# A question-answering system for aircraft pilots’ documentation

Alexandre ARNOLD
Authors are in alphabetical order. Gérard DUPONT11footnotemark: 1
Félix FURGER11footnotemark: 1
Catherine KOBUS11footnotemark: 1
François LANCELOT11footnotemark: 1
Airbus AI Research
<EMAIL_ADDRESS>

(2020)

###### Abstract

The aerospace industry relies on massive collections of complex and technical
documents covering system descriptions, manuals or procedures. This paper
presents a question answering (QA) system that would help aircraft pilots
access information in this documentation by naturally interacting with the
system and asking questions in natural language. After describing each module
of the dialog system, we present a multi-task based approach for the QA module
which enables performance improvement on a Flight Crew Operating Manual (FCOM)
dataset. A method to combine scores from the retriever and the QA modules is
also presented.

##  1 Introduction

The aerospace industry relies on large collections of documents covering
system descriptions, manuals or procedures. Most of these are subjected to
dedicated regulation and/or have to be used in the context of safety of life
scenarios such as cockpit procedures for pilots. This documentation, called
Flight Crew Operating Manual (FCOM) incorporates aircraft manufacturer
guidance on how to use the systems on-board the aircraft for enhanced
operational safety, as well as for increased efficiency. Overall it can be
seen as several PDF documents which amounts for several thousand pages in
total for each aircraft type.

A user looking for specific information in response to a given situation in
this large corpus has to spend a lot of time navigating and searching through
the documents. For example, pilots can sometimes have difficulties in finding
known items in a constrained time [Arnold et al., 2019].

Search technologies are a way to address the structural complexity; however,
they come with their own limitations. Most of the time, it is the user’s
responsibility to define their search needs through specific query syntax and
refine the query until the right information is uncovered. This is known as
the difficulty of articulating information needs [Ying-Hsang and J, 2008]
[Wittek et al., 2016]. For simple queries that have a ready-made answer in the
document, this is not always a difficult problem. However, for the
understanding of complex procedures or for troubleshooting system errors, it
can lead to multiple queries thus a cumbersome search experience for the user.

The recent advances in natural language understanding and interactive search
system have attempted to reduce cognitive overhead. The use of natural
language conversation with the system can alleviate the users’ need to
understand the system’s query syntax or document structure. Coupled with high-
performing speech-to-text systems, it can even reduce the dependency to
physical inputs to free users hand, allowing better multitasking. In this
direction, conversational search agents appears to be a promising approach.
For a more complete review on data driven dialog systems, please refer to
[Serban et al., 2015] or [Radlinski and Craswell, 2017] for a promising
framework for conversational search.

The release of pretrained language models like BERT [Devlin et al., 2019]
enabled a boost in performance of lots of NLP downstream tasks, in question
answering (QA) in particular. It would not have been possible without the
emergence of QA datasets too, like SQuAD 2.0 [Rajpurkar et al., 2018a] or CoQA
[Reddy et al., 2019]. Now, best performing systems on those datasets are
mainly based on BERT fine-tuning approaches and are nearly approaching human
performance.

The main contribution of this work is to present a complete dialog pipeline
that allows pilots to access information by naturally interacting with the
system. We introduce a multi-task approach for the QA module, which improves
the QA performance on a Flight Crew Operating Manual (FCOM) dataset. A method
to combine scores from the retriever and the QA modules is also detailed.

This paper is organized as follows : in section 2, the overall architecture
with its different components is detailed, with a focus on the multi-task
approach for QA 2.3. Section 3 describes the dataset used in our experiments
and the results obtained. Section 4 presents a summary of the main findings
and future prospects.

While this paper focus on the system side of the study, the reader interested
by interactive user study that was executed using this system, is invited to
read the dedicated paper [Liu et al., 2020].

##  2 System description

We have developed a prototype system to address the evaluation objective of
determining the relationship between the types of search tasks and the
perceived usefulness of search. The system was built around three main
components (inspired by the DrQA proposal from [Chen et al., 2017]):

  * •

A dialog engine (based on RASA platform [Bocklisch et al., 2017]) handling the
conversation and identifying user’s intents;

  * •

A retriever : a search engine (based on Solr [Turnbull and Berryman, 2016])
where the documents collection is indexed following the BM25F relevance
framework [Robertson and Zaragoza, 2009];

  * •

A QA engine, based on a fined-tuned BERT large model [Devlin et al., 2019]. A
multi-task setup was used for the fine-tuning: one task is the classical QA
task (detecting the span of text) on SQUAD 2.0 dataset [Rajpurkar et al.,
2018a]; the other is a classification task (i.e. whether the answer to the
question is contained or not in the document extract). This QA module is
described more deeply in the section 2.3.

On top of these, additional capabilities to process speech inputs and produce
speech outputs are available as an alternative to the traditional textual
input. Figures 1 and 2 offer an overview of the whole architecture.

The whole system is made available through a reactive web interface enabling
conversation and document exploration (See Figure 3). It was deployed in a
cloud environment and made available to users through a tablet.

Figure 1: Overview of the prototype architecture. Figure 2: Overall
architecture with the dialog model as an orchestrator. Figure 3: Screenshot of
the prototype showing conversation on the left, search result panel in the
center and document view on the right.

###  2.1 The dialog engine

We used the open-source Rasa framework for the dialog engine [Bocklisch et
al., 2017], comprising:

  * •

Natural language understanding: recognizing high-level intent/entities from
raw user utterances (e.g. ”greeting”, ”positive/negative feedback” or
”question”)

  * •

Dialog policy: predicting the next best action (an utterance or a custom
action) based on current dialog state, including last recognized intent

Both components above were trained with machine learning pipelines provided in
Rasa, based on natural language and story examples: the former maps user
utterances to predefined intents/entities, the latter gives typical dialog
scenarios to learn & generalize from (to avoid building manually the whole
conversation state machine).

The core ”skill” of our dialog engine focuses on recognizing any generic
question from the user, mapping it to the ”question” intent and predicting the
trigger of a custom action (written in Python) which calls the retriever & QA
systems described in next sections to provide an answer. Examples of natural
language questions were built by combining open QA dataset questions with in-
house examples more related to our pilots’ documentation context. We also
integrated positive/negative feedback intents to be able to handle user
reactions after providing an answer: in case of negative feedback, a custom
action is triggered to propose the best answer from the document ranked just
below the one currently suggested.

A chitchat ”skill” (i.e. another sub-part of our conversational system) was
added to the core one, containing more than 50 typical small talk intents &
responses to make the dialog appear more human-like: ”greeting”, ”goodbye”,
”thanking”… Some chitchat user utterances might be in question form, but are
usually learnt not to be confused with the generic ”question” intent mentioned
above with enough training data.

###  2.2 The retriever

The first component consists of an information retrieval system which follows
the now classic architecture of most recent QA systems. It allows to filter
the overall document collection 1) to exclude non relevant documents and 2)
reduce the considered set of documents to a size that is compatible with the
foreseen response time.

First the document collection has been extracted from its original XML format
which includes simultaneously semantic and presentation tagging. The isolation
of each individual procedure has been done at this level as well as the
extraction of metadata such as unique identifier, applicability scope and
classification in the hierarchical ATA chapters111https://av-
info.faa.gov/sdrx/documents/JASC_Code.pdf. This allowed to define the minimal
granularity of the collection. The text content has then been pre-processed to
eliminate inconsistent formatting issues and improve quality of the terms
indexed such as resolving abbreviations meanings, adapting the numerical
representation of units and flattening table content to ensure headers and
legends are correctly indexed.

We compared different indexing scheme and the BM25F from [Robertson and
Zaragoza, 2009] allowed to offer the best performances (compared to tf/idf).

###  2.3 The QA system

####  2.3.1 BERT fine-tuning approach

The QA module in the pipeline is an extractive QA approach, where the model
extracts a span of text from a document to answer a natural language question.
The QA model is obtained by fine-tuning a BERT language model [Devlin et al.,
2019] for an extractive QA task on the SQuAD 2.0 dataset [Rajpurkar et al.,
2018b].

Almost all QA datasets are made of long documents that cannot fit in a
standard transformer model. Hence, for a given question and a document to
consider, the document is first decomposed into $n$ smaller passages; those
passages are provided, with the question to answer, as input to the QA engine;
this step results in $n$ predictions, that need to be aggregated to get a
final answer for the given question/document pair. The different steps are
summarized in figure 4.

The input 5 provided to the QA model contains tokens from both the question
and the passage, some special tokens and eventually some padding tokens (that
enables to reach the model’s maximum sequence length if needed).

Formally, we define a training set instance as a triple $(c,s,e)$, where $c$
is a context of a given size ($max\\_seq\\_len\in\\{384,512\\}$ in this study)
of wordpiece ids, corresponding to the question, to the passage considered, to
some special tokens and eventually some padding.
$(s,e)\in[0,max\\_seq\\_length]^{2}$ respectively refer to the start and end
of the target answer span when the answer span is contained in the passage;
they are both equal to $0$ otherwise.

During inference, the predictions for each passage need to aggregated to
extract, from the document, the answer (or not) to the question. If the best
prediction for each passage is no answer, then the final prediction is no
answer. Otherwise, the final prediction is built by picking up the answer span
kind prediction with the highest score.

The FARM framework 222https://github.com/deepset-ai/FARM was used to fine-
tuned the BERT models on the QA task. More details on the approach can be
found in this blog post 333https://towardsdatascience.com/modern-question-
answering-systems-explained-4d0913744097.

Figure 4: QA engine pipeline Figure 5: Input to the QA model (with a size
equal to the maximum sequence length).

####  2.3.2 Multi-task approach

Multi-Task Learning (MTL) aims at boosting the overall performance of each
individual task by leveraging useful information contained in multiple related
tasks. It has shown great success in Natural Language Processing (NLP). The
main idea of MTL is to leverage useful information contained in multiple
related tasks to improve the generalization performance of all the tasks [Yu
and Qiang, 2017]. Multi-task learning has been successfully used in many
applications from machine learning, from natural language processing [R. and
J., 2008] and speech recognition [Deng L., 2013] to computer vision [Girshick,
2015], etc.

With the MTL approach and following the formalism proposed in the technical
note [Alberti et al., 2019], a training set instance becomes a $4$-tuple
$(c,s,e,t)$ where $c$ is a context of a given size
($max\\_seq\\_len\in\\{384,512\\}$ in this study) of wordpiece ids,
corresponding to the question, to the passage considered, to some special
tokens and eventually some padding. $(s,e)\in[0,max\\_seq\\_length]^{2}$
respectively refer to the start and end of the target answer span. $t$ is the
tag associated to the sample with two possible values : $t=\mathbf{SPAN}$ if
the answer to the question is in the passage considered,
$t=\mathbf{NO\\_SPAN}$ otherwise. One can note this is streamlined version
compared to the one proposed by [Alberti et al., 2019] which proposed 5 labels
for $t$ and showed significant improvements.

Adding this classification head to the network, that tries to predict if it is
able to answer a question given the considered passage, should help the
overall performance of the QA engine.

During training, the losses of both tasks, question answering on one hand and
classification on the other hand, are summed. During inference, as for the
question answering task, for a question and a given document, all results from
all the samples of question/passage have to be aggregated to get at the end a
unique classification : ”$\mathbf{SPAN}$” if the answer is in the document,
”$\mathbf{NO\\_SPAN}$”. For this step, we went for a basic approach, that
consists in taking the classification tag from the passage with the highest
score for the classification head.

##  3 Experiments

In this study, different BERT models are fine-tuned on the SQuAD 2.0 dataset
[Rajpurkar et al., 2018b] using the open-source FARM, framework that easily
allows to fine-tune BERT model on classical downstream tasks like Question
Answering. The different hyper-parameters are synthesized in the table 1. The
code, that enables to fine-tune BERT models, with or without multi-tasking,
can be found in the following repository 444https://github.com/ckobus/FARM.

| BERT large
---|---
learning rate | $3\cdot 10^{-5}$
epochs | 2
max. seq. length | $\\{384,512\\}$
cased | True
batch size | $12$
gradient accumulation | $4$
Table 1: Values of different hyperparameters used for BERT models fine-tuning.

###  3.1 Data

The BERT models, fine-tuned, with or without fine-tuning, on the SQuAD 2.0
dataset [Rajpurkar et al., 2018b] are evaluated on both the SQuAD 2.0
development set and on a in-house FCOM dataset built through a crowd-source
procedure. Domain experts, knowledgeable about the FCOM or even authors of
FCOM procedures, were asked to construct a set of questions/answers,
highlighting precisely answers’s location in the document. More than $300$
questions were collected but with a large proportion of questions for which
the answer is contained in a table or or graphical parts of the document.

Figure 6: Example of a task.

The dataset was thus filtered so that it only contains questions for which
answer is a span of text in the document (see figure 6 for an example). The
final version of the FCOM dataset, limited to SQUAD2 type Q/A, contains $114$
questions. The objective was to focus first on this type of mostly factoid
questions. This dataset is of course much too limited for any domain training
or even fine-tuning. The goal was here to use it for evaluation only to assess
applicability of pre-trained literature models to the specificities of the
aerospace domain and documents.

The distribution of the questions, answers, and context length of the FCOM
dataset can be seen in figure 7. The average length of questions (in terms of
word tokens) is less than 9, while the average length of answers is 11. The
average length of context is 200 but some context sentences can be long (3072
words for example).

Figure 7: Distribution of the question, answer, and context length.

###  3.2 QA performance

The results obtained with the different BERT large models (with or without
whole word masking) on the 2 SQuAD datasets and on the FCOM internal dataset
are reported in terms of Exact Match (EM) and F1 scores, in table 2. The
results for the DrQA model (only the document reader part - see [Chen et al.,
2017]) have been reproduced and are presented for comparison.

Multiple conclusions can be drawn from those results :

  * •

While the multi-task approach brings limited improvements on the SQuAD 2.0
dataset, the improvement is quite significant in terms of both $EM$ and $F1$
scores on the FCOM dataset, for which the baseline performance is also quite
low compared to the SQuAD dataset; this result is expected because the FCOM
dataset is made of complex aeronautical documents, with a technical vocabulary
and specific phraseology, that is obviously not found in the SQuAD 2.0
dataset, on which the BERT model is fine-tuned; the multi-task approach
enables to get in average $18\%$ and $15\%$ respectively in terms of $EM$ and
$F1$ on the FCOM dataset;

  * •

The maximum sequence length hyperparameter has a limited impact on the SQuAD
dataset whereas it has a significant one on the FCOM dataset. The paragraphs
in the FCOM dataset are larger (in average $200$ words and some paragraphs can
contain more than $3k$ words) than the ones in the SQuAD datasets. Having a
larger maximum sequence length enables the QA model to capture a larger
context, which helps it in getting a better aggregated answer in the end;

  * •

The best performance are obtained with the whole word masking version of BERT
large model, which confirms the trend in the NLP community [Cui et al., 2019].
Hence, this wmm version of the model is used for the experiments about
retiever and QA scores combination in the following section 3.3.

Model | max_seq_length | Multi-task | SQUAD 2.0 | FCOM
---|---|---|---|---
|  |  | EM | F1 | EM | F1
DrQA | NA | No | $34.25$ | $39.21$ | $28.95$ | $48.57$
BERT large | $384$ | No | $77.54$ | $80.55$ | $21.92$ | $30.08$
Yes | $76.52$ | $79.86$ | $24.56$ | $34.40$
$512$ | No | $76.17$ | $79.27$ | $23.68$ | $33.62$
Yes | $76.70$ | $79.93$ | $29.82$ | $39.38$
BERT large (whole word masking) | $512$ | No | $81.60$ | $84.28$ | $29.82$ | $40.77$
Yes | 82.10 | 84.98 | 35.08 | 46.14
Table 2: QA systems evaluation on SQUAD, SQUAD 2.0 (development set) and on
FCOM datasets, without and with multi-tasking. DrQA relies on RNN so no
specific limit on sequence length and it was designed for the original SQUAD
(with no rejection) on which it obtains $69.08$ and $78.47$ in EM and F1.

###  3.3 Retriever and QA scores combination

In order to improve document ranking as given by the retriever alone (Solr),
we propose to also leverage the confidence score returned by the QA system
(BERT) for the document’s best answer span. The intuition is that a high
confidence in best answer span should partly reflect the document’s relevance
with regard to the question asked, thus an indication of a higher ranking. We
first introduce two simple baselines to combine retriever and QA scores into
one, which will serve to re-rank documents by sorting in ascending order:

  * •

Simple combination 1: retriever and QA scores are multiplied (multiplication
is preferred over addition here, since both scores have different scales and
orders of magnitude)

  * •

Simple combination 2: retriever and QA z-scores, computed from absolute
scores, are added (z-scores give a normalized relative scoring taking into
account other results in the set, thus might be more meaningful than raw
scores to know how good a document or answer is compared to the others)

Additionally we introduce a more advanced XGBoost combination, predicting the
re-ranking score using XGBoost, an efficient machine learning framework based
on gradient boosting. This has the ability to learn more complex combinations
than the linear interpolation proposed in [Yang et al., 2019]. Also we propose
to take as input features not only the raw scores of the retriever and QA
system, but their respective z-scores as well (as mentioned in the Simple
combination 2). The XGBoost model is trained with 100 rounds on 80% of the
FCOM dataset and tested on the remaining 20% (this in-house dataset includes
both the expected document and answer span in the ground truth). We compare
the different ranking systems with the nDCG@10 metric (normalized Discounted
Cumulative Gain accumulated at rank position 10), which is common for document
ranking evaluations.

Results are shown in table 3; they were obtained with the multi-task QA model,
fine-tuned from the BERT large model obtained with whole word masking and with
a maximum sequence size of $512$ (which gave the best results overall 2). For
the XGBoost part, more rounds of training (e.g. 300) or additional input
features (e.g. original document rank, question length…) did not improve
performance further - on the latter, it was even detrimental because of over-
fitting effects.

Ranker |  | Mean nDCG@10
---
(on test set)
Retriever alone | 0.86
QA system alone | 0.68
Simple combination 1 | 0.75
Simple combination 2 | 0.83
XGBoost combination | 0.97
Table 3: Evaluation of document ranking methods on FCOM dataset.

An example of the XGBoost document re-ranking is shown in figure 8 for the
question ”What is max crosswind for landing?”. For each scoring configuration
(pure document ranking vs XGBoost reranking), the top 10 answers are listed
with horizontal bars representing their scores. The right document containing
the answer, highlighted in red, is correctly ranked 1st with XGBoost, as
opposed to 5th with the original retriever ranking on this example.

Many possible re-ranking could be explored to improve the performance of this
component . However these might necessitate larger training corpora. Given the
small size of the FCOM dataset (only few hundreds Q/A pairs), it was chosen
not to focus too much on this single component of the system.

Figure 8: Example of XGBoost document re-ranking benefit

###  3.4 System overall performance

An interactive user experimentation was proposed to confirm the positive
perceptions of the offline performances measured. Our evaluation objective was
to _determine the relationship between the search tasks in the typical flight
operation scenarios and the perceived usefulness of the system for task
completion_ (see relevant literature on information-seeking strategies and
perceived usefulness of information resources [H et al., 2015, Freund, 2013,
Hertzum and Simonsen, 2019, Yuelin and J, 2008]).

The experiment was conducted in the environment of a flight simulator (ENAC
BIGONE A320/A330 cockpit simulator) within the ACHIL platform. The setting was
intended to create an environment that can elicit the information needs of
participants, as suggested in simulated work task situations [Pia and David,
2016]. The subjects were given access to a tablet - similar to the ones used
by the pilot in flight - to access the Flight Crew Operating Manual (FCOM)
through one of the two systems: our system, called Smart Librarian (SL) and
electronic flight bag (EFB), which is the system used currently by pilots and
is basically pdf viewer with a keyword search functionality.

Several search tasks with different complexity were designed for the user
experiment. Specifically, the easy task involves fact-finding (see figure 6)
while the hard task requires a higher level of understanding of the problems
and/or some cognitive reasoning for answering the questions. In easy search
tasks, the problem description contains relevant words that can be used to
craft the ”best question” pointing to a unique procedure (or document unit)
that contains the solution. By contrast, in hard search tasks, the problem
description does not contain any words matching the ”best question” and the
subject will need to rephrase the problem. Moreover, the user needs to explore
at least two document units to find the answer.

One of the conclusion is that our system (SL) was more helpful than the
current system (EFB) for the more complex search tasks. A complete description
of this user study including data analysis and conclusion about the experiment
are detailed in a dedicated paper [Liu et al., 2020].

##  4 Conclusion and Perspectives

In this paper, we proposed a multi-task approach for Question Answering, which
enables to improve QA engine’s performance especially in a technical domain
like the aeronautic one. We also proposed an innovative way to combine scores
from both the retriever and the QA module, which enables a combination of the
two modules.

There are a number of obvious improvements we might consider; a first obvious
one is to get more in-domain data, that will allow us to further fine-tune the
QA engine on in-domain data.

With respect to the multi-task approach, the way the predictions are
aggregated to get the final classification tag is a baseline and can be
improved.

An other point of improvement is related to tables handling; FCOM
documentations are technical and contain a lot of tables. QA engines should be
able to handle those tables and extract answer from them if needed. There are
already promising works around that topic [Pengcheng et al., 2020] [Herzig et
al., 2020] and around the recent Natural Question dataset [Kwiatkowski et al.,
2019].

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# Cavity-mediated collective emission from few emitters in a diamond membrane

Maximilian Pallmann† Physikalisches Institut, Karlsruhe Institute of
Technology (KIT), Wolfgang-Gaede Str. 1, 76131 Karlsruhe, Germany  Kerim
Köster† Physikalisches Institut, Karlsruhe Institute of Technology (KIT),
Wolfgang-Gaede Str. 1, 76131 Karlsruhe, Germany  Yuan Zhang  Institute of
Quantum Materials and Physics, Henan Academy of Sciences, Zhengzhou 450046,
China  School of Physics and Microelectronics, Zhengzhou University, Zhengzhou
450052, China  Julia Heupel  Institute of Nanostructure Technologies and
Analytics (INA), Center for Interdisciplinary Nanostructure Science and
Technology (CINSaT), University of Kassel, Heinrich-Plett-Straße 40, 34132
Kassel, Germany  Timon Eichhorn  Physikalisches Institut, Karlsruhe Institute
of Technology (KIT), Wolfgang-Gaede Str. 1, 76131 Karlsruhe, Germany  Cyril
Popov  Institute of Nanostructure Technologies and Analytics (INA), Center for
Interdisciplinary Nanostructure Science and Technology (CINSaT), University of
Kassel, Heinrich-Plett-Straße 40, 34132 Kassel, Germany  Klaus Mølmer  Niels
Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark  David
Hunger  Physikalisches Institut, Karlsruhe Institute of Technology (KIT),
Wolfgang-Gaede Str. 1, 76131 Karlsruhe, Germany

###### Abstract

${\dagger}$ These authors contributed equally.

When an ensemble of quantum emitters couples to a common radiation field,
their polarizations can synchronize and a collective emission termed
superfluorescence can occur. Entering this regime in a free-space setting
requires a large number of emitters with a high spatial density as well as
coherent optical transitions with small inhomogeneity. Here we show that by
coupling nitrogen-vacancy (NV) centers in a diamond membrane to a high-finesse
microcavity, also few, incoherent, inhomogeneous, and spatially separated
emitters - as are typical for solid state systems - can enter the regime of
collective emission. We observe a super-linear power dependence of the
emission rate as a hallmark of collective emission. Furthermore, we find
simultaneous photon bunching and antibunching on different timescales in the
second-order auto-correlation function, revealing cavity-induced interference
in the quantized emission from about fifteen emitters. We develop theoretical
models for mesoscopic emitter numbers to analyze the behavior in the Dicke
state basis and find that the population of collective states together with
cavity enhancement and filtering can explain the observations. Such a system
has prospects for the generation of multi-photon quantum states, and for the
preparation of entanglement in few-emitter systems.

††preprint: AIP/123-QED

##  I Introduction

Figure 1: Experimental setup and involved physics. a, An NV center ensemble in
a diamond membrane coupled to a fiber-based microcavity, where the NV centers
are excited incoherently and emit collectively. b, Simplified level scheme of
the NV center. The zero phonon line (ZPL) $\ket{g}\leftrightarrow\ket{e}$ is
resonant with the cavity, leading to selective Purcell enhancement of the
transition. The system is excited off-resonance via the phonon sideband (green
arrow). c, Photograph of the cavity setup, showing the fiber micro mirror
mounted in a steel needle, and the diamond membrane on a planar mirror. d,
Schematic drawing of the inhomogeneously broadened ensemble ZPL, consisting of
many individual transitions from which a few are overlapping with the narrow
cavity resonance. e, Exemplary level scheme of collective Dicke states for ten
identical two-level emitters where the quantum jumps due to the incoherent
pumping (green arrows), dephasing (black arrows), individual and collective
decay (red arrows) are indicated.

Collective emission of quantum emitters is an intriguing behavior that can
arise from the coupling of closely spaced atomic dipoles, where one of the
most prominent features is a time-delayed superradiant pulse that grows
quadratically in intensity with the emitter number dicke_coherence_1954 . It
has been subject of numerous studies andreev_collective_1980 ;
gross_superradiance_1982 ; raimond_statistics_1982 ; scully_super_2009 ;
cong_dicke_2016 ; raino_superfluorescence_2018  and currently spurs great
interest for possible applications, such as superradiant lasers
meiser_prospects_2009 ; bohnet_steady-state_2012 , quantum memories
ferioli_storage_2021 ; shen_subradiant_2022 ; lei_many-body_2022 ;
rastogi_superradiance-mediated_2022  and quantum metrology
paulisch_quantum_2019 .

Notably, even incoherent excitation can lead to spontaneous synchronization of
emitters, an effect called superfluorescence (SF) bonifacio_quantum_1971 ;
bonifacio_cooperative_1975 . This requires emitters with sufficient optical
coherence and a high spatial density such that the average distance between
two emitters is smaller than the transition wavelength. Reaching this regime
can be challenging in a solid state environment, as emitters in dense
ensembles typically suffer from effects like inhomogeneous and homogeneous
broadening and spectral diffusion davies_jahn-teller_1981 ; tamarat_stark_2006
; wolters_measurement_2013 . Recently, superfluorescence from perovskite
quantum dot superlattices could be observed, which showed narrow ensemble
emission lines under cryogenic conditions raino_superfluorescence_2018 .
Surprisingly, also under strong phonon broadening at room temperature, high-
density ensembles of nitrogen vacancy centers in nanodiamonds showed
signatures of superfluorescence bradac_room-temperature_2017 . In both cases,
a high density and a large number of emitters were crucial for the emergence
of collective emission. These requirements can be relaxed by coupling the
emission to a high-finesse microcavity, which acts as a single-mode common
light field that can couple also distant emitters and provides full spatial
coherence Auffeves_few_2011 ; Leyman_2015 . Furthermore, the spontaneous
emission rate is increased due to the Purcell effect purcell_resonance_1946 ,
and photons are funneled into a spectrally narrow cavity resonance
grange_cavity-funneled_2015 , which improves their spectral coherence.

In an idealized picture, a system of $N$ identical emitters can be described
in a basis of collective Dicke states $|J,M\rangle$ analogous to a number of
spin $1/2$ systems that form a collective spin $J=0,1/2...N/2$ with a
projection $M=-J...J$ that represents the overall number of excitations $J+M$.
As shown in Fig. 1e, the collective decay preserves $J$ and leads to quantum
jumps between states with different $M$, while dephasing and individual decay
cause quantum jumps also to states with different $J$. The relative rates of
the jumps are illustrated by the thickness of the arrows, and their precise
values can be found in Ref. Zhang_2018 ; Shammah_2018 . Overall, dephasing
tends to reduce $J$ and lead to the population of subradiant states close to
the lower boundary of the triangle. Due to their suppressed collective
coupling to the cavity mode, these states promote excitation to higher levels
under continuous pumping. These, in turn, can decay via the emission of
multiple photons, leading to photon bunching Auffeves_few_2011 ;
Bhatti_super_2015 , see Fig. 1e. This has been observed for ultracold atoms
hennrich_transition_2005 , and for larger numbers of solid state emitters
Wiersig_Direct_2009 ; Jahnke_2016 ; bradac_room-temperature_2017 ;
raino_superfluorescence_2018 .

Here, we report the observation of collective emission from a small ensemble
of nitrogen-vacancy (NV) centers in diamond which display inhomogeneous and
homogeneous broadening. They are dispersed over a few-micron large area in a
diamond membrane and are resonantly coupled to a microcavity, see Fig. 1.
Remarkably for such sample properties, we observe a super-linear increase of
the emission rate with increasing incoherent excitation, indicating the onset
of collective emission. On the other hand, we observe the hallmark of
quantized emission of individual emitters with anti-bunching in the second-
order correlation function. Furthermore, on a short time scale, we observe a
sharp bunching peak. Notably, all these features are solely visible for the
coherent zero-phonon-line (ZPL) photons at low temperature, while they are
absent for the incoherent phonon side band (PSB) and at room temperature. To
analyze the findings, we develop two complementary theoretical models that can
treat the radiation of mesoscopic emitter numbers coupled to a cavity. Thereby
we can reproduce the observations and obtain a detailed insight into the
contribution of the population of collective Dicke states and the influence of
the emitter number, dephasing, and inhomogeneity.

##  II Cavity-enhanced Fluorescence

We incorporate NV centers in a thin membrane into a fully tunable, fiber-based
microcavity with a quality factor of $Q=4.7\times 10^{5}$ at the wavelength of
the ZPL of the NV centers, see Fig. 1a,b and c. The cavity linewidth is much
narrower than the inhomogeneously broadened ZPL of the ensemble (Fig. 1d),
such that a small number of resonantly coupling NV centers is spectrally
selected. These emitters couple with a Purcell-enhanced rate to a single
cavity mode.

Figure 2: Cavity-enhanced fluorescence of NV centers. a, PL spectrum of an
ensemble in the cavity (red) and in a confocal microscope (blue). The inset
illustrates the decreased ZPL count rate as the cavity resonance is detuned
from the transition by slightly changing the mirror separation. b, Cavity-
enhanced spectroscopy of the ensemble ZPL, obtained by recording fluorescence
spectra while step-wise changing the cavity length. The integrated counts
reveal the inhomogeneous ZPL with a FWHM linewidth of
$\delta\nu_{\mathrm{inh}}\approx 300$ GHz. c, Comparison of ZPL and PSB
fluorescence count rates, coupled to the same spatial cavity mode, in
dependence of the excitation laser power. The ZPL features a super-linear
increase, a characteristic of superfluorescent emission. The inset shows
exponents obtained under different conditions like cavity length or lateral
position.

We study the cavity-enhanced fluorescence by recording photoluminescence (PL)
spectra under off-resonant excitation. Therefore, the cavity fiber was
laterally positioned such that the fundamental Gaussian mode shows an air-like
character janitz_fabry-perot_2015  at
$\lambda=$637\text{\,}\mathrm{n}\mathrm{m}$$ with a finesse of
$\mathcal{F}=12000$. When tuning a cavity resonance to the ZPL, the resulting
spectrum shows a series of sharp cavity resonances, with a dominant line at
the ZPL wavelength (Fig. 2a). Within the spectrally broad phonon sideband, a
large number of additional fundamental and higher order transverse cavity
modes is visible. The measurement also illustrates the improved ratio of ZPL
and phonon side band (PSB) photons, as compared to a confocal measurement at
room temperature. This results from the Purcell effect, which leads to an
increased emission rate and thereby an improved branching ratio, a higher
emission directionality, leading to a better collection efficiency, and the
suppression of off-resonant fluorescence by cavity spectral filtering. We
remark that while for the cavity-enhanced ZPL emission, only about fifteen
emitters contribute due to spectral filtering, all emitters (several hundred)
spatially located within the cavity mode contribute to the resonances in the
phonon side band, such that the ZPL enhancement is in fact more than an order
of magnitude larger than is visible in Fig. 2a. By applying a step-wise
increasing voltage to the z-piezo and thus controlling the cavity length, the
cavity resonance is tuned across the ensemble ZPL, see Fig. 2b. Since the
spectral cavity linewidth of
$\delta\nu_{\mathrm{cav}}=$1.0\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ is
much narrower than the inhomogeneously broadened ZPL, we can obtain a high-
resolution spectrum. The summed counts reveal the spectral distribution of ZPL
transitions, yielding a full width at half maximum (FWHM) of
$\delta\nu_{\mathrm{inh}}\approx$300\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$.
When the cavity resonance frequency matches the center frequency of the ZPL,
an up to 4-fold increase of fluorescence counts can be observed.

In a next step, we measure the power dependence of the fluorescence spectra
under off-resonant excitation and compare the partially coherent emission into
the ZPL with the red-shifted incoherent emission into the phonon side band.
While the power dependence for the PSB emission matches the expected linear
scaling when far from saturation, the trend for the ZPL emission clearly
exhibits a super-linear behaviour following a power-law dependence $I\sim
P^{k}$ with an exponent as high as $k=1.42$, see Fig. 2c. This is reproducible
under various conditions like different cavity length or lateral position on
the sample (inset of Fig. 2c) . The super-linear behaviour was only observed
at cryogenic temperatures ($T=11$ K), in accordance with the prediction from a
theoretical model (see below and SI). For an ensemble of superradiant emitters
under pulsed excitation, one expects up to $k=2$. The lower value observed in
the experiment can be explained by the inhomogeneity and the pure dephasing of
the emitters, which leads to a population of Dicke states towards low $J$
values. These states decay at a lower rate, which corresponds to a smaller
value for $k$.

Figure 3: Temporal photon statistics of NV fluorescence. a, Second-order
autocorrelation function of the emitted photons. The cavity enhanced
fluorescence (red data points) reveals bunching on a sub-ns timescale, as
opposed to the fluorescence recorded in a confocal microscope (blue data
points). The PSB emission in the cavity setup (grey data points, shifted by
0.02 for clarity) does not show any bunching or antibunching. b, Enlarged view
of the bunching around $\tau=0$ together with the measured instrument response
function. A modeled $g^{(2)}(\tau)$ function is convolved with the instrument
response to show the agreement with the background-corrected experimental
data. c, Antibunching time constants for different excitation laser
intensities, extracted from the respective correlation functions. The fits
correspond to a rate equation model, and allow one to infer the effective
Purcell factor $F_{P,eff}=1.1\pm 0.3$. d, From the measured antibunching
amplitude, one can extract the number of contributing emitters $N=15\pm 3$.
For $N$ individually decaying emitters, one expects a thermal bunching
amplitude of $2(1-1/N)$ (solid grey line), which coarsely agrees with our
measured value (orange data point). The prediction of our model is shown for
comparison (dashed grey line). e, A set of $g^{(2)}$ functions for increasing
temperature reveals that the bunching amplitude is robust against further
dephasing.

##  III Photon statistics

Further insight can be obtained by studying the photon statistics of the
radiation emitted from the cavity. Therefore, we use a Hanbury-Brown and Twiss
(HBT) setup and record the second-order autocorrelation function
$g^{(2)}(\tau)$. An exemplary $g^{(2)}(\tau)$ of the cavity-enhanced ZPL
fluorescence is shown in Fig. 3a, where three contributions can be fitted by
the function

$\displaystyle
g^{(2)}(\tau)=1-\left(a+b\right)\,e^{-\frac{|\tau|}{\tau_{1}}}+b\,e^{-\frac{|\tau|}{\tau_{2}}}+\phi_{c,\tau_{3}}(\tau).$
(1)

Here, $a$ and $\tau_{1}$ ($b$ and $\tau_{2}$) describe the single-emitter
antibunching (bunching) amplitude and time constant that are linked to the NV
center excited (shelving) state lifetime.

We can extract the cavity-shortened excited state lifetime and thus the
Purcell factor from the antibunching time constant $\tau_{1}(I)$, which
depends on the excitation rate $\eta$ and thus the laser intensity $I$.
Therefore, we perform a power-dependent set of measurements of the
$g^{(2)}$-function and evaluate $\tau_{1}(I)$, see Fig. 3c. Extrapolating to
zero excitation intensity yields the excited state lifetime. The fit function
$\tau_{1}(I)=\tau_{1}(0)/\left(1+\sigma I\tau_{1}(0)\right)$ can be deduced
from simplified two-level rate equations, where $\sigma I=\eta$ is the
excitation rate from the ground to the excited state. With this analysis, the
excited state lifetime of the cavity-coupled NV center ensemble was estimated
as $\tau_{\mathrm{c}}=7.6\pm 0.9$ ns. A second set of power-dependent
$g^{(2)}$ functions was taken in a confocal microscope, from which the excited
state lifetime without cavity-enhancement was estimated as
$\tau_{\mathrm{0}}=15.8\pm 0.8$ ns. The increased lifetime compared to the
bulk value of 13 ns can be explained by the presence of the nearby diamond-
silica interface for shallowly implanted NV centers as studied here, which
leads to a modified local dielectric environment that reduces the decay rate
radtke_nanoscale_2019 ; zahedian_modeling_2023 . From the ratio of the cavity-
enhanced and free-space lifetime, we can extract the effective Purcell factor
to be $F_{\mathrm{P,eff}}=\tau_{\mathrm{0}}/\tau_{\mathrm{c}}-1=1.1\pm 0.3$,
which corresponds to an ideal Purcell factor
$F_{\mathrm{P,ideal}}=F_{\mathrm{P,eff}}/\xi=36\pm 11$ with the Debye-Waller
factor $\xi=0.03$ for NV centers faraon_resonant_2011 . This value is in the
range of expected Purcell factors for the given cavity finesse, mode volume,
dipole orientation and random spectral cavity-emitter overlap.
From the amplitude of the antibunching dip, we can deduce the number of
emitters that couple to the cavity and thus dominantly contribute to the ZPL
signal. Therefore, we follow the prediction for $N$ independent emitters,
which is given by $g^{(2)}(0)=1-1/N$. The antibunching contrast $a$ is
furthermore reduced by the purity $p$, which we define as the fraction of the
collected cavity radiation that originates from ZPL emission. Since a large
number of spectrally detuned NV centers additionally contribute to the
collected cavity resonance via their phonon sideband, they introduce
uncorrelated background and thus reduce the purity. We can infer $p$ by
measuring the cavity-collected PSB background when detuning the cavity from
the ensemble ZPL (see Fig. 2b) and obtain $p=0.5$. We use $a/p^{2}=1/N$ to
obtain $N$. In the case of the measurement shown in Fig. 3a, this analysis
yields $N=15\pm 3$ emitters coupling to the cavity. For comparison, we observe
a flat $\mathrm{g^{(2)}}$ function when selectively collecting the cavity-
coupled PSB emission, and an antibunching contrast of $a\approx 0.01$ in a
measurement of the same sample region with a confocal microscope at room
temperature. The latter contrast is smaller compared to the cavity ZPL
measurement since no spectral filtering is present and all excited emitters
contribute. However, the spatial excitation and collection modes of the
confocal microscope are smaller than the cavity mode cross section, such that
fewer emitters are contributing than in the cavity under PSB collection. The
antibunching contrast yields an estimate of hundred emitters in the confocal
point-spread function and - scaled to the mode area of the cavity - several
hundred emitters in the spatial cavity mode. This relatively high number
explains why we do not observe any $\mathrm{g^{(2)}}$-contrast in the cavity-
filtered PSB.

We finally turn to the additional bunching peak close to zero time delay. We
observe a Gaussian shape described with $\phi_{c,\tau_{3}}(\tau)$ in Eq. 1
with a FWHM of $\approx 400$ ps. This time constant is entirely limited by the
instrument response of the HBT setup, originating from the timing jitter of
the APDs. A measurement with a pulsed laser is shown in Fig. 3b. In the limit
of weak absorption, the fastest possible time scale for emission from the
cavity is given by the cavity intensity decay time of
$T_{cav}=\delta\nu_{cav}^{-1}=100$ ps. This value is also in accordance with
our model considering emitters with additional dephasing that couple
collectively to a cavity mode (see Fig. 4c and SI). Correcting for the
incoherent background from uncoupled NV centers and deconvolving the measured
bunching peak allows us to estimate the intrinsic peak value of the bunching
(Fig. 3b). We obtain a value of $g^{(2)}(0)=1.7$, which is close to an
analytical estimate for interfering independent emitters Auffeves_few_2011 ,
$g^{(2)}(0)=2(1-1/N)$ for our extracted emitter number $N=15\pm 3$. This
thermal limit describes individually decaying but identical emitters and
neglects possible contributions from collective states. We compare the
prediction for $g^{(2)}(0)$ from our model to the analytical expression in
Fig. 3d, and only a minor difference is visible. In contrast, no bunching is
observed in the cavity-enhanced PSB at any temperature, as well as in the
confocal measurement at room temperature.

Finally, to study the effect of dephasing more explicitly, we perform
measurements of the $g^{(2)}$ function as a function of temperature (Fig. 3e).
We observe a decrease of the antibunching amplitude, in accordance with the
prediction of our model when assuming a temperature-dependent dephasing rate
as measured in fu_observation_2009 . We note that at elevated temperature,
also an increased number of emitters are expected to contribute to the signal.
This is due to the increased homogeneous linewidth, which leads to a growing
spectral overlap of detuned emitters with the cavity mode. The temperature-
dependent collective bunching amplitude remains approximately constant, even
when approaching room-temperature. However, when accounting for the changing
background contribution, an overall decrease results.

The measurements show that while the bunching contrast is not very sensitive
to collective effects in the presence of dephasing, the observed signatures
can not be described solely by individually decaying emitters. Together with
the super-linear power dependence of the cavity emission, the observations
support the picture that the Purcell-enhanced emission into a single cavity
mode synchronizes emission if strong enough cavity coupling and sufficient
coherence is present.

##  IV Numerical Model

Figure 4: Modeling of collective emission. a-d Results for an ensemble with
identical NV centers. a Population of Dicke states under continuous incoherent
pumping for 40 NV centers. While population peaks at the lower-left boundary
of the triangle, also higher-lying collective states are populated. b,
Radiation rate as a function of the ratio $\eta/\gamma$ between the pumping
and decay rate, shown for increasing number of NV centers $N$. c Second-order
correlation function $g^{(2)}(\tau)$ for different values of $N$ and
$\eta/\gamma=2$. d Corresponding decay and rise time of $g^{(2)}(\tau)$. e,f,
Results for NV centers with different frequencies (sub-ensembles). e,
Influence of a Gaussian distributed frequency inhomogeneity of 500 NV centers
with a width $w$ on the radiation rate. f, Influence of the frequency detuning
$\delta$ on the radiation rate, and the radiation weighted by the frequency
fluctuation (black line, amplified by 10 times), treated by a Gaussian
distribution of $\delta$ with a width $300$ GHz.

For a deeper analysis, we develop numerical models for small ensembles of
emitters coupled to a cavity. We study first the case of identical NV centers
to explain the main physics (Fig. 4a-d), and then we introduce sub-ensembles
of NV centers with different transition frequencies to examine the influence
of the frequency inhomogeneity and fluctuations (Fig. 4e,f). For the sake of
simplicity, we focus here on the transition between the $m_{s}=0$ levels of
the triplet states ${}^{3}A_{2},^{3}E$, because it dominates the radiation
under continuous illumination. In our theoretical model (see Methods section),
we treat this transition as two-level system in the presence of decay,
dephasing, incoherent pumping with rates $\gamma,\chi,\eta$ and coupled by a
strength $g$ to an optical cavity with a frequency $\omega_{c}$ and a loss
rate $\kappa$. All these processes are described by a quantum master equation
(QME). For the homogeneous system, all the NV centers are identical and they
can be well described with Dicke states $|J,M\rangle$, and the QME can be
solved in the basis of these states Zhang_2018 ; Shammah_2018 . Under the
incoherent pumping with rate $\eta$, the excited ensemble emits radiation.
Since our system is in the weak coupling regime, we can eliminate
adiabatically the cavity to obtain the radiation rate
$I_{\mathrm{rad}}=\Gamma\sum_{J,M}|A_{J,M}^{-}|^{2}\rho_{MM}^{J}$, where
$\Gamma\approx 4g^{2}/\kappa$ is the cavity-mediated Purcell-enhanced decay,
the coefficient $A_{JM}^{-}=\sqrt{(J+M)(J-M+1)}$ arises from the action of the
collective lowering operator, and $\rho_{MM}^{J}$ is the total population of
the degenerate Dicke state manifold with same quantum numbers $J$ and $M$. As
$\eta$ increases, the steady state of the ensemble rises along the lower
boundary of the Dicke states with decreasing values of $J$ and
$M=-J,-J+1,-J+2,...$ (Fig. 4a), which leads to the increased radiation.
Furthermore, as the number of NV centers increases, the number of excitations
in the system increases, and the scaling of radiation changes from linear to
super-linear (Fig. 4b). By applying the quantum regression theorem, we have
also computed the second-order auto-correlation function $g^{(2)}(\tau)$. In
general, this function does not change qualitatively with increasing pumping
rate (Fig. S3 of SI), and thus, here, we focus on the influence of the number
of NV centers $N$ for the maximal pumping rate $\eta=2\gamma$ in Fig. 4c.
While the $g^{(2)}(\tau)$ function shows a pronounced dip for a single NV
center, it shows a sharp bunching peak around zero delay time $\tau=0$ for
$N\geq 2$, and an anti-bunching dip for longer time. The quantitative analysis
of the time constants is illustrated in Fig. 4d. The decay time of the peaks
(the rising time of the dips) reduces to about 0.23 ns (1.37 ns) with
increasing $N$. Note that the decay time increases slightly when $N$ changes
from 2 to 3. For $\eta=2\gamma$ and $N=40$, the Dicke state $|J=4,M=-4\rangle$
has the largest population (Fig. 4a), and the singly (doubly) excited state
$|J=4,M=-3(-2)\rangle$ has a decay rate of
$\Gamma|A_{(4,-3)}^{-}|^{2}=8\Gamma,(\Gamma|A_{(4,-2)}^{-}|^{2}=18\Gamma)$.
Since the inverse of these rates $1/(8\Gamma)\approx 0.33$ ns and
$1/(18\Gamma)\approx 0.15$ ns are close to the decay time 0.23 ns, the sharp
anti-bunching peak of the $g^{(2)}(\tau)$ function should be associated with
the populated Dicke states. When introducing additional dephasing at a value
equal to the cavity decay rate, we observe a reduced bunching peak time
constant of $110$ ps, closely matching the cavity decay rate of $100$ ps, (see
SI). This motivates the choice of the time constant in the model for Fig. 3b.
At the same time, the peak maximum (dip minimum) increases and saturates
gradually to a value slightly above 2 (below 1) for larger $N$. While the dip
minimum matches closely with the analytical prediction, the bunching peak
remains systematically higher than given by the analytical expression and is
shown in Fig.3d. With the above results, we have confirmed that the non-linear
scaling and the sharp bunching peaks, as observed in the experiment, are
caused by the cavity-mediated collective radiation of the NV centers.

To reflect more closely the experimental conditions, we study the influence of
frequency inhomogeneity and fluctuations Bychek_superradiant_2023 . We divide
the total NV centers into multiple sub-ensembles with different frequencies,
and assume that the number of NVs as function of the transition frequency
follows a Gaussian distribution with a mean $\mu$ and a linewidth
$w=2\sqrt{2\mathrm{ln}(2)}\sigma$ (standard deviation $\sigma$). Although we
cannot solve the QME of such system with the density matrix technique due to
the exponentially increased Hilbert space, we can solve it with the second-
order mean-field theory, and derive semi-analytical expressions for the
quantities characterizing the system steady-state. In our simulations, we
consider 500 NV centers, as estimated from the experimentally measured
$g^{(2)}$ function. If the Gaussian distribution is resonant with the cavity
and has a similar width $(\mu=\omega_{c},w=\kappa)$, the radiation rate shows
obvious super-linear and linear scaling for $\eta/\gamma\leq 1.2$ and
$\eta/\gamma>1.2$, respectively (Fig. 4e). However, as $w$ increases to
$5\kappa$, the super-linear scaling becomes less pronounced. If $w$ increases
further to $20\kappa$, the radiation rate decreases, and approaches gradually
the linear scaling. If the Gaussian distribution with a fixed width
$w=3\kappa$ is detuned from the cavity by an increasing amount
$\delta=\mu-\omega_{c}$, the radiation rate reduces slightly for smaller
detuning $|\delta|\leq 3\kappa$, but dramatically for much larger detuning
$|\delta|>3\kappa$ (Fig. 4f). Furthermore, by assuming that the probability of
the detuning $\delta$ follows also a Gaussian distribution with a large
linewidth $w\approx 300$ GHz, as estimated from Fig. 2b, we can weigh the
above results with this probability to obtain the averaged radiation (black
line in Fig. 4f), and observe the persistence of the super-linear scaling. To
evaluate the influence of the frequency inhomogeneity and fluctuation on the
$g^{(2)}$ function, we adopted an indirect approach, and verify firstly that
the radiation is affected by dephasing of NV centers in a similar way as by
the frequency inhomogeneity (see SI), which permits us to account for the
inhomogeneity in the model of identical NV centers with a single dephasing
rate Wu_2021 ; Julsgaard_2012 . The sharp bunching peak of the
$g^{2}$-function is preserved in the presence of large dephasing and frequency
fluctuation, which is accord with the fact that such behavior appears already
for the system with as few as two NV centers (Fig. 1c). In the end, we note
that the observed bunching shoulder of the $g^{(2)}(\tau)$ function at longer
times is known to originate from the singlet excited levels
${}^{1}A_{1},^{1}E$ of NV centers, and thus improved models including also
these levels should be able to capture all these characteristics. Due to the
complexity of the models, we defer their discussion to a more detailed
theoretical paper.

##  V Conclusions

In summary, we demonstrate that incoherent, inhomogeneous and spatially
separated emitters can yield signatures of collective emission, enabled by the
coupling to a common cavity mode. We have focused on the nonlinear scaling and
the photon bunching of the incoherently pumped NV centers. It would be
interesting to further explore the collective radiation from population-
inverted NV centers, or from a system with more NV centers, which might enter
the strong coupling regime. Cavity-induced synchronization can furthermore
lead to non-trivial collective behavior even for emitters outside the cavity
linewidth Debnath_2019 . Collective emission may be used to generate multi-
photon quantum states munoz_emitters_2014  and can be a promising route to
produce multi-qubit entangled states Santos_2022 ; Roulet_2018 . It will be an
intriguing direction to explore cooperative behavior in the cavity quantum
electrodynamics regime that has recently been demonstrated for two emitters in
cavities Reimann_2015 ; Neuzner_2016 ; Kim_2018 ; evans_photon-mediated_2018 ;
lukin_two_2023  and expand it to a controlled number of few emitters.

## Appendix A: Experimental setup

The sample used in this experiment is an electronic grade single crystal CVD-
grown diamond with a [100] surface orientation. The sample was initially cut
to have an average thickness of $40\pm 10$ µm and a thickness variation of
$<1$ µm/mm. Afterwards, the sample was shallowly implanted by the manufacturer
with nitrogen ions, leading to a high density of NV centers ($\sim$300 NV/µm2)
at a depth of 10 nm. The sample was further cut to have a footprint of
$1.5\times 1.5$ mm2 and bonded onto a dielectric mirror by van der Waals
bonding. Afterwards, the diamond was machined by inductively-coupled plasma
reactive ion etching, creating a membrane with a minimal thickness of 3.5 µm
and a surface roughness as low as 0.3 nm rms in an area of $4\times 4$ µm2
Heupel_2020 .

The sample is integrated into a fiber-based microcavity, consisting of a
macroscopic plane mirror which holds the diamond and a laser-machined fiber
mirror. We excite the NV centers incoherently using a continuous wave laser
with $\lambda_{exc}=532$ nm through the free space port of the cryostat. The
same optical path is used to filter and collect fluorescence light.

A more detailed description of the optical setup can be found in the SI.

## Appendix B: Theoretical model

For the system with multiple NV sub-ensembles, the dynamics is described by
the quantum master equation (QME):

$\begin{split}&\frac{\partial}{\partial
t}\hat{\rho}=-\frac{i}{\hbar}\left[\hat{H}_{tls}+\hat{H}_{c}+\hat{H}_{tls-c},\hat{\rho}\right]-\kappa\mathcal{D}[\hat{a}]\hat{\rho}\\\
\
&-\sum_{k=1}^{N_{ene}}\sum_{i=1}^{N_{k}}\left\\{\eta_{k}\mathcal{D}[\hat{\sigma}_{k,i}^{+}]\hat{\rho}+\gamma_{k}\mathcal{D}[\hat{\sigma}_{k,i}^{-}]\hat{\rho}+\frac{\chi_{k}}{2}\mathcal{D}[\hat{\sigma}_{k,i}^{z}]\hat{\rho}\right\\}\end{split}$
(2)

The Hamiltonian
$\hat{H}_{tls}=\hbar\sum_{k=1}^{N_{ene}}(\omega_{k}/2)\sum_{i=1}^{N_{k}}\hat{\sigma}_{k,i}^{z}$
describes the NV centers as two-level systems with the transition frequency
$\omega_{k}$ and the Pauli operator $\hat{\sigma}_{k,i}^{z}$, where the labels
$k,i$ indicate the individual sub-ensemble and the individual NV center. The
Hamiltonian $\hat{H}_{c}=\hbar\omega_{c}\hat{a}^{+}\hat{a}$ describes the
optical cavity with frequency $\omega_{c}$, the photon creation $\hat{a}^{+}$
and annihilation operators $\hat{a}$. The Hamiltonian
$\hat{H}_{NV-c}=\hbar\sum_{k=1}^{N_{ene}}g_{k}\sum_{i=1}^{N_{k}}\left\\{\hat{\sigma}_{k,i}^{+}\hat{a}+\hat{a}^{+}\hat{\sigma}_{k,i}^{-}\right\\}$
describes the NVs-optical cavity coupling with a strength $g_{k}$, and the
raising and lowering operators $\hat{\sigma}_{k,i}^{+}$,
$\hat{\sigma}_{k,i}^{-}$. The Lindblad terms describe the cavity photon loss,
the individual decay, dephasing, pumping of the two-level systems with rates
$\kappa$, $\gamma_{k}$, $\chi_{k}$, $\eta_{k}$, respectively, and the
superoperator is defined as
$\mathcal{D}[\hat{o}]\hat{\rho}=\frac{1}{2}\left\\{\hat{o}^{+}\hat{o},\hat{\rho}\right\\}-\hat{o}\hat{\rho}\hat{o}^{+}$
(for any operator $\hat{o}$). The second order auto-correlation function can
be computed with the expression
$g^{(2)}(\tau)=\braket{\hat{a}^{+}(t_{ss})\hat{a}^{+}(t_{ss}+\tau)\hat{a}^{-}(t_{ss}+\tau)\hat{a}^{-}(t_{ss})}/\braket{\hat{a}^{+}(t_{ss})\hat{a}^{-}(t_{ss})}$,
where $t_{ss}$ refers to the time for the system reaching steady-state. The
numerator can be transformed into the expression
$\mathrm{tr}\left\\{\hat{a}^{+}\hat{a}^{-}\hat{\varrho}(\tau)\right\\}$ in the
Schrödinger picture, where the ancillary operator $\hat{\varrho}(\tau)$
satisfies also the QME as $\hat{\rho}(\tau)$ with however the initial
condition $\hat{\varrho}(0)=\hat{a}^{-}\hat{\rho}(t_{ss})\hat{a}^{+}$
according to quantum regression theorem.

To describe the ideal system with identical NV centers, we remove the label
$k$ in the above model, and further eliminate adiabatically the optical cavity
to achieve an effective master equation, which differs from the original one
by replacing $\hat{H}_{c}+\hat{H}_{tls-c}$ with
$\hat{H}_{shift}=\hbar(\delta\omega/2)\sum_{i=1}^{N}\hat{\sigma}_{i}^{z}$, and
$\kappa\mathcal{D}[\hat{a}]\hat{\rho}$ with
$\Gamma\mathcal{D}[\sum_{i=1}^{N}\hat{\sigma}_{i}^{-}]\hat{\rho}$, where
$\delta\omega=2g^{2}(\omega_{c}-\omega_{0})/((\omega_{c}-\omega_{0})^{2}+\left((\kappa+\eta+\gamma)/2+\chi\right)^{2})$
and
$\Gamma=g^{2}\kappa/((\omega_{c}-\omega_{0})^{2}+\left((\kappa+\eta+\gamma)/2+\chi\right)^{2})$
are the cavity-induced frequency shift and Purcell-enhanced decay rate. To
proceed, we introduce the collective operators
$\hat{J}_{x}=\frac{1}{2}\sum_{i=1}^{N}(\hat{\sigma}_{i}^{-}+\hat{\sigma}_{i}^{+})$,
$\hat{J}_{y}=\frac{i}{2}\sum_{i=1}^{N}(\hat{\sigma}_{i}^{-}-\hat{\sigma}_{i}^{+})$,
$\hat{J}_{z}=\frac{1}{2}\sum_{i=1}^{N}\hat{\sigma}_{i}^{z}$,
$\hat{J}^{2}=\sum_{i=x,y,z}(\hat{J}_{i})^{2}$, and define the Dicke states
from the eigen-value problems $\hat{J}^{2}|J,M\rangle=J(J+1)|J,M\rangle$ and
$\hat{J}_{z}|J,M\rangle=M|J,M\rangle$. By expressing the Dicke states for $N$
two-level systems as the product states of $N-1$ two-level systems and one
two-level system (using the Clebsch-Gordan coefficients), one can identify the
dissipative effects in the collective Dicke state picture induced by the
incoherent pumping, decay, and dephasing of individual two-level systems
Zhang_2018 ; Shammah_2018  (Fig. 1e). In this way, we can derive and then
solve the equation for the density matrix elements $\rho_{MM^{\prime}}^{J}(t)$
within the Dicke states space (the density matrix is diagonal in $J$). From
$\rho_{MM^{\prime}}^{J}(t)$, we can extract the population of the Dicke states
$P_{JM}(t)=\rho_{MM}^{J}(t)$, the radiation rate
$I_{rad}(t)=\Gamma\langle\hat{J}^{+}\hat{J}^{-}\rangle(t)=\Gamma\sum_{J,M}|A_{JM}^{-}|^{2}P_{JM}(t)$,
and the second-order auto-correlation function
$g^{(2)}(\tau)=\langle\hat{J}^{+}(t_{ss})\hat{J}^{+}(t_{ss}+\tau)\hat{J}^{-}(t_{ss}+\tau)\hat{J}^{-}(t_{ss})\rangle/(I_{rad}(t_{ss})/\Gamma)^{2}$,
where the numerator can be calculated in a similar way as stated in the
previous paragraph. In Sec. S1 of the Supporting Information, we provide the
corresponding codes based on the QuTip package Julsgaard_2012 .

For the system with multiple NV sub-ensembles, we adopt a second-order mean-
field approach to derive the equation $\frac{\partial}{\partial
t}\langle\hat{o}\rangle=\text{tr}\\{\hat{o}(\frac{\partial}{\partial
t}\hat{\rho})\\}$ for the mean value
$\langle\hat{o}\rangle=\text{tr}\\{\hat{o}\hat{\rho}\\}$ from the QME, apply
the approximation
$\langle\hat{o}\hat{p}\hat{q}\rangle\approx\langle\hat{o}\rangle\langle\hat{p}\hat{q}\rangle+\langle\hat{p}\rangle\langle\hat{o}\hat{q}\rangle+\langle\hat{q}\rangle\langle\hat{o}\hat{p}\rangle-2\langle\hat{o}\rangle\langle\hat{p}\rangle\langle\hat{q}\rangle$
to truncate the equation hierarchy to obtain a closed set of equations and
assume the NV centers in each sub-ensemble to be identical. Furthermore, we
consider the equations at the steady-state, and derive the semi-analytical
expressions for the intracavity photon number
$\langle\hat{a}^{\dagger}\hat{a}\rangle\approx
i\sum_{k=1}^{N_{\text{ene}}}(N_{k}g_{k}/\kappa)(\langle\hat{\sigma}_{ki}^{+}\hat{a}\rangle-\langle\hat{a}^{\dagger}\hat{\sigma}_{ki}^{+}\rangle)$
, the population differences
$\langle\sigma_{ki}^{z}\rangle\approx\left[-2ig_{k}(\langle\sigma_{ki}^{+}\hat{a}\rangle-\langle\hat{a}^{\dagger}\sigma_{ki}^{+}\rangle)+\eta_{k}-\gamma_{k}\right]/(\gamma_{k}+\eta_{k}),$
and the NV-NV correlations
$\langle\sigma_{ki}^{+}\sigma_{k^{\prime}i^{\prime}}^{-}\rangle\approx(\omega_{k}^{*}-\omega_{k^{\prime}})^{-1}(g_{k}\langle\sigma_{ki}^{z}\rangle\langle\sigma_{k^{\prime}i^{\prime}}^{+}\hat{a}\rangle^{*}-g_{k^{\prime}}\langle\sigma_{k^{\prime}i^{\prime}}^{z}\rangle\langle\sigma_{ki}^{+}\hat{a}\rangle)$
and the NV-photon correlations
$\langle\sigma_{ki}^{+}\hat{a}\rangle\approx(\omega_{ki}^{*}-\omega_{c})^{-1}[g_{k}(\langle\hat{a}^{\dagger}\hat{a}\rangle\langle\sigma_{ki}^{z}\rangle-\langle\sigma_{ki}^{+}\sigma_{ki^{\prime}}^{-}\rangle)+\sum_{k^{\prime}\neq
k}^{N_{\text{ene}}}N_{k^{\prime}}g_{k^{\prime}}\langle\sigma_{ki}^{+}\sigma_{k^{\prime}i^{\prime}}^{-}\rangle+\frac{g_{k}}{2}(\langle\sigma_{ki}^{z}\rangle+1).$
By taking the complex conjugate of the last equation, we can also obtain the
expressions for $\langle\hat{a}^{\dagger}\sigma_{ki}^{+}\rangle.$ In these
expressions, the complex frequencies are defined as
$\omega_{k}=\omega_{k}-i(\gamma_{k}+\eta_{k})+\chi_{k},\quad\omega_{c}=\omega_{c}-i\kappa/2.$
In practice, we insert the first three expressions into the last one, and then
solve the self-consistent equations for the NV-photon correlations with the
FindRoot command in Mathematica; see the codes in Sec. S1 of the SI.

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