# Advising Agent for Service-Providing Live-Chat Operators

Aviram Aviv3 Yaniv Oshrat1 Samuel A. Assefa2 Tobi Mustapha2 Daniel Borrajo2 On
leave from Universidad Carlos III de Madrid (consultant). Manuela Veloso2 On
leave from Carnegie Mellon University. Sarit Kraus1 1Department of Computer
Science, Bar-Ilan University
2JP Morgan AI Research, New York
3Department of Mathematics, Bar-Ilan University
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Call centers, in which human operators attend clients using textual chat, are
very common in modern e-commerce. Training enough skilled operators who are
able to provide good service is a challenge. We suggest an algorithm and a
method to train and implement an assisting agent that provides on-line advice
to operators while they attend clients. The agent is domain-independent and
can be introduced to new domains without major efforts in design, training and
organizing structured knowledge of the professional discipline. We demonstrate
the applicability of the system in an experiment that realizes its full life-
cycle on a specific domain and analyse its capabilities.

##  1 Introduction

In modern e-commerce, many business services are provided via the Internet.
Not only new web-oriented enterprises use this option, but traditional ones as
well have moved relevant services to the digital medium. For example, banks
are increasingly closing their physical branches and moving services, formerly
provided only face-to-face, to the internet Nam et al. (2016). There are many
actions that customers can perform by themselves via the Internet, without
human intervention, either by a self-service application or using a
conversational chatbot. But when customers want to perform actions that do not
have an online solution yet, or when they fail to do it by themselves, they
still need to approach the bank’s customer service and seek human help.

There are several communication channels between the customers and the call
center employees (operators). The first method is using a telephone call. This
method gives the customer the full attention of someone capable of helping,
but at the same time it forces the operator to wait for the customer’s
reactions. In many of these calls, the customer needs to perform actions he is
not familiar with, making the operator wait idly for the customer to finish.
Since the operator attends only one customer at a time, this approach wastes
time that could be better utilized. With the rise of the Internet, another
approach for call centers emerged, using a text-based chat service. This
method obviates the constraint of giving one customer the full attention of
the operator, as it parallels the service. Instead of talking with one
customer at a time, the operator interacts with 2-4 customers in parallel,
switching between customers instead of waiting for the customer’s reactions.

While this approach has its advantages, it also raises some challenges for the
human operator to deal with; as the number of tasks that the operator has to
perform simultaneously grows, so may her stress. The operator also needs to
prioritize the tasks, keep track of each individual’s information while
assisting different clients, and provide help without making any client wait
too long.

We propose to mitigate these challenges by assisting the human operator by
creating an advising agent. This kind of agent works alongside the operator
during the chat session and suggests on-line advice to help the operator deal
with the situation. To the best of our knowledge, we are the first to use an
advising agent to cope with these challenges. However, building an advising
agent and training it to the specific service domain can be a long and
expensive process that requires both domain expertise and system engineering
knowledge.

##  2 Related Work

In recent years, many companies have started to develop chatbots for the task
of customer service Nuruzzaman and Hussain (2018). There is much work
regarding the design and deployment of such bots in various domains Okuda and
Shoda (2018). It is evident that at the current phase chatbots cannot fully
replace human workers, and when a bot detects it cannot help the customer, it
refers the customer to human help. Li et al. Li et al. (2020) and Liu et al.
Liu et al. (2020) explore the challenge of detecting this kind of situation in
various domains.

###  2.1 Agents that Support Human Actions

Intelligent agents that support humans in their complex activities need to be
able to predict the user’s behavior and decisions Rosenfeld et al. (2016);
Kraus (2016); Rosenfeld and Kraus (2018). This is a difficult task because of
an extensive set of factors that influence human decision-making and behavior,
including inherent differences between individuals and groups of individuals
Bruine de Bruin et al. (2007), which make the prediction of an individual’s
decisions and behavior even more challenging.

There are numerous previous predictions of human behavior and decisions in
agent-human interactions. Azaria et al. Azaria et al. (2015) developed CARE
(automobile Climate control Advisor for Reducing Energy consumption) – an
agent that uses two models, finds a compromise between them and offers it to
the driver, who chooses whether or not to accept it.

Rosenfeld et al. Rosenfeld et al. (2017) developed automated agents that can
assist a single operator to better manage a group of robots, showing that an
agent can significantly improve the performance of a team of an operator and
ten low-cost mobile robots. This work also compared two approaches of advice:
one looks for the advice that will have the best impact on the current
situation, and the other searches for advice that will lead to better results
in the near future.

###  2.2 Method of Advice Provision

When it comes to advising in repeated human interaction environments, several
methods were used in the literature. Rosenfeld et al. Rosenfeld et al. (2017)
directly estimated the reward for every possible advice and recommended a
piece of advice that maximizes the reward the user will get if the provided
advice is chosen. Elmalech et al. Elmalech et al. (2015) suggested that the
agent will try to find a compromise between maximizing rewards and user
acceptance. The common ground of these advising agents is that they all advise
in order to maximize a certain reward function. This kind of advising
mechanism tends to recommend non-intuitive advice that the operators tend to
reject, making them ineffective Carroll et al. (1988).

###  2.3 Learning How to Provide Advice

In recent years, companies began keeping records of their workers’ actions and
their interactions with the customers due to low digital storage costs Katal
et al. (2013). This accumulation of information was mostly used for basic
performance analysis but with the improvement of machine learning capabilities
this abundance called for new usesEsposito et al. (2015).

Argall et al. Argall et al. (2009) coined the term “learning from
demonstration” (LfD) for the practice of using human actions and decisions as
the base of the learning process (though there are several other terms in the
literature). The LfD approach is used in a large variety of fields (e.g, Wong
et al. (2018); Floyd et al. (2017)). In our context, Levy and Sarne Levy and
Sarne (2016) combined LfD and advice provision as they created an agent that
used the way people act in a specific scenario in order to guess what they
would do in similar situations.

Even though there are many ways in the literature to generate conversational
data Crook and Marin (2017); Majumdar et al. (2019); Madotto et al. (2020), we
focus on using human-human conversations for the learning process of the agent
because they better reflect the real-life scenario Williams and Young (2007)
and hopefully help in generating more intuitive advice.

##  3 Modus Operandi and Life-Cycle of the Agent

Our research goal is to develop a domain-independent, data-driven algorithm
for an automated system that will assist in the operators’ training process
and daily activities, will help new and experienced operators, and will advise
the operators about possible actions and relevant information during textual
chat interactions with customers in real-time. Implementing the automated
system (i.e., the agent) in a call center will reduce daily workload and
improve the interaction with the customers from the human operator’s point of
view. It will improve the system’s service efficiency and reduce the time
needed to help each customer.

###  3.1 Agent’s Life-Cycle

The process of building an advising-agent for a new domain is performed in
three phases, as follows:

  1. 1.

The Apprentice Phase (Phase 1) – experienced human operators serve human
customers regarding the new domain of service. The operators tag the
information they find important (in real time or afterwards). The collected
data is fed to the learning process (as detailed in Section 3.2). There is no
agent assistance in this phase. Section 4.1 elaborates about the
experimentation of this phase.

  2. 2.

The Novice-Advisor Phase (Phase 2) – this phase contains both data collection
(for the improvement of the agent’s capabilities) and service to clients: the
agent works alongside a non-experienced human operator who attends clients,
and it simultaneously advises and learns. For advising the human operator, the
agent uses the tagging to predict the message that the operator should send or
an action it should perform, and offers it to the operator. The operator may
use this advice or not, as suits her.

In addition, the data collected in this phase may be fed into the agent’s
machine learning model in order to improve its tagging and advising
capabilities. This feeding may be performed daily, weekly, monthly or in any
batch form that is suitable to the managers of the service. Additional details
regarding this phase are presented in Section 4.2.

  3. 3.

The Expert-Advisor Phase (Phase 3) – The agent works alongside a non-
experienced human operator and provides advice based on former tags and a
learned model. The agent is not engaged in further learning, since its
capabilities have already reached an adequate level. This phase is the final
and steady state of the agent in the current domain.

A rollback from Phase 3 to previous phases may be performed if new
circumstances emerge in reality. The system can be returned to Phase 1 or to
Phase 2 (according to managers’ preferences), collect data (i.e. tagged chat
conversations) and feed them to the machine learning model. When reaching the
desired level of advice, the agent may be advanced again to Phase 3, and so
forth.

###  3.2 The Learning Process

In order to provide advice, the agent relies on a predictive model learned
from observations of the domain: Operators conduct chat sessions with clients
and attend to their needs. During the chat sessions, the operators tag the
vital information they used to reach the satisfactory outcomes (there is no
tagging in advance). Each session’s tag list is turned into an information
vector. Each time a new tag is added, the vector’s current version is saved to
be used later in the learning process as an information vector.

####  3.2.1 Building the Information Vector

We build the information vector as follows: At first, we take the $n$ most
common features that operators marked in the data and sort them in
alphabetical order (the label list).

Notation remark: We write $X_{t}$ as the information vector after $t$ pieces
of information (that is, $X$ at time $t$), and $X[t]$ for the value of $X$ at
index $t$. Whenever we mention tag $i$, we refer to the value of the tag list
at index $i$.

We define two vectors of size $n$. The first one is:

$V[i]:=\begin{cases}word,&\text{{\footnotesize if a known word was tagged as
label i}}\\\ "unknown",&\text{{\footnotesize if an unknown word was tagged as
label i}}\\\ "-",&\text{{\footnotesize if no input was labeled as label
i}}\end{cases}$

and the second one is:

$W[i]:=\begin{cases}1,&\text{if there is an input labeled as label i}\\\
0,&\text{otherwise}\end{cases}$

the vector $X$ is a concatenation of ($V,W$). Figure 1 demonstrates the
building of the information vector.

Figure 1:  The process of building an information vector as the chat between
an operator and a client proceeds.

####  3.2.2 Advice Types

There are three types of advice that we wish to provide: (1) Topic acquisition
– questions the operator should ask the client in order to acquire information
she needs in order to help him; (2) Resolution – data the operator should
provide to the client as a response to his query; and (3) Useful information –
data the operator may need in order to provide answers, such as relevant
websites, calculations, etc.

####  3.2.3 Advice Providing Process

During the call with the customer, whenever the operator uses a website
operation or finds out new information about the customer, useful information
is marked under a suitable label (the feature), and the customer information
vector $X$ is updated accordingly.

For each advice type $i$ of the three mentioned above, advice is provided by
taking the feature vector $X_{t}$ and inserting it into a machine learning
module $F_{i}$ that tries to find the best set of advice $A$ for the current
situation ($A_{t}$). The module uses $k$ pairs ${D_{1}\dots D_{k}}$ of past
experiences $D_{j}$=($X_{j}$,$A_{j}$), where $X_{j}$ is an information vector
at time $j$ and $A_{j}$ is the respective set of advice, in order to find a
set that maximizes the chance to be the most used set of advice in the past
similar situations: $P(A_{t}=A|X_{t},D_{1},...,D_{k})$

For the learning algorithm, we wanted to find an algorithm with the ability to
work efficiently on several domains and handle messy and conflicting data. The
first model that came to mind was Random Forest, a model that works well but
cannot fully utilize the vast amount of data usually available in such
domains. To deal with this problem, we thought of using neural networks. That
idea also worked pretty well, but an architecture that works on one domain
might fail to learn on another. With all that in mind, we decided to combine
them as an ensemble method of neural networks Hansen and Salamon (1990) where
each network takes the information known about a customer at a certain time
and outputs the recommended set of advice for the situation. Each network in
the ensemble was trained on a subset of the data and had a random number of
layers of an arbitrary length, as shown in Algorithm 1.

Result: a list of trained neural networks

nets=$\emptyset$

while  _length(nets) <ensembleSize_ do

num=GetRandomNumber()

if  _num >0.5_ then

trainSet=getRandomSubSet(trainData)

else

trainSet=getBalancedSubSet(trainData)

net=GenerateRandomNeuralNetwork()

train(net,trainSet)

$P_{net}$=accuracy(net,testData)
if  _$P_{net}$ >$P_{threshold}$_ then

nets $\xleftarrow[]{\text{add}}$ net

Algorithm 1 Training the ensemble:

The final set of advice is chosen using a majority voting variation (as shown
in Algorithm 2). We also tested this method against other variations of Random
Forest (LGBM Ke et al. (2017) and regular Random Forest) and other crowd
related algorithms (SVM and KNN). This method outperformed the others in an
80:20 cross-validation where the target label needed to be in the top 2
recommendations (the ensemble reached 87% accuracy, regular and gradient
boosted Random Forests with 84%, KNN with 83%, neural network with 77% and SVM
with 70%). We chose this metric because there can be a large variation based
on the operator’s preferences, even in a small amount of data.

Result: Final recommendations

results=$\emptyset$

X=getData()

for  _net in ensemble_ do

results$\xleftarrow[]{\text{add}}$ prediction(X)

bestOptions=twoMostCommonOptions(results)

finalRecommendations=$\emptyset$

if  _rankOf(bestOptions[0]) >firstOptionThreshold _ then

finalRecommendations $\xleftarrow[]{\text{add}}$ bestOptions[0]

if  _rankOf(bestOptions[1]) >secondaryOptionThreshold _ then

finalRecommendations $\xleftarrow[]{\text{add}}$ bestOptions[1]

return finalRecommendations

Algorithm 2 Using the ensemble:

As we can see, the agent can recommend one set, recommend a combination of two
sets, or remain silent (when $\emptyset$ is chosen or when
finalRecommendations is empty).

##  4 Experiment

Our experiment was designed to test whether implementing the suggested system
will yield better performance of the operators. For this purpose we chose a
domain, set up a working environment and recruited participants to play the
roles of operators and clients in various configurations. At the end of each
session the operators filled out questionnaires to quantify their opinions
regarding different aspects of the performance. We hereby describe the setup
and course of the experiment. The results will be presented in Section 5.

The domain on which we chose to perform our experiment is students loans in
the US. Customers who are interested in understanding their options in getting
such loans, either for themselves or for their relatives (usually a son or a
daughter), call the information center and chat with the operator. In some
cases, the customers know what the relevant data is, and they can provide it
to the operator right away. Nevertheless, in many cases the customers are not
familiar with the parameters that define their entitlement to a loan, and they
should be guided. For example, in the US men are required to register in the
Selective Service System in order to be entitled to a federal loan. Many
loaners are not aware of this requirement, and informing them of it, or of
other parameters that might affect their ability to get a loan of the sum they
need, is very beneficial. Good service by the operator should clarify these
issues in order to enable the customer to exhaust his rights. Hence, there is
much room for accurate advice to the operator in this process. Since Phase 3
in our model is the steady state working mode, we performed our field
experience on phases 1 and 2 which implement the building of the model.

###  4.1 Phase 1 – the Apprentice Phase

As mentioned above, the goal of this phase is to provide the agent with
labeled data regarding our domain by listening to sessions in which
experienced human operators chat with clients. This phase was implemented in
our experiment by recruits that played the operators and the customers. The
operators were thoroughly briefed and trained about the domain and the service
they should provide to customers. At the end of this preparation stage the
recruits were at the level of an experienced operator in the domain of student
loans. The customer received storyboards, each with character information
(profession, university, savings, financial status etc.) and objectives to
achieve (loan options, pre-specified information about the loans etc.).

The chat between the customers and the operators was performed using a textual
chat application. The operators used a computer where half of the screen is a
“WhatsApp web” interface with a special browser extension that knows when the
operator switches between two conversations (as a single operator attended 2-3
customers simultaneously), and allows the operator to mark words. The other
half of the screen shows a website which presents relevant information and
enables the operator to perform common calculations by clicking on pre-defined
buttons (see Figure 2).

Figure 2:  A screenshot of the operator interface. The left-hand side is the
”WhatsApp” chat interface that enables the operator to switch easily between
clients. On the right-hand side the operator can tag information in the
conversation and use calculators. The red rectangle (bottom-left) presents the
advice that the agent offers to the operator.

The subjects played multiple client-operator sessions. In these sessions there
was no participation of an assisting agent, and only the human operators and
the human clients took part. During the sessions, in addition to collecting
relevant information from the clients and answering clients’ questions, the
operators also tagged phrases in the chat. They were asked to tag (by marking
words on the screen) any information that they considered relevant to the loan
issue. Each tag contained a category (e.g. university name) and a value for
that category (e.g. UCLA, MIT, Columbia University). Operators were neither
told nor limited regarding what categories of tags they can mark. They saw
what categories were tagged and used before, but did not see their values.
They could add additional tags as needed.

In this phase of the experiment, 4 subjects took part as operators and one
subject played the clients. In total there were 76 sessions, and in each of
them a single operator attended 2-3 simulated clients.

###  4.2 Phase 2 – the Novice-Advisor Phase

The goal of the Novice-Advisor Phase is twofold: To assist operators in their
work, as well as to collect additional data for the improvement of the agent.
In the experiment, our main goal was to evaluate the helpfulness of the agent
we built in Phase 1.

In our experiment this phase was implemented using recruits from the AI course
for undergraduate students in XXX University (full name not mentioned due to
anonymity) as clients, and paid recruits from the general population as
operators or clients. Each operator played two sessions: one with an agent’s
assistance and the other without it. Half of the operators played the assisted
session first and the unassisted session second, and the other half vice
versa.

At the end of each session, we asked the participants who played as operators
to fill out a NASA-TLX questionnaire Hart et al. (1988), which is an
assessment tool for comparing the workload of different tasks (a summary of
the NASA-TLX questionnaire can be found in the supplementary material). These
opinions were analyzed in order to evaluate the performance of the agent and
its contribution to the performance of the operators. The findings of the
analysis are presented in Section 5.

At this point we had 23 operators who played 46 sessions: 15 of the 23
operators attended 2 clients simultaneously and the remaining 8 operators
attended 3 clients simultaneously. The tagging of the text was done manually
during the sessions by the operators.

####  4.2.1 The Tagging Problem

The tagging of the chat conversations is essential for the agent in order to
follow the line of conversation and provide proper advice. Our preliminary
design was to tag the chat by the human operator, in real-time, during the
session. Unfortunately, we found out that the operators of Phase 2 managed to
perform the tagging well while attending 2 clients, but when they needed to
attend 3 clients simultaneously the workload was too heavy, and they could not
tag the conversation properly; as the session proceeded, there was much less
tagging or none at all. As a result the ability of the agent to provide advice
weakened. This situation called for a change.

In order to perform good tagging even in loaded sessions, we introduced an
automated tagging mechanism. We took the raw data in real-time and made the
agent use it directly, a common notion in goal-oriented dialogue systems, and
chatbots in general Serban et al. (2015). We used a machine learning module
that follows the messages in real-time and outputs annotations for the
advising agent. The module that we chose is a combination of two sub-models,
as follows: We denote a network consisting of a BERT Devlin et al. (2019)
embedding layer with a linear layer on top as a BERTLL. At first, a BERTLL
predicts what information categories the message may contain. For each
category that the first model predicted, another BERTLL predicted what tag the
message contains. We chose to use this combination after it reached a maximum
$F_{1}$ score of $0.72$ and was seen to generalize well in practice. It also
outperformed a gradient boosted Random Forest (that reached an $F_{1}$ score
of $0.7$), a single BERTLL for all the labels (that reached a maximum $F_{1}$
score of $0.5$) and a large variety of neural network based models that were
far from reaching a $0.5$ $F_{1}$ score. Implementing the automated tagging
mechanism relieves the operator from the tagging task, and enables her to
concentrate on the sole task of attending the clients.

Another improvement in the experiment method (relative to the original design)
was the introducing of bots as clients in this phase: Instead of human
subjects playing the role of clients, we deployed bots that were built using a
combination of two strategies: a rule-based approach, and a learning approach.
The first approach followed the spirit of early chatbots, such as Eliza
Weizenbaum (1966). Based on the previous interaction with the operator, the
bot would randomly generate answers to operator’s questions, or questions for
asking the operator. The second approach used BERT to learn how to perform the
interaction. We found out that these two models (knowledge-based and learning-
based) complemented each other quite well in overcoming their respective
disadvantages. This change was made because the use of bots instead of human
subjects made the experiment much easier to perform, since we needed to
recruit and to brief only the operators, and the influence of the agent on the
clients was not examined in this research.

After implementing the aforementioned changes, we performed the experiment of
Phase 3, this time with each operator attending 3 clients simultaneously. In
this improved design we did not encounter an excessive load on the operators
since the tagging was done automatically by the agent. We had 14 operators
playing two sessions each (again half of the operators played the assisted
session first and the unassisted session second, and the other half vice
versa). Together with the 15 operators who attended 2 clients simultaneously,
we had 29 operators, and each of them played 2 sessions. The demographic data
regarding the subjects in the experiment is presented in the supplementary
material.

##  5 Results

###  5.1 Operators’ Opinions

The participants who played the role of operators filled out TLX
questionnaires. There are six categories in this questionnaire which are
factored into the total TLX grade (as elaborated on in the complementary
material). We compared the grades regarding sessions that were played with the
agent’s assistance to the grades regarding sessions that were played without
the agent’s assistance. We specified five categories:

  1. 1.

First Session - only the first session of every operator.

  2. 2.

Total Sessions - all of the sessions (both the first and the second) of all of
the operators.

  3. 3.

First with agent - only the sessions of operators whose first session was
played with an agent’s assistance.

  4. 4.

First without agent - only sessions of operators whose first session was
played without an agent’s assistance.

We also divided the data into sessions with two simultaneous customers and
sessions with three simultaneous customers, and looked at these groups
separately.

Figure 3 presents the results of the TLX questionnaires in the aforementioned
categories for sessions in which two clients were attended simultaneously by a
single operator. It can be seen that total workload (Total TLX) and temporal
demand decreased in all cases when having the agent by the operator’s side as
compared to not having the agent. All data presented in Figure 3 are
statistically significant ($p<0.05$), except columns marked with an asterisk
which have $p<0.07$, and columns with two asterisks ($p<0.5$).

Figure 3:  TLX questionnaires data (lower is better).

###  5.2 Time Performance

We presumed that a good performance of the agent would be manifested in
providing the service in shorter time, and with less idle time during the
session. Figure 4 presents the time performance data of the operators in three
categories:

  1. 1.

Total session time - the average length of a full session, including clients’
time, operator’s time and idle time.

  2. 2.

Maximal waiting time - the maximal time a client had to wait for an operator’s
response.

  3. 3.

Total waiting time - the average total time a client spent waiting for an
operator’s responses during a session.

Figure 4:  Time performance data (in minutes, decimal notation).

In all categories (except one) the times are shorter when the agent assisted
the operator than when it did not, and in most of the categories the reduction
is greater than 10%. It implies that the use of an agent alongside the
operator reduces the time needed for the session in general and the time spent
by the client idly waiting for the operator to respond. Nevertheless, the data
was not found statistically significant in most of the categories.

###  5.3 Learning Effectiveness of Phase 2

The Apprentice Phase (Phase 1) is, naturally, crucial to the building of the
preliminary knowledge base of the agent. Nevertheless, we wondered whether the
Phase 2 actually improves the capabilities of the agent, or if it is
superfluous and we may skip it and go straight to the final stage (Phase 3).
In order to answer this question we compared the performance of the tagging
model in two configurations: The first one was based on data collected in
Phase 1 only, while the second one was based on data collected in both Phase 1
and Phase 2. We found that the performance of the second model (precision –
$78\%$, recall – $75\%$, $F_{1}$-score – $72\%$) was better than the
performance of the first model (precision – $65\%$, recall – $60\%$,
$F_{1}$-score – $58\%$). This result indicates that although the data of Phase
1 alone suffices to provide basic assistance to human operators, expanding it
with the data of Phase 2 significantly improves the tagging capability and, as
a result, the quality of the agent’s performance.

##  6 Conclusions, Discussion and Future Work

In this paper we introduced an algorithm and a method to implement an advising
agent that assists operators who attend clients in a call center using chat
conversations. The main advantage of this method is its adaptability – the
agent is domain-independent and can be fitted to a new domain with relatively
little effort and short time. Training the agent does not require long design,
domain analysis or rule-definition. In an existing human call center, the
agent only needs tagged conversation logs with clients and experienced human
operators in order to build all of the required knowledge.

Integrating the results of the role-playing experiment, we see that operators
who were assisted by the agent enjoyed a lower cognitive load in attending
their clients, with less effort and less time-pressure. Time is used in a more
efficient way, as sessions are shorter and less time is spent on idle waiting.
This trend is evident both in the objective measure of time to perform a
mission (Section 5.2) and in the subjective views of the participants who
played as operators (Section 5.1).

There are several issues that still need to be examined. One such issue is an
optimization of the process of adjusting the agent to a new domain. We found
that the Novice-Advisor Phase (Phase 2) indeed improves the performance of the
agent, and therefore the 3-stage process that was suggested is justified.
However, the optimal conditions for switching from Phase 2 to Phase 3 still
need to be determined.

Another issue is the possibility for an operator to attend to a larger number
of clients simultaneously when having the agent’s assistance. We performed
experiments attending 2 and 3 clients because this was the customary situation
for a single operator. However, an operator might be able to attend more than
3 clients simultaneously by having an agent by her side. The feasibility of
this option should be tested as well.

It should be noted that this research study was designed to examine the
effects of the assisting agent on the assisted operators. A differently
designed experiment may explore the influence of the agent on the service from
the clients’ perspective.

## Acknowledgement

This paper was prepared for informational purposes in part by the Artificial
Intelligence Research group of JPMorgan Chase & Coȧnd its affiliates (“JP
Morgan”), and is not a product of the Research Department of JP Morgan. JP
Morgan makes no representation and warranty whatsoever and disclaims all
liability, for the completeness, accuracy or reliability of the information
contained herein. This document is not intended as investment research or
investment advice, or a recommendation, offer or solicitation for the purchase
or sale of any security, financial instrument, financial product or service,
or to be used in any way for evaluating the merits of participating in any
transaction, and shall not constitute a solicitation under any jurisdiction or
to any person, if such solicitation under such jurisdiction or to such person
would be unlawful.

##  Appendix A The NASA TLX Questionnaire

The NASA TLX questionnaire Hart et al. (1988) is an assessment tool for
comparing the workload of different tasks. While it is subjective to the
worker’s perspective, it is a commonly used benchmark in many papers Hart
(2006); Seker (2014); Miyake (2020). The final score is calculated using
grades the worker gave to the following categories:

Mental demand - How mentally demanding was the task?

Physical demand - How physically demanding was the task?

Temporal Demand - How hurried or rushed was the pace of the task?

Performance - How successful was the worker in accomplishing what he/she was
asked to do?

Effort - How hard did the worker have to work to accomplish that level of
performance?

Frustration - How insecure, discouraged, irritated, stressed, and annoyed was
the worker?

All of the grades above are ranged from 0 to 100, where 0 means the parameter
didn’t affect the workload, and 100 means that the parameter heavily
contributed to the workload. In the context of our experiment, as the goal is
to ease the load the operators feel during their work, ”lower is better” –
lower grades mean that the load felt by the operators is lower.

##  Appendix B Subjects Demographic Data

Herewith is the demographic data of the human subjects who took part in Phase
2 of our experiment. As explained in the paper, the experiment was performed
in two stages: In the first stage, operators attended 2 clients simultaneously
(the operators’ demographic data is presented in Fig. 5), and human subjects
played the role of clients (their demographic data is presented in Fig 6). In
the second stage, operators attended 3 clients simultaneously (these operators
demographic data is presented in Fig 7), and we used bots as clients
(therefore there is no relevant demographic data).

Figure 5:  Demographic data of the subjects who played the role of operators
in the first stage of the experiment – each operator attended 2 clients
simultaneously.

Figure 6:  Demographic data of the subjects who played the role of clients in
the the first stage of the experiment.
Figure 7:  Demographic data of the subjects who played the role of operators
in the second stage of the experiment – each operator attended 3 clients
simultaneously .

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# Droplet-superfluid compounds in binary bosonic mixtures

M. Nilsson Tengstrand<EMAIL_ADDRESS>S.M. Reimann
Division of Mathematical Physics and NanoLund, Lund University, SE-221 00
Lund, Sweden

###### Abstract

While quantum fluctuations in binary mixtures of bosonic atoms with short-
range interactions can lead to the formation of a self-bound droplet, for
equal intra-component interactions but an unequal number of atoms in the two
components, there is an excess part that cannot bind to the droplet. Imposing
confinement, as here through periodic boundary conditions in a one-dimensional
setting, the droplet becomes amalgamated with a residual condensate. The
rotational properties of this compound system reveal simultaneous rigid-body
and superfluid behavior in the ground state and uncover that the residual
condensate can carry angular momentum even in the absence of vorticity. In
contradiction to the intuitive idea that the superfluid fraction of the system
would be entirely made up of the excess atoms not bound by the droplet, we
find evidence that this fraction is higher than what one would expect in such
a picture. Our findings are corroborated by an analysis of the elementary
excitations in the system, and shed new light on the coexistence of
localization and superfluidity.

In the field of ultra-cold atomic quantum gases it was suggested early-on
Bulgac (2002); Bedaque  _et al._ (2003); Hammer and Son (2004) that quantum
effects beyond mean-field (BMF) may lead to self-bound droplets of fermionic
or bosonic atoms. For weakly-interacting single-component Bose gases, quantum
fluctuations alone are often too small to play any significant role.
Nevertheless, for binary or dipolar Bose gases the interactions may be
adjusted such that the different main contributions to the mean-field (MF)
energy nearly cancel out, leaving only a small residual term that can be tuned
to equilibrate with the BMF part of the total energy. A self-bound dilute
boson droplet may then form Petrov (2015); Petrov and Astrakharchik (2016),
with curious properties originating from its genuine quantum many-body nature.
Although originally proposed for binary Bose gases Petrov and Astrakharchik
(2016), the first experimental observations of droplets stabilized by the Lee-
Huang-Yang (LHY) quantum fluctuations Lee  _et al._ (1957) were made for
strongly dipolar atoms such as Dysprosium Kadau  _et al._ (2016); Ferrier-
Barbut  _et al._ (2016a, b); Schmitt  _et al._ (2016) and Erbium Chomaz  _et
al._ (2016). Here, a scenario similar to the binary case Staudinger  _et al._
(2018) arises due to the peculiarities of the dipolar interactions Wächtler
and Santos (2016a, b); Macia  _et al._ (2016); Bisset  _et al._ (2016);
Baillie  _et al._ (2017). Experiments with binary bosonic mixtures of
potassium Cabrera  _et al._ (2018); Semeghini  _et al._ (2018); Cheiney  _et
al._ (2018) or hetero-nuclear mixtures D’Errico  _et al._ (2019); Guo  _et
al._ (2021) followed soon after (for brief reviews on droplet formation, see
Refs. Böttcher  _et al._ (2021); Luo  _et al._ (2021)). In low-dimensional
systems, quantum fluctuations may be enhanced, facilitating and stabilizing
the droplet formation process Petrov and Astrakharchik (2016); Astrakharchik
and Malomed (2018); Parisi  _et al._ (2019), and the dimensional crossover has
been discussed in Refs. Zin  _et al._ (2018); Ilg  _et al._ (2018); Lavoine
and Bourdel (2021). Recent work on binary self-bound states in 1D or quasi-1D
also investigated corrections beyond LHY Ota and Astrakharchik (2020), applied
the quantum Monte-Carlo method Parisi  _et al._ (2019); Parisi and Giorgini
(2020), used the Bose Hubbard model Morera  _et al._ (2020, 2021a) or
formulated an effective quantum field theory Chiquillo (2018). It has been
shown that microscopic pairing or dimer models agree with variational
approaches Hu  _et al._ (2020); Morera  _et al._ (2021b). Collective
excitations Cappellaro  _et al._ (2018); Tylutki  _et al._ (2020) and
thermodynamic properties Ota and Astrakharchik (2020); De Rosi  _et al._
(2021) were also studied.
Droplets may also form in systems with inter-component asymmetry. One obvious
realization is heteronuclear mixtures, see e.g. Refs. Ancilotto  _et al._
(2018); D’Errico  _et al._ (2019); Minardi  _et al._ (2019); Guo  _et al._
(2021); Mistakidis  _et al._ (2021). Another interesting scenario arises when
the intra-component interactions are equal, but the components have different
numbers of particles. Then, the excess particles in the larger component
cannot bind to the droplet Petrov (2015), but instead form a uniform
background into which the droplet is immersed Mithun  _et al._ (2020). In a
similar line of thought, but for non-equal interactions within the components,
it was recently suggested Naidon and Petrov (2021) that a mixed phase may
coexist with a non-amalgamated gaseous component, where partial miscibility is
caused by BMF contributions leading to so-called “bubble” phases with
similarities to the droplet self-bound states.

In this Letter, we set focus on the case of asymmetric components confined in
a one-dimensional trap with periodic boundary conditions. For equal intra-
species interactions but different numbers of particles in the two components,
with increasing coupling strength the translational symmetry of the uniform
system is broken. A localized droplet forms, stabilized by quantum
fluctuations, which coexists with a uniform residual condensate of excess
atoms that cannot bind to the droplet, but are kept together by the
confinement. We demonstrate that this asymmetric system, although with just a
single droplet unlike what is typically seen in dipolar supersolids Böttcher
_et al._ (2021), simultaneously exhibits rigid-body and superfluid properties.
The non-classical rotational inertia (NCRI) reveals that the motion of the
droplet at low velocities is not only that of a classical rigid body but is
accompanied by the response of the non-droplet atoms moving in a direction
opposite to the motion of the rigid body. Importantly, this response is found
to exist for infinitesimal rotations, thus having a profound impact on the
degree of superfluidity of the system. The number of atoms contributing to the
formation of a vortex is larger than that of the residual condensate,
coinciding with the NCRI fraction. Our findings are corroborated by an
analysis of the lowest excitation modes in the compound system.
The energy density for a uniform binary Bose-Bose mixture in one dimension
with equal masses and short-ranged interactions, including BMF corrections,
equals Petrov and Astrakharchik (2016)

$\mathcal{E}=\frac{g}{2}(n_{1}^{2}+n_{2}^{2})+(\delta
g-g)n_{1}n_{2}-\frac{2m^{1/2}g^{3/2}}{3\pi\hbar}(n_{1}+n_{2})^{3/2},$ (1)

where $n_{i}$ are the densities of each component and $m$ the mass of a single
particle. Here we have set the intraspecies coupling constants to be equal,
$g_{11}=g_{22}=g$, and introduced $\delta g=g+g_{12}$, where $g_{12}$ is the
interspecies coupling constant. The first two terms in Eq. (1) constitute the
MF energy density and the last term accounts for the first correction beyond
mean field. The energy density Eq. (1) is valid provided
$\eta=\sqrt{mg/n\hbar^{2}}\ll 1$, ensuring weak interactions (here we have
assumed similar order of magnitudes for the densities $n_{1}\sim n_{2}\sim
n$), and that $\delta g$ is small in the sense $\delta g/g\sim\eta$. For a
finite-size system the extended coupled Gross-Pitaevskii equations
corresponding to the energy density Eq. (1) are

$\displaystyle\mathrm{i}\hbar\partial_{t}\psi_{i}=$
$\displaystyle-\frac{\hbar^{2}}{2m}\partial_{xx}\psi_{i}+g|\psi_{i}|^{2}\psi_{i}+(\delta
g-g)|\psi_{j}|^{2}\psi_{i}$ (2)
$\displaystyle-\frac{m^{1/2}g^{3/2}}{\pi\hbar}(|\psi_{i}|^{2}+|\psi_{j}|^{2})^{1/2}\psi_{i}~{},$

where $i,j=1,2$ and $i\neq j$. We impose periodic boundary conditions
$\psi_{i}(x)=\psi_{i}(x+2\pi R)$, enforcing a confinement of length $2\pi R$.
This is a good approximation for a ring of radius $R$ whenever bending effects
may be neglected, i.e. when the transversal confinement length is much smaller
than $R$. (Such binary 1D ring systems have been extensively studied, both
experimentally Ryu  _et al._ (2007); Ramanathan  _et al._ (2011); Wright  _et
al._ (2013); Ryu  _et al._ (2013); Beattie  _et al._ (2013); Eckel  _et al._
(2014); Guo and Pfau (2020); Nicolau  _et al._ (2020) and theoretically
Smyrnakis  _et al._ (2009, 2012); Anoshkin  _et al._ (2013); Smyrnakis  _et
al._ (2014); Abad  _et al._ (2014); Muñoz Mateo  _et al._ (2015); Roussou  _et
al._ (2018); Chen  _et al._ (2019); Polo  _et al._ (2019); Ögren  _et al._
(2021)).
From here on, we use dimensionless units such that $\hbar=m=R=1$. The order
parameter is normalized to the number of particles in each component according
to $\int_{0}^{2\pi}|\psi_{i}(x)|^{2}\mathrm{d}x=N_{i}$. The ground state is
obtained by solving Eq. (2) numerically with a split-step Fourier method in
imaginary time. To analyze the system in a rotating frame the term
$-\Omega\hat{L}\psi_{i}$ is added to the right side of Eq. (2), where
$\hat{L}=-\mathrm{i}\partial_{x}$, which is then solved in the same manner. In
order to find the ground state for a fixed value of the angular momentum
$L=L_{1}+L_{2}$, where
$L_{i}=\int_{0}^{2\pi}\psi^{\ast}_{i}\hat{L}\psi_{i}\mathrm{d}x$, we consider
the quantity $\tilde{E}=E+(C/2)(L-L_{0})^{2}$ Komineas  _et al._ (2005), where
$E$ is the total energy corresponding to Eq. (2). By minimizing $\tilde{E}$
for sufficiently large values of the constant $C>0$, the obtained ground state
will have angular momentum $L\approx L_{0}$ since the critical point of
$\tilde{E}$ is $L=L_{0}-\partial_{L}E/C$, which is a minimum whenever
$C>-\partial_{LL}E$. Introducing $\lambda=\delta g/g$, $N=N_{1}+N_{2}$ and
$\nu=N_{2}/N_{1}$, we illustrate our findings below by fixing $\lambda=0.01$
and $N=5000$. The asymmetry parameter is restricted to $\nu\geq 1$.
We first investigate the density distributions of the two components, starting
with the symmetric case where both components of the mixture are equal,
$\nu=1$. The upper panel of Fig. 1 shows the corresponding densities for
different values of the interaction parameter $g$ for a slowly rotating
system. For high enough values of $g$ the ground state has the form of a
droplet in agreement with previous findings in one dimension Petrov and
Astrakharchik (2016); Astrakharchik and Malomed (2018). As $g$ is decreased
the extent of the droplet increases, which results in a transition to a
uniform state. We understand this by considering the bulk density of a flat-
top droplet $n_{0}=8g/(9\pi^{2}\lambda^{2})$ Petrov and Astrakharchik (2016),
which implies a droplet extent $\sim N/(2n_{0})$. When this droplet size is
much smaller than the ring length the periodic boundary conditions do not
significantly affect the system. As $g$ is decreased, the droplet size
eventually becomes comparable to the ring length for $g\sim 9\pi
N\lambda^{2}/32$. This results in a transition from a droplet to a uniform
system and is a consequence of the periodic boundary conditions. (See also the
discussion of boundary effects in Cui and Ma (2021)). For low values of $g$,
the situation is similar also in the asymmetric case, where both components
display uniform behavior, as shown for $\nu=3$ in the lower panel of Fig. 1.
With increasing interaction strength $g$ the translation symmetry is
eventually broken, and the ground state density for the component with more
particles changes to what appears to be a mix between a droplet and a uniform
medium, while the other component displays a normal droplet solution. The
droplet coexists with a uniform background, since the excess particles in the
second component can not bind to it Petrov (2015). Deviating from $\nu=1$
results in an increase in the ratio of MF to BMF energy, eventually causing
the former to dominate the total energy. Thus, unlike the symmetric case where
droplet formation takes place where the MF and BMF terms are of similar orders
of magnitude, for the asymmetric system we have BMF effects such as displayed
in Fig. 1 even though the MF contribution can be much larger than the BMF one.
We note that solving Eq. (2) without the BMF contribution ceteris paribus
leads to uniform solutions in the regimes where non-uniform ones were obtained
with the full Eq. (2) for our choice of parameters.

Figure 1: (Color online) Condensate densities $|\Psi_{i}|^{2}$ (red, blue) and
velocities $\phi_{i}^{\prime}$ (black lines) of the two components in the
rotating frame where $\Omega=0.1$. The upper panel shows the symmetric case,
$\nu=1$. (a) $|\Psi_{1}|^{2}=|\Psi_{2}|^{2}$ for $g=0.5$ (red), being
homogeneous along the ring. (b) as in (a) but for $g=1.2$, showing droplet
formation. The lower panel shows the asymmetric case for $\nu=3$ for the same
parameters as in the upper panel. The axis to the right in both panels shows
the corresponding condensate velocities $\phi_{i}^{\prime}$.

We are here interested in the properties of this mixed-phase system. The
ground-state energy $E(\ell)$ as a function of the angular momentum per
particle $\ell$ is shown in Fig. 2 for different degrees of asymmetry $\nu$.
It takes the form of a single parabola for $\nu=1$, corresponding to the
rigid-body rotation of the droplet in the ring. For the asymmetric case with
$\nu>1$, different parabolae appear to intersect. Intriguingly, the structure
of $E(\ell)$ is similar to the one found for a dipolar toroidal system in
three dimensions in the supersolid phase Nilsson Tengstrand  _et al._ (2021).
With this analogy in mind, we model the system by considering $N_{c}$
particles taking angular momentum as a classical rigid body under rotation and
$N_{v}$ particles taking angular momentum only in terms of vorticity, with
$N=N_{c}+N_{v}$.

Figure 2: (Color online) Energy as a function of angular momentum per particle
for $\nu=1,2,3,4,5$ as indicated by the legend, where $g=1.5$.

The energy cost for adding a vortex with $s$-fold quantization to the
condensate can be determined by assuming an order parameter for the vortex
component on the form $\propto\hbox{e}^{\mathrm{i}sx}$, where the integer $s$
is the angular momentum per particle in the vortex-carrying component, leading
to an energy cost equal to $N_{v}s^{2}/2$. Since the solid-like part of the
system carries angular momentum according to $N_{c}\ell_{c}^{2}/2$, where
$\ell_{c}$ is its angular momentum per particle and
$N\ell=N_{c}\ell_{c}+N_{v}s$, the different energetic branches dependent on
angular momentum can be written

$E_{s}/N=\frac{1}{2}\left[\frac{(\ell-
f_{v}s)^{2}}{1-f_{v}}+f_{v}s^{2}\right],$ (3)

where we have defined the fraction of particles carrying vorticity
$f_{v}=N_{v}/N$. The branches $E_{s}$ and $E_{s+1}$ intersect at $\ell=s+1/2$,
i.e., having no vortex is energetically favorable for $\ell~{}<~{}0.5$, having
a singly-quantized one for $0.5<\ell<1.5$ and so on. The energies in Eq. (3)
obtained within our model predict that the functional form of the dispersion
relation should be parabolae intersecting at half-integer values of $\ell$, in
accordance with the numerical results displayed in Fig. 2. Importantly, this
suggests that the asymmetric system can exhibit properties of a solid and
superfluid simultaneously, with rigid-body rotation and quantized vorticity
coexisting. Unlike ring-shaped dipolar supersolids, which display a similar
behavior under rotation Roccuzzo and Ancilotto (2019); Nilsson Tengstrand  _et
al._ (2021), here in the case of isotropic short-range interactions the ground
state density does not exhibit any repeating crystalline structure. Since the
branches $E_{s}$ have minima at $\ell=f_{v}s$, there is a possibility for the
system to exhibit metastable superflow. These minima exist in the ground state
energy $E(\ell)$ only if they occur on the interval where the corresponding
branch $E_{s}(\ell)$ is the lowest in energy, thus giving a criterion for the
existence of a metastable persistent current related to an $s$ times multiply-
quantized vortex according to $f_{v}>(2s-1)/(2s)$.
It is tempting to identify the amount of particles belonging to the vortex
component with the number $N_{2}-N_{1}$, intuitively imagining that $N_{1}$
particles from each component bind as a droplet and thus act in a solid-body
fashion while the excess particles take the role of a background superfluid
capable of carrying vorticity. If this were true, then
$f_{v}=(N_{2}-N_{1})/N=(\nu-1)/(\nu+1)$, in contradiction with the numerically
obtained positions of the minima in Fig. 2 which were predicted to occur at
$\ell=f_{v}s$. To give an example, for $\nu=3$ the first minimum occurs at
$\ell\approx 0.63$ while $(\nu-1)/(\nu+1)=0.5$. To investigate this obvious
discrepancy, we compute the non-classical rotational inertia fraction for each
component Leggett (1998)

$f_{i}=1-\lim_{\Omega\to 0}\frac{L_{i}}{N_{i}\Omega}~{}.$ (4)

The total NCRI fraction is defined as $f=(N_{1}f_{1}+N_{2}f_{2})/N$, and we
plot the numerically obtained results in Fig. 3 for $\nu=1$ and $\nu=3$.
Interestingly, for $\nu=1$ there is a discontinuous jump from $f=1$ to a value
that is non-zero, signaling that even for the symmetric droplet system there
are parameter values such that the motion is not entirely classical. As $\nu$
is increased the discontinuity decreases until it eventually disappears, as
exemplified for $\nu=3$ in Fig. 3. We see that the total NCRI fraction for the
asymmetric system not only differs from $(\nu-1)/(\nu+1)$ but is also
dependent on $g$.

Figure 3: (Color online) Non-classical rotational inertia (NCRI) fraction as a
function of interaction strength $g$ for $\nu=1$ (orange; the discontinuity is
indicated by the bullets and connecting dashed line), and for $\nu=3$ with
$f_{1},f_{2}$ and $f$ as indicated in the legend. (Note that $f=f_{1}=f_{2}$
for the symmetric system).

Let us now study the motion of the condensate under rotation by examining the
condensate velocity $\phi_{i}^{\prime}(x)$ (where $\phi_{i}$ is the phase of
the order parameter) which can be written

$\phi_{i}^{\prime}(x)=\Omega+\frac{L_{i}-N_{i}\Omega}{2\pi n_{i}(x)}~{},$ (5)

see Appendix A. The velocities for some parameters are plotted in Fig. 1 where
in particular panels (b) and (d) show how the velocity for both components in
the symmetric system and the first component in the asymmetric one are equal
to $\Omega$ throughout most of the ring, reflecting the solid-like movement.
Away from the bulk of the droplet where the density is small, the second term
in Eq. (5) becomes significant and the velocity thus deviates from $\Omega$ in
a manner depending on $L_{i}-N_{i}\Omega$. For small $\Omega$ this difference
is negative, resulting in a change of sign for the velocity as displayed in
Fig. 1(b) and (d). If the density is negligible in the region where the
velocity deviates from $\Omega$ this will have little effect, but if the
droplet instead occupies most of the ring, as is the case close to the
transition point between the uniform and droplet phases, this results in parts
of the system moving in a direction opposite to the rest of the condensate.
The NCRI fraction will consequently differ from zero, explaining why $f$ for
the symmetric system and $f_{1}$ for the asymmetric system do not immediately
fall to zero at the transition point. For the second component in the non-
uniform asymmetric case the velocity has opposite signs inside and outside the
droplet region, indicating that the movement of the droplet is accompanied by
a response of the background medium, which moves in the other direction. Since
this response exists also for infinitesimal rotations, it affects the results
based on the definition in Eq. (4), increasing it compared to if there had
been no response flow of the background. Curiously, this implies that the
interpretation of $f$ as the fraction of particles that stay at rest as the
container is set to slowly rotate is not a correct one. (Interestingly, a
similar type of response by the background has been noticed also for dipolar
condensates in the supersolid phase Roccuzzo  _et al._ (2020); Nilsson
Tengstrand  _et al._ (2021)). To connect to the results in Fig. 2, we compute
the corresponding NCRI fractions and find that $f\approx 0.47$, $f\approx
0.63$, $f\approx 0.72$ and $f\approx 0.78$ for $\nu=2,3,4,5$, respectively.
These data agree well with the positions of the minima of $E(\ell)$,
suggesting that $f_{v}\approx f$, i.e., that the fraction of particles related
to vortex formation is larger than that of the residual condensate and
coincides with the NCRI fraction. Finally, we investigate the spectra of
collective excitations. Following the usual procedure we linearize the
extended Gross-Pitaevskii equation Eq. (2) around the ground state
$\psi_{0,i}$ and write

$\psi_{i}(x,t)=\mathrm{e}^{-\mathrm{i}\mu_{i}t}\left[\psi_{0,i}(x)+u_{i}(x)\mathrm{e}^{-\mathrm{i}\omega
t}+v_{i}^{\ast}(x)\mathrm{e}^{\mathrm{i}\omega t}\right]~{},$ (6)

keeping terms up to first order in the Bogoliubov amplitudes $u_{i}$ and
$v_{i}$. Here $\mu_{i}$ is the chemical potential and $\omega/(2\pi)$ is the
frequency of oscillation of $u_{i}$ and $v_{i}$. Due to the periodic boundary
conditions imposed on the system, we expand the amplitudes in plane waves
according to $u_{i}(x)=\mathrm{e}^{\mathrm{i}kx}u_{i,k}(x)$ and
$v_{i}(x)=\mathrm{e}^{\mathrm{i}kx}v_{i,k}(x)$ allowing us to solve the
resulting Bogoliubov-de Gennes equations for fixed $k$ (see Appendix B for
more details). Fig. 4 shows the lowest modes as a function of $k$ for the
symmetric ($\nu=1$) and asymmetric case (here for $\nu=4$) in the non-uniform
regime. In both cases there are three gapless modes, which we characterize by
looking at the density and phase fluctuations of the system, $\Delta
n_{i,k}=|u_{i,k}+v_{i,k}|^{2}$ and $\Delta\phi_{i,k}=|u_{i,k}-v_{i,k}|^{2}$,
respectively Wu and Griffin (1996); Roccuzzo and Ancilotto (2019). Two of
these modes are imaginary, indicating a dynamical instability of the system.
As $g$ is decreased and we approach the uniform regime these modes harden,
implying a higher degree of instability closer to the phase boundary. With
increased $g$ the modes instead soften until they become vanishingly small and
the system thus becomes stable. The softer of the imaginary modes is at $k=0$
a phase mode, and remains so for $\nu=1$ at all $k$. For $\nu=4$, the
characteristics of this mode changes as $k$ is increased, and the component
with more particles is instead dominated by fluctuations in the density while
the component with fewer particles still is related to fluctuations in the
phase. The harder of the imaginary modes is in all cases a density mode,
corresponding to a center of mass excitation of the droplet.

Figure 4: (Color online) Spectra of elementary excitations for $\nu=1$ (left
panel) and $\nu=4$ (right panel) as a function of $k$ for $g=1.4$. The inset
shows a zoom-in of the two lowest (green) modes (which are purely imaginary).
The (gray) dashed lines show the analytical solutions in the homogeneous
regime for $|p|\geq 3$ as discussed in the text for the corresponding
parameter values.

The single real gapless mode is a phase mode that is related to the
superfluidity of the system. For the symmetric case, it becomes soft rapidly
as the NCRI goes to zero. With increasing $\nu$ the superfluid character of
the condensate is enhanced, hardening this mode such that it may no longer be
the lowest-lying one, resulting in both crossings and avoided crossings with
the gapped modes, see the right panel of Fig. 4. The lowest-lying gapped modes
of the pure droplet system are characterized by their insensitivity to $k$,
the first one being the breathing mode Tylutki  _et al._ (2020). For the
asymmetric system we find that the lowest gapped mode is still the breathing
mode, followed by higher modes with linear slopes that form a zig-zag pattern
higher up in the spectrum. We can understand this structure from the solution
of the Bogoliubov-de Gennes equations of the uniform condensate, which may be
obtained analytically (see Appendix B). In this case the low-lying part of the
spectrum is dominated by solutions on the form $\omega_{pk}\approx
p^{2}/2+G+pk$, where $p$ is an integer and $G$ a number that depends only on
the interaction strengths and particle numbers but not on $p$ or $k$. Figure 4
clearly shows how the non-uniform system exhibits a structure that is similar
to the uniform one, but displaced to higher energies. This is a clear
indication for the increased importance of the non-droplet superfluid
background higher up in the excitation spectrum.
In conclusion, we have studied the effects of quantum fluctuations beyond just
droplet formation of a one-dimensional binary Bose mixture with short-range
interactions in a 1D confinement with periodic boundary conditions. Although
the BMF contribution for asymmetric configurations may be small compared to
the MF one, it can cause a transition to a system where translational symmetry
is broken. Examining the rotational properties and collective excitations we
found that droplet and superfluid characteristics co-exist. Through energetic
considerations and from calculating the non-classical rotational inertia we
demonstrated that the non-droplet part may not be simply identified as the
total superfluid, due to its reaction to the droplet’s movement. Investigating
the collective excitations revealed the existence of a single real gapless
phase mode that softens with decreased NCRI, tying it to the degree of
superfluidity of the system. Further superfluid signatures were observed in
the spectrum of the non-uniform asymmetric system at higher energies where the
structure became qualitatively similar to that of a homogeneous one. Although
the present study focused on 1D, the results are expected to hold also in the
quasi-1D limit for a sufficiently strong transversal trapping. As mentioned
above, droplets have been experimentally observed for binary 39K Cabrera  _et
al._ (2018); Semeghini  _et al._ (2018), or hetero-nuclear mixtures D’Errico
_et al._ (2019); Guo  _et al._ (2021), and have also been confined in a
quasi-1D optical waveguide Cheiney  _et al._ (2018). A ring-shaped superfluid
similar to what was envisioned here should thus be straight-forward to realize
experimentally. Annular flow with long lifetime was realized in anharmonic
traps Guo  _et al._ (2020), and ring traps with widely tunable parameters were
recently reported in Ref. de Goër de Herve  _et al._ (2021), although for a
single-component BEC. In the light of these recent experiments, the
realization of toroidal droplet-superfluid compounds is a realistic endeavor,
opening up for new insight regarding the coexistence of solid-like and
superfluid properties.

Acknowledgments. This research was financially supported by the Knut and Alice
Wallenberg Foundation, and the Swedish Research Council.

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##  Appendix A Condensate Velocity

The time-independent coupled extended Gross-Pitaevskii equations in the
rotating frame are

$\displaystyle\mu_{i}\psi_{i}=$
$\displaystyle-\frac{1}{2}\psi^{\prime\prime}_{i}+g|\psi_{i}|^{2}\psi_{i}+(\delta
g-g)|\psi_{j}|^{2}\psi_{i}$ (7)
$\displaystyle-\frac{g^{3/2}}{\pi}(|\psi_{i}|^{2}+|\psi_{j}|^{2})^{1/2}\psi_{i}-\Omega\hat{L}\psi_{i},$

where $\mu_{i}$ is the chemical potential and it is assumed that $i\neq j$ as
in the main text. Writing
$\psi_{i}(x)=\tilde{\psi}_{i}(x)\mathrm{e}^{\mathrm{i}\phi_{i}(x)}$ and
inserting into Eq. (7) we find

$\displaystyle\mu_{i}\tilde{\psi}_{i}=$
$\displaystyle-\frac{1}{2}\tilde{\psi}_{i}^{\prime\prime}+\frac{1}{2}\tilde{\psi}_{i}(\phi_{i}^{\prime})^{2}+g\tilde{\psi}_{i}^{3}+(\delta
g-g)\tilde{\psi}_{j}^{2}\tilde{\psi}_{i}$ (8)
$\displaystyle-\frac{g^{3/2}}{\pi}(\tilde{\psi}_{i}^{2}+\tilde{\psi}_{j}^{2})^{1/2}\psi_{i}-\Omega\tilde{\psi}_{i}\phi_{i}^{\prime}$
$\displaystyle+\mathrm{i}\left(-\tilde{\psi}_{i}^{\prime}\phi_{i}^{\prime}-\frac{1}{2}\tilde{\psi}_{i}\phi_{i}^{\prime\prime}+\Omega\tilde{\psi}_{i}^{\prime}\right).$

Using the fact that $\tilde{\psi}_{i}$ and $\phi_{i}$ are real we separate Eq.
(8) into two equations by identifying real and imaginary terms, where the
imaginary part yields the equation

$\frac{1}{2}\tilde{\psi}_{i}\phi_{i}^{\prime\prime}+\tilde{\psi}_{i}^{\prime}\phi_{i}^{\prime}=\Omega\tilde{\psi}_{i}^{\prime}.$
(9)

This equation can be solved for $\phi_{i}^{\prime}(x)$ in terms of
$\tilde{\psi}_{i}(x)$, with the solution

$\phi_{i}^{\prime}(x)=\Omega+\frac{C_{i}}{n_{i}(x)},$ (10)

where $C_{i}$ is an integration constant and
$n_{i}=~{}|\psi_{i}|^{2}=~{}\tilde{\psi}_{i}^{2}$. Since the angular momentum
can be written $L_{i}=\int n_{i}(x)\phi_{i}^{\prime}(x)\mathrm{d}x$ we finally
obtain

$\phi_{i}^{\prime}(x)=\Omega+\frac{L_{i}-N_{i}\Omega}{2\pi n_{i}(x)},$ (11)

which is the expression for the condensate velocity used in the main text.

##  Appendix B Bogoliubov-de Gennes Equations

The extended Gross-Pitaevskii equation Eq. (4) of the main text is linearized
around the ground state $\psi_{0,i}$ by writing

$\psi_{i}(x,t)=\mathrm{e}^{-\mathrm{i}\mu_{i}t}\left[\psi_{0,i}(x)+u_{i}(x)\mathrm{e}^{-\mathrm{i}\omega
t}+v_{i}^{\ast}(x)\mathrm{e}^{\mathrm{i}\omega t}\right]$ (12)

and keeping terms up to first order in the Bogoliubov amplitudes $u_{i}$ and
$v_{i}$ (here $\omega/2\pi$ is the frequency of oscillation of $u$ and $v$).
Due to the periodicity of the system we expand the amplitudes in plane waves
$u_{i}(x)=\mathrm{e}^{\mathrm{i}kx}u_{i,k}(x)$ and
$v_{i}(x)=\mathrm{e}^{\mathrm{i}kx}v_{i,k}(x)$, resulting in the Bogoliubov-de
Gennes equations

$\mathbf{A}_{k}\mathbf{v}_{k}=\omega_{k}\mathbf{v}_{k},$ (13)

where
$\mathbf{v}_{k}=\begin{pmatrix}u_{1,k}(x)&v_{1,k}(x)&u_{2,k}(x)&v_{2,k}(x)\end{pmatrix}^{\mathrm{T}}$
and

$\mathbf{A}_{k}=\begin{pmatrix}X_{12,k}&Y_{1}&Z&Z\\\ -Y_{1}&-X_{12,k}&-Z&-Z\\\
Z&Z&X_{21,k}&Y_{2}\\\ -Z&-Z&-Y_{2}&-X_{21,k}\end{pmatrix}$ (14)

with

$\displaystyle X_{ij,k}$
$\displaystyle=-\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}-\mathrm{i}k\frac{\partial}{\partial x}+\frac{k^{2}}{2}-\mu_{i}$ (15)
$\displaystyle+(2g-3\alpha)\psi_{0,i}^{2}+(\delta g-g-2\alpha)\psi_{0,j}^{2}$
$\displaystyle Y_{i}$ $\displaystyle=(g-\alpha)\psi_{0,i}^{2}$ $\displaystyle
Z$ $\displaystyle=(\delta g-g-\alpha)\psi_{0,1}\psi_{0,2}$

and

$\alpha=\frac{g^{3/2}}{2\pi(\psi_{0,1}^{2}+\psi_{0,2}^{2})^{1/2}}.$ (16)

Here $\alpha$ is the contribution due to beyond mean-field effects and we have
assumed $\psi_{0,i}$ to be real. In the homogeneous regime the BdG equations
can be solved analytically by writing Gao and Cai (2019)

$\mathbf{v}_{k}(x)=\sum_{p=-\infty}^{\infty}\frac{e^{\mathrm{i}px}}{\sqrt{2\pi}}\tilde{\mathbf{v}}_{pk},$
(17)

where
$\mathbf{\tilde{v}}_{pk}=\begin{pmatrix}\tilde{u}_{1,pk}&\tilde{v}_{1,pk}&\tilde{u}_{2,pk}&\tilde{v}_{2,pk}\end{pmatrix}^{\mathrm{T}}$.
By substituting into Eq. (13) it can be seen that
$\mathrm{e}^{\mathrm{i}px}\tilde{\mathbf{v}}_{pk}/\sqrt{2\pi}$ is an
eigenvector with eigenvalue $\omega_{pk}$ satisfying

$\tilde{\mathbf{A}}_{pk}\tilde{\mathbf{v}}_{pk}=\omega_{pk}\tilde{\mathbf{v}}_{pk},$
(18)

where

$\tilde{\mathbf{A}}_{pk}=\begin{pmatrix}\tilde{X}_{1,pk}&Y_{1}&Z&Z\\\
-Y_{1}&-\tilde{X}_{1,pk}&-Z&-Z\\\ Z&Z&\tilde{X}_{2,pk}&Y_{2}\\\
-Z&-Z&-Y_{2}&-\tilde{X}_{2,pk}\end{pmatrix},$ (19)

and $\tilde{X}_{i,pk}=(p+k)^{2}/2+(g-\alpha)n_{i}$ with $n_{i}=N_{i}/(2\pi)$.
The solutions to Eq. (18) are

$\displaystyle\omega_{\pm,pk}^{2}=\frac{(p+k)^{2}}{4\pi}\bigg{[}\pi(p+k)^{2}+(g-\alpha)N$
(20) $\displaystyle\pm\sqrt{(g-\alpha)^{2}(N_{1}-N_{2})^{2}+4(\delta
g-g-\alpha)^{2}N_{1}N_{2}}\bigg{]},$

which in the limit $g\gg\delta g,\alpha$ can be written

$\omega_{+,pk}^{2}\approx\frac{(p+k)^{2}}{4\pi}\bigg{[}\pi(p+k)^{2}+2(g-\alpha)N-4(\delta
g-2\alpha)\frac{N_{1}N_{2}}{N}\bigg{]}$ (21)

and

$\omega_{-,pk}^{2}\approx\frac{(p+k)^{2}}{4\pi}\bigg{[}\pi(p+k)^{2}+4(\delta
g-2\alpha)\frac{N_{1}N_{2}}{N}\bigg{]}.$ (22)

From these two types of solutions we observe that there is a separation of
scales, where the low-lying part of the spectrum is dominated by the
$\omega_{-,pk}$ branches. For sufficiently large $|p|$, $k\in[0.0,0.5]$ and
$\omega_{-,pk}>0$, we have the approximate form

$\omega_{-,pk}\approx\frac{p^{2}}{2}+(\delta g-2\alpha)\frac{N_{1}N_{2}}{\pi
N}+pk,$

which on the interval $k\in[0.0,0.5]$ takes the form of a zig-zag pattern as
discussed in the main text and displayed in Fig. 5 of the paper.