y)&\text{ if }y_{\eta}^{-}\leq y\leq
y_{\eta}^{+}\\\ u_{T}(x,y_{\eta}^{-})&\text{ if }y\leq
y^{-}_{\eta}\end{cases}$

Let $L_{\eta}$ be the inclusion length related to $\mathcal{T}_{\eta}$ and let
$S>T+L_{\eta}$. Define

$\mathcal{I}_{S}=\mathbb{Z}^{d}\cap\Big{[}0,\Big{\lfloor}\frac{S}{T+L_{\eta}}\Big{\rfloor}-1\Big{)}^{d}$

and, for every $\ell\in\mathcal{I}_{S}$, choose

$\tau_{\ell}\in\big{(}(T+L_{\eta})\ell+[0,L_{\eta}]^{d})\big{)}\cap\mathcal{T}_{\eta},$

and the related $z_{\ell}$. We then define a new test function by

$u_{S}(x,y)=\begin{cases}\widetilde{u}_{T}(x-\tau_{\ell},y-z_{\ell})&(x,y)\in\big{(}\tau_{\ell}+(0,T)^{d}\big{)}\times(-h,h)\\\
0,&\text{otherwise},\end{cases}$

for every $(x,y)\in(0,S)^{d}\times(-h,h)$.

Note that $u_{S}$ is equal to zero on the set

$Q_{S}\coloneqq\Big{(}(0,S)^{d}\setminus\bigcup_{\ell\in\mathcal{I}_{S}}\tau_{\ell}+(0,T)^{d}\Big{)}\times(-h,h).$

We have

$|Q_{S}|\leq
2hS^{d}\bigg{(}1-\Big{(}\frac{T}{T+L_{\eta}}-\frac{T}{S}\Big{)}^{d}\bigg{)}.$

We can now estimate $g_{A}(S)$ using (2.11) and (2)

$\begin{split}2S^{d}g_{A}(S)&\leq\int_{(0,S)^{d}\times(-h,h)}f\big{(}x,y,A+\nabla_{x}u_{S}(x,y),\partial_{y}u_{S}(x,y)\big{)}\,\mathrm{d}x\,\mathrm{d}y\\\
&=\sum_{\ell\in\mathcal{I}_{S}}\int_{(\tau_{\ell}+(0,T)^{d})\times(-h,h)}f\big{(}x,y,A+\nabla_{x}\widetilde{u}_{T}(x-\tau_{\ell},y-z_{\ell}),\partial_{y}\widetilde{u}_{T}(x-\tau_{\ell},y-z_{\ell})\big{)}\,\mathrm{d}x\,\mathrm{d}y\\\
&\mkern 90.0mu+\int_{Q_{S}}f(x,A)\,\mathrm{d}x\,\mathrm{d}y\\\
&\leq\sum_{\ell\in\mathcal{I}_{S}}\int_{(0,T)^{d}\times(-h,h)}f\big{(}\tau_{\ell}+x,z_{\ell}+y,A+\nabla_{x}{u}_{T}(x,y),\partial_{y}{u}_{T}(x,y)\big{)}\,\mathrm{d}x\,\mathrm{d}y\\\
&\mkern 90.0mu+\Big{\lfloor}\frac{S}{T+L_{\eta}}\Big{\rfloor}^{d}\frac{2\beta
C_{T}}{\alpha|\log(\delta/\eta)|}+\Big{\lfloor}\frac{S}{T+L_{\eta}}\Big{\rfloor}^{d}2T^{d}\beta(\delta+\eta)+\int_{Q_{S}}f(x,A)\,\mathrm{d}x\,\mathrm{d}y\end{split}$

Now, exploiting the almost-periodicity of $f$ we get

$\begin{split}2S^{d}g_{A}(S)&\leq\Big{\lfloor}\frac{S}{T+L_{\eta}}\Big{\rfloor}^{d}\bigg{(}\int_{(0,T)^{d}\times(-h,h)}f\Big{(}x,y,\big{(}A+\nabla_{x}{u}_{T}(x,y)\big{|}\partial_{y}{u}_{T}(x,y)\Big{)}\,\mathrm{d}x\,\mathrm{d}y\\\
&\mkern
90.0mu+\eta\int_{(0,T)^{d}\times(-h,h)}1+\Big{|}\big{(}A+\nabla_{x}u_{T}(x,y)\big{|}\partial_{y}u_{T}(x,y)\big{)}\Big{|}^{p}\,\mathrm{d}x\,\mathrm{d}y\bigg{)}\\\
&\mkern 90.0mu+\Big{\lfloor}\frac{S}{T+L_{\eta}}\Big{\rfloor}^{d}\frac{2\beta
C_{T}}{\alpha|\log(\delta/\eta)|}+\Big{\lfloor}\frac{S}{T+L_{\eta}}\Big{\rfloor}^{d}2T^{d}\beta(\delta+\eta)+\int_{Q_{S}}f(x,A)\,\mathrm{d}x\,\mathrm{d}y\end{split}$

Using the p-growth conditions

$\begin{split}2S^{d}g_{A}(S)&\leq\Big{(}1+\frac{\eta}{\alpha}\Big{)}\Big{\lfloor}\frac{S}{T+L_{\eta}}\Big{\rfloor}^{d}{\int_{(0,T)^{d}\times(-h,h)}f\Big{(}x,y,\big{(}A+\nabla_{x}{u}_{T}(x,y)\big{|}\partial_{y}{u}_{T}(x,y)\Big{)}\,\mathrm{d}x\,\mathrm{d}y}\\\
&+2\eta
hT^{d}\Big{\lfloor}\frac{S}{T+L_{\eta}}\Big{\rfloor}^{d}+\Big{\lfloor}\frac{S}{T+L_{\eta}}\Big{\rfloor}^{d}\frac{2\beta
C_{T}}{\alpha|\log(\delta/\eta)|}+\Big{\lfloor}\frac{S}{T+L_{\eta}}\Big{\rfloor}^{d}2T^{d}\beta(\delta+\eta)\\\
&\mkern 90.0mu+2\beta
hS^{d}\bigg{(}1-\Big{(}\frac{T}{T+L_{\eta}}-\frac{T}{S}\Big{)}^{d}\bigg{)}\big{(}1+|A|\big{)}^{p}\end{split}$

We now exploit $C_{T}/(2T^{d})\leq g_{A}(T)+1/T$

$\begin{split}g_{A}(S)&\leq\Big{(}\frac{T}{T+L_{\eta}}\Big{)}^{d}\Big{(}1+\frac{\eta}{\alpha}\Big{)}\Big{(}1+\frac{2\beta}{\alpha|\log(\delta/\eta)|}\Big{)}\Big{(}g_{A}(T)+\frac{1}{T}\Big{)}\\\
&+\Big{(}\eta
h+\beta(\delta+\eta)\Big{)}\Big{(}\frac{T}{T+L_{\eta}}\Big{)}^{d}+\beta
h\bigg{(}1-\Big{(}\frac{T}{T+L_{\eta}}-\frac{T}{S}\Big{)}^{d}\bigg{)}\big{(}1+|A|\big{)}^{p}\end{split}$

Taking the limit, first as $S\to\infty$ and then as $T\to\infty$

$\limsup_{S\to+\infty}g_{A}(S)\leq\Big{(}1+\frac{\eta}{\alpha}\Big{)}\Big{(}1+\frac{2\beta}{\alpha|\log(\delta/\eta)|}\Big{)}\liminf_{T\to+\infty}g_{A}(T)+\eta
h+\beta(\delta+\eta)$

We first send $\eta\to 0$ and obtain

$\limsup_{S\to+\infty}g_{A}(S)\leq\liminf_{T\to+\infty}g_{A}(T)+\beta\delta,$

and then conclude by the arbitrariness of $\delta$. ∎

Eventually, it suffices to remark that Proposition 2.2 implies that $f_{0}(A)$
is independent on the subsequence $\\{\varepsilon_{k}\\}$, so that we
simultaneously have the existence of the limit as $\varepsilon\to 0$ and the
representation (1.12), which concludes the proof of the theorem.

Acknowledgments. Andrea Braides is a member of GNAMPA, INdAM.

## References

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# Illinois Express Quantum Network for Distributing and Controlling
Entanglement on Metro-Scale

Wenji Wu1, Joaquin Chung2, Gregory Kanter3, Nikolai Lauk4, Raju Valivarthi4,
Russell R. Ceballos1,5, Cristián Peña1,4, Neil Sinclair4,6, Jordan M. Thomas7,
Ely M. Eastman7,
Si Xie4, Rajkumar Kettimuthu2, Prem Kumar7, Panagiotis Spentzouris1, Maria
Spiropulu4
1Fermi National Accelerator Laboratory
2Argonne National Laboratory
3NuCrypt LLC
4California Institute of Technology
5Chicago State University
6Harvard University
7Northwestern University

###### Abstract

We describe an implementation of a quantum network over installed fiber in the
Chicago area. We present network topology and control architecture of this
network and illustrate preliminary results for quantum teleportation and
coexistence of quantum and classical data on the same fiber link.

###### Index Terms:

quantum networks, quantum communication, entanglement distribution,
entanglement swapping, quantum and classic networks co-existence

##  I Introduction

Analogous to classical optical networks, eventually quantum optical networks
will be used to interconnect quantum processors including quantum computers
and sensors. While it is attractive to leverage as much of the traditional
fiber-optic infrastructure as possible, quantum signals have different
characteristics that, for instance, preclude their use with optical amplifiers
or electrical termination before the end-point. Additionally, they are very
sensitive to loss and even miniscule levels of added noise. As such, we expect
quantum networks to require custom engineering while also benefiting from as
much compatibility with traditional networks as is practical.

The Illinois Express Quantum Network (IEQNET) is a program for developing
metro-scale quantum networking over deployed optical fiber infrastructure.
While new technologies, such as quantum repeaters, will be needed to realize
the long-term goals of quantum networking, IEQNET focuses on leveraging
currently available technology (with provision for future upgrades as
technology develops). Two of the prime needs in quantum information systems
are distributing entanglement and using entanglement to perform quantum state
teleportation. These needs should be met while simultaneously allowing higher-
power classical signals to share the same fiber, both for the purposes of
enabling communications to support quantum applications and for independent
coexistence of high data-rate classical channels. In this paper we will
introduce our architecture to realize these functions over a metro-scale
network, focusing for simplicity on entanglement distribution, and discuss
some of the major control and management issues that need to be addressed to
enable reliable network operation. While there have been several proof-of-
concept demonstrations of deployed quantum communications and networking, over
both free space and fiber, in various locations around the globe, see e.g.
Refs. [1, 2, 3, 4, 5, 6, 7] for an overview, there is a need to advance
entanglement-based technologies to be more integrated into the (classical)
networking framework beyond that of previous demonstrations. This includes
scaling to more users, reaching longer link distances, allowing coexisting
quantum and classical data channels on the same fiber links, as well as
optimizing and automating the software control of network operations such as
synchronization.

The rest of the paper is organized as follows. In Section II we describe
IEQNET’s architecture, design, and implementation. Section III provides
details on experiment demonstration. Section 4 presents our conclusions and
future work.

##  II IEQNET Architecture, Design, and Implementation

Figure 1: IEQNET topology

###  II-A IEQNET Topology, Architecture, and Design

IEQNET consists of multiple sites that are geographically dispersed in the
Chicago metropolitan area with sites at Northwestern University (NU), Fermi
National Accelerator Laboratory (FNAL), Argonne National Laboratory (ANL), and
a Chicago-based data center (StarLight). Each site has one or more quantum
nodes (Q-nodes), which can communicate data and generate and/or measure
quantum signals. Q-nodes will be connected to software-defined-networking
(SDN) enabled optical switches through optical fibers. The optical switches
are further connected among one another to form a meshed all-optical network.
IEQNET contains three logically independent quantum local area networks (see
Figure 1): Q-LAN1 at FNAL, Q-LAN2 at NU and StarLight, and Q-LAN3 at ANL. The
Q-LANs are connected by dedicated communication channels and additional dark
fibers between FNAL, ANL, and StarLight. In addition, IEQNET has one or
multiple shared entangled photon sources (EPS’s) and Bell state measurement
nodes (BSM nodes), which are also connected to SDN-enabled optical switches
through optical fibers. An EPS generates entangled photon pairs at N
wavelengths, allowing a maximum of $N/2$ user-pairs to simultaneously share
bipartite entangled photons. A BSM node performs Bell state measurements and
local qubit operations for incoming photon pairs.

Inspired by the Internet architecture, IEQNET implements a similar layered
quantum networking architecture, which describes how quantum network functions
are vertically composed to provide increasingly complex capabilities. IEQNET’s
layered quantum networking architecture relies on four key vertical layers.

  * •

Quantum physical layer deals with the physical connectivity of two
communicating quantum nodes. It defines quantum channel frequencies, signal
rates, photon pulses used to represent quantum signals, etc.

  * •

Quantum link layer is the protocol layer in Q-nodes that handles the
transmission of quantum signals and messages across quantum channels.

  * •

Quantum networking layer performs wavelength routing and assignment in optical
networks to establish quantum paths between quantum nodes.

  * •

Quantum service layer provides quantum services, such as entanglement
distribution and quantum teleportation to users and applications.

Mechanisms for generation, synchronization, and measurement of quantum states
in networks are crucial for realizing a multitude of quantum information
applications. They can allow high-quality distribution of entanglement
throughout the network—provided errors are identified, their magnitude
estimated, and steps taken for their correction. Orchestration and control
mechanisms are especially important for performing advanced quantum
communication tasks, such as quantum teleportation and entanglement swapping,
which are based on quantum interference and thus are more susceptible to
dynamical processes in a network environment, such as polarization rotations
in the fiber channels or electronic control drift from local clock mismatches.
IEQNET uses a centralized control approach in which SDN controllers monitor
the status of key infrastructure plane metrics (e.g., loss on fiber links,
status of optical switches, etc.). IEQNET’s control and management software
performs such functions as time synchronization, optical path routing and
wavelength assignment for quantum and classical channels, channel calibration
and optimization, and error detection and feedback.

  * •

Time Synchronization. Synchronizing remote locations for distribution of
entanglement and their use in subsequent applications is crucial for quantum
networking. For the fiber channels, this is done by distributing clock pulses
in the same fiber as the quantum signals, where permitted. To limit the effect
of Raman scattering, which generates spurious photons over a broad range of
wavelengths, we set the wavelength of the clock light to be at longer
wavelengths compared to that for the quantum signals.

  * •

Routing is a fundamental network function, and multihop networks require a
means of selecting paths through the network. IEQNET’s underlying quantum
network is a WDM-based all-optical network. We will use SDN technology to
perform traditional routing and wavelength assignment (RWA) [8] to establish
paths between Q-nodes in IEQNET’s quantum physical layer. RWA is typically
formulated as a multi-commodity ﬂow problem, an NP-hard problem that is
typically solved with heuristic algorithms. For IEQNET, we use the following
shortest-path RWA (SP-RWA) algorithm as the baseline. We represent the network
by an undirected graph $G(V,E)$, where V represents the set of nodes in the
graph (Q-nodes, BSM, EPSs, and optical switches) and E represents the set of
edges in the graph (optical links). Each edge in the graph (optical link in
the network) has the following characteristics that contribute to the
computation of the edge’s metric/weight: link length, total number of
wavelengths, number of wavelengths available, attenuation, etc. Each node in
the graph may also contribute to the edge’s metric/weight with attributes such
as insertion loss, polarization-dependent loss (PDL), and polarization mode
dispersion (PMD).

  * •

Quantum Channel Calibration & Optimization. The single-photon nature of
quantum communication signals makes them extremely sensitive to noise on the
quantum channels. In addition, as mentioned above, protocols such as
teleportation require indistinguishability in spectral, temporal, spatial, and
polarization properties of the two photons arriving at the BSM node. IEQNET
employs several active and automated quantum-channel calibration and
optimization mechanisms to minimize quantum-channel loss, reduce background
noise, and compensate for polarization and delay drifts. HOM measurement and
calibration is used to ensure quantum indistinguishability. The HOM signal
provides feedback to compensate for the photons’ relative time-of-flight,
ensuring stable operation. Active polarization measurement and calibration
using coexisting classical signals, such as clock pulses, is used to
compensate for polarization drifts in fibers.

###  II-B IEQNET Control Plane Implementation

Figure 2: (a) IEQNET control plane implementation and (b) Q-NET Server block
diagram

We are working on implementing and deploying an SDN-based, logically
centralized control plane for IEQNET. As illustrated in Figure 2, a logically
centralized Q-NET server will coordinate all activities in the network. This
server will manage and schedule various quantum network resources (Q-nodes,
EPS, BSM-nodes, and channels) to perform key control and management functions
of quantum networking services, such as entanglement distribution or quantum
teleportation.

A web portal will authenticate, authorize, and audit users and applications,
and allow them to access IEQNET services. For example, for an entanglement
distribution service request, the following information will be conveyed to
the Q-NET server via the web portal: the credentials of the task submitter,
the Q-nodes involved, and the entanglement distribution requirements, such as
qubit type, rate, duration, etc. The Q-NET server will use this information to
schedule and broker resources for the task. In addition, users will be able to
browse the quantum network topology or monitor the system/site status via the
web portal. An SDN agent will keep track of the quantum topology and traffic
status with the aid of SDN controllers. It will also be responsible for
reliably updating SDN-enabled switch rules, as requested by the Q-NET server,
to assign paths for quantum, clock, and classical signals/messages. SDN
controllers are open-source network operating systems, such as ONOS. The SDN
agent will access SDN controllers through northbound APIs. One or multiple SDN
agents will be deployed, depending on the size of the quantum network.

In this design, the Q-NET server communicates with other entities through a
message queuing telemetry transport (MQTT) based message bus. Such a control
plane design offers flexibility, robustness, and scalability.

###  II-C IEQNET Control Protocol Suite

In this section we describe the IEQNET control protocol suite, which is
composed of two main protocols: quantum network resource and topology
discovery protocol (see Figure 3) and the protocol for handling entanglement
distribution requests (see Figure 4).

Figure 3 shows a sequence diagram of the discovery protocol, which starts by
making each quantum networking resource (e.g., Q-Nodes, EPS, BSM-nodes, and
switches) load their configuration. The next step is for the SDN agent to
discover the network topology through the SDN controller’s southbound API. As
all-optical switches are passive devices and thus classical active mechanism
for topology discovery will not work, we have extended ONOS’s link discovery
service to build the topology from configuration loaded in a tag field on each
optical port’s configuration. Quantum network resources register to the Q-NET
server by sending their features and connectivity information. The Q-NET
server will request the topology from the SDN agent and will subsequently ask
the SDN agent to verify the connectivity information provided by individual
quantum resources. Once the topology has been verified, the Q-NET server will
build a topology graph based on the updated topology and present it to users
through the Web portal. This discovery protocol keeps running as quantum
network resources can come and go. Furthermore, the SDN agent has the
capability to notify the Q-NET server of topology changes asynchronously.

Figure 3: Quantum network resource and discovery protocol

Figure 4 illustrates how IEQNET handles a user entanglement distribution
request. The protocol starts with a user requesting entanglement distribution
between Q-Node1 and Q-Node2. The Q-NET server will analyze this request and
choose an EPS that meets the requirements specified by the user. Upon
acceptance of the request, the Q-NET server will execute path routing and
wavelength assignment and will establish the paths among involved entities via
the SDN agent. Q-NET server will notify Q-Node1 and Q-Node2 when paths are
established and initiate path verification, which involves a series of active
probes from EPS to Q-Nodes (and vice versa) using both classical and quantum
light. After path verification, the Q-NET server will initiate calibration and
optimization processes (as described in Section II-A) for the requested
service. Once all entities send the READY signal to the Q-NET server, the
entanglement distribution process starts. Q-Nodes will collect measurements
until they have a long enough ebit string for the upper layer application. At
that point they will send the END signal to the Q-NET server to stop
entanglement distribution. Periodically during entanglement distribution, the
Q-NET server will re-initiate the calibration and optimization processes.
After the Q-NET server stops the EPS, all measurements will be stored at the
Q-NET server and the user will be able to access them through the Web portal.

Figure 4: The protocol for handling entanglement distribution requests

##  III Experiment Demonstration

In this section we present quantum teleportation and coexistence results
obtained at experiments performed at both Q-LAN1 and Q-LAN2 of IEQNET. These
systems are used to implement and test the IEQNET architecture and processes.
We employ time-bin entangled photons to perform teleportation over 44 km of
fibers at 1536 nm and achieve above 90% teleportation fidelities with a semi-
autonomous system that can sustain stable operation via modern data-
acquisition systems and integrated feedback mechanisms. We also discuss the
commissioning results of a clock distribution system, coexisting in the same
optical fiber as the quantum channel, and where we measured a time jitter of
$\sim$5 ps. These coexisting clock distribution systems will pave the way
towards synchronizing multiple nodes in remote locations and thus allowing
more complex quantum protocols. Finally, we present preliminary measurements
that support network designs with simultaneously coexisting quantum and
classical signals on the same fiber connecting the Q-nodes of the network. Our
experiments and data analysis are guided and supported by a phenomenological
model which can be quickly compared with experimental data. We discuss the
model and its utilization in our results.

###  III-A Quantum Teleportation of Time-bin Qubits and coexistence with
telecommunication O-band classical signals at Q-LAN1

Using fiber-coupled devices, including state-of-the-art low-noise
superconducting nanowire single photon detectors and off-the-shelf optics, the
Q-LAN1 systems achieve quantum teleportation of time-bin qubits at the
telecommunication wavelength of 1536.5 nm [9]. We measure teleportation
fidelities of $>90\%$ that are consistent with an analytical model of our
system, which includes realistic imperfections. To demonstrate the
compatibility of our setup with deployed quantum networks, we teleport qubits
over 22 km of single-mode fiber while transmitting qubits over an additional
22 km of fiber. Our systems, which are compatible with emerging solid-state
quantum devices, provide a realistic foundation for a high-fidelity quantum
internet with practical devices. This is accomplished using a compact setup of
fiber-coupled devices, including low-dark-count single photon detectors and
off-the-shelf optics, allowing straight-forward reproduction for multi-node
networks.

To illustrate network compatibility, teleportation is performed with up to 44
km of single-mode fiber between the qubit generation and the measurement of
the teleported qubit, and is facilitated using semi-autonomous control,
monitoring, and synchronization systems, with results collected using scalable
acquisition hardware. Our system, which operates at a clock rate of 90 MHz,
can be run remotely for several days without interruption and yield
teleportation rates of a few Hz using the full length of fiber. Our qubits are
also compatible with erbium-doped crystals, e.g. Er:Y2SiO5, that is used to
develop quantum network devices like memories and transducers [10, 11, 12].
The 1536.5 nm operating wavelength is within the low-loss (C-band)
telecommunication window for long-haul communication and where a variety of
off-the-shelf equipment is available. We also develop an analytical model of
our system, which includes experimental imperfections, predicting that the
fidelity can be improved further towards unity by well-understood methods
(such as improvement in photon indistinguishability). Our demonstrations
provide a step towards a workable quantum network with practical and
replicable nodes, such as the ambitious U.S. Department of Energy quantum
research network envisioned to link the U.S. National Laboratories.

The Q-LAN1 fiber-based experimental system, summarized in the diagram of
Figure 5, allows us to demonstrate a quantum teleportation protocol in which a
photonic qubit (provided by Alice) is interfered with one member of an
entangled photon-pair (from Bob) and projected (by Charlie) onto a Bell-state
whereby the state of Alice’s qubit can be transferred to the remaining member
of Bob’s entangled photon pair. Up to 22 (11) km of single mode fiber is
introduced between Alice and Charlie (Bob and Charlie), as well as up to
another 11 km at Bob, depending on the experiment. All qubits are generated at
the clock rate, with all of their measurements collected using a data
acquisition (DAQ) system. The measured teleportation fidelities with and
without are presented in Figure 6. Going further, the collaboration is
currently working towards time-bin entanglement swapping [13]. Preliminary
results in terms of indistinguishability have been obtained, in the newly
commissioned Fermilab FCC Q-node, when interfering the two entanglement photon
sources and performing a Hong-Ou-Mandel analysis [14].

Another aspect of quantum networking addressed by the Q-LAN1 experiments is
clock distribution. Since photons are identified by recording their times of
generation and detection, each with respect to a local (node-based) clock,
such variations can lead to misidentification of photons. To avoid this, the
variations can be accounted for by adjusting the phases of each local clock
using strong optical pulses that are co-transmitted with individual photons.
We developed such a clock distribution system, in which O-band light is
directed to two independent nodes to adjust their local clocks while C-band
photon pairs, originating from a source based on spontaneous parametric down-
conversion, are co-transmitted. We find that the clock distribution system can
allow for high-fidelity qubit distribution despite the presence of (Raman)
noise. Furthermore, we observe little additional timing jitter between clocks
at the central and end nodes, suggesting our method can be used for high-rate
networks. The deployment of such a system will pave the road towards
implementing important multi-node quantum functions towards scalable networks.

Figure 5: Schematic diagram of the quantum teleportation system consisting of
Alice, Bob, Charlie, and the data acquisition (DAQ) subsystems. One cryostat
is used to house all SNSPDs, it is drawn as two for ease of explanation.
Detection signals generated by each of the SNSPDs are labelled 1-4 and
collected at the TDC, with 3 and 4 being time-multiplexed. All individual
components are labeled in the legend, with single-mode optical fibers
(electronic cables) in grey (green), and with uni- and bi-chromatic (i.e.
unfiltered) optical pulses indicated.

Figure 6: Quantum teleportation fidelities for $|e>_{A}$, $|l>_{A}$, and
$|+>_{A}$, including the average fidelity. The dashed line represents the
classical bound. Fidelities using quantum state tomography (QST) are shown
using blue bars while the minimum fidelities for qubits prepared using
$|n=1>$, $F^{d}_{e}$, $F^{d}_{l}$, and $F^{d}_{+}$, including the associated
average fidelity $F^{d}_{a}vg$, respectively, using a decoy state method (DSM)
is shown in grey. Panels a) and b) depict the results without and with
additional fiber, respectively. Uncertainties are calculated using Monte-Carlo
simulations with Poissonian statistics

###  III-B Coexistence Studies of Quantum-Entangled and Classical Light on
Real-World Installed Fiber at Q-LAN2

Future quantum networks will leverage the traditional fiber optical
infrastructure as much as possible to gain the cost efficiency in deploying
such networks. This means that quantum signals will likely coexist with
classical communications that carry either information related (e.g.,
synchronization signals or quantum-protocol related data) or unrelated (e.g.,
independent high-rate data communication) to the quantum channel. Anticipating
this need, there have been many experimental studies of quantum-classical
coexistence in standard telecom fibers using attenuated laser light as proxy
for quantum signals. Most such studies have been in the context of quantum key
distribution (QKD), and rightfully so, because QKD is the most advanced
application of quantum communications that is closest to real-world deployment
[15]. However, quantum network functions beyond QKD, such as teleportation,
quantum repeaters, etc., will require the distribution of quantum entanglement
in real-world scenarios. Entangled light is not only required for such quantum
communication functions, but it also has an inherent robustness to background
noise due to the use of coincidence detection. In fibers carrying multiplexed
quantum and classical signals, the dominant background noise is expected to be
Raman-scattering from the copropagating classical light channels. One approach
to suppressing such Raman noise is to place the quantum signals in the 1310-nm
telecom O-band and the classical signals far away in the traditional 1550-nm
telecom C-band [16], which ensures that the sub-photon-level quantum signals
are far detuned ($\sim$35 THz) on the anti-Stokes side of the much stronger
classical light. In addition, by tightly filtering the quantum signals in time
and frequency domains, we have shown in recent experiments (described below)
that polarization-entangled quantum light can co-propagate with milliwatt-
level classical light over 45 km of installed underground fiber that presents
about 20 dB of loss to the quantum signals. Such classical light levels are
consistent with modern high-rate communication channels, and far exceed
another recent entangled and classical light coexistence experiment that used
C-band for both the quantum and classical channels [17].

In our experiments, the use of O-band/C-band quantum/classical wavelength
allocation allows us to achieve much higher co-propagating classical powers
(about 7 dBm) while still maintaining high visibility ($>$71%) in polarization
entanglement two-photon interference. Figure 7(a) shows a schematic of our
experiment. We send one photon of the polarization-entangled photon pair over
a 45.6 km loop of underground installed fiber. The underground fiber link
connects the Quantum Communications Laboratory at Northwestern University in
Evanston to the Starlight Communications Facility located on the Northwestern
Campus in Chicago (link distance of 22.8 km) where we loop the co-propagating
light back to the Evanston laboratory for characterization with polarization
analyzers and low-dark-count superconducting nanowire single photon detectors
(SNSPDs). The measured loss in the underground fiber link is 0.43 dB/km at
1310 nm, which is higher than expected in modern fibers typically used in
laboratory experiments, making the equivalent loss closer to a 60 km distance
if newer fibers were used.

We generate the quantum signal via cascaded second harmonic generation-
spontaneous parametric down conversion (c-SHG-SPDC) in a single periodically-
poled lithium-niobate waveguide (PPLN) [18]. The waveguide is phase matched
for SHG of the 1320 nm pump pulse train at 417 MHz repetition rate with 80 ps
pulse-width. We place the waveguide inside a polarization Sagnac loop to
generate polarization entangled photon pairs. The pump light entering the loop
is split by a polarizing beam splitter (PBS) into two counter-propagating
directions. The c-SHG-SPDC generates broad-bandwidth quantum amplitudes for
photon pairs centered around 1320 nm in both directions, and upon
recombination at the PBS the two-photon quantum state becomes polarization-
entangled. We separate the signal/idler photons into bands using a standard
coarse wavelength division multiplexer (CWDM). The CWDM outputs 20 nm wide
bands with center wavelengths of 1310 and 1330 nm. We send the 1310 nm channel
over the 45.6 km underground fiber link while keeping the 1330 nm channel
locally. For the classical channel, we amplify the C-band light from a phase
modulated (PM) CW laser at 1550.1 nm with an erbium-doped fiber amplifier
(EDFA) and multiplex it into the underground fiber to co-propagate with the
O-band quantum signal. We phase modulate the C-band light to broaden its
spectrum in order to emulate a data channel and inhibit stimulated Brillouin
scattering. At the receiver, we demultiplex out the C-band return light and
filter the signal and idler with 100 GHz bandpass (BP) filters at 1306.5 and
1333.5 nm, respectively.

We detect both the signal and idler photons by SNSPDs, which are followed by a
time-tagging correlation detection system. We apply an electronic delay
between the two channels to account for the fiber delay and perform
coincidence measurements. We use a pair of polarization analyzers consisting
of a quarter-wave plate (QWP), a half-wave plate (HWP), and a PBS to make
arbitrary projections of the incoming entangled-photon polarizations. Figure
7(b) shows two-photon interference fringes after the transmitted photon has
propagated alongside 6.8 dBm of C-band launch power, where a visibility of 77%
is observed in the HV basis and 74% in the DA basis. Both values are $>$71%
and thus fall in the nonclassical regime of two-photon interference. Figure
7(c) shows a plot of the coincidence-to-accidental ratio (CAR) as a function
of the co-propagating C-band power for an entangled pair rate higher than in
Figure 7(b). CAR quantifies the signal-to-noise ratio of the coincidence
detection of photon pairs, where CAR $>$6 is desired for most applications.
CAR falls as the co-propagating power is increased, as expected. Given the
$\sim$0.5 ns temporal correlation window, frequency filtering to $<$5 GHz
could presumably reduce Raman noise by another factor of $\sim$20, which would
allow for even higher co-propagating powers ($>$10 dBm) to be used.

Figure 7: (a) Experimental diagram. (b) Coincidence counts as a function of
relative HWP angle between signal and idler polarization analyzers for a co-
propagating classical power of $P_{C-band}=6.8~{}dBm$. Data in blue is for the
horizontal-vertical (HV) basis and that in yellow is for the diagonal-
antidiagonal (DA) basis. (c) CAR (blue) and single photon count rates (orange)
as a function of C-band power (KCPS = kilo-counts per second)

The above results clearly demonstrate that O-band/C-band quantum/classical
wavelength allocation along with temporal and spectral filtering are useful
noise mitigation methods for coexistence scenarios in fiber-optic quantum
networking. Since future quantum networks will require entanglement
distribution, we expect that the noise mitigation methods presented here will
be useful when integrating quantum network functionality into the installed
fiber optical infrastructure.

##  IV Conclusion

We have developed a network architecture which aids in developing free-
running, practical, quantum network controls and demonstrating metro-scale
quantum operations including entanglement distribution and eventually quantum
teleportation. We have performed experiments on quantum teleportation and
coexistence of quantum and classical information on deployed fiber. Our
complementary approaches of distributing quantum and classical signals in the
telecommunication C- and O-band, or vice versa, allow greater network
flexibility and testing under varied conditions and use cases. For example,
quantum communication in the C-band is suitable for long-haul links, while the
utility of qubits in the O-band are not restricted by noise from strong
classical light. Importantly, these two approaches are not mutually exclusive
but can be linked using quantum teleportation. This will ensure quantum
information can be routed between IEQNET Q-LANs, opening the door for real-
world testing of hybrid quantum physical layer architectures.

## Acknowledgment

IEQNET is funded by the Department of Energy’s Advanced Scientific Computing
Research Transparent Optical Quantum Networks for Distributed Science program,
but no government endorsement is implied. Fermilab is managed by Fermi
Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S.
Department of Energy, Office of Science, Office of High Energy Physics.

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