# Studying Retrievability of Publications and Datasets in an Integrated
Retrieval System

Dwaipayan Roy  Indian Institute of Science Education and Research,
KolkataIndia<EMAIL_ADDRESS>,  Zeljko Carevic  GESIS – Leibniz
Institute for the Social Sciences, CologneGermany<EMAIL_ADDRESS>and
Philipp Mayr  GESIS – Leibniz Institute for the Social Sciences,
CologneGermany<EMAIL_ADDRESS>
(2022)

###### Abstract.

In this paper, we investigate the retrievability of datasets and publications
in a real-life Digital Library (DL). The measure of _retrievability_ was
originally developed to quantify the influence that a retrieval system has on
the access to information. Retrievability can also enable DL engineers to
evaluate their search engine to determine the ease with which the content in
the collection can be accessed. Following this methodology, in our study, we
propose a system-oriented approach for studying dataset and publication
retrieval. A speciality of this paper is the focus on measuring the
accessibility biases of various types of DL items and including a metric of
_usefulness_. Among other metrics, we use Lorenz curves and Gini coefficients
to visualize the differences of the two retrievable document types
(specifically datasets and publications). Empirical results reported in the
paper show a distinguishable diversity in the retrievability scores among the
documents of different types.

Retrievability, Dataset Retrieval, Interactive IR, Diversity

††journalyear: 2022††copyright: acmcopyright††conference: The ACM/IEEE Joint
Conference on Digital Libraries in 2022; June 20–24, 2022; Cologne,
Germany††booktitle: The ACM/IEEE Joint Conference on Digital Libraries in 2022
(JCDL ’22), June 20–24, 2022, Cologne, Germany††price: 15.00††doi:
10.1145/3529372.3530931††isbn: 978-1-4503-9345-4/22/06

##  1\. Introduction

With the availability of various types of information from different sources,
sometimes with different modalities, the necessity of federated or integrated
search systems (Adali and Emery, 1995; Arguello, 2012) has become a prominent
research topic in information retrieval and digital library. Textual data
still remains the predominant type among them and significant research has
been conducted in the domain of textual document retrieval. Among the rest,
recent research on dataset retrieval (Kunze and Auer, 2013) has become
increasingly important in the (interactive) information retrieval and digital
library community. One of the reasons is undoubtedly the massive amount of
research datasets available. However, the underlying characteristics of
dataset retrieval also contribute to the attention in this area. One often
mentioned characteristic is the increased complexity of datasets over
traditional document retrieval. While the latter is well known and adequately
studied, datasets often include more extensive material and structures that
are relevant for retrieval. This may involve the raw data, descriptions of how
the data was collected, taxonomic information, questionnaires, codebooks, etc.
Recently, numerous studies have been conducted to further identify the
characteristics of dataset retrieval. These studies include the observation of
data retrieval practices (Krämer et al., 2021), interviews and online
questionnaires (Kern and Mathiak, 2015; Friedrich, 2020) and transaction log
analysis (Kacprzak et al., 2017; Carevic et al., 2020).

In this paper, we follow a system-oriented approach for studying dataset
retrieval. By employing the measure of retrievability (Azzopardi and Vinay,
2008), we aim to gain insights into the particularities of dataset retrieval
in comparison to traditional document retrieval. The measure of retrievability
was originally developed to quantify the influence that a retrieval system has
on the access to information. In a simplified way, retrievability represents
the ease with which a document can be retrieved given a particular IR system
(Azzopardi and Vinay, 2008). The measure of retrievability can be utilised for
several use cases. Retrievability can, for instance, be employed to identify
potential biases in the underlying corpus. Retrievability can also enable
digital library (DL) engineers to evaluate their search engine to determine
the ease with which the content in the collection can be accessed.

In this paper, we investigate the retrievability of various types of documents
in the integrated DL GESIS Search (see Section 4), focusing on datasets and
publications in the same system. The assumption followed here is that in an
ideal ranking system111In this paper, by _ranking system_ or, _IR system_ , we
refer to _a system_ containing a corpus together with the retrieval model to
be used to search on that corpus., the retrievability of each indexed item
(dataset or other publication) is equally distributed. Likewise, a discrepancy
to this assumption may reveal an inequality between the items in a collection
caused by the system. By employing a measure of retrievability, we expect to
gain further insight into the characteristics of dataset retrieval compared to
traditional document retrieval.

### Research questions

Considering the real-life digital library GESIS Search, in this paper, we
address the following research questions:

  * •

RQ1: In the integrated search system with various types of items, can we
observe any prior bias of accessibility of documents from a particular type?

  * •

RQ2: Can we formalize this type-accessibility bias utilizing the concept of
document retrievability?

  * •

RQ3: How diverse are the retrievability score distributions in the different
categories of documents in our integrated search system?

In sum, our contributions are as follows: 1) we utilize the retrievability
measure to better understand the diversity of accessing datasets in comparison
to publications with real-life queries from a search log; 2) building on
retrievability, we propose the measurement of usefulness, which represents
implicit relevance signals observed for datasets and publications. Our
understanding of bias follows the argumentation provided in (Wilkie and
Azzopardi, 2017) where bias denotes the inequality between documents in terms
of their retrievability within the collection. Bias can be observed when a
document is overly or unduly favoured due to some document features (e.g.
length, term distribution etc.) (Wilkie and Azzopardi, 2014).

The rest of the paper is organized as follows. We first present related work
in Section 2. A formal introduction of the concept of retrievability is
presented in Section 3. The integrated search system GESIS Search along with
the motivation of our retrievability study is presented in Section 4. Section
5 discusses the empirical results and analysis of the outcome of the
experimentation before introducing the novel concept of usefulness in Section
6 along with the experimental study of usefulness. We conclude the paper in
Section 7 highlighting the contributions and findings of the paper.

##  2\. Related work

Retrievability as a measure was proposed in (Azzopardi and Vinay, 2008). In
the work, the authors demonstrate the utility of retrievability on two TREC
collections using query based sampling (Callan and Connell, 2001a).
Retrievability has been primarily used to detect bias in ranking systems. For
instance, (Samar et al., 2018) employ retrievability to research the effect of
bias across time for different document versions (treated as independent
documents) in a web archive. Their results show a ranking bias for different
versions of the same document. Furthermore, the study confirms a relationship
between retrievability and findability measured by Mean Reciprocal Rank (MRR).
They follow the assumption that the lower a document’s retrievability score
the more difficult it is to find the document. Another application of the
retrievability measure can be found in patent or legal document retrieval
which provides a unique use case due to its recall oriented application. In
both studies, (Bashir and Rauber, 2009a, b) look at document retrievability
measurements and argue that a single retrievability measure has several
limitations in terms of interpretability. In (Bashir and Rauber, 2009a) they
try to improve accessibility measurement by considering sets of relevant and
irrelevant queries for each document. In this way, they try to simulate recall
oriented users. In addition, they plot different retrievability curves to
better spot the gaps between an optimal retrievable system and the tested
system. In (Bashir and Rauber, 2009b) they analyze the bias impact of
different retrieval models and query expansion strategies. Their experiments
show that clustering based document selection for pseudo-relevance feedback is
an effective approach for increasing the findability of individual documents
and decreasing the bias of a retrieval system.

A study on the query list generation phase for determining the measure of
retrievability is presented in (Bashir and Rauber, 2011). The study addresses
two central problems when determining retrievability: 1) query selection and
2) query characteristics identification. It is argued that the query selection
phase is usually performed individually without well-accepted criteria for
query generation. Hence their goal is to evaluate how far the selection of
query subsets provides an accurate approximation of retrieval bias. The second
shortcoming is addressed by determining retrievability bias considering
different query characteristics. In their experiments they recognise that
query characteristics influence the increase or decrease of retrievability
scores. As recognised in (Bashir and Rauber, 2011), the majority of
retrievability experiments employ simulated queries to determine
retrievability. To study the ability of the retrieval measure in detecting a
potential retrievability bias using real queries issued by users, (Traub et
al., 2016) conducted an experiment on a newspaper corpus. Their study confirms
the ability to expose retrievability bias within a more realistic setting
using real world queries. A comparison of simulated and real queries with
regard to retrievability scores further shows considerable differences which
indicates a need for an improved construction of simulated queries.

The concept of usefulness was first introduced in (Cole et al., 2009) as a
criterion to determine how well a system is able to solve a user’s information
need. In (Hienert and Mutschke, 2016), it has been operationalised within a
log-based evaluation approach to determine the usefulness of a search term
suggestion service. In (Carevic et al., 2018), usefulness has been
operationalised to determine the effects of contextualised stratagem browsing
on the success of a search session.

Recently, a considerable amount of research has been carried out concerning
the characteristics of dataset retrieval. A comprehensive literature review on
dataset retrieval practices is provided in (Gregory et al., 2019) focusing on
dataset retrieval practices in different disciplines. Research in this area
covers, for instance, the analysis of information-seeking behaviour during
dataset retrieval through observations (Krämer et al., 2021), questionnaires
and interviews (Kern and Mathiak, 2015; Friedrich, 2020), and transaction-log
studies (Kacprzak et al., 2017; Carevic et al., 2020).

In their work in (Kern and Mathiak, 2015), the authors investigated the
requirements that users have for a dataset retrieval system. Their findings on
dataset retrieval practices suggest that users invest greater effort during
relevance assessment of a dataset. They conclude that the selection of a
dataset is a much more important decision compared to the selection of a piece
of literature. This results in high demands on metadata quality during the
dataset retrieval. The complexity of assessing the relevance of a dataset is
also highlighted in (Krämer et al., 2021). Besides topical relevance, access
to metadata as well as documentation about the dataset plays a crucial role. A
query log analysis from four open data portals is presented in (Kacprzak et
al., 2017). Their study indicates differences between queries issued towards a
dataset retrieval system and queries in web search. In a subsequent study
(Kacprzak et al., 2018), the extracted queries are further compared to queries
generated from a crowdsourcing task. The intuition and focus of this work is
to determine whether queries issued towards a data portal differ from those
collected in a less constrained environment (crowdsourcing).

##  3\. Retrievability

The _retrievability of a document_ in a collection, as defined by Azzopardi
and Vinay, is the concept of systematically measuring the possibility of
retrieval of the document by _any query_ , or in other words, it indicates of
how easily the information within the collection can be accessed with a given
retrieval model (Azzopardi and Vinay, 2008). Formally, given a collection
$\mathsf{C}$ of documents, and $\mathsf{Q}$, a set of all possible queries for
which the answers are expected to be present in $\mathsf{C}$, the
retrievability of a document $d$ ($d\in\mathsf{C}$) is defined as:

(1) $r(d)=\sum_{q\in\mathsf{Q}}w_{q}\cdot f(k(d,q),c)$

In Equation 1, $w_{q}$ is the weight of the query $q$, and $k(d,q)$ specifies
the rank of document $d$ after querying with $q$ with some retrieval model.
The function $f(k(d,q),c)$ is a boolean function indicative of whether
$k(d,q)$ (the rank of document $d$) is within a rank cut-off $c$.
Specifically, the function $f(\cdot)$ returns $1$ if $k(d,q)\leq c$, and $0$
otherwise, as presented in Equation 2.

(2) $f(k(d,q),c)=\begin{cases}1,&\text{if $k(d,q)\leq c$}.\\\
0,&\text{otherwise}.\end{cases}$

Informally, the retrievability score $r(d)$ of a document $d$ can also be
realized as the number of queries for which $d$ is ranked within rank-cutoff
$c$.

The query weight ($w_{q}$ in Equation 1) may depend on the _popularity_ of the
query which can be treated as a uniform constant for all the queries in
$\mathsf{Q}$ to sidestep the effect of query bias (as presented in (Azzopardi
and Vinay, 2008)); that is, all queries are considered to have equal weights.
The approximated retrievability score of document $d$ will then be a discrete
value $x$ indicating the number of queries for which $d$ is retrieved within
rank $c$. Certainly, this is a simplifying assumption and the queries
submitted to a search system in practice vary vastly both in terms of
_popularity_ as well as _difficulty_ (Carmel et al., 2006). Constructing
$\mathsf{Q}$, the set of all queries, expects the utilization of a query log
in absence of which a query-based sampling method (Callan and Connell, 2001b)
can be applied to randomly populate $\mathsf{Q}$. Considering a fairly sized
collection of documents, there can be infinitely many distinct queries that
can be answered by various documents in the collection. Hence, the
construction of $\mathsf{Q}$ based on either query log or random sampling of
terms from the collection are only some practical approximations that we can
adapt in order to realize the concept of retrievability of documents in a
collection.

The second factor of the per-query component in Equation 1 is defined to be a
boolean function the output of which depends solely on the rank at which
document $d$ is retrieved. Increasing the value of the rank cut-off ($c$)
broadens the domain of documents retrieved which will positively influence the
retrievability scores of more documents. Note that, being selected by a
retrieval model for some queries does not ensure the relevance of the document
which can only be comprehended by human judgements.

##  4\. Retrievability in an integrated retrieval system

Figure 1. Screenshot of GESIS Search showing result sets for research data and
publications.

We define an _integrated search system_ as a system that searches multiple
sources of different types and integrates the output in a unified
framework222This is similar to the concepts of aggregated search (Lalmas,
2011) or federated search (Arguello, 2012).. The retrieval in such a system
requires sophisticated decision making considering the various modalities in
documents in the collection of data.

Following Equation 1, the retrievability score of documents is dependent on
the other documents in the collection333Here, we are considering the employed
retrieval function as constant.: considering a rank-cut $c$, the rank of a
document under consideration can be greater than $c$ ($>c$) due to the
documents, taking the top $c$ positions, being more relevant or duplicate
(Nikkhoo, 2011). Another factor that can influence the retrievability score of
a document is the popularity; a trendy document will be retrieved multiple
times by the users over time. In case of an integrated search engine, where
the documents belong to various categories, some particular types could be
having higher chances than the others in terms of being retrieved. In general,
there can be some disparity in the number of documents of various categories
being retrieved which can be a result of popularity bias in the collection.
This type of popularity bias can impede the satisfaction of the information
need of a user, and in turn, can affect the performance of the system.
Traditional methods of query performance prediction (Hauff et al., 2010;
Cronen-Townsend et al., 2002), in this scenario, can not be effectively
applied.

In this paper, we are going to study the diversity in retrievability scores
for different categories of document in the integrated search system GESIS
Search444https://search.gesis.org (Hienert et al., 2019). To the best of our
knowledge, this is the first attempt to formalize this type of diversity
utilizing the concepts of retrievability.

##  5\. Experimental Study

As presented in Section 4, we use the integrated search system with various
categories of documents in this work. In this section, we start with
describing the data that we have used in the work along with different
statistics of the data; this will be followed by the experimental evaluation
of the study.

###  5.1. Datasets

To identify possible differences between publication and dataset retrieval
concerning their retrievability, we use the integrated search system GESIS
Search containing a total of 830K indexed records. A screenshot showing the
interface of GESIS search is presented in Figure 1. The indexed records are
divided into six categories based on their types, covering more than 118K
_publications_ and 64K _research datasets_ (also referred to as _datasets_).
Given a query, the system returns six search result pages (SERP) corresponding
to each of the categories. The segregation of the SERP enables us to study the
retrievability of the different types. In this study, we concentrated on the
types _publications_ and _datasets_ independently.

In the integrated search system, the interaction of the users with the system
is logged and stored in a database. A total of more than 40 different
interaction types are stored covering, for instance, searches (queries),
record views and export interactions etc. (Hienert et al., 2019). The export
of a record belongs to an umbrella of categories including various
interactions such as bookmarking, downloading or citing. These interactions
are specifically useful for the application of implicit relevance feedback as
they indicate a relevance of a record that goes beyond a simple record view.
The interaction log of the search system provides the basis for our analysis
in Section 5.4 (and later in Section 6.2). The queries contained in the log
(more than 1.2 M) form the basis for determining the retrievability of
documents. This ensures realistic queries in $\mathsf{Q}$ of Equation 1 as
opposed to the simulated queries used in (Azzopardi and Vinay, 2008) or (Traub
et al., 2016). The data used in this study was extracted from the interaction
log of the integrated search system between July 2017 and December 2021.
Statistics regarding the extracted interactions utilized in our study can be
found in Table 1.

Record type | Size |  #queries (unique) |  avg. query length | #exports
---|---|---|---|---
Publication | 113,000 |  626,335 (276,553) |  15.4 | 8,146
Dataset | 64,000 |  602,581 (187,644) |  20.5 | 24,310
Table 1. Statistics of the extracted information.

###  5.2. Measuring retrievability in a collection

In a traditional relational databases, there can be a one-to-one
correspondence between a query and a record. In other words, each record
stored in R-DBMS can be distinctly selected by a specific query raised to the
system. Also the different queries that can be submitted and addressed by such
systems are finitely tractable which depends on the number of records stored.
However, considering the nature of _free-text query_ , infinitely many queries
can be answered by a traditional information retrieval system through the
selection of documents containing the responses. Also due to the notion of
relevance grade (extremely relevant, loosely relevant etc.), and the necessity
of having a ranked list, certain documents have more chances of being
retrieved than the others.

Considering a traditional document collection $\mathsf{C}$, all the documents
are not equally important to a query, hence paving the need to have a ranked
retrieval. Now given a set of all possible queries $\mathsf{Q}$, some
documents in $\mathsf{C}$ will be relevant to more number of queries
(depending on the topical coverage of the document) than others which can be
measured by the concept of retrievability (see Section 3). Formally with the
notion of retrievability, some documents will be having higher $r(d)$ in a
collection, resulting in an unequal distribution of retrievability scores.

Similar types of inequalities are observed in economics and social sciences,
and they are traditionally measured using the Gini coefficient or Lorenz curve
(Gastwirth, 1972) which measures the statistical dispersion in a
distribution555Lorenz curve and Gini coefficient are popular in economics to
measure of wealth disparity in a community/country..

Mathematically, the Gini coefficient (G) of a certain value $v$ in a
population $\mathcal{P}$ can be defined as:

(3) $G=\frac{\sum_{i=1}^{N}{(2*i-N-1)*v(i)}}{N\sum_{j=1}^{N}{v(j)}}$

where $N$ is the size of the population and $v(i)$ specifies the value of
$i^{th}$ item in $\mathcal{P}$. The value of Gini coefficient in the
population will be between $0$ and $1$ and is proportional to the inequality
inherent in the population: higher value of $G$ indicates greater disparity
and vice versa. In other words, a value of $G$ equal to $0$ in Equation 3
indicates that all the items in the population are equally probable to be
selected where as higher values of $G$ specify a bias implying that only
certain items will be selected.

Rank | Publication |  | Dataset
---|---|---|---
cutoff | $\mu$ | geo-$\mu$ | $\sigma^{2}$ | $\sigma$ |  | $\mu$ | geo-$\mu$ | $\sigma^{2}$ | $\sigma$
10 | 54.86 | 15603.2 | 124.91 | 24.26 |  | 103.27 | 144018.77 | 379.5 | 20.63
20 | 103.95 | 38350.5 | 195.83 | 50.8 |  | 176.63 | 317920.25 | 563.84 | 38.21
30 | 152.49 | 66770.93 | 258.4 | 79.17 |  | 247.01 | 540999.45 | 735.53 | 56.93
40 | 200.36 | 100471.29 | 316.97 | 108.51 |  | 315.71 | 794970.82 | 891.61 | 76.73
50 | 247.66 | 134928.78 | 367.33 | 138.47 |  | 383.59 | 1080000.55 | 1039.23 | 97.57
100 | 477.78 | 352835.83 | 594 | 291.22 |  | 708.76 | 2724762.92 | 1650.69 | 214.74
Table 2. The mean (both arithmetic and geometric), variance and standard
deviation of the retrievability values when the rank-cutoff is varied.

(a) Changes in mean of r(d)

(b) Changes in geometric-mean of r(d)

(c) Changes in variance of r(d)

(d) Changes in standard deviation of r(d)

Figure 2. Graphical representation of the change in various statistical
measures of the observed distribution of retrievability scores. The mean,
geometric-mean, variance and standard deviation of the distribution of
retrievability scores of publication (in blue) and dataset (in orange) are
presented.

###  5.3. Experimentation

As explained in Section 3, the retrievability of a document is a measurement
of how likely the document will be retrieved by _any_ query submitted to the
system666By a system, we are referring to the organization of the collection,
along with a retrieval model to be used for retrieval for a given query..
Hence, the study of retrievability in a collection of documents requires
rigorous retrieval with a set of diversified queries to cover all topics
discussed in the collection. In other words, the retrievability of the
documents should be calculated considering all sorts of queries submitted to
the system. However, an infinite number of queries are possible to be answered
by a collection for free-text queries. To cover all the topics, a traditional
approximation is to simulate a set of queries randomly, accepting the risk of
erratic queries not aligned with the real scenario (Azzopardi and Vinay, 2008;
Traub et al., 2016). With the availability of a query log, the process of
query generation can be made more formalized and streamlined to consider the
actual queries submitted by real users. For the study reported in this
article, we utilize the query log presented in Section 5.1.

As reported in the earlier study, the retrievability distribution in a
collection depends on the employed retrieval model (Azzopardi and Vinay,
2008). Following the findings by Azzopardi and Vinay, we use BM25 as the
retrieval model (Sparck Jones et al., 2000). Particularly, we use the
implementation available in Elasticsearch777https://www.elastic.co/ which use
Lucene888https://www.lucene.apache.org/ as the background retrieval model.
Following Equation 1, the retrievability of a document depends on the
selection of the rank cutoff value ($c$) - a rank threshold to indicate how
deep in the ranked list are we going to explore before finding that document.
Considering the model employed for retrieval and the set of all queries
$\mathsf{Q}$ as fixed, $c$ is the only parameter in calculating the
retrievability. For a query $q$, setting a lower value to $c$ will restrict
less number of documents being considered retrievable because $f(k(d,q),c)$
will be $1$ only if $k(d,q)\leq c$ (see Equation 2). Having a higher value of
$c$ will allow more documents to be considered retrievable reducing the
overall inequality. In this study, we have varied the value of $c$ in the
range $10$ to $100$ in steps of $10$ and have analyzed the observations which
is reported in the next section.

(a) Publication

(b) Dataset

Figure 3. The Lorenz curve with the retrievability (rank cutoff set to 100).
The straight line going through the origin (in black) indicate the equality,
that is, when all the documents are equally retrievable.

###  5.4. Observation and analysis

We start this section with describing different statistical properties of the
retrievability distribution of documents (both publication and dataset) when
the value of $c$ is varied. The mean ($\mu$), geometric mean (geo-$\mu$),
variance ($\sigma^{2}$), and standard deviation ($\sigma$) of the
retrievability score distribution on different types (publication and dataset)
are given in Table 2. On varying the value of $c$ from $10$ to $100$, we
observe a drastic $770\%$ change in the mean retrievability score in case of
publication while for datasets, the change in the mean is around $580\%$.
Similar observations are recorded for variance and standard deviation as well
when computed using the distribution of $r(d)$ on publication and dataset with
different $c$ values. The variations are presented graphically in Figure 2.
From the table, we can conclude that most of the statistical measurements
(specifically mean, variance, and standard deviation) are higher for the
datasets than publications. One contrasting observation can be made from
Figure 2b where the variation in geometric mean (geo-$\mu$) is presented.
Unlike the other three statistics, the geometric mean is seen to be higher for
publication than dataset across all values of $c$. Comparing geometric-mean
values with the other statistics, we can perceive that for some dataset items,
the retrievability values are extensive (for the popular datasets); at the
same time, there are number of datasets items with poor $r(d)$ values (for
those which are rarely cited/used). The first category of datasets are
contributing to the high average $r(d)$, which is consistent across different
$c$ values, while the datasets of the second category cause the geometric-mean
to fall. This is also in line with the observed higher values of variance (and
standard deviation) for datasets that indicates diverse $r(d)$ scores. As
opposed to that, we witness a similar behaviour of the publications in terms
of both mean and its geometric counterpart.

To examine the imbalance in the $r(d)$ scores across the documents of a
category, in Figure 3 we plot the Lorenz curve with the $r(d)$ scores computed
separately for publications and datasets. The Lorenz curve shows that
retrievability of datasets is more imbalanced than publications. The value of
$c$ is set to $100$ in this plot however similar trends are observed with the
lower values of $c$ as well. As discussed in Section 3, the retrievability
score of documents escalate with higher values of $c$; consequently, the
overall retrievability-balance of the collection also changes positively. To
empirically see the variations in terms of imbalance, Gini coefficients ($G$)
attained at different values of $c$ are presented for both the category of
documents in Table 3 which is also graphically displayed in Figure 4. From the
table, it can be noticed that the fall of $G$ for publications is more than
$19\%$ while there is a $10\%$ change in the $G$ value for datasets. This
confirms that the inequality of documents being searched for is higher in case
of datasets than publications.

Additionally, we report the percentage of total documents (respectively of
type, publication and dataset) being retrieved while changing $c$ in Table 3.
Note that, more than 95% of documents of type publication are retrieved within
the top $10$ retrieved documents; in contrast, just above 76% datasets are
retrievable within the same rank cut-off. Increasing the value of $c$, it is
noticed that more than $98\%$ documents are retrievable within the top $100$
ranked document by all the queries for both publication and dataset. The
significant change in the percentage of retrieved documents of type dataset
indicates that searching for datasets is more complex than publications; a
deeper ranked list traversal might be essential to find a relevant dataset.

Figure 4. The change in Gini coefficient when the rank cut-off is varied in the range from 10 to 100. The blue line indicates the publication while dataset is specified by the orange curve. Rank | Gini coefficient |  | Retrieved
---|---|---|---
cutoff | Publication | Dataset |  | Publication | % | Dataset | %
10 | 0.6230 | 0.7975 |  | 105795 | 95.02 | 48934 | 76.19
20 | 0.5863 | 0.7793 |  | 109483 | 98.33 | 55528 | 86.45
30 | 0.5639 | 0.7661 |  | 110413 | 99.17 | 58429 | 90.97
40 | 0.5481 | 0.7562 |  | 110796 | 99.51 | 60025 | 93.45
50 | 0.5359 | 0.7479 |  | 110987 | 99.68 | 60991 | 94.96
100 | 0.5008 | 0.7159 |  | 111269 | 99.93 | 62966 | 98.03
Table 3. Change in Gini coefficient when the rank cut off is increased. Also
the number and percentage of documents retrieved of type Publication and
Dataset is presented.

##  6\. From Retrievability to Usefulness

Usefulness was introduced in (Cole et al., 2009) designed initially as a
criterion for the evaluation of interactive search systems. The _usefulness_
of a document can be defined as how often the document is retrieved and
_exported_ (see Section 5.1) by the end user. Of course the concept of
usefulness can only reliably be recognized by relevance judgements submitted
by the user for a given query and the relevance of a document may also depend
on the perspective of the user which may vary across users. Without an
explicit relevance judgement, the approximation of usefulness or importance of
documents can not be reliably accomplished. Considering the availability of
this information from the query log, we can define the usefulness of a
document ($u(d)$) by the following equation:

(4) $u(d)=\sum_{q\in\mathsf{Q}}w_{q}\cdot g(d,q)$

In Equation 4, the weight of the query ($w_{q}$) can be defined in a similar
way as defined in retrievability (Equation 1). The usefulness of a document
may also depend on the difficulty of the query (Carmel et al., 2006; Carmel
and Yom-Tov, 2010)999A query can be considered as _difficult_ if the top
ranked documents are mostly non-relevant in which scenario, the user has to go
deep down the ranked list to get the document addressing the query (Carmel and
Yom-Tov, 2010).. A document $d$ should be considered more useful if it is
retrieved and consumed following a query $Q$ than any other document, say
$d^{\prime}$ with an associated query $Q^{\prime}$ which is relatively easier
than $Q$ (i.e. $difficulty(Q)>difficulty(Q^{\prime})$). Hence, we extend the
definition of the weight of the query taking into account a difficulty factor
in Equation 5.

(5) $w^{\prime}_{q}=w_{q}*h(q)$

where the function $h(q)$ represents the difficulty of the query $q$. The
function $g(\cdot)$ in Equation 4 indicates usefulness in terms of relevance
of the document $d$ for the query $q$. Mathematically, $g(\cdot)$ can be
defined as follows:

(6) $g(d,q)=rel(d,q)$

The function $rel(d,q)$ in Equation 6 indicates the relevance of $d$ for the
query $q$. It will work similar to the way function $f(k(d,q),c)$ is defined
in Equation 2 if a binary relevance, (that is $d$ can be either relevant -
$rel(d,q)=1$, or non-relevant - $rel(d,q)=0$ to the query $q$) is considered.

The usefulness of a document cannot be realised until it is retrieved by some
retrieval model even though it is relevant. Given a pair of documents $d_{1}$
and $d_{2}$ having the same relevance score, a document, say $d_{1}$, should
be considered more useful than the other ($d_{2}$) if $d_{1}$ is ranked higher
than $d_{2}$ in the ranked list produced by a retrieval model. Hence, the
relevance factor of Equation 6 can be extended considering the rank of the
document as in the following Equation:

(7) $g(d,q)=\frac{1}{k(d,q)}*rel(d,q)$

where the function $k(\cdot)$ indicates the rank of document $d$ after
performing a retrieval with query $q$. Based on Equation 7, we can assert that
the usefulness and the rank of the document are inversely proportional to each
other; the chance of being consumed by the end user will decrease if the
document $d$ is being retrieved at lower rank. Note that, the function
$g(\cdot)$ in Equation 7 is dependent on the retrieval model that is used to
retrieve $d$ for $q$; the rank-order of a pair of documents in two ranked
lists can be different if the corresponding retrievals are performed using
different retrieval models.

###  6.1. Experimentation

As presented and argued earlier in Section 6, the signal of document
consumption by the user after submitting a query is essential in order to
compute the usefulness of the documents. We utilize the information stored in
the interaction-log of the integrated search system as the indication of
document consumption by the user. Particularly, the usefulness is determined
on the basis of implicit relevance feedback from the export interactions (see
Section 5.1). A rank cut-off value ($c$) of $10$ is considered; that is, the
top $10$ documents of the SERP are considered as viewed/consumed. Utilizing
the difficulty of the query in order to compute the usefulness has been left
as part of future work; the difficulty of the query is kept as constant
($h(q)$ in Equation 5 set to $1$) in this study.

(a) Publication

(b) Dataset

Figure 5. The Lorenz curve with usefulness. The straight line going through
the origin (in black) indicate the _equality_ , that is, when all the
documents are equally useful. The blue line specifies (Figure 5a) the
publication while dataset is indicated by the orange curve (Figure 5b).

###  6.2. Observation and analysis

The experimental results on the integrated system are graphically presented in
Figure 5 where a pair of Lorenz curves are displayed with the usefulness of
the documents of type publication and dataset. In Figure 5a, the usefulness
distribution of publications is presented indicating that publications are
close to being equally distributed. In comparison, the similar distribution of
datasets (presented in Figure 5b) is observed to be more skewed with an
evident inclination towards certain items being more useful. The corresponding
Gini coefficient of the two distributions (for publications and datasets) is
presented in Table 4. The value of $G$ for the usefulness of dataset
distribution is seen to be almost three times greater than the publication.
This observation clearly highlights that a few datasets are more useful than
the rest, whereas the usefulness distribution of the publications is
considerably close to being uniform.

| Publication | Dataset
---|---|---
Gini coefficient | 0.2594 | 0.7466
Table 4. The Gini coefficient computed with the distribution of usefulness of
the publication and dataset. A higher Gini coefficient (upper bound 1.0)
indicates an uneven distribution of usefulness.

##  7\. Conclusion

In this paper, we have studied the retrievability of documents from different
types belonging to an integrated, real-life digital library, and have observed
noticeable differences between documents from the publication and dataset
categories. In response to RQ1, we have seen significant popularity-bias of
certain items being retrieved than the others. Particularly it has been shown
that certain items from the dataset category are more likely to be retrieved
than the other items of the same category. In contrast, the retrievability
scores of documents from the publication type are more evenly distributed. For
the RQ2, the intra-document selection bias is formalized using the common
measures of Lorenz curve and Gini coefficient. In response to RQ3, we have
observed that the distribution of document retrievability is more diverse for
the dataset as compared to publications. This can be again attributed to the
popularity bias of certain items in the dataset category.

Additionally, we have proposed a measurement of _usefulness_ of documents
based on the signal of document consumption by the users after submitting a
query to the system. From the experimentation, we have noticed that the
usefulness of documents from the publication category, in general, is
substantially greater than the usefulness of documents from the datasets.
Comparing the intra-category usefulness, a vast diversity is reported for
datasets which signifies that certain documents are considerably more _trendy_
in terms of being retrieved and satisfying the information need in case of
searching in datasets than publications.

#### Acknowledgement

This work was funded by DFG under grant MA 3964/10-1, the Establishing
Contextual Dataset Retrieval - transferring concepts from document to dataset
retrieval (ConDATA) project, http://bit.ly/Condata.

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# Quantum photonics in triangular-cross-section nanodevices in silicon carbide

Sridhar Majety 111Corresponding author<EMAIL_ADDRESS>Department of
Electrical and Computer Engineering, University of California, Davis, CA
95616, USA  Victoria A. Norman  Department of Physics, University of
California, Davis, CA 95616, USA  Liang Li  Department of Electrical and
Computer Engineering, University of California, Davis, CA 95616, USA  Miranda
Bell  Department of Materials Science and Engineering, University of
California, Davis, CA 95616, USA  Pranta Saha  Department of Electrical and
Computer Engineering, University of California, Davis, CA 95616, USA  Marina
Radulaski  Department of Electrical and Computer Engineering, University of
California, Davis, CA 95616, USA

###### Abstract

Silicon carbide is evolving as a prominent solid-state platform for the
realization of quantum information processing hardware. Angle-etched
nanodevices are emerging as a solution to photonic integration in bulk
substrates where color centers are best defined. We model triangular cross-
section waveguides and photonic crystal cavities using Finite-Difference Time-
Domain and Finite-Difference Eigensolver approaches. We analyze optimal color
center positioning within the modes of these devices and provide estimates on
achievable Purcell enhancement in nanocavities with applications in quantum
communications. Using open quantum system modeling, we explore emitter-cavity
interactions of multiple non-identical color centers coupled to both a single
cavity and a photonic crystal molecule in SiC. We observe polariton and
subradiant state formation in the cavity-protected regime of cavity quantum
electrodynamics applicable in quantum simulation.

##  1 Introduction

Color center photonics has been on the rise in explorations of quantum
information processing. Initially, the negatively charged nitrogen vacancy in
diamond was explored as a solid-state qubit that can be individually
initialized, manipulated and measured with high fidelity even at room
temperature [1, 2]. Owing to the lack of availability of wafer-scale
substrates and color center emission at telecommunications wavelengths, new
material hosts have been explored as potential defect hosts. Among them,
silicon carbide (SiC) gained attention as a substrate whose color centers have
long spin coherence times [3, 4, 5], excellent brightness [6] and
manipulations of nuclear spin [7, 8], and are suitable for applications in
quantum computing [9, 10, 8], quantum photonics [11, 12], quantum
communications [13, 6, 14, 15], quantum sensing [16, 17, 18, 19, 20], quantum
metrology [21, 22] and quantum simulation [23, 24, 25, 26, 27]. In addition,
SiC has benefits like large bandgap, high thermal conductivity, strong second-
order non-linearity, infra-red emitting color centers, and has a decades-long
industrial presence.

It has been well studied in atomic [28] and quantum dot [29, 30] systems that
optical resonators enhance emitter properties, from Purcell emission
enhancement and coupling to a desired mode to polariton physics and cavity
Quantum Electrodynamics (cQED). These effects have been used in high-speed
quantum light generation, all-optical switching, creation of hybridized light-
matter states, and more. The advantages of integrated photonics with color
centers lie in scalability granted by the combination of the solid-state
platform and the nearly identical optical properties of ensembles [31],
inaccessible to the previously studied systems.

A key requirement for nanophotonic integration is the ability to grow or place
high quality, high refractive index thin films on a contrasting substrate
which can facilitate high optical confinement in nanostructures. For example,
the ability to bond silicon on insulator in a scalable way launched silicon as
a strong material platform for photonics applications [32]. Although
commercial scale high quality crystalline SiC wafers grown using homoepitaxy
(MBE, CVD) or sublimation have transferred into color center research, for
example by _in situ_ doping of vanadium centers in 4H-SiC [33], this approach
does not provide an undercutting layer needed to generate freestanding
photonic devices. On the other hand, growth of 3C-SiC on silicon has been
established in MEMS research [34], but has not lent itself well for color
center formation in nanoscale films due to the lattice mismatch damage [35].
These substrate limitations have been circumvented via photo-electrochemical
etching [36] and SiC thinning and bonding to an insulator [37]. While these
approaches successfully integrated and manipulated color centers in a
nanocavity, the fabrication processes have been limited in terms of either
type or size of the substrate, requiring a specific doping profile or a
limited chip area.

Another bulk substrate nanofabrication approach, initially championed in
diamond [38], has been emerging in silicon carbide: the angle-etching method
[39]. In contrast to the more traditional (rectangular) profile, the photonic
devices created via angle-etch have a triangular cross-section. Such SiC
processing methods are currently based on the Faraday-cage assisted etching;
still, wafer scale processes could be implemented using ion beam etching [40].
Triangular device geometry resolves the problem of the unavailability of
heteroepitaxial thin film color center substrates because it can be
implemented in bulk substrates of any polytype or doping level. However, due
to the novelty of the approach, not much is known about the optical properties
and parameters related to the color center integration into angle-etched SiC
devices.

This paper studies triangularly shaped nanophotonic devices in SiC suitable
for the development of quantum communication and quantum simulation hardware.
Nitrogen vacancy and vanadium impurity in 4H-SiC are of particular interest in
this work, since their emission wavelengths are in the telecommunications
range which makes them useful candidates for long-distance quantum information
distribution. Using Lumerical MODE and FDTD software packages, we examine the
TE and TM polarized modes in the near infrared (NIR) part of the spectrum
supported in triangular waveguides with variable etch-angles, characterizing
their effective index of refraction and the depth of the mode maximum. We
design high $Q$/$V$ ratio photonic crystal cavities and analyze their
resonance parameters and potential Purcell enhancement for variable triangular
profiles. We extend this design into a photonic crystal molecule with coupled
cavity array integration in sight. Finally, using open quantum system
modeling, we explore emitter-cavity interactions of multiple non-identical
color centers coupled to both a single cavity and a photonic crystal molecule,
observing polariton and subradiant states formation.

##  2 SiC color center emission properties

Silicon carbide hosts a variety of color centers across its polytypes. Most
prominently studied centers in 4H-SiC polytype are shown in Table 1. As solid-
state emitters, color centers interact with the host lattice which modifies
their optical properties. The proportion of radiative emission occurring
through the zero-phonon transition compared to the transitions to all vibronic
levels is given by the Debye-Waller (DW) factor $\xi$. Integration into high
quality factor ($Q$) and small mode volume ($V$) cavities can rebalance this
ratio via the Purcell effect. The radiative emission enhancement is given by
the Purcell factor:

$F_{P}=\frac{3}{4\pi^{2}}\bigg{(}\frac{\lambda}{n}\bigg{)}^{3}\bigg{(}\frac{Q}{V}\bigg{)}\xi.$

The Purcell term for any position of the emitter scales with
$\left|E/E_{\textnormal{max}}\right|\cos(\phi)$, where $\phi$ is the angle
between the electric field and the emitter dipole moment. A photonic crystal
cavity (PCC) enabling large Purcell enhancement ensures a high emission rate
of indistinguishable photons needed for the implementation of quantum
entanglement schemes [41].

Defect | ZPL wavelength (nm) | DW factor | Emission lifetime
---|---|---|---
V1’ (V${}_{\textnormal{Si}}^{-}$) [42] | 858 | 19% | (5.6 $\pm$ 1.2) ns
V1 (V${}_{\textnormal{Si}}^{-}$) [43] | 861 | 8% | (5.5 $\pm$ 1.4) ns
V2 (V${}_{\textnormal{Si}}^{-}$) [43, 44] | 918 | 9% | 6 ns
Cr4+ [45] | 1070 | 75% (30K) | 145.8 $\pm$ 9.7 $\mu$s
V${}_{\textnormal{C}}$V${}_{\textnormal{Si}}$0 (kk) [46, 20] | 1130 | (5.3 $\pm$ 1.1)% | (14 $\pm$ 3) ns
V${}_{\textnormal{Si}}$NC [47] | 1176 - 1243 | - | 2.2 ns
V4+ ($\alpha$) [48] | 1279 | 25% | 167 ns
V4+ ($\beta$) [48] | 1335 | 50% | 41 ns
Table 1: Emission parameters for various color centers in 4H-SiC.
V${}_{\textnormal{Si}}^{-}$, V${}_{\textnormal{C}}$V${}_{\textnormal{Si}}$0,
V${}_{\textnormal{Si}}$NC, Cr4+, and V4+ stand for silicon vacancy, divacancy,
nitrogen vacancy, chromium impurity, and vanadium impurity respectively. V1
and V1’ are ZPLs of hexagonal site (h) silicon vacancy, V2 is the ZPL of cubic
site (k) silicon vacancy. A DW factor for
V${}_{\textnormal{Si}}$N${}_{\textnormal{C}}$ has not been reported in
literature and the upper and lower values of the ZPL range come from the hk
and kh orientations respectively. V${}^{4+}(\alpha)$ (V${}^{4+}(\beta)$)
likely corresponds to the h (k) site [48]. The temperature associated with
each DW factor measurement is 300K unless otherwise specified.

##  3 Triangular cross-section SiC waveguide

A waveguide uses total internal reflection to confine propagating light in the
high refractive index material. The triangular cross-section waveguide modeled
in this study using Finite-Difference Eigensolver (Lumerical MODE) was
parametrized by the wavelength to the top-width ratio $\lambda$/$d$ and the
half-angle $\alpha$. The fundamental TE and TM modes were studied for
$0.5\leq\lambda/d\leq 2$ and $30^{\circ}\leq\alpha\leq 40^{\circ}$ with
obtained mode profiles shown in Figure 1.

Figure 1: Electric field intensity profiles $|E|^{2}$ of TE and TM polarized
modes supported by triangular-cross-section SiC waveguides with
$0.5\leq\lambda/d\leq 2$ and $30^{\circ}\leq\alpha\leq 40^{\circ}$. A mesh
dimension of 10 nm was used in the x, y, z directions and a width of 0.754
$\mu$m was used. Color bar is the same as in Figure 4.

To find the optimal positioning of the color center in the waveguide, we
analyzed the depth $h$ at which the electric field intensity maximum
($|E|^{2}_{\textnormal{max}}$) of a mode is positioned relative to the
waveguide’s top plane. As $\lambda$/$d$ increases,
$|E|^{2}_{\textnormal{max}}$ for the fundamental TE (TM) polarized mode shifts
towards (away from) the top waveguide surface i.e., the depth $h$ decreases
(increases). This behavior is consistent for all studied half-angle values
$\alpha$, as shown in Figure 2a. For higher values of $\lambda$/$d$, the mode
becomes less confined and the evanescent losses into the surrounding medium
dominate, as can be observed in mode profiles in Figure 1. These observations
are consistent with the trend in the effective refractive index
($n_{\textnormal{eff}}$) values shown in Figure 2b. The effective refractive
index for a mode is given by the ratio of propagation constant of that mode
($\beta$) to the wavenumber in vacuum ($k_{0}$). When
$n_{\textnormal{eff}}\sim 1.5$, significantly below the SiC refractive index
$n=2.6$, $|E|^{2}_{\textnormal{max}}$ for the TM mode lies outside the
waveguide and results in a lossy propagation.

Figure 2: Normalized depth $h/d$ of the electric field intensity maximum
$|E|^{2}_{\textnormal{max}}$ and the effective index of refraction of TE and
TM modes supported in a triangular-cross-section SiC waveguide, as a function
of the geometry parameters.

##  4 Triangular cross-section SiC photonic crystal cavity

Photonic crystal cavities are nanostructures with periodic variation of
refractive index which localize photons to sub-wavelength mode volumes. Their
use in crystalline silicon carbide has been explored for 2D geometries in
smart-cut 6H-SiC [49] and heteroepitaxially grown 3C-SiC on Si [50, 35], which
do not host high quality color centers, as well as in the 2D heterostructure
geometry [51] and rectangular and trapezoidal 1D cavities [52, 39, 37] in
4H-SiC substrates that do, or in principle could, host color centers. The
quality factors of devices in these studies was up to 630,000 in structures
without emitters and 19,300 for the structures with emitters. In this study,
we turn to triangular nanobeam photonic crystal cavities with elliptical air
holes whose major axis radius is tapered along the device, which have proved
successful in diamond color center research in achieving ultra-high Q-factors
and ultra-small mode volumes [53, 54]. As shown in Figure 3, the cavity has a
width $d=1.94a$ and consists of 30 through holes on either side of the center
of the cavity with uniform spacing between the holes, lattice constant
$a=387.8$ nm. The minor radius of the holes is $0.292a$ and to minimize
scattering and maximize Q-factor, the major radius is quadratically varied
from $r{{}_{1}}=0.486a$ to $r{{}_{30}}=0.292a$ over 30 holes. The beam has a
triangular cross-section with a half-angle $\alpha$ = $None$.

Figure 3: Triangular nanobeam cavity structure and the parabolic variation of
its hole’s major radius $r$ along the length of the beam.  Figure 4: a) The
top and the cross-sectional profile of the fundamental resonant mode. b-d)
Top-view profile of the first three higher order modes in the cavity.

The optical response of the triangular nanobeams was modeled using the Finite-
Difference Time-Domain method in the Lumerical FDTD software package. The
electric field profile of the fundamental mode is shown in Figure 4a. The
fundamental cavity resonance occurs at $\lambda=1324$ nm with a quality factor
of $Q\sim 1.05\times 10^{7}$ and mode volume of $V\sim 2(\lambda/n)^{3}$.
Higher order TE-polarized longitudinal modes are also supported in the cavity,
as shown in Figure 4b-d, where the cross-sectional profile is representative
of the profiles of the electrical field nodes in all modes. Cavities with
half-angle $30^{\circ}\leq\alpha\leq 40^{\circ}$ were simulated while
maintaining the lattice constant and radii of the holes. It was observed that
the resonant wavelength of the fundamental mode decreased with increasing
$\alpha$, whereas the Q-factor and mode volume remained almost the same, as
shown in Figure 5. The impact of a changing lattice constant was studied
across a range of 100 nm, which tuned the resonant wavelength from 1200 nm to
1500 nm, as shown in Figure 5. The Q-factor and mode volume of the fundamental
resonance for varying values of $a$ remained nearly constant (less than a
factor of 2 for $Q$ and within $10\%$ for $V$). Thus, for different
technological implementations of the inclines in angled etching, this tapered-
hole design of a photonic cavity can be used to integrate various SiC color
centers in the telecommunications wavelength range with strong Purcell
enhancement of $F_{max}$ in the range of $2.7\times 10^{5}$ \- $5.3\times
10^{5}$ for DW factor $\xi=1$.

Figure 5: Dependence of the resonant wavelength $\lambda$, the quality factor
$Q$ and the mode volume $V/(\lambda/n)^{3}$ on the half-angle $\alpha$ of the
triangular cross-section and the lattice constant in the triangular nanobeam
SiC cavity.

Triangular photonic crystal geometry lends itself well toward the 1D coupled
cavity array fabrication. To study the coupling between the neighboring
cavities, a photonic crystal molecule was designed by stacking two photonic
crystal cavities side-by-side with a reduced number of holes in the interior
($M<30$) as shown in Figure 6a. The major axis radius of the M holes is
tapered using the same parabolic variation as the 30 holes with the M largest
holes kept and 30-M holes discarded. The coupling between the two cavities
causes the resonant mode splitting. We observe in the transmission spectra
that as the separation between the two cavities is reduced ($M$ decreases),
the coupling strength between the cavities represented in the wavelength
separation between the two resonances increases [55]. By integrating emitters
into the photonic crystal molecule, the designed structures can be employed
for investigation of cQED effects [56].

Figure 6: a) Top-view profile of the resonant mode of the photonic molecule.
Color bar is the same as in Figure 4. b) Variation of $r$ along the length of
the beam. c) Transmission spectrum of a photonic crystal molecule connected
via $2M$ holes, in comparison with a single cavity case.

##  5 Multi-emitter cavity QED in SiC photonic devices

Strong coupling between light and matter lies at the crux of many applications
of optics, including all-optical quantum gates for quantum computing [57, 58]
and quantum simulations of strongly correlated condensed matter systems [59].
While cQED physics has been realized in single quantum dot systems [60, 61]
and single defects in semiconductors [62], the integration of multiple quantum
emitters into cavities has been suggested as a method of accessing the strong
coupling regime for emitters with smaller cavity-emitter coupling terms [31].
In a multi-emitter system, the $N$ emitters that are coupled individually to
the same cavity will effectively behave as a single emitter with emitter-
cavity coupling enhanced by a factor of $\sqrt{N}$. In order to determine
whether this stronger coupling effect will occur in the triangular cavity
devices described above, the quantum master equation was solved using QuTiP,
an open source python package for simulating open quantum systems[63].

The Tavis-Cummings model describes such a multi-emitter system and is given by
the Hamiltonian:

$H_{TC}=\Omega
a^{\dagger}a+\sum_{i=1}^{N}\left[\omega_{i}\sigma_{i}^{+}\sigma_{i}^{-}+g_{i}\left(a^{\dagger}\sigma_{i}^{-}+\sigma_{i}^{+}a\right)\right],$

where $\Omega$ and $\omega_{i}$ are the cavity frequency and the emitter
frequencies, respectively, $a$ and $\sigma_{i}^{-}$ represent the cavity
annihilation operator and the $i$-th emitter lowering operators. Parameters
$g_{i}$ refer to the cavity-emitter coupling rates whose maximal value can be
calculated by the equation:

$g_{max}=\sqrt{\frac{3\pi c^{3}\xi}{2\tau\omega^{2}n^{3}V}},$

and depends on the positioning of the emitter relative to the electric field
profile of the cavity by the relationship
$g=g_{max}\left|E/E_{max}\right|\cos{\phi}$. $E$ is the electric field
intensity at the color center, and $\phi$ is the angle between the orientation
of color center dipole and the electric field. As detailed above, the mode
volume of the simulated triangular photonic crystal cavity is $V\sim
2(\lambda/n)^{3}$, and the values of the optical lifetime $\tau$ and the
Debye-Waller factor $\xi$ are dependent upon the choice of the color center
(see Table 1). The calculated values of $g/2\pi$ in SiC are: $3.41$ GHz for
V1’ (V${}_{\textnormal{Si}}^{-}$), $2.23$ GHz for V1
(V${}_{\textnormal{Si}}^{-}$), $2.27$ GHz for V2
(V${}_{\textnormal{Si}}^{-}$), $0.04$ GHz for Cr4+, $1.14$ GHz for
V${}_{\textnormal{C}}$V${}_{\textnormal{Si}}^{0}$ (kk), $0.72$ GHz for V4+
($\alpha$), and $2.04$ GHz for V4+ ($\beta$). The loss in the system is
simulated via Lindblad dissipators using loss parameters $\kappa$ for cavity
and $\gamma$ for each emitter [64].

Figure 7: Open quantum system simulations of the Tavis-Cummings model for a
single cavity coupled to $N=4$ emitters. The cavities are resonant at
$\omega/2\pi=0$. The emitter frequencies are randomly selected from a Gaussian
distribution centered at $\omega=0$ with a full-width half-maximum of
$\delta$. In the case $\delta=0$, all the emitters are resonant at $\omega=0$.
The emitter frequencies in each panel are randomly generated, independently of
those in other panels. The cavity and the emitter loss rates are
$\kappa/2\pi=10$ GHz, and $\gamma/2\pi=0.17$ GHz.

A suitable range of $g/2\pi$ values was selected for simulations, and the
resulting transmission spectra of a single cavity coupled to $N=4$ emitters
are shown in Figure 7. The emission frequencies of the emitters are randomly
generated from a Gaussian distribution centered at the cavity resonance with a
standard deviation $\delta/2$, where $\delta$ represents the inhomogeneous
broadening in the color center ensemble. The splitting of the transmission
spectrum into two separate polariton peaks separated by $2g\sqrt{N}$ confirms
that the system is in the strong cQED coupling regime for both $g/2\pi$ values
of 3 GHz and 5 GHz. Figure 7 shows that the expected cavity-emitter polariton
peak splitting persists for values of $\delta\lesssim g\sqrt{N}$. Furthermore,
the linewidth of the split emitter-cavity polariton peaks is
$\left(\kappa+\gamma\right)/2$, which indicates these systems exist within the
cavity-protection domain wherein the emitter ensemble is protected from
inhomogeneous broadening caused by the dephasing [65].

Photonic quantum simulation proposals [66] rely on the use of coupled cavity
arrays in the strong coupling regime of cQED. Here, we explore the smallest
such system of $n=2$ coupled nanocavities, as modeled in Figure 8, and in
relation to the designed photonic crystal molecule in Figure 6. In order to
explore the physics of such a system, the Tavis-Cummings model can be expanded
upon by adding a cavity coupling term to form the Tavis-Cummings-Hubbard (TCH)
model:

$H_{TCH}=\sum_{i=1}^{n}\left\\{\Omega_{i}a_{i}^{\dagger}a_{i}+\sum_{j=1}^{N_{i}}\left[\omega_{i,j}\sigma_{i,j}^{+}\sigma_{i,j}^{-}+g_{i,j}\left(a_{i}^{\dagger}\sigma_{i,j}^{-}+\sigma_{i,j}^{+}a_{i}\right)\right]\right\\}-\sum_{i=1}^{n-1}J_{i,i+1}\left(a_{i}^{\dagger}a_{i+1}+a_{i+1}^{\dagger}a_{i}\right)$

where $n$ is the number of cavities, $\omega_{c,i}$, $N_{i}$ and $a_{i}$ are
the resonant frequency of, the number of emitters in, and the annihilation
operator of the $i$-th cavity. The resonant frequency of the $j$-th emitter in
the $i$-th cavity is $\omega_{i,j}$ and it has the annihilation operator
$\sigma_{i,j}^{-}$. The coupling rate between neighboring cavities is given by
$J_{i,i+1}$.

Figure 8: Open quantum system simulations of the Tavis-Cummings-Hubbard model
for $n=2$ cavities coupled to $N_{i}=2$ emitters each at a rate $g/2\pi=5$
GHz, with the same values for $\kappa$ and $\gamma$ as in Figure 7. The
emitter frequencies in each panel are randomly generated, independently of
those in other panels. The peaks in the all-resonant ($\delta=0$) cases are
found at the approximate eigenvalues of the lossless strong coupling case:
$\omega=\pm g\sqrt{2}\pm J$ for the $g>J$ case (orange), and $\omega=\pm
2g^{2}/J,\pm 2g^{2}/J+J$ for the $g<J$ cases (blue and green). The linewidth
of these peaks in the case $\delta<J$ is given by
$\left(\kappa+\gamma_{i}\right)/2$ which is indicative of the cavity
protection regime wherein the emitter ensemble is protected from the
inhomogeneous broadening by the cavity. The intermediate peaks correspond to
the subradiant states and are present even in the $J/2\pi=200$ GHz simulations
(green) though, due to scale, they are not easily visible in the plots.

The transmission spectra of $n=2$ cavities with $N_{i}=2$ emitters coupled to
each cavity for $g/2\pi=5$ GHz were simulated in QuTiP with and without
broadening in emitter frequencies and the results are shown in Figure 8. We
extracted values of the photon hopping rate $J$ by fitting a two-peak
Lorentzian to the plots in Figure 6; the splitting of polariton peaks in
strongly coupled systems is directly proportional to the coupling strength.
The $J/2\pi=1$ GHz and $J/2\pi=200$ GHz photonic crystal molecule cQED plots
exhibit the same cavity protection linewidth signature and peak splitting as
the single cavity simulations. The structure of the $J/2\pi=10$ GHz set of
simulations merits some closer discussion; the eigenvalues of the lossless TCH
model when $J<g$ in a fully resonant case are $\omega=\pm 2g^{2}/J$ and
$\omega=\pm J+2g^{2}/2$, as seen in the $\delta/2\pi=0$ GHz, $J/2\pi=10$ GHz
plot in Figure 8. This same set of peaks is maintained for the values of
$\delta\lesssim g\sqrt{N_{i}}$, which means that light-matter hybridized
states continue even with the introduction of modest, but realistic,
inhomogeneous broadening into the system. These results demonstrate that the
designed triangular cross-section devices are strong candidates for quantum
information and simulation platforms. In addition, the inhomogeneously
broadened emitter systems exhibit subradiant peaks, which have been studied in
single-cavity systems and found to facilitate coherence conducive to high-
purity quantum light generation [67].

##  6 Discussion

Our modeling results provide insights into color center integration with
triangular SiC photonic devices, as well as quantum optical phenomena in
cavity QED systems. This geometry is of great interest because it overcomes
the limitations of color center platforms requiring implementation in bulk
substrates.

Triangular-cross-section waveguides are found to efficiently guide the light
emission from color centers embedded at optimal depths. The waveguides support
propagation of both TE and TM polarized modes, thus serving a variety of color
center crystalline orientations. Fabrication imperfections in the curvature of
the apex affect only the modes present in the lower half of the triangular
cross-section. Similar designs fabricated in diamond have successfully
combined triangular waveguide geometry with suitable grating couplers for
efficient collection of light into a high numerical aperture objective, as
well as tapered down for coupling to an optical fiber [68]. They can also be
designed for integration with the emerging superconducting nanowire single-
photon detectors [69].

The modeled triangular nanobeam photonic crystal cavities have high $Q$/$V$
ratio with promising applications in high-speed and indistinguishable photon
generation. For the same triangular geometry, altering of the lattice constant
is found to tune the resonant wavelength without reduction in the quality
factor, meaning that many different high-performing devices can be
simultaneously fabricated on the same chip. Interestingly, in literature on
diamond and SiC nanobeam devices, the fabricated quality factors fell 2-3
orders of magnitude short of their theoretical predictions [54, 39]. The
reduced performance can be attributed to fabrication errors such as mask
alignment imperfections [39]. This indicates that additional efforts in
nanofabrication process development need to be invested to realize the full
application potential of these designs. Purcell enhancement of the zero-phonon
line emission would play a crucial role in realizing quantum systems that
require two-photon interference and quantum entanglement, such as quantum
repeaters [70] and cluster-entangled states [71, 72].

The open quantum system modeling of a single nanocavity and a photonic crystal
molecule integrating non-identical color centers confirmed that cavity-
protection effects can support polariton physics even in the presence of
realistic inhomogeneous broadening in SiC structures. Extensions to larger
coupled-cavity-arrays would provide novel proposals for photonic quantum
simulation and quantum state transfer.

Our simulations of triangular cross-section waveguides and cavities in 4H-SiC
demonstrate that this geometry can achieve similar figures of merit (e.g. Q
$\sim 10^{6}$) as other state-of-the-art geometries in both 4H- and 3C-SiC
polytype. Since, the thickness and width of these devices are related through
the etch angle, global scaling is possible without affecting the figures of
merit as much. The benefit to our approach is that 4H-SiC can host color
centers without damaging the lattice unlike 3C-SiC and the geometry is highly
scalable using angle RIE etching unlike some of the more complex 2D photonic
crystal cavities. In fact, triangular cross-section nanophotonics have already
been successfully fabricated in diamond; the work already performed in diamond
has illuminated some shortfalls in nanofabrication that need to be overcome
before we can fully realize our simulated results. The prior experiments and
our work in this paper pave the way toward high quality, color center
integrated, highly scalable nanophotonic devices in 4H-SiC.

##  7 Acknowledgments

This work is supported by the National Science Foundation (CAREER-2047564).
The authors thank Richard Scalettar and Jesse Patton for fruitful discussions
on Tavis-Cummings-Hubbard physics.

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