# A Combinatorial Proof of the Existence of Dense subsets in $\mathbb{R}$
without the “Steinhaus” like property.

Arpan Sadhukhan  Eindhoven University of Technology

The Steinhaus theorem states that for any positive measurable set $S$, the
difference set $S-S$ contains an open interval around the origin [1]. It is
somewhat natural to ask whether there exists a subset $P\subseteq\mathbb{R}$
such that neither $P-P$ nor $P^{c}-P^{c}$ contains any interval in
$\mathbb{R}$. Clearly, by the Steinhaus theorem we know that if $P$ is
Lebesgue measurable, then $P$ must have measure $0$, in which case we know
$P^{c}-P^{c}$ has an open interval around $0$. Consequently, such sets must
not be Lebesgue measurable. Also, in the domain of nonmeasurable subsets of
$\mathbb{R}$, it is not entirely clear that such sets will exist. In this note
we prove the existence of such sets which are also dense. The idea for the
proof is similar to [2]. Somewhat related, but entirely different results, are
found in [3].

######  Theorem 1.

There exists a dense subset $P\subseteq\mathbb{R}$ such that neither
difference set $P-P$ nor $P^{c}-P^{c}$ contains any interval in $\mathbb{R}$.

###### Proof.

Let $S=\\{2N+\frac{2k+1}{3^{n}}:|2k+1|<3^{n}\>\text{and}\>N,k\>\text{are
integers},n\in\mathbb{N}\\}$. Thus $S$ is dense in $\mathbb{R}$. Define an
infinite graph $G$ on $\mathbb{R}$ such that two vertices $a,b\in\mathbb{R}$
are adjacent iff $|a-b|\in S$. Now if $G$ has an odd cycle
$(v_{1},v_{2},\ldots v_{2n+1})$, then it is easy to see
$0=|v_{1}-v_{1}|=2N^{\prime}+\frac{2k^{\prime}+1}{3^{n^{\prime}}}$ where
$N^{\prime},k^{\prime}$ are integers and $n^{\prime}\in\mathbb{N}$, which is a
contradiction. Hence, $G$ is bipartite. Let $G=\bigcup_{\alpha\in
J}G_{\alpha}$ where $G_{\alpha},\alpha\in J$, represents the connected
components of $G$ and $J$ is the index set. Using the axiom of choice, choose
a vertex $v_{\alpha}\in G_{\alpha}\>\forall\alpha\in J$. Let $P_{\alpha}$ be
the subset of vertices of $G_{\alpha}$ such that the shortest path from each
element of $P_{\alpha}$ to $v_{\alpha}$ is of even length. Let
$P=\bigcup_{\alpha\in J}P_{\alpha}$. Observe that the sets $P$ and $P^{c}$ are
dense in $\mathbb{R}$. Also observe that no two vertices in $P$ and $P^{c}$
are adjacent as $G$ has no odd cycles. Now, without loss of generality, assume
$D=P-P$ contains an interval $I\subseteq\mathbb{R}$. Hence, $D\cap S$ is non-
empty, which is a contradiction. This completes the proof.

∎

## References

  * [1] Tao, T. (2011). An Introduction to Measure Theory. Providence, RI: American Mathematical Society.
  * [2] Thomas, R. (1985). A Combinatorial Construction of a Nonmeasurable Set. Amer. Math. Monthly. 92(4): 421–422. doi.org/10.1080/00029890.1985.11971638
  * [3] Bartoszewicz, A., Filipczak, M. (2017). Remarks on sets with small differences and large sums. J. Math. Anal. Appl. 456(1): 245–250. doi.org/10.1016/j.jmaa.2017.06.087

MSC: Primary 28A05, Secondary 05C15


# Nonadiabatic nonlinear non-Hermitian quantized pumping

Motohiko Ezawa  Department of Applied Physics, The University of Tokyo, 7-3-1
Hongo, Tokyo 113-8656, Japan  Natsuko Ishida  Research Center for Advanced
Science and Technology, The University of Tokyo, 4-6-1 Komaba, Tokyo 113-8656,
Japan  Yasutomo Ota  Research Center for Department of Applied Physics and
Physico-Informatics, Keio University, 3-14-1 Hiyoshi, Japan  Satoshi Iwamoto
Research Center for Advanced Science and Technology, The University of Tokyo,
4-6-1 Komaba, Tokyo 113-8656, Japan

###### Abstract

We analyze a quantized pumping in a nonlinear non-Hermitian photonic system
with nonadiabatic driving. The photonic system is made of a waveguide array,
where the distances between adjacent waveguides are modulated. It is described
by the Su-Schrieffer-Heeger model together with a saturated nonlinear gain
term and a linear loss term. A topological interface state between the
topological and trivial phases is stabilized by the combination of a saturated
nonlinear gain term and a linear loss term. We study the pumping of the
topological interface state. We define the transfer-speed ratio
$\omega/\Omega$ by the ratio of the pumping speed $\omega$ of the center of
mass of the wave packet to the driving speed $\Omega$ of the topological
interface. It is quantized as $\omega/\Omega=1$ in the adiabatic limit. It
remains to be quantized for slow driving even in the nonadiabatic regime,
which is a nonadiabatic quantized pump. On the other hand, there is almost no
pump for fast driving. We find a transition in pumping as a function of the
driving speed.

##  I Introduction

Topological insulator is a prominent idea in contemporary physics[1, 2]. A
typical example is a quantum Hall effect or a Chern insulator in a two-
dimensional system described by the Chern number. The Thouless pump is a
dynamical counter part of a Chern insulator[3, 4, 5, 6], where the Chern
number is defined in the space-time variable. A pumped charge per one cycle is
quantized. Especially, a topological-edge state is pumped in quasicrystal
model[7, 8], Rice-Mele model[9], and the Su-Schrieffer-Heeger (SSH) model[10].

Photonic systems provide us with an ideal playground of topological
physics[11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. Various topological
phases are realized in photonic crystal by modulating the hopping amplitude
spatially. A simplest example is the SSH model[22, 23, 24, 25, 26].
Especially, a large area topological interface laser is theoretically proposed
by using the topological interface state of the SSH model[27, 28]. The
Thouless pumping is realized by using spatially modulated waveguides[7, 29,
30, 8, 31], where the hopping amplitude between waveguides are spatially
modulated by modulating the distances between the adjacent waveguides.
Dynamics is governed by the Schrödinger equation[32, 33], where the direction
$z$ of the waveguide acts as time $t$. Nonlinear Thouless pumping has been
studied in photonic systems[34, 35, 36, 37, 38]. Recently, pumping by
modulating the topological interface state of the SSH model is proposed in a
linear Hermitian system[39], which is not the Thouless pumping.

Non-Hermicity[40, 41, 42] and nonlinearity[43, 44, 45, 46, 47] naturally
arises in topological photonics, which has expanded the field of topological
physics starting from condensed matter physics. Photon loss is effectively
well described by a non-Hermitian loss term. On the other hand, the gain has a
saturation, which is described by a nonlinear term. Stable laser emission
occurs in the presence of both of these terms[48, 49]. The interplay of non-
Hermicity and nonlinearity is interesting[50, 51, 52].

The Thouless pumping is valid only in the adiabatic limit. The pumped charge
is not quantized in the nonadiabatic regime[53, 54]. It is interesting if a
pumping can be quantized in the nonadiabatic regime in general.

In this paper, we study a nonadiabatic quantized pumping in a nonlinear non-
Hermitian photonic system. The basic structure is described by the SSH model,
which has the topological and trivial sectors. The interface state emerges at
the interface between these two sectors. We investigate the pumping of the
state by changing the pumping velocity, and find a transition to occur between
a quantized charge pumping and no pumping. We define the transfer-speed ratio
by the ratio $\omega/\Omega$, where $\omega$ is the pumping speed of the wave
packet while $\Omega$ is the driving speed of the topological interface. The
transfer-speed ratio 1 means that the wave packet perfectly follows the
driving. In slow pumping, the wave function of the interface state perfectly
follows the modulation, and hence the transfer-speed ratio is 1. On the other
hand, in fast pumping the wave function collapses because it cannot follow the
modulation, and hence, the transfer-speed ratio is zero.

Figure 1: Illustration of modulated photonic waveguide. (a) The modulation is
given to the hopping parameter $\kappa_{A}$ according to Eq.(1) with
$n_{\text{IF}}(z)=N/2$ at $z=0$. (b) The basic structure is the SSH chain made
of $N$ sites. (c) The modulation is given in the $n$-$z$ plane by Eq.(1) with
$n_{\text{IF}}(z)=N/2+\Omega(z-z_{0})$, as indicated by a black line. Here,
$N=20$.

Figure 2: Wave packet in quench dynamics in bird’s eye view and in the ($n,z$)
plane. (a1), (b1) $\Omega=0$; (a2), (b2) $\Omega=\kappa/8$; (a3), (b3)
$\Omega=\kappa/4$; (a4), (b4) $\Omega=\kappa/2$; (a5), (b5) $\Omega=\kappa$;
(a6), (b6) $\Omega=2\kappa$; (a7), (b7) $\Omega=4\kappa$; (a8), (b8)
$\Omega=8\kappa$. We have used a chain with a length $N=800$. We have set
$\eta=10,\xi=20$, $\lambda=0.5,\gamma=0.1$ and $\chi=1$. The evolution is
calculated in the range $0\leq z\leq 200$, where $z_{0}=100$. Green lines
represent the interface line (14).

##  II Model

We consider an array of spatially modulated waveguides[7, 29, 8, 31, 14, 55],
as illustrated in Fig.1. The basic structure is described by the SSH chain
made of $N$ sites as in Fig.1(b), which is a bipartite system. We take $N$ to
be an even integer. The hopping parameter $\kappa_{A,n}$ has a site
dependence, while $\kappa_{B}$ does not. They are given by

$\kappa_{A,n}\left(z\right)=\kappa\left(1+\lambda\tanh\frac{n-n_{\text{IF}}\left(z\right)}{\xi}\right),\quad\kappa_{B}=\kappa,$
(1)

where $n_{\text{IF}}\left(z\right)$ is the position of the interface as a
function of $z$; $\lambda>0$ and $\xi>0$ represent the interface modulation
amplitude and the interface width, respectively. Small (large) $\xi$
represents a sharp (smooth) interface. The function $\kappa_{A,n}-\kappa$ is
illustrated in Fig.1(a) with the choice of $n_{\text{IF}}\left(z\right)=N/2$.
We set $\xi=20$, $\lambda=0.5$, $\kappa=1$, $\gamma=0.1$, $\chi=1$ and
$\eta=10$ in numerical studies, unless otherwise stated.

The system is governed by[48]

$i\frac{d\psi_{n}}{dz}=\sum_{nm}M_{nm}\psi_{m}-i\gamma\left(1-\chi\frac{\left(1-\left(-1\right)^{n}\right)/2}{1+\left|\psi_{n}\right|^{2}/\eta}\right)\psi_{n},$
(2)

with a site-dependent hopping matrix representing the SSH model,

$\displaystyle M_{nm}\left(z\right)$
$\displaystyle=\left(\begin{array}[]{cccccc}0&\kappa_{A,n}\left(z\right)&0&0&0&\cdots\\\
\kappa_{A,n}\left(z\right)&0&\kappa_{B}&0&0&\cdots\\\
0&\kappa_{B}&0&\kappa_{A,n}\left(z\right)&0&\cdots\\\
0&0&\kappa_{A,n}\left(z\right)&0&\kappa_{B}&\ddots\\\
0&0&0&\kappa_{B}&0&\ddots\\\
\vdots&\vdots&\vdots&\ddots&\ddots&\ddots\end{array}\right),$ (9)

where $\psi_{n}$ is the amplitudes at the site $n$, where $n=1,2,3,\cdots,N$;
$\gamma$ represents the constant loss in each waveguide; $\gamma\chi$
represents the amplitude of the optical gain via stimulated emission induced
only at the odd site; $\eta$ represents the saturation parameter of nonlinear
gain[48]. All these parameters are positive. The system turns out to be a
linear model in the limit $\eta\rightarrow\infty.$ On the other hand, $\gamma$
controls the non-Hermicity, where the system is Hermitian for $\gamma=0$. In
Eq.(2) we measure $z$ in units of $1/\kappa$ and the loss parameter $\gamma$
in units of $\kappa$, where $\kappa$ is defined in Eq.(1).

The explicit equations for a finite chain with length $N$ follow from Eq.(2)
as

$\displaystyle i\frac{d\psi_{2n-1}}{dz}=$
$\displaystyle\kappa_{B}\psi_{2n-2}+\kappa_{A,n}\left(z\right)\psi_{2n}$
$\displaystyle-i\gamma\left(1-\frac{\chi}{1+\left|\psi_{2n-1}\right|^{2}/\eta}\right)\psi_{2n-1},$
(10) $\displaystyle i\frac{d\psi_{2n}}{dz}=$
$\displaystyle\kappa_{B}\psi_{2n+1}+\kappa_{A,n}\left(z\right)\psi_{2n-1}-i\gamma\psi_{2n}.$
(11)

We solve this set of equations together with the boundary condition

$\psi_{n}\left(z=0\right)=\delta_{n,n_{\text{IF}}}.$ (12)

It is convenient to regard $z$ as time $t$. Then, Eqs.(10) and (11) are the
Schrödinger equations describing a quench dynamics starting from the interface
site by giving an input to it with the initial condition (12). We consider a
case where

$n_{\text{IF}}\left(z\right)=N/2$ (13)

for the relaxation process $z<z_{0}$, and

$n_{\text{IF}}\left(z\right)=N/2+\Omega\left(z-z_{0}\right)$ (14)

for the driving process $z>z_{0}$. See Fig.1(c) for the modulation of the
hopping parameters with this choice of Eqs.(13) and (14). In what follows, we
may occasionally regard $z$ as time $t$. The pumping is said adiabatic only
for $\Omega\ll\kappa$, which means that any finite driving is nonadiabatic.

##  III Nonadiabatic regime

We start with a review of the stationary solution in the case of $\Omega=0$,
where the interface $n_{\text{IF}}\left(z\right)$ is given by the constant
(13) in the hopping parameter (1) for all $z$. This case was studied
previously, where the stationary solution is found to be[28]

$\displaystyle\Psi_{A}\left(n\right)$
$\displaystyle\varpropto\exp\left[-\frac{\kappa\lambda}{2\xi}\left(1+c_{2}\right)(n-n_{\text{IF}}\left(z\right))^{2}\right],$
$\displaystyle\Psi_{B}\left(n\right)$
$\displaystyle\varpropto-i\frac{x}{\eta}\frac{c_{2}\kappa\lambda}{\xi}\exp\left[-\frac{\kappa\lambda}{2\xi}\left(1+c_{2}\right)(n-n_{\text{IF}}\left(z\right))^{2}\right],$
(15)

together with (13) for all $z$, where $c_{2}$ is a certain constant: See
Eq.(78) in the reference[28]. This is a generalization of the Jackiw-Rebbi
solution to a nonlinear non-Hermitian model. The quench dynamics was also
examined, where the wave packet spreads from the delta function (12) and
reaches the stational distribution (15) as in Fig.2(a1) and (b1). Namely, the
interface state (15) is formed in the relaxation process ($z<z_{0}$).

We investigate the driving process, where the interface
$n_{\text{IF}}\left(z\right)$ is modulated as in Eq.(14) for $z>z_{0}$. In the
adiabatic approximation ($\Omega\simeq 0$), the wave packet is expected to
perfectly follow the interface state (15) together with (14). This is indeed
the case, as numerically checked in Fig.2(a2) and (b2) for $\Omega=\kappa/8$.

What is unexpected is the numerical result for larger values of $\Omega$. For
slow driving $\Omega<\kappa$, the state maintains its form as in
Fig.2(a1)$\sim$(a5), whose center perfectly follows the sample modulation
(14), as shown by the green line in Fig.2(b1)$\sim$(b5). On the other hand,
for fast driving $\Omega>\kappa$, the wave packet collapses and spreads, as
shown in Fig.2(b6)$\sim$(a8) and (b6)$\sim$(b8).

We show the snapshot at $z=100$, $150$ and $200$ for $\Omega=\kappa/2$, $1$
and $4$ in the system with $z_{0}=100$ in Fig.3. The wave packet does not
change its form for $\Omega=\kappa/2$ and $1$, as shown in Fig.3(a) and (b),
which leads to the quantized pumping. On the other hand, the wave packet is
collapsed for $\Omega=4\kappa$ as shown in Fig.3(c), where the pumping does
not occur.

Figure 3: Spatial profile of the wave packet for (a) slow driving
$\Omega=\kappa/2$, (b) the critical driving $\Omega=\kappa$ and (c) fast
driving $\Omega=4\kappa$. The red curves are the spatial profile at $z=100$,
the green curves are that at $z=150$ and the blue curves are that at $z=200$,
where $z_{0}=100$.

Figure 4:  (a) Maximum value of the wave packet as a function of $z$. (b) The
maximum value of the wave packet at $z=200$ as a function of $\Omega/\kappa$
in the logarithmic scale (log${}_{2}\Omega/\kappa$).

Figure 5: (a) Evolution of the mean position $\left\langle
x\left(z\right)\right\rangle$. The pumping velocity $\omega$ is determined
from the slop of the line for each $\Omega/\kappa$. The horizontal axis is $z$
with $0\leq z\leq 200$. (b) Ratio $\omega/\Omega$ as a function of $\Omega$.
The inset is an enlarged curve in the vicinity of
$(\Omega/\kappa,\omega/\Omega)=(1,1)$. The horizontal axis is in the
logarithmic scale (log${}_{2}\Omega/\kappa$).

We show the $z$ evolution of the maximum value of the wave packet in Fig.4(a).
The maximum value increases in the relaxation process and reaches a stational
value. It maintains as it is for slow driving ($\Omega<\kappa$). On the other
hand, it decreases as soon as fast driving ($\Omega>\kappa$) is started at
$z=100$. We plot the maximum value at $z=200$ in Fig.4(b). It is constant for
slow driving and decreases for fast driving.

We have shown the existence of the critical value $\Omega=\kappa$ for the
driving velocity. In order to determine this critical value more clearly, we
calculate the expectation value of the mean position by

$\left\langle
x\left(z\right)\right\rangle\equiv\frac{\sum_{n}\left(n-n_{\text{IF}}\left(0\right)\right)\left|\psi_{n}\left(z\right)\right|^{2}}{\sum_{n}\left|\psi_{n}\left(z\right)\right|^{2}}.$
(16)

We show the $z$ evolution of $\left\langle x\left(z\right)\right\rangle$ for
various $\Omega$ in Fig.5(a). It is almost zero in the relaxation process
($z<z_{0}$). On the other hand, it increases almost linearly in the driving
process ($z>z_{0}$). Namely, $\left\langle x\left(z\right)\right\rangle$ is
well approximated by

$\left\langle x\left(z\right)\right\rangle=\omega\left(z-z_{0}\right).$ (17)

We define the pumping velocity $\omega$ at each $\Omega$ by this formula.

We then calculate numerically the ratio $\omega/\Omega$ of the pumping speed
$\omega$ to the driving speed $\Omega$. We plot $\omega/\Omega$ as a function
of $\Omega$ in Fig.5(b), where it is found that the pumping speed is almost
identical to the driving speed, i.e., $\omega=\Omega$ for $\Omega\leq\kappa$.
It means that the wave packet follows the modulation
$n_{\text{IF}}\left(z\right)$. On the other hand, $\omega$ decreases as the
increase of $\Omega$. It means that the modulation is too fast for the wave
packet to follow it. The transition occurs near $\Omega\simeq\kappa$, where
the system is nonadiabatic.

Figure 6: (a) Linear Hermitian model ($\gamma=0$), (b) Linear non-Hermitian
model ($\gamma=0.1$, $\eta=\infty)$. and (c) non-Hermitian with the loss term
and without the gain term ($\gamma=0.1$, $\chi=0)$. We have set
$\Omega=\kappa/2$.

The quantized pumping in the nonadiabatic regime requires both non-Hermicity
and nonlinearity. The wave packet rapidly collapses for the Hermitian system
($\gamma=0$) as in Fig.6(a). This is because the delta function contains
components not only from the topological interface state but also from many
bulk states, where the bulk states spreads rapidly over the sample[47, 28].
The wave packet exponentially grows and does not reach the stationary solution
for the linear non-Hermitian system ($\gamma\neq 0,$ $\eta=\infty)$ as in
Fig.6(b). The wave packet rapidly decreases if there is the loss term without
the gain term ($\gamma\neq 0,\chi=0)$ as in Fig.6(c).

##  IV Discussion

We have shown that the nonlinear non-Hermitian pump is quantized even in the
nonadiabatic regime. The combination of the saturated gain and linear loss
greatly stabilizes the wave packet, and hence, it is robust even for
nonadiabatic driving. It is highly contrasted to the Thouless pump, which is
broken in the nonadiabatic regime[53, 54]. Our work provides an example that
the interplay among non-Hermicity and nonlinearity gives an intriguing
phenomena in the nonadiabatic regime.

We compare the maximum driving speed $\Omega_{\text{max}}$ in the nonadiabatic
regime with those in the previous work[10], where the system is Hermitian and
linear. It is of the order of $\Omega_{\text{max}}/\kappa=30/5000\eqsim 0.006$
in the Thouless pumping based on the Rice-Mele model, and it is of the order
of $\Omega_{\text{max}}/\kappa=31/400\eqsim 0.075$ in the pumping of the
topological interface state. On the other hand, it is of the order of
$\Omega_{\text{max}}/\kappa=1$ in the present mechanism. Hence, the ratio is
much faster than the previous ones. This is because that the wave packet is
stabilized by the combination of the non-Hermicity and nonlinearity.

Femto-second laser writing waveguide[14, 56, 57] or semiconductor
waveguide[58, 59, 60] are used in experiments. Typical distance between
waveguides is 15$\mu m$ (5$\mu m$) and the length of the waveguide is 500mm
(50$\mu m$) for a femto-second laser writing[14] (semiconductor waveguide[60])
waveguide. We have set $N=800$, which corresponds to 12mm for a femto-second
laser writing waveguide and 4mm for a semiconductor waveguide, in numerical
simulations.

M.E is supported by CREST, JST (Grants No. JPMJCR20T2) and Grants-in-Aid for
Scientific Research from MEXT KAKENHI (Grant No. 23H00171). N. Ishida is
supported by the Grants-in-Aid for Scientific Research from MEXT KAKENHI
(Grants. No. JP21J40088). Y. Ota is supported by the Grants-in-Aid for
Scientific Research from MEXT KAKENHI (Grants. Nos. 22H01994 and 22H00298). S.
Iwamoto is supported by CREST, JST (Grants No. JPMJCR19T1) and the Grants-in-
Aid for Scientific Research from MEXT KAKENHI (Grants. Nos. 22H00298 and
22H01994).

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# With or Without Permission: Site-Specific Augmented Reality for Social
Justice

Rafael M.L. Silva  0000-0001-8376-0416<EMAIL_ADDRESS>University of
WashingtonUSA ,  Ana María Cárdenas Gasca  0000-0003-4842-4729
<EMAIL_ADDRESS>UC Santa BarbaraUSA ,  Joshua A. Fisher
0000-0001-5628-5138<EMAIL_ADDRESS>Ball State UniversityUSA ,  Erica
Principe Cruz  0000-0003-0992-0527<EMAIL_ADDRESS>Carnegie Mellon
UniversityUSA ,  Cinthya Jauregui<EMAIL_ADDRESS>0009-0008-2860-3264 Santa
Clara UniversityUSA ,  Amy Lueck  0000-0002-0432-2280<EMAIL_ADDRESS>Santa
Clara UniversityUSA ,  Fannie Liu  0000-0002-5656-3406<EMAIL_ADDRESS>JPMorgan Chase & Co.USA ,  Andrés Monroy-Hernández  0000-0003-4889-9484
<EMAIL_ADDRESS>Princeton UniversityUSA and  Kai Lukoff
<EMAIL_ADDRESS>0000-0001-5069-6817 Santa Clara University500 El Camino
RealUSA95053

(2024)

###### Abstract.

Movements for social change are often tied to a particular locale. This makes
Augmented Reality (AR), which changes how people perceive their surroundings,
a promising technology for social justice. Site-specific AR empowers activists
to re-tell the story of a place, with or without permission of its owner. It
has been used, for example, to reveal hidden histories, re-imagine problematic
monuments, and celebrate minority cultures. However, challenges remain
concerning technological ownership and accessibility, scalability,
sustainability, and navigating collaborations with marginalized communities
and across disciplinary boundaries. This half-day workshop at CHI 2024 seeks
to bring together an interdisciplinary group of activists, computer
scientists, designers, media scholars, and more to identify opportunities and
challenges across domains. To anchor the discussion, participants will each
share one example of an artifact used in speculating, designing, and/or
delivering site-specific AR experiences. This collection of artifacts will
inaugurate an interactive database that can inspire a new wave of activists to
leverage AR for social justice.

augmented reality, spatial justice, design justice, social justice

††journalyear: 2024††copyright: rightsretained††conference: Extended Abstracts
of the CHI Conference on Human Factors in Computing Systems; May 11–16, 2024;
Honolulu, HI, USA††booktitle: Extended Abstracts of the CHI Conference on
Human Factors in Computing Systems (CHI EA ’24), May 11–16, 2024, Honolulu,
HI, USA††doi: 10.1145/3613905.3636283††isbn: 979-8-4007-0331-7/24/05††ccs:
Human-centered computing Human computer interaction (HCI)††ccs: Human-centered
computing Mixed / augmented reality††ccs: Applied computing Arts and
humanities

##  1\. Introduction

Activism is often tightly connected with the places, spaces, and physical
environments related to a cause (Silva et al., 2022). With the rise of
immersive technologies on everyday devices like smartphones, activists have
created new ways of digitally augmenting physical locations to support social
causes, leveraging the affordances of Augmented Reality (AR). Accordingly,
activists have created AR experiences that re-evaluate the histories
manifested in the social, cultural, and built environments (Rucker, 2021;
Hahn, 2020; Abebe et al., 2022). These include the Monument App, which invites
users to place digital monuments of people of color at specific sites (Fig.
1). In another example, SkytypingAR (Times, 2020) enabled activists to
coordinate the display of protest messages condemning immigration policies
above detention and correction centers. While such digital activism holds
value in of itself, there is even precedent for it to serve as a ‘prototype’
for changes that are later realized in the physical world (Schroeder et al.,
2023).

Figure 1. An Example of a ‘Final Artifact.’ The Kinfolk App invites users to
place digital monuments of people of color at specific locations. The project
emerged from an artist and activist campaign to advocate for the removal of
the statue of Christopher Columbus at Columbus Circle in New York City
(McWhirter et al., 2021). Here, Toussaint L’Ouverture, the leader of a Haitian
slave rebellion and revolutionary movement, is shown superimposed over that
statue of Columbus, leveraging augmented reality to envision a more just
future.

Toussaint L’Ouverture is superimposed over Columbus Circle, a bustling traffic
circle in New York City

However, open questions remain about when and how activists should leverage AR
for social justice. For instance, when should activists work with existing
powers, such as AR platforms that participate in surveillance capitalism, and
when should they circumvent them? What methods can interdisciplinary
collaborators use to co-design site-specific AR experiences? And how can
activists reach wider audiences and maintain their creations?

This workshop seeks to address these questions by centering artifacts used in
speculating, designing, and delivering AR experiences for social justice.
Since it is difficult to explain AR experiences in words alone (Bolter et al.,
2021), these artifacts will ground the workshop discussions in concrete
examples. One intended outcome of the workshop is an interactive database that
will showcase these artifacts, serving as a repository that inspires future
activists. Artifacts might include:

  * •

A Speculative Artifact such as a sketch, diagram, or description of an
imagined future AR experience or creator tool (e.g., Fig. 2).

  * •

A Process Artifact such as a site map, an early prototype, a memorandum of
understanding with a community partner, or a snippet of code (e.g., Fig. 3).

  * •

A Final Artifact such as a 3D model, a soundscape, or a short video of the
completed experience (e.g., Fig 1).

The resulting collection of artifacts will facilitate discussion of a wide
range of visions, methods, and results in the creation of AR experiences for
social justice.

Drawing from Design Justice practice and theory (Costanza-Chock, 2020), we see
community engagement as the key to identifying locations of interest,
connecting to existing social movements, and developing appropriate resources.
We aim to translate our academic efforts into tools and knowledge that enable
community-led praxis of AR for social justice. As AR technology goes
mainstream, it becomes crucial to craft a future where AR does not become
another hegemonic force but a tool to embolden prosocial change.

Figure 2. An example of a ‘Speculative Artifact.’ The Memory Layers
application, a depiction of sensitive narratives, was developed in
collaboration with the Museo de la Memoria, a human rights museum in Colombia
(Cárdenas Gasca et al., 2022). The research team and museum employees co-
created sketches of three different modes of user interaction, focusing on
adapting existing exhibitions and collections to an AR format. This
exploration delved into a design space for AR applications, considering users,
technology, and objectives.

Three rows of sketches, each displaying a different AR interaction. First row:
interacting with a wall on the phone to show a mural in AR. Second row:
multiple people scanning markers on a large map on the floor. Third row:
person waving their arms to select between stories that are displayed on the
wall.

##  2\. Workshop Aims

This workshop has three main aims:

  * •

Build community around social justice AR. Connect creators and researchers
working in this emerging domain.

  * •

Interdisciplinary discussion. Bring together activists, computer scientists,
designers, media scholars, and more to identify opportunities and challenges
for social justice AR across domains.

  * •

Develop an interactive database of artifacts. Given the difficulty of
accessing site-specific AR experiences, a repository of exemplars can serve as
a resource for future social justice AR projects.

##  3\. Background and Motivation

###  3.1. Opportunities and Challenges

Place-based action has always been an integral aspect of social justice work
and, as a technology tightly couple with physical world, AR introduces
interesting affordances for this type of initiatives. In recent years, the
widespread adoption of smartphones with AR technology has spurred activists to
digitally augment places to support their causes. Practitioners can now
overlay digital content onto the environment to embody their message. For
example, users of the BeHere/1942 mobile AR exhibition app (Fujihata, [n. d.])
can witness the physical space of a Los Angeles block overlaid with a scene
from the Japanese-American internment. GPS-enabled applications, such as the
Emmett Till Memory Project (instute of museum and library services, [n. d.]),
harness the power of location to reveal neglected stories. By preserving the
sites and narratives of the Till lynching, this project offers users an
immersive connection to specific locales, emphasizing their historical
significance.

The creators of these apps are confronted with a selection of frameworks and
tools that influence not only the user experience but also touch upon
political and ethical matters. Activists must consider what it means to use
technologies that could be antithetical to their causes or introduce privacy,
accessibility, and maintenance concerns. This workshop aims to start an
interactive database that shares and categorizes annotated examples in order
to help activists navigate this landscape.

(a) To create the Thámien Ohlone AR Tour at Santa Clara University, the team
tried a new co-design exercise called “landmark-based affinity diagramming”
(Lukoff, 2023). Here, members of the Muwekma Ohlone Native American Tribe
discuss where to place a media asset on the map.

Two members of the Muwekma Ohlone lean over a table with a giant map, pointing
and discussing where to place a Post-it note as part of this co-design
exercise.

(b) The team printed a 10 x 6 foot map of the historical campus, the site of
the AR tour. Pink post-its are stories/messages the tribe wanted to share,
e.g. “Hopes for a future Ohlone garden at the University.” Blue post-it notes
are media assets, e.g. 3D models of tribal artifacts, recordings of songs,
that were then displayed at that location in AR.

A hand places a Post-it note with hand-written text on a giant map of the
Santa Clara University Campus. Many other Post-it notes have already been
placed all over the map.

Figure 3. An Example of a ‘Process Artifact’

###  3.2. Outcomes and Future Speculations

As a process and action, the practice of social justice using AR seeks to
address a complex social challenge or issue. While there are no easy solutions
to systemic issues, activists might use AR to achieve justice-oriented
outcomes. These outcomes can be characterized by goals of transformation,
recognition, reciprocity, enablement, distribution, and accountability
(Dombrowski et al., 2016). Practitioners’ use of AR along the trajectories of
these goals do this work through both digital media and non-digital social
justice practices. The effort is heterotopic, as are the outcomes, which can
exist in both physical and lived spaces as well as those of media
representation, AR or otherwise (Schranz, 2014).

For example, the Movers and Shakers (Kinfolk, 2017) produced and developed The
Monument App (2021) to place AR statues of important women, people of color,
and LGBTQ+ icons in places of prominence around New York City (Fig. 1).
Additionally, they released a curriculum to be used with public schools in the
city along with the mobile app. The work’s outcome provides reciprocity
(recompense for those that deserve justice); and enablement (facilitating the
growth of people to develop their own capacities (Hidalgo, 2015)) in its
effort to decolonize the history of the city.

Advocates using AR for social justice may also understand their outcomes
through the lens of spatial justice, or “the fair and equitable distribution
in the space of socially valued resources and the opportunities to utilize
them” (Soja, 2013), particularly for supporting liberatory narratives and
experiences. Outside of AR, digital technologies like games have been used for
liberatory speculative design, mediating and generating new realities (Coulton
et al., 2016; Tynes et al., 2023; Coopilton, 2022). Marginalized communities
could similarly use AR to alter the physical world with a new layer of reality
and imagine liberatory places and futures, developing visions and strategies
of how they can manifest and shape them. Communities could use AR to explore
answers to questions like “what could this building become if we repurposed it
to empower our community as technology creators,” or “how would we build and
sustain a utopia where we are free from oppression?”

We invite workshop participants to speculate on the future outcomes of AR for
spatial justice, and the actualization of digital efforts into tangible
outcomes, physical or otherwise. As part of this process, scholars and
practitioners should consider: accessibility, i.e., how to achieve outcomes
when augmentations may not be cohesive or shared across community populations
(Manfredini, 2023); sustainability, as activists need to grapple with
corporate infrastructures that support ubiquitous AR (Manfredini, 2019); and
community engagement regarding who is involved in these AR experiences and how
they engage with them (Barba and MacIntyre, 2011; Oh et al., 2016).

##  4\. Organizers

The organizers have a diverse range of experience in creating and researching
AR projects for social justice. Their combined expertise includes scholarship
in different academic disciplines, work at industry leaders in AR, membership
in and partnerships with historically marginalized communities, and editorship
of a book on Augmented and Mixed Reality for Communities (Fisher, 2021). They
also have a track record of running successful workshops (e.g., Digital
Wellbeing at CHI 2019 (Cecchinato et al., 2019) and Dark Patterns at CHI 2021
(Lukoff et al., 2021)). Most of the organizers are early career researchers
and all are eager to build community around this emerging area of research and
practice.

Rafael M. L. Silva (corresponding organizer) is a PhD candidate in Human
Centered Design and Engineering at the University of Washington. His research
explores the design, use, and social contextualization of emerging
technologies (i.e., IoT and Immersive Media). He is particularly interested in
understanding how technology for civic engagement can be leveraged by people
in their everyday lives in order to build more just futures.

Ana María Cárdenas Gasca is a PhD student of the Expressive Computation Lab at
the University of California Santa Barbara. Her research focuses on
developing, informing, and studying technologies for spatial storytelling in
the context of memorialization and documentation of Human Rights violations.
She has collaborated with the Museo de la Memoria in Colombia, exploring AR
applications for presenting victims’ testimonials.

Joshua A. Fisher is an assistant professor in the Center for Emerging Media
Design and Development at Ball State University. His research focuses on
utilizing emerging media for and with communities for interactive storytelling
and expression. He is the XR Chair on the board for the Association for
Research in Digital Interactive Narratives.

Erica Principe Cruz is a PhD student of the Center for Transformational Play
and Human-Computer Interaction Institute at Carnegie Mellon University. Her
research explores how counterspaces that center the joy, rest, and healing of
marginalized people can be co-created across realities. She is synthesizing
strategies for cultivating counterspaces as games and AR/VR/XR experiences.

Cinthya Jauregui is a masters student in Engineering Management and Leadership
at Santa Clara University. Her work focuses on Human Computer Interaction. She
is the Project Lead for the Thámien Ohlone AR Tour, where she facilitates co-
designing with Muwekma Ohlone tribal leaders, humanities scholars, and
computer science researchers to tell the story of Muwekma Ohlone past,
present, and future through emerging AR technologies.

Amy J Lueck is an associate professor of rhetoric and composition at Santa
Clara University, where her research and teaching focus on histories of
rhetorical instruction and practice, feminist historiography, cultural
rhetorics, and rhetorical memory studies. Since 2018 she has been
collaborating with Muwekma Ohlone and Ohlone tribal members on public-facing
projects that use digital media to unsettle the patterns of Indigenous erasure
that her research documents and to help sponsor the diverse cultural rhetorics
practices of Ohlone youth.

Fannie Liu is a VP Applied Research Lead on the Global Tech Applied Research
AR/VR team at JPMorgan Chase & Co. Her research involves the design of novel
social experiences that leverage immersive technologies to promote
communication and well-being. Previously, she was a Research Scientist at
Snap, where she was the PI for research on the use of AR for activism.

Andrés Monroy-Hernández is an assistant professor in Princeton’s Department of
Computer Science and an associated faculty in Princeton’s Center for
Information Technology Policy. His research focuses on social computing,
leveraging technologies such as AR and others. Previously, he led a research
team at Snap focused on social AR.

Kai Lukoff is an assistant professor of Computer Science and Engineering at
Santa Clara University. He is part of an interdisciplinary team of Muwekma
Ohlone tribal leaders, humanities scholars, and computer science researchers
who are creating an AR walking tour of the Native American history of Mission
Santa Clara. He is developing a toolkit of AR resources that empower educators
and storytellers around the world to develop ‘counter-tours’ that challenge
hegemonic narratives of cultural heritage sites.

##  5\. Publication of Workshop Proceedings

Accepted workshop papers will be published in arXiv arXiv, an open-access,
online repository of research papers. This follows the recommendation to
publish position papers in the field of computer science (Hertzmann, 2023) and
the practice of previous workshop proceedings published at the CHI Conference
that have used report numbers (Fuentes et al., 2019; Tigwell et al., 2019).
This will ensure a stable archive that makes it easy for scholars to find and
reference the workshop papers in the future.

##  6\. Workshop Activities

The core activities of this workshop are interactive sessions in small- and
medium-sized groups (Table 1). The workshop will begin with a round of
lightning introductions. In Session 1 (Artifact discussions), each participant
will have a chance to share and discuss their artifact in a small group. In
Session 2 (Working across boundaries), medium-sized groups will discuss: (a)
Interdisciplinary collaboration; (b) Partnering with community groups; (c)
Navigating site permissions; and (d) Additional topics suggested by
participants. In Session 3 (Accessibility, sustainability, and impact),
medium-sized groups will discuss (a) Democratizing creator tools; (b) Scaling
and maintaining impact; (c) Designing an artifact database; and (d) Additional
topics suggested by participants. Instead of one-way presentations, workshop
activities are intended to facilitate active dialogue between participants.

##  7\. Hybrid Format and Asynchronous Engagement

This workshop will be held as a half-day hybrid workshop with 25-35
participants. This format was selected in order to enable meaningful
participation from a wide audience, including those who are unable to travel
to Hawaii or have chosen not to travel due to environmental or social justice
concerns. Organizers are committed to creating a rewarding experience with
interaction between all participants.

There will be a shared online document with a section for each group in each
workshop session. This section will include: group name, participants, links
to relevant materials (e.g., artifacts), three initial discussion questions
proposed by the group host, and space for note-taking. These collaborative
notes serve as: (1) a record of the workshop discussions; and (2) asynchronous
materials in case technical or accessibility issues prohibit synchronous
engagement.

Participants will be assigned to groups in advance of the workshop based on
their preferences in a pre-workshop survey. This will enable the organizers to
accommodate preferences, manage group size, and arrange the physical
conference room accordingly. For small groups (about 4-participants), we will
assign groups to different corners of the room and request they use one laptop
per group to facilitate discussion between in-person and virtual participants.
For medium groups (about 8-participants), we plan to set up four tables in the
room, each with a single laptop connected to a wide angle webcam and an
external monitor. Virtual participants will be shown on the external monitor.
In-person participants will sit in a semi-circle so that they can be seen by
the wide angle webcam and view the monitor.

A pre-workshop survey will ask participants about accessibility requirements
(e.g., transcription). Organizers will plan the workshop around these
requirements and request support from the CHI Workshop Chairs as necessary.

Time |  Activity |  Description
---|---|---
9:00 |  Welcome |  Organizers introduce themselves and the agenda
9:15 |  Lightning introductions |  Participants introduce themselves and share one slide with an example of an artifact from a social justice AR experience that connects to their work (one-minute per participant)
9:45 |  Session 1A: Artifact discussions |  Breakout rooms with four participants. One participant serves as a host and briefly presents their artifact (about 3-minutes) for discussion (about 12-minutes). After sessions 1A-1D, each participant will have served as the host for one breakout room.
10:00 |  Short break |
10:10 |  Session 1B: Artifact discussions |
10:25 |  Session 1C: Artifact discussions |
10:40 |  Session 1D: Artifact discussions |
10:55 |  Coffee break |
11:15 |  Session 2: Working across boundaries |  Group discussions (about 8 participants)
11:45 |  Group sharing |  Rapporteur from each group shares Session 2 highlights
11:55 |  Short break |
12:05 |  Session 3: Accessibility, sustainability, and impact |  Group discussions (about 8 participants)
12:35 |  Group sharing |  Rapporteur from each group shares Session 3 highlights
12:50 |  Next steps |  Conclusion and sign up participants to contribute to post-workshop plans
12:55 |  End of formal activities |
13:00 |  In-person social event |  Optional lunch
|  Virtual social event |  Optional online hangout
Table 1. Draft schedule for the workshop

##  8\. Post-Workshop Plans

The primary outcome of this workshop will be an interactive collection of
site-specific social justice AR projects. Each entry will feature speculative
artifact(s), process artifact(s), and/or final artifacts. At present, a major
challenge for designers is the lack of awareness of past work in site-specific
AR. Such experiences are extremely difficult to access because (1) they
require the user to be in a specific location; and (2) the technology behind
AR is rapidly changing, so a platform or device that is supported today may
not be supported tomorrow.

The starting point for this collection will be the annotated artifacts that
workshop authors are required to include as a part of their submission. At the
workshop, participants will be encouraged to identify further projects and
artifacts to include, as well as relevant tags to explore this collection.
After the workshop, the workshop organizers and interested participants will
create a public website that highlights these projects. Users will be able to
explore projects on a global map and via category tags such as technology
(e.g., webAR, iOS app, Android app), site information (e.g, museum, monument,
neighborhood), and identity (e.g., Black, Indigenous, women), and project
metadata (e.g., Authors, Venue, Social Media). New entries will be accepted
via an online form. This online collection is inspired by the Locomotion Vault
(Di Luca et al., 2021) and Haptipedia (Seifi et al., 2019, 2020), community-
sourced interactive databases that have resulted in CHI publications and serve
as a resource for the wider community.

To disseminate the results of the workshop, the organizers will also publish a
blog post of highlights on the UX Collective (uxdesign.cc), a Medium
publication with over 400,000 followers, mostly design practitioners. One of
the organizers on this proposal did this for a previous workshop at CHI 2019
and it has served as a popular summary of the workshop discussion (Lukoff and
Munson, 2019). A similar blog post will share a summary of our discussions
about AR for social justice.

##  9\. Call for Participation

This half-day hybrid workshop will bring together an interdisciplinary group
of researchers to discuss site-specific augmented reality (AR) for social
justice. AR offers unique affordances, such as a strong tie to the physical
environment and an ability to circumvent certain permissions when needed, that
align with many forms of activism. Yet open questions remain about why, when,
and how activists should leverage AR for social justice.

Applicants should submit a 1-6 page position paper (excluding references) as a
single-column manuscript using the ACM Conference Proceedings Primary Article
Template. Submissions should start by presenting one specific artifact. This
could be a speculative artifact, such as a sketch or diagram of an imagined AR
experience. It could be a process artifact, such as a labeled site map or
early prototype. Or it might be a final artifact, such as a soundscape or a
video clip of a completed experience. Authors may select and discuss an
artifact created by someone else, it does not have to be their own work.

The remainder of the position paper should discuss the artifact as it relates
to the themes of the workshop and prior work. The artifacts will not only
guide our discussions at the workshop but also form the first entries in an
anticipated interactive database of social justice AR resources that will be
created as an outcome of the workshop.

Submissions will be reviewed by the workshop organizers based on quality,
relevance, and diversity. Accepted papers will be published as workshop
proceedings on arXiv. At least one author of each accepted submission must
attend the workshop and all participants must register for both the workshop
and at least one day of the conference.

  * •

Website: https://ar4socialjustice.org

  * •

Date: Sunday, 12 May, 2024

  * •

Time: 9am-1pm HST (Hawaii Standard Time) (tbc)

  * •

Format: Hybrid (in-person and virtual)

  * •

Submit applications to: http://tinyurl.com/ar4socialjustice

  * •

Submission deadline: 1 March, 2024

  * •

Acceptance notification: 22 March, 2024

###### Acknowledgements.

The opinions, findings, and conclusions or recommendations expressed in this
material are those of the author(s) and do not necessarily reflect the views
of JPMorgan Chase & Co. or its affiliates. This paper was prepared for
informational purposes with contributions from the Global Technology Applied
Research center of JPMorgan Chase & Co. This paper is not a product of the
Research Department of JPMorgan Chase & Co. or its affiliates. Neither
JPMorgan Chase & Co. nor any of its affiliates makes any explicit or implied
representation or warranty and none of them accept any liability in connection
with this paper, including, without limitation, with respect to the
completeness, accuracy, or reliability of the information contained herein and
the potential legal, compliance, tax, or accounting effects thereof. This
research was funded in part by the NSF ER2-Ethical and Responsible Research
Program (Award: 2026286).

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