# What’s Kooking? Characterizing India’s Emerging Social Network, Koo

Asmit Kumar Singh1§, Chirag Jain1§, Jivitesh Jain2§, Rishi Raj Jain1§,
Shradha Sehgal2§, Tanisha Pandey1§, Ponnurangam Kumaraguru2# # Major part of
this work was done while Ponnurangam Kumaraguru was a faculty at IIIT-Delhi.
1Indraprastha Institute of Information Technology, Delhi, India
2International Institute of Information Technology, Hyderabad, India
{asmit18025, chirag17041, rishi18304<EMAIL_ADDRESS>{jivitesh.jain<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Social media has grown exponentially in a short period, coming to the
forefront of communications and online interactions. Despite their rapid
growth, social media platforms have been unable to scale to different
languages globally and remain inaccessible to many. In this paper, we
characterize Koo, a multilingual micro-blogging site that rose in popularity
in 2021, as an Indian alternative to Twitter. We collected a dataset of 4.07
million users, 163.12 million follower-following relationships, and their
content and activity across 12 languages. We study the user demographic along
the lines of language, location, gender, and profession. The prominent
presence of Indian languages in the discourse on Koo indicates the platform’s
success in promoting regional languages. We observe Koo’s follower-following
network to be much denser than Twitter’s, comprising of closely-knit
linguistic communities. An N-gram analysis of posts on Koo shows a
#KooVsTwitter rhetoric, revealing the debate comparing the two platforms. Our
characterization highlights the dynamics of the multilingual social network
and its diverse Indian user base.

###### Index Terms:

Social Computing, Data Mining, Online Social Media, Computational Social
Science, Koo App, Twitter

§§footnotetext: Equal contribution. Arranged in the alphabetical order of the
first name.

##  I Introduction

With 4.66 billion users worldwide,111https://bit.ly/3d2VzRB Accessed 11 March
2021. the Internet has become a mainstream medium for social interaction and
information dissemination. Social media is an integral part of the Internet,
as it enables users and organizations worldwide to connect, socialize, and
express themselves with ease to a large audience. With 560 million Internet
users, India ranks second in the world. However, English is the first language
for only 0.02% of the Indian population,222https://bit.ly/3vtwWno Accessed 11
March 2021. thereby creating a language barrier for the majority of the
population [1]. The diverse set of languages spoken in India raises the need
for multilingual social media platforms in Indian languages.

Recognising this need, Koo is a recent attempt at building a multilingual
Indian social network. Formerly known as Ku Koo Ku, it was launched in March
2020 as an Indian alternative to the popular social networking service,
Twitter. Originally available in Kannada, Koo allows for and encourages
discourse in Indian languages and currently supports 9 other Indian languages
apart from English.333At the time of data collection, support for three of
these was still under development. It plans to include more Indian languages
soon.

Koo gained popularity during August 2020, when it won the Government of
India’s Aatmanirbhar Bharat App Innovation Challenge
Award.444https://bit.ly/2RWGI3w Accessed 12 March 2021. During the Indian
Farmers’ protest in January 2021, Twitter entered a week-long standoff with
the Indian Government, over their refusal to block accounts that the Indian
Government claimed were spreading misinformation.555https://bit.ly/3zwWaUS
Accessed 21 March 2021. Consequently, several Indian Government
ministers,666https://bit.ly/3gyWWZ7 Accessed 21 March 2021. officials and
agencies777https://bit.ly/3xuECY1 Accessed 21 March 2021. created accounts on
Koo. There has been a general promotion of Koo amongst Government
organizations since this event, sparking a rise in the platform’s popularity.
It again saw an increase in the influx of users when Koo became the first
platform to agree to abide by the Government of India’s Information Technology
(Intermediary Guidelines and Digital Media Ethics Code) 2021 [2], while
Twitter and Facebook resisted them. Koo has even announced its entry into
Nigeria, with the Nigerian Government creating an account on the platform,
following the ban of Twitter in the country.888https://bit.ly/2RXEaCi Accessed
14 June 2021.

Koo uniquely positions itself as an alternative to mainstream online social
networks like Twitter. A cardinal reason for its popularity in the Indian
context is its multilingual support that allows for more inclusivity. The
language barrier is broken, and more people can now join social media and
express and share their opinions in the language that they choose.
Additionally, Koo has received support from the Indian and Nigerian government
agencies, contributing to its acclaim. With the ever-increasing political
discourse on social media, Koo becomes an important platform from a political
standpoint.

The rich diversity of Indian languages on the platform coupled with its sudden
rise and steady expansion in popularity and political backing motivate the
need to understand what goes on inside such a unique platform. To understand
this, we provide, to the best of our knowledge, the first characterization of
the Koo social network. We do this by investigating the following research
questions:

  1. 1.

RQ1: What are the characteristics and demographic of Koo users? When did they
join the platform?

  2. 2.

RQ2: What kind of content is posted on the platform? Which languages are most
popular?

  3. 3.

RQ3: What are the network properties of Koo and how do they differ from
Twitter’s? What communities are formed on the platform?

We were able to collect data for 4.07 million users out of the total 4.7
million999As of 16th March 2021, https://bit.ly/3d2VcGH users and 163 million
follower-following edges in our data (Section III). In Section IV, we analyse
the user demographic - we found that even though the users on the platform are
predominantly male, female reported users are more active, and have more
average followers and average likes. We also found that the influx of users
surged on the platform in August 2020 when it won the Atmanirbhar Bharat App
Innovation Challenge, and in the starting months of 2021 when the government
promoted the App. On analysing the content in Section V, we saw that the most
popular language on Koo is Hindi, followed by English, Kannada and Telugu. We
also observed a Koo vs Twitter rhetoric on the platform and support for the
Indian political party BJP. We saw that major political figures were one of
the most mentioned users on Koo. On comparing the network properties of Koo
and Twitter (Section VI), we found Koo to have a dense, well connected network
with a higher clustering coefficient. Furthermore, we observed distinct
communities of users on Koo, which are based on languages, with English
speaking users more centrally placed and having connections to Hindi and
Kannada users.

Through our work, we make the following contributions:

  1. 1.

Perform an extensive characterization of the new Indian social network Koo, in
terms of its user demographic and content.

  2. 2.

Present the first dataset of users, their connections, and content on Koo.

  3. 3.

Study the network and communities formed on this multilingual platform.

##  II Related Work

In this section, we review previous work on social media analysis, in
particular the multilingual nature of platforms.

There has been a vast amount of research on popular social media, Twitter.
This includes its characterization in its initial years [3] and the
documentation of its steady user growth [4]. With the growth of the platforms,
the amount of research increased as well, with new topics of interest
emerging, such as fake news and misinformation [5] [6], automated bots [7],
hate speech [8] [9], etc. Some works have explored Indian languages on
Twitter, with many using it as a corpus of indic-NLP research [10] [11] or
studying the topics of discourse in the Indian subcontinent [12] [13]. Our
work differs substantially from these as we focus on a platform dedicated to
promoting Indian languages, with a predominantly Indian user base.

There have been studies exploring the emergence of new social platforms with
unique properties such as Whisper for its anonymity [14], TikTok for its short
videos [15], Twitch for mixed media [16], and so on. Some Twitter alternative
alt-right platforms like Gab [17] [18] and Parler [19] have also been analysed
in the past. Other social media platforms in local languages have also been
studied, like VKontakte in Russia [20] and Weibo in China [21].

Multiple studies look at the multilingual aspect of other social platforms as
well like blogs [22] and reviews [23]; but most of these cases find the
English language to be dominant over the other languages. Our study focuses on
a platform that has a native language to be dominant. Agarwal et al. [24] do
study the qualities of an Indian multilingual social network, Sharechat.
However, it focuses mainly on image-based content posted by the user. Koo is
majorly a text-based platform, thereby making an inquiry of language aspects
even more pertinent. Apart from the content posted, our work also lays
emphasis on the user characteristics and network of Koo and draw a comparison
of the platform’s properties to that of Twitter. The choice of a novel
platform and the context behind its rapid rise in popularity differentiates
our work from the remaining.

##  III Methodology

We describe next our data collection methodology. We start by presenting an
overview of the platform.

###  III-A Overview of Koo platform

Similar to Twitter, Koo allows logged-in users to share microblog posts known
as “koos”. While signing up, users can choose their display name, handle,
language and other personal details such as gender, marital status, and birth
date. Once logged-in, users can post koos, which can at most be 400 characters
long; whereas, Twitter allows tweets to be 280 characters at maximum. Users
can also comment on others’ koos or re-share (also known as “rekoo”) them with
or without adding a comment. Koo provides transliteration support while typing
in native Indian scripts. The platform’s user interface, the users’ feed, and
the list of recommended accounts to follow changes according to their chosen
language, enabling the formation of linguistic communities on the platform.

###  III-B Data Collection

We collected data pertaining to the users’ profiles, the follower-following
network, and content (koos, rekoos, comments, likes, and mentions) posted on
the platform, from Koo’s public API that serves the Koo web and mobile
applications. Data collection started on 26 February 2021 and lasted until 11
March 2021. Table I summarises statistics of our proposed dataset.

Koo’s stark resemblance to Twitter also invites a comparison between user
behaviour on the two platforms. Hence, we created a dataset of Koo and Twitter
user IDs that correspond to the same entity. This dataset can be used for
automated identity resolution tasks and cross-platform analysis of Koo and
Twitter.

We make our dataset public in adherence to FAIR principles as described in
Section III-C.

Entity | Count
---|---
User profiles | 4,061,735
Follower-following relationships | 163,117,465
Koos | 7,339,684
Rekoos | 2,828,158
Rekoo with Comments | 413,955
Comments | 4,793,492
TABLE I: Statistics of the dataset we collected from Koo.

####  III-B1 Koo Data Collection

We collected and analyzed data about the general discourse on the platform and
the indulgent users, without restricting ourselves to certain trends,
hashtags, or topics. Analyzing close-to-complete networks enables a
significant understanding of the entire platform [25]. We used the follower-
following network to discover and collect users recursively using the snowball
methodology, essentially traversing the network graph in a breadth-first
manner, similar to Kwak, et al. [3] and Zannettou, et al. [17]. We seeded the
search with Koo’s official language accounts (Table II) which are the first
accounts shown to a new user, and the list of popular accounts that Koo
recommends users to follow.101010https://www.kooapp.com/people These were
chosen because of their large number of followers. We then collected the
(previously undiscovered) followers and followees of these accounts and
repeated the process for their follower-followee network. This approach is
bound to leave out singletons and isolated communities; however, we
hypothesize that they would not constitute a large share of the active users
on the platform. We preferred this over enumerating and querying the API for
all possible user IDs because of the significantly larger number of API
requests the latter entails, which may have overloaded Koo’s servers. We then
used the user IDs to collect the users’ profiles and the content (koos,
rekoos, comments, likes, and mentions) generated by them, by querying the
appropriate endpoints.

---
TABLE II: Koo’s official language accounts. Koo’s Hindi account has the
highest number of followers. Koo handles are written in the specific
languages.

####  III-B2 Twitter Data Collection

To create the Koo-Twitter user ID dataset, we considered two sets of users:
those with verified Koo accounts and those who had listed their Twitter
handles on their Koo profile. For the verified users, we manually curated
corresponding Twitter handles. We queried Twitter’s Users Search
API111111https://api.twitter.com/1.1/users/search.json for the users’ names
and found probable matches, manually annotating each one. 1,030 verified Koo
handles were distributed equally among 10 annotators, who were asked to verify
the corresponding Twitter handles of the users. According to a set of
guidelines prepared by the authors, the annotators looked for similarity in
profile pictures, profile information, and the content the users posted on the
two platforms. All the annotations were verified by two annotators and
ambiguous cases were dropped from the dataset. Out of the 1,030 verified Koo
users in our dataset, we found matching Twitter profiles for 872 of them, 499
of which are verified on Twitter as well.

For the second case, we used Twitter handles provided by the users themselves.
We removed duplicate handles and queried Twitter’s Users Show
API121212https://api.twitter.com/1.1/users/show.json to eliminate invalid
usernames. In all, we make public 38,711 Koo and Twitter user IDs that
correspond to the same entity.

###  III-C Adherence to FAIR Dataset Principles

The gathered data consists of publicly available information about a social
network, gathering and examining which would provide significant insights into
the platform’s characteristics. Our dataset also conforms to the FAIR
principles. In particular, the dataset is “findable”, as it is shared
publicly.131313https://precog.iiit.ac.in/resources.html This dataset is
also“accessible”, given the format used (CSV) is popular for data transfer and
storage. This file format also makes the data “interoperable”, given that most
programming languages and softwares have libraries to process CSV files.
Finally, the dataset is “reusable”, as the included README file explains the
data files in detail. The data was collected through public API endpoints of
Koo, adhering to their privacy policy.141414https://www.kooapp.com/privacy.
Accessed 12 March 2021. The data we collected was stored in a central server
with restricted access and firewall protection. All experiments shown in this
paper were performed on this dataset.

##  IV RQ1 : User Characteristics and Demographics

We analyze the demographics of users on Koo using their profile information.
It should be noted that such user-entered information might not always be
accurate and should be dealt with cautiously.

Figure 1: Weekly user creation timeline with language distribution, for 10
most popular languages. We observe peaks in users joining around August 2020,
when Koo won the Aatmanirbhar Bharat App Innovation Challenge, and 10 February
2021, when MeitY tweeted about Koo. Kannada language was popular during the
initial stages of Koo, following which Hindi language took over from August
2020. Many English users started joining from February 2021. The inset graph
represents the lower frequency language users (languages except for Hindi and
English).

###  IV-A User Onboarding

The platform is reported to host 4.7 million users at the time of writing. Our
dataset has approximately 4 million users, of which, 1.9 million joined Koo in
the first two months of 2021 alone. Figure 1 shows the user creation timeline
across various languages. We find that Koo was predominantly used by Kannada
users in its initial stages, presumably because it was launched in Bangalore
where Kannada is the vernacular dialect. Surges in influx of users can be seen
in August 2020 - around the time of the Aatmanirbhar Bharat App Innovation
Challenge Award. The user base expanded to other Indian languages, with Hindi
users joining the platform in large numbers. February 2021 saw a huge spike in
users with almost 200,000 of the users signing up on the days around 10
February 2021, just after the Ministry of Electronics and Information
Technology, Government of India tweeted about the Koo app. As the debate of
#KooVsTwitter trended on Twitter, many English users joined the Koo platform.
The inset graph of Figure 1 shows the distribution of languages other than
Hindi and English - highlighting the prominence of Kannada, Telugu, and Tamil
amongst Indian languages.

###  IV-B Gender Distribution

Of the 18.1% of the users who specified their gender on their profile, 92.1%
identify as male (699,083), with only 7.5% users identifying as female
(58,996) and 0.36% as others (3,236). However, Female users are more active,
in terms of the number of average likes (103.6) and average rekoos (21.7) they
do, as compared to other genders. Figure 2a also shows that Female have more
followers (632.9) on average as compared to male users with an average of
117.0 followers and users identifying with the other category with an average
of 283.45 followers. Male users, on average, produce fewer koos (5.9) and
follow fewer people on average (84.8) than the other two categories, as
visible in Figures 2a and 2b. Figure 2c shows that the median age of is
similar for all gender categories at 28 years. More male users identify
themselves as single (12.0%), more female users as married (7.6%) and more
users of the other category as divorced (3.0%), as shown in Figure 2d. Table
III shows the follower-following patterns between the genders. We observe that
for all genders male users contribute to major proportion of the followers and
following.

Gender | Followers |  | Following
---|---|---|---
Male |  Female |  Others |  |  Male |  Female |  Others
Male |  87.90% |  11.14% |  0.95% |  |  75.61% |  23.56% |  0.81%
Female |  92.12% |  7.32% |  0.54% |  |  83.19% |  16.20% |  0.59%
Others |  91.59% |  7.84% |  0.56% |  |  84.85% |  14.61% |  0.52%
TABLE III: Gender wise follower-following distribution. Male users contribute
to the majority of followers and following across all the genders.

###  IV-C Language and Location

Koo allows users to choose their location from a list of Indian cities. As
shown in Figure 3, in the 75,091 user profiles with location information,
Bengaluru appears most frequently with 141,469 users, presumably because of
the platform being headquartered in Bengaluru and being initially available in
Kannada. Bengaluru also appears as an example help text over the location
field on both the Android and iOS mobile applications.

Despite Kannada being the first language on the platform, it is not the most
popular, as it is outnumbered by Hindi, which was the language for 44.2% and
51.2% of the user base and total posts, respectively. Hindi was followed by
English, which constituted to 23.8 % of the users and 25.9% of content (see
Table IV). The popularity of Hindi over English demonstrates a degree of
success of the platform in promoting discourse in Indian languages. The
distribution of users across languages closely mirrors the number of followers
of Koo’s official language accounts, apart from English (see Table II). This
may be because the language account corresponding to the user’s language is
the first account whose post appears in a user’s feed.

|  |  |
---|---|---|---

(a) Gender vs Content

(b) Gender vs Follower

(c) Gender vs Age

(d) Gender vs Marital status

Figure 2: Distribution of the user meta-data across genders. 18.1% of total
users specified their gender, $N=761,315$, of which 92.1% are male, 7.55% are
female and 0.362% belong to the others category. (a) The box plot corresponds
to the 25th, 50th and 75th quartile. The iOS App Store policy mentions a
minimum age of 12. Hence, we discard all ages below 12 (1.2%). Median age of
28 is observed across all users. (b) Female users have much higher average
followers as compared to the other two categories (c) Female users are more
active in terms of likes and rekoos as compared to the other two categories.
(d) 12% of male users are single which is much higher than the other two
categories, while higher proportion of other category users are divorced.
Figure 3: Top 10 current locations mentioned by users in the metadata,
N=75,091. The y-axis is in logarithmic scale. Bengaluru is the most frequent
location. Koo is headquartered in Bengaluru and was originally available in
Kannada, which is Bengaluru’s local language.

###  IV-D Bio and Professional details

Koo allows the users to add and edit their profile bio. Figure 4 shows the
wordcloud of the user profile bios on Koo. We see the occurrence of words
pertaining to state and national identities like “Marathi”, “Tamil”,
“Bengali”, “Indian”, “Bharat” etc. Users are also allowed to add and edit
professional details. Figure 5 shows the top occurring work titles and
educational qualifications of the users, with “Student” being the most common
work title (61,778 users) and “MBA” being the most common educational
qualification (4,988 users).

Language | Number of Users | Percentage of Users | Number of Posts | Percentage of Posts
---|---|---|---|---
Hindi | 1,795,411 | 44.2030 | 3,755,829 | 51.1715
English | 968,271 | 23.8388 | 1,907,993 | 25.9955
Kannada | 711,049 | 17.5060 | 818,679 | 11.1541
Telugu | 259,171 | 6.3807 | 359,874 | 4.9031
Marathi | 183,073 | 4.5072 | 242,803 | 3.3080
Gujarati | 64,829 | 1.5960 | 126,853 | 1.7283
Tamil | 48,285 | 1.1887 | 77,981 | 1.0624
Bangla | 31,211 | 0.7684 | 49,172 | 0.6699
Malayalam | 318 | 0.0078 | 257 | 0.0035
Assamese | 46 | 0.0011 | 113 | 0.0015
Punjabi | 43 | 0.0010 | 43 | 0.0005
Oriya | 27 | 0.0006 | 87 | 0.0011
TABLE IV: Language distribution on the platform. Hindi is by far the most
popular language, for both user profiles as well as posting contributing to
44.20% of the user profile languages and 51.17% of the posts.

Figure 4: Wordcloud for user bios. Presence of words pertaining to state and
national identities such as “Tamil”, “Marathi”, “Gujarati” and “Indian” can be
seen.

|
---|---

(a) Work Title.

(b) Educational Qualifications.

Figure 5: Top 10 occurring work titles and educational qualifications. (a) A
total of N = 584,352 users mentioned their title, with “student” being the
highest mentioned title with 61,778 occurrences (b) A total of N = 118,474
users mentioned educational qualifications, with “MBA” being the most common
qualification with 4,988 mentions.  Figure 6: Weekly post creation timelines
with language distribution, for top 10 used languages. A peak around 10
February 2021 is observed, when MeitY promoted Koo on Twitter. Posts were
majorly in Kannada language during the initial days of Koo, following which
Hindi posts took over from August 2020. Posts in English language spiked
around February 2021. The inset graph represents the lower frequency language
posts (languages, except for English and Hindi).

##  V RQ2 : Content Analysis

###  V-A User-Generated Content

We have a total of 7,339,684 koos in our dataset. Figure 6 shows the post
generation timeline of Koo with the language distribution. This displays a
very similar behaviour to the user creation timeline. A major spike in posting
is observed around 10 February 2021, especially in English koos, when the
Government of India’s Ministry of Electronics and Information Technology
posted a tweet promoting Koo, and many prominent government figures started
joining the platform. The Kannada and Hindi language posts follow the same
pattern as the user joining distribution. Similar to other social networks
[26], the posting activity on Koo gradually increases throughout the day,
attaining its peak in the evening at around 2100 hours IST.

###  V-B Media Distribution

Koo allows for multiple types of media such as images, videos, links, gifs and
text to be shared. Figure 7 shows the distribution of these media types across
various languages on the platform. Being a microblogging platform, more than
50% of the content on the platform appears as text, followed by images, gifs,
links, video and audio. Most languages follow the same general trend, with a
stark exception of Kannada, showing 40% of the content being shared through
gifs.

Figure 7: Media share in posts per language.

###  V-C Hashtag Analysis

Hashtags in social media have become a way for users to build communities
around topics and promote opinions. We extract the top occurring hashtags from
the user posts and plot them as a wordcloud in Figure 8. We see hashtags like
“kooforindia”, “bantwitter”, and “koovstwitter”, which project a sentiment of
competition between Twitter and Koo, and promote the Koo platform. Hashtags
like “indiawithmodi”, “atmanirbharbharat”, “modi”, “modistrikesback”, and
“bjp”, which are associated with the Bharatiya Janata Party (BJP) are also
present. Figure 9 shows a network of the 100 most frequently occurring
hashtags, with the edges indicating two hashtags that occur in the same post.
Although the graph has a low modularity score (0.329), hashtags related to one
topic fall in the same category when the graph is clustered based on
modularity [27].

Figure 8: Wordcloud for top 25 hashtags in posts. Hashtags like #koo,
#Koovstwitter, and #bantwitter show the competitive sentiment between the two
platforms.

###  V-D N-gram Analysis

In order to get an insight into the popular conversations going on the
platform, we plot the top occurring uni-grams and bi-grams in the content of
the posts (Figure 11). We observe an overwhelming number of Hindi n-grams.
Through both uni-grams and bi-grams, we observe the existence of Hindu-centric
words like jy99wFrAm (“jaishreeram”) and rAm-rAm (“ram-ram”) that allude to
“Lord Ram”, a major deity in Hinduism. There is also a mention of many Indian
religious leaders like rAmpAl-jF (“rampal-ji”), st-99wF (“sant-shri”), and jF-
mhArAj (“ji-maharaj”)151515Words in brackets are the transliterated forms of
the original Hindi text.. This indicates that some of the discussions on Koo
are around religion and that people may be using slogans like jy99wFrAm
(“jaishreeram”) as a symbol of their religious faith.

Figure 9: A network of the 100 most frequently occurring hashtags, where an
edge represents co-occurrence of hashtags in a post. Colors indicate
clustering based on modularity. The entire graph has a modularity score of
0.329.

###  V-E Mentions and Likes

The number of mentions and likes of a particular user are useful indicators to
study a user’s engagement and popularity on a platform [25]. Table VI shows
the top 10 most mentioned users on the platform, of which “republic”, which is
an Indian news channel comes at the top with 16,041 mentions. Notably,
Republic also has an editorial partnership with
Koo.161616https://bit.ly/35nZFPJ Accessed 12 March 2021. Ravi Shankar
Prasad171717Minister of Law and Justice, Electronics and Information
Technology and Communications, Government of India. is the next most mentioned
user with 12,991 mentions. Many prominent ministers and political figures like
Piyush Goyal181818Minister of Railways, Commerce & Industry, Consumer Affairs
and Food & Public Distribution, Government of India. and Sambit
Patra191919Official spokesperson of the Bharatiya Janata Party in India. are
also present in the list of top mentioned users. Table VI shows the top 10
users with the most liked posts on the platform. Ravi Shankar Prasad is the
most liked user with 435,752 likes. Both English and Hindi pages for Republic
News channel are present in the top 10 list.

##  VI RQ3 : Koo’s user network and communities

We analyzed Koo’s following network, looking for characteristics of the
community forming as the young platform grows. The network has a noticeably
high local clustering coefficient of 0.561, that represents how well connected
the neighbourhood of a vertex is. This indicates a strong modular structure in
the network, presumably due to Koo only catering to audiences from a single
country. In contrast, Twitter, which caters to worldwide audiences, only had
an average local clustering coefficient of 0.072 during its early years in
2009 [28], indicating much weaker communities.

We see that around 90% of the users have less than a hundred followers.
However, a few users with an extremely high number of followers (of the order
of $10^{6}$) skew the distribution. The distribution of the number of
followees follows a similar pattern. Further, users with a high number of
followers tend to only have a small number of followees and vice-versa,
indicating the presence of influential and popular accounts that are followed
by a majority of other users. This may, in part, be due to new users being
shown a list of popular accounts to follow, on top of their feed.

Figure 10 shows the following network between verified users on Koo. We see
distinct communities of users based on language, possibly because users
interact more with others whose language they can understand. English-speaking
users are more centrally placed in the network, with connections to both Hindi
and Kannada speakers. Verified accounts in the two Indian languages, on the
contrary, do not have many follower-followee relationships.

Handle | Mentions
---|---
republic | 16,041
ravishankarprasad | 12,991
kisanektamorcha | 11,010
ErpENlk_BArt | 9,366
piyushgoyal | 9,127
mayank | 7,588
leledirect.com | 7,045
aprameya | 6,390
sambitpatra | 5,742
khushbookapoor | 5,693
TABLE V: Users with most number of mentions. News channel Republic has the
highest number of mentions (16,041), followed by Ravi Shankar Prasad (12,991).

Handle | Likes
---|---
ravishankarprasad | 435,752
piyushgoyal | 395,674
republic | 357,745
ErpENlk_BArt | 296,039
meghupdates | 277,495
rinki | 257,266
sawatimehera | 239,962
chouhanshivraj | 185,354
narendramodiforyou | 181,271
anandranganathan | 168,518
TABLE VI: Users with most number of likes. Ravi Shankar Prasad is the most
liked user on Koo with 435,752 likes, while Republic takes the third place.

##  VII Discussion

Koo’s tagline, “The Voices of India”, captures the essence of the platform,
i.e., support for Indian languages, large Indian user base, and homegrown
development. The recent surge in popularity of the platform, presence of
multilingual content, and linguistic communities makes the study of Koo
interesting. We release the first-ever dataset of Koo and characterize the
platform based on the users, content posted, and the network. We show the
formation of tight communities based on language, as well as the massive
popularity of Indian politicians, news media agencies and government
organizations on the network. We note that female users are more active,
despite being present in smaller numbers. The higher presence of Hindi than
English on Koo indicates a degree of success in promoting discourse in Indian
languages. Kannada, Tamil, Telugu and Marathi also constitute a considerable
portion of the content and activity on the platform. We observe a Koo vs
Twitter rhetoric with hashtags such as “#koovstwitter“ and “#bantwitter“
trending. Koo is still in its nascent stages and is being developed to include
more languages - Gujarati, Malayalam, Oriya, Punjabi, and Assamese. As Koo
grows in size, it should build convenient mechanisms with which researchers
can collect and work on their data, which can prove useful for research in
social computing and Indian languages.

Figure 10:  Following graph of verified users on Koo. Edges are directed from
the follower to the followee. Node size is proportional to in-degree, while
colors indicate language. Names are only shown for a few prominent accounts.
Singletons are not shown.

|
---|---

(a) Unigram Wordcloud.

(b) Bigram Wordcloud.

Figure 11: Word Clouds for user posts. Both unigrams and bigrams show
substantial Hindu religion centric content.

##  VIII Limitations and Future Work

This paper performs an exploratory analysis of the characteristics of Koo,
presenting a novel dataset and uncovering valuable insights. Although our
dataset contains a substantial portion of the user-base of Koo, it does have
the limitation that it leaves out singletons and isolated communities. For
future work, this multilingual data can act as a corpus for research in Indian
languages. A study of posting trends by the same users on Koo and Twitter
could reveal if users use the two platforms for different purposes, or if
their activity on one mirrors that on the other. Our dataset of corresponding
Koo and Twitter user IDs makes this data collection convenient. An analysis of
who gains popularity on Koo, as well as the presence of bot accounts could
reveal more about the inter-user interactions on the platform.

## Acknowledgements

The authors would like to thank the annotators for their help and contribution
for creating the dataset. The authors would also like to thank Nidhi Goyal,
Samiya Caur and Shivangi Singhal for their helpful reviews and comments on the
initial draft versions.

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# Thermalization and disentanglement with a nonlinear Schrödinger equation

Eyal Buks<EMAIL_ADDRESS>Andrew and Erna Viterbi Department of
Electrical Engineering, Technion, Haifa 32000, Israel

###### Abstract

We study a recently proposed modified Schrödinger equation having an added
nonlinear term. For the case where a stochastic term is added to the
Hamiltonian, the fluctuating response is found to resemble the process of
thermalization. Disentanglement induced by the added nonlinear term is
explored for a system made of two coupled spins. A butterfly-like effect is
found near fully entangled states of the spin-spin system. A limit cycle
solution is found when one of the spins is externally driven.

##  I Introduction

In 1935 Schrodinger_807  Schrödinger has identified a self-inconsistency in
the quantum to classical transition process Penrose_4864 ; Leggett_939 ;
Leggett_022001 , which became known as the problem of quantum measurement.
This problem is related to the phenomenon of quantum entanglement. Exploring
possible mechanisms of disentanglement, may help resolving this long-standing
problem.

Processes such as disentanglement require nonlinear time evolution. A variety
of nonlinear terms Geller_2111_05977  that can be added to the Schrödinger
equation have been explored before Weinberg_336 ; Weinberg_61 ; Doebner_397 ;
Doebner_3764 ; Gisin_5677 ; Kaplan_055002 ; Munoz_110503 . In most previous
proposals, the purpose of the added nonlinear terms is to generate a
spontaneous collapse Bassi_471 .

Here we explore a recently proposed modified Schrödinger equation having an
added nonlinear term Buks_355303  (see section II). The proposed equation can
be constructed for any physical system having Hilbert space of finite
dimensionality, and it does not violate unitarity of the time evolution.

The effect of the added term on the dynamics of a single spin 1/2 is studied
in section III. The spin’s response to an applied fluctuating magnetic field
is found to mimic the process of thermalization Diosi_451 ; Molmer_524 ;
Dalibard_580 ; Semin_063313  (see section IV). Disentanglement induced by the
nonlinear term is explored with two coupled spins (see section V). A
butterfly-like effect is found near fully entangled spin-spin states.

The system can become unstable when one spin is externally driven (see section
VI). Limit cycle solutions for the modified Schrödinger equation are found in
the instability region. The instability of the modified Schrödinger equation
is closely related to an instability found with a similar spin-spin system
Levi_053516 , when the equations of motion generated by the standard
Schrödinger equation are analyzed using the mean field approximation
breuer2002theory ; Drossel_217 ; Hicke_024401 ; Klobus_034201 .

##  II The modified Schrödinger equation

Let $\mathcal{H}$ be a time-independent Hermitian Hamiltonian of a given
physical system. Consider a modified Schrödinger equation for the state vector
$\left|\psi\right\rangle$ given by Buks_355303

$\frac{\mathrm{d}}{\mathrm{d}t}\left|\psi\right\rangle=\left(-i\hbar^{-1}\mathcal{H}+\gamma_{\mathrm{D}}M_{\mathrm{D}}\right)\left|\psi\right\rangle\;,$
(1)

where $\mathrm{d}/\mathrm{d}t$ is a time derivative. In the added (to the
standard Schrödinger equation) term $\gamma_{\mathrm{D}}M_{\mathrm{D}}$, the
rate $\gamma_{\mathrm{D}}$ is a positive coefficient, and the operator
$M_{\mathrm{D}}$ is derived from a given non-zero state vector
$\left|\Psi\right\rangle$ according to (the state vector
$\left|\Psi\right\rangle$ is not required to be normalized)

$M_{\mathrm{D}}=-\sqrt{\frac{\left\langle\Psi\right.\left|\Psi\right\rangle}{1-\left\langle\mathcal{P}\right\rangle}}\left(\mathcal{P}-\left\langle\mathcal{P}\right\rangle\right)\;,$
(2)

where the projection operator $\mathcal{P}$ is given by

$\mathcal{P}=\frac{\left|\Psi\right\rangle\left\langle\Psi\right|}{\left\langle\Psi\right.\left|\Psi\right\rangle}\;,$
(3)

and the expectation value $\left\langle\mathcal{P}\right\rangle$ is given by

$\left\langle\mathcal{P}\right\rangle=\frac{\left\langle\psi\right|\mathcal{P}\left|\psi\right\rangle}{\left\langle\psi\right.\left|\psi\right\rangle}=\frac{\left|\left\langle\Psi\right.\left|\psi\right\rangle\right|^{2}}{\left\langle\Psi\right.\left|\Psi\right\rangle\left\langle\psi\right.\left|\psi\right\rangle}\;.$
(4)

The modified Schrödinger equation yields a modified master equation for the
pure state density operator
$\rho=\left|\psi\right\rangle\left\langle\psi\right|$ given by (note that
$M_{\mathrm{D}}^{{\dagger}}=M_{\mathrm{D}}$ and
$\mathcal{H}^{{\dagger}}=\mathcal{H}$)

$\frac{\mathrm{d}\rho}{\mathrm{d}t}=\frac{\left[\mathcal{H},\rho\right]}{i\hbar}+\gamma_{\mathrm{D}}\left(\rho
M_{\mathrm{D}}+M_{\mathrm{D}}\rho\right)\;.$ (5)

Note that $\left(\mathrm{d/d}t\right)\operatorname{Tr}\rho=0$ provided that
$\operatorname{Tr}\rho=1$ (i.e. $\left|\psi\right\rangle$ is normalized) [see
Eq. (2), and note that $\left\langle
O\right\rangle\equiv\operatorname{Tr}\left(\rho O\right)$ for an arbitrary
observable $O=O^{{\dagger}}$], and that
$\left(\mathrm{d/d}t\right)\operatorname{Tr}\rho^{2}=0$ provided that
$\rho^{2}=\rho$ [note that $\left\langle M_{\mathrm{D}}\right\rangle=0$, see
Eq. (2)]. Henceforth it is assumed that $\left|\psi\right\rangle$ is
normalized, and that $\rho^{2}=\rho$. The modified master equation (5) yields
a modified Heisenberg equation given by

$\frac{\mathrm{d}\left\langle
O\right\rangle}{\mathrm{d}t}=\frac{\left\langle\left[O,\mathcal{H}\right]\right\rangle}{i\hbar}+\gamma_{\mathrm{D}}\left\langle
M_{\mathrm{D}}O+OM_{\mathrm{D}}\right\rangle\;,$ (6)

where $O=O^{{\dagger}}$ is a given observable that does not explicitly depend
on time.

##  III One spin

As an example, consider a spin 1/2 particle. The $2\times 2$ density matrix
$\rho$ is expressed as

$\rho=\frac{1+\bm{k}\cdot\bm{\sigma}}{2}\;,$ (7)

where $\bm{k}=\left(k_{x},k_{y},k_{z}\right)$ is a real vector, and
$\bm{\sigma}=\left(\sigma_{x},\sigma_{y},\sigma_{z}\right)$ is the Pauli
matrix vector

$\sigma_{x}=\left(\begin{array}[c]{cc}0&1\\\
1&0\end{array}\right),\;\sigma_{y}=\left(\begin{array}[c]{cc}0&-i\\\
i&0\end{array}\right),\;\sigma_{z}=\left(\begin{array}[c]{cc}1&0\\\
0&-1\end{array}\right)\;.$ (8)

The Hamiltonian $\mathcal{H}$ is assumed to be given by
$\hbar^{-1}\mathcal{H}=\bm{\omega\cdot\sigma}$, where
$\bm{\omega}=\left(\omega_{x},\omega_{y},\omega_{z}\right)$ is a constant real
vector. With the help of the identity
$\left(\bm{\sigma}\cdot\bm{a}\right)\left(\bm{\sigma}\cdot\bm{b}\right)=\bm{a}\cdot\bm{b}+i\bm{\sigma}\cdot\left(\bm{a}\times\bm{b}\right)$,
where $\bm{a}$ and $\bm{b}$ are three-dimensional vectors, one finds that [see
Eq. (6), and note that
$\operatorname{Tr}\sigma_{x}=\operatorname{Tr}\sigma_{y}=\operatorname{Tr}\sigma_{z}=0$,]

$\frac{\mathrm{d}\bm{k}}{\mathrm{d}t}=2\left(\bm{\omega}\times\bm{k}\right)+\gamma_{\mathrm{D}}\left\langle
M_{\mathrm{D}}\bm{\sigma}+\bm{\sigma}M_{\mathrm{D}}\right\rangle\;.$ (9)

Consider the case where $\left|\Psi\right\rangle$ is taken to be a normalized
eigenvector of $\bm{\hat{s}\cdot\sigma}$, where
$\bm{\hat{s}}=\left(s_{x},s_{y},s_{z}\right)$ is a constant real unit vector,
and the corresponding eigenvalue is $+1$ (i.e.
$\left\langle\Psi\right.\left|\Psi\right\rangle=1$,
$\bm{\hat{s}\cdot\hat{s}}=1$ and
$\bm{\hat{s}\cdot\sigma}\left|\Psi\right\rangle=\left|\Psi\right\rangle$). For
this case $\sqrt{\left(1-\bm{\hat{s}}\cdot\bm{k}\right)/2}\left\langle
M_{\mathrm{D}}\bm{\sigma}+\bm{\sigma}M_{\mathrm{D}}\right\rangle=\left(\bm{\hat{s}}\cdot\bm{k}\right)\bm{k}-\bm{\hat{s}}$
[see Eq. (2)], and thus (compare with Refs. Kowalski_1 ; Fernengel_385701 ;
Kowalski_167955 )

$\frac{\mathrm{d}\bm{k}}{\mathrm{d}t}=2\bm{\omega}\times\bm{k}+\gamma_{\mathrm{D}}\frac{\left(\bm{\hat{s}}\cdot\bm{k}\right)\bm{k-\hat{s}}}{\sqrt{\frac{1-\bm{\hat{s}}\cdot\bm{k}}{2}}}\;.$
(10)

The following holds
$\bm{\hat{s}}=\bm{\hat{s}}_{\parallel}+\bm{\hat{s}}_{\perp}$, where the
parallel $\bm{\hat{s}}_{\parallel}$ and perpendicular $\bm{\hat{s}}_{\perp}$
(with respect to $\bm{k}$) components are given by
$\bm{\hat{s}}_{\parallel}=\left(\bm{k}\cdot\bm{k}\right)^{-1}\left(\bm{k}\cdot\bm{\hat{s}}\right)\bm{k}$
and
$\bm{\hat{s}}_{\perp}=-\left(\bm{k}\cdot\bm{k}\right)^{-1}\bm{k}\times\left(\bm{k}\times\bm{\hat{s}}\right)$
[recall the vector identity
$\bm{A}\times\left(\bm{B}\times\bm{C}\right)=\left(\bm{A}\cdot\bm{C}\right)\bm{B}-\left(\bm{A}\cdot\bm{B}\right)\bm{C}$].
Thus, for the case where $\left|\bm{k}\right|=1$ (i.e. $\bm{k}\cdot\bm{k}=1$)
Eq. (10) becomes

$\displaystyle\frac{\mathrm{d}\bm{k}}{\mathrm{d}t}$
$\displaystyle=2\bm{\omega}\times\bm{k}-\gamma_{\mathrm{D}}\frac{\bm{\hat{s}}_{\perp}}{\sqrt{\frac{1-\bm{\hat{s}}_{\parallel}\cdot\bm{k}}{2}}}$
$\displaystyle=\left(2\bm{\omega}+\gamma_{\mathrm{D}}\frac{\bm{\hat{s}}\times\bm{k}}{\sqrt{\frac{1-\bm{\hat{s}}\cdot\bm{k}}{2}}}\right)\times\bm{k}\;.$
(11)

The above result (11) indicates that the radial component of
$\mathrm{d}\bm{k/}\mathrm{d}t$ vanishes [i.e.
$\left(\mathrm{d}\bm{k/}\mathrm{d}t\right)\cdot\bm{k}=0$] on the surface of
the Bloch sphere (i.e. when $\left|\bm{k}\right|=1$).

By multiplying Eq. (10) by
$\bm{\hat{\omega}}=\bm{\omega}/\left|\bm{\omega}\right|$ one finds that

$\frac{\mathrm{d}k_{\parallel}}{\mathrm{d}t}=\gamma_{\mathrm{D}}\frac{\bm{\hat{s}}\cdot\left(\left(\bm{k}\cdot\bm{\hat{\omega}}\right)\bm{k-\hat{\omega}}\right)}{\sqrt{\frac{1-\bm{\hat{s}}\cdot\bm{k}}{2}}}\;,$
(12)

where $k_{\parallel}=\bm{k}\cdot\bm{\hat{\omega}}$, hence
$\mathrm{d}k_{\parallel}/\mathrm{d}t=0$ for $\bm{k}=\pm\bm{\hat{\omega}}$. For
the case where $\bm{\hat{\omega}}=\bm{\hat{z}}$ Eq. (12) becomes

$\frac{\mathrm{d}k_{\parallel}}{\mathrm{d}t}=\gamma_{\mathrm{D}}\frac{\left(s_{x}k_{x}+s_{y}k_{y}\right)k_{z}+s_{z}\left(k_{z}^{2}-1\right)}{\sqrt{\frac{1-\bm{\hat{s}}\cdot\bm{k}}{2}}}\;.$
(13)

When $\gamma_{\mathrm{D}}\ll\left|\bm{\omega}\right|$ the dynamics is
dominated by the term $2\bm{\omega}\times\bm{k}$ in Eq. (10), which gives rise
to spin precession. For this case the averaged value of the term
$s_{x}k_{x}+s_{y}k_{y}$ in Eq. (13) is nearly zero. Note that $k_{z}^{2}-1\leq
0$, hence for this case $\bm{k}\rightarrow+\bm{\hat{\omega}}$
($\bm{k}\rightarrow-\bm{\hat{\omega}}$) in the limit $t\rightarrow\infty$ when
$s_{z}<0$ ($s_{z}>0$). The plot shown in Fig. 1(a) is obtained by numerically
integrating the modified Schrödinger equation (10) for the case where
$\bm{\omega}$ is parallel to the $\bm{\hat{z}}$ direction, and
$\gamma_{\mathrm{D}}/\left|\bm{\omega}\right|=0.25$. As can be seen from Eq.
(11), to first order in $\gamma_{\mathrm{D}}/\left|\bm{\omega}\right|$ the
$\bm{k}$ fixed point is located at
$\pm\left(\bm{\hat{\omega}}+\left(\gamma_{\mathrm{D}}/\left|\bm{\omega}\right|\right)\left(2\left(1-\bm{\hat{s}}\cdot\bm{\hat{\omega}}\right)\right)^{-1/2}\bm{\hat{s}}\times\bm{\hat{\omega}}\right)$
[see the red star symbol in Fig. 1(a)].

Figure 1: One spin. The black solid line is obtained by numerically
integrating of the one-spin modified Schrödinger equation (10). The blue solid
(dashed) line connects the origin and the point $\bm{\hat{s}}$
($-\bm{\hat{s}}$), and $\bm{\omega}$ is parallel to the $\bm{\hat{z}}$
direction. (a) The ratio $\gamma_{\mathrm{D}}/\left|\bm{\omega}\right|=0.25$.
The red star symbol represents the analytical approximation for the $\bm{k}$
fixed point given by
$\pm\left(\bm{\hat{\omega}}+\left(\gamma_{\mathrm{D}}/\left|\bm{\omega}\right|\right)\left(2\left(1-\bm{\hat{s}}\cdot\bm{\hat{\omega}}\right)\right)^{-1/2}\bm{\hat{s}}\times\bm{\hat{\omega}}\right)$,
which is valid when $\gamma_{\mathrm{D}}\ll\left|\bm{\omega}\right|$. (b) The
ratio $\gamma_{\mathrm{D}}/\left|\bm{\omega}\right|=25$. The red star symbol
represents the analytical approximation for the $\bm{k}$ fixed point given by
$-\bm{\hat{s}}+2\left(\left|\bm{\omega}\right|/\gamma_{\mathrm{D}}\right)\bm{\hat{s}}\times\bm{\hat{\omega}}$,
which is valid when $\gamma_{\mathrm{D}}\gg\left|\bm{\omega}\right|$.

In the opposite extreme case of
$\gamma_{\mathrm{D}}\gg\left|\bm{\omega}\right|$, the dynamics is dominated by
the term proportional to $\gamma_{\mathrm{D}}$ in Eq. (10). Note that
$\left(\bm{\hat{s}}\cdot\bm{k}\right)\bm{k-\hat{s}}=0$ for
$\bm{k}=\pm\bm{\hat{s}}$, and that the term proportional to
$\gamma_{\mathrm{D}}$ in Eq. (10) gives rise to attraction (repulsion) of
$\bm{k}$ to the point $-\bm{\hat{s}}$ ($+\bm{\hat{s}}$). Consider the case
where $\bm{k}=-\bm{\hat{s}}+\bm{\delta}$, and where
$\bm{\hat{s}}\cdot\bm{\delta}=0$ ($\bm{\delta}$ is considered as
infinitesimally small). To first order in $\left|\bm{\delta}\right|$ Eq. (10)
yields

$\frac{\mathrm{d}\bm{\delta}}{\mathrm{d}t}=2\bm{\omega}\times\left(-\bm{\hat{s}}+\bm{\delta}\right)-\gamma_{\mathrm{D}}\bm{\delta}+O\left(\left|\bm{\delta}\right|^{2}\right)\;.$
(14)

hence the $\bm{k}$ point
$-\bm{\hat{s}}+2\left(\left|\bm{\omega}\right|/\gamma_{\mathrm{D}}\right)\bm{\hat{s}}\times\bm{\hat{\omega}}$
is nearly a stable fixed point when
$\gamma_{\mathrm{D}}\gg\left|\bm{\omega}\right|$ [see the red star symbol in
Fig. 1(b)].

##  IV Spin in thermal equilibrium

The effect of coupling between the spin and its environment can be accounted
for using the modified Schrödinger equation (1) provided that a fluctuating
magnetic field is added Slichter_Principles . Consider the case where the spin
Hamiltonian $\mathcal{H}$ is given by
$\hbar^{-1}\mathcal{H}=\bm{\omega\cdot\sigma}$, where
$\bm{\omega}=\omega_{0}\bm{\hat{z}}+\left(\omega_{x},\omega_{y},\omega_{z}\right)$,
where $\omega_{0}$ is a constant, and where $\omega_{x}\left(t\right)$,
$\omega_{y}\left(t\right)$ and $\omega_{z}\left(t\right)$ represent the effect
of a fluctuating magnetic field. The following is assumed to hold
$\left\langle\omega_{x}\right\rangle=\left\langle\omega_{y}\right\rangle=\left\langle\omega_{z}\right\rangle=0$,
where $\left\langle{}\right\rangle$ denotes time averaging (i.e. the
fluctuating field has a vanishing averaged value), and the correlation
function
$\left\langle\omega_{i}\left(t\right)\omega_{j}\left(t^{\prime}\right)\right\rangle$
is given by

$\left\langle\omega_{i}\left(t\right)\omega_{j}\left(t^{\prime}\right)\right\rangle=\delta_{ij}\omega_{\mathrm{s}}^{2}\exp\left(-\frac{\left|t-t^{\prime}\right|}{\tau_{\mathrm{s}}}\right)\;,$
(15)

where both the variance $\omega_{\mathrm{s}}^{2}$ and the correlation time
$\tau_{\mathrm{s}}$ are positive constants, and where
$i,j\in\left\\{x,y,z\right\\}$. The added fluctuating magnetic field gives
rise to longitudinal $T_{\mathrm{s}1}^{-1}$ and transverse
$T_{\mathrm{s}2}^{-1}$ relaxation rates given by [see Eqs. (17.274) and
(17.275) of Ref. Buks_QMLN ]

$\frac{1}{T_{\mathrm{s}1}}=\frac{2\omega_{\mathrm{s}}^{2}\tau_{\mathrm{s}}}{1+\omega_{0}^{2}\tau_{\mathrm{s}}^{2}}\;,$
(16)

and

$\frac{1}{T_{\mathrm{s}2}}=\frac{1}{2T_{\mathrm{s}1}}+\omega_{\mathrm{s}}^{2}\tau_{\mathrm{s}}\;.$
(17)

The effect of a fluctuating magnetic field is demonstrated by the plot shown
in Fig. 2. The time evolution of $\bm{k}$ is evaluated by numerically
integrating the modified Schrödinger equation (1) with added fluctuating
magnetic field. The parameters used for the calculation are listed in the
figure caption. The Wiener-Khinchine theorem is employed to relate the given
correlation function (15) to the power spectrum, which, in turn, is used to
derive the variance of Fourier coefficients of $\omega_{x}$, $\omega_{y}$ and
$\omega_{z}$. The variance values are employed for generating random Fourier
coefficients, which in turn, allow generating the functions
$\omega_{x}\left(t\right)$, $\omega_{y}\left(t\right)$ and
$\omega_{z}\left(t\right)$ in a way consistent with Eq. (15).

Figure 2: Thermal equilibrium. The red line represents the static magnetic
field direction, and the black line the Bloch vector $\bm{k}$. In this
calculation $\omega_{0}=10$, $\gamma_{\mathrm{D}}=5$,
$\bm{\hat{s}}=\bm{\hat{z}}$, $\omega_{\mathrm{s}}^{2}=10$ and
$\tau_{\mathrm{s}}=5$.

To account for the effect of the fluctuating field, a longitudinal relaxation
term proportional to $T_{\mathrm{s}1}^{-1}$ is added to Eq. (13). Consider the
case where
$T_{\mathrm{s}1}^{-1}\ll\gamma_{\mathrm{D}}\ll\left|\bm{\omega}\right|$ and
$\bm{\hat{s}}=\bm{\hat{z}}$. For this case Eq. (13) has a steady state
solution given by
$k_{\parallel}=-1+1/\left(1+2\gamma_{\mathrm{D}}T_{\mathrm{s}1}\right)$ [the
term proportional to $s_{x}k_{x}+s_{y}k_{y}$ in Eq. (13) has been disregarded,
since it is assumed that $\gamma_{\mathrm{D}}\ll\left|\bm{\omega}\right|$].
The corresponding effective temperature $T_{\mathrm{eff}}$ is given by

$T_{\mathrm{eff}}=\frac{\hbar\omega_{0}}{2k_{\mathrm{B}}\tanh^{-1}\left(1-\frac{1}{1+2\gamma_{\mathrm{D}}T_{\mathrm{s}1}}\right)}\;,$
(18)

where $k_{\mathrm{B}}$ is the Boltzmann’s constant.

##  V Two spins

Consider a two spin 1/2 system in a pure state $\left|\psi\right\rangle$ given
by
$\left|\psi\right\rangle=a\left|++\right\rangle+b\left|+-\right\rangle+c\left|-+\right\rangle+d\left|--\right\rangle$.

###  V.1 Single spin angular momentum

The single-spin angular momentum (in units of $\hbar/2$) vector operators are
denoted by $\bm{S}_{1}=\left(S_{1x},S_{1y},S_{1z}\right)$ and
$\bm{S}_{2}=\left(S_{2x},S_{2y},S_{2z}\right)$, and the total spin angular
momentum is $\bm{S}=\bm{S}_{1}+\bm{S}_{2}=\left(S_{x},S_{y},S_{z}\right)$. A
given single-spin linear operator of spin 1 (2) is represented by the $4\times
4$ matrix $\sigma_{0}\otimes K$ ($K\otimes\sigma_{0}$) where $\otimes$ denotes
the Kronecker tensor product, $\sigma_{0}$ is the $2\times 2$ identity matrix,
and where $K$ is the $2\times 2$ matrix representation of the given single-
spin operator. The matrix representations of $\bm{S}_{1}\cdot\bm{\hat{u}}_{1}$
and $\bm{S}_{2}\cdot\bm{\hat{u}}_{2}$, where
$\bm{\hat{u}}_{1}=\left(\sin\theta_{1}\cos\varphi_{1},\sin\theta_{1}\sin\varphi_{1},\cos\theta_{1}\right)$
and
$\bm{\hat{u}}_{2}=\left(\sin\theta_{2}\cos\varphi_{2},\sin\theta_{2}\sin\varphi_{2},\cos\theta_{2}\right)$
are unit vectors, are thus given by [see Eq. (8)]

$\displaystyle\bm{S}_{1}\cdot\bm{\hat{u}}_{1}$
$\displaystyle\dot{=}\left(\begin{array}[c]{cccc}\cos\theta_{1}&0&\sin\theta_{1}e^{-i\varphi_{1}}&0\\\
0&\cos\theta_{1}&0&\sin\theta_{1}e^{-i\varphi_{1}}\\\
\sin\theta_{1}e^{i\varphi_{1}}&0&-\cos\theta_{1}&0\\\
0&\sin\theta_{1}e^{i\varphi_{1}}&0&-\cos\theta_{1}\end{array}\right)\;,$ (23)

and

$\displaystyle\bm{S}_{2}\cdot\bm{\hat{u}}_{2}$
$\displaystyle\dot{=}\left(\begin{array}[c]{cccc}\cos\theta_{2}&\sin\theta_{2}e^{-i\varphi_{2}}&0&0\\\
\sin\theta_{2}e^{i\varphi_{2}}&-\cos\theta_{2}&0&0\\\
0&0&\cos\theta_{2}&\sin\theta_{2}e^{-i\varphi_{2}}\\\
0&0&\sin\theta_{2}e^{i\varphi_{2}}&-\cos\theta_{2}\end{array}\right)\;.$ (29)

With the help of Eqs. (LABEL:S1*u1) and (LABEL:S2*u2) and the normalization
condition $aa^{\ast}+bb^{\ast}+cc^{\ast}+dd^{\ast}=1$ one finds that

$\left|\left\langle\bm{S}_{1}\right\rangle\right|^{2}=\left|\left\langle\bm{S}_{2}\right\rangle\right|^{2}=1-4\left|ad-
bc\right|^{2}\;.$ (31)

Note that the normalization condition implies that $\left|ad-bc\right|^{2}\leq
1/4$. In standard quantum mechanics the term $ad-bc$ is time-independent,
provided that the spins are decoupled [see Eq. (8.121) of Ref. Buks_QMLN ].
The term $\left|ad-bc\right|$ can be extracted from the partial transpose
$\rho^{\mathrm{T}1}$ ($\rho^{\mathrm{T}2}$) of the spin-spin density operator
with respect to spin 1 (2) using the relation
$\det\rho^{\mathrm{T}1}=\det\rho^{\mathrm{T}2}=-\left|ad-bc\right|^{4}$
Peres_1413 .

Consider the case where
$\left\langle\bm{S}_{1}\right\rangle=\left\langle\bm{S}_{2}\right\rangle=0$.
For this case the following holds $\left\langle
S_{1z}\right\rangle=a^{\ast}a+b^{\ast}b-c^{\ast}c-d^{\ast}d=0$, $\left\langle
S_{2z}\right\rangle=a^{\ast}a-b^{\ast}b+c^{\ast}c-d^{\ast}d=0$, $\left\langle
S_{1+}\right\rangle=2\left(a^{\ast}c+b^{\ast}d\right)=0$ and $\left\langle
S_{2+}\right\rangle=2\left(a^{\ast}b+c^{\ast}d\right)=0$, where
$S_{n\pm}=S_{nx}\pm iS_{ny}$ and $n\in\left\\{1,2\right\\}$. The conditions
$\left\langle S_{1z}\right\rangle=\left\langle S_{2z}\right\rangle=0$ imply
that $a^{\ast}a=d^{\ast}d$ and $b^{\ast}b=c^{\ast}c$, whereas the conditions
$\left\langle S_{1+}\right\rangle=\left\langle S_{2+}\right\rangle=0$ yield
$a^{\ast}/d=-b^{\ast}/c=-c^{\ast}/b$. Hence the state vector
$\left|\psi\right\rangle$ for this case has the form

$\displaystyle\left|\psi\right\rangle$
$\displaystyle=\frac{\cos\frac{\theta_{\psi}}{2}e^{-\frac{i\phi_{\alpha}}{2}}}{\sqrt{2}}\left|++\right\rangle+\frac{\sin\frac{\theta_{\psi}}{2}ie^{-\frac{i\phi_{\beta}}{2}}}{\sqrt{2}}\left|+-\right\rangle$
$\displaystyle+\frac{\sin\frac{\theta_{\psi}}{2}ie^{\frac{i\phi_{\beta}}{2}}}{\sqrt{2}}\left|-+\right\rangle+\frac{\cos\frac{\theta_{\psi}}{2}e^{\frac{i\phi_{\alpha}}{2}}}{\sqrt{2}}\left|--\right\rangle\;,$
(32)

where $\theta_{\psi}$, $\phi_{\alpha}$ and $\phi_{\beta}$ and are real. Note
that $ad-bc=1/2$ for the state (32).

The operator $R$ is defined by [note that
$\bm{S}_{1}\cdot\bm{S}_{2}=\bm{S}_{2}\cdot\bm{S}_{1}$, see Eqs. (LABEL:S1*u1)
and (LABEL:S2*u2)]

$R=\bm{S}_{1}\cdot\bm{S}_{2}-\left\langle\bm{S}_{1}\right\rangle\cdot\left\langle\bm{S}_{2}\right\rangle\;,$
(33)

and its expectation value $\left\langle R\right\rangle$ is given by

$\left\langle
R\right\rangle=4\left(\left|ad\right|^{2}-\left|bc\right|^{2}\right)-4\operatorname{Re}\left(\left(b^{\ast}{}^{2}+c^{\ast}{}^{2}\right)\left(ad-
bc\right)\right)\;.$ (34)

Note that $\left\langle R\right\rangle=0$ when $ad-bc=0$.

###  V.2 Single spin purity

The single-spin purity $P$ is given by $P=1-2\left|ad-bc\right|^{2}$ [see Eq.
(8.642) of Ref. Buks_QMLN ]. It is bounded by $1/2\leq P\leq 1$ [recall that
$\left|ad-bc\right|^{2}\leq 1/4$]. In terms of the purity $P$, Eq. (31) reads
$\left|\left\langle\bm{S}_{1}\right\rangle\right|^{2}=\left|\left\langle\bm{S}_{2}\right\rangle\right|^{2}=2\left(P-1/2\right)$.

###  V.3 Spin-spin disentanglement

Spin-spin disentanglement is generated by the term proportional to
$\gamma_{\mathrm{D}}$ in the modified Schrödinger equation (1) provided that
the bra vector $\left\langle\Psi\right|$ is taken to be given by Buks_355303 ;
Wootters_2245

$\left\langle\Psi\right|=d\left\langle++\right|-c\left\langle+-\right|-b\left\langle-+\right|+a\left\langle--\right|\;.$
(35)

Note that $\left\langle\Psi\right|$ (35) is normalized provided that
$\left|\psi\right\rangle$ is normalized, and that
$\left\langle\Psi\right.\left|\psi\right\rangle=2\left(ad-bc\right)$ [compare
with Eq. (31)].

The plots shown in Fig. 3 are based on numerical integration of the spin-spin
modified Schrödinger equation (1) with $\mathcal{H}=0$, and with
$\left\langle\Psi\right|$ given by Eq. (35). In the panels labelled by the
number ’1’ (’2’), the single-spin purity is initially low $P\simeq 1/2$ (high
$P\simeq 1$), i.e. initially$\ \left|ad-bc\right|^{2}\simeq 1/4$ ($\left|ad-
bc\right|^{2}\simeq 0$). The three-dimensional plots labelled by the letter
’a’ (’b’) display the time evolution of $\left\langle\bm{S}_{1}\right\rangle$
($\left\langle\bm{S}_{2}\right\rangle$). In these plots, red straight lines
are drawn between the origin and the initial value of
$\left\langle\bm{S}_{n}\right\rangle$, whereas green lines represent time
evolution of $\left\langle\bm{S}_{n}\right\rangle$, where
$n\in\left\\{1,2\right\\}$. The plots labelled by the letters ’c’ and ’d’
display the time evolution of $P$ and $\left\langle R\right\rangle$ [see Eq.
(33)], respectively. For both cases ’1’ and ’2’,
$\left|\left\langle\bm{S}_{1}\right\rangle\right|^{2}=\left|\left\langle\bm{S}_{2}\right\rangle\right|^{2}\rightarrow
1$ [i.e. $\left|ad-bc\right|^{2}\rightarrow 0$ and $P\rightarrow 1$, see Eq.
(31)] in the long time limit $t\rightarrow\infty$, i.e. entanglement vanishes
in this limit.

Figure 3: Spin-spin disentanglement. In the panels labelled by the numbers ’1’
(’2’), the single-spin purity is initially $P\simeq 1/2$ ($P\simeq 1$). The
(initial value) time evolution of $\left\langle\bm{S}_{1}\right\rangle$ and
$\left\langle\bm{S}_{2}\right\rangle$ is indicated by red (green) lines in the
plots labelled by the letter ’a’ and ’b’, respectively. The single spin purity
$P$ and $\left\langle R\right\rangle$ are shown as a function of time $t$ in
the plots labelled by the letters ’c’ and ’d’, respectively.

The case where initially $P\simeq 1/2$ [see Fig. 3 (a1), (b1), (c1) and (d1)]
is explored by numerically integrating the modified Schrödinger equation (1)
with the initial condition (before normalization) of
$\left|\psi\right\rangle=\left|\psi_{0}\right\rangle+\epsilon\left|\psi_{\mathrm{p}}\right\rangle$,
where
$\left|\psi_{0}\right\rangle=\left|\mathrm{B}_{0,0}\right\rangle=2^{-1/2}\left(\left|+-\right\rangle-\left|-+\right\rangle\right)$
is the singlet Bell state (which is invariant under single-spin basis
transformation, and which is fully entangled), $\epsilon$ is a small positive
number, and
$\left|\psi_{\mathrm{p}}\right\rangle=\alpha\left|++\right\rangle+\beta\left|+-\right\rangle+\gamma\left|-+\right\rangle+\delta\left|--\right\rangle$
is normalized. For this case, to first order in $\epsilon$ initially
$\left\langle S_{1+}\right\rangle=-\left\langle
S_{2+}\right\rangle=2^{1/2}\left(\delta-\alpha^{\ast}\right)\epsilon$ and
$\left\langle S_{1z}\right\rangle=-\left\langle
S_{2z}\right\rangle=2^{-1/2}\left(\beta+\beta^{\ast}+\gamma+\gamma^{\ast}\right)\epsilon$,
i.e. initially
$\left\langle\bm{S}_{1}\right\rangle=-\left\langle\bm{S}_{2}\right\rangle$.
The plots in Fig. 3 (a1), (b1), (c1) and (d1) demonstrate that both
$\left\langle\bm{S}_{1}\right\rangle$ and
$\left\langle\bm{S}_{2}\right\rangle$ increase in magnitude with time while
remaining nearly anti-parallel to each other as they both approach the Bloch
sphere surfaces. The red star symbols in Fig. 3(a1) and (b1) indicate the
initial values of
$\left\langle\bm{S}_{1}\right\rangle/\left|\left\langle\bm{S}_{1}\right\rangle\right|$
and
$\left\langle\bm{S}_{2}\right\rangle/\left|\left\langle\bm{S}_{2}\right\rangle\right|$.
As the plots in Fig. 3(a1) and (b1) demonstrate, time evolution leaves these
normalized values nearly unchanged, provided that $\epsilon\ll 1$ (the value
of $\epsilon=0.1$ has been used for generating the plots).

The case shown in Fig. 3 (a1), (b1), (c1) and (d1) demonstrates strong
dependency of the long time value of $\left|\psi\right\rangle$ on its initial
value. This dependency, which becomes extreme when $\left|\psi\right\rangle$
is initially fully entangled, resembles the butterfly effect. In the limit
$\epsilon\rightarrow 0$, angular momentum is conserved by the modified
Schrödinger equation, provided that
$\left|\psi_{0}\right\rangle=\left|\mathrm{B}_{0,0}\right\rangle$. This can be
attributed to the fact that
$S_{x}\left|\mathrm{B}_{0,0}\right\rangle=S_{y}\left|\mathrm{B}_{0,0}\right\rangle=S_{z}\left|\mathrm{B}_{0,0}\right\rangle=0$.
Conservation of the angular momentum $\bm{\hat{z}}$ component is obtained when
the initial state is the Bell triplet state
$\left|\mathrm{B}_{1,0}\right\rangle=2^{-1/2}\left(\left|+-\right\rangle+\left|-+\right\rangle\right)$,
which is also fully entangled, and for which
$S_{z}\left|\mathrm{B}_{1,0}\right\rangle=0$.

The case where initially $P\simeq 1$ is demonstrated by Fig. 3 (a2), (b2),
(c2) and (d2). For this case both $\left\langle\bm{S}_{1}\right\rangle$ and
$\left\langle\bm{S}_{2}\right\rangle$ are initially close to the Bloch sphere
surfaces. Consequently, the time evolution of $\left|\psi\right\rangle$
towards a fully product state does not significantly change
$\left\langle\bm{S}_{1}\right\rangle$ and
$\left\langle\bm{S}_{2}\right\rangle$ (time evolution is represented by the
green lines).

##  VI Instability

Consider a system composed of two spins 1/2. The first spin, which is labelled
as ’$\mathrm{a}$’, has a relatively low angular frequency
$\omega_{\mathrm{a}}$ in comparison with the angular frequency
$\omega_{\mathrm{b}}$ of the second spin, which is labelled as ’$\mathrm{b}$’,
and which is externally driven. The angular momentum vector operator of
particle $\mathrm{a}$ ($\mathrm{b}$) is denoted by $\bm{S}_{\mathrm{a}}$
($\bm{S}_{\mathrm{b}}$). The Hamiltonian $\mathcal{H}$ of the closed system is
given by

$\mathcal{H}=\omega_{\mathrm{a}}S_{\mathrm{az}}+\omega_{\mathrm{b}}S_{\mathrm{bz}}+\frac{\omega_{1}\left(S_{\mathrm{b+}}+S_{\mathrm{b-}}\right)}{2}+V\;,$
(36)

where the driving amplitude and angular frequency are denoted by $\omega_{1}$
and $-\omega_{\mathrm{p}}=\omega_{\mathrm{b}}-\Delta$, respectively ($-\Delta$
is the driving detuning), the operators $S_{\mathrm{a\pm}}$ are given by
$S_{\mathrm{a\pm}}=S_{\mathrm{ax}}\pm iS_{\mathrm{ay}}$, and the rotated
operators $S_{\mathrm{b\pm}}$ are given by
$S_{\mathrm{b\pm}}=\left(S_{\mathrm{bx}}\pm iS_{\mathrm{by}}\right)e^{\mp
i\omega_{\mathrm{p}}t}$. The coupling term is given by
$V=g\hbar^{-1}\left(S_{\mathrm{a+}}+S_{\mathrm{a-}}\right)S_{\mathrm{bz}}$,
where $g$ is a coupling rate. In a rotating frame, the matrix representation
of the transformed Hamiltonian $\mathcal{H}^{\prime}$ is given by
$\mathcal{H}^{\prime}\dot{=}\hbar\Omega$, where the $4\times 4$ matrix
$\Omega$ is given by

$\Omega=\left(\begin{array}[c]{cccc}\frac{\omega_{\mathrm{a}}+\Delta}{2}&\frac{\omega_{1}}{2}&\frac{g}{2}&0\\\
\frac{\omega_{1}}{2}&\frac{\omega_{\mathrm{a}}-\Delta}{2}&0&-\frac{g}{2}\\\
\frac{g}{2}&0&\frac{-\omega_{\mathrm{a}}+\Delta}{2}&\frac{\omega_{1}}{2}\\\
0&-\frac{g}{2}&\frac{\omega_{1}}{2}&\frac{-\omega_{\mathrm{a}}-\Delta}{2}\end{array}\right)\;.$
(37)

Disentanglement is generated by the modified Schrödinger equation (1) by
choosing the bra vector $\left\langle\Psi\right|$ to be given by Eq. (35). The
expectation values of $\left\langle\bm{S}_{1}\right\rangle$ and
$\left\langle\bm{S}_{2}\right\rangle$, which are shown in Fig. 4(a) and in
Fig. 4(b), respectively, are calculated by numerically integrating the
modified Schrödinger equation (1). For the plot shown in Fig. 4, for which the
Hartmann–Hahn matching condition $\omega_{\mathrm{a}}=\omega_{\mathrm{R}}$
Hartmann1962 ; Yang_1 ; Levi_053516  is assumed to be satisfied, where
$\omega_{\mathrm{R}}=\sqrt{\omega_{1}^{2}+\Delta^{2}}$ is the Rabi angular
frequency, both $\left\langle\bm{S}_{1}\right\rangle$ and
$\left\langle\bm{S}_{2}\right\rangle$ undergo a limit cycle (LC). The
instability responsible for the LC was studied in Ref. Levi_053516 , in which
the equations of motion generated by the Hamiltonian (36) were treated using
in the mean-field approximation. This example demonstrates the connection
between the mean-field approximation and disentanglement.

Figure 4: Dipolar LC. The expectation values of
$\left\langle\bm{S}_{1}\right\rangle$ and
$\left\langle\bm{S}_{2}\right\rangle$ are shown in (a) and (b), respectively.
The modified Schrödinger equation (1), with the Hamiltonian (36) and the bra
vector (35) is numerically integrated. The parameters used for the calculation
are $\gamma_{\mathrm{D}}=10^{3}$, $\omega_{\mathrm{a}}=10^{2}$,
$\omega_{1}=-\Delta=2^{-1/2}\omega_{\mathrm{a}}$ and $g=0.2$. Fluctuating
magnetic field with parameters $\omega_{\mathrm{s}}^{2}=1$ and
$\tau_{\mathrm{s}}=0.05$ is added [see Eq. (15)].

##  VII Summary

In summary, both processes of thermalization and disentanglement can be
modeled using a recently proposed modified Schrödinger equation. The added
nonlinear term can give rise to instabilities and LC solutions. On the other
hand, it remains unclear whether quantum mechanics can be self-consistently
reformulated based on the proposed modified Schrödinger equation. Future study
will be devoted to exploring candidate formalisms.

##  VIII Acknowledgments

This work was supported by the Israeli science foundation, the Israeli
ministry of science, and by the Technion security research foundation. We
thank Michael R. Geller for helpful discussions.

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