# AI and Generative AI for Research Discovery and Summarization

Mark E. Glickman
Department of Statistics
Harvard Univeristy
<EMAIL_ADDRESS>Department of Statistics, Harvard University,
Cambridge, MA, USA, 02138 Yi Zhang
Department of Statistics
Harvard University
<EMAIL_ADDRESS>
###### Abstract

AI and generative AI tools, including chatbots like ChatGPT that rely on large
language models (LLMs), have burst onto the scene this year, creating
incredible opportunities to increase work productivity and improve our lives.
Statisticians and data scientists have begun experiencing the benefits from
the availability of these tools in numerous ways, such as the generation of
programming code from text prompts to analyze data or fit statistical models.
One area that these tools can make a substantial impact is in research
discovery and summarization. Standalone tools and plugins to chatbots are
being developed that allow researchers to more quickly find relevant
literature than pre-2023 search tools. Furthermore, generative AI tools have
improved to the point where they can summarize and extract the key points from
research articles in succinct language. Finally, chatbots based on highly
parameterized LLMs can be used to simulate abductive reasoning, which provides
researchers the ability to make connections among related technical topics,
which can also be used for research discovery. We review the developments in
AI and generative AI for research discovery and summarization, and propose
directions where these types of tools are likely to head in the future that
may be of interest to statistician and data scientists.

Keywords: Abductive reasoning, Hallucination, Literature discovery, Manuscript
abstraction, Research discovery

##  1 Introduction

One of the most important ways for practicing statisticians and data
scientists to remain viable in their work is being able to learn and apply
unfamiliar methods. Doing so requires the ability to perform literature
searches, become familiar with quantitative subfields, and learn about
methodological advances. Fortunately, in the current age of Artificial
Intelligence (AI), new tools are being developed at a rapid rate that can
assist research in ways that were unthinkable a short time ago. The scope of
AI technology is advancing so quickly, it is difficult to keep track of the
areas of application and of the tools becoming available. As an example,
ChatGPT Plus, the premium version of the widely used generative AI large
language model (LLM) tool by OpenAI, established 3rd-party plugins in March
2023, with a total of 11 plugins (Chincha,, 2023), and has since grown to 1037
plugins as of mid-December 2023 (ScriptByAI,, 2023). Keeping track of the
functionality of all of these plugins, not to mention the explosion of other
AI tools that are becoming available, has become a full-time job.

Until relatively recently, statistical researchers have relied mostly on
digital libraries such as JSTOR (www.jstor.org), which has for decades
archived influential journals in statistics, including Journal of the American
Statistical Association (JASA), Journal of the Royal Statistical Society -
Series B (JRSS-B), Biometrika, the Annals of Statistics, and many more.
However, with options for publication that go way beyond the JSTOR collection,
other avenues have opened up. Many statistical researchers have relied on
Google Scholar searches (scholar.google.com) to find relevant articles based
on provided keywords. Based on key word input, Google Scholar ranks candidate
documents using criteria such as the relevance of the work, where it was
published, and the degree to which the work has been cited by other documents
(Beel and Gipp,, 2009). One of the key limitations of Google Scholar, however,
is the lack of a LLM front end to interpret nuanced inputs.

The following example, which we return to throughout this manuscript,
demonstrates a limitation of non-LLM search tools. Consider the problem in
which a pairwise distance or dissimilarity matrix of $n$ objects has been
computed, and it is of interest to generate $n$ vectors each of specified
dimension $d$ (that is, $d$-dimensional embeddings) whose pairwise Euclidean
distances correspond, at least approximately, to the initial pairwise distance
matrix. Such a procedure is known as classical multidimensional scaling
(Torgerson,, 1952; Gower,, 1966), and is more commonly known among
psychometricians than statisticians and data scientists. Performing a Google
search using the keywords “generating Euclidean vectors from a pairwise
distance matrix” returns the search results in Figure 1.

Figure 1: Google search results based on the keywords “generating Euclidean
vectors from a pairwise distance matrix.”

In every search result item on the first page (and several subsequent pages),
Google returned links to methods for computing pairwise distance matrices
based on Euclidean vectors, which is the opposite of what was requested.
Similarly, a Bing web search resulted in similar sets of results on computing
distance matrices from vectors, though Bing did provide a link for a
StackExchange question-and-answer mentioning the creation of vectors from a
distance matrix, but without providing the correct answer (StackExchange,,
2013).

The difficulty with this particular type of search query is that the parser
may not be able to distinguish whether the query is about computing distance
matrices or about determining vectors that produce given distance matrices.
Additionally, the search engine may not be able to distinguish whether web
pages are addressing the question of interest or a question that involves a
reconfiguration of the words in the search request. Given that many more sites
on the web are concerned with computing distance matrices from vectors, a much
more conventional task, a search engine is more likely in the face of the
uncertainty about the search query to produce results that address the more
commonly queried topic. The role of AI in this type of search is to improve
both on making sense of the question’s intent, and to have a more concrete
“understanding” of website content.

This article reviews the current landscape of AI tools available to
statistical and data science researchers to perform literature searches, to
synthesize topics from disparate sources, and to improve the acquisition of
knowledge for a data science audience. We describe some of the basic functions
of existing tools available to researchers, but our emphasis is on a high-
level explanation of how AI tools can enhance work. These tools are constantly
updating and improving, so it is only possible to provide a snapshot in time
of their basic functioning. We conclude with our hypotheses about how
developments in AI tools for data science research will evolve, given the
technological accomplishments seen up to this point.

##  2 Hallucination issues

In November 2022, the chatbot ChatGPT was made publicly available by OpenAI
and probably changed the world forever. Early indications were that, despite
being trained on an enormous corpus of data that was only current as of 2021
(as of writing, ChatGPT is trained on data up through April 2023), it would
eventually replace web search tools like Google, at least to search for
information recorded no later than 2021. The availability of ChatGPT inspired
other companies working on generative AI to make their products available to
the public as well, proliferating a number of generative AI chatbots.

Traditionally, web searches have been the standard for information retrieval,
offering results based on indexed web pages. However, tools like ChatGPT,
powered by LLMs, represent a new paradigm. They provide conversational and
context-aware responses, which is a departure from the list of links and
summaries typical of standard web searches. Despite this innovative approach,
ChatGPT and similar AI tools do face a unique challenge not commonly
encountered in traditional web searches: the issue of “hallucination.” In the
context of AI, hallucination typically refers to the generation of information
that is either factually incorrect, irrelevant, or nonsensical, yet is often
delivered with a high degree of confidence. This presents a unique challenge
in ensuring the reliability and accuracy of the information provided by these
AI systems.

As widely covered in news reports (e.g., Weise and Metz,, 2023) and detailed
in technical papers (e.g., Achiam, J. et al.,, 2023), LLM-based chatbots,
including ChatGPT, consistently encounter hallucinations. This issue goes
beyond simple technical glitches, striking at the fundamental principles of
how these models are trained and function. The comprehensive survey paper by
Ji et al., (2023) offers an in-depth examination of this phenomenon. It
presents a detailed analysis of the various aspects and implications of
hallucinations in language models, providing a comprehensive overview of how
and why hallucinations occur in AI systems. We refer interested readers to
this paper for a more thorough understanding.

A particularly notable case of hallucination in ChatGPT, which has been
observed in multiple instances, is its tendency to fabricate references or
cite non-existent sources. For example, in a study investigating the frequency
of AI hallucinations in research proposals generated by ChatGPT, it was
discovered that out of 178 references cited, 69 did not have a Digital Object
Identifier (DOI), and 28 of these were found to be completely non-existent
(Alkaissi and McFarlane,, 2023). Similarly, another study evaluating the
quality of the answers and the references provided by ChatGPT for medical
questions, found that out of the 59 references included in the primary
analysis, 41 (69%) were fabricated. Moreover, among the remaining 18 valid
references, several had issues: 3 contained minor citation inaccuracies, and 5
had major citation errors (Gravel et al.,, 2023).

Such hallucinations are not the exception, but rather the norm. We requested
ChatGPT to provide five recent papers on “multidimensional scaling” from the
past decade. The search results are presented in Figure 2. While these results
initially seem credible, a closer look found that two of the five papers
(citations 1 and 4) listed actually do not exist. Moreover, among the three
valid references, each contains citation inaccuracies, including nonexistent
DOIs, incorrect publication years, and in one case (citation 2), an incorrect
journal name.

Figure 2: Response of ChatGPT to provide citations for five papers on
multidimensional scaling from the last ten years.

This pattern of hallucinations in ChatGPT’s output significantly undermines
the reliability of AI-generated information, especially when used as a search
tool for academic content where precision and factual accuracy are not merely
preferred, but essential. Consequently, while LLMs like ChatGPT can augment
traditional search methods by providing quick, comprehensive, and sometimes
insightful summaries, their use must be complemented with an appreciation of
their limitations.

Current efforts are focusing on developing more sophisticated training
methods, such as using verified and fact-checked datasets, and implementing
advanced algorithms capable of cross-referencing and validating information
against trusted sources. To this end, chatbots like ChatGPT Plus, which uses
the GPT-4 (Achiam, J. et al.,, 2023) LLM, have incorporated plugins that
specifically search for scholarly content and grab the actual publication
details rather than let the LLM hallucinate document details on its own.
Moreover, there is a growing recognition of the importance of human oversight
in the use of LLMs. Integrating expert review can provide an additional layer
of verification. Researchers and users are best served by approaching the LLM
output with a critical eye, verifying against primary sources and employing a
combination of methods for thorough research.

##  3 Abductive reasoning

The process of inferring the most believable explanation based on a given set
of observations or statements is known as abductive reasoning (Peirce,, 1935;
Peirce et al.,, 1973). Typically, potential explanations may not be spelled
out a priori, so unlike statistical inferential settings in which a set of
potential explanations is usually listed, some notion of creativity is
typically associated with the abductive process. A classic example of
abductive reasoning is when a patient mentions to their medical provider
various symptoms from which they are suffering. The provider then needs to
(abductively) reason the likeliest explanation of the symptoms before
proposing treatment.

Abductive reasoning is also an essential skill in research discovery, one that
is practiced by quantitative researchers all the time. For example, a
researcher may have an idea for a novel approach to a problem, and wants to
know whether such an approach already exists and has been developed and
studied, or about related methods that address the problem the researcher is
attempting to solve. Another situation is where a researcher may suspect that
an algorithm, statistical procedure, or computational method may already
exist, but is unaware of the conventional name of the procedure. These tasks
are instances of abductive reasoning; the details of an algorithm or process
(the “observations”) are described, and then the conventional description that
most plausibly describes the process (the “explanation”) is identified.

These abductive research discovery tasks are now being successfully carried
out using LLMs. It is a remarkable feat that LLMs can mimic various aspects of
logical reasoning, despite not being trained to do so. Evidence from recent
work (Wei et al., 2022a, ; Wei et al., 2022b, ; Suzgun et al.,, 2022) suggests
that the more highly parameterized the LLM, the greater the likelihood for the
LLM to exhibit reasoning ability in different forms. Examination of the
ability of LLMs to reason successfully have produced mixed results (Qiao et
al.,, 2022; Guo et al.,, 2023; Huang and Chang,, 2022; Davis,, 2023), though
many of the documented conclusions may already be out of date. Bang et al.,
(2023) assessed ChatGPT’s abductive reasoning ability based on a challenge
dataset (Bhagavatula et al.,, 2019) that tests abductive natural language
inference. The authors noted that ChatGPT achieved 86.7% accuracy, and that
ChatGPT performed better at deductive and abductive reasoning than inductive
reasoning. Pareschi, (2023) has shown that advanced LLMs like GPT-4 show
promise in abductive reasoning in complex situations.

We have found that chatbots such as ChatGPT using GPT-4 can be effective
abductive reasoners to discover existing methods based on descriptions. As a
simple example, suppose we are interested in learning about methods that can
be used to compare the distribution of a binary variable across different
groups, as an analog to ANOVA for quantitative outcomes. Using the prompt
“What methods exist for testing whether the distribution of a binary variable
across different groups is the same, like how ANOVA is often used for
quantitative outcomes?”, ChatGPT responds with a summary of common methods
that include brief descriptions of chi-square tests, Fisher’s Exact test,
logistic regression, the Mantel-Haenszel test, and a few others that are also
relevant. For a researcher who may not have been familiar with these standard
approaches, ChatGPT’s response provides an answer that can be used as a
starting point for follow-up.

The earlier example on generating embeddings given a pairwise distance matrix
is even more compelling, especially given the failure of commonly used search
engines to direct the user to appropriate websites. When prompted with, “How
do I generate Euclidean vectors from a given pairwise distance matrix? Explain
the basic approach,” ChatGPT responds with “Generating Euclidean vectors from
a given pairwise distance matrix is a task that involves multidimensional
scaling (MDS). The goal is to find a set of points in a Euclidean space such
that the distances between points in this space are as close as possible to
the distances given in the pairwise distance matrix.” It then provides the
correct algorithm to construct the embeddings.

##  4 Literature discovery

Literature discovery and comparative analysis of research literature often
come up as some of the most time-consuming areas when conducting research. As
we discuss below, generative AI has been making a big impact in these areas.

###  4.1 Standalone web-based literature search tools

A literature search engine with a similar purpose to Google Scholar and serves
as the back-end for several AI-enhanced tools is Semantic Scholar
(www.semanticscholar.org). This free literature search tool is popular with
developers given its reliable API. The Semantic Scholar engine, which focuses
on scientific literature searches, has the same limitations on input as Google
Scholar, but relies more heavily on citation networks and collaborative
filtering to identify relevant documents. It can also fine-tune literature
searches by allowing the user to specify which documents in its initial search
were relevant and which were not, allowing better results after iterating the
search. While not strictly an AI tool on its own, Semantic Scholar, along with
other recent literature search tools we describe below, has begun enhancing
its utility by adding generative AI features. For example, a search on
Semantic Scholar accompanies many suggested documents with a brief, typically
one-sentence, summary of the contents of the work. These distillations are
created using GPT-3 (Brown et al.,, 2020) style parsing of an article’s
abstract, body, conclusion and title (Cachola et al.,, 2020). At the time of
writing, Semantic Scholar has a database of over 200 million papers over which
searches can be performed.

Over the past year, many standalone web-based tools have entered the market as
options for conducting research literature searches. The main distinction
between these tools and ones like Google Scholar and Semantic Scholar is the
use of LLMs to interpret the research prompt provided by the user. This
feature may be used to improve the relevance of search queries, or provide
downstream output based on the form of the prompt. One of the most popular
tools is Consensus (https://consensus.app). Early in 2023, the engine had a
fairly restricted database of articles that focused on only six areas of
scientific inquiry, but more recently Consensus has expanded to the Semantic
Scholar database. When a user enters a prompt, Consensus performs a search of
the non-stop words in the prompt to information in the titles and abstracts of
all the documents in the database. If the user asks a question as their
prompt, Consensus aims to supply an LLM-generated answer based on the results
of the search.

Another popular standalone tool is Assistant by Scite
(https://scite.ai/assistant). Like Semantic Scholar, Scite’s Assistant
performs a key word search to generate candidate works relevant to the search
query. The main strength of Scite is its ability to sort the results by a more
complex citation algorithm (Nicholson et al.,, 2021). Because Scite is focused
on citation networks and algorithms, the search tool can provide forward
citations of articles on the search list results as easily as it can construct
traditional backward citations. Much of the use for Scite is on finding
articles that may support or contrast the search prompt. This may be more
appropriate to fields with domain-specific queries, but probably not so
directly useful for statistical or data science research.

A standalone web-based literature search tool that we explore below in more
detail is Elicit (Kung,, 2023), which can be accessed at https://elicit.com/.
Elicit also uses Semantic Scholar as its database, and finds relevant
documents that are semantically related to the research question. The
summaries from Elicit include one-sentence summaries of each suggested work
(like those directly produced by Semantic Scholar), along with a GPT-3 created
overall paragraph summary of the top papers in the list. The user can interact
with individual articles by clicking on them, revealing more information about
the article. A useful feature is the ability to view critiques of the work
(determined through a semantic analysis) by forward-cited articles.

As an example of the use of Elicit, we asked it, “What methods can be used to
perform multidimensional scaling based on a pairwise distance matrix?” The
results were relevant, but most were older references. We used a now-
deprecated feature of Elicit, called “brainstorming”, to generate the useful
follow-up question “What are the most commonly used algorithms for
multidimensional scaling?” This produced a more useful set of articles, the
results of which can be seen in Figure 3.

Figure 3: Results of query to Elicit.

The article in the list by Yang et al., (2006) seemed to be worthy of follow-
up, and upon selecting that article, we obtained the summary seen in Figure 4.

Figure 4: Information provided by Elicit on selected article.

Again, we note that in addition to the abstract summary, the possible
critiques and other citations can help in evaluating the usefulness of the
referenced article. These standalone tools have other helpful features, such
as the ability to further explore the text content of suggested articles via
natural language processing (NLP) analyses, export articles and meta-data, and
to save and organize previous sessions. It should be noted that these
standalone web-based tools are moving in the direction of being fee-based, and
the feature selections described here may be out of date as the development of
these tools quickly evolve.

###  4.2 Standalone web-based literature mapping tools

AI-powered engines like Litmaps (www.litmaps.com) and ResearchRabbit
(www.researchrabbit.ai) simplify the challenging task of identifying potential
gaps in existing literature by visualizing the connections among different
academic publications, helping to highlight overlooked or underexplored study
areas. Litmaps is an innovative tool designed to streamline the process of
literature review for researchers and academics. Utilizing citation searches
tailored to databases, Litmaps reveals papers related to a specific topic and
then visually maps out the relationships among these studies, providing a
cohesive overview of the scientific landscape. This interactive tool generates
literature maps that not only help researchers find relevant papers and
understand the connections among them, but also facilitate the creation of a
customized repository to aid in the literature review process.

As a simple application, we return to our previous example of generating
embeddings from a distance matrix. By inputting the keywords “multidimensional
scaling”, Litmaps promptly displays a list of relevant articles and papers.
The user can then mark preferred articles and proceed to click on the
“Generate Seed Map” button to initiate a literature review, as shown in Figure
5.

Figure 5: Results of search in Litmaps based on keywords.

To illustrate, we selected the published book Borg and Groenen, (2005) as a
starting point, at which point Litmaps then generates a seed map, finding
papers that either cite or are cited by this foundational piece. In the top
left corner of the map, the paper on “Local Linear Embedding” (Roweis and
Saul,, 2000) is displayed as seen in Figure 6, which has received numerous
citations reflecting its impact in dimension reduction.

Figure 6: A display of the relevant literature on “local linear embedding”
based on the generated seed map.

Building upon this newly selected article and its associated seed map, the
“Discovery” feature in Litmaps now accommodates multiple seed inputs, piecing
together an extensive discovery map that encompasses a wider array of related
literature. This map is organized such that our initial selections form the
core in the inner circle, while the outer circle highlights the most relevant
additional works. The results are illustrated in Figure 7.

Figure 7: Results of the “Discovery” feature in Litmaps.

These visualization tools, serving as a kind of “Spotify” for academia, have
significantly expanded the possibilities for literature discovery, especially
in identifying papers that cross multiple topics or disciplines. They simplify
the process of finding related papers and enhance the ability to efficiently
access a broad spectrum of academic knowledge. Importantly, these tools can
also uncover hidden, cross-disciplinary connections that might be missed when
specialists concentrate exclusively on their own fields.

###  4.3 Within-ChatGPT plugin tools/Custom GPTs

In addition to standalone applications, ChatGPT plugins have extended the
functionality of the base ChatGPT model, enabling it to perform specialized
tasks related to research discovery and summarization.

One popular tool is the ScholarAI plugin (scholarai.io), which is tailored to
help researchers access a wide range of peer-reviewed articles, journals, and
conference papers. It simplifies the process of finding relevant literature by
allowing searches based on specific queries, such as keywords, authors, or
topics, and then retrieves abstracts or even full texts (when available) from
various academic publications. A key feature of ScholarAI is its ability to
provide concise summaries of these papers. This is particularly useful for
researchers who need a quick understanding of a study without reading the
entire document, or even an article’s abstract. Additionally, ScholarAI
includes citation management tools, which help users save and organize the
most relevant papers for their research.

To illustrate the functionality of the ScholarAI plugin, we again consider our
previous example. We asked the plugin to find relevant papers on the topic of
“local linear embedding”.

Figure 8: Search results on “local linear embedding” using the ScholarAI
plugin.

As shown in Figure 8, the plugin returned several articles based on these
keywords, each with a summary, citation counts, and source links. Notably, it
included Roweis and Saul, (2000), the highly cited Science paper which was
also discovered previously through the Litmaps App. ScholarAI provided a brief
abstract of this paper, which allows us to quickly grasp the main ideas and
importance of the paper. In addition to this key publication, the plugin also
identified other relevant studies, each summarized with essential details.
This enables us to get a quick understanding of each paper’s contributions
without needing to read the full texts.

In addition to the standard ChatGPT plugins, OpenAI has recently been
encouraging users to create and share their own “custom” GPTs in the
marketplace. Since the launch of OpenAI’s new GPT store, there has been a
significant increase in the demand for custom GPTs across various fields and
domains. Simply speaking, a custom GPT is a version of ChatGPT that is
preloaded with specific knowledge and instructions for interacting with users.
For example, we have noticed the recent launch of ResearchGPT
(www.researchgpt.com), a custom GPT created by Consensus, which provides a
seamless integration of ChatGPT’s conversational capabilities with access to
the Semantic Scholar database of research papers, making it arguably more
useful for scientific research than using the standalone Consensus website.
This GPT can perform in-depth searches in academic databases, provide brief
summaries and answers based on science, help write content with accurate
references, and even assist in creating introductions for academic papers.

###  4.4 Specific tools for summarizing and abstracting manuscripts

ChatGPT is increasingly recognized as an effective tool for summarizing
research papers. In general, it does well in analyzing a whole research paper
and then producing a concise summary. However, its functionality has been
previously limited by the requirement for manual text input, and only recently
does it support direct file uploads. ChatGPT plugin tools such as AskYourPDF
and ChatwithPDF, and even ScholarAI which has been enhanced to read pdf files,
were more specifically developed to enable direct handling of (multiple) pdf
files, providing a more flexible option for researchers who need to engage
closely with their lengthy documents and data. These tools can provide short,
clear summaries of long articles, making them easier to understand. This saves
time and helps to clearly show the main points of each paper, making them more
digestible.

Although ChatGPT and its associated plugins are effective for summarizing
general written content, unfortunately, they encounter difficulties when it
comes to more technical research papers, especially those heavy in mathematics
and data analysis. The design of ChatGPT, which is more suited for natural
language, can struggle to accurately summarize and abstract important details
from the technical parts of these papers. This issue is particularly
noticeable in areas such as statistical methodology, where crucial theorems
and technical specifics are frequently presented through intricate
mathematical expressions. In such cases, ChatGPT is better at offering a basic
overview rather than a detailed technical analysis.

As an example, we explored the accuracy of details about Yang et al., (2006)
summarized by ChatGPT. With the ScholarAI plugin loaded, ChatGPT presented a
brief summary of the initial sections of the paper. One detail in the ChatGPT
summary was “The authors propose a divide-and-conquer approach, dividing the
matrix into smaller submatrices for MDS, then combining the solutions for the
complete $n\times n$ matrix.” We followed up with the question, “How are the
results from performing MDS on the smaller submatrices combined to form a
complete solution?” The response by ChatGPT can be seen in Figure 9.

Figure 9: Summary extraction from Yang et al., (2006).

The answer by ChatGPT is mostly a rephrasing of the first three paragraphs of
text in Section 3 of the manuscript. It is mostly accurate, which may not be
surprising given that it is essentially a rephrasing of the body of the paper.
However, ChatGPT makes a mistake that appears in items 1 and 5 of its
response. ChatGPT refers to the dimensions of the submatrices being of size
$n_{p}\times n_{p}$, and that $n_{p}$ points have been sampled from $D_{i}$.
In fact, the manuscript refers to the dimensions of the submatrices as
$\frac{n}{p}\times\frac{n}{p}$, and refers to $\frac{n}{p}$ points sampled
from $D_{i}$, so ChatGPT misread the quotients as subscripts. Otherwise, the
technical details appear accurate.

The ability of ChatGPT to summarize research papers varies depending on the
content. It currently appears to be useful for text-focused papers but less
effective for those with significant technical or mathematical details.

##  5 Discussion

Generative AI technology has made enormous strides in such a short time that
it should not be surprising that the capabilities of the tools we described
are evolving rapidly. On almost a biweekly basis, we receive email updates
about new and upcoming features of the tools we reviewed in this article.
Given this dynamic landscape, attempting to encapsulate the full spectrum of
features in a static publication would be futile. Even though the state of the
art in AI for use in research discovery and exploration is impressive, there
remains significant potential for further enhancement. Insightful predictions,
challenges, and possible directions for AI development have been articulated
by Morris, (2023) drawing on input from scientists in various fields, and by
Brodnik et al., (2023) in the area of mechanics. In this context, we offer our
perspective on the probable trajectories of AI advancement, aiming to further
streamline and empower the processes of literature discovery and synthesis.

The tools that currently exist to discover research references are remarkable,
yet the scope for enhancing the databases and repositories that underpin these
research discovery tools is substantial. We anticipate that as the tools
improve, larger catalogs of articles and books would become discoverable. A
major challenge, however, lies in navigating copyright issues when accessing
literature behind paywalls, such as non-open access journals. This obstacle
underscores the need for innovative solutions. Importantly, however, tools
could be developed that no longer only searched conventional formats such as
articles and books, but would include videos (like lectures or talks),
podcasts, slide presentations, and other non-traditional methods of
dissemination that could greatly benefit researchers. This is especially
valuable for those in the early stages of exploring fields within statistics
and data science, who might prefer absorbing information through these
alternative, more engaging formats.

In addition to expanding the resources for research discovery, the burgeoning
field of extracting and synthesizing information from multiple sources is
poised for considerable advancement, complementing the expansion of research
discovery resources. On a smaller scale, having reliable tools that analyze
multiple documents can help to distinguish the unique contributions among
them, as well as to detect common patterns or methodological trends. On a
larger scale, the ability to synthesize large collections of documents on a
related topic can lead to creating review articles that would enable
researchers to delve into and learn new domains within statistical scientific
inquiry with greater ease and depth.

By integrating the content analysis of large collections of published research
with impact measures like citation indices, it is possible to discern emerging
trends of topics in statistics and data science. This information could guide
statistical researchers in strategically choosing their future areas of focus.
Using the same source of data, a more speculative development would be the
ability to determine from, say, an article’s abstract whether the article is
likely to be highly cited in the future. In principle, this could be achieved
through careful modeling of article citation indices as a function of an
abstract’s contents, providing a predictive tool for gauging the potential
influence of research works.

Another type of tool that could take advantage of both large collections of
published research along with AI analysis of text relates to improving
citations within manuscripts. For example, having a tool that analyzes a draft
version of a manuscript and suggests alternative and possibly more
foundational or highly-cited references instead of ones currently in the
manuscript could be enormously helpful. Perhaps even more ambitiously, it is
within the realm of possibilities that a tool could analyze a draft manuscript
and find instances of text that should include proper citations if they do not
already exist, flagging these for the researcher’s attention. Given the amount
of time researchers spend on tracking down relevant citations, off-loading
this task to an AI tool would make publishing research more efficient.

Finally, the success of AI in language translation (Doherty,, 2016; Tran et
al.,, 2018; Gülçehre et al.,, 2015) suggests its potential application in a
unique academic challenge: translating specialized terminologies across
various quantitative sub-fields in statistics and data science. Researchers
often encounter difficulties when trying to reconcile differing terminologies
used in classical statistics, computer science, machine learning,
econometrics, and psychometrics. The development of AI tools designed to
translate technical language across disciplines could serve as a crucial
bridge, facilitating better understanding and integration of research across
these areas. Such tools would enable researchers to more easily comprehend and
utilize work that might currently be challenging to interpret, thus
encouraging interdisciplinary collaboration and innovation.

It is important to acknowledge that the future directions of AI in research
discovery and summarization outlined here are speculative. However, we believe
that the foundational work for these developments is already in place, making
significant advancements a matter of “when,” not “if.” The rapid pace of AI
progress in recent years is nothing short of astonishing, with tangible
impacts already felt by researchers in their professional lives. We anticipate
that in the near future, AI tools for research tasks will have evolved to such
an extent that researchers can dedicate their efforts to what truly matters:
engaging in in-depth research. This shift promises to transform the research
landscape, allowing researchers to focus more on innovation and less on
administrative tasks.

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# On a possibility of baryonic exotica††thanks: Presented at the 2nd
Jagiellonian Symposium on Fundamental and Applied Subatomic Physics, June
3-11, 2017, Kraków, Poland.

Michał Praszałowicz  M. Smoluchowski Institute of Physics, Jagiellonian
University,
ul. S. Łojasiewicza 11, 30-348 Kraków, Poland.

###### Abstract

Models based on chiral symmetry predict pentaquarks that have relatively low
masses. We briefly review both theoretical and experimental status of exotica
in the light sector. Next, shall show how to extend chiral models to baryons
with one heavy quark and show that one expects exotica also in this case.
Finally, we interpret recently discovered by the LHCb Collaboration five
$\mathit{\Omega}^{*}_{c}$ resonances in terms of regular and exotic
excitations of the ground state $\mathit{\Omega}_{c}$.

14.20.Lq, 12.38.Lg, 12.39.Hg

##  1 Introduction: Chiral Quark-Soliton Model

The Chiral Quark-Soliton Model ($\chi$QSM) is based on an old argument by
Witten [1], which says that in the $N_{c}\rightarrow\infty$ limit ($N_{c}$
stands for number of colors), $N_{c}$ relativistic valence quarks generate
chiral mean fields represented by a distortion of a Dirac sea that in turn
interact with the valence quarks themselves (for a review see Ref.[2]). In
this way, a self-consistent configuration called a _soliton_ is formed. In
Fig. 1 (a) we plot schematic pattern of light quark energy levels
corresponding to this scenario. It is assumed that the mean fields exhibit so
called hedgehog symmetry, which means that neither quark spin ($\bm{S}_{q}$)
nor quark isospin ($\bm{T}_{q}$) are ”good” quantum numbers. Instead a grand
spin $\bm{K}=\bm{S}_{q}+\bm{T}_{q}$ is a ”good” quantum number. The lowest
valence level has $K^{P}=0^{+}$.

Figure 1: Schematic pattern of light ($u$ and $d$) quark levels in a self-
consistent soliton configuration. In the left panel all sea levels are filled
and $N_{c}$ (=3 in the figure) valence quarks occupy the $K^{P}=0^{+}$ lowest
positive energy level. In the middle panel, one valence quark has been
stripped off, and the soliton has to be supplemented by a heavy quark not
shown in the figure. In the right panel, a possible excitation of a sea level
quark, conjectured to be $K^{P}=1^{-}$, to the valence level is shown, and
again the soliton has to couple to a heavy quark. Levels for strange quarks
that exhibit different filling pattern are not shown.

In order to project out spin and isospin one has to rotate the the soliton,
both in flavor and configuration spaces. These rotations are then quantized
semiclassically and the collective Hamiltonian is computed. The model predicts
rotational baryon spectra that satisfy the following selection rules:

  * •

allowed SU(3) representations must contain states with hypercharge
$Y^{\prime}=N_{c}/3$,

  * •

the isospin $\bm{T}^{\prime}$ of the states with $Y^{\prime}=N_{c}/3$ couples
with the soliton spin $\bm{J}$ to a singlet: $\bm{T}^{\prime}+\bm{J}=0$.

Figure 2: Lowest lying SU(3) flavor representations allowed by the constraint
$Y^{\prime}=1$. The first exotic representation, $\overline{\mathbf{10}}$,
contains the explicitly exotic pentaquark states $\mathit{\Theta}^{+}$,
$\mathit{\Xi}^{+}$ and $\mathit{\Xi}^{--}$ and non-exotic nucleon- and sigma-
like states.

In the case of light positive parity baryons the lowest allowed
representations are $\mathbf{8}$ of spin 1/2, $\mathbf{10}$ of spin 3/2, and
also exotic $\overline{\mathbf{10}}$ of spin 1/2 with the lightest state
corresponding to the putative $\mathit{\Theta}^{+}(1540)$. They are shown in
Fig. 2. Chiral models in general predict that pentaquarks are light [3, 4] and
– in some specific models – narrow [4].

After the first enthusiastic announcements of the discovery of pentaquarks in
2003 by LEPS [5] and DIANA [6] collaborations, the experimental evidence for
the light exotica has been questioned (see e.g. [7]). Nevertheless, both DIANA
[8] and LEPS [9] upheld their original claims after performing higher
statistics analyses. The report on exotic $\mathit{\Xi}$ states (see Fig. 2)
by NA49 [10] from 2004, to the best of my knowledge, has not been questioned
so far, however the confirmation is still strongly needed.

Another piece of information on $\overline{\mathbf{10}}$ comes from the $\eta$
photo-production off the nucleon. Different experiments confirm the narrow
structure at the c.m.s. energy $W\sim 1.68$ GeV observed in the case of the
neutron, whereas no structure is observed on the proton (see Fig. 27 in the
latest report by CBELSA/TAPS Collaboration [11] and references therein). The
natural interpretation of this ”neutron puzzle” was proposed already in 2003
in Ref. [12]. There one assumes that the narrow excitation at $W\sim 1.68$ GeV
corresponds to the non-exotic penta-nucleon resonance belonging to
$\overline{\mathbf{10}}$. Indeed, the SU(3) symmetry forbids photo-excitation
of the proton member of $\overline{\mathbf{10}}$, while the analogous
transition on the neutron is possible. This is due to the fact that photon is
an SU(3) $U$-spin singlet, and the $U$-spin symmetry is exact in the SU(3)
symmetric limit. An alternative interpretation is based on a partial wave
analysis in terms of the Bonn-Gatchina approach [13]. There is an ongoing
dispute on the interpretation of the ”neutron puzzle” (for the latest
arguments see Ref. [14]).

##  2 Heavy Baryons in the Chiral Quark-Soliton Model

In a recent paper [15] following [16] we have extended the $\chi$QSM to
baryons involving one heavy quark. In this case the valence level is occupied
by $N_{c}-1$ light quarks (see Fig 1 (b)) that couple with a heavy quark $Q$
to form a color singlet. The lowest allowed SU(3) representations are shown in
Fig. 3. They correspond to the soliton in representation in
$\overline{\mathbf{3}}$ of spin 0 and to ${\mathbf{6}}$ of spin 1. Therefore,
the baryons constructed from such a soliton and a heavy quark form an SU(3)
antitriplet of spin 1/2 and two sextets of spin 1/2 and 3/2 that are subject
to a hyper-fine splitting. The next allowed representation of the rotational
excitations corresponds to the exotic $\overline{\mathbf{15}}$ of spin 0 or
spin 1 [17]. The spin 1 soliton has lower mass and when it couples with a
heavy quark, it forms spin 1/2 or 3/2 exotic multiplets that should be hyper-
fine split similarly to the ground state sextets by $\sim 70$ MeV.

The rotational states described above correspond to positive parity and are
clearly seen in the data [17]. Negative parity states are generated by soliton
configurations with one light quark excited to the valence level from the
Dirac sea (Fig. 1 (c)). The selection rules for excited quark solitons can be
summarized as follows [18]:

  * •

allowed SU(3) representations must contain states with hypercharge
$Y^{\prime}=(N_{c}-1)/3$,

  * •

the isospin $\bm{T}^{\prime}$ of the states with $Y^{\prime}=(N_{c}-1)/3$
couples with the soliton spin $\bm{J}$ as follows:
$\bm{T}^{\prime}+\bm{J}=\bm{K}$, where $\bm{K}$ is the grand spin of the
excited level.

Figure 3: Lowest lying SU(3) flavor representations allowed by the constraint
$Y^{\prime}=2/3$. The first exotic representation, $\overline{\mathbf{15}}$,
contains the putative pentaquark states $\mathit{\Omega}_{c}$ with
$\mathit{\Omega}_{c}^{0}$ marked in red.

The first allowed SU(3) representation for one quark excited soliton is again
$\overline{\mathbf{3}}$, Fig. 3, with $T^{\prime}=0$, which for $K=1$ is
quantized as spin 1. The coupling of a heavy quark results in two hyperfine
split antitriplets that are indeed seen in the data [17]. The hyperfine
splitting parameter is in this case $\kappa^{\prime}/m_{c}\sim 30$ MeV. Next
possibility is flavor $\mathbf{6}$ with $T^{\prime}=1$, which may couple with
$K=1$ to $J=0,1$ and $2$ resulting in 5 hyperfine split heavy sextets: two
$1/2^{-}$, two $3/2^{-}$ and one $5/2^{-}$ (see Tab. 1).

##  3 Possible interpretation of the LHCb $\mathit{\Omega}_{c}^{0}$ resonances

In a very recent paper the LHCb Collaboration announced five
$\mathit{\Omega}^{0}_{c}$ states with masses in the range of $3-3.2$ GeV [19].
The simplest possibility would be to associate them with the five sextets
described at the end of Sect. 2. We have shown, however, in [17] that this
scenario fails, as can be seen from Table 1.

$J$ | $S^{P}$ |  $M$ [MeV] |  $\kappa^{\prime}/m_{c}$ [MeV] |  $\Delta_{J}$ [MeV]
---|---|---|---|---
0 | ${1}/{2}^{-}$ | 3000 | – | –
1 | ${1}/{2}^{-}$ | 3050 | 16 | 61
${3}/{2}^{-}$ | 3066
2 | ${3}/{2}^{-}$ | 3090 | 17 | 47
${5}/{2}^{-}$ | 3119
Table 1: $\chi$QSM scenario where all LHCb $\mathit{\Omega}_{c}^{0}$ states
are assigned to the excited sextets. This assignment requires hyperfine
splitting which is almost two times smaller than in the
$\overline{\mathbf{3}}$ case and relation $\Delta_{2}=2\Delta_{1}$ derived in
[17] is badly broken. Here $\Delta_{J}$ is the mass difference between states
of given $J$ and $J-1$ before hyper-fine splitting. Figure 4: LHCb spectrum
[19] with the assignment described in the text. Two narrow $1/2^{+}$ and
$3/2^{+}$ states marked with arrows are interpreted as
$\overline{\mathbf{15}}$ pentaquarks.

In the second scenario proposed in [17], we have interpreted three LHCb states
as quark excitations of the ground state sextets, shown in Fig. 4 as vertical
lines. Two remaining sextet excitations have higher mass and are above the
threshold for the decays into charm mesons. They can be, therefore, wide and
the branching ratio to $\mathit{\Xi}^{+}_{c}+\mathit{K}^{-}$ final state may
be small. This would explain why they are not seen by LHCb. On the other hand,
two remaining $\mathit{\Omega}_{c}^{0}$ peaks are in this scenario interpreted
as rotational excitations corresponding to the exotic
$\overline{\mathbf{15}}$. As such, they are isospin triplets and should decay
not only to $\mathit{\Xi}^{+}_{c}+\mathit{K}^{-}$ but also to
$\mathit{\Xi}_{c}^{0}+\mathit{K}^{-}$ or
$\mathit{\Xi}_{c}^{+}+\bar{\mathit{K}}^{0}$ and
$\mathit{\Omega}_{c}+\mathit{\pi}$ final states. This scenario is, therefore,
very easy to confirm or falsify. Moreover, they are very narrow with widths
around 1 MeV, and the $\chi$QSM provides a mechanism that suppresses
pentaquark decays both in the light sector and in the present approach to
heavy baryons [4, 17].

Summarizing, let us stress that despite many ”null findings” there is still an
experimental support for light and narrow pentaquarks. Using the ideas of the
$\chi$QSM, we have proposed an interpretation of recently discovered
$\mathit{\Omega}_{c}^{0}$ states in terms of quark and and rotational
excitations of the ground state charmed baryons, the latter corresponding to
the pentaquarks.

## Acknowledgemens

This note is based on Refs. [15, 17] where more complete list of references
can be found. I would like to thank H.C. Kim, M.V. Polyakov and G.S. Yang for
a fruitful collaboration.

## References

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# Synchronization in optically-trapped polariton Stuart-Landau networks

S. L. Harrison  School of Physics and Astronomy, University of Southampton,
Southampton, SO17 1BJ, UK.  H. Sigurdsson  School of Physics and Astronomy,
University of Southampton, Southampton, SO17 1BJ, UK.  Skolkovo Institue of
Science and Technology, Novaya St. 100, Skolkovo 143025, Russian Federation.
P. G. Lagoudakis  School of Physics and Astronomy, University of Southampton,
Southampton, SO17 1BJ, UK.  Skolkovo Institue of Science and Technology,
Novaya St. 100, Skolkovo 143025, Russian Federation.

###### Abstract

We demonstrate tunable dissipative interactions between optically trapped
exciton-polariton condensates. We apply annular shaped nonresonant optical
beams to both generate and confine each condensate to their respective traps,
pinning their natural frequencies. Coupling between condensates is realized
through the finite escape rate of coherent polaritons from the traps leading
to robust phase locking with neighboring condensates. The coupling is
controlled by adjusting the polariton propagation distance between neighbors.
This permits us to map out regimes of both strong and weak dissipative
coupling, with the former characterized by clear in-phase and anti-phase
synchronization of the condensates. With robust single-energy occupation
governed by dissipative coupling of optically-trapped polariton condensates,
we present a system which offers a potential optical platform for the
optimization of randomly connected $XY$ Hamiltonians.

Introduction. Studies on instabilities, synchronization, and pattern formation
in systems of limit-cycle oscillators appear in many scientific disciplines
such as hydrodynamics, biological ensembles, neuronal networks, nonlinear
optics, Josephson junctions, and coupled Bose-Einstein condensates Cross and
Hohenberg (1993); Acebrón  _et al._ (2005); Matheny  _et al._ (2019). In the
regime of strong light-matter coupling, condensates of microcavity exciton-
polaritons (herein polaritons) are found to follow similar oscillatory
dynamics due to their nonlinear and dissipative physics Kavokin  _et al._
(2011). The condensation of polaritons Deng  _et al._ (2010), attributed to
their bosonic nature and very light effective mass, has given rise to a
powerful experimental platform to investigate nonlinear and out-of-equilibrium
physics at the macroscopic quantum level and even at room-temperature Plumhof
_et al._ (2014).

The dynamics between multiple coupled polariton condensates, denoted by a
complex number $c_{n}$, are can be described using a discretized version of
the driven-dissipative Gross-Pitaevskii equation (dGPE) Lagoudakis  _et al._
(2010); Stępnicki and Matuszewski (2013); Rayanov  _et al._ (2015); Kalinin
and Berloff (2019),

$i\frac{dc_{n}}{dt}=\Big{[}\Omega_{n}+\alpha|c_{n}|^{2}\Big{]}c_{n}+\sum_{\langle
nm\rangle}J_{nm}c_{m}.$ (1)

where $\Omega_{n}=\omega_{n}+i(p_{n}-\gamma_{n})$, $\alpha=g-iR$, and
$J_{nm}=|J_{nm}|e^{i\beta_{nm}}$ denote the complex self-energy of each
condensate, its non-linearity, and coupling to nearest neighbors respectively.
Physically, $\gamma_{n},p_{n}>0$ denote the condensate linear losses and gain
respectively. For a single condensate the evolution of its density $|c|^{2}$
coincides with that of the Landau equation, describing the dynamics of
disturbances in the laminar flow of fluids,
$\partial_{t}|c|^{2}=k_{1}|c|^{2}+k_{2}|c|^{4}$, where $k_{1,2}$ are real
constants Landau (1965); Stuart (1960). When connections are present,
$J_{nm}\neq 0$, Eq. (1) can be regarded as a discretized form of the complex
Ginzburg-Landau equation Hakim and Rappel (1992) describing a system of
coupled limit-cycle oscillators labeled as _Stuart-Landau networks_. We note
that the complex Ginzburg-Landau equation differs from the driven-dissipative
Gross-Pitaevskii equation in its historical origin and intent Aranson and
Kramer (2002).

Interestingly, recent studies on optical networks of limit-cycle oscillators
have found that there exists a regime with a strong attractor in phase space,
where the relative difference between the arguments of the oscillators,
$\theta_{nm}=\arg{(c_{n}^{*}c_{m})}$, correlates with the ground state of the
$XY$ Hamiltonian Lagoudakis and Berloff (2017); Berloff  _et al._ (2017);
Kalinin and Berloff (2018a), and the Ising-Hamiltonian Inagaki  _et al._
(2016a, b). However, in order for an optical system to work in this “minimal
spin energy” regime, the natural frequencies of the oscillators $\omega_{n}$
need to be resonant with each other, and $J_{nm}$ should be imaginary valued
to ensure that dissipative coupling between oscillators fixes a definite phase
relationship Kalinin and Berloff (2018b). From a practical viewpoint, the
relative phases in a desynchronized network of limit-cycle oscillators would
average to zero over time. Thus, the phase information cannot be extracted in
any setup relying on time-average measurements. It therefore becomes
paramount, in order to successfully extract the phase configurations
$\theta_{nm}$, that one possesses enough control over the networks parameters
for it to remain synchronized such that phase readout is possible.

In this paper, we experimentally demonstrate and analyze an optical system of
limit-cycle oscillators with tunable couplings $J_{nm}$ and fixed global
natural frequencies $\omega_{n}=\omega$ using optically confined exciton-
polariton condensates. We demonstrate clear regimes of synchronization between
two condensates and map these regimes to the weights of the $XY$ Hamiltonian.
We corroborate our findings by numerically solving both the continuous and
discretized version of the driven-dissipative Gross-Pitaevskii equation, and
benchmark the dGPE’s performance in finding the $XY$ ground state against
system uncertainties.

The optically trapped condensates are formed by exciting a semiconductor
microcavity with a non-resonant laser pump profile shaped into rings
Askitopoulos  _et al._ (2013, 2015). The ring-shaped pumps start building up
trapped polaritons which at a critical power form a condensate in the minimum
of the pump potential. Because of their non-equilibrium nature, polaritons can
diffuse away from their pumping spots, transforming their potential energy
into kinetic energy. Such a flow of coherent polaritons Schmutzler  _et al._
(2015); Su  _et al._ (2018); Töpfer  _et al._ (2020), with tunable cavity in-
plane momentum, then leads to interference and robust phase locking between
spatially separated condensates Wouters (2008); Baas  _et al._ (2008); Eastham
(2008); Christmann  _et al._ (2014); Ohadi  _et al._ (2016).

The resulting phase locking can be detected by observation of interference
fringes in the cavity real-space and/or reciprocal-space photoluminescence,
which has been achieved today over more than one hundred microns Töpfer  _et
al._ (2020). More importantly, optically trapped polariton condensates show
coherence time which exceeds the cavity lifetime by 3 orders of magnitude
Askitopoulos  _et al._ (2019), increasing the scalability of the system to
phase lock far beyond that of the optically pumped regime. Moreover, by
scaling to a condensate network interspersed with optically imprinted
variable-height potential barriers Alyatkin  _et al._ (2019), we propose a
robust platform on which to imprint nearly arbitrary weights belonging to an
$XY$ Hamiltonian into the polariton system for heuristic optical ground state
searching through nonlinear transients.

Experiment. We experimentally realise the trapped polariton condensates using
a strain compensated $2\lambda$ GaAs planar microcavity sample containing 3
pairs of InGaAs quantum wells sandwiched between another InGaAs quantum well
pair, as described in Cilibrizzi  _et al._ (2014). The sample is held in a
cold finger cryostat at $\sim$4 K and is non-resonantly pumped at a cavity-
detuning of around -5meV by right-circularly polarised light from a continuous
wave (CW) Ti:Sapphire laser, blue-detuned in energy to a minimum above the
reflectivity stopband ($\lambda=780$ nm). To avoid heating the sample, we form
a quasi-CW beam with the use of an acousto-optic modulator, at a 10 KHz
repetition rate and 5% duty cycle, to modulate the amplitude of the beam
periodically. The annular shape of the beam is achieved using a spatial light
modulator (SLM) displaying a phase-modulating hologram (see Supplementary
Material for method of hologram generation). The beam is focused on the
surface of the sample using two lenses and a high numerical aperture objective
(NA$=0.4$). The photoluminescence emission is collected through the same
objective and an $808$ nm long pass filter is used to cut out the excitation
beam. The beam is also spectrally resolved with a 1800 grooves/mm grating in a
750 mm spectrometer, centered at 857 nm.

Results and Discussion. Above threshold power, polaritons condense into a
phase coherent trap ground state at the pump center. In Fig. 1(a, b) we show
the real-space and reciprocal-space condensate photoluminescence respectively
for two phase locked condensates as evidenced by the clear formation of
interference fringes. The radial outflow of coherent polaritons from their
pumping spots corresponds to the faint outer ring seen in reciprocal-space
whereas the brighter central region corresponds to polaritons localized in the
traps. In Fig. 1(d) we plot the integrated horizontal line-profile in
reciprocal space, taken over separation distances (i.e., the real space
distance between the ring centers) from 15 $\upmu$m to 35 $\upmu$m for rings
$d_{\text{trap}}=9.4$ $\upmu$m in diameter. The interference fringes indicate
that the condensates are phase locked, where a bright or dark central fringe
shows even (in-phase) and odd (anti-phase) parity respectively.

Figure 1: (a) Real- and (b) reciprocal-space condensate photoluminescence for
two annular traps at $d=50$ $\upmu$m separation. Integrated horizontal line
profiles are shown by white lines. (c) Energies of the condensates for varying
separation. Red triangles and blue pentagrams correspond to uncoupled left and
right condensate respectively. Black circles correspond to coupled
condensates. Error bars show FWHM of the energy linewidth. Zero energy
corresponds to 0.796 meV above the bottom of the lower polariton dispersion.
(d) Integrated horizontal line-profiles from reciprocal-space
photoluminescence (i.e., at $k_{y}=0$) for varying trap separation. Dashed
white marks $k_{x}=0$.

The observed phase locking means that the coupling between condensates cannot
be negligible and should therefore result in normal mode splitting, where new
energies of the system are shifted away from the bare energies of the
uncoupled system. Surprisingly for the trapped condensates studied here, we
observe that the energy of the coupled system stays within the linewidth of
the polaritons [see Fig. 1(c)]. This observation is made more clear by
considering the interacting condensates in the linear regime as a zero-detuned
two-level system with states $|c_{1}\rangle$ and $|c_{2}\rangle$ and energy
$\omega_{1,2}=\omega$. Coupling between the states is realized with an
operator of the form $\hat{\mathcal{J}}=Je^{i\beta}\hat{\sigma}_{1}$, where
$J>0$, and the Hamiltonian becomes,

$\mathcal{H}=\omega\hat{\sigma}_{0}+\hat{\mathcal{J}}=\begin{pmatrix}\omega&Je^{i\beta}\\\
Je^{i\beta}&\omega\end{pmatrix}.$ (2)

Here, $\hat{\sigma}_{n}$ are the Pauli matrices. The resulting even and odd
parity eigenmodes of Eq. (2) (corresponding to in-phase and anti-phase
locking), written $\Psi=(1,\pm 1)^{T}/\sqrt{2}$, will have eigenfrequencies
$\omega_{A,B}=\omega\pm 2Je^{i\beta}$. It is clear if $\beta=\pm\pi/2$ then
both modes are degenerate in real frequency but are split by $\Delta=i2J$ in
imaginary frequency (i.e., their linewidths are different). During
condensation, a state of definite parity will form corresponding to the
eigenstate with a larger imaginary part in its energy. Physically, it
corresponds to increased scattering from the reservoir of uncondensed
polaritons, and consequently becomes populated during the transient process of
condensation. Therefore, even though the real energy splitting is within the
linewidth of the system [Fig. 1(c)], the impact of the dissipative splitting
is not negligible as evidenced by the clear regions of interference fringes,
indicating condensation into a definite parity [Fig. 1(d)].

As can be seen in Fig. 1(d), regions of clear interference fringes appear
periodically as a function of separation distance with intermediate transition
regions of no clear parity. This periodic behavior stems from the fact that
away from the pumped rings the polariton flow is dictated by solutions of the
time-independent cylindrical wave equation (i.e., the Helmholtz equation)
which are given by the Hankel functions Wouters  _et al._ (2008). This results
in $J_{nm}$ spiraling in the complex plane to smaller values with increasing
polariton outflow momentum and distance between traps Lagoudakis and Berloff
(2017); Berloff  _et al._ (2017). When the coupling is dominantly imaginary
($\beta=\pm\pi/2$) then fringes appear clearly due to deterministic
condensation into the highest gain mode. When the coupling is dominantly real
($\beta=0,\pi$) then both parity modes are degenerate in gain and
stochastically condense, where by phase-locking can occur with either even or
odd parity for each realisation of the system, rather than both at once. With
a camera exposure time of $\sim$1 ms, multiple realisations are measured with
each experimental shot, with half randomly forming in even parity states, and
the other in odd. This results in the blurring seen in Fig. 1(d) as both
parity states are realised, smearing out the interference fringes in the shot-
to-shot averaged measurements of the experiment.

Figure 2: Simulations obtained through the full numerical solution of the
2DGPE in real-space (a,b) and reciprocal-space (c,d) density profiles of two
interfering steady state polariton condensates. In-phase locking is observed
at separation distance of 23.2 $\upmu$m (a,c), and anti-phase locking at 26.6
$\upmu$m (b,d). All plots are shown on the same normalized color scale. Black-
contoured surface in (a,b) illustrated the trap potentials formed by the
incident laser profiles. White lines in (c,d) show the horizontal line-
profiles of the reciprocal-space density. (e) Horizontal reciprocal-space
density line-profiles shown for varying trap separation distances.

The above findings are corroborated by numerical simulations using the two-
dimensional driven-dissipative Gross-Pitaevskii equation (2DGPE) [see
Supplementary Material]. In agreement with experiment, the energy of the
simulated condensate wavefunction maintains, on average, a value around $\sim
0.782$ meV above the bottom of the lower polariton branch, and parity
switching is seen in both real-space and reciprocal-space condensate profiles
as the separation distance varies (see Fig. 2(a-d) for two different steady
state examples). In Fig. 2(e), the horizontal line-profile in reciprocal space
is plot for two ring traps of $d_{\text{trap}}=7$ $\upmu$m diameter with
separation distances going from 10 to 40 $\upmu$m. This figure is built up by
averaging the line profile over 20 simulation realisations, each starting from
a different random background noise. In agreement with experiment, we see
smeared-out interference fringes in the transition regions when the
dissipative coupling is weak (i.e., coupling becomes $\beta\approx 0,\pi$). We
note that we do not apply a time-dependent stochastic treatment of the 2DGPE.
Consequently, the simulation [Fig. 2(e)] shows a much sharper transition from
one parity to next as opposed to the extended blurred regions seen in
experiment.

Figure 3: (a) Experimental reciprocal-space photoluminescence as shown in Fig
1(d), with dashed white line to show $k=0$ $\upmu$m-1. (b) Colormap shows the
spectral intensity (real energy) from simulation of two condensates as a
function of distance $d$ using Eq. (1) and with coupling as defined by Eq.
(3). Blue circles show the steady state relative phase $\theta_{12}$ from same
simulation. Solid and dashed white lines correspond to the real and imaginary
parts of Eq. (3) respectively. Parameters: $k_{c}=1.04$ $\upmu$m-1,
$d_{\text{trap}}=9.4$ $\upmu$m, $\mu=0.28$ meV ps2 $\upmu$m-2, $\omega_{n}=0$
and $\gamma_{n}^{-1}=5.5$ ps, $p_{n}/\gamma_{n}=1.65$, $R=0.005$ ps-1, and
$g/R=0.02$.

We now describe the observations using Eq. (1) which can be derived by
adiabatically eliminating the dynamics of the exciton reservoirs feeding the
condensates Keeling and Berloff (2008) and applying a tight binding method for
the localized dissipative condensates Stępnicki and Matuszewski (2013);
Kalinin and Berloff (2019). The coupling is taken proportional to the Hankel
function (see Supplementary Figures 2,3),

$J_{nm}=|J_{nm}|e^{i\beta_{nm}}=J_{0}e^{i\phi}H_{0}^{(1)}[k(d_{nm}-d_{\text{trap}})].$
(3)

Here $J_{0}$ is the magnitude of the coupling strength, $\phi$ is a phase
adjustment parameter to match experiment, $k=k_{c}+i\mu\gamma_{n}/2\hbar
k_{c}$ where $k_{c}$ is the outflow polariton momentum Wouters  _et al._
(2008), $\mu$ is the polariton mass, $d_{nm}$ is the separation distance
between condensate $n$ and $m$, and $d_{\text{trap}}$ is the trap diameter.

The condensation of two interacting polariton condensates is then simulated
using Eq. (1) and the resulting spectral intensity (real energy) for
$p_{n}>\gamma_{n}$ is plotted in Fig. 3(b) as a function of separation
distances varying from 15 $\upmu$m to 32 $\upmu$m. The blue circles denote the
relative phase between condensates $\theta_{12}=\arg{(c_{1}^{*}c_{2})}$
showing step-function regions of in-phase and anti-phase locking. In Fig. 3(a)
we show a section from Fig. 1(d) for comparison. The spectrum shows
discontinuous jumps where the imaginary (dissipative) part of the coupling
$J_{nm}$ (dashed white curve) changes sign. This corresponds to the lowest
threshold condensate mode switching parities. The solid white curve denotes
the real part of $J_{nm}$. The results show that the system of two coupled
condensates follows robustly the highest gain mode dictated by the imaginary
part of $J_{nm}$.

By additionally modulating the phase of the coupling ($\beta_{nm}$) through
the use of optically generated potential-barriers Alyatkin  _et al._ (2019),
the couplings $J_{nm}$ between adjacent condensates can be programmed to have
nearly arbitrary values of magnitude $|J_{nm}|$, with phases chosen as
$\beta_{nm}\approx\pm\pi/2$. This then allows the design of a synchronized
random network of dissipative coupled limit-cycle oscillators for simulation
of the $XY$ Hamiltonian Lagoudakis and Berloff (2017); Berloff  _et al._
(2017); Kalinin and Berloff (2018a). Applying Eq. (1), the principle idea is
starting with $p_{n}-\gamma_{n}$ negative enough that $c_{n}=0$ is the only
stable solution of the network. Physically, this scenario corresponds to
condensates being pumped below threshold. By adiabatically increasing $p_{n}$
(slowly raising the pump power), this fixed point eventually becomes unstable
and the system undergoes a nonlinear transient process (Hopf bifurcation) to a
“condensed” steady state $|c_{n}|>0$ whose phase configuration correlates with
that of the $XY$ ground state.

In a comparable method to Kalinin and Berloff (2018b, c), we verify the
performance of Eq. (1) and test it against a global classical optimizer, the
Basin Hopping (BH) method Wales and Doye (1997), at finding the $XY$
Hamiltonian ground state of a randomly connected closed chain [see Fig. 4(d)],
$\mathcal{H}_{XY}=-\sum_{nm}\text{Im}{(J_{nm})}\cos{(\theta_{nm})}$. Since no
cavity system is ideal, the robustness of the dGPE is additionally
investigated by deviating $\beta_{nm}$ from the ideal values of $\pm\pi/2$. We
also investigate the effects of the ratio of the two nonlinearities $g/R$
where $g$ is responsible for shifting the real energy of each condensate [see
Eq. (1)]. Results on a fully connected random etwork of condensates is given
in Supplementary Figure 4. Illustrative phase configuration for a network of
10 randomly connected spins after minimising the $XY$ Hamiltonian via the dGPE
and Basin Hopping method is shown in Supplementary Figure 5.

Figure 4: (a-c) Average ratio of minimised $XY$ Hamiltonian energy between the
dGPE and BH method, $\eta=E_{\text{GPE}}/E_{\text{BH}}$, for 20 random
realizations of couplings $J_{nm}$. Performance is shown for different network
sizes $N$, values of $g/R$, and $\sigma$. (d) shows schematic of the randomly-
connected continuous-chain lattice tested with dGPE and BH. (e) Elapsed
computation time taken, $\tau$, to find a local minimum energy phase
configuration using the dGPE and BH methods.

For each set of undirected couplings $J_{nm}$, we define the energy found by
the BH method and the dGPE as $E_{\text{BH}}$ and $E_{\text{GPE}}$
respectively. In Fig. 4(a-d) we plot a histogram of
$\eta=E_{\text{GPE}}/E_{\text{BH}}$ averaged over 20 different random coupling
configurations for different numbers of condensates $N$ in the network. Figure
4(a) shows the results in an ideal case where the phases of the couplings are
randomly chosen either $\beta_{nm}=\pm\pi/2$. In Fig. 4(b,c) we plot the
performance with deviation in the couplings defined as $J_{nm}=|J_{nm}|(e^{\pm
i\pi/2}\pm\sigma)$ where the plus-minus signs are randomly chosen separately.
The deviation $\sigma>0$ then corresponds to the dissipative coupling
obtaining a small real part which can desynchronize the network of
condensates.

With $g/R=0.0001$, the performance does not drop below $\eta=0.96$ and we see
that whilst $g\leq R$, $\eta$ remains above 0.78 and does not vary
significantly as $\sigma$ changes. Increasing $g$ beyond this point
considerably reduces the accuracy of the dGPE. The computation time for each
method is also shown in Fig. 4(e) over different sized systems, where each
minimization method is implemented for a single coupling configuration on a
single core of the same Intel(R) Xeon(R) W3520 @ 2.67GHz CPU. The
computational time taken by the dGPE increases by just 2.5 seconds as the
system size is scaled by a factor of 10, while time taken by the classical BH
method scales more than three orders of magnitude.

Conclusions. We have demonstrated robust synchronization between optically-
trapped polariton condensates which is attributed to a dissipative coupling
mechanism arising from the condensates mutual interference. The coupled
condensate system does not show a measurable normal mode splitting due to the
linewidth of the polaritons, yet at the same time displays ability to
synchronize at separation distances where dissipative coupling is dominant.
The single-frequency operation of the system is critical in order to read out
the relative phase information between interacting condensates in time-average
measurements. It therefore offers a way to implement the recently proposed
gain-dissipative Stuart-Landau networks for ultrafast simulation of randomly
connected spin Hamiltonians in the optical regime Lagoudakis and Berloff
(2017); Berloff  _et al._ (2017); Kalinin and Berloff (2018a).

Acknowledgements. The authors acknowledge technical support from Mr. Julian
Töpfer and Dr. Ioannis Chatzopoulos. PGL acknowledges useful discussions with
Prof. Nikolay A. Gippius for recognizing the importance of controlling the
natural frequencies of ballistically expanding coupled condensates. The
authors acknowledge the support of the UK’s Engineering and Physical Sciences
Research Council (grant EP/M025330/1 on Hybrid Polaritonics), the use of the
IRIDIS High Performance Computing Facility, and associated support services at
the University of Southampton.

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Supplemental Material

## Numerical Spatiotemporal Simulations

The dynamics of polariton condensates can be modelled via the mean field
theory approach where the condensate order parameter $\Psi(\mathbf{r},t)$ is
described by a 2D semiclassical wave equation often referred as the
generalised Gross-Pitaevskii equation coupled with an excitonic reservoir
which feeds non-condensed particles to the condensate Wouters and Carusotto
(2007). The reservoir is divided into two parts: an active reservoir
$n_{\mathrm{A}}(\mathbf{r},t)$ belonging to excitons which experience bosonic
stimulated scattering into the condensate, and an inactive reservoir
$n_{\mathrm{I}}(\mathbf{r},t)$ which sustains the active reservoir Lagoudakis
_et al._ (2010, 2011).

$\displaystyle i\frac{\partial\Psi}{\partial t}$
$\displaystyle=\left[-\frac{\hbar\nabla^{2}}{2m}+\frac{G}{2}(n_{\mathrm{A}}+n_{\mathrm{I}})+\frac{U}{2}|\Psi|^{2}+\frac{i}{2}\left(\xi
n_{\mathrm{A}}-\gamma\right)\right]\Psi,$ (S1) $\displaystyle\frac{\partial
n_{\mathrm{A}}}{\partial t}$
$\displaystyle=-\left(\Gamma_{\mathrm{A}}+\xi|\Psi|^{2}\right)n_{\mathrm{A}}+Wn_{\mathrm{I}},$
(S2) $\displaystyle\frac{\partial n_{\mathrm{I}}}{\partial t}$
$\displaystyle=-\left(\Gamma_{\mathrm{I}}+W\right)n_{\mathrm{I}}+P(\mathbf{r}).$
(S3)

Here, $m$ is the effective mass of a polariton in the lower dispersion branch,
$U$ is the interaction strength of two polaritons in the condensate, $G$ is
the polariton-reservoir interaction strength, $\xi$ is the rate of stimulated
scattering of polaritons into the condensate from the active reservoir,
$\gamma$ is the polariton decay rate, $\Gamma_{{\mathrm{A,I}}}$ is the decay
rate of active and inactive reservoir excitons respectively, $W$ is the
conversion rate between inactive and active reservoir excitons, and
$P(\mathbf{r})$ is the non-resonant CW pump profile.

We perform numerical integration of Eqs. (S1), (S2) and (S3) in time using a
linear multistep method in time and spectral methods in space. The polariton
mass and lifetime are based on the sample properties: $m=0.28$ meV ps2
$\upmu$m-2 and $\gamma=\frac{1}{5.5}$ ps-1. We choose values of interaction
strengths typical of InGaAs based systems: $\hbar U=7$ $\upmu$eV $\upmu$m2,
$G=10U$. The non-radiative recombination rate of inactive reservoir excitons
is taken to be much smaller than the condensate decay rate
($\Gamma_{\mathrm{I}}=0.01\gamma$), whereas the active reservoir is taken
comparable to the condensate decay rate $\Gamma_{\mathrm{A}}=\gamma$ due to
fast thermalisation to the exciton background Wouters  _et al._ (2008). The
final two parameters are then found by fitting to experimental results where
we use the values $\hbar\xi=99$ $\upmu$eV $\upmu$m2, and $W=0.035$ ps-1.

## Annular Pump Profiles

The pump profiles consist of two annular traps, each written as
$P(r)=P_{0}e^{-(r-r_{0})^{2}/2\sigma^{2}}$ where $P_{0}$ denotes the pump
power, $r$ sweeps radially from the centre of each pump, $r_{0}$ marks the
trap radius and $\sigma$ corresponds to a $2\;\mathrm{\upmu m}$ experimentally
diffraction-limited full width at half maximum of each annulus.

Experimentally, the annular shape of the beam is achieved using a spatial
light modulator (SLM) displaying a phase-modulating hologram. The hologram is
created using the mixed-region amplitude-freedom (MRAF) algorithm Pasienski
and DeMarco (2008), and adjusted to balance the condensate intensities
Nogrette  _et al._ (2014). The laser photo-luminescence profile of a pump used
to trap a single polariton condensate in its ground state is shown in S1.

Figure S1: Real-space laser photo-luminescence of 12 $\upmu$m diameter pump,
including horizontal and vertical line profiles through the centre of the pump
profile, shown by solid white lines.

## Hankel function polariton outflow

In S2 we fit a zeroth-order Hankel function of the first kind (magenta
circles) to a steady state solution of Eqs. (S1)-(S3) (solid blue curve) for a
annular shaped pump geometry (black dotted line). The results show that
outside of the pump spot the steady state condensate assumes the solution of
the Helmholtz equation as expected.

Figure S2: Cross-section of the nonresonant pump profile (black dotted line),
real part of the numerically obtained condensate wavefunction from 2DGPE
simulations, and a fitted zeroth-order Hankel function of the first kind
$\Psi(\mathbf{r})=Ae^{i\phi}H_{0}^{(1)}[k(r-r_{0})]$, where $A$ and $\phi$ are
real valued fitting parameters. Radial coordinate corresponds to $r$. Here
$k=k_{c}+i\kappa$ where $k_{c}=0.96$ $\upmu$m-1 and $\kappa=m\gamma/2\hbar
k_{c}$. The parameter $r_{0}=11$ $\upmu$m is adjusted to the point where
particles have escaped the trap.

Moreover, we verify the validity of approximating the coupling with a Hankel
function [see Eq. (3) in main text] by calculating the overlap integral
between two condensates using the 2DGPE steady state solution of a single
condensate,

$J=\int\psi^{*}(\mathbf{r}-\mathbf{d})V(\mathbf{r})\psi(\mathbf{r})d\mathbf{r}.$
(S4)

Here $\psi(\mathbf{r})$ is the numerically obtained steady state condensate
wavefunction for a single pump system by solving the 2DGPE, $V(\mathbf{r})$ is
its corresponding optical trap, and $\mathbf{d}$ is the separation between two
such neighboring wavefunctions. The results of the integration as a function
of separation distance $|\mathbf{d}|$ are shown in S3 where we fit a zeroth-
order Hankel function of the first kind (red circles and blue squares) to the
values obtained from Eq. (S4). The results show that the precise details of
the pump shape are not necessary and that a qualitative analytical form to the
coupling between condensates can be obtained by considering their wavefunction
shape outside the pumped potential.

Figure S3: Real and imaginary part from Eq. (S4) in reply letter for
wavefunctions separated by a distance $d=|\mathbf{d}|$. As the magnitude of
the distance is decreased the value of the integral expectedly decreases as
the overlap diminishes. The circle and square markers are a fit to the data
using a Hankel function $J=J_{0}e^{i\phi}H_{0}^{(1)}[k(d-d_{\text{trap}})]$.
Here $k=k_{c}+i\kappa$ where $k_{c}=0.96$ $\upmu$m-1, $\kappa=m\gamma/2\hbar
k_{c}$, and $d_{\text{trap}}=10$ $\upmu$m is the diameter of the trap used in
this simulation.

## Densely Connected Polariton Graph

In addition to the closed sparsely-connected chain studied in the main text,
we also compare the robustness of the dGPE to BH for a densely and randomly
connected polariton graph of $N$ condensates [S4(a-d)]. We plot
$\eta=E_{dGPE}/E_{BH}$ for a range of realistic and unrealistic polariton-
polariton interactions strengths and include a small percentage, $\pm\sigma$
of non-dissipative coupling to each value of $J_{nm}$, where $\pm$ is chosen
randomly for each spin site. The minimisation of this all-to-all connected toy
model, though unrealistic, shows that the dGPE is able to minimise any lattice
configurations. An example of the minimised phases of the dGPE and BH is shown
in S5.

Figure S4: (a-d) measure of $\eta=E_{dGPE}/E_{BH}$ for a range of polariton-
polariton interaction strengths $g/R$, and fraction of non-dissipative
coupling strength $\sigma$ for a range of $N$ densely and randomly coupled
spin. (e-h) Average standard deviation $\Delta$ ($M=20$) between the minimized
spins using the dGPE and BH methods, again for a range of $g/R$ and $\sigma$
for different $N$.

The average standard deviation between the dGPE and the BH is written:

$\Delta=\frac{1}{M}\sum_{m=1}^{M}\text{min}\sqrt{\frac{1}{2N}(\mathbf{x}_{\text{dGPE}}-\mathbf{x}_{\text{BH}}^{(\pm)})^{\dagger}(\mathbf{x}_{\text{dGPE}}-\mathbf{x}_{\text{BH}}^{(\pm)})},$
(S5)

where $\mathbf{x}_{\text{dGPE}}=\\{e^{i\theta_{n}}\\}_{n=1}^{N}$ and
$\mathbf{x}_{\text{BH}}^{(\pm)}=\\{e^{\pm i\theta_{n}^{\prime}}\\}_{n=1}^{N}$
are the complex state vectors coming from each method with angles (phases)
$\theta_{n}$ and $\theta_{n}^{\prime}$ respectively. The global gauge is fixed
by rotating the state vectors such that $\theta_{1},\theta_{1}^{\prime}=0$ in
each method. The min operation is added since the $XY$ Hamiltonian is
invariant by an overall sign factor, i.e.,
$\mathcal{H}_{XY}=-\sum_{nm}\text{Im}{(J_{nm})}\cos{[\pm(\theta_{n}-\theta_{m})]}$.
The integer $M$ denotes the number of coupling realizations in the ensemble
average (number of different networks tested).

In S4(e-h), we plot a histogram for $M=20$ realizations of random couplings
$J_{nm}$ for $N$ condensates in the network. S4(e) shows the results in an
ideal case where the phases of the couplings are randomly chosen either
$\beta_{nm}=\pm\pi/2$. In S4(f-h) we plot the performance with deviation in
the couplings defined as $J_{nm}=|J_{nm}|(e^{\pm i\pi/2}\pm\sigma)$ where the
plus-minus signs are randomly chosen separately. The deviation $\sigma>0$ then
corresponds to the dissipative coupling obtaining a small real part which can
desynchronize the network of condensates. The results show that difference
between the BH and the dGPE states increases when both $g$ and $\sigma$
increase. This then corresponds to the system becoming desynchronized.

Figure S5: Relative phases of 10 randomly and densely connected spins
minimising the $XY$ Hamiltonian achieved by (top) the dGPE and (bottom) the
classical Basin Hopping method. In the minimisation shown, $g/R=0.0001$ and
$\sigma=0\%$.

## Supplementary References

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