# Robot Patrol: Using Crowdsourcing and Robotic Systems to Provide Indoor
Navigation Guidance to The Visually Impaired

Ike Obi<EMAIL_ADDRESS>Purdue University401 N Grant StreetWest
LafayetteIndiana47907USA ,  Ruiqi Wang<EMAIL_ADDRESS>Purdue
University401 N Grant StreetWest LafayetteIndiana47907USA ,  Prakash Shukla
<EMAIL_ADDRESS>Purdue University401 N Grant StreetWest
LafayetteIndiana47907USA and  Byung-Cheol Min<EMAIL_ADDRESS>Purdue
University401 N Grant StreetWest LafayetteIndiana47907USA

(2023)

###### Abstract.

Indoor navigation is a challenging activity for persons with disabilities,
particularly, for those with low vision and visual impairment. Researchers
have explored numerous solutions to resolve these challenges; however, several
issues remain unsolved, particularly around providing dynamic and contextual
information about potential obstacles in indoor environments. In this paper,
we developed Robot Patrol, an integrated system that employs a combination of
crowdsourcing, computer vision, and robotic frameworks to provide contextual
information to the visually impaired to empower them to navigate indoor spaces
safely. In particular, the system is designed to provide information to the
visually impaired about 1) potential obstacles on the route to their indoor
destination, 2) information about indoor events on their route which they may
wish to avoid or attend, and 3) any other contextual information that might
support them to navigate to their indoor destinations safely and effectively.
Findings from the Wizard of Oz experiment of our demo system provide insights
into the benefits and limitations of the system. We provide a concise
discussion on the implications of our findings.

Accessibility, Robotics, Indoor Navigation, Civic Engagement

††price: 15.00††isbn: 978-1-4503-XXXX-X/18/06††ccs: Computer systems
organization Embedded systems††ccs: Computer systems organization
Redundancy††ccs: Computer systems organization Robotics††ccs: Networks Network
reliability

##  1\. Introduction

Navigation in an indoor environment requires awareness of continually evolving
dynamic and contextual situations like closed elevators, obstacles, cleaning
by janitors, crowd levels, noise, wet floors, and other detrimental obstacles.
Although sighted users are typically able to navigate these kinds of
situations with ease, people with visual impairments struggle a lot in these
kinds of environments. Hence, highlighting the need to support them with
reliable and near real-time information about obstacles or events on the route
to their indoor destination and ultimately to empower them to navigate indoor
spaces safely, independently, and effectively. We implicate that combining the
strengths of crowdsourcing, computer vision, and robotic systems is a
pragmatic approach to generate, validate and circulate contextual information,
such as obstacles, public events, in indoor environments. Specifically, the
crowd provides the initial contextual information, then robots can patrol the
environment periodically to validate and update it via a deep learning-based
vision system, providing up-to-date contextual guidance to the visually
impaired.

To promote the adoption of the Robot Patrol platform among members of the
public, we designed crowdsourcing and gamification approaches. These methods
will facilitate the recruitment, engagement, and retention of sighted
participants that will provide contextual information about obstacles in the
indoor spaces in their proximity. The framework is designed such that users
are rewarded with points and badges for every step of their participation in
providing information to the systems. They are also to be notified if a
visually impaired person uses their information (report) and finds it helpful.
Furthermore, the platform also integrated robotic and computer vision
frameworks into the system to perform the following tasks: 1) verify the
information provided by the participants or members of the public 2)
investigate potential obstacles or warnings on the path the blind will pass,
and 3) provide feedback of their verification task to the system. The
validation and update tasks are very time and labor-consuming. Therefore,
using a robot instead of the crowd to do this can 1) reduce human workload and
increase interest, etc. 2) provide the ability to continuously update/make the
environments more likely to be up-to-date. In all, this approach is designed
to provide relevant information to support people with low vision and visual
impairment to navigate indoor spaces meaningfully, safely, and effectively. We
further conducted a wizard of oz experiment with our system by deploying it in
a controlled space to explore its workability, benefits, and limitations.

Altogether, our exploratory research project contributes insights on the
potential implications of employing crowdsourcing, computer vision, and
robotic frameworks to provide contextual information and guidance to the
visually impaired to empower them to navigate indoor spaces safely. In the
next section, we discuss related works and connect our project to previous
researches within the field, particularly, in the areas of providing indoor
navigation support to the visually impaired.

##  2\. Related Works

Various researchers have explored challenges of developing indoor navigation
systems for the visually impaired (Dias, 2014; Kulyukin et al., 2006; Galatas
et al., 2011; Jeamwatthanachai et al., 2019; Chen et al., 2019; Unhelkar et
al., 2015; Burger et al., 2020). Jeamwatthanachai et. al. (Jeamwatthanachai et
al., 2019), investigated the challenges blind people face while navigating
indoors, and surfacing those existing technologies are insufficient. Dias
(Dias, 2014) explored approaches for empowering blind people to navigate
indoors safely and independently. Their project employed robotics,
crowdsourcing, path-planning and multimodal interfaces (Dias, 2014) only for
mapping WiFi locations and landmarks and however, did not extend to providing
‘real-time’ dynamic and contextual information about indoor spaces to the low
vision and visually impaired users. Kulyukin (Kulyukin et al., 2006; Chen et
al., 2019; Unhelkar et al., 2015), and several other researchers have also
demonstrated how robots could be used for wayfinding in the indoor environment
for the visually impaired.

Similarly, prior work by researchers has explored the approach of using
crowdsourcing to support the activities of the visually impaired (Hoonlor et
al., 2015; Xiao et al., 2015; Rice et al., 2013). Hoonlor et al. (Hoonlor et
al., 2015) designed a mobile app to enable the visually impaired obtain
information about consumer products by accessing images in their database
crowdsourced by users. Findings from their research revealed that the visually
impaired users have positive reviews of the system. Rice et al. (Rice et al.,
2013) identified that the inability to add real-time information to maps
affects people with disability and visually impairment. Findings from their
research revealed that crowdsourcing framework is a suitable approach for
adding information to maps as a way of supporting persons with disability.
Xiao et al. (Xiao et al., 2015) also developed an assistive framework for
context-aware navigation and as a means of supporting people with visual
impairment. Results from their pilot study revealed the suitablity of
crowdsourcing framework to providing indoor navigation guidance to the
visually impaired. Altogether, the works of these authors highlight that
employing crowdsourcing frameworks enables organizations to distribute tasks
among a larger network of users and participants than is typically feasible in
traditional methods.

##  3\. Approaches

The Robot Patrol project has four main sections, including 1) the system
(software application), 2) the crowdsourcing framework, 3) the robotic system,
4) the low vision and visually impaired users. The system mediates the
connection between the participants, the robot system, and the visually
impaired users. The flow map [1] highlights how the information flows from the
participants and members of the public to the robot operating system to verify
the information provided before being sent back to the system for final
delivery to the visually impaired.

###  3.1. System

The system represents the central point that connects all the other sections
of the project and consists of two parts: interface and database. The
interface is where the participants provide information about obstacles in
their location that might affect the navigation of the low vision and visually
impaired users. The interface is also where the visually impaired users
request and receive information about the safety-levels of their desired path
to their indoor destination. Once participants provide information about the
presence of an obstacle via the interface, the information is logged in the
database and then synced with the robot system to enable the robot to verify
the continued existence of the obstacle at the location specified by the
participant. The database is where obstacle and event information are stored
and from where necessary data is transmitted to the robot system for
verification. Information in the database is divided into $events$ \- which
represents information provided by users about ongoing events,
$events\\_verified$ which represents information about verification of events
by the robot. $Obstacles$ contains information about detrimental objects in
indoor environments provided by participants and members of the public to the
system, $obstacles\\_verified$ contains information about obstacles in indoor
environments that has been verified by the robot. The database also contains
information about registered and verified $users$, $users\\_guests$, and
$users\\_maintenance$, with guests and maintenance representing unregistered
users and maintenance staff respectively. This categorization allows for the
potential rating of information provided by the different user categories for
reliability. The system syncs with the robot operating system using Google
Drive API. After the verification of events or obstacles by the robot, the
updated information is synced with the database from where the information is
then provided to visually impaired with time stamp of when the information was
obtained. Figure 1 highlights how the different components of the system
interconnect with each other. Our demo was built with Python and MySQL. And
our front-end was a low-fidelity interactive prototype (See Fig. 2 as at the
time of this publication.

Figure 1. How each component of the platform connects to the central system.
Figure 2. Front-end prototype.

###  3.2. Robot System

The patrol robot part consists of three parts: navigation system, vision
system, and internet system. (See Fig. 3), and these three systems will be
integrated into the Robot Operating System (ROS) (Quigley et al., 2009). The
objective of patrol robot system is to verify and update events and obstacles
in the environment. We describe these parts in the subsections below.

Figure 3. Robot system framework.

####  3.2.1. Internet System

The function of internet system is to exchange data with other parts in the
assistive system, receiving mission message and sending back updated
environment information including events and obstacles. The mission message is
stored in a TXT file with the name of Mission Message, sent by Crowd-sourcing
database through Google Drive Sync API. The received message consists of the
reported events and obstacles which need to be verified: the information of
events is the tuple: (#Event, number, Event keyword, Semantic Location) and
the one of obstacles is the tuple: (#Obstacle, number, Obstacle type, Obstacle
number, Semantic Location). The Mission Message file will be overwritten and
updated by the Crowd-sourcing database every event. We adapted the NLTK
toolkit in Python, to process the text information in the TXT file. The
semantic location of event and obstacle is then sent to navigation system,
activating the robot to depart for the corresponding check point. The event
keyword, obstacle number and type are conveyed to vision system to verify the
event and obstacle by comparing with vision recognition results at
corresponding check point.

Then based on the vision recognition results from vision system and also
confirmation of location from the navigation system, the internet system will
generate another TXT file with the name of Update, which can be extracted by
Crowd-sourcing database through Google Drive Sync API. The file consists of
verified event information: (#Event, number, result), where if the event still
exist the robot returns 1 in result, otherwise, 0, and updated obstacle lists:
(#Obstacle, number, Obstacle type, Obstacle number, Semantic Location). The
Update file will be overwritten and updated by the internet system every time
when a patrol task is completed. Based on the Update file, the crowd-sourcing
database system can know if the reported events are still going on and the
obstacle level of the current environment, and then update them to the low-
vision users.

####  3.2.2. Navigation System

The navigation system aims at providing the patrol path planning and location
of the robot. For this goal, we adapt ROS Navigation Package, which provides
SLAM and path planning algorithms, based on which the robot will be able to
locate itself and plan the patrol routine.

The robot map is separated into several semantic parts based on corridors and
corners to make the map syncing between the main system and robot system
easier. Each semantic area has one checkpoint, where the robot’s camera can
capture a full and clear scenario of the area in order to recognize potential
obstacles. In addition to this regular set of checkpoints, we also set another
set of checkpoints for event checking, where the robot can see the event
scenario clearly. The regular and event checkpoints correspond to
$ob\\_location$ and $ev\\_location$ in the internet system respectively. For
instance, the robot map of experimental site is divided into nine semantic
areas and is labeled with nine regular checkpoints (red) and five event
checkpoints (green). This semantic map can provide semantic location labels,
e.g., $corridor\\_5$ or $corner\\_2$, for vision system, reflecting the
location of the robot and thus the location of recognized obstacles, and the
location for robot to check reported events, for example, for the event that
elevator #1 is under maintenance, we set the checkpoint, which is facing the
elevator, so that the robot can see the situation of the elevator, e.g., there
is a maintenance sign in front of it (See Fig. 4).

Figure 4. Illustration of semantic areas in robot map.

####  3.2.3. Vision system

The vision system oversees the real-time recognition of obstacles and events
in the environment. To this end, we utilize YOLO v3 (Redmon and Farhadi,
2018), which is proven and has a good performance in real-time recognition, as
our deep learning model for this exploratory study. 11 kinds of items, which
are typical obstacles for low-vision people in school environments, including:
table, sofa, chair, telephone booth, trash can, hand-washer rod, warning
signal, shelf, chair with desk, people, and door, are regarded as the targets
of recognition. We collected 100 sample images for each type to build a
database including total 1,100 pictures for training. The trained model
achieved 100% accuracy on testing data set with 550 static images (See Fig.
5). However, although the model successfully recognized most of items during
real-time testing, it could not achieve 100% accuracy in real-time recognition
due to the light condition, image deforming and object overlapping. We
consider adding more samples in different angle and lighting conditions and in
overlapping situations to further explore improving accuracy in real-time
recognition.

Figure 5. Illustration of recognition results on static images.

For obstacle update, the robot will reach each regular checkpoint to conduct
real-time recognition. After recognition, the model can return the type,
number, and bounding box of the recognized objects. Based on the returned
information combined with the semantic location labels from navigation system,
the vision system can update all the obstacles along the patrol path using a
tuple: (Obstacle type, Obstacle number, Semantic Location). For event
verification, the robot will get to the event checkpoints mentioned in Mission
Message file and perform recognition tasks. For $class\\_waiting$ event, if
the robot recognizes more than two people at the checkpoint, then it will
report the event is still going on and return 1 for $ev\\_result$, otherwise,
0. And for the event of $elevator\\_repair$, if the robot detects a
maintenance sign ahead the elevator, it will return 1 for or $ev\\_result$,
otherwise, 0. Then after finishing all recognition tasks, the vision system
will send the information of event verification and obstacle update back to
internet system.

###  3.3. Crowdsourcing and Gamification

Crowdsourcing is a distributed problem-solving methodology that involves using
a network of users to complete a task or resolve a problem (Brabham, 2013). In
crowdsourcing, the community of users often work together asynchronously to
achieve a predefined goal. We are employing the crowdsourcing framework to
recruit, engage, and retain participants who will provide information about
the obstacles in an indoor environment to enable the visually impaired users
to navigate the space meaningfully and safely. The success of our system
relies heavily on the effectiveness of our crowdsourcing and gamification
strategies. Hence, our goal involves ensuring that the process of
participating is very pleasurable and entertaining to ensure that participants
continually engage with the system and also invite people within their local
network to participate. To achieve this goal, we inserted gamification and
pleasurability loops into our crowdsourcing flowchart as seen in Fig. 6. We
describe the gamification process below.

Figure 6. Gamification of the process.

####  3.3.1. Gamification

Games are a great way for people to express their competitive nature (Hamari
et al., 2014). When embedded within a system as microgames, it motivates
people to participate in the events before and after the games so that they
can win their prizes. In this context, it will allow participants to gain the
double benefit of engaging in an altruistic act and also getting rewarded for
helping out.

####  3.3.2. Gamification as an Incentive Mechanism for Sustaining Robot
Patrol

A participant gains a point anytime they log into the app with their user
account details. The goal of rewarding the users anytime they log in to the
app are two-fold 1) to motivate them to return to the app after they complete
their task and 2) to encourage them to participate as registered users instead
of guests. Furthermore, if the participant selects the option of reporting an
obstacle, they also earn a point. The goal is to make the reporting of an
obstacle feel like a real-life game that people typically get rewarded for
after completing a task. If the participant completes the process of reporting
an obstacle, then they gain 5 points for their efforts. When they complete all
the related tasks, they are shown a leaderboard that helps them to visualize
the impact of their effort and how it might have helped the visually impaired
to navigate indoor spaces meaningfully. They will also be shown the feedback
from the visually impaired with their comment of how the information that was
provided by the participant helped them to get to their destination
meaningfully and safely. Furthermore, at the end of specified period, the
participant with the highest points and badges gets a recognition and reward
from the designated authorities for the assistance and contribution to the
independence of the visually impaired community.

###  3.4. Visually Impaired Users

The visually impaired users are the main beneficiaries of the information that
will be provided by participants and verified by the robot. This information
to be sent to the visually impaired will contain contextual guidance,
including information about the severity of the obstacle ranging from low, to
middle and high. Presenting this severity information contextually instead of
numbers reduces the cognitive load that might arise when the information is
presented in numbers and forcing the visually to process information in a way
that is relatively difficult for them, thus, making the information less
meaningful for them. Furthermore, the information is being presented using the
audio and haptic modalities to ensure that the visually impaired users have
the freedom to determine how they want to consume the information that is
being presented to them.

##  4\. Evaluation and Results

In this section we focus on limitations and highlight benefits in the
concluding section. We conducted a wizard of oz experiment to investigate the
benefits and limitations of our approach. This test was conducted across two
dimensions - testing the robot system and testing the crowdsourcing framework.
Both tests were conducted in a controlled area that we had secured for the
experiment. For testing the robot system, we placed multiple obstacles in two
different locations within the controlled area and then simulated the process
of creating and sending a report about a potential obstacle to the robot
system for verification. The robot was then deployed to investigate the
obstacles, verify their continued existence, and then report an update back to
the system. For testing the crowdsourcing framework, one of the researchers
for this project played the role of a participant and reported information
about a closed elevator to the system. Subsequently, we simulated the role of
a visually impaired who has been notified that the elevator has been closed to
explore how the visually impaired participants might engage with the
information. Conducting this test with visually impaired participants would
have been the ideal scenario, however, we employed this testing session as a
first-level assessment of our prototype that would allow us to refine the
system and then recruit visually impaired users for a more robust testing.

Results from the testing session revealed areas of improvement for the future
iteration and scaling of the system. They include 1) Object recognition
challenges: findings from our experiment revealed that the model could not
achieve 100% accuracy in real-time recognition due to objects overlapping each
other, light conditions, and image deforming; 2) Algorithms: findings from the
testing session revealed the need to explore the suitable algorithms that
would be used in determining the location of the participant to allow them to
report obstacles more accurately, we had used semantic mapping for this demo,
but this would be difficult to scale up; 3) Quality of information from the
public: there is a need to ensure that accurate and good-intentioned
information is being provided to the robot system for verification to ensure
efficient use of resources; 4) Data Privacy: another concern was about the
privacy of the information of the planned destination of the participants; and
5) test limitation: there were also concerns about the testing scenario since
visually impaired users rarely travel to unfamiliar areas without prior
training. In all, we view these findings as learning opportunities that would
provoke the research towards new directions.

##  5\. Conclusion and Future work

In this project, we developed Robot Patrol, an integrated system that employs
a combination of crowdsourcing and robotic frameworks to provide contextual
information to the visually impaired to empower them to navigate indoor spaces
safely. We conducted an experiment to test the feasibility of the system.
Future work involves conducting user-testing with visually impaired users and
implementing the platform with a universal design framework based on findings
from the testing sessions. That is to say that, although the platform would be
developed for the visually impaired, other users should also be able to employ
the platform to obtain information about events and obstacles within their
vicinity. For example, since the start of the pandemic, facility managers face
an increased workload. They have to perform extra cleaning and maintenance of
potentially indoor spaces in addition to their traditional role of monitoring
and maintaining order. This system can as well support facility managers and
provide them with up-to-date information and contextual guidance on spaces
that need their attention and the severity of the potential scenario, without
them actually having to visit the spaces beforehand, among other potential use
cases of the system in indoor environments. In all, this was an insightful
project that will continue to evolve.

###### Acknowledgements.

Thanks to the building administrators at our home institution for providing us
with construction signs and space for conducting the testing session.

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# Accessing the degree of Majorana nonlocality with photons

L. S. Ricco<EMAIL_ADDRESS>Science Institute, University of Iceland,
Dunhagi-3, IS-107, Reykjavik, Iceland  V. K. Kozin  Science Institute,
University of Iceland, Dunhagi-3, IS-107, Reykjavik, Iceland  Department of
Physics, ITMO University, St. Petersburg 197101, Russia  A. C. Seridonio  São
Paulo State University (Unesp), School of Engineering, Department of Physics
and Chemistry, 15385-000, Ilha Solteira-SP, Brazil  São Paulo State University
(Unesp), IGCE, Department of Physics, 13506-970, Rio Claro-SP, Brazil  I. A.
Shelykh  Science Institute, University of Iceland, Dunhagi-3, IS-107,
Reykjavik, Iceland  Department of Physics, ITMO University, St. Petersburg
197101, Russia

###### Abstract

We explore the tunneling transport properties of a quantum dot embedded in an
optical microcavity and coupled to a semiconductor-superconductor one-
dimensional nanowire (Majorana nanowire) hosting Majorana zero modes (MZMs) at
their edges. Conductance profiles reveal that strong light-matter coupling can
be employed to distinguish between the cases of highly nonlocal MZMs,
overlapped MZMs and quasi-MZMs. Moreover, we show that it is possible to
access the degree of Majorana nonlocality (topological quality factor) by
changing the dot spectrum through photon-induced transitions tuned by an
external pump applied to the microcavity.

Introduction.— Over the past decade, a huge effort in both theoretical and
experimental fields has been performing in the quest for an unquestionable
signature of exotic ‘half-fermionic’ states, the so-called Majorana zero-modes
(MZMs) in quasi-one dimensional hybrid semiconductor-superconductor nanowires
with strong spin-orbit coupling subject to external magnetic field, termed as
Majorana nanowires [1, 2, 3, 4]. In such devices, each isolated ‘half-
fermionic’ MZM appears at one of the opposite nanowire ends when the bulk of
the system undergoes a topological phase transition [5, 6, 7].

In tunneling spectroscopy measurements performed through a Majorana nanowire
[8, 9, 10, 11, 12, 13, 4, 7, 14], the emergence of a quantized zero-bias peak
(ZBP) robust to changing of relevant system parameters such as magnetic field
and gate voltages is considered as a strong evidence supporting the emergence
of the isolated MZMs. However, it has been extensively demonstrated that other
physical mechanisms such as the formation of trivial zero-energy Andreev bound
state (ABS) at wire ends due to inhomogeneous smooth confining potentials [15,
16, 17, 18, 19, 20, 21, 22, 23, 24] and disorder-induced bound states [25, 21,
26, 24] can produce a robust quantized ZBP, leading to an ambiguity of the MZM
signature.

The emergence of a trivial zero-energy ABS can be mathematically described as
resulting from two half-fermionic states with some spatial separation between
them. In this scenario, a quantized ZBP may arise if one of these half-
fermionic states couples to a tunneling spectroscopy probe stronger than the
other [19, 20, 14]. This kind of trivial ABS formed by two half-fermionic
states with small spatial separation between them has been dubbed as partially
separated ABS (ps-ABS) [19] or quasi-Majoranas (quasi-MZM) [20].

Thus, despite some recent advances, distinguishing ZBPs resulting from genuine
topological MZMs, trivial quasi-MZMs and disorder still remains a challenge
[21, 23, 22, 14, 24]. One possible way to experimentally distinguish between
the ambiguous signatures of MZMs and quasi-MZMs is to measure the degree of
Majorana nonlocality as proposed by Prada et al. [27, 28], which provides an
estimate of the localization of the MZMs wavefunctions along the nanowire.
This quantity can be accessed by means of a quantum dot (QD) working as a
local tunneling spectroscopy probe at one of the ends of a Majorana nanowire
[29, 30, 11, 12]. The degree of Majorana nonlocality is defined as
$\eta^{2}=(\lambda_{c,R}/\lambda_{c,L})$, where $\lambda_{c,L}$
($\lambda_{c,R}$) is the coupling between the dot and the left (right) MZM
[see Fig. 1(a)]. Equivalently, this quantity provides a topological quality
factor $q=1-(\lambda_{c,R}/\lambda_{c,L})$ of the nanowire [31, 32]. Highly
nonlocal MZMs are characterized by $\eta\rightarrow 0$ ($q\rightarrow 1$),
while $\eta\rightarrow 1$ ($q\rightarrow 0$) for a trivial quasi-MZM.
Experimentally [27, 12], the degree of Majorana nonlocality (quality factor)
can be accessed by measuring the corresponding energies of anticrossing
patterns appearing in tunneling conductance profiles as functions of both
applied bias-voltage through the QD-nanowire and QD gate-voltage.

In the present work, we show that light-matter coupling can be employed as an
alternative to differentiate between topological MZMs and a quasi-MZMs (ps-
ABS) in tunneling conductance experiments. We consider the system
schematically shown in Fig. 1(a): a QD placed inside a single-mode optical
cavity with frequency $\omega_{0}$, which is tuned in resonance with valence-
to-conduction band optical transition, i.e, $\Omega=\omega_{0}$, see Fig.
1(b). The conductance through the dot can be accessed by metallic source-drain
leads. Moreover, the QD couples with both left and right MZMs hosted at the
edges of the Majorana nanowire. We demonstrate, that conductance profiles
through the QD as functions of both bias voltage and mean photon occupation
adjusted by external pump reveal distinct patterns for the cases of highly
nonlocal MZMs, overlapped MZMs and quasi-MZMs. We also show that it is
possible to access the degree of Majorana nonlocality (topological quality
factor) in the conductance profile by tuning the mean photon occupation
instead of the QD gate-voltage as in the Prada´s original proposal [27, 28,
31].

Figure 1: (a) The sketch of the considered system: A quantum dot (QD) embedded
inside a single-mode optical cavity with frequency $\omega_{0}$. An electron
in the conduction level of the dot couples with both the left ($\gamma_{R}$)
and the right ($\gamma_{L}$) Majorana zero-modes (MZMs) located at the
opposite ends of the Majorana nanowire, with strengths $\lambda_{c,L}$ and
$\lambda_{c,R}$, respectively. The overlap between the wavefunctions of MZMs
is given by $\varepsilon_{M}$. The tunneling conductance through the QD can by
probed by means of source and drain metallic leads symmetrically coupled ($V$)
to the dot. The energy of the cavity photons is brought in resonance with
interband transition in the dot $\Omega$. (b) The scheme of a QD energy
levels: the valence level $\omega_{v}$ is far below the Fermi level while the
conduction level $\omega_{c}$ is above $E_{F}$. Strong coupling with cavity
photons results in a Rabi splitting of the levels $\Omega_{R}$, which is
determined by the oscillator strength of the optical transition and the
geometry of the cavity.

The model.— The full Hamiltonian which describes the system of Fig. 1 reads
($\hbar=1$):

$\mathcal{H}=\mathcal{H}_{ph}+\mathcal{H}_{QD}+\mathcal{H}_{int}+\mathcal{H}_{lead}+\mathcal{H}_{MZMs},$
(1)

where $\mathcal{H}_{ph}=\omega_{0}c^{\dagger}c,$ is the Hamiltonian of a
single-mode optical cavity, with $c^{\dagger},(c)$ being creation and
annihilation operators of the cavity photons with energy $\omega_{0}$. The
Hamiltonian of the QD reads
$\mathcal{H}_{QD}=\omega_{c}d_{c}^{\dagger}d_{c}+\omega_{v}d_{v}^{\dagger}d_{v}$,
where $d_{v}$ and $d_{c}$ describe the electrons in the valence and conduction
levels of the QD with energy $\omega_{c}$ and $\omega_{v}$, respectively, and
$\Omega=\omega_{c}-\omega_{v}$ is the energy difference between these levels.
The interaction between the cavity and the QD is given by [33]

$\mathcal{H}_{int}=-\Omega_{R}(d_{v}^{\dagger}d_{c}c^{\dagger}+d_{c}^{\dagger}d_{v}c),$
(2)

where $\Omega_{R}$ is the Rabi splitting strength.

The source (S) and drain (D) metallic leads and their coupling with the QD are
described by

$\mathcal{H}_{lead}=\sum_{\bm{k},\alpha}\varepsilon_{\bm{k}}^{\alpha}c_{\bm{k},\alpha}^{\dagger}c_{\bm{k},\alpha}+\sum_{\bm{k},\alpha,l}V_{l}(c_{\bm{k},\alpha}^{\dagger}d_{l}+\text{h.c}),$
(3)

where $c_{\bm{k},\alpha}^{\dagger}$ ($c_{\bm{k},\alpha}$) creates
(annihilates) an electron in the lead $\alpha=S/D$, with wave-number $\bm{k}$,
energy $\varepsilon_{\bm{k}}^{\alpha}=\epsilon_{\bm{k}}-\mu_{\alpha}$ and
chemical potential $\mu_{\alpha}$. The bias-voltage through the QD is defined
as $eV=\mu_{S}-\mu_{D}$. The parameter $V_{l}$ represents the coupling
strength of the conduction (${l=c}$) and valence (${l=v}$) levels of the QD
with the leads. Once we are considering the situation in which the valence
level is far below the Fermi level $(\omega_{v}\ll E_{F},E_{F}=0)$ [Fig.
1(b)], one can assume $V_{v}=0$ and $V_{c}=V$ in Eq. (3) without loss of
generality. Introducing even and odd linear combinations of the states of the
leads $c_{\bm{k},e}$,$c_{\bm{k},o}$, i.e. performing the unitary
transformation according to
$c_{\bm{k},S}=\left(c_{\bm{k},o}+c_{\bm{k},e}\right)/\sqrt{2}$ and
$c_{\bm{k},D}=\left(c_{\bm{k},e}-c_{\bm{k},o}\right)/\sqrt{2},$ the odd states
become decoupled from the dot, and the Hamiltonian of Eq. (3) transforms into
[34]:

$\mathcal{H}_{lead}=\sum_{\bm{k},a=e,o}\epsilon_{\bm{k}}c_{\bm{k},a}^{\dagger}c_{\bm{k},a}+\sqrt{2}V\sum_{\bm{k}}(c_{\bm{k},e}^{\dagger}d_{c}+\text{h.c}).$
(4)

The effective Hamiltonian which describes the ‘half-fermionic’ states
corresponding to MZMs at the ends of the Majorana nanowire coupled to the QD
reads [27, 31]:

$\displaystyle\mathcal{H}_{MZMs}$ $\displaystyle=$
$\displaystyle\imath\varepsilon_{M}\gamma_{L}\gamma_{R}+\lambda_{c,L}(d_{c}-d_{c}^{\dagger})\gamma_{L}$
(5) $\displaystyle+$
$\displaystyle\imath\lambda_{c,R}(d_{c}+d_{c}^{\dagger})\gamma_{R},$

where $\gamma_{L,R}=\gamma_{L,R}^{\dagger}$ represent the MZMs at the opposite
ends of the Majorana nanowire with $\varepsilon_{M}$ being the overlap
strength between them [3]. The hybridization between the left and right MZMs
with the conduction level of the dot is given by $\lambda_{c,L}$ and
$\lambda_{c,R}$, respectively, and one can safely assume, that
$\lambda_{v,L(R)}=0$ for the situation of $\omega_{v}\ll E_{F}$ considered by
us.

The Hamiltonian 5 can be rewritten in terms of a fermionic operator $f$ [1, 3]
by considering the fermionic representation of Majorana operators
$\gamma_{L}=(f^{\dagger}+f)/\sqrt{2}$ and
$\gamma_{R}=\imath(f^{\dagger}-f)/\sqrt{2}$, where $f$ obeys the standard
fermionic anticommutation relations. Hence, Eq. (5) becomes into
$\mathcal{H}_{MZMs}=\varepsilon_{M}f^{\dagger}f+(t_{c}d_{c}f^{\dagger}+\Delta_{c}d_{c}f+\text{h.c.}),$
with $t_{c}=(\lambda_{L}-\lambda_{R})/\sqrt{2}$ and
$\Delta_{c}=(\lambda_{L}+\lambda_{R})/\sqrt{2}$.

Conductance through the QD.— The application of a bias-voltage $eV$ between
source and drain leads, leads to the onset of the current through the system,
and, according to the Landauer-type formula [35, 36], at low temperatures
($T\rightarrow 0$) the conductance through the QD reads:

$G(eV)=\left(\frac{e^{2}}{h}\right)\pi\Gamma\rho_{c}(eV)$ (6)

where $e^{2}/h$ is the quantum of conductance and $\Gamma=2\pi V^{2}\rho$
represents the effective broadening introduced by the coupling between the QD
and the even conduction operator of the leads [Eq. (4)], with a constant
density of states $\rho$ [37, 36], valid within the the wide-band limit
approximation. The local density of states of the dot is given by [36]:

$\rho_{i}(\omega)=-\frac{1}{\pi}\text{Im}\langle\langle
d_{c};d_{c}^{\dagger}\rangle\rangle_{\omega},$ (7)

where $\langle\langle
d_{c};d_{c}^{\dagger}\rangle\rangle_{\omega}\equiv\bm{G}^{c}(\omega)$ is the
retarded Green’s function in the spectral domain $\omega$ [38], which can be
calculated by successive applications of equation-of-motion technique [36, 38,
39] together with an appropriate truncation scheme 111See the Supplemental
Material for details concerning the equation of motion technique and the
truncation scheme adopted. This gives

$\bm{G}^{c}(\omega)=\frac{\bm{g}_{0}(\omega)}{1+\bm{g}_{0}(\omega)\left[\imath\Gamma-\Sigma_{ph}^{c}(\omega)-\Sigma_{MZMs}(\omega)\right]},$
(8)

where $\bm{g}_{0}(\omega)=1/(\omega+\imath\delta-\omega_{c})$ is the bare
Green’s function of the QD conduction level [36, 41, 42], the term
$\imath\Gamma$ describes the corresponding broadening introduced by the
coupling with metallic leads and

$\Sigma_{ph}^{c}(\omega)=\frac{\Omega_{R}^{2}\langle\nu\rangle}{(\omega+i\delta-\omega_{0}-\omega_{v})}$
(9)

is the self-energy associated to the valence-to-conduction band transition
induced by the photonic field of the optical cavity, with
$\langle\nu\rangle=\langle n_{c}\rangle+N_{ph}$ being the mean photon
occupation, which depends on the number of excitations ($\langle
n_{c}\rangle$) in the QD and photons ($N_{ph}$) in the microcavity, which can
be tunned by an external optical pump.

The part of the self-energy associated with the direct coupling between the QD
conduction level and MZMs reads:

$\Sigma_{MZMs}(\omega)=\kappa_{1}(\omega)+(t_{c}\Delta_{c})^{2}\kappa_{0}(\omega)K(\omega),$
(10)

where
$\kappa_{0}(\omega)=[(\omega+i\delta+\varepsilon_{M})^{-1}+(\omega+i\delta-\varepsilon_{M})^{-1}]$,
$\kappa_{1}(\omega)=[t_{c}^{2}(\omega+i\delta-\varepsilon_{M})^{-1}+\Delta_{c}^{2}(\omega+i\delta+\varepsilon_{M})^{-1}]$,
$\tilde{\kappa}_{1}(\omega)=[\Delta_{c}^{2}(\omega+i\delta-\varepsilon_{M})^{-1}+t_{c}^{2}(\omega+i\delta+\varepsilon_{M})^{-1}]$,
$K(\omega)=\kappa_{0}(\omega)/(\omega+\imath\Gamma+\omega_{c}-\tilde{\Sigma}_{ph}^{c}(\omega)-\tilde{\kappa}_{1}(\omega))$.
If a QD is decoupled from the optical cavity ($\Omega_{R}=0$),
$\tilde{\Sigma}_{ph}^{c}(\omega)=0$ and Eq. (10) is reduced to the well-known
expression for the self-energy associated to the leaking of a single MZM into
the QD for $\lambda_{c,R}=0$ [30, 11, 29, 43].

The presence of $\tilde{\Sigma}_{ph}^{c}(\omega)$ in the expression for
$K(\omega)$ in the self-energy defined in Eq. (10) means that the MZMs somehow
‘feel’ the photonic field, although there is no direct coupling between cavity
photons and the Majorana nanowire.

Figure 2:  (a) Conductance through the QD [Eq. (6)] as a function of bias-
voltage $eV$ describing the case of highly nonlocal MZMs
($\varepsilon_{M}=\lambda_{c,R}=0$, $\eta=0$) for increasing number of the
excitations in the system $\langle\nu\rangle$, which can be tuned by external
pump (the curves are offset along the $y$ axis for the better viewing). One
clearly sees the appearance of additional peaks due to the Rabi splitting. (b)
Zoom of the conductance around $eV=0$. (c) Conductance through the QD at zero-
bias $eV=0$ for increasing $\langle\nu\rangle$. (d) Colormap showing the
conductance behavior as a function of both the $eV$ and $\langle\nu\rangle$,
for a range of $eV$ corresponding to gray region of panel (a).

Results and Discussion.— In what follows, we analyze the conductance through
the QD [Eq. (6)] as a function of bias-voltage $eV$, in presence of a photonic
field in the cavity [Fig. 1(a)]. We take the effective broadening
$\Gamma=$40\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$ as the energy unit [30,
8, 44, 45] and use typical parameters for QDs embedded in microcavities [46],
taking $\omega_{v}\ll E_{F}$,
$\omega_{c}=100\Gamma\approx$968\text{\,}\mathrm{GHz}$$,
$\Omega_{R}=2.5\Gamma=$0.1\text{\,}\mathrm{meV}$$
($\approx$24\text{\,}\mathrm{GHz}$$). The energy of a cavity mode is taken in
resonance with the excitonic transition in the QD,
$\omega_{0}=\Omega=35.1\times 10^{3}\Gamma\approx$1.4\text{\,}\mathrm{eV}$$
($338\text{\,}\mathrm{THz}$).

Figure 3:  (a) Conductance through the QD [Eq. (6)] as a function of bias-
voltage $eV$ describing the case of overlapped MZMs localized at the nanowire
ends ($\varepsilon_{M}=10\Gamma$, $\lambda_{c,R}=0$) for increasing numbers of
photon occupation $\langle\nu\rangle$. For a better viewing, the curves are
offset along the $y$ axis. (b) Same curves depicted in panel (a), but
considering $eV$ only near zero-bias without offset along $y$ axis. (c)
Conductance through the QD at zero-bias $eV=0$ for increasing number of
$\langle\nu\rangle$. (d) Colormap showing the conductance behavior as a
function of both the $eV$ and $\langle\nu\rangle$, for a range of $eV$
corresponding to gray region of panel (a).

Figure 4: (a) Conductance through the QD [Eq. (6)] as a function of bias-
voltage $eV$ describing the case of quasi-MZMs
($\lambda_{c,R}\gg\varepsilon_{M}$) for increasing number of photon occupation
$\langle\nu\rangle$. For a better viewing, the curves are offset along the $y$
axis. (b) Same curves depicted in panel (a), but considering $eV$ only near
zero-bias without offset along $y$ axis. (c) Conductance through the QD at
zero-bias $eV=0$ for increasing number of $\langle\nu\rangle$ considering
distinct values of $\lambda_{c,R}$ . (d) Colormap showing the conductance
behavior as a function of both the $eV$ and $\langle\nu\rangle$, for a range
of $eV$ corresponding to gray region of panel (a). Dashed light-green lines
indicate the energies $\epsilon_{QD}^{+}$ and $\epsilon_{MZMs}^{+}$ of
anticrossing points employed to estimate the degree of Majorana nonlocality
$\Omega_{M}=\epsilon_{MZMs}^{+}/\epsilon_{QD}^{+}$, as proposed by Prada et
al. [27]. For the data shown in (d), $\Omega_{M}=0.35$.

Fig. 2 describes the conductance as a function of $eV$ when the QD is coupled
only with the left MZM ($\lambda_{c,L}=40\Gamma$), corresponding to the
situation of highly nonlocal MZMs ($\lambda_{c,R}=\varepsilon_{M}=0$,
$\eta=0$), for several values of $\langle\nu\rangle$. This ideal condition of
true topological MZMs is expected for clean disorder-free nanowires [26, 24],
long enough to avoid the overlap between the Majorana wavefunctions located at
the opposite ends [22]. In the absence of the cavity photons
($\langle\nu\rangle=0$), a robust ZBP with $G(0)=0.5e^{2}/h$ appears, as it is
shown by the solid red lines in Figs. 2(a) and (b), characterizing the ‘half-
fermionic’ nature of an isolated MZM leaking into the QD [30, 47, 11, 45]. The
conductance peak localized at $eV\approx 100\Gamma$ corresponds to the
conduction level of the dot renormalized by the finite QD-left MZM coupling.

For a finite photon occupation $\langle\nu\rangle=500$ [solid blue line, Fig
2(a)], optically-induced transitions between valence and conduction levels of
the QD come into play, splitting the single peak associated with the QD
conduction level into two polariton peaks located at $eV\approx 50\Gamma$ and
$eV\approx 150\Gamma$. The higher is the value of $\langle\nu\rangle$, the
bigger is the distance between the polariton peaks [Fig. 2(a), dash-dotted
green and dotted black lines]. This behavior resembles the Rabi splitting for
an individual dot inside a cavity, for which
$2\Omega_{R}\sqrt{\langle\nu\rangle}$ [48] [Fig. 1(b)]. Note, however, that
for sufficiently high values of $\langle\nu\rangle$ additional polariton peaks
stemming from indirect MZM to photon coupling of lower amplitude appear (see
black and magenta curves).

The amplitude of the peak at zero-bias remains unchanged with increase of the
number of the photons [Figs 2(b)]. This robustness is characteristic for the
situation of highly-nonlocal MZMs [27, 24, 21, 22]. In the same time, the
width of the ZBP is monotonously increasing with increase of the number of the
cavity photons, which means that the latter affect the effective lifetime of
the electronic states of the dot and MZMs.

In Fig. 2(c), the conductance behavior at zero-bias as a function of
$\langle\nu\rangle$ is shown for several values of $\lambda_{c,L}$. A well-
defined plateau of $0.5e^{2}/h$, independent on the mean photon occupation,
characterizes the robustness of the ZBP for the cases of strong QD-nanowire
coupling $\lambda_{c,L}\geq\Gamma$ [magenta stars and red circles]. For
smaller values of $\lambda_{c,L}$, the plateau is destroyed, as can be seen in
the green line with triangles of Fig. 2(c), corresponding to
$\lambda_{c,L}=10^{-2}\Gamma$. When the QD is totally decoupled from the
Majorana nanowire [solid purple line, $\lambda_{c,L}=0$], the conductance at
$eV=0$ exhibits a single resonance at $\langle\nu\rangle=1600$, which
corresponds to the crossing between the photon-induced lower polariton peak
and Fermi energy.

The colormap of Fig. 2(d) summarizes the conductance behavior as a function of
both $eV$ and $\langle\nu\rangle$, corresponding to a range of bias-voltage
within the gray region of Fig. 2(a). One can notice that the photon-induced
peak above $eV=0$ have its amplitude reduced with the co-emergence of a
conductance peak symmetrically localized below $eV=0$ and a ZBP with unchanged
$0.5e^{2}/h$ height pinned at zero-bias. Direct comparison between profile of
Fig. 2(d) and the corresponding results for highly nonlocal MZMs found by
Prada et al. [27] and others [11, 31, 12, 49] shows that the increasing of the
mean photon occupation (i.e. increasing the pump intensity) is a fast indirect
way of tuning the QD energy without applying any gate-voltage due to
renormalization of the QD spectrum in the strong light-matter coupling regime,
which bridges Quantum Optics and the physics of MZMs.

Fig. 3 shows the profiles of conductance through QD coupled to the Majorana
nanowire for the case of MZMs still localized at opposite wire ends, but with
finite overlap between them ($\varepsilon_{M}=10\Gamma\gg\lambda_{c,R}$),
which may describe the situation of shorter nanowires [22]. For a null photon
occupation, $G(eV)$ [Eq. (6)] is characterized by two near zero-energy peaks
at $eV\approx\pm\varepsilon_{M}$ [Figs. 3(a) and (b), dashed red lines] coming
from the ZBP splitting due to the finite overlap [29] and a third peak at
$eV\approx 100\Gamma$, corresponding to the QD conduction level renormalized
by the QD-nanowire coupling [Figs. 3(a), dashed red line]. As the mean photon
occupation is increased, in analogy to what happened for nonoverlapped MZMs,
the peak corresponding to the QD conduction level splits into two polariton
peaks [Fig. 3(a), solid blue line], with the value of this splitting depending
on $\langle\nu\rangle$. It can be seen that the increase of the mean photon
occupation strongly affects the pattern of near zero-energy peaks associated
with overlapped MZMs [see Fig. 3(b)]. First, they become closer to each other,
and coalesce into a single peak at zero-bias with height of $e^{2}/h$ for
$\langle\nu\rangle=1600$. Further increase of the number of cavity photons
leads to the reappearance of the splitting.

The conductance profile at $eV=0$ as a function of the mean photon occupation
is shown in Fig. 3(c). One can notice that the zero-bias conductance plateau
corresponding to nonverlapped and highly nonlocal MZMs [solid red line] is
destroyed even by a small overlap [dashed blue line]. For bigger values of the
overlap strength $\varepsilon_{M}$, $G(eV=0)$ as a function of
$\langle\nu\rangle$, exhibits a single peak at $\langle\nu\rangle=1600$,
showing the behavior similar to those corresponding to a QD embedded in an
optical cavity and decoupled from the nanowire [Fig. 2(a), solid purple line].
This similarity characterizes the single-fermion nature of the state formed
from the overlap between the two half-fermionic MZMs.

The conductance through the QD as a function of both $eV$ and photon
occupation, corresponding to the $eV$-range within the gray area in Fig. 3(a),
is shown in Fig. 3(d). One can see that the photon-induced transitions in the
QD renormalize its spectrum, giving rise to a ‘bowtie-like’ shape, which
resembles those found by Prada et al. [27].

In Fig. 4 we display the case of quasi-MZMs (a ps-ABS), which corresponds to
MZMs which have lost their nonlocal feature ($\eta\rightarrow 1$). This
situation describes a scenario where the right MZM is displaced towards to
left edge of the nanowire, leading to a partial separation between the MZMs
and hence a finite overlap between the wavefunction of the right MZM and the
QD ($\lambda_{c,R}\gg\varepsilon_{M}$) [27, 20]. The emergence of this quasi-
MZMs with partial separation can result from inhomogeneous confining
potentials in the nanowire [28, 22, 24, 21]. Similarly to Fig. 3(a)-(b), we
can notice two near-zero-energy conductance peaks at
$eV\approx\pm\lambda_{c,R}$ for the case of zero photon occupation [Fig.
4(a)-(b), dashed red line], which points that the quasi-MZMs leak into the dot
with different intensities ($\lambda_{c,L}\gg\lambda_{c,R}$) and form a state
of a regular fermionic character.

The peak corresponding to the QD conduction level at $eV\approx 100\Gamma$ is
again splitted into two polariton peaks in presence of cavity photons, which
also affects the position of the near-zero-energy conductance peaks [Fig.
4(b)]. But distinct from the previous case of overlapped MZMs localized at the
nanowire edges [Fig. 3(b)], there is no coalescence into a single peak at
$eV=0$ for certain value of $\langle\nu\rangle$.

Fig. 4(c) displays the evolution of zero-bias conductance as a function of
$\langle\nu\rangle$ for increasing values of coupling between the right-MZM
and the QD. For the ideal situation of highly nonlocal MZMs [solid red line],
one can see a ZBP plateau associated to the isolated MZM which has leaked into
the dot. A plateau is still present for a tiny value of $\lambda_{c,R}$
[dashed orange line], but with a reduction of the ZBP height
($G(eV=0)<0.5e^{2}/h$). However, a little enhancement of $\lambda_{c,R}$
[dotted green and dash-dotted blue lines] drops the zero-bias conductance to
almost zero for any value of $\langle\nu\rangle$.

The behavior of the conductance through the QD as a function of $eV$ and
photon occupation $\langle\nu\rangle$ is shown in Fig. 4(d). One can clearly
notice that the near-zero-energy peaks corresponding to quasi-MZMs slightly
move away from each other as photon-induced peak [upper branch in Fig. 4(d)]
is driven towards $eV=0$ and then start to approach each other again as the
photon occupation is increased. This behavior yields a pattern that resembles
the ‘diamond-like’ profile reported by Prada et al. [27]. In this work, it was
suggested, that the degree of Majorana nonlocality can be extracted from the
anticrossing points between the QD level and the quasi-MZMs levels by
application of a QD-gate voltage [27, 12].

Similarly, in the geometry considered by us, one can also extract the
information about the degree of Majorana nonlocality from the photon-induced
conductance profile of Fig. 4(d). The dot spectrum now is optically, and not
electrically, and the corresponding anticrossing points
$\epsilon^{+}_{\text{QD}}$ and $\epsilon^{+}_{\text{MZM}}$ are indicated by
dashed green lines. As originally proposed by Prada et al. [27], the degree of
Majorana nonlocality in tunneling conductance profiles [12] is given by
$\Omega_{M}^{2}\approx\eta^{2}=\epsilon^{+}_{\text{MZM}}/\epsilon^{+}_{\text{QD}}$,
valid when $\varepsilon_{M}\ll\lambda_{c,L},\lambda_{c,R}$. For the parameters
corresponding to Fig. 4(d), $\Omega_{M}\approx 0.35$, or equivalently, a
topological quality factor of $\approx 0.88$ [31, 32], thus indicating that
the MZMs are not well-localized at the nanowire edges, giving rise to quasi-
MZMs.

Conclusions.— In summary, we have theoretically explored the effects of strong
light-matter coupling on transport properties of the hybrid device, consisting
on quantum dot embedded inside a single-mode optical cavity and coupled to a
Majorana nanowire. Conductance profiles as functions of the bias-voltage and
mean photon occupation number controlled by an external pump revealed distinct
shapes for the cases of highly nonlocal MZMs, overlapped MZMs localized at
opposite nanowire edges and quasi-MZMs (ps-ABS). This makes possible to access
the degree of Majorana nonlocality (topological quality factor) all optically,
by means of the tuning of the polariton energies in the dot.

Acknowledgments.— LSR, VKK and IAS acknowledge support from Icelandic Research
Fund (project “Hybrid polaritonics”) and Russian Science Foundation tion
(project 20-12-00224). ACS acknowledges support from Brazilian National
Council for Scientific and Technological Development (CNPq), grant
305668/2018-8.

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Supplemental Material

##  I Conduction band Green’s function derivation

The quantum dot conduction band Green’s function
$\bm{G}^{c}(\omega)\equiv\langle\langle
d_{c};d_{c}^{\dagger}\rangle\rangle_{\omega}$ [Eq. (8) in the main text] can
be derived through successive applications of the equation-of-motion (EOM)
technique [1, 2]. For retarded Green’s functions in the spectral domain, the
EOM reads

$(\omega+\imath\delta)\bm{G}_{A_{i},B_{j}}(\omega)=[A_{i},B_{j}^{\dagger}]_{+}+\langle\langle[A_{i},\mathcal{H}];B_{j}^{\dagger}\rangle\rangle_{\omega},$
(S1)

where $\bm{G}_{A_{i},B_{j}}(\omega)\equiv\langle\langle
A_{i};B_{j}^{\dagger}\rangle\rangle_{\omega}$ is the retarded Green’s function
in the notation adapted from Zubarev [2, 3], $\delta\rightarrow 0^{+}$ is a
positive infinitesimal number, $A_{i}$ and $B_{j}$ are operators belonging to
the Hamiltonian $\mathcal{H}$ of the system [Eq. (1) in the main text] and
$[\cdots,\cdots]_{+}$ is the standard anticommutation relation for fermions
[1]. Considering $A_{i}=B_{j}=d_{c}$:

$(\omega-\omega_{c}+\imath\delta)\bm{G}^{c}(\omega)=1+\sqrt{2}V\sum_{\bm{k}}\langle\langle
c_{\bm{k},e};d_{c}^{\dagger}\rangle\rangle-\Omega_{R}\langle\langle
d_{v}c;d_{c}^{\dagger}\rangle\rangle_{\omega}-t_{c}\langle\langle
f;d_{c}^{\dagger}\rangle\rangle_{\omega}-\Delta_{c}\langle\langle
f^{\dagger};d_{c}^{\dagger}\rangle\rangle_{\omega},$ (S2)

with $t_{c}=(\lambda_{L}-\lambda_{R})/\sqrt{2}$,
$\Delta_{c}=(\lambda_{L}+\lambda_{R})/\sqrt{2}$ and
$f=(\gamma_{L}+\imath\gamma_{R})/\sqrt{2}$ is a complex fermionic operator
built from combination of left ($\gamma_{L}$) and right ($\gamma_{R}$) MZMs
[4], and can acquire a nonlocal feature depending on the distance between
these ‘half-fermionic’ Majorana states. As noticed in Eq. (S2), the high-order
Green’s function $\langle\langle
d_{v}c;d_{c}^{\dagger}\rangle\rangle_{\omega}$ arises due to the light-matter
coupling term given by $\mathcal{H}_{int}$ [Eq. (2) in the main text].
According to the EOM:

$(\omega+\imath\delta)\langle\langle
d_{v}c;d_{c}^{\dagger}\rangle\rangle_{\omega}=[d_{v}c,d_{c}^{\dagger}]_{+}+\langle\langle[d_{v}c,\mathcal{H}];d_{c}^{\dagger}\rangle\rangle_{\omega}=\langle\langle[d_{v}c,\mathcal{H}];d_{c}^{\dagger}\rangle\rangle_{\omega},$
(S3)

once $[d_{v}c,d_{c}^{\dagger}]_{+}=0$ and the commutation relation

$[d_{v}c,\mathcal{H}]=\omega_{0}d_{v}c+\omega_{v}d_{v}c-\Omega_{R}(d_{v}d_{v}^{\dagger}d_{c}+c^{\dagger}cd_{c}).$
(S4)

Thus, Eq. (S3) becomes

$(\omega-\omega_{0}-\omega_{v}+\imath\delta)\langle\langle
d_{v}c;d_{c}^{\dagger}\rangle\rangle_{\omega}=-\Omega_{R}\langle\langle
d_{v}d_{v}^{\dagger}d_{c};d_{c}^{\dagger}\rangle\rangle_{\omega}-\Omega_{R}\langle\langle
c^{\dagger}cd_{c};d_{c}^{\dagger}\rangle\rangle_{\omega},$ (S5)

where new high-order Green’s functions $\langle\langle
d_{v}d_{v}^{\dagger}d_{c};d_{c}^{\dagger}\rangle\rangle_{\omega}$ and
$\langle\langle c^{\dagger}cd_{c};d_{c}^{\dagger}\rangle\rangle_{\omega}$
arises. In order to find an analytical expression for $\langle\langle
d_{v}c;d_{c}^{\dagger}\rangle\rangle_{\omega}$ and avoid cumbersome
calculations, we prevent the emergence of more high-order Green’s functions in
the EOM process by considering the following truncation:

$\langle\langle
d_{v}d_{v}^{\dagger}d_{c};d_{c}^{\dagger}\rangle\rangle_{\omega}=\langle
d_{v}d_{v}^{\dagger}\rangle\langle\langle
d_{c};d_{c}^{\dagger}\rangle\rangle_{\omega}=(1-\langle
d_{v}^{\dagger}d_{v}\rangle)\langle\langle
d_{c};d_{c}^{\dagger}\rangle\rangle_{\omega}=\langle
d_{c}^{\dagger}d_{c}\rangle\langle\langle
d_{c};d_{c}^{\dagger}\rangle\rangle_{\omega},$ (S6)

wherein we considered the one-electron assumption $\langle
d_{c}^{\dagger}d_{c}\rangle+\langle d_{v}^{\dagger}d_{v}\rangle=1$ valid for
all times [5] and

$\langle\langle
c^{\dagger}cd_{c};d_{c}^{\dagger}\rangle\rangle_{\omega}=\langle
c^{\dagger}c\rangle\langle\langle
d_{c};d_{c}^{\dagger}\rangle\rangle_{\omega}.$ (S7)

Without loss of generality, in the truncation scheme of Eqs. (S6) and (S7) we
turned the high-order Green’s functions of Eq. (S5) into the conduction level
Green’s function $\langle\langle d_{c};d_{c}^{\dagger}\rangle\rangle_{\omega}$
modulated by the associated mean number of excitations in the QD $\langle
d_{c}^{\dagger}d_{c}\rangle=\langle n_{c}\rangle$ and the mean number of
photons within the optical cavity $\langle c^{\dagger}c\rangle=N_{ph}$,
respectively. Thus, Eq. (S5) reads

$(\omega-\omega_{0}-\omega_{v}+\imath\delta)\langle\langle
d_{v}c;d_{c}^{\dagger}\rangle\rangle_{\omega}=-\Omega_{R}(\langle
n_{c}\rangle+N_{ph})\langle\langle
d_{c};d_{c}^{\dagger}\rangle\rangle_{\omega}.$ (S8)

Considering that the $\langle n_{c}\rangle+N_{ph}=\langle\nu\rangle$ gives the
mean photon occupation in the cavity, Eq. (S8) turns into

$\langle\langle
d_{v}c;d_{c}^{\dagger}\rangle\rangle_{\omega}=\frac{-\Omega_{R}\langle\nu\rangle}{(\omega-\omega_{0}-\omega_{v}+\imath\delta)}\langle\langle
d_{c};d_{c}^{\dagger}\rangle\rangle_{\omega}.$ (S9)

Substituting Eq. (S9) into (S2), we find

$(\omega-\omega_{c}+\imath\delta)\bm{G}^{c}(\omega)=1+\sqrt{2}V\sum_{\bm{k}}\langle\langle
c_{\bm{k},e};d_{c}^{\dagger}\rangle\rangle+\Sigma_{ph}^{c}(\omega)\bm{G}^{c}(\omega)-t_{c}\langle\langle
f;d_{c}^{\dagger}\rangle\rangle_{\omega}-\Delta_{c}\langle\langle
f^{\dagger};d_{c}^{\dagger}\rangle\rangle_{\omega},$ (S10)

where
$\Sigma_{ph}^{c}(\omega)=\frac{\Omega_{R}^{2}\langle\nu\rangle}{\omega-\omega_{0}-\omega_{v}+i\delta}$
[Eq. (9) in the main text] is the self-energy associated to the valence-to-
conduction level transition in the quantum dot induced by the single-mode
photonic field of the cavity.

At this point, the Green’s function of Eq. (S10) that mixes the even
conduction operator $c_{\bm{k},e}$ from the leads with the operator $d_{c}$
from the dot should be calculated. According to the EOM procedure, this
Green’s function reads

$\langle\langle
c_{\bm{k},e};d_{c}^{\dagger}\rangle\rangle_{\omega}=\frac{\sqrt{2}V}{\omega+\imath\eta-\epsilon_{\bm{k}}}\langle\langle
d_{c};d_{c}^{\dagger}\rangle\rangle_{\omega}$ (S11)

Substitution of the expression above into Eq. (S10) yields

$(\omega-\omega_{c}+\imath\delta)\bm{G}^{c}(\omega)=1+\Sigma_{lead}(\omega)\bm{G}^{c}(\omega)+\Sigma_{ph}^{c}(\omega)\bm{G}^{c}(\omega)-t_{c}\langle\langle
f;d_{c}^{\dagger}\rangle\rangle_{\omega}-\Delta_{c}\langle\langle
f^{\dagger};d_{c}^{\dagger}\rangle\rangle_{\omega},$ (S12)

where
$\Sigma_{\text{lead}}(\omega)=\sum_{\bm{k}}\frac{2V^{2}}{\omega+\imath\eta-\epsilon_{\bm{k}}}$
is the self-energy due to the coupling between the QD and the even conduction
operators of metallic leads [see Eq.(4) of main text]. In the wide-band limit,
$\text{Re}[\Sigma_{\text{lead}}(\omega)]\rightarrow 0$ and this self-energy is
reduced to $-i\Gamma$, which is independent of $\omega$, with $\Gamma=2\pi
V^{2}\rho$ [6] as defined in the main text. Thus, Eq. (S12) reads

$(\omega-\omega_{c}+\imath\delta)\bm{G}^{c}(\omega)=1-\imath\Gamma\bm{G}^{c}(\omega)+\Sigma_{ph}^{c}(\omega)\bm{G}^{c}(\omega)-t_{c}\langle\langle
f;d_{c}^{\dagger}\rangle\rangle_{\omega}-\Delta_{c}\langle\langle
f^{\dagger};d_{c}^{\dagger}\rangle\rangle_{\omega}$ (S13)

Also through application of EOM [Eq. (S1)], the Green’s functions of Eq. (S10)
which mix the operator of the dot conduction level with the fermionic operator
built from Majorana modes are given by

$\langle\langle
f;d_{c}^{\dagger}\rangle\rangle_{\omega}=\frac{-t_{c}\langle\langle
d_{c};d_{c}^{\dagger}\rangle\rangle_{\omega}+\Delta_{c}\langle\langle
d_{c}^{\dagger};d_{c}^{\dagger}\rangle\rangle_{\omega}}{\omega-\varepsilon_{M}+\imath\delta}$
(S14)

and

$\langle\langle
f^{\dagger};d_{c}^{\dagger}\rangle\rangle_{\omega}=\frac{t_{c}\langle\langle
d_{c}^{\dagger};d_{c}^{\dagger}\rangle\rangle_{\omega}-\Delta_{c}\langle\langle
d_{c};d_{c}^{\dagger}\rangle\rangle_{\omega}}{\omega+\varepsilon_{M}+\imath\delta}.$
(S15)

Hence, Eq. (S13) reads

$(\omega-\omega{}_{c}+i\delta)\bm{G}^{c}(\omega)=1-\imath\Gamma\bm{G}^{c}(\omega)+\Sigma_{ph}^{c}(\omega)\bm{G}^{c}(\omega)+\kappa_{1}(\omega)\bm{G}^{c}(\omega)-t_{c}\Delta_{c}\kappa_{0}(\omega)\langle\langle
d_{c}^{\dagger};d_{c}^{\dagger}\rangle\rangle_{\omega}$ (S16)

with

$\kappa_{1}(\omega)=\frac{t_{c}^{2}}{\omega-\varepsilon_{M}+\imath\delta}+\frac{\Delta_{c}^{2}}{\omega+\varepsilon_{M}+\imath\delta}\,\,\,\,\text{and}\,\,\,\,\kappa_{0}(\omega)=\frac{1}{\omega-\varepsilon_{M}+\imath\delta}+\frac{1}{\omega+\varepsilon_{M}+\imath\delta}.$
(S17)

In Eq. (S16), one can notice the presence of the retarded Green’s function
$\langle\langle d_{c}^{\dagger};d_{c}^{\dagger}\rangle\rangle_{\omega}$
associated to the superconductivity in the quantum dot induced by the coupling
with the Majorana nanowire [7, 8, 9, 10] and, according to the EOM technique,
is given by

$(\omega+\omega_{c}+\imath\delta)\langle\langle
d_{c}^{\dagger};d_{c}^{\dagger}\rangle\rangle_{\omega}=-\imath\Gamma\langle\langle
d_{c}^{\dagger};d_{c}^{\dagger}\rangle\rangle_{\omega}+\tilde{\Sigma}_{ph}^{c}(\omega)\langle\langle
d_{c}^{\dagger};d_{c}^{\dagger}\rangle\rangle_{\omega}+t_{c}\langle\langle
f^{\dagger};d_{c}^{\dagger}\rangle\rangle_{\omega}+\Delta_{c}\langle\langle
f;d_{c}^{\dagger}\rangle\rangle_{\omega},$ (S18)

where
$\tilde{\Sigma}_{ph}^{c}(\omega)=\frac{\Omega_{R}^{2}\langle\nu\rangle}{\omega+\omega_{0}+\omega_{v}+\imath\delta}$
as defined in the main text. Substituting Eqs. (S14) and (S15) into Eq. (S18),
we find

$\langle\langle
d_{c}^{\dagger};d_{c}^{\dagger}\rangle\rangle_{\omega}=-t_{c}\Delta_{c}K(\omega)\bm{G}^{c}(\omega),$
(S19)

with

$K(\omega)=\frac{\kappa_{0}(\omega)}{\omega+\imath\Gamma+\omega_{c}-\tilde{\Sigma}_{ph}^{c}(\omega)-\tilde{\kappa}_{1}(\omega)}\,\,\,\,\text{and}\,\,\,\,\tilde{\kappa}_{1}(\omega)=\frac{t_{c}^{2}}{\omega+\varepsilon_{M}+\imath\delta}+\frac{\Delta_{c}^{2}}{\omega-\varepsilon_{M}+\imath\delta}.$
(S20)

Straightforward substitution of Eq. (S19) into Eq. (S16) yields

$\bm{G}^{c}(\omega)=\bm{g}_{0}(\omega)+\bm{g}_{0}(\omega)\left[\Sigma_{ph}^{c}(\omega)+\Sigma_{MZMs}(\omega)-\imath\Gamma\right]\bm{G}^{c}(\omega),$
(S21)

as the Dyson equation [1, 11] for the quantum dot conduction band, with the
self-energy [Eq. (10) in the main text]

$\Sigma_{MZMs}(\omega)=\kappa_{1}(\omega)+(t_{c}\Delta_{c})^{2}\kappa_{0}(\omega)K(\omega)$
(S22)

responsible for renormalizing the dot energy spectrum due to the coupling with
the MZMs located at the nanowire. The presence of
$\tilde{\Sigma}_{ph}^{c}(\omega)$ within $K(\omega)$ [Eq. (S20)] in the
resulting self-energy of Eq. (S22) reveals that although the Majorana nanowire
is decoupled from the optical cavity, the MZMs are indirectly affected by the
photon-induced transitions in the quantum dot.

Finally, by isolating $\bm{G}^{c}(\omega)$ in Eq. (S21), we can write the
expression for the Green’s function of the quantum dot conduction band as

$\bm{G}^{c}(\omega)=\frac{\bm{g}_{0}(\omega)}{1+\bm{g}_{0}(\omega)\left[\imath\Gamma-\Sigma_{ph}^{c}(\omega)-\Sigma_{MZMs}(\omega)\right]},$
(S23)

which is the same relation of Eq. (8) in the main text.

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