# Harnessing quantum emitter rings for efficient energy transport and trapping

R. Holzinger  Institute for Theoretical Physics, University of Innsbruck,
Technikerstrasse 21a, A-6020 Innsbruck, Austria  J. S. Peter  Department of
Physics, Harvard University, Cambridge, Massachusetts 02138, USA  Biophysics
Program, Harvard University, Boston, Massachusetts 02115, USA  S. Ostermann
Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
H. Ritsch  Institute for Theoretical Physics, University of Innsbruck,
Technikerstrasse 21a, A-6020 Innsbruck, Austria  S. F. Yelin  Department of
Physics, Harvard University, Cambridge, Massachusetts 02138, USA

(August 28, 2024)

###### Abstract

Efficient transport and harvesting of excitation energy under low light
conditions is an important process in nature and quantum technologies alike.
Here we formulate a quantum optics perspective to excitation energy transport
in configurations of two-level quantum emitters with a particular emphasis on
efficiency and robustness against disorder. We study a periodic geometry of
emitter rings with subwavelength spacing, where collective electronic states
emerge due to near-field dipole-dipole interactions. The system gives rise to
collective subradiant states that are particularly suited to excitation
transport and are protected from energy disorder and radiative decoherence.
Comparing ring geometries with other configurations shows that that the former
are more efficient in absorbing, transporting, and trapping incident light.
Because our findings are agnostic as to the specific choice of quantum
emitters, they indicate general design principles for quantum technologies
with superior photon transport properties and may elucidate potential
mechanisms resulting in the highly efficient energy transport efficiencies in
natural light-harvesting systems.

In quantum optics, ordered quantum emitter lattices with subwavelength spacing
have emerged as a resourceful platform for near-term quantum technologies [1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Here long-range interactions between
light-induced dipoles lead to highly modified optical properties of the
quantum emitter ensemble, including Dicke superradiance [14] and the emergence
of collective long-lived subradiant states [15, 16]. Applications range from
single photon switch gates [17] to enhanced single photon detection for
biomedical applications [18, 19] and topological edge state lasing [20, 21].
Likewise, uncovering design principles underlying biological systems and
applying this understanding to synthetic systems is crucial for near-term
quantum technologies. Ring geometries of quantum emitters promise to enhance
single photon sensing, transport, storage, and light generation in engineered
nanoscale systems [22, 16, 23, 24]. In photosynthetic energy transfer, as it
occurs in nature, organisms utilize ring-shaped antennae that increase the
photon scattering cross-section of a single reaction center: the site where
photosynthesis takes place. This transfer process occurs at near unit
efficiency, and understanding the mechanisms behind this remarkable feat is an
outstanding scientific challenge [25, 26, 27, 25, 28, 29, 30, 31, 32, 33, 34,
35].

Figure 1: Lattices of nanoscopic quantum emitter rings. (a) Each ring is
composed of two-level quantum emitters with resonance frequency $\omega_{0}$
and separation $d<\lambda_{0}$, where $\lambda_{0}=\omega_{0}/c$ is the
wavelength of light. The excited state $|e\rangle$ spontaneously decays with
rate $\Gamma_{0}$ to the ground state $|g\rangle$ and the emitters are coupled
via long-range dipole-dipole interactions with nearest-neighbor coupling
strength $J$. Emitters acting as donor and acceptor are shown in yellow and
the acceptor features an additional trapping state to which excitations
irreversibly decay with rate $\Gamma_{\mathrm{T}}$. (b) More detailed sketch
illustrating the inter-ring separation $d_{R}$. The ring radius $R$ and the
emitter spacing $d$ are related via $d=2R\sin(\pi/N_{\mathrm{R}})$, with
$N_{R}$ emitters per ring. (c) Excitation transport efficiency according to
Eq. 1 for a chain of 10 rings and various $N_{R}$. Parameters:
$d/\lambda_{0}=0.05$, $d_{R}/d=0.9$, $\Gamma_{\mathrm{T}}/\Gamma_{0}=2$ and
$\Delta=0$.

Taking inspiration from biological systems, we examine the long-range
excitation transport between a donor and an acceptor emitter through a lattice
of quantum emitter rings [Fig. 1(a)]. As a main result, we show that efficient
excitation transport at low trapping rates preferentially occurs for ring
geometries, as compared to other lattices. This property has important
consequences for devising artificial light harvesting and transport systems,
and may be relevant for understanding the excellent excitation transport
capabilities of biological systems [36]. We also highlight that for ring
lattices, the trapping of light at an acceptor site under low-light conditions
is enhanced by many orders of magnitude as compared to other geometries and to
independent emitters. By choosing an optimal detuning for the donor and
acceptor with respect to the lattice, radiative losses are strongly
suppressed, and excitations are protected during the transport by subradiance
[16, 15]. While the influence of the excitation trapping rate on the transport
efficiency in other geometries has been explored in other works [37, 38, 39,
40], as have certain design principles for bio-inspired artificial solar-
harvesting devices [27, 25, 28, 41, 42, 43, 44, 45], our findings specifically
highlight the advantages of the rotationally symmetric ring geometry. This
special feature of ring configurations is particularly intriguing for its
close connection to natural photosynthetic complexes found in biological
systems. Our work therefore opens the possibility of exploiting quantum
effects in bio-inspired configurations of quantum emitters for near-term
optical technologies that enable quantum-enhanced light-matter coupling on the
nanoscale.

Figure 2: Band structure, edge states, and excitation transport for a chain of
rings. (a) Eigenmodes of the effective Hamiltonian
$\hat{\mathcal{H}}_{\mathrm{lattice}}$ in Eq. (2a) can be cast into $N_{R}=9$
energy bands with an angular momentum projection $|m|$ using Eq. (9) and
translation along the ring chain axis is given by $\tilde{d}=2R+d_{R}$, the
spacing between adjacent ring centers. (Decay rates of all eigenmodes are
color coded). A band gap emerges with two edge states residing inside with an
energy separation $\Delta_{\mathrm{gap}}^{(0,1)}$ to the nearest lower/upper
band edge respectively. (b) The minimal energy gap
$\Delta_{\mathrm{gap}}^{(0,1)}$ to the nearest lower/upper band edge
normalized by the maximal nearest-neighbor coupling $J$ shows distinct maxima
as a function of $d_{R}$. These maxima correspond to an optimal transport
efficiency $\eta_{t}$ as shown in (e) for $N_{R}=8,9$. Furthermore, smaller
emitter numbers per ring $N_{R}$ are effected more by randomly rotated rings
except for even $N_{R}$. (The average was taken over 50 random realizations).
(c) For $N_{R}=9$ edge states appear with maximum amplitude on the left/right
end of the ring chain and a superradiant decay rate for decreasing inter-ring
spacings $d_{R}/d$. Notably, topological edge states have been observed in
zigzag chains of gold nano-rings [20, 21] with lasing from the edge rings. (d)
With decreasing inter-ring spacing $d_{R}$ edge states become more pronounced
with distinct minima where the bulk amplitude vanishes leading to a
suppression of excitation transport. (e) Excitation transport between a donor
and acceptor site paced in the center of the edge rings. The transport
efficiency $\eta_{t}$ is evaluated after a time $t\Gamma_{0}=150$ with the
donor/acceptor detuning $\Delta$ optimized for maximum transport. Suppression
appears when the edge states become too pronounced with vanishing amplitude in
the bulk rings as shown in (d). (f) Time dynamics of the excitation transport
process between a donor and acceptor site with $\Delta=0$. The eigenstate
fidelity $|\langle\Psi(t)|\Psi_{\mathrm{eig}}\rangle|^{2}$ for $N_{R}=9$
demonstrates the importance of edge states at early times. Parameters:
$d=0.05\lambda_{0}$, $\Gamma_{\mathrm{T}}/\Gamma_{0}=1$ in (e)-(f) and 10
rings in (b)-(f).

As a paradigmatic quantum optical model to simulate excitation energy
transport, we consider a one-dimensional lattice of $M$ rotationally symmetric
rings, each composed of $N_{R}$ identical two-level emitters with a ground
state $|g\rangle$ and an excited state $|e\rangle$. The two states are
connected via the transition operator $\hat{\sigma}_{n}=|g_{n}\rangle\langle
e_{n}|$ for the $n$-th emitter. Additional emitters acting as donor and
acceptor sites are placed in the center of two rings at either end of the
lattice, as illustrated in Fig. 1(a). The transport efficiency between these
two sites is the core quantity of interest and is defined as

$\displaystyle\eta_{t}=\Gamma_{\mathrm{T}}\int_{0}^{t}dt^{\prime}\langle\Psi(t^{\prime})|\hat{\sigma}^{\dagger}_{\mathrm{a}}\hat{\sigma}_{\mathrm{a}}|\Psi(t^{\prime})\rangle.$
(1)

Here $t$ is the integration time over which excitation can accumulate in the
trap state [see Fig. 1(a)]. $\eta_{t}$ can take values between 0 and 1, where
0 corresponds to no transport at all and 1 identifies maximal transport
efficiency. $\hat{\sigma}^{\dagger}_{a}\hat{\sigma}_{a}=|e_{a}\rangle\langle
e_{a}|$ corresponds to the projector onto the excited state of the acceptor.
The trap population accumulates over time and reaches a steady state value at
large times $t$ when the total excited state population is either dissipated
via radiative losses or accumulated in the trap. The transition frequencies
and decay rates of the ring emitters are assumed to be equal and given by
$\omega_{0}=2\pi c/\lambda_{0}$ and $\Gamma_{0}$, respectively, whereas the
donor/acceptor transitions may be detuned by
$\Delta=\omega_{\mathrm{d,a}}-\omega_{0}$ with respect to the ring emitter
frequencies. We assume $\omega_{\mathrm{d}}=\omega_{\mathrm{a}}$ for the
remainder of this work. The acceptor features an extra trapping channel
through which excitations are extracted from the system at a rate
$\Gamma_{\mathrm{T}}$. Furthermore, the quantum emitters are confined in the
$x$-$y$ plane with intra-ring separation $d=2R\sin(\pi/N_{R})$ and inter-ring
separation $d_{R}$, where $R$ is the ring radius, as illustrated in Fig. 1(b).
To reduce the number of free paramters, all dipole emitters are assumed to be
circular polarized, namely $(1,i,0)^{T}/\sqrt{2}$. However, qualitatively
similar results can be obtained for geometries consisting of linear polarized
emitters.
We model the system within the Born-Markov approximation [15], and only
consider the quantum emitter’s internal degrees of freedom. Furthermore, we
assume the weak excitation regime, where at most a single excitation is
present in the system (see Methods), and therefore the system can be described
(in the rotating frame with $\omega_{0}$) by the non-Hermitian Hamiltonian
$\hat{\mathcal{H}}_{\mathrm{eff}}=\hat{\mathcal{H}}_{\mathrm{ad}}+\hat{\mathcal{H}}_{\mathrm{lattice}}+\hat{\mathcal{H}}_{\mathrm{int}}$.
Here
$\hat{\mathcal{H}}_{\mathrm{ad}}=(\Delta-\frac{i}{2}\Gamma_{0})(\hat{\sigma}^{\dagger}_{\mathrm{a}}\hat{\sigma}_{\mathrm{a}}+\hat{\sigma}^{\dagger}_{\mathrm{d}}\hat{\sigma}_{\mathrm{d}})-\frac{i}{2}\Gamma_{\mathrm{T}}\hat{\sigma}^{\dagger}_{\mathrm{a}}\hat{\sigma}_{\mathrm{a}}$
is the bare Hamiltonian of the donor and acceptor,
$\hat{\mathcal{H}}_{\mathrm{lattice}}$ describes the emitters in the ring
lattice, and $\hat{\mathcal{H}}_{\mathrm{int}}$ describes the interaction
between the ring emitters and the donor/acceptor,

$\displaystyle\hat{\mathcal{H}}_{\mathrm{lattice}}$
$\displaystyle=\sum_{n,m}\Big{(}J_{nm}-i\frac{\Gamma_{nm}}{2}\Big{)}\hat{\sigma}^{\dagger}_{n}\hat{\sigma}_{m},$
(2a) $\displaystyle\hat{\mathcal{H}}_{\mathrm{int}}$
$\displaystyle=\sum_{n;k=\mathrm{a},\mathrm{d}}\Big{(}J_{nk}-i\frac{\Gamma_{nk}}{2}\Big{)}(\hat{\sigma}^{\dagger}_{n}\hat{\sigma}_{k}+\hat{\sigma}^{\dagger}_{k}\hat{\sigma}_{n}).$
(2b)

All emitters interact via vacuum-mediated dipole-dipole interactions in free
space. The pairwise coherent and dissipative interactions are given by
$J_{nm}=-3\pi\Gamma_{0}/k_{0}\operatorname{Re}(G_{nm})$ and
$\Gamma_{nm}=6\pi\Gamma_{0}/k_{0}\operatorname{Im}(G_{nm})$ respectively, with
$G_{nm}$ being the free space Green’s function (see Methods). The Green’s
function depends only on the separations between the emitters and their dipole
orientation. The time evolution of the system is described by the effective
Hamiltonian in Eq. (2) via the Schrödinger equation
$i\partial_{t}|\Psi(t)\rangle=\hat{\mathcal{H}}_{\mathrm{eff}}|\Psi(t)\rangle$.
Since the Hamiltonian is non-Hermitian, the amplitude of the wavefunction for
the quantum emitters decreases with time, which is a direct manifestation of
the dissipative nature of the system.

As discussed in previous works [22, 15, 16], a subwavelength-spaced ring of
quantum emitters exhibits guided eigenmodes that are extremely subradiant,
exhibiting an exponentially increasing lifetime,
$\tau\Gamma_{0}\sim\mathrm{exp}(N_{R})$, of a single excitation [22]. Aside
from the bright symmetric superposition state, the fields of the remaining
eigenmodes vanish at the center of the ring due to symmetry. Thus, they are
decoupled from any emitter at the center. Here we demonstrate that a
donor/acceptor at the center of the ring that is dipole-dipole coupled to the
symmetric ring mode can form a subradiant state with a majority of the
excitation concentrated in the donor/acceptor. We start by analyzing a single
ring of $N_{R}$ emitters with a single donor in the center. For a single ring,
where the dipole orientations preserve the discrete rotational invariance, the
collective eigenmodes of the effective Hamiltonian are spin waves of the form
$|\Psi_{m}\rangle=\hat{S}_{m}^{\dagger}|G\rangle$, where

$\displaystyle\hat{S}_{m}=\frac{1}{\sqrt{N_{R}}}\sum_{j=1}^{N_{R}}e^{im\varphi_{j}}\hat{\sigma}_{j}$
(3)

and $|G\rangle$ denotes all emitters in the ground state. Here
$\varphi_{j}=2\pi j/N_{R}$ is the angle between neighboring emitters along the
ring and $m=0,\pm 1,\cdot\cdot\cdot,\lceil\pm(N_{R}-1)/2\rceil$ is the angular
momentum of the collective mode. The associated energy shifts and decay rates
of these spins waves are given by
$\tilde{J}_{m}=\sum_{j}e^{im\varphi_{j}}J_{1j}$ and
$\tilde{\Gamma}_{m}=\sum_{j}e^{im\varphi_{j}}\Gamma_{1j}$, respectively. In
such a configuration, all ring emitters couple equally to the central donor,
which restricts the spectrum to the $m=0$ mode. This system features two
eigenstates $|\Psi_{\pm}\rangle$, that are symmetric/anti-symmetric
superpositions of the symmetric ring mode and the central donor. The anti-
symmetric state can be extremely subradiant depending on the detuning $\Delta$
of the donor with respect to the ring emitters, resulting in a vanishingly
small net dipole strength [24]. This leads to an optimal detuning
$\Delta_{\mathrm{sub}}\approx
J_{\mathrm{d}}(\tilde{\Gamma}_{0}-\Gamma_{0})-\tilde{J}_{0}$ that maximizes
the subradiance of the donor with an effective decay rate
$\Gamma_{\mathrm{eff}}/\Gamma_{0}\lesssim 10^{-3}$ (see Methods). Here,
$J_{\mathrm{d}}$ is the coherent coupling between the donor and a ring
emitter.
Likewise, a chain of quantum emitter rings features a rich collective
eigenmode structure. In particular, the subradiance of the eigenmodes protects
the excitations from radiative decoherence and leads to efficient excitation
transport [16, 46]. As shown above, eigenmodes of a rotationally symmetric
ring carry angular momentum $m$. Similarly, the eigenmodes of a linear chain
of quantum emitters carry linear momentum $k$ [15, 5]. This leads to an ansatz
wavefunction for the eigenmodes of a ring chain, $|\Psi_{m,k}\rangle$, with an
angular and linear quasi-momentum pair $(m,k)$, associated eigenenergies
$\omega_{0}+J_{\mathrm{m,k}}$ and decay rates $\Gamma_{\mathrm{m,k}}$ [47]
(details provided in the Methods). The translational distance between adjacent
ring centers along the chain is given by $\tilde{d}=2R+d_{R}$. Fig. 2(a) shows
the energy bands for $N_{R}=9$ and $d_{R}/d=0.9$. The band structure exhibits
a nontrivial topology with a non-zero Zak phase
$\varphi=i\int_{BZ}dk\,\langle\Psi_{m=0,k}|\partial_{k}|\Psi_{m=0,k}\rangle$
[48] as well as gapped edge states between the energy bands of the $m=0$ and
$|m|=1$ eigenmodes. The edge states emerge for decreasing inter-ring spacings
$d_{R}$, illustrated in Fig. 2(c), and become more pronounced until a critical
spacing of $d_{R}/d\approx 0.58$ and $d_{R}/d\approx 0.34$ for $N_{R}=8,9$
emitters per ring, respectively. The edge states are energetically degenerate
and detuned by $\Delta_{\mathrm{gap}}^{(0,1)}$ from the lower/upper band edge
respectively. Fig. 2(b) shows the minimum distance of the edge states to the
nearest band edge as a function of the inter-ring spacing $d_{R}$.
Topologically protected edge states are crucial for resilient excitation
transport in disordered systems [4] and
$\mathrm{min}(\Delta_{\mathrm{gap}}^{(0,1)})$ can serve as a figure of merit
in this regard. Specifically for lattices of rings, the band gap remains
finite in the presence of rotational disorder and exhibits a distinct maximum,
e.g. $d_{R}/d\approx 0.9$ for $N_{R}=9$, as shown in Fig. 2(b). Edge states
also become superradiant at the critical distance where excitation transport
is surpressed, as is shown in Fig. 2(c). This points to the possibility of
edge mode lasing, already observed in gold nano-rings arranged in zigzag
chains [20, 21, 49].
Figs. 2 (e) and (f) demonstrate the fundamental influence of the edge states
on the transport dynamics between a donor and acceptor site for 10 rings with
$N_{R}=9$. At the critical spacings, transport is either completely or
strongly suppressed because the edge states possess no amplitudes in the bulk
rings. Conversely at other spacings $d_{R}$, edge states are crucial during
the early times of the transport process, as demonstrated by the eigenstate
fidelities $\mathcal{F}(t)=|\langle\Psi(t)|\Psi_{\mathrm{eig}}\rangle|^{2}$
for $N_{R}=9$. Qualitatively similar results hold for other $N_{R}$. Indeed,
edge states have been thoroughly studied in dimerized chains ($N_{R}=2$),
which reproduces a long-range generalization of the well-known
Su–Schrieffer–Heeger (SSH) model [21]. A more complete discussion of the
emergence of edge and corner states [49] in two-dimensional ring lattices is
briefly discussed in Methods and warrants further study.

Figure 3: (a)-(c) Excitation transport in quantum emitter rings between a
donor and acceptor site. Scan over the number of emitters per ring $N_{R}$ and
donor/acceptor detuning $\Delta$ for 10 rings with decreasing inter-ring
spacings $d_{R}$ after a time $t\Gamma_{0}=150$. Efficient transport emerges
only with $N_{R}\geq 6$, irrespective of $d_{R}$. The excited state population
in the donor can get trapped in a subradiant state involving the ring
surrounding the donor and follows the detuning $\Delta_{\mathrm{sub}}$ (white
dashed line) derived in the main text. For 9-fold symmetric rings the
donor/acceptor detuning, that optimizes transport, is given by $\Delta\approx
0$ for all inter-ring spacings. Additional parameters are:
$\Gamma_{\mathrm{T}}=2\Gamma_{0}$, $d=0.05\lambda_{0}$. Figure 4: Excitation
transport in ring geometries exhibits superior robustness against disorder.
Comparison of transport efficiency in (a) chain and hexagonal lattices and in
the absence of a lattice (grey line), (b) honeycomb, and (c) ring lattices as
a function of the trapping rate $\Gamma_{\mathrm{T}}$ and frequency disorder.
Lattice emitter frequencies are randomly fluctuating by $\delta\omega$ around
the resonance frequency $\omega_{0}$. The donor-acceptor distance is
approximately the wavelength of light $\lambda_{0}$. Also shown in the dashed-
dotted lines is the case of a donor-acceptor pair separated by $d$ in the
absence of any lattice. Although frequency disorder decreases the long-range
transport capacity, it prevails remarkably well even at large frequency
fluctuations. In particular, the ring lattices exhibit high transport
efficiencies (close to 90$\%$) over a wide range of trapping rates as compared
to the other geometries. At trapping rates much below the magnitude of the
coherent transfer rate $J$, ring-based lattices are superior to any other
lattice in our study. Additional parameters: $d_{R}/d=0.9$,
$d/\lambda_{0}=0.06$, $J/\Gamma_{0}\approx-8.4$, $t\Gamma_{0}=150$. An average
over 25 random realizations with standard deviation $\delta\omega$ was
performed in all plots. Donor/acceptor detunings in (a) $\Delta=0$, (b)
$\Delta=4.5\Gamma_{0}$, and (c) $\Delta=-\Gamma_{0}$.

We now focus on the excitation transport dynamics and discuss the time
evolution of a single initially excited donor. We find that excitation
transport is optimized at particular donor/acceptor detunings, and that
efficient transport occurs only for ring emitter numbers $N_{R}\geq 6$. In
particular, rings with 8-, 9- and 10-fold symmetry seem to be most optimal.
This is particularly intriguing because [50, 51] 8-, 9- and 10-fold rings, the
most abundant type occuring in natural light harvesting antennae [52], show
the highest resilience when rings are randomaly rotated with respect to each
other. In Fig. 3 the donor excited state populations and the trap populations
$\eta_{t}$ are shown after a time $t\Gamma_{0}=150$ for a chain of 10 rings
with various inter-ring spacings $d_{R}$. The detuning $\Delta$ where the
donor excited state population is maximzed (i.e., most subradiant), follows
the optimal detuning $\Delta_{\mathrm{sub}}$ for the single ring case. Here,
the donor excitation largely remains trapped in the subradiant state discussed
above, even for small inter-ring spacings.

Figure 5: Ring lattices exhibit the most efficient energy trapping under weak
light illumination and small trapping rates. A comparison of the steady state
effective trapping rate
$\Gamma_{\mathrm{T}}\langle\hat{\sigma}_{a}^{\dagger}\hat{\sigma}_{a}\rangle_{\mathrm{st}}/(4\Omega_{0}^{2})$
between different lattice geometries under continuous coherent driving with
rate $\Omega_{0}/\Gamma_{0}=10^{-3}$. The drive is modeled by a Gaussian beam
with waist $w$ centered at the acceptor emitter and is on-resonance with the
lattice emitter frequencies $\omega_{0}$ (see Eq. 4). The effective trapping
rate is normalized by the single emitter trapping rate driven on resonance
(see main text). We study a hexagonal and chain lattice in (a) with $20$ and
$8\times 9$ emitters respectively, a honeycomb lattice in (b) with 130 lattice
emitters, a chain of 5 rings in (c), and a $3\times 3$ hexagonal ring lattice
in (d). The donor/acceptor distance is approximately the wavelength of light
$\lambda_{0}$ in all lattices. Strikingly, only the ring-based lattices are
orders of magnitude more efficient at trapping incoming light for trapping
rates below the nearest-neighbor coupling rate, namely when
$\Gamma_{\mathrm{T}}\ll|J|$. Conversely, the honeycomb and hexagonal lattices
are orders of magnitude less efficient in the same regime with the honeycomb
lattice not deviating significantly from the single acceptor case. The linear
chain (grey dashed) and the free space transfer (solid grey) shown in (a) is
more efficient at trapping rates $\Gamma_{\mathrm{T}}>|J|$. Additional
parameters: $d/\lambda_{0}=0.06$, $J/\Gamma_{0}\approx-8.4$ and donor/acceptor
detunings in (a) $\Delta/\Gamma_{0}=-18$ (hexagonal), $\Delta=0$ (chain), (b)
$\Delta/\Gamma_{0}=-20$, (c) $\Delta/\Gamma_{0}=-3.85$, (d)
$\Delta/\Gamma_{0}=-4.63$ with $d_{R}/d=0.9,\ N_{R}=9$, in (c) and (d).

A crucial element of excitation energy transport is robustness against energy
disorder. We provide a comparison of ring lattices with other lattice
geometries, including the influence of static frequency disorder in the
lattice emitters. This is achieved by taking emitter frequencies $\omega_{m}$
from a Gaussian distribution around the unperturbed emitter frequency
$\omega_{0}$ with a standard deviation $\delta\omega$ and adding the term
$\sum_{m}(\omega_{m}-\omega_{0})\hat{\sigma}^{\dagger}_{m}\hat{\sigma}_{m}$ to
the Hamiltonian in Eq. (2). The donor/acceptor detuning $\Delta$ remains
unchanged and is chosen such that unperturbed excitation transport is
maximized. Fig. 4 shows various geometries, many of which also have been
studied previously for atoms trapped in optical lattices [5, 10, 12, 46, 4,
22]. The nearest-neighbor distance is kept at $d=0.06\lambda_{0}$ and the
donor-acceptor distance at $\sim\lambda_{0}$ to establish a uniform comparison
between the different geometries. Figs. 4(a) and 4(b) show a hexagonal lattice
with $\Delta=0$ and a honeycomb lattice with $\Delta=4.5\ \Gamma_{0}$. In Fig.
4 (c) a 1D ring chain and 2D hexagonal ring lattice with $N_{R}=9$ are shown
with $\Delta=\Gamma_{0}$. The fluctuation in the emitter frequencies
$\delta\omega$ is set to $|J|/4$ and $|J|/2$, where $J\approx-8.4\Gamma_{0}$.
Altogether, the different geometries show a similar reduction in the maximal
transport efficiencies under disorder but behave quite differently in the
range of trapping frequencies $\Gamma_{\mathrm{T}}$ where maximal transport
occurs. Whereas the hexagonal lattice exhibits peak transport at a trapping
rate far above the optical decay rate, namely at $\Gamma_{\mathrm{T}}/|J|\sim
2$, the ring lattices demonstrate efficient transport over a large range
$\Gamma_{\mathrm{T}}^{\mathrm{opt}}/|J|\sim 0.01-1$, even in the moderately
disordered case. Qualitatively similar conclusions also apply to ring lattices
with $N_{R}\neq 9$. Just as importantly, the ring lattices in Fig. 4(c) show
significant transport enhancement for $\Gamma_{\mathrm{T}}\ll|J|$ compared to
the independent case where no lattice is present. In summary, the ring
lattices show significantly better transport capability and robustness against
disorder.
So far we have assumed, that a single donor is initially excited, and we have
quantified the transport behavior by calculating the fraction of the
excitation that accumulates in a trap state via Eq. (1) after a waiting time
$t$. However, in many realistic scenarios, a perfectly excited donor is rather
unlikely, and emitters close to the donor will be excited too. This motivates
the study of the trapping rate at which the excitation ends up in the trap
state under continuous coherent illumination in the form of a Gaussian laser
beam with finite beam waist $w$. The continuous coherent drive is modeled by

$\displaystyle\hat{\mathcal{H}}_{\mathrm{laser}}=\Omega_{0}\sum_{i}\exp{\Big{(}-\frac{|\vec{r}_{d}-\vec{r}_{i}|^{2}}{2w^{2}}\Big{)}}(\hat{\sigma}^{\dagger}_{i}+\hat{\sigma}_{i}),$
(4)

where $\Omega_{0}$ is the laser Rabi frequency, $\vec{r}_{d}$ is the position
of the donor with the sum including all emitters. The driving rate of the
laser is kept small ($\Omega_{0}\ll\Gamma_{0}$) to ensure that the system
stays in the single-excitation regime and the model remains valid. As a figure
of merit for energy transport efficiency, we define
$\Gamma_{\mathrm{T}}\langle\hat{\sigma}_{a}^{\dagger}\hat{\sigma}_{a}\rangle_{\mathrm{st}}/(4\Omega_{0}^{2})$,
as the steady-state trapping rate at the acceptor emitter. The effective
trapping rate is normalized by the trapping rate of a single acceptor, given
by
$\sigma_{0}\Gamma_{0}\Gamma_{\mathrm{T}}/(\Gamma_{0}+\Gamma_{\mathrm{T}})^{2}$,
where $\sigma_{0}=6\pi/k_{0}^{2}$ is the single emitter scattering cross-
section [24]. In Fig. 5 a hexagonal lattice (a), a honeycomb lattice (b), and
a hexagonal ring lattice with $N_{R}=9$ (c) are compared for two laser beam
waists under continuous driving. By choosing a beam waist of
$w/\lambda_{0}=0.3$, most of the incoming light is focused around the donor
emitter while the acceptor emitter remains mostly undriven. For
$w/\lambda_{0}=3$ the whole lattice is uniformly driven—a scenario more
applicable to deeply subwavelength lattices under illumination from a non-
directional light source. Natural light-harvesting antennae in purple bacteria
offer an example [25]. In all cases, the donor/acceptor detuning $\Delta$ is
chosen optimally such that the trapping rate is maximized. We find that the
ring lattice is many orders of magnitude more efficient in trapping incident
light as compared to both the triangular and honeycomb lattices as well as the
single emitter at trapping rates much below the nearest-neighbor coherent
transfer rate $J$. In particular for $\Gamma_{\mathrm{T}}/|J|\lesssim 0.01$
the ring lattice exhibits an almost $100\times$ higher trapping efficieny
compared to an independent emitter and the other lattices.

In conclusion, we have demonstrated intriguing optical properties of quantum
emitter ring lattices, including the emergence of topological edge states.
Furthermore, we have shown based on general symmetry principles that ring
lattices form a superior platform for transporting and trapping excitations.
We have also elucidated the guiding principles that govern optimal
donor/acceptor detunings, trapping rates, and geometric arrangements with
robustness against static energy disorder. Under more realistic conditions of
weak coherent light illumination, we have shown that ring lattices are orders
of magnitude more efficient at trapping the absorbed light when the trapping
rate is much smaller than the nearest-neighbor coherent coupling rate. This
result is thought-provoking since natural light-harvesting systems also
operate with trapping mechanisms that are orders of magnitude slower than the
coherent transfer time between neighboring chromophores [53, 54]. Measurements
performed on natural light-harvesting complexes show that the coherent energy
transfer time between neighboring chromophores during photosynthesis is of the
order of $\sim 0.1-10$ ps whereas the trapping time in the reaction center is
typically $\sim 0.1-10$ ns [54, 36, 53]. For a pre-existing trapping
structure, this could provide an explanation why nature utilizes ring
geometries as a moderating mechanism to trap absorbed sun light in reaction
centers. Other studies have focused on molecular emitters in ambient
conditions with vibrational degrees of freedom and multiple decoherence
channels [55, 56]. The impact of these additional effects on the results
presented here are an exciting avenue for future research [57, 58, 59].
Nevertheless, our results suggest that there exist general and platform-
agnostic design principles that govern the efficient transport of excitation
energy at the nano scale. These geometrical considerations may have played a
role in evolutionary design and warrant further study.

Acknowledgments – R. H. acknowledges funding from the Austrian Science Fund
(FWF) doctoral college DK-ALM W1259-N27. S.O. is supported by the Harvard
Quantum Initiaive (HQI). S.F.Y. thanks the AFOSR and the NSF (through the CUA
PFC and QSense QLCI).

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## Methods

## Free space Green’s function

The free-space Green’s function
$\textbf{G}(\textbf{r},\textbf{r'},\omega_{0})$ is the solution to the
electromagnetic wave equation in vacuum, and obeys

$\displaystyle\boldsymbol{\nabla}\times\boldsymbol{\nabla}\times\textbf{G}(\textbf{r},\textbf{r'},\omega_{0})-\frac{\omega_{0}^{2}}{c^{2}}\textbf{G}(\textbf{r},\textbf{r'},\omega_{0})=\delta(\textbf{r}-\textbf{r'})\mathbb{1}_{3},$
(5)

where $\mathbb{1}_{3}$ is the identity matrix in Cartesian coordintes. The
Green’s function describes the field at point r due to a normalized,
oscillating dipole at r’. The emitters are assumed to be fixed in space,
allowing a treatment of the emitters as quantum point dipoles instead of
dynamical variables. Writing
$\textbf{G}(\textbf{r}_{n},\textbf{r}_{m},\omega_{0})=\textbf{G}(\textbf{r}_{nm},\omega_{0})$,
the Green’s function is given by

$\displaystyle\textbf{G}(\textbf{r}_{nm},\omega_{0})=\frac{e^{ik_{0}r_{nm}}}{4\pi
k_{0}^{2}r^{3}_{nm}}\Big{[}(k_{0}^{2}r_{nm}^{2}+ik_{0}r_{nm}-1)\mathbb{1}_{3}-(k_{0}^{2}r^{2}_{nm}+3ik_{0}r_{nm}-3)\frac{\boldsymbol{r}_{nm}\otimes\boldsymbol{r}_{nm}}{r_{nm}^{2}}\Big{]},$
(6)

where $r_{nm}=\lvert\mathbf{r}_{nm}\rvert$ and
$k_{0}=2\pi/\lambda_{0}=\omega_{0}/c$ is the wave number corresponding to the
atomic transition energy, and $c$ is the speed of light in free-space. The
rates for coherent and dissipative interactions between emitter $n$ and $m$
are given respectively by

$\displaystyle J_{nm}$
$\displaystyle=-\frac{3\pi\Gamma_{0}}{k_{0}}\operatorname{Re}(G_{nm})$ (7)
$\displaystyle\Gamma_{nm}$
$\displaystyle=\frac{6\pi\Gamma_{0}}{k_{0}}\operatorname{Im}(G_{nm}),$ (8)

where we have defined
$G_{nm}=\boldsymbol{p}^{*}_{n}\cdot\mathbf{G}(\mathbf{r}_{nm},\omega_{0})\cdot\boldsymbol{p}_{m}$
and $\boldsymbol{p}_{n}$ is the transition dipole matrix element vector for
atom $n$, which we set to $\boldsymbol{p}_{n}=(1,i,0)^{T}/\sqrt{2}$ (circular
polarization) througout the main text. For a single emitter, evaluating Eq. 8
reproduces the vacuum emission rate $\Gamma_{nn}=\Gamma_{0}$ , where
$\Gamma_{0}=\omega_{0}^{3}|\boldsymbol{\mu}|^{2}/3\pi\hbar c^{3}$, and
$|\boldsymbol{\mu}|$ is the dipole moment strength. The single-emitter energy
shift $J_{nn}$ in Eq. 7 formally yields a divergence and is set to zero, as it
would simply renormalize the resonance frequency $\omega_{0}$.

## Wavefunction ansatz for a ring chain

Here we show the wavefunction ansatz that is used to create the energy band
structure. As shown in the main text, one can assign an angular momentum
number $m$ to each eigenmode of a rotationally symmetric ring. Similarly, the
eigenmodes of a linear 1D chain of quantum emitters carry linear momentum $k$
[15]. This suggests, that the eigenmodes of a chain of rings can be understood
in terms of wavepackets with a combination of angular and linear momentum
$(m,k)$, which thus serves as a label of the eigenmodes [47]. The ansatz is
given by $|\Psi_{m,k}\rangle=\hat{S}_{m,k}^{\dagger}|g\rangle$ with

$\displaystyle\hat{S}_{m,k}=\frac{1}{\sqrt{N}}\sum_{j=1}^{N_{R}}\sum_{j^{\prime}=0}^{M-1}e^{im\varphi_{j}}e^{ik\tilde{d}j^{\prime}}\hat{\sigma}_{j+j^{\prime}N_{R}},$
(9)

where $N=M\times N_{R}$, $M$ is the number of rings,
$\varphi_{j}=2\pi(j-1)/N_{R}$, and $\tilde{d}=2R+d_{R}$ is the distance
between neighboring ring centers. To order the eigenmodes into energy bands
with linear momentum $k$ and angular momentum $m$, we calculate the overlap
fidelities of the exact numerical eigenstates $|\Psi\rangle$ of
$\hat{\mathcal{H}}_{\mathrm{lattice}}$ with the ansatz in Eq. 9, namely
$|\langle\Psi_{m,k}|\Psi\rangle|^{2}$. The fidelities exhibit peaks centered
around a particular linear momentum $k$ and angular momentum projection $|m|$
the numerical eigenmodes can then be ordered into energy bands. The ansatz
only allows us to distinguish energy bands with angular momentum projection
$|m|$ because the ansatz cannot discriminate between $\pm m$ except for $m=0$,
which is unique. Alternatively, one can obtain the energy bands for an
infinite chain of rings by block diagonalizing
$\hat{\mathcal{H}}_{\mathrm{lattice}}$, taking the individual rings as unit
cells and invoking Bloch’s theorem [4, 60]. However, here the bands cannot be
split into their angular momentum projection, since the individual rings are
numerically diagonalized.

## Ring with a central emitter

Subwavelength rotationally symmetric rings of emitters feature eigenmodes that
are extremely subradiant, but whose fields vanish at the center of the ring
due to rotational symmetry. They are thus decoupled from any emitter at the
center. The center only couples to the symmetric ring mode and the effective
Hamiltonian can be rewritten as

$\displaystyle\hat{\mathcal{H}}_{\mathrm{eff}}=\Big{(}\Delta-\frac{i\Gamma_{0}}{2}\Big{)}\hat{\sigma}_{\mathrm{d}}^{\dagger}\hat{\sigma}_{\mathrm{d}}+\Big{(}\tilde{J}_{0}-\frac{i\tilde{\Gamma}_{0}}{2}\Big{)}\hat{S}_{0}^{\dagger}\hat{S}_{0}+\sqrt{N_{R}}\Big{(}J_{d}-\frac{i\Gamma_{d}}{2}\Big{)}(\hat{S}_{0}^{\dagger}\hat{\sigma}_{\mathrm{d}}+\hat{S}_{0}\hat{\sigma}_{\mathrm{d}}^{\dagger}),$
(10)

where $\tilde{J}_{0}$ and $\tilde{\Gamma}_{0}$ are the collective couplings of
the symmetric ring mode, ${J}_{d}$ and ${\Gamma}_{d}$ are the couplings
between the central donor and a ring emitter and $\Gamma_{0}$ is the
spontaneous decay rate of a single emitter. The field created by the symmetric
ring mode at the donor position is the same as that of a single dipole with
dipole moment strength enhanced by a factor of $\sqrt{N_{R}}$. For the single
excitation manifold, one finds the eigenmodes by diagonalizing the $2\times 2$
matrix resulting from projecting Eq. 10 into the subspace spanned by the basis
$\\{|\Psi_{0}\rangle,|d\rangle\\}$, with
$|\Psi_{0}\rangle=S_{0}^{\dagger}|G\rangle$ and
$|d\rangle=\hat{\sigma}_{d}^{\dagger}|G\rangle$. The corresponding eigenvalues
are given by

$\displaystyle\lambda_{\pm}(\Delta)=\frac{1}{2}\Bigg{(}\tilde{J}_{0}-\Delta-\frac{i}{2}{(\tilde{\Gamma}_{0}+\Gamma_{0})}\mp\sqrt{\Big{[}\tilde{J}_{0}+\Delta-\frac{i}{2}{(\tilde{\Gamma}_{0}-\Gamma_{0})}\Big{]}^{2}+4\Gamma_{0}N_{R}\Big{[}J_{d}-\frac{i}{2}{\Gamma_{d}}\Big{]}^{2}}\Bigg{)}.$
(11)

For $d\leq\lambda_{0}/3$ the eigenvalue $\lambda_{-}(\Delta)$ corresponds to
the asymmetric superposition eigenmode between donor and symmetric ring mode
with a majority of the excitation concentrated in the donor. Varying the donor
detuning $\Delta$ varies the decay rate of the eigenmode, given by $-2\
\mathrm{Im}\lambda_{-}(\Delta)$, and leads to a detuning
$\Delta_{\mathrm{sub}}$ that minimizes the decay rate. For $d\ll\lambda_{0}$,
the optimal detuning can be approximated by

$\displaystyle\Delta_{\mathrm{sub}}\approx
J_{d}(\tilde{\Gamma}_{0}-\Gamma_{0})-\tilde{J}_{0}.$ (12)

For $N_{R}=9$, $\tilde{J}_{0}\approx(N_{R}-1)J_{d}$ which yields
$\Delta_{\mathrm{sub}}=0$. For a small total dipole moment, the central donor
needs to inherit a larger fraction of the excitation, since the symmetric ring
mode features a collective diple moment that is enhanced by a factor of
$\sqrt{N_{R}}$. In the ideal Dicke case with $J_{nm}=0$ and
$\Gamma_{nm}=\Gamma_{0}$ for all $n,m$, the excitation fraction in the donor
is given by
$\langle\Psi_{-}|\hat{\sigma}_{\mathrm{d}}^{\dagger}\hat{\sigma}_{\mathrm{d}}|\Psi_{-}\rangle={N_{R}}/({N_{R}+1})$,
while the excitation fraction in each of the ring emitters is given by
$1/(N_{R}(N_{R}+1))$. The anti-symmetric state is given by
$|\Psi_{-}\rangle=(\sqrt{N_{R}}\hat{\sigma}^{\dagger}_{\mathrm{d}}-\hat{S}^{\dagger}_{0})|g\rangle/\sqrt{N_{R}+1}$,
but due to finite emitter separations and contributions from the dipole-dipole
energy shifts, the actual anti-symmetric state deviates from the ideal case
[24]. This ideal scenario can be realized experimentally, for instance with
artificial atoms interacting via a single-mode 1D waveguide [61], but without
transport capabilities due to the absence of coherent interactions.

## Role of the trapping rate

Here we briefly discuss the influence of the trapping rate $\Gamma_{T}$ on
excitation transport (for a more complete analysis, see ref. [37]). We show
that a large enough trapping rate leads to suppression of excitation
transport, exhibiting the well-known quantum Zeno effect [62]. Conversely, a
trapping rate that is too small reduces excitation transport as well, leading
to an optimal rate $\Gamma_{\mathrm{T}}$. Furthermore, we show that the
optimal trapping rate is closely related to the group velocity
$v_{g}(k)=|\partial_{k}J_{m,k}|$ of particular collective modes in a specific
band with angular momentum projection $|m|$. The physical interpretation is
quite intuitive since the optimal trapping rate is determined by the velocity
with which the excitation is reaching the acceptor site, and trapping it too
fast leads to the quantum Zeno effect.

Figure 6:  Excitation transport emerges in a limited range of trapping rates.
(a) A numerical scan for the transport efficiency $\eta_{t}$ over $N_{R}$ and
the trapping rate for 10 rings in a chain. The donor/acceptor detuning
$\Delta$ is optimized for each $N_{R}$. (b) For ordered lattices with
subwavelength spacing, the optimal trapping rate is dictated by the group
velocity $v_{g}$ of the collective eigenmode, resonant with the donor/acceptor
frequency. Additional parameters: $d_{R}=d$ for $N_{R}$ even,
$d_{R}=\sqrt{3}/2\ d$ for $N_{R}$ odd and in (a) and (b) a fixed ring radius
$R/\lambda_{0}=0.08$.

First, one can already observe the quantum Zeno effect for a single quantum
emitter, which features a ground state, an excited state, and an additional
trap state $|t\rangle$ to which the excited state decays irreversibly with
rate $\Gamma_{\mathrm{T}}$. Now we add a coherent drive to the system via
$\Omega_{0}(\hat{\sigma}^{\dagger}+\hat{\sigma})$. It follows that the
effective rate at which excitation is accumulating in the trap state is given
by $4\Omega_{0}^{2}\Gamma_{\mathrm{T}}/(\Gamma_{0}+\Gamma_{\mathrm{T}})^{2}$
in steady state. This effective trapping rate features an optimum at a
trapping rate $\Gamma_{\mathrm{T}}^{\mathrm{opt}}=\Gamma_{0}$. This is the
well-known quantum Zeno effect for a single driven quantum emitter. In Fig.
6(a) a numerical scan of the transport efficiently $\eta_{t}$ after a time
$t\Gamma_{0}=150$ is shown as a function of the ring emitter number $N_{R}$
and the trapping rate for 10 rings. Noteably $N_{R}=8,9,10$ allows for the
smallest trapping rate $\Gamma_{\mathrm{T}}$ where transport is still
substantial. In Fig. 6(b) the energy bands ($|m|=2)$ around $J_{m,k}=0$ are
shown and the corresponding group velocities $v_{g}$ plotted. The optimal trap
rate is approximately given by the group velocity with which the
donor/acceptor emitters are resonant,

$\displaystyle\Gamma_{\mathrm{T}}^{\mathrm{opt}}\approx v_{g}/\tilde{d},$ (13)

where $\tilde{d}=2R+d_{R}$ is the separation between neighbouring ring
centers. The group velocity is determined numerically at $J_{k}=0$,
corresponding to the detuning $\Delta$ of the donor/acceptor emitters. An
interesting consequence of Eq. 13 is that by controlling the number of
emitters $N_{R}$ per ring, therefore by extension $\tilde{d}$ and the
donor/acceptor detuning $\Delta$, a lower optimal trapping rate can be
engineered while keeping a high transport efficiency.

## The single excitation subspace

The analysis in the main text is performed in the single-excitation sector for
the electronic degrees of freedom. This allows us to rewrite the effective
Hamiltonian in non-Hermitian form as in the main text. The time dynamics
follow the Schrödinger equation,
$i\partial_{t}|\Psi(t)\rangle=\hat{\mathcal{H}}_{\mathrm{eff}}|\Psi(t)\rangle$
with $\hbar=1$, and the expectation value of observable $\hat{\mathcal{O}}$
becomes ${\mathcal{O}}=\langle\Psi(t)|\hat{\mathcal{O}}|\Psi(t)\rangle$. The
advantage is that instead of describing the state of the quantum system by a
density matrix of size $N\times N$ for $N$ emitter, the system is described in
terms of state vectors of size $N$. This makes the numerical analysis feasible
for larges system sizes. The general state vector can be written as

$\displaystyle|\Psi(t)\rangle=\sum_{m=1}^{N}\alpha_{m}(t)|g,g,...e_{m},...\rangle=\sum_{m=1}^{N}\alpha_{m}(t)\hat{\sigma}_{m}^{\dagger}|G\rangle,$
(14)

with time-dependent coefficients $\alpha_{m}(t)$ and
$|G\rangle\equiv|g,g,...,g\rangle$ denotes all emitters in the ground state.

## Superradiant corner states in two-dimensional ring lattices

The emergence of edge states in a one-dimensional chain of quantum emitter
rings is analyzed in the main text. Furthermore, these edge states become
superradiant with emission occurring at the ends of the chain. This might hint
at the possibility of edge state lasing, observed in chains of golden nano-
ring resonators [20, 21]. Here we show that two-dimensional lattices of rings
feature intriguing states, such as 2D edge states and corner states [63],
illustrated in Fig. 7. For $N_{R}=8$ with decreasing inter-ring spacings
$d_{R}$, corner states become more pronounced, exhibiting a transition to
superradiance for $d_{R}\lesssim d$.

Figure 7:  Emerging single-excitation edge and corner states [49] in one- and
two-dimensional quantum emitter ring lattices (a) Edge states for a 1D chain
of rings ($N_{R}=8$) of various spacings. At the critical spacing
$d_{R}/d\approx 0.58$, the edge states become strongly localized at the edges
with vanishing amplitude in the bulk rings. For (b) square lattices and (c)
hexagonal lattices with $N_{R}=8$, corner states become more pronounced and
exhibit a transition to superradiant decay. Future studies might show, that
these states could exhibit directional coherent light emission. Superradiant
corner state lasing in two-dimensional lattices could have the additional
advantage of a larger spatial range in absorbing pump light as opposed to edge
state lasing in one-dimensional ring chains [21]. Emitter spacing is
$d/\lambda_{0}=0.05$ in all plots.

The single-excitation corner and edge states are obtained numerically with the
quantum optical model described in the main text taking long-range dipole-
dipole interactions ($\sim d^{-1}$) into account. By contrast, the observed
photonic corner states in kagome lattices [49] are described via a tight-
binding model with exponentially decreasing tunneling strengths
$\sim\mathrm{exp}(-d)$ between nearest-neighbor sites. The open quantum system
framework described here allows the studying of radiative properties and
photon statistics, namely coherent light emission of corner states. We find
that edge states with emitter excitation along the whole edge of the 2D
lattice are always subradiant, suggesting superradiance in the low-excitation
regime as a unique feature of corner states.


# Pilot Study Suggests Online Media Literacy Programming Reduces Belief in
False News in Indonesia

Pamela Bilo-Thomas
University of Louisville
<EMAIL_ADDRESS>Clark Hogan-Taylor
Moonshot
<EMAIL_ADDRESS>Michael Yankoski
University of Notre Dame
<EMAIL_ADDRESS>Tim Weninger
University of Notre Dame
<EMAIL_ADDRESS>
###### Abstract

Amidst the threat of digital misinformation, we offer a pilot study regarding
the efficacy of an online social media literacy campaign aimed at empowering
individuals in Indonesia with skills to help them identify misinformation. We
found that users who engaged with our online training materials and
educational videos were more likely to identify misinformation than those in
our control group (total $N$=1000). Given the promising results of our
preliminary study, we plan to expand efforts in this area, and build upon
lessons learned from this pilot study.

## Introduction

While the use of targeted misinformation campaigns by state actors is well
documented (Keller et al. 2020, Zannettou et al. 2019, Bradshaw & Howard
2018), ordinary citizens often both consume and spread (mis)information
through their well-meaning online activity, which can inadvertently help to
spread falsehoods to their friends and followers. Many users of online and
social media systems are not aware of how misinformation is generated or
spread, and thus they may unwittingly participate in the acceleration or
distribution of these materials. Furthermore, the more exposed to
misinformation a person becomes, the more likely they are to spread it, for
repeated exposure leads to an increased belief of this information (Guess et
al. 2020). Ultimately, healthy democracies depend on a well-informed public,
the very possibility of which is undermined by this new deluge of digital
misinformation (Lewandowsky et al. 2012).

In response to these challenges, we present the results of a small pilot study
of a media literacy campaign in Indonesia. This pilot study is the first step
in a long-running program to understand the effects of digital misinformation
on individuals in developing digital economies – those who may not be
sufficiently equipped to understand and navigate perils that proliferate on
social media platforms.

There are many active projects seeking effective ways to stop this spread of
misinformation, from new AI-based misinformation detection warning systems
(Yankoski et al. 2020), to “inoculation” style misinformation games such as
Harmony Square (Roozenbeek & van der Linden 2020), to fact checking
organizations that meticulously comb through available evidence to affirm or
debunk claims, such as Snopes.com. But there is no silver bullet for this
accelerating problem: the adversarial nature of misinformation content
creation processes means that AI-based misinformation detection systems will
likely encourage the development of more sophisticated misinformation
campaigns designed to elude the latest generation of AI-based detection
systems(Anderson & Rainie 2017). Furthermore, fact-checking efforts may
backfire by actually increasing the visibility of the very falsehoods they are
seeking to debunk. For even if exposure to misinformation comes by seeing a
piece of misinformation proven false, once a person has been exposed it is
very difficult to turn them back to a state where they have never heard of it
(Lewandowsky et al. 2012).

Many view media literacy and fact checking as the obvious solution(s) to
misinformation and fake news (Kim & Walker 2020). For example, watching others
be corrected on social media can reduce beliefs in misperceptions (Bode et al.
2020), and techniques such as gamification Chang et al. (2020), inoculation
theory, and prebunking can reduce a user’s susceptibility to misinformation
Roozenbeek & van der Linden (2020), Lee (2018). Indeed, social media platforms
like Facebook and Twitter are rolling out fact checking systems (Fowler 2020)
and Google is promoting its Be Internet Awesome media literacy program (Seale
& Schoenberger 2018). However, others view these efforts with warranted
suspicion. danah boyd (2017) argues that “thorny problems of fake news and the
spread of conspiracy theories have, in part, origins in efforts to educate
people against misinformation” because media literacy campaigns naturally ask
the savvy information consumer to question the narratives that are presented
to them. A study by Craft et al. (2017) showed that higher levels of news
literacy predicted a lower endorsement of conspiracy theories. In contrast,
Jones-Jang et al. (2021) found that only information literacy, which was
measured by a skills test, predicted a user’s ability to recognize fake news
stories; whereas a user’s media, news, and digital literacy, which are self-
reported metrics, were not predictive of a user’s ability to recognize fake
news stories.

In this article we offer preliminary results from our pilot study in combating
the growing phenomenon of online misinformation: a social media literacy
education campaign strategy that seeks to empower new digital arrivals in the
Republic of Indonesia with increased ability to identify misinformation. After
an assessment of the media landscape in Indonesia, we created 6 animated
videos and 2 live-action videos demonstrating various lessons in online social
media literacy. These lessons were presented as advertisements on YouTube,
Google, Facebook, and Twitter and backed by http://literasimediasosial.id.
Building off of other work such as Harmony Square (Roozenbeek & van der Linden
2020) and Bad News (Basol et al. 2020), which are each games in which players
learn how misinformation spreads online, we ask the following research
question: Does online media literacy content lead to an increased ability to
identify misinformation? In other words, can we teach people to do their own
fact checking, which then would result in a potentially scalable solution to
solve the global problem of misinformation? We measure the effectiveness of
our campaign by asking visitors to our website via a phone survey to rate the
accuracy of true, misleading, and false headlines. The results from this pilot
study suggest that there is a modest increase in the ability to determine
whether a story is real or not after engaging our media literacy lessons.

### Why Indonesia?

Indonesia is a large democracy with a rapidly growing Internet user base and
social media penetration. As of 2019, approximately 68% of Indonesia’s 270
million citizens were online, a figure that represents a dramatic increase
from the approximately 43% of the population just four years prior in 2015. By
2025 an estimated 89% of Indonesians will be online. As of 2019, 88%, 84%,
82%, and 79% of Indonesian internet users self-reported use of YouTube,
WhatsApp, Facebook, and Instagram, respectively (Statista 2020). Because of
this, many Indonesian citizens are new to the Internet. We position our work
in this country to measure the effectiveness of this media literacy approach
to a population which has had less exposure to the Internet, as compared to a
Western audience. Additionally, this work provides important insights into the
problems of online misinformation and propaganda, specifically within
Southeast Asia, which remains an understudied region. Specifically, this
context allows us to perform our research in a geographic area which is both
in the Muslim world and in a country that is subjected to the Chinese sphere
of influence. While similar previous work, such as Harmony Square and Bad News
Game, was done targeting a Western audience, this study specifically targets
another cultural context.

Many political events are frequently met with substantial coordinated
misinformation campaigns (known as “hoaxes” in the Indonesian context). For
example, in the 2018 Jakarta mayoral election, recent reports have indicated
that the election campaigns paid as much as 280 USD per month to individuals
who would promote messages from a particular candidate on social media Lamb
(2018). Additionally, digital misinformation played a significant and highly
divisive role in the national election in April of 2019 (Theisen et al. 2021).
Protests against the results of the election resulted in six deaths and the
temporary suspension of access to social media platforms by the Indonesian
government (BBC 2019).

As more Indonesians gain reliable access to the Internet and begin to use
online social media platforms, it is possible that those who are new to the
Internet have not yet fully developed the media literacy skills needed to
distinguish between trustworthy and false news sources, and may therefore be
vulnerable to manipulation through misinformation. Because Indonesia is a
relatively young democracy, traditional democratic institutions like the press
may not be as robust as those institutions in more established democracies
(Bennett & Livingston 2018). Furthermore, the susceptibility of the voting
population to misinformation campaigns may also pose a threat to the stability
of the democratic institutions of the nation itself.

### Assessing the Online and Social Media Landscape in Indonesia

Our initial work focused on identifying legitimate news stories, propaganda,
and disinformation, and the popular narratives and hashtags that were used in
these stories and across these domains. For this we used a methodology
developed by Moonshot, a company which uses social media monitoring, digital
campaigns, and interventions to counter online harms.

This methodology uses a combination of desk research, consultation and
workshops with subject-matter experts, open-source intelligence methods, and
computer systems to identify words, phrases, tropes, slogans, memes, slang,
and other indicators of engagement, in multiple languages and across search,
social media, image boards, forums, apps, and other discursive online spaces
as necessary. Exact sources vary depending on the subject matter and for this
deployment in February 2019 we focused on words and phrases indicative of
intent to engage with disinformation on Google Search, YouTube, Facebook, and
Twitter.

The fundamental output of this methodology is a clear understanding of how
interest in a given online harm manifests online. Depending on the exact
methods of collection, this can be broken down by time, platform, aggregate
user location, age and gender, subcategories of harm and different levels of
risk. This understanding is then used to inform the creation of campaigns
designed to reduce the impact of that harm, as was the case for this project.

Analysis of the resulting dataset revealed four common themes of
disinformation specific to the Indonesian context: (1) anti-Chinese, (2) anti-
Communist, (3) Islamic chauvinism, and (4) political smears. This assessment
of the social media and search engine landscape in Indonesia was used to
inform the specific content used in the social media literacy campaign, based
upon the common themes of disinformation that were identified.

## Methods

### Social Media Literacy Intervention

We developed a preliminary social media literacy education website
(http://literasimediasosial.id) consisting of six informative lessons with
several short educational videos and context-specific slogans that encourage
social media behavior characterized by an increased ability to identify
misinformation for widespread online delivery. This Learn To Discern (L2D)
(Murrock et al. 2018) approach, developed by IREX, builds communities’
resilience to state-sponsored misinformation, inoculates communities against
public health misinformation, promotes inclusive communities by empowering its
members to recognize and reject divisive narratives and hate speech, improves
young people’s ability to navigate increasingly polluted online spaces, and
enables leaders to shape decisions based on facts and quality information
(Vogt 2021). L2D is a purely demand-driven approach to media literacy,
encouraging participants to increase self-awareness of their own media
environments and the forces or factors that affect the news and information
that they consume. L2D is traditionally presented as a training program and it
typically includes a package of in-person activities, online games, distance-
learning courses, public service announcements, and other methods that are
tailored to the needs of social media users. Examples of L2D lessons are shown
in Fig. 2. For this campaign several short explainer videos were created. The
full-length videos are hosted on YouTube
(https://www.youtube.com/channel/UCU7jNlA-4gH3cxsFV6_xkjw) and direct users to
http://literasimediasosial.id. Shortened versions of these animated and live
action videos were also created to capture various aspects of the media
literacy curriculum in short commercial-length snippets.

### Delivering Media Literacy Content to Social Media Users

Whereas previous media literacy efforts have been shown to be effective when
conducted via in-person classroom settings (Hobbs & Frost 2003), our goal was
to use the digitized lessons and explainer videos to directly reach users –
not just students – within the social media platform itself. The Redirect
Method (Helmus & Klein 2018) is a methodology created by Moonshot and Jigsaw
in 2016, originally to fight against violent extremism. It is now used by
Moonshot to counter a range of online harms by reaching individuals who
perform searches which indicate intent to engage in, or that they are affected
by, harmful content on social media or on search engines, with positive,
alternative content.

Just as other companies use advertisements on social media and search engines
to sell material products to an audience defined by, in whole or in part, the
keywords they are searching for (for instance, a vacuum cleaner company might
bid for advertising space from users who are searching for cleaning products),
the Redirect Method places ads in the search results and social media feeds of
users who are searching for pre-identified terms that we have associated with
disinformation.

The Redirect Method can be extensively tailored to platform requirements and
campaign goals, but at its core are three fundamental components: the
indicators of risk (e.g. keywords); the 9 advertisements triggered by the
indicators; and the content to which users are redirected by the
advertisements.

### The Indicators of Risk

We curated an extensive gazette of keywords and phrases indicative of a desire
to engage with harmful content – in this case, disinformation. The keywords
for this campaign were created through a combination of desk research,
consultation, and workshops with independent and local subject-matter experts
and fact-checking organizations (some of whom shared their own keywords), and
continuous mining for new keywords throughout the course of the project. The
database currently contains more than 1,000 individual keyword phrases in
Bahasa Indonesia.

Of the keywords identified for the campaign as indicative of intent to engage
with disinformation, the three most-searched for were as follows:

  1. 1.

“jokowi pki”, which translates to, “Jokowi [President Joko Widodo] is a member
of PKI [the communist party of Indonesia]”

  2. 2.

“PDIP adalah topeng pki”, which translates to, “PDIP [the Indonesian
Democratic Party of Struggle and the party to which the President belongs] is
a mask of the PKI”

  3. 3.

“9 naga pendukung jokowi”, which translates to “the nine dragons support
Jokowi”. The nine dragons refers to an age-old conspiracy that the capital
city of Indonesia and its politicians are controlled by an underworld of nine
Chinese mafia bosses. It is a myth based on the racist trope of “ethnic-
Chinese control”, used to instill fear and distrust.

### The Advertisements

Figure 1: An English-language version of one of the advertisements used in
this deployment of the Redirect Method, visible only to users who were
otherwise searching for disinformation (actual ad text was in Bahasa
Indonesia).

The fundamental purpose of these advertisements is the same as those used in
the commercial sector, i.e., to entice people to click on them. The difference
in the case of this deployment of the Redirect Method is that social gain
takes the place of commercial gain, with users who were initially seeking
disinformation being taken instead to content designed to improve their media
literacy

Figure 2: Ad impressions on Google Search for disinformation content.
Disinformation-related searches in Indonesia peaked during significant
political and social events, such as the 2019 student riots, and the beginning
of the COVID-19 pandemic.

Using Google Ads to display advertisements, e.g., Fig. 1, in response to
disinformation-related keyword searches means we can gather data on the number
of times our advertisements were triggered (‘impressions’), thereby gaining an
insight into the volume of searches for disinformation. When plotted by time,
as in Fig 2., we can also check for correlations in the search data with
relevant and notable offline events.

### The Content

Figure 3: (A) Sample of media literacy lessons. These lessons translate as:
What is misinformation?, Check Your Emotions, and Evaluating Written Content.
Each lesson contained a short video and simple phrases which promote
responsible social media behavior. (B) Snippet from the lesson on “Checking
your Emotions.” The translation is: “Stop: Take a moment to pay attention to
the image; Ask: How am I feeling right now?; State it: Express these feelings
to yourself.” We found that many of the visitors were spending less than a
minute on our website, so it was necessary to condense the lessons into short,
concise bullet points and present them like an advertisement.

Our social media literacy and search engine campaign was launched in August
2019 on Facebook, Twitter, Instagram, Google, and YouTube and concluded in
April 2020. In total, the campaign generated 3.4 million impressions
approximately equally distributed over 7 different video lessons. Those
impressions resulted in 72,976 unique page views of the campaign website.
Because of the wide reach of the advertisement campaign, we cannot be certain
that individuals in our control group did not see our content. However,
because 72,976 unique user sessions page views represents approximately 0.03%
of Indonesia’s estimated 183 million online users, we feel fairly confident
that most of the control group users did not see our media literacy content,
though we do not rule out this possibility. A sample of the media literacy
lessons is illustrated in Fig. 3.

Overall, we received 3,444,398.2 impressions (as calculated by each platform).
Of those, 72,976 users clicked through to the website. In Table 1, we show how
long each user spent on our site based upon the platform that the user was
coming from. Additional campaign statistics are described in the Table A1 in
the Appendix.

### Phone Surveys

After the conclusion of our social media literacy campaign, a team of
interviewers based in Jakarta, Indonesia, conducted a computer-assisted
telephone interview (CATI) for 1,000 successful interviews. The surveys were
nationally representative proportionate to 2019 census data estimates at the
province-level of location, age group, and gender. The main interview language
of the survey was Bahasa Indonesia, but the interviewer team was able to
switch to other languages such as Balinese and Javanese if requested by the
interviewee.

The research protocol and questionnaire were approved by the internal ethics
committee at the University of Notre Dame (#18-11-5009). Data is anonymized,
but demographic information is included. Data and full cross tabulations can
be found at https://www.geopoll.com/misinformation-indonesia/

Visitors to the media literacy education website were asked to be included in
a phone survey in exchange for approximately $.50 USD equivalent in local
currency in phone credit. 331 individuals agreed and provided their phone
number, and of those, 94 completed the CATI survey. This constitutes the
treatment group. The control group was comprised of 996 respondents to
verified random digit dialing phone surveys.

The treatment group was both older and skewed male as compared to the control,
as shown in Tables A2 and A3 in the Appendix. We asked participants about
general demographic information, inquired about their media consumption
habits, and also queried their ability to correctly judge the veracity of news
headlines that we presented to them. We acknowledge that the treatment group
is likely to have self-selection bias. In addition, users who were redirected
to the website were targeted because of their propensity to search for
misinformation subjects, which presents another avenue for data bias. However,
we believe that as a pilot study, this data provides important information
about the efficacy of media literacy campaigns which invites further study.
During data collection the survey team made approximately 4500 phone calls
that resulted in a 21% response rate. The margin of error for the survey at
the 95% confidence level is +/- 3.10%.

## Findings

To judge the effectiveness of the social media literacy campaigns, each survey
participant was asked if they recognized two true, one misleading, and one
false headline. If so, we asked them to rate the accuracy of that headline on
a Likert scale from very accurate to very inaccurate. Our goal was to examine
the differences between how the control and treatment groups rated the
accuracy of headlines. An improvement in the treatment group’s ability to
correctly identify false, misleading, and real stories when compared to the
control group would demonstrate that our campaign had an effect on a user’s
ability to analyze the news headlines they encountered.

Finding 1: Users who encountered the media literacy content were more likely
to identify false news than those that did not.

Figure 4: 58.3 percent of our treatment group was able to identify the misinformation as either very or somewhat inaccurate, compared with 42.4 percent of the control group. We supply the number of respondents that gave an accuracy rating for each headline. For this plot, we averaged together the number of individuals who replied to our real headlines, since each person was given two real headlines to rate. Headline | Group | $N$ | Mean Rank | $\rho$ | Significance | Remarks
---|---|---|---|---|---|---
Real | Control | 1113 | 3.36 | 0.470 | 0.871 | Not Sig.
Treatment | 120 | 3.49
Misleading | Control | 321 | 3.00 | 0.497 | 0.520 | Not Sig.
Treatment | 48 | 3.04
False | Control | 273 | 2.71 | 0.569 | 0.081 | Sig. at 90% confidence
Treatment | 36 | 2.44
Table 1: Treatment users were statistically better at identifying
misinformation headlines compared to the control group at 90% confidence.

Our preliminary findings from this pilot study suggest that our media literacy
intervention was positively correlated to a respondent’s ability to identify
misinformation. Treatment group users in our pilot study were 15.9% more
likely to identify a misinformation headline as either very or somewhat
inaccurate. These results are significant at 90 percent confidence
(Mann–Whitney $\rho$ = 0.569, $N$ = 309, $p$ = 0.081 one-tailed). Figure 4
shows the results of the Likert-scale responses and Table 2 shows the
statistical breakdown of the responses. Because the treatment group consisted
of individuals who were actively searching for misinformation, these results
are even more encouraging because they suggest that our campaign had a
measurable effect even on those who are familiar with and are actively
searching for misinformation content. We plan to expand upon this pilot
study’s encouraging results in our future research, as we acknowledge the
limitations of such a small sample size.

Across both the control and treatment groups, accuracy ratings were very
similar for the different headline types. Interestingly, as shown in Fig. 4,
it appears that relatively few individuals in the treatment group had a
neutral opinion of the misinformation story, as compared to the other groups
and headline types. This suggests that our media literacy campaign might have
persuaded those who were not sure about the validity of the story that the
narrative was false, via the new skills they acquired from the website. This
is supported by the observation that the truthfulness rating of the
misinformation headlines was similar between the treatment and control groups.
More research is needed to understand the effect of media literacy campaigns
on these fence-sitters -– those who have neutral feelings about the headlines
that they see, and are unable to tell if these stories are true or false.

In summary, we found that our social media literacy intervention is associated
with an increase in the ability of users to correctly identify misinformation.
However, despite these findings, participants in our social media literacy
campaign did not show an increased ability to accurately identify misleading
or real news stories. This phenomenon presents an intriguing area for future
research.

Finding 2: False news did not spread as broadly in Indonesia as real news.

Heard of Story? | Real ($N$ = 1865) | Misleading ($N$ = 875) | False ($N$ = 870) | Total
---|---|---|---|---
Yes | 1332 (71.4%) | 401 (45.8%) | 333 (38.2%) |  $N$ = 2066
No | 533 (28.5%) | 474 (54.1%) | 537 (61.7%) |  $N$ = 1544
Table 2: Awareness of news stories is positively correlated with its veracity
$\chi^{2}$(2, $N$ = 3,610) = 327.60 $p<$ 0.001

Interestingly and rather unexpectedly, we found that the headline accuracy was
positively correlated with its reach. These results, shown in Table 2, suggest
that misinformation does not spread quite as easily as real news stories in
Indonesia – however, further research is needed to understand this phenomenon.
In future work, we plan to expand the number of headlines that we supply to
our survey respondents.

Our results suggest that our media literacy campaign had an effect on the
ability of individuals to accurately label hoaxes as misinformation. However,
further research is needed to answer some of the questions that our findings
raise. For instance, we see that large numbers of respondents have neutral
feelings about news stories. While our campaign focused on identifying
misinformation, perhaps future work can promote trust in legitimate news
sources, which would create statistically significant differences in the
abilities of media literacy campaign participants to identify real headlines
as such, not simply to identify misinformation.

## Conclusions

In this work, we present findings from our early pilot study on the efficacy
of a media literacy campaign in Indonesia in helping users identify
misinformation headlines. These results indicate that 58.3 percent of users
who visited our media literacy website and engaged with the education content
were able to correctly identify misinformation headlines as inaccurate,
compared to our 42.4 percent of the control group. These results suggest that
it is possible to use a media literacy approach to help citizens spot and
identify misinformation.

This work presented a small, limited study to understand the misinformation
landscape in Indonesia, and demonstrates the feasibility of conducting a
larger research project of this nature. Since our results were positive, we
plan on expanding this study in several ways. First, we will introduce a
gamification approach - analogous to Harmony Square and Bad News game – into
our media literacy education materials. Second, we will enlarge the sample
size and test the significance of the results on larger groups. Third, we will
track information such as how long a user stayed on the site and engaged with
the material. Fourth, we will ask more questions about Internet usage, such as
how long a user has been using the Internet. Fifth, we will ask users about
more disinformation headlines, since currently we only ask them about the
veracity of one headline. With these additional considerations, we will create
and deploy a broader study which will give us even more insights into the
efficacy of media literacy campaigns.

We present these preliminary results to others that might be considering
larger, more comprehensive studies in this area. We hope that our work will
help those who are designing such studies, and implement the lessons learned
from our study to further study media literacy, especially in developing
countries or for new digital arrivals.

## Acknowledgements

We would like to acknowledge and thank Carolina Rocha da Silva, Rachel
Fielden, Joel Turner, Tavian MacKinnon, and Walter Scheirer, Joshua Macleder,
and Anders Mantius for their assistance on this project. The funding for this
research study was supported by USAID Cooperative Agreement #7200AA18CA00059

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Platform | Impressions | Clicks | Avg. Duration (sec) | Click Through Rate
---|---|---|---|---
Google | 252,511.0 | 6,877 | – | 4.0%
Facebook | 568,452.3 | 22,747 | 14.17 | 4.0%
Instagram | 525,292.8 | 2,315 | 27.54 | 0.4%
Twitter | 294,793.4 | 2,265 | 35.20 | 0.8%
Youtube | 1,803,348.8 | 38,772 | 0.62 | 2.2%
Table A1: Reach and viewership statistics of media literacy campaign Age |  Treatment ($N$ = 94) |  Control ($N$ = 906) |  Total ($N$ = 1000)
---|---|---|---
15-24 | 13 (13.8%) | 201 (22.1%) |  $N$ = 214
25-35 | 23 (24.4%) | 305 (33.6%) |  $N$ = 328
36-45 | 18 (19.1%) | 216 (23.8%) |  $N$ = 234
46-55 | 21 (22.3%) | 123 (13.5%) |  $N$ = 144
56-65 | 15 (15.9%) | 48 (5.2%) |  $N$ = 63
66+ | 4 (4.2%) | 13 (1.4%) |  $N$ = 17
Table A2: Treatment group is older than our control group $\chi^{2}$(5, $N$ = 1,000) = 29.66 $p<$ 0.001) Gender |  Treatment ($N$ = 94) |  Control ($N$ = 906) |  Total ($N$ = 1000)
---|---|---|---
MALE | 72 (76.5%) | 428 (47.2%) |  $N$ = 500
FEMALE | 22 (23.4%) | 478 (52.7%) |  $N$ = 500
Table A3: Treatment group is more male than our control group $\chi^{2}$(1, $N$ = 1,000) = 29.35 $p<$ 0.001) Urban/Rural? |  Treatment ($N$ = 93) |  Control ($N$ = 901) |  Total ($N$ = 994)
---|---|---|---
Urban | 46 (49.4%) | 404 (44.8%) |  $N$ = 450
Rural | 47 (50.5%) | 497 (55.1%) |  $N$ = 544
Table A4: Treatment group is similar to our control group based upon urban/rural identity $\chi^{2}$(1, $N$ = 994) = 0.727 $p$ = 0.393) Religion |  Treatment ($N$ = 91) |  Control ($N$ = 889) |  Total ($N$ = 990)
---|---|---|---
Muslim | 87 (95.6%) | 827 (91.1%) |  $N$ = 914
Christian | 4 (4.3%) | 58 (6.4%) |  $N$ = 62
Other | 0 (0.0%) | 13 (1.4%) |  $N$ = 13
None | 0 (0.0%) | 1 (0.1%) |  $N$ = 1
Table A5: Treatment group is similar to our control group based upon religious
identity $\chi^{2}$(3, $N$ = 990) = 2.09 $p$ = 0.553)