#  $\mathcal{PT}$-symmetry effects in measurement-based quantum thermal
machines

Jonas F. G. Santos<EMAIL_ADDRESS>Faculdade de Ciências Exatas e
Tecnologia, Universidade Federal da Grande Dourados, Caixa Postal 364,
Dourados, CEP 79804-970, MS, Brazil  Centro de Ciências Naturais e Humanas,
Universidade Federal do ABC, Avenida dos Estados 5001, 09210-580, Santo André,
São Paulo, Brazil  Pritam Chattopadhyay<EMAIL_ADDRESS>AMOS and
Department of Chemical and Biological Physics, Weizmann Institute of Science,
Rehovot 7610001, Israel

###### Abstract

Measurement-based quantum thermal machines are fascinating models of
thermodynamic cycles where measurement protocols play an important role in the
performance and functioning of the cycle. Despite theoretical advances,
interesting experimental implementations have been reported. Here we move a
step further by considering a measurement-based quantum thermal machine model
where the only thermal bath is structured with $\mathcal{PT}$-symmetric non-
Hermitian Hamiltonians. Theoretical results indicate that
$\mathcal{PT}$-symmetric effects and measurement protocols are related along
the cycle. Furthermore, tuning the parameters suitably it is possible to
improve the absorbed heat and the extracted work, operating in the Otto limit
for the efficiency, provided we consider a quasi-static cycle. Our model also
allows switching the configuration of the cycle, engine, or refrigerator,
depending on the strength of the measurement protocol.

##  I Introduction

The development of classical thermodynamics is a solid ground for the
theoretical and practical study of thermal machines. For example, using the
association between one thermal engine and one refrigerator it is possible to
show the second law of classical thermodynamics and that no design of
classical machines is able to overcome the Carnot bounds Callenbook . As the
systems involved in the fabrication of machines get miniaturized, the working
substance or the refrigerant system being the spin (or spins) of an atom
(atoms) Peterson2018 ; Henao2018 ; Lisboa2022  or the vibrational modes of an
ion Rossnagel2016 ; Maslennikov2019 , quantum fluctuations become relevant in
the proper understanding of the energy transfer and the entropy production
during any cyclic operation Esposito2009 ; Campisi2011 . Quantum thermal
machines (QTM) Scovil1959 ; Quan2007 ; Feldmann2000 ; Gelbwaser2013 ;
Misra2022 ; Mukhopadhyay2018 ; Camati2019 ; Singh2020 ; chand1 ; chand2 ;
Pritam2020  are the grounds for the so-called quantum thermodynamics. By
investigating QTM we can study quantum effects such as entanglement and
coherence and how they affect the performance of QTM. It is also possible to
understand and elaborate quantum fluctuation theorems Esposito2009 ;
Campisi2011 ; Jarzynski2004 ; Seifert2012  as well as thermodynamic
uncertainty relations Timpanaro2019 ; Lee2021 ; Sacchi2021 ; Pritam2019 ;
Pritam2021 ; Pritam2020 ; Pritam2021a . In recent years, quantum measurements
have also played a crucial role in designing new QTM models, as they can
change the state of a system and modify its internal energy Yi2017 ;
Elouard2018 ; Elouard2017 . Measurement protocols are non-unitary operations
acting on a quantum system. Due to this aspect, many recent QTM’s replace one
thermalization process with a measurement protocol and the performance becomes
dependent on how the measurement is implemented.

Protocols involving quantum measurements are vast in the literature
Wiseman2009 , and they are the basis of quantum mechanics theory. To get an
interpretation of the non-ideal measurement, Hendrikx Muynck2001  in their
work considered Ramsay’s experiment to explore the generalized measurement
protocols. While in projective measurements the system state is completely
collapsed in a particular eigenstate after the measurement, in the so-called
weak measurement protocols Clerk2010 ; Svensson2013  the system is only
slightly perturbed resulting in many applications in quantum information
Kim2012 . On the other hand, generalized measurements are designed such that
one can control the action on the quantum system and, depending on the
parameter strength involved, it is possible to vary between weak and
projective measurements. Generalized measurements are well described by the
positive operator-valued measurements (POVMs). There is a growing interest in
designing QTM where at least one of the steps is performed by quantum
measurements, where it plays the role of non-unitary processes, thus relating
quantum measurements to entropy production and irreversibility. In particular,
recently an experimental implementation of QTM using generalized measurements
in nuclear magnetic resonance setup was reported Lisboa2022 .

The theory of quantum mechanics supports an interesting generalization well-
known as $\mathcal{PT}$-symmetric quantum mechanics, which mathematically
means that the property of Hermiticity for the observables is relaxed and
replaced by the $\mathcal{PT}$-symmetric conditions, i.e., the operators have
to fulfill the conditions of invariance by spatial reflection (parity
$\mathcal{P}$) and time reversal $\mathcal{T}$ in order to assure real
eigenvalues and thus they can represent physical systems Bender1998 ;
Bender2015 . From the seminal paper by Bender and Boettcher Bender1998 , it
became evident that this new class of Hamiltonians, so-called
$\mathcal{PT}$-symmetric Hamiltonians, could have a huge set of applications
in quantum physics, for instance, in fluctuation relations theorems
Deffner2015 ; Zeng2017 ; Gardas2016 , in quantum optics and photonics systems
Ruter2010 ; Regensburger2002 ; Gao2021 ; Xue2022 . In open quantum systems,
Ref. Bartlomiej2016  showed that the decoherence dynamics are considerably
modified if the system-environment interaction is built by
$\mathcal{PT}$-symmetric Hamiltonians Duarte2018 ; Dey2019 . Furthermore, Ref.
Santos2021  proposed a thermal reservoir model with $\mathcal{PT}$-symmetric
Hamiltonians based on a collisional model such that modifying the strength of
the $\mathcal{PT}$-symmetry is sufficient to switch the configuration from the
engine to refrigerator. Although there exists the study of
$\mathcal{PT}$-symmetry properties in quantum thermodynamics, their
applications in thermodynamic cycles are still unclear for general cycles
besides the standard quantum Otto cycle.

In this work, we extend the study of measurement-based thermal machines models
by assuming that the only thermal bath in the cycle is structured through
$\mathcal{PT}$-symmetric Hamiltonians, while the working substance is assumed
to be described by a Hermitian Hamiltonian. This assumption guarantees that
the advantage in the performance comes from the thermal bath or of the
measurement protocol. Since the measurement mechanism playing the role of a
thermal reservoir introduces a new class of quantum thermal machines, the
introduction of a second non-trivial degree of freedom may be interesting to
control the energy flow along the cycle and then modify its structure and
performance. With this in mind, we propose a measurement-based cycle in which
the working substance is a single harmonic oscillator, for instance, the
vibration mode of a trapped ion, with one of the non-unitary processes a
measurement protocol, whereas the second is the interaction with a thermal
reservoir modeled by $\mathcal{PT}$-symmetric Hamiltonians via collisional
model. This work is organized as follows. In section II we review the main
properties of $\mathcal{PT}$-symmetric quantum mechanics and generalized
measurements for continuous variable systems. Section III is dedicated to
describe the proposed cycle in detail and discuss its consequences. The
conclusion and final remarks are drawn in section IV.

##  II $\mathcal{PT}$-symmetric quantum mechanics and generalized measurements

###  $\mathcal{PT}$-symmetric quantum mechanics

We briefly revisit the main aspects of the $\mathcal{PT}$-symmetric quantum
mechanics. In order to represent physical observables, quantum mechanics
imposes that operators have to be Hermitian, $A=A^{\dagger}$, such that they
have a complete set of eigenstates and real spectra. As evidenced in the
pioneering work Bender1998 , non-Hermitian operators that are simultaneously
invariant under parity $\mathcal{P}$ and time reversal $\mathcal{T}$
symmetries also fulfill the conditions to represent physical observables. This
property is well-known as $\mathcal{PT}$-symmetry and for a non-Hermitian
Hamiltonian $H\left(q_{i},p_{i}\right)$ with $i=1...,N$ and with eigenstates
$|\psi\left(t\right)\rangle$, it implies
$\left[H\left(q_{i},p_{i}\right),\mathcal{PT}\right]=0$ as well as
$\mathcal{PT}|\psi\left(t\right)\rangle=|\psi\left(t\right)\rangle$. Under
this condition, the Hamiltonian is called $\mathcal{PT}$-symmetric and
invariant under the transformations

$\displaystyle\mathcal{PT}q_{i}\left(\mathcal{PT}\right)^{-1}$
$\displaystyle=$ $\displaystyle-q_{i},$
$\displaystyle\mathcal{PT}p_{i}\left(\mathcal{PT}\right)^{-1}$
$\displaystyle=$ $\displaystyle p_{i}$
$\displaystyle\mathcal{PT}i\left(\mathcal{PT}\right)^{-1}$ $\displaystyle=$
$\displaystyle-i.$ (1)

Apart from the real spectra of a non-Hermitian Hamiltonian fulfilling the
$\mathcal{PT}$-symmetry condition, its connection with its Hermitian partner
is realized through the similarity transformation Mostafazadeh2004 ; Fring2016

$h\left(q_{i},p_{i}\right)=\eta H\left(q_{i},p_{i}\right)\eta^{-1},$ (2)

with $h\left(q_{i},p_{i}\right)$ the Hermitian partner of
$H\left(q_{i},p_{i}\right)$ and $\eta=\eta\left(q_{i},p_{i}\right)$ is the
Dyson map with the property $\eta\eta^{-1}=\mathbb{I}$.
$\mathcal{PT}$-symmetric Hamiltonians satisfying Eq. (2) together with the
Hermitian condition guarantee the quasi-Hermiticity relation, $\Theta
H\left(q_{i},p_{i}\right)=H^{\dagger}\left(q_{i},p_{i}\right)\Theta$, with
$\Theta=\eta^{\dagger}\eta$ the metric operator in order to ensure the
probability conservation Fring2016 ; Luiz2020 . For Hermitian Hamiltonians,
$\Theta=\mathbb{I}$. Finally, expected values of observables of
$\mathcal{PT}$-symmetric non-Hermitian Hamiltonians and their Hermitian
partner are linked through the relation
$\langle\phi\left(t\right)|O|\phi\rangle=\langle\psi\left(t\right)|\mathcal{O}|\psi\rangle$,
where $\mathcal{O}$ and $O$ are the non-Hermitian and Hermitian partner
observables, respectively, and
$|\phi\left(t\right)\rangle=\eta^{-1}|\psi\left(t\right)\rangle$ Luiz2020 .

### Generalized measurements

Recent studies concerning measurement protocols in quantum mechanics assume
generalized measurement as the most general set of measurements Wiseman2009 .
They range from the class of projective to weak measurements, depending on
their strength. Generalized measurements of such a observable $A$ with
eigenvalues $a_{\alpha}$ are characterized by Hermitian measurement operators
$M_{\alpha}=M_{\alpha}^{\dagger}$ such that
$\sum_{\alpha}M_{\alpha}^{2}=\mathbb{I}$ Yi2017 . The state of the system
after the measurement in the non-selective case has the form
$\rho^{M}=\sum_{\alpha}M_{\alpha}\rho M_{\alpha}^{\dagger}$. Quantum thermal
machines powered by generalized measurements have been recently reported in
Lisboa2022 ; Behzadi2021 .

Generalized measurements have been used in quantum thermal machine models with
qubit playing the role of working substance in Ref. Lisboa2022 ; Yi2017 ;
Behzadi2021 . For continuous variable systems, we can perform Gaussian
measurements on the position or momentum operators such that
$M_{\alpha}=\left(2\pi\sigma^{2}\right)^{-1/4}\text{exp}\left[-\left(q-\alpha\right)^{2}/\left(4\sigma^{2}\right)\right]$
for the position case, with $\alpha$ the measured position and $\sigma^{2}$ is
the variance of the measurement apparatus and characterizes its precision.
Here the measurement operators satisfy the normalization condition $\int
d\alpha\,M^{2}_{\alpha}=\mathbb{I}$. Note that for $\sigma^{2}\rightarrow 0$
we have infinite precision in the measurement which is in agreement with a
projective measurement protocol. Also, these measurement operators correspond
to the homodyne probability density associated with a measurement of the
position Serafinibook . It is important to stress the non-unitary aspect of
this generalized measurement, such that after the measurement protocol the
system in general absorbs or releases an amount of heat, whenever the
Hamiltonian and the measurement operators do not commute. It is interpreted as
heat because the Hamiltonian is kept fixed during the process.

##  III $\mathcal{PT}$-symmetry in a based-measure quantum heat engine

A measurement-based quantum thermal machine with $\mathcal{PT}$-symmetric
effects as the only thermal bath is explored. The cycle is illustrated in Fig.
(1) and it is built following the notions of Ref. Yi2017 , i.e., the unitary
processes are performed quasi-statically. The working substance is a single-
mode quantum harmonic oscillator, described by a Hermitian Hamiltonian. The
cycle is structured with two unitary and two non-unitary processes. In the
unitary process, (the first and third strokes) changes in the frequency
$\omega(t)$ of the working substance are implemented quasi-statically. On the
other hand, the non-unitary processes are the position measurement process
(second stroke) and the complete thermalization with a thermal bath (fourth
stroke). We incorporate $\mathcal{PT}$-symmetry in the thermal bath through a
collisional model, where each particle is composed of a single harmonic
oscillator with Hamiltonian Santos2021 ; Behzadi2021

$H^{\mathcal{PT}}=\frac{p^{2}}{2m}+\frac{m\omega^{2}q^{2}}{2}+2i\omega\epsilon
p\,q.$ (3)

Note that the above Hamiltonian is $\mathcal{PT}$-symmetric and then the
eigenvalues are real. We use the Dyson map
$\eta=\text{exp}\left[\epsilon/\left(m\omega\hbar\right)p^{2}\right]$ to
obtain the Hermitian counterpart such that

$h=\frac{\mu^{2}}{2m}p^{2}+\frac{m\omega^{2}q^{2}}{2}+\hbar\omega\epsilon,$
(4)

with $\mu^{2}\equiv\left(1+4\epsilon^{2}\right)$ and we vanish any
$\mathcal{PT}$-symmetry signature by setting $\epsilon=0$.

In the simplest collisional model (CM) Abah2020 , the thermal bath is modeled
by a large number of non-interacting particles (ancillary systems), with each
of them prepared in a thermal state. The assumption that the ancillas are non-
interacting assures a Markovian dynamics of the working substance Chiara2018 ;
Ciccarello2017 . Furthermore, the CM implies that the working substance
interacts with just one ancilla at a time interval $\delta t$, after which the
ancilla $i$ is discarded and a new ancilla $i+1$ is brought to interact. By
repeating this method many times and doing $\delta t\rightarrow 0$ the CM
results in the Lindblad master equation Breuer2002

$\frac{d\rho}{dt}=-i\left[H_{S},\rho\right]+\gamma\left(N+1\right)\mathcal{D}[a]\rho+\gamma
N\mathcal{D}\left[a^{\dagger}\right]\rho,$ (5)

where $H_{S}=\hbar\omega_{S}\left(a^{\dagger}a+1/2\right)$ is the Hamiltonian
of the working substance, $\mathcal{D}\left[o\right]=o^{\dagger}\rho
o-\left(\rho o^{\dagger}o+o^{\dagger}o\rho\right)/2$ is the Lindblad operator,
and $N$ is the average number of photons associated with the thermal bath.
Using Hamiltonian in Eq. (4) to build thermal states in which each ancilla is
prepared, we can write them in the Fock basis as
$\rho^{\text{th}}=\sum_{n}\left[N^{n}/\left(N+1\right)^{N+1}\right]|n\rangle\langle
n|$, with $N=\left(e^{\beta\hbar\omega\mu}-1\right)^{-1}$, with
$\beta=1/\left(k_{B}T\right)$ the inverse temperature of the thermal
reservoir. Note that it is possible to define an effective temperature
$\beta_{c}^{\text{eff}}=\mu\beta_{c}$, similar to a squeezed thermal bath, in
which tuning $\mu$ implies a larger or smaller temperature in the thermal
reservoir.

The second non-unitary stroke is a position measurement process which is a
Gaussian operation, such that since the state before the measurement is
Gaussian, the state after it remains Gaussian. We choose the measurement
operators to be
$M_{\alpha}=\left(2\pi\sigma^{2}\right)^{-1/4}\text{exp}\left[-\left(x-\alpha\right)^{2}/\left(4\sigma^{2}\right)^{2}\right]$
Yi2017 . The whole cycle is described by the following strokes.

Figure 1: Illustration of the model. A single-temperature quantum thermal
machine where the thermal reservoir is designed employing a
$\mathcal{PT}$-symmetric Hamiltonian. The position measurement process is a
Gaussian operation such that the state after it is still a Gaussian state.

First stroke. The working substance is detached from the thermal reservoir and
from the measurement apparatus. The frequency is quasi-statically changed from
$\omega_{1}$ to $\omega_{2}$, such that there no transition between different
eigenstates. During this process, the populations of the state remains
constant. The work associated with this stroke is
$W_{1}=\text{Tr}\left[\rho_{\tau_{1}}\left(\omega_{2}\right)H\left(\omega_{2}\right)\right]-\text{Tr}\left[\rho_{0}\left(\omega_{1}\right)H\left(\omega_{1}\right)\right]$.

Second stroke. A measurement apparatus is attached to the working substance
such that its position is measured. Using the position measurement operator
$M_{\alpha}$ mentioned above, we measure the position $\alpha$ with precision
given by $\sigma^{2}$. The post-measurement state is
$\rho^{M}\left(\omega_{2}\right)$ and the energy exchanged with the
measurement apparatus is $\langle
Q_{2}^{M}\rangle=\text{Tr}\left[\rho^{M}\left(\omega_{2}\right)H\left(\omega_{2}\right)\right]-\text{Tr}\left[\rho_{\tau_{1}}\left(\omega_{2}\right)H\left(\omega_{2}\right)\right]$.
This energy exchange is defined as heat because the Hamiltonian is kept fixed
during this process.

Third stroke. The working substance is again detached from the measurement
apparatus and the frequency changes back from $\omega_{2}$ to $\omega_{1}$.
During this process, the populations of the state remains constant. The work
associated to this process is
$W_{3}=\text{Tr}\left[\rho_{\tau_{3}}\left(\omega_{1}\right)H\left(\omega_{1}\right)\right]-\text{Tr}\left[\rho^{M}\left(\omega_{2}\right)H\left(\omega_{2}\right)\right]$.

Fourth stroke. In order to close the cycle, the working substance is brought
to interact with the thermal bath for a sufficiently long time such that the
asymptotic state is reached, i.e.,
$\rho_{\tau_{4}}\left(\omega_{1}\right)=\rho_{0}\left(\omega_{1}\right)$. The
heat during this process is $\langle
Q_{4}\rangle=\text{Tr}\left[\rho_{\tau_{4}}\left(\omega_{1}\right)H\left(\omega_{1}\right)\right]-\text{Tr}\left[\rho_{\tau_{3}}\left(\omega_{1}\right)H\left(\omega_{1}\right)\right]$.

Noticing that all the processes involved in the quasi-static cycle are
Gaussian and the initial state is also a Gaussian state, we can use the Fock
basis to directly compute the net work $\langle W_{\text{net}}\rangle=\langle
W_{1}\rangle+\langle W_{3}\rangle$ as well as the heat associated with the
non-unitary processes. We get the results (more details in the Appendix)

$\displaystyle\langle W_{\text{net}}\rangle$ $\displaystyle=$
$\displaystyle\frac{\hbar\mu}{2}\left(\omega_{2}-\omega_{1}\right)\left(1-\frac{1}{\left(2\pi\sigma^{2}\right)^{1/2}}\right)\coth\left(\frac{\beta_{c}\hbar\omega_{1}\mu}{2}\right),$
$\displaystyle\langle Q_{2}^{M}\rangle$ $\displaystyle=$
$\displaystyle\frac{\hbar\omega_{2}\mu}{2}\left(\frac{1}{\left(2\pi\sigma^{2}\right)^{1/2}}-1\right)\coth\left(\frac{\beta_{c}\hbar\omega_{1}\mu}{2}\right),$
$\displaystyle\langle Q_{4}\rangle$ $\displaystyle=$
$\displaystyle\frac{\hbar\omega_{1}\mu}{2}\left(1-\frac{1}{\left(2\pi\sigma^{2}\right)^{1/2}}\right)\coth\left(\frac{\beta_{c}\hbar\omega_{1}\mu}{2}\right).$
(6)

For the cycle to operate as an engine (refrigerator), the thermodynamic
quantities have to satisfy $\langle W_{\text{net}}\rangle<0$, $\langle
Q_{2}^{M}\rangle>0$, and $\langle Q_{4}\rangle<0$ ($\langle
W_{\text{net}}\rangle>0$, $\langle Q_{2}^{M}\rangle<0$, and $\langle
Q_{4}\rangle>0$). From the results in (6) we obtain the efficiency
$\eta=-\langle W_{\text{net}}\rangle/\langle
Q_{2}^{M}\rangle=1-\omega_{1}/\omega_{2}$ for the engine regime and the
coefficient of performance $\text{COP}=\langle Q_{4}\rangle/\langle
W_{\text{net}}\rangle=\omega_{1}/\left(\omega_{2}-\omega_{1}\right)$. These
results show that the proposed quantum machine operates in the Otto limit.
This behavior is expected since we do not have the production of quantum
coherence in the unitary processes Camati2019 , such that the quantum friction
along the cycle is zero. Besides the efficiency and coefficient of
performance, the important aspect is how the quantum machine behavior depends
on the $\mathcal{PT}$-symmetry and the position measurement process composing
the cycle.

Figure 2: Thermodynamics quantities. (a) Heat absorbed by the working
substance during the measurement process (second stroke) and (b) net work as a
function of the $\mathcal{PT}$-symmetric parameter $\mu$. Black (blue) curves
are for $\sigma=0.1$ ($\sigma=0.2$)). Solid lines are set for $\mu=1$,
representing a cycle without $\mathcal{PT}$-symmetry in the thermal reservoir.

In order to better understand this behavior, Fig. (2) depicts the heat
exchanged during the measurement process and the net work as a function of the
$\mathcal{PT}$-symmetric parameter $\mu$, fixing the value of $\sigma=0.1$
($\sigma=0.2$) in the black (blue) curves. To compare with the scenario where
we have a standard thermal reservoir, we define the solid lines for $\mu=1$.
We first note that for these values of $\sigma$, the cycle operates as an
engine for any values of $\mu$. However, it can be noted that the larger the
value of $\mu$, the larger the absorbed heat and net work. An interesting
point is that the $\mathcal{PT}$-symmetry of the thermal bath and the
measurement process are dependent on each other. This is because the
$\mathcal{PT}$-symmetric property appears exactly in the part of the
Hamiltonian which does not commute with the measurement operator. Thus a
momentum measurement operator would not affect the thermodynamic quantities as
observed in Fig. (2).

Figure 3: Thermodynamics quantities. Heat absorbed/released by the working
substance during the measurement process (red curve) and net work (black
curve) as a function of the measurement parameter $\sigma$. We have set
$\mu=10$, $(\omega_{c},\omega_{h},\beta)=(1.0,2.0,0.2)$.

Figure (3) shows the thermodynamic quantities as a function position
measurement parameter $\sigma$ by keeping fixed $\mu$. We observe that tuning
the measurement precision results in switching the quantum machine regime from
refrigerator to engine. For $\sigma^{2}\rightarrow 0$ we have an infinite
precision (projective measurement protocol) and in this case, the machine
operates as an engine. However, if the measurement process is such that the
working substance is only weakly modified, then the cycle operates as a
refrigerator. This is an interesting aspect of generalized measurement in
quantum thermal machines which are not present in quantum cycles fueled by
projective measurements. The value of $\sigma$ where the cycle regime switches
is $\sigma_{s}=1/\sqrt{2\pi}$. The fact that $\sigma_{s}$ does not depend on
the temperature is because the cycle contains only one thermal bath.

The proposed quantum thermal machine is theoretical in nature. However, some
recent advances should be highlighted in order to indicate that this cycle can
be experimentally performed or simulated in the future. Optomechanical systems
are interesting platforms where continuous variables can be tested. For
example, $\mathcal{PT}$-symmetric dynamics have been considered in
optomechanical systems Xu2021 ; Longhi2016 . In particular, Ref. Xu2016
proposes to generate a $\mathcal{PT}$-symmetric dynamics employing two-coupled
optomechanical systems, which mimics exactly the collisional model we have
considered in the thermal bath. On the other hand, the measurement process
(second stroke), has been investigated in Ref. Clarke  in which the position
operator is measured. These advances show that an implementation of the
present cycle could be possible using optomechanical systems.

##  IV Conclusion

We have considered a measurement-based quantum thermodynamic cycle in which
the only thermal bath is built employing $\mathcal{PT}$-symmetric non-
Hermitian Hamiltonians. With this, we introduced two non-trivial parameters in
the cycle performance, the $\mathcal{PT}$-symmetry signature through the Dyson
map and the measurement parameter. We stress that from our results, the
relevant effect caused by the $\mathcal{PT}$-symmetry signature is to
introduce an effective temperature in the thermal bath.

The detailed analysis of the cycle showed that fixing the measurement position
precision, the larger the $\mathcal{PT}$-symmetry aspect in the only thermal
bath, the larger the absorbed heat and the net work. On the other hand, for a
fixed value of $\mu$, varying the position measurement precision implies the
conversion from the refrigerator to the engine regime. This interesting aspect
sheds some light on the role played by generalized measurements in quantum
thermodynamics protocols. But from Ref. Camati2019  and from our
thermodynamics quantities it is clear that this new degree of freedom does not
affect the efficiency or the COP for the quasi-static case. As a future
perspective, the finite-time dynamics in the unitaries processes could be
investigated to study how transitions between the eigenstates modifies the
$\mathcal{PT}$-symmetry and measurement protocol effects.

Finally, we have seen some advances in the experimental scenario, in
particular optomechanical systems, which could be useful to realize or
simulate the present quantum cycle. We hope our work can help to contribute to
unveiling the role played by $\mathcal{PT}$-symmetric non-Hermitian
Hamiltonians in quantum thermodynamics.

## Data availability statement

All data that support the findings of this study are included in the article.

## Acknowledgments

J. F. G. Santos acknowledges São Paulo Research Grant No. 2019/04184-5,
Universidade Federal da Grande Dourados and Universidade Federal do ABC for
the support.

## Appendix

We here detail the results in Eqs. (6). Under the collisional model
assumptions in the main text, the initial state of the working substance is

$\rho_{0}=2\sinh\left[\frac{\beta\hbar\omega_{1}\mu}{2}\right]e^{-\beta\hbar\omega_{1}\mu\left(n+1/2\right)}.$
(7)

At $t=0$ the Hamiltonian is
$H\left(0\right)=\frac{\hbar\omega_{1}\mu}{2}\left(1+2a^{\dagger}a\right)$,
such that the initial energy is

$\displaystyle U_{0}$ $\displaystyle=$ $\displaystyle
Tr\left[\rho_{0}H\left(0\right)\right]$ (8) $\displaystyle=$
$\displaystyle\frac{\omega_{1}\hbar}{2}\mu\coth\left(\frac{\beta\hbar\omega_{1}\mu}{2}\right).$

After the first stroke the Hamiltonian is
$H_{\tau_{1}}=\frac{\hbar\omega_{2}\mu}{2}\left[1+2a^{\dagger}a\right]$ and
$\rho_{\tau_{1}}=\rho_{0}$, resulting in the energy

$\displaystyle U_{1}$ $\displaystyle=$ $\displaystyle
Tr\left[\rho_{\tau_{1}}H\left(\tau_{1}\right)\right]$ (9) $\displaystyle=$
$\displaystyle\frac{\omega_{2}\hbar\mu}{2}\coth\left(\frac{\beta\hbar\omega_{1}\mu}{2}\right).$

The second stroke is a position measurement process, such that the state after
the measurement is

$\displaystyle\rho^{M}$ $\displaystyle=$ $\displaystyle 2A^{2}\int d\alpha
e^{-2\left(x-\alpha\right)^{2}/\left(4\sigma^{2}\right)}\sinh\left[\frac{\beta\hbar\omega_{1}\mu}{2}\right]e^{-\beta\hbar\omega_{1}\mu\left(n+1/2\right)}$
(10) $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{\left(\pi\sigma^{2}\right)^{1/2}}\sinh\left[\frac{\beta\hbar\omega_{1}\mu}{2}\right]e^{-\beta\hbar\omega_{1}\mu\left(n+1/2\right)}.$

This results in the following internal energy after the measurement

$\displaystyle U_{2}$ $\displaystyle=$ $\displaystyle
Tr\left[\rho_{\tau_{2}}H_{\tau_{2}}\right]$ (11) $\displaystyle=$
$\displaystyle\frac{1}{\left(2\pi\sigma^{2}\right)^{1/2}}\frac{\omega_{2}\hbar\mu}{2}\coth\left(\frac{\beta\hbar\omega_{1}\mu}{2}\right),$

and the internal energy at the end of the third stroke

$\displaystyle U_{3}$ $\displaystyle=$ $\displaystyle
Tr\left[\rho_{\tau_{3}}H_{\tau_{3}}\right]$ (12) $\displaystyle=$
$\displaystyle\frac{1}{\left(2\pi\sigma^{2}\right)^{1/2}}\frac{\omega_{1}\hbar\mu}{2}\coth\left(\frac{\beta\hbar\omega_{1}\mu}{2}\right).$

The thermodynamic quantities, i.e., the net work, the heat during the
measurement, and the heat exchanged with the thermal reservoir are given
respectively by

$\displaystyle\langle W_{\text{net}}\rangle$ $\displaystyle=$
$\displaystyle\langle U_{1}\rangle-\langle U_{0}\rangle+\langle
U_{3}\rangle-\langle U_{2}\rangle$ $\displaystyle\langle Q_{2}^{M}\rangle$
$\displaystyle=$ $\displaystyle\langle U_{2}\rangle-\langle U_{1}\rangle$
$\displaystyle\langle Q_{4}\rangle$ $\displaystyle=$ $\displaystyle\langle
U_{4}\rangle-\langle U_{3}\rangle.$ (13)

and we obtain the expressions in the main text.

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relevance score distribution for each of our real-world datasets. Note that
the nDCG used in our QREM experiments does not expect relevance scores to lie
within a particular range, and that the relevance scores obtained by
preprocessing YT are not strictly bounded, but they are guaranteed to _mostly_
lie between $0$ and $1$, whereas the relevance scores obtained by
preprocessing NF are directly cosine similarities, rescaled to lie between $0$
and $1$. As the relevance scores of the NF datasets are very concentrated, the
quality threshold $q$ hardly constrains our rewiring options when solving QREM
on NF graphs.

Table 9. Expected initial exposure $\nicefrac{{f(G)}}{{n}}$ of nodes in $G$ for the YouTube datasets.  | $d$ | $\alpha$ | $\chi$ | $c_{B1}$ | $c_{B2}$ | $c_{R1}$ | $c_{R2}$
---|---|---|---|---|---|---|---
YT-100k | 5 | 0.05 | S | 6.318 | 8.129 | 4.335 | 2.518
|  |  | U | 6.506 | 8.475 | 4.486 | 2.637
|  | 0.10 | S | 3.251 | 4.245 | 2.303 | 1.491
|  |  | U | 3.357 | 4.412 | 2.377 | 1.530
|  | 0.20 | S | 1.694 | 2.234 | 1.245 | 0.900
|  |  | U | 1.737 | 2.297 | 1.272 | 0.908
| 10 | 0.05 | S | 6.387 | 8.316 | 4.440 | 2.688
|  |  | U | 6.605 | 8.701 | 4.623 | 2.842
|  | 0.10 | S | 3.355 | 4.417 | 2.395 | 1.584
|  |  | U | 3.466 | 4.590 | 2.482 | 1.647
|  | 0.20 | S | 1.750 | 2.317 | 1.290 | 0.938
|  |  | U | 1.796 | 2.382 | 1.324 | 0.961
| 20 | 0.05 | S | 6.983 | 9.026 | 4.880 | 3.014
|  |  | U | 7.372 | 9.444 | 5.153 | 3.195
|  | 0.10 | S | 3.606 | 4.716 | 2.582 | 1.722
|  |  | U | 3.749 | 4.874 | 2.687 | 1.801
|  | 0.20 | S | 1.844 | 2.429 | 1.359 | 0.989
|  |  | U | 1.894 | 2.486 | 1.398 | 1.022
YT-10k | 5 | 0.05 | S | 4.198 | 5.597 | 2.992 | 1.926
|  |  | U | 4.173 | 5.785 | 3.040 | 2.066
|  | 0.10 | S | 2.330 | 3.217 | 1.734 | 1.244
|  |  | U | 2.401 | 3.369 | 1.801 | 1.315
|  | 0.20 | S | 1.309 | 1.854 | 1.019 | 0.806
|  |  | U | 1.353 | 1.922 | 1.053 | 0.834
| 10 | 0.05 | S | 5.093 | 6.729 | 3.641 | 2.377
|  |  | U | 5.712 | 7.576 | 4.101 | 2.704
|  | 0.10 | S | 2.729 | 3.743 | 2.027 | 1.448
|  |  | U | 2.958 | 4.063 | 2.203 | 1.584
|  | 0.20 | S | 1.450 | 2.046 | 1.125 | 0.883
|  |  | U | 1.525 | 2.152 | 1.185 | 0.932
| 20 | 0.05 | S | 6.185 | 8.186 | 4.405 | 2.820
|  |  | U | 6.741 | 8.987 | 4.819 | 3.094
|  | 0.10 | S | 3.120 | 4.285 | 2.310 | 1.625
|  |  | U | 3.306 | 4.569 | 2.460 | 1.741
|  | 0.20 | S | 1.577 | 2.228 | 1.222 | 0.949
|  |  | U | 1.638 | 2.324 | 1.275 | 0.996

##### Presence of safe nodes

In 3 and 1, we established that if a graph $G$ has at least $\Lambda^{+}$ safe
nodes, where a node $i$ is safe if its exposure
$\mathbf{e}_{i}^{T}\mathbf{F}\mathbf{c}$ is $0$ and $\Lambda^{+}$ is the
maximum degree of an unsafe node in $G$, then we can approximate $f_{\Delta}$
up to a factor of $(1-\nicefrac{{1}}{{e}})$. In Fig. 17, we demonstrate that
under _all_ our cost functions, this applies to _all_ YT graphs and roughly
_two thirds_ of the NF graphs, with the notable exception of news articles on
the topic of January 6 (i.e., content reporting on the Capitol riot). Hence,
our theoretical approximation guarantee mostly holds also in practice.

Table 10. Number of news outlets $|C|$ and number of videos $|V|$ in each of our NELA-GT-2021 datasets, for each unique combination of assigned costs.  |  |  |  |  | NF-Jan6 | NF-Cov19 | NF-All
---|---|---|---|---|---|---|---
$c_{B1}$ | $c_{B2}$ | $c_{R1}$ | $c_{R2}$ | $|C|$ | $|V|$ | $|V|$ | $|V|$
0 | 0 | 0.0 | 0.0 | 4 | 147 | 631 | 993
0 | 0 | 0.2 | 0.0 | 69 | 3188 | 17794 | 29021
0 | 0 | 0.4 | 0.0 | 18 | 1463 | 3986 | 6549
0 | 1 | 0.6 | 0.0 | 1 | 0 | 0 | 0
0 | 1 | 0.6 | 1.0 | 40 | 3920 | 19398 | 32303
1 | 1 | 0.6 | 0.5 | 73 | 1533 | 8804 | 15549
1 | 1 | 0.8 | 0.5 | 112 | 1497 | 6436 | 14337
1 | 1 | 1.0 | 0.5 | 24 | 237 | 983 | 1987
Table 11. Initial expected total exposure to harm $f(G)$, as well as total segregation and maximum segregation from Fabbri et al. (Fabbri et al., 2022), on each of our NF graphs with $\alpha=0.05$, $\chi=\textbf{U}$, and $\mu=1.0$ for segregation computations under $c_{R1}$ and $c_{R2}$.  | $d$ | $c$ | $f(G)$ | Total Seg. | Max. Seg.
---|---|---|---|---|---
NF-Jan06 | 5 | $c_{B1}$ | 51 940 | 6 896 | 19.99
$c_{B2}$ | 118 506 | 25 038 | 19.99
$c_{R1}$ | 104 948 | 259 | 2.04
$c_{R2}$ | 92 536 | 8 555 | 19.99
10 | $c_{B1}$ | 50 682 | 5 342 | 5.79
$c_{B2}$ | 114 141 | 18 151 | 19.99
$c_{R1}$ | 103 482 | 250 | 1.47
$c_{R2}$ | 88 800 | 6 172 | 10.33
20 | $c_{B1}$ | 50 319 | 4 889 | 4.05
$c_{B2}$ | 113 738 | 16 194 | 6.50
$c_{R1}$ | 103 445 | 246 | 1.21
$c_{R2}$ | 88 579 | 5 782 | 2.93
NF-Cov19 | 5 | $c_{B1}$ | 264 781 | 54 699 | 19.99
$c_{B2}$ | 635 867 | 201 579 | 19.99
$c_{R1}$ | 520 763 | 1 075 | 2.19
$c_{R2}$ | 503 477 | 81 903 | 19.99
10 | $c_{B1}$ | 252 294 | 44 313 | 19.99
$c_{B2}$ | 618 645 | 157 105 | 19.99
$c_{R1}$ | 513 982 | 1 038 | 1.86
$c_{R2}$ | 492 498 | 65 199 | 19.99
20 | $c_{B1}$ | 248 708 | 38 597 | 19.99
$c_{B2}$ | 617 014 | 134 998 | 19.99
$c_{R1}$ | 513 113 | 1 020 | 1.48
$c_{R2}$ | 492 660 | 56 696 | 19.05
NF-All | 5 | $c_{B1}$ | 520 092 | 128 388 | 19.99
$c_{B2}$ | 1 111 742 | 383 640 | 19.99
$c_{R1}$ | 890 383 | 3 013 | 19.99
$c_{R2}$ | 851 697 | 136 356 | 19.99
10 | $c_{B1}$ | 496 667 | 103 825 | 19.99
$c_{B2}$ | 1 089 690 | 307 444 | 19.99
$c_{R1}$ | 880 536 | 2 666 | 15.10
$c_{R2}$ | 841 357 | 111 298 | 19.99
20 | $c_{B1}$ | 480 186 | 88 983 | 19.99
$c_{B2}$ | 1 076 287 | 260 998 | 19.99
$c_{R1}$ | 874 785 | 2 187 | 6.90
$c_{R2}$ | 836 194 | 96 895 | 19.99
Figure 16. Distributions of relevance scores $\mathbf{R}[i,j]$ for each source
node $i\in V$ and the top $100$ nodes $j$ considered as potential rewiring
targets, for each of our real-world datasets in the QREM setting. The
relevance distributions are much more skewed in the YT datasets than in the NF
datasets.

(a) NF

(b) YT

Figure 17. Fraction (color) and number (annotation) of safe nodes in each of
our real-world graphs with degrees $d\in\\{5,10,20\\}$, under our four cost
functions $c\in\\{c_{B1},c_{B2},c_{R1},c_{R2}\\}$. In all YT graphs and
roughly two thirds of the NF graphs, the precondition of 3 holds, such that
the greedy algorithm can approximate $f_{\Delta}$ up to a factor of
$(1-\frac{1}{e})$.

##### Impact of cost-function noise

To see how errors in harmfulness assessment might impact our ability to rewire
edges effectively, we investigate the behavior of our exposure objective under
noise in the cost function. In particular, we assess how the distribution of
node exposures shifts when we change the original cost vector $\mathbf{c}$ to
a cost vector $\mathbf{c}^{\prime}$ by either swapping the cost of a randomly
chosen harmful node with that of a randomly chosen benign node (_cost swaps_),
or setting the cost $c_{i}$ of a randomly chosen node $i$ to its opposite,
i.e., $1-c_{i}$ (_cost flips_). Illustrating the results on the YT-100k
dataset in Fig. 18, we observe that as expected—and _by_ _construction_ —,
node exposures are generally sensitive to individual cost assignments.
However, the median impact of moderate cost-function noise on node exposure
levels is close to zero, and the most extreme cost fluctuations occur for
nodes whose _observed_ exposure decreases as compared to their _actual_
exposure. The latter might lead Gamine to undervalue some highly exposed nodes
in its rewiring considerations, but this risk is unavoidable when dealing with
noisy data. In contrast to prior work, Gamine uses an exposure objective that
depends on the cost assignments of _all_ nodes in the graph. Overall, our
experiments with node-level cost-function noise demonstrate that this
objective decays rather smoothly—not only in theory but also in practice.

(a) Cost flips

(b) Cost swaps

Figure 18. Distribution of differences between node exposures before and after
the introduction of noise, shown for the YT-100k dataset with $d=5$,
$\alpha=0.05$, and $\chi\in\\{\mathbf{U},\mathbf{S}\\}$, as measured under
$c_{R1}$. _Negative_ values signal that adding noise _decreased_ the exposure
to harm of a particular node.

##  Appendix F Further Experiments

###  F.1. Impact of Modeling Choices

In the main text, we only demonstrated the impact of the quality threshold $q$
on the performance of Gamine. Here, we further discuss the performance impact
of the regular out-degree $d$, the absorption probability $\alpha$, the shape
of the probability distribution over out-edges $\chi$, and the cost function
$c$.

##### Impact of regular out-degree $d$

Since the impact of individual edges on the objective function decreases as
$d$ increases, for a given budget $r$, we expect Gamine to reduce our
objective more strongly for smaller values of $d$. This is exactly what we
find, as illustrated in Fig. 19, and the pattern persists across absorption
probabilities $\alpha$, probability shapes $\chi$, quality thresholds $q$, and
cost functions $c$.

##### Impact of absorption probability $\alpha$ and out-edge probability
distribution shape $\chi$

For smaller random-walk absorption probabilities $\alpha$, we obtain longer
random walks and thus higher exposure to harmful content, and for
$\chi=\mathbf{S}$, some edges are traversed particularly often. Thus, given a
constant budget $r$, we expect Gamine to achieve a _larger_ decrease of $f$
for smaller $\alpha$, and an initially _faster_ decrease on graphs with skewed
out-edge probability distributions. Again, this is what we find, as depicted
in Fig. 20.

##### Impact of cost function $c$

As the binary cost function $c_{B1}$ (used also in (Fabbri et al., 2022) on a
prior version of the data from (Ribeiro et al., 2020)) labels only videos from
Alt-Right, Alt-Lite, and Intellectual Dark Web (IDW) channels as harmful
($c_{B1}=1$) and all other videos as benign ($c_{B1}=0$), whereas all other
cost functions also assign positive cost to videos from anti-feminist channels
(Incel, MGTOW, MRA, and PUA) (cf. Table 7), we expect Gamine to perform
strongest under $c_{B1}$. As exemplified in Fig. 21, this is exactly what we
observe, and the pattern persists across regular out-degrees $d$, absorption
probabilities $\alpha$, distribution shapes $\chi$, and quality thresholds
$q$. Interestingly, we also consistently observe that Gamine is roughly
equally strong under the binary cost function $c_{B2}$ and the real-valued
cost function $c_{R1}$, and weakest under the real-valued cost function
$c_{R2}$. As $c_{R1}$ and $c_{R2}$ differ only in how they assign costs to
videos from IDW and anti-feminist channels, with $c_{R1}$ ($c_{R2}$) placing
IDW to the _right_ (_left_) of anti-feminist channels, this means that
reducing the exposure to harm is _harder_ when we consider the IDW more benign
than anti-feminist communities, _even though_ there are more IDW videos in
YT-100k than videos from all anti-feminist communities combined.

(a) Absorption probability $\alpha=0.05$

(b) Absorption probability $\alpha=0.1$

Figure 19. Performance of Gamine for out-regular degrees $d\in\\{5,10,20\\}$,
run with $q=0.0$ under $c_{B1}$ on YT-100k with $\chi=\mathbf{U}$. The smaller
the out-degree, the stronger Gamine.

(a) Cost function $c_{R1}$

(b) Cost function $c_{B1}$

Figure 20. Performance of Gamine for absorption probabilities
$\alpha\in\\{0.05,0.1,0.2\\}$ and out-edge probability distribution shapes
$\chi\in\\{\mathbf{U},\mathbf{S}\\}$, run with $q=0.5$ on YT-100k with $d=5$.
The smaller the absorption probability, the stronger the performance of
Gamine, and our objective function drops faster when the out-edge probability
distribution is skewed.

(a) Quality threshold $q=0.99$

(b) Quality threshold $q=0.0$

Figure 21. Performance of Gamine under cost functions
$c\in\\{c_{B1},c_{B2},c_{R1},c_{R2}\\}$, run on YT-100k with $d=5$,
$\alpha=0.05$, and $\chi=\mathbf{U}$. Gamine is strongest under the binary
cost function $c_{B1}$, weakest under the real-valued cost function $c_{R2}$,
and roughly equally strong under the binary $c_{B2}$ and the real-valued
$c_{R1}$.

###  F.2. Scalability

####  F.2.1. Precomputations

In Fig. 5 in the main text, we showed that Gamine’s individual edge rewirings
scale approximately linearly in practice, whereas MMS’s individual edge
rewirings scale quadratically. In Fig. 22, we additionally show that
precomputations add approximately linear overhead for Gamine and somewhat
unpredictable, at times quadratic overhead for MMS. This could be due to two
factors. First, the relevance precomputations for MMS are slightly more
complicated than for Gamine. Second, one part of MMS’s precomputations not
present in Gamine is a matrix inverse approximation via power iteration. This
computation is quadratic in the number of _harmful_ nodes, as MMS considers
only these nodes as transient states.

(a) Gamine on REM

(b) Gamine and MMS on QREM

Figure 22. Empirical scaling of precomputations for Gamine and MMS, computed
with $\alpha=0.05$, $\chi=\mathbf{U}$, $c_{B1}$, and $q=0.0$ (REM) resp.
$0.99$ (QREM). We depict scaling for REM as a function of $n$, with a linear
regression fitted across datasets, and scaling for QREM as a function of $m$,
where we connect the observations stemming from different regular out-degrees
of the same dataset for NF-Cov19, NF-All, and YT-10k. Precomputations for REM
are much faster than for QREM, and while Gamine scales approximately linearly,
MMS scales somewhat unpredictably.  Figure 23. Empirical scaling of Gamine as
a function of the quality threshold $q$, run on YT-100k with $\alpha=0.05$,
$\chi=\mathbf{U}$, and $c_{B1}$, for $d\in\\{5,10,20\\}$. The larger $q$, the
faster Gamine.

####  F.2.2. Impact of quality threshold

In addition to Gamine’s scaling behavior as a function of $n$ and $m$, for
QREM, we would like to understand how the scaling behavior of our method
depends on the quality threshold $q$. To this end, we run Gamine on each of
our YT-100k datasets with $q\in\\{\nicefrac{{x}}{{100}}\mid 0<x<100,x\mod
5=0\\}\cup\\{0.99\\}$. Since increasing $q$ eliminates rewiring candidates, we
hope to see the running time decrease as $q$ increases, and we expect a larger
acceleration on graphs with higher (regular) out-degrees. As reported in Fig.
23, this is precisely what we find—and the dependence on $q$ is particularly
small for our sparser YT-100k datasets.

(a) SH, $n=100$

(b) SU, $n=100$

(c) SU, $n=1\,000$

Figure 24. Correlation of the $\hat{\Delta}$ and $\Delta$ values for the $100$
candidates $(i,j,k)$ with the largest $\Delta$, in $10$ rewiring rounds on
synthetic graphs with $\alpha=0.05$, $\beta=0.7$, and $\chi=\textbf{U}$, under
our binary costs $c_{B}$. $\Delta$ and $\hat{\Delta}$ are almost perfectly
correlated.

(a) Pearson’s Correlation

(b) Spearman’s Correlation

Figure 25. Pearsons’ product-moment correlation and Spearman’s rank
correlation between $\Delta$ and $\hat{\Delta}$, in $10$ rewiring rounds on
synthetic graphs with $\alpha=0.05$, $\beta=0.7$, and $\chi=\textbf{U}$, under
binary costs. Both correlations are almost perfect across all rounds, and they
are even closer to $1$ for the larger synthetic graphs than for the smaller
ones.

###  F.3. Impact of Using $\hat{\Delta}$ Instead of $\Delta$

Having confirmed in the main text that Gamine scales linearly not only in
theory but also in practice (cf. Fig. 5), we would like to ensure that moving
from $\Delta$ to $\hat{\Delta}$, which enables this scalability, has little
impact on the quality of our results. To this end, we investigate the
relationship between $\Delta$ and $\hat{\Delta}$ on the smallest instances of
our synthetic graphs, SU and SH. As illustrated in Fig. 24, $\Delta$ and
$\hat{\Delta}$ are almost perfectly correlated, and Fig. 25 shows that this
holds not only for the top-ranked candidates but for all candidates, under
both product-moment correlation and, more importantly, rank correlation. Thus,
we are confident that our reliance on $\hat{\Delta}$, rather than $\Delta$, to
select greedy rewirings hardly degrades our results.

###  F.4. Performance on the NELA-GT Datasets

Whereas on the YT datasets, $100$ rewirings with Gamine reduce the expected
total exposure to harm by $50\%$ while guaranteeing recommendations still
$95\%$ as relevant as the original recommendations (Fig. 2), the reduction we
achieve on the NF datasets is more moderate. As illustrated in Fig. 26, our
best result here is a reduction of the expected total exposure to harm by
about $30\%$, again under a $95\%$ quality guarantee. Notably, changing the
quality threshold $q$ has a smaller impact on the NF than on the YT datasets,
and sometimes it has no performance impact at all. In fact, for the NF-Jan06
graphs involved in Fig. 26, $q=0.95\equiv 0.9\equiv 0.5$ (and hence, we only
draw the line for $q=0.95$). This indicates that unlike on the YT datasets, on
the NF datasets, Gamine is actually affected by the restriction of rewirings
to the $100$ most relevant candidates, which we implement for all real-world
datasets (cf. Section 6.1.2 and also Section E.2, Fig. 16).

(a) Cost function $c_{R2}$

(b) Cost function $c_{B1}$

Figure 26. Performance of Gamine on our NF datasets with $d=5$, $\alpha=0.05$,
and $\chi=\textbf{U}$. Varying $q$ on the NF-Jan06 dataset has a much smaller
performance impact than what we observed on the YT-100k dataset, and the
relative reduction in the expected total exposure after $100$ rewirings is at
the level of what we achieve after $10$ rewirings on the YT-100k dataset (cf.
Fig. 2).

(a) Quality threshold $q=0.99$

(b) Quality threshold $q=0.5$

Figure 27. Performance of Gamine and MMS when measured under $c_{B1}$ by the
maximum segregation or the total segregation from Fabbri et al. (Fabbri et
al., 2022), or by the total exposure as defined in Eq. 3, run on NF-Jan06 with
$d=5$, $\alpha=0.05$, and $\chi=\mathbf{U}$. Gamine outperforms MMS on the
exposure objective, but MMS reduces its segregation objective more strongly.
Counterintuitively, MMS achieves a _stronger_ reduction of its segregation
objective under a _stricter_ quality threshold.

As illustrated in Fig. 27, on the NF-Jan06 dataset under $c_{B1}$, Gamine
still outperforms MMS on our exposure objective, but MMS achieves a much
stronger relative reduction of its segregation objective. However, MMS
counterintuitively reduces its objective function more strongly under a
_stricter_ quality threshold—a behavior we never observe with Gamine under our
exposure objective. As given the same recommendation sequence $\mathbf{r}_{i}$
at node $i$, a rewiring $(i,j,k)$ that is $q$-permissible under $q=0.99$ is
also $q$-permissible under $q=0.5$, this suggests that MMS is highly dependent
on its trajectory and sometimes requires greedily suboptimal choices to obtain
the best possible result after $r$ rewirings.

Moreover, the promising performance we observe for MMS on NF-Jan06 under
$c_{B1}$ does not carry over to NF-Cov19 and NF-All, or even to other cost
functions on NF-Jan06: On NF-Cov19 and NF-All under $c_{B1}$, and on NF-Jan06
under $c_{B2}$ or $c_{R2}$ with binarization threshold $\mu=1.0$, MMS cannot
reduce its segregation objective at all, _even though_ the starting value of
the maximum segregation is exactly the same as for NF-Jan06 under $c_{B1}$
(cf. Section E.2, Table 11). On NF-Jan06 under $c_{R2}$ with binarization
threshold $\mu=1.0$, MMS stops after four rewirings with a reduction of
$25\%$, but the maximum segregation is already miniscule from the start. Thus,
our experiments on NF data confirm our impression from the main paper that MMS
less robust than Gamine.

###  F.5. Edge Statistics of Rewired Edges

In the main text, we reported that Gamine frequently rewires edges $(i,j)$
with a comparatively large sum of in-degrees. To corroborate this claim, in
Fig. 28, we compare the distribution of normalized in-degree sums for edges
rewired by Gamine to that of all edges. We find that the distribution of in-
degree sums for edges rewired by Gamine has a higher median than the
distribution of in-degree sums for all edges, and that the former is generally
shifted toward higher in-degree sums as compared to the latter.

(a) NF, $q=0.0$

(b) YT, $q=0.5$

Figure 28. Distributions of normalized in-degree sums
$\frac{\delta^{-}\\!(i)+\delta^{-}\\!(j)}{m}$ for edges rewired by Gamine
under $c_{B1}$ vs. all edges, on real-world graphs with $d=5$, $\alpha=0.05$,
and $\chi=\mathbf{U}$. Gamine tends to rewire edges with larger in-degree
sums.


# Optical Efficiency Measurements of Large Area Luminescent Solar
Concentrators

Tomi K. Baikie  James Xiao  Bluebell Drummond  Neil C. Greenham
<EMAIL_ADDRESS>
Akshay Rao<EMAIL_ADDRESS>[

###### Abstract

Luminescent solar concentrators (LSCs) are able to concentrate both direct and
diffuse solar radiation and this ability has led to great interest in using
them to improve solar energy capture when coupled to traditional photovoltaics
(PV). In principle, a large area LSC could concentrate light onto a much
smaller area of PV, thus reducing costs or enabling new architectures.
However, LSCs suffer from various optical losses which are hard to quantify
using simple measurements of power conversion efficiency. Here, we show that
spatially resolved photoluminescence quantum efficiency measurements on large
area LSCs can be used to resolve various losses processes such as out-
coupling, self-absorption via emitters and self-absorption from the LSC
matrix. Further, these measurements allow for the extrapolation of device
performance to arbitrarily large LSCs. Our results provide insight into the
optimization of optical properties and guide the design of future LSCs for
improved solar energy capture.

Uni of Cam] Cavendish Laboratory, J.J. Thomson Avenue, University of
Cambridge, Cambridge, CB3 OHE, UK

##  1 Introduction

Measurements of the efficiency of luminescent solar concentrators (LSCs) are
problematic owing to difficulties in determining re-absorbance, reflectivity,
PV characteristics and coupling efficiency, and practical considerations
arising from the physical size of the LSC. In the context of LSCs coupled to
solar cells, there is great interest in quantitatively establishing the
optical and system efficiencies as this provides a means to determine
potential improvements in LSC materials and design 1, 2, 3.

Recent attempts to standardize reporting for LSC device performance are vital,
to allow for the direct comparison between different LSC technologies 4 and
highlight the importance of clearly delineating the metrics used to describe
the performance of the LSC, typically optical efficiency and power conversion
efficiency 5. However, the reported power conversion efficiency,
$\eta_{\text{dev}}$, may reveal little about the performance of the waveguide
itself, since it convolves other factors such as the optical coupling to the
solar cell and the properties of the solar cell itself. Therefore,
measurements of complete device performance without further quantitative
measurements on the optical properties of the LSC itself, will not directly
aid our understanding as to which materials and designs are effective, to what
extent, and why. It is therefore instructive to understand the optical
performance of the LSC itself, before coupling to PV, as this provides a
metric to compare LSCs and an understanding of the loss mechanisms in the LSC
6, 7, 8, 9, 10. In this context, measuring the spatially dependent internal
quantum efficiency could unravel loss mechanisms of the LSC, as these
mechanisms are a function of photon pathlength within the LSC (see Figure 1).

The internal quantum efficiency, $\eta_{\text{int}}$, of an LSC is defined by
Equation 1,

$\eta_{\text{int}}=\frac{\textrm{number of photons emitted from
edges}}{\textrm{number of photons absorbed by the LSC}}.$ (1)

Equation 1 may be reported for a narrow or broad wavelength range of
illumination. Writing Equation 1 in terms of the photon count is most relevant
for LSC efficiency, as this directly relates to the number of photogenerated
carriers. By measuring the photoluminescence as a function of excitation
position, we may determine the $\eta_{\text{int}}$ for arbitrarily large LSCs,
and outline how improvements offered by specific technologies will impact LSC
efficiencies.

Figure 1:  The laser illumination (in blue) is rastered across the LSC (in
grey). Loss mechanisms within the LSC, such as those depicted, have
characteristic length scales over which they operate. Measurement of the
internal quantum efficiency at different positions of illumination will allow
these loss mechanisms to be quantified as a function of LSC size.

###  1.1 Measuring $\eta_{\text{int}}$

To determine the $\eta_{\text{int}}$, a standard technique involves the use of
an integrating sphere 11, 12, 13. The integrating sphere ($r=25$ cm, Lisun
Instruments, in more detail see Methods 4.2) contains a coating of a diffusely
reflecting material, in our case barium sulphate (Pro-Lite Technology), to
ensure that light is redistributed isotropically over the sphere interior
regardless of the angle of emission 14.

The integrating sphere was calibrated using a NIST-traceable quartz tungsten
halogen lamp (Newport 63976 200QC OA) to ensure that both the spectral
dependence of reflectance of the sphere is considered as well as any opaque
material applied to the sides of the LSC inside the sphere. This results in
multiple wavelength-dependent calibration files, detailed in SI Section 1.4.

To determine $\eta_{\text{int}}$, we follow a revised de Mello method, where a
collimated laser beam is directed into the sphere, impinging the LSC 11. As in
Figure 2A, for the first measurement (measurement A) the sphere is empty and
laser light alone is measured. The spectral integral of the laser in
measurement A is termed $I_{a}$. For the second measurement (measurement B)
the LSC is placed inside the sphere and moved out of the beam path so the
laser, $I_{b}$, impinges on the sphere wall. Here, only $\mu$, the fraction of
incident laser light scattered by the sphere wall and absorbed by the sample
will contribute to the laser spectral integral $E_{b}$. For measurement C, the
laser is now directed onto the sample and care is taken to ensure the sample
is oriented such that reflected laser light from the surface of the sample is
directed into the sphere. The spectral integral of the photoluminescence and
laser is given by $E_{c}$ and $I_{c}$, respectively. For the fourth
measurement, measurement D, opaque material is applied to the edges of the LSC
while the laser and sample orientation are the same as in measurement C. The
opaque material will prevent emission from the edges of the LSC, leaving only
photoluminescence from the front and back faces, given by $E_{d}$. Therefore
the efficiency $\eta_{d}$ from measurement D will be the contribution of the
entire LSC, $\eta_{c}$, minus the edges, $\eta_{\text{int}}$, contribution,
i.e. $\eta_{d}=\eta_{c}-\eta_{\text{int}}$.

Figure 2B depicts the measured spectra from a series of four measurements,
with the sharp peak at $405$ nm corresponding to the laser excitation, with
the broad profiles at $650$ nm corresponding to the emission of the LSC. As
detailed in SI Section 1.1, as long as the LSC is strongly absorbing at the
laser wavelength, laser fluctuations are small and calibration corrects for
laser absorbance by the opaque material, the expression for internal
efficiency simplifies to

$\eta_{\text{int}}=\frac{E_{c}-E_{d}}{I_{a}A},$ (2)

where $A$ is the fractional absorption given by $A=(1-{I_{c}}/{I_{b}})$.

###  1.2 LSC Size and Self Absorption

The size of the LSC relative to the integrating sphere introduces a further
set of requirements on experimental design. The smallest possible radius of
the integrating sphere will produce the highest radiance within the sphere and
improve the signal-to-noise ratio of luminescence detection. However, an
unavoidable consequence of integrating spheres is the reabsorption of emitted
light, which will introduce error in the resulting measured
$\eta_{\text{int}}$, which is dependent on the relative geometry between LSC
and sphere, as well as the concentration of the chromophore.

To determine the error in $\eta_{\text{int}}$ due to secondary photon
reabsorption, we determined the fraction of photons which may contribute to an
erroneous signal after being emitted by the LSC. To study how relative sizes
of the LSC and integrating sphere relate to $\eta_{\text{int}}$ error, we
determined the probability of a photon interacting with the LSC before a
photon is measured. The number of times a photon will on average bounce before
detection, known as the sphere multiplier, was determined analytically (see SI
Section 1.5.1 for details). We determined that a broad range of sphere
multipliers approximated a steady state solution for the sphere (see SI
Section 1.5.1 for details).

We then utilised a Monte Carlo ray tracing algorithm to determine how many
photons will interact with the LSC as a function of sphere radius and LSC
size. The simulation was run over the number of bounces determined by the
sphere multiplier for given LSC and sphere dimensions (see SI Section 1.5.1
for details). We then measured the emission and absorption spectra of the LSC
face outside the sphere, identifying the region of overlap between absorption
and photoluminescence. Finally, we analytically determined the probability of
reflection or transmission and the associated pathlength of photons impinging
isotropically on the LSC, which allowed us to determine what portion of
emitted photons may be reabsorbed.

From these probabilities, we can quantify the relative error in the
$\eta_{\text{int}}$ measurement as a function of sphere radius and LSC size
for a specific chromophore and concentration. Figure 2C plots the probability
of a photon emitted by the LSC colliding with the LSC for different sphere
radii and LSC sizes. Surprisingly, a larger integrating sphere relative to the
LSC dimensions does not give a meaningful improvement to the experimental
error arising from reabsorption. This is because the average number of photon
bounces before detection increases with LSC size, and thus the probability of
interaction with the LSC also increases. Typically, the secondary reabsorption
error in an $\eta_{\text{int}}$ measurement is dominated by the spectral
overlap for all but the smallest LSCs. Minimizing the spectral overlap is a
fundamental design goal for LSCs and so, rather usefully, the accuracy of this
measurement will increase as LSC chromophores improve 15.

A detailed analysis of uncertainties and error propagation is given in SI
Section 1.6. However, we draw attention here to a few considerations that can
have a large impact on the reliability of the measured $\eta_{\text{int}}$.
LSCs with an exceptionally low optical density at the excitation wavelength
may have an unacceptable level of accuracy using the presented method and may
wish to consider the method presented by Yang et al., and determine the
$\eta_{\text{dev}}$ alone 6. Additionally, laser fluctuations between the
measurements A to D can have a dramatic effect on the calculated
$\eta_{\text{int}}$, and as such care should be taken to ensure laser
stability. In our case, a power meter (Thorlabs PM16-130) was mounted in the
LSC to ensure laser stability before measuring. Laser fluctuations of even 1%
to 5% over the 4 measurements can induce 50% fluctuations in the recorded
$\eta_{\text{int}}$ for low-absorbance samples.

Figure 2:  A \- In all measurements, a laser beam is directed into the
integrating sphere through a fibre optic coupling. For measurement A the
sphere is empty and laser light alone is measured. Measurement B the LSC is
placed inside the sphere and moved out of the beam path so the laser impinges
on the sphere wall. For measurement C the laser is now directed onto the
sample. For measurement D opaque material is applied to the edges of the LSC.
B \- The spectral results of the 4 measurements to determine
$\eta_{\text{int}}$. The emission spectra have been magnified for clarity. C
\- Photons emitted from LSC intersecting with the LSC before being detected as
a function of sphere radius and LSC size.

###  1.3 Spatially Resolved Photoluminescence

Figure 3: A \- The laser beam is rastered across the LSC in the integrating
sphere. The pathlength is determined by the average distance for photons to
travel from the laser spot to the uncovered edge. B \- Photoluminescence as a
function of effective pathlength from one edge of the perylene LSC as the
laser beam is rastered across the LSC. C \- Photoluminescence as a function of
effective pathlength from the front surface alone when all edges of the LSC
are covered.

Of particular interest in LSC design is $\eta_{\text{int}}$ as a function of
the size, or geometric gain, of the LSC. Geometric gain is defined as the
ratio between the area of the absorbing face area to the total side area of
the LSC perpendicular to illumination. The optical efficiency,
$\eta_{\text{int}}$, is unlikely to remain constant with an increasing
geometric gain due to additional losses associated with photoluminescence
reabsorption or scattering within the LSC 16.

By rastering the illumination point over the LSC, as depicted in Figure 3A, we
can determine $\eta_{\text{int}}$ at each point on the LSC, and hence
$\eta_{\text{int}}$ as a function of geometric gain within the same
experimental setup for arbitrary large LSCs. We recorded photoluminescence as
a function of effective pathlength for the square $10\text{ by}10\text{ cm}$
by $3\text{ mm}$ perylene red LSC (see Methods 4.1 for details). Two spatially
resolved measurements are required, the first where 3 edges of the LSC are
covered (Figure 3B) and another where all edges are covered (Figure 3C). The
incident laser power ($I_{a}$) and the absorption, as in Equation 3, are the
same, so $\eta_{\text{int}}$ can be determined using Equation 2. By rastering
the point of illumination across the LSC away from the edge of emission, we
increase the effective pathlength a photon must travel before it is emitted
from the LSC edge.

We distinguish effective pathlength from the actual pathlength travelled by
the photon. The pathlength is typically defined as the real-space distance the
photon travels within the LSC. This is best determined from LSC ray-tracing
simulations 17. However, we define the effective pathlength here as the
average distance from the point of illumination to the emission edge. Although
the effective pathlength does not reflect the actual distance the photon
travels, it is meaningful in LSC design as it provides a measurable distance
over which photon loss occurs. Assuming an isotropic emitter, the effective
pathlength, $\overline{l}$ is therefore defined by the length of the paths,
$d$, along the angle of acceptance, divided by the angle of acceptance,

$\overline{l}=\frac{\int_{\theta_{\min}}^{\theta\max}d(\theta)\mathrm{d}\theta}{\int_{\theta_{\min}}^{\theta\max}1\mathrm{~{}d}\theta}.$
(3)

The analytical solution to Equation 3 is trivial for rectangular LSCs,
although the solution is rather lengthy and is therefore detailed in SI
Section 1.7. To determine $\eta_{\text{int}}$ for LSCs of arbitrary size, the
photoluminescence must be corrected by a geometric factor to account for the
solid angle subtended from the point of illumination to the uncovered edge for
the size of LSC. Full derivations of the solid angle correction for arbitrary
forms of LSCs are given SI Section 1.8.

In Figure 3B a spectral shift is readily observed as high-energy photons
become redder photons due to chromophore reabsorption and emission within the
LSC. The decay in photoluminescence intensity as a function of pathlength
arises from host matrix reabsorption, non-unity PLQE of the chromophore and
emission into non-waveguiding modes, known as escape cone losses and also from
the change in solid angle of the emission edge as the illumination point is
moved across the LSC. Figure 3C, where all edges are covered and only emission
from the top surface is recorded, is then a measure of the photoluminescence
from the escape cone. It is not sufficient to use measurement D in Figure 2A,
as the probability of re-absorption and hence escape cone loss may also be a
function of distance from the edge. By subtracting the measurement with one
edge uncovered by the measurement with all edges obscured, we are left with
the photoluminescence coming from the unobscured edge as a function of
pathlength from the emitting edge. By extrapolation, we can now determine LSC
edge photoluminescence as a function of LSC size, even beyond the size of the
measured LSC.

###  1.4 Data Analysis

Figure 4:  A&B \- SVD decomposition of the spatially resolved
photoluminescence for the perylene LSC, highlighting the major spectral and
spatial components of the SVD analysis. Solid lines in B are the recovered fit
using the presented analysis method. C \- Analysis model extending the
modelled edge photoluminescence for large LSCs.

We model spatial dependence of LSC photoluminescence by assuming the
photoluminescence follows a sum of weighted exponentials, following Beer’s
law,

$PL(\overline{l},\lambda)=\sum_{i}A_{i}(\lambda)\ \mathrm{e}^{-\overline{l}\
\alpha_{i}},$ (4)

where $i$ corresponds to the size of the basis expansion used to describe the
data, $A_{i}(\lambda)$ denotes the coefficient related to each wavelength and
$\alpha_{i}$ is the absorption coefficient associated with each exponential
decay. We conduct a simultaneous analysis of the photoluminescence spectra
traces at all wavelengths by using singular value decomposition (SVD) where we
reduce the spatially dependent photoluminescence data to its primary
components.

SVD facilitates the interpretation of observed spatially resolved
photoluminescence by reducing the dimensionality of the problem. Let matrix
$\mathrm{D}$ describe the measured photoluminescence at the measured
pathlengths where $\mathrm{D}$ is an $m\times n$ real matrix with $m>n$, where
$m$ is the number of pathlengths sampled, $n$ is the number of wavelengths
recorded, then $\mathrm{D}$ can be written in the form

$\mathrm{D}=\mathrm{US}\mathrm{V}^{\mathrm{T}}.$ (5)

where $\mathrm{U}$ has dimensions of $m\times m,\mathrm{S}$ has $m\times n,$
and $\mathrm{V}$ has $n\times n$. $\mathrm{U}$ and $\mathrm{V}$ are unitary,
so that $\mathrm{U}^{\mathrm{T}}\mathrm{U}=\mathrm{I}$ and
$\mathrm{V}^{\mathrm{T}}\mathrm{V}=\mathrm{I}$ (where the two identity
matrices may have different dimensions). $\mathrm{S}$ has entries only along
the diagonal, known as singular values. The weighted left singular vectors
(wLSV) are given by $\mathrm{US}$.

Decomposing the spatial data in this way has a useful interpretation;
$\mathrm{U}$ is the matrix of left singular vectors giving the spatial
dependence of the signal, $\mathrm{V}^{T}$ is the matrix giving the spectral
dependence of the signal as plotted in Figure 4A and B, respectively. The goal
is to determine what subset of the data is required to adequately describe the
full dataset. The best practice in choosing SVD components is to target a
minimally descriptive model, using the smallest possible set of components to
describe the data 18.

Adapting Beer’s law (Equation 4) into matrix form, and using the reduced
weighted left singular vectors in place of the full data matrix, we can write

$(\mathrm{US})_{n}=\mathrm{E}(\vec{\alpha})\mathrm{x},$ (6)

where (US)n represents the matrix of chosen weighted left singular vectors.
Here $\mathrm{E}$ is the design matrix, which is an exponential function of
the absorption coefficients vector, $\vec{\alpha}$. Array $\mathrm{x}$
corresponds to the coefficient, $A_{i}$, for each weighted left singular
vector. The problem becomes for what vector of absorption coefficients,
$\vec{\alpha}$, is Equation 6 best satisfied, which can be solved efficiently
using any good numerical solver, by solving the associated least squares
problem,

$\min_{\vec{\alpha}}\left\|(US)_{n}-E(\vec{\alpha})x\right\|_{2}^{2},$ (7)

where the subscript refers to the Euclidean norm. The residue is then the norm
of the square of all the differences. If the number of exponentials is not
sufficient to describe the measured data, this suggests a number of absorption
coefficients higher than the number of components detected by SVD. The number
of components need not be equal to the number of spectrally distinct
components present 19.

From the SVD analysis, we find two components are sufficient to describe the
data in the case of the perylene red LSC. Figure 3C plots the extrapolated
edge photoluminescence returned by the model to larger LSCs than measured,
here up to a $25$ cm pathlength, corresponding to a $\sim 40\text{ by}40\text{
cm}$ LSC. As we have now obtained the edge photoluminescence spectra for LSC
of arbitrary size, we may now predict $\eta_{\text{int}}$ for arbitrary large
LSCs.

##  2 Results and Discussion

Figure 5:  A \- $\eta_{\text{int}}$ from the measurements are by dots and the
solid line is determined from the analysis model. B \- External quantum
efficiency (EQE) of two different solar cells. The green line is the solar
cell which has been laser cut to match the edge of the LSC. The purple line is
the solar cell fully intact but largely obscured other than an area to match
the edge surface area of the LSC. C \- Solid lines give $\eta_{\text{dev}}$
determined from the analysis model, and data points are 5 similar LSCs coupled
to the solar cells and $\eta_{\text{dev}}$ determined from I-V curves.

The measured and extrapolated $\eta_{\text{int}}$ as a function of LSC size
for the perylene red LSC and a standard 3T19 LSC (see Methods 4.1 for details)
are given Figure 5A. Here, the recorded photoluminescence as a function of
pathlength (Equation 3) has been corrected for the angle subtended (see SI
Section 1.8), supposing that the illumination point is the centre of the
imagined LSC. The measured $\eta_{\text{int}}$ (blue dots) and the modelled
$\eta_{\text{int}}$ (black line), are plotted, with good agreement between the
two. The black line also extends $\eta_{\text{int}}$ to LSC sizes far beyond
what is practical to place into an integrating sphere. The model reproduces
the characteristic inflection point observed in LSCs which exhibit self-
absorption 16. Notably, this is predicted by the model for the perylene red
LSC even before the inflection is reached. This method may be readily extended
to include the effect of back reflectors or mirrors.

Methods for experimentally determining $\eta_{\text{dev}}$ are well
established in solar cell and LSC literature 4, 5, 6, 20, 21, 22. However,
significant variation of reported $\eta_{\text{dev}}$ exists in the literature
for similar LSCs, as $\eta_{\text{dev}}$ is highly dependent on the nature of
the attached solar cell 23, 4. Extreme care must be taken in
$\eta_{\text{dev}}$ measurements to ensure that there is no direct
illumination of the solar cells by the light source and to minimize reflection
of light initially transmitted through the LSC. Coupling the solar cell to the
LSC as well as identifying the active area also induces significant scope for
systematic error which is difficult to determine. Various methods are used for
attaching solar cells to an LSC, including refractive index matching optical
tape, or index matching solutions and epoxy 16, 21.

The external quantum efficiency of the system can be easily determined for
arbitrary sized LSCs by adapting the model if the external quantum efficiency
(EQE) of the solar cell is known (see SI Section 1.10). However, with a little
effort and some assumptions, it is also possible to approximate
$\eta_{\text{dev}}$ (Equation 9) as a function of LSC size from the spatially
resolved photoluminescence. The incident optical power on the LSC surface,
assuming standard terrestrial illumination, is the integral of the terrestrial
solar spectrum, $\text{AM1.5}(\lambda)$, in Watts per metre squared, over the
active area of the LSC. The output power can be then calculated from the short
circuit current which is integral of the EQE of the side-mounted solar cell,
the edge photoluminescence corrected by some photon conservation factor, times
$V_{\text{OC}}$ and $FF$,

$\eta_{\text{dev}}(G)=\frac{V_{\text{OC}}\
FF\overbrace{\int_{0}^{\infty}\lambda/hc\ \text{AM1.5}(\lambda)\
\text{Abs}(\lambda)\ \text{d}\lambda\ \int_{0}^{\infty}\ q\
\eta_{\text{int}}(G)\ \text{PL}(G,\lambda)\ \text{EQE}(\lambda)\
\text{d}\lambda}^{\propto\ I_{SC}}}{\int_{0}^{\infty}\ \text{AM1.5}(\lambda)\
\text{d}\lambda},$ (8)

where $\text{PL}(G,\lambda)$ is the normalised photoluminescence
$\left(\int_{0}^{\infty}\text{PL}(G,\lambda)\text{d}\lambda=1\right)$ in
photons per second per nanometer. The absorption, $\text{Abs}(\lambda)$, is
given by $\text{Abs}(\lambda)=1-10^{-A_{\text{abs}}(\lambda)}$ where
$A_{\text{abs}}$ is the absorbance.

The values to use for the open circuit voltage $V_{\text{OC}}$ and the $FF$ in
Equation 8 may be determined from either direct measurement or using a
suitable diode model. Utilising an appropriate diode model, it is possible to
relate the $V_{\text{OC}}$ and $FF$ of the solar cell measured directly under
AM1.5, to the emitted photon flux and its spectra. However, in our case, we
determined that $FF=0.48$ and $V_{\text{OC}}=0.62\ V$ by directly measuring
from the PV attached to the emitting edge of the LSC (see Methods 4.3 for
details). To use Equation 8, we make the explicit assumption here that the EQE
nor $\eta_{int}$ are strongly dependent on the photon flux, although this may
be relaxed by measurement of either as a function of power. However, more
difficult to determine is the extent that the $FF$ and $V_{\text{OC}}$ change,
which limits over what range Equation 8 is valid. Equation 8 supposes that all
the light leaving the LSC will make it to the PV, whereas in fact some may be
reflected at the PV interface or coupling optics. However, as long as changes
in the photon flux are not much greater than $\pm 30$ % this should make a
negligible change to $FF$ or $V_{\text{OC}}$ chosen (see SI Section 1.11 for
details). Particularly poor couplings of PV to LSCs should not use Equation 8.

Equation 8 is plotted for the LSC in Figure 5C, which represents a perfect
case, neglecting coupling losses and concentration effects on the EQE of the
PV cell. We measured $\eta_{\text{dev}}$ for both LSCs using the taped solar
cell. Comparing measurements made from IV measurements to the theoretical
$\eta_{\text{dev}}$ reveals our coupling losses and voltage losses are as high
as 15% of $\eta_{\text{dev}}$. Values of 20% have been previously anticipated
4. This highlights further difficulties in relating $\eta_{\text{dev}}$
directly to the optical performance of the LSC.

We highlight here the importance of providing the EQE as a function of
wavelengths of the attached solar cell. Figure 5B highlights the difference
between where the solar cell has been laser cut to match the side edge of the
LSC and where it has been taped to match the active area of the LSC. The laser
treatment results in a decreased EQE (Figure 5) compared to the PV where the
active area has been taped to match the size of the LSC. If the EQE of the
solar cell is not given, $\eta_{dev}$ says little to the effectiveness of the
LSC.

##  3 Conclusion

Notwithstanding the method presented here, for full devices, where the LSC is
coupled to PV, we believe it is vital that researchers carry out the standard
reporting of $\eta_{\text{dev}}$, as only this figure will allow the community
to track the meaningful impact of LSCs and allow for comparisons to the wider
PV literature 4. Further, without providing the EQE of the solar cell as a
function of wavelength, it is impossible to deconvolve $\eta_{\text{dev}}$ and
$\eta_{\text{int}}$. Although $\eta_{\text{dev}}$ remains the figure of merit,
we caution the difficulties in using $\eta_{\text{dev}}$ as a design tool when
considering the optical properties of the LSC as coupling the PV to the LSC
may obfuscate $\eta_{\text{int}}$.

Performance of the optical properties of the LSC are of paramount research
interest in LSC research. As such $\eta_{\text{int}}(G)$ is valuable guide
where to spend efforts to improve performance. We consider that the major
advantage of the proposed method is that it provides an accurate means of
determining both the $\eta_{\text{int}}$ and maximum potential
$\eta_{\text{dev}}$ as a function for arbitrary LSC size and shape, within one
system. The method allows the experimentalist to outline reasonable
$\eta_{\text{int}}$ and $\eta_{\text{dev}}$ for large scale window sized LSCs,
which cannot be realistically produced with typical laboratory facilities.
Using spatially resolved photoluminescence measurements is possible to
visualise the losses, and easily determine efficiency benefits arising from
different technologies, such as different back reflectors or optimise for
potential improvements arising from different solar cell technologies, if
their EQE is known. Further, it is trivial to control photon flux, which is of
importance for future LSC technologies24. The global analysis method reported
here has the advantage compared to previously reported spatially resolved
methods that we need not approximate the self-absorption ratio or assume a
single peak wavelength of emission 16, 25, 26. The authors hope that spatially
resolved photoluminescence measurements may lead to the visualisation of more
complex loss channels and provide future insights to improve LSC efficiency.

##  4 Methods

###  4.1 LSC Manufacture

We utilise two LSCs throughout this paper. In an effort to introduce an easily
reproducible standard, we utilised a commercially available acrylic known as
3T19, often referred to as Lava Orange, which is manufactured by Lucite
International and is often sold under the Perspex brand. The commercial LSC is
widely available and relatively affordable, on the order of 2 USD per 10 cm2.
We purchased 3 LSCs from three resellers, which were laser cut and polished to
form a 10 x 10 x 0.3 cm LSC. 3T19 was consistent across 3 different suppliers
(see SI Section 1.5.3). The advantages of the 3T19 LSC are its
reproducibility, robustness, longevity, ease of cleaning and ubiquity.
However, although the dye concentration remains the same across the suppliers
sampled, no published information is available on this value or the structure
of the emitting dye. Therefore, to have control over the luminophore, we also
manufactured an LSC using perylene red (CAS 123174-58-3, Tokyo Chemical
Industries).

A stock solution of monomer was prepared by mixing 80% lauryl methacrylate
(96%, 500 ppm MEHQ inhibitor, CAS 142-90-5, Merck) and 20% ethylene glycol
dimethacrylate (98%, 90–110 ppm MEHQ inhibitor, CAS 97-90-5) with ($0.10\pm
0.025$)% UV initiator 2,2-dimethoxy-2-phenylacetophenone, CAS 24650-42-8,
Merck) by weight under ambient atmosphere and degassing in a vacuum chamber.
The mixtures were then placed in between two glass sheets with a PTFE spacer
resulting in dimensions of $10$ cm by $10$ cm by $2$ mm. The mixture was then
injected between the glass sheets, and exposed to $385$ nm LEDs (Wicked
Engineering, CUREbox) for 5 minutes, before being left overnight in the dark.

###  4.2 Integrating Sphere Details

An optical fibre (Andor SR-OPT-8019) leads from the sphere to a grating
spectrograph (Andor Kymera-328i) and detectors (Andor iDus 420 and iDus InGaAs
1.7). Immediately in front of the optical fibre port is a baffle, also coated
with barium sulfate, preventing direct illumination of the optical fibre, and
one-bounce illumination of the optic fibre. This arrangement sets geometric
conditions on the size and placement of the baffle with respect to the size of
the LSC (SI Section 1.2). The laser (Thorlabs L405G1, profile and stability
details in SI Section 1.3) is coupled directly to the sphere and mounted on a
temperature-controlled stage (Thorlabs LDM56). Coupling optics were supplied
by Thorlabs and modified in-house to fit the ports of the integrating sphere.

###  4.3 Solar Simulator and Power Measurement Details

A solar simulator (Unisim, TS-SpaceSystems) was used which replicates AM1.5G.
Silicon solar cells from SunPower (California, United States) rated at 22%
efficiency were coupled to the LSC using refractive index matching tape (3M,
USA). For demonstration purposes, we also used laser solar cells from Solar
Made (Colorado Springs, USA), highlighting differences that the properties of
the attached solar cells can make on reported $\eta_{\text{dev}}$. Diode
characteristics of the PV cells are obtained by connecting them with gold
Kelvin clips to a LabView-controlled Keithley 2400 Digital SourceMeter. The
load was then varied to generate an I–V curve.

$\eta_{\text{dev}}$ was then determined from the ratio of the electric power
from the side attached PV cell ($P_{LSC}$) to the incident power on the area
of the LSC exposed to light ($P_{in}$), typically AM1.5,

$\eta_{\text{dev}}=\frac{P_{\text{LSC}}}{P_{\text{In}}}=\frac{I_{\text{SC}}V_{\text{OC}}}{P_{\text{In}}}FF\text{
where }FF=\frac{I_{\text{MP}}V_{\text{MP}}}{I_{\text{SC}}V_{\text{OC}}},$ (9)

and $I_{\text{SC}}$ is the short circuit current of the attached PV,
$V_{\text{OC}}$ is the open circuit voltage, $FF$ is the fill factor and
$V_{\text{MP}}$ $I_{\text{MP}}$ are the max power points.

T.K.B. gives thanks to the Centre for Doctoral Training in New and Sustainable
Photovoltaics for financial support. We acknowledge financial support from the
EPSRC and the Winton Program for the Physics of Sustainability. This project
has received funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (Grant
agreement no. 758826).

### Author Contributions

TKB and AR conceived of the presented idea. TB developed the experimental
arrangement and performed the computations presented. JX advised on the
$\eta_{IQE}$ uncertainty analysis. BD assisted in the integrating sphere
calibration and spectrometer alignment. NCG and AR supervised the work.

The authors declare no competing financial interest.

The supplementary information contains further information on integrating
sphere design rules and calibration. It also includes detailed error
propagation and mathematical derivations referred to in the text. All code for
the simultaneous analysis is available on Github URL TO BE ADDED. Measured
data reported in the main text is available on the Cambridge Data Repository
URL TO BE ADDED.

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