# Metaknowledge Extraction Based on Multi-Modal Documents

Shukan Liu∗, Ruilin Xu∗, Boying Geng†, Qiao Sun, Li Duan, and Yiming Liu ∗
Contributed equally and should be considered as co-first authors.†
Corresponding author (e-mail: boying_geng@126.com).Shukan liu was with the
School of computer science and engineering, Southeast University, Nanjing,
211189, China; and the School of Electronic Engineering, Naval University of
Engineering.
E-mail<EMAIL_ADDRESS>Ruilin Xu and Yiming Liu were with the Graduate
School, Naval University of Engineering, Wuhan, 430033, China; and the School
of Electronic Engineering, Naval University of Engineering.
E-mail<EMAIL_ADDRESS>and<EMAIL_ADDRESS>Boying Geng, Qiao
Sun, and Li Duan were with the School of Electronic Engineering, Naval
University of Engineering.

###### Abstract

The triple-based knowledge in large-scale knowledge bases is most likely
lacking in structural logic and problematic of conducting knowledge hierarchy.
In this paper, we introduce the concept of metaknowledge to knowledge
engineering research for the purpose of structural knowledge construction.
Therefore, the Metaknowledge Extraction Framework and Document Structure Tree
model are presented to extract and organize metaknowledge elements (titles,
authors, abstracts, sections, paragraphs, etc.), so that it is feasible to
extract the structural knowledge from multi-modal documents. Experiment
results have proved the effectiveness of metaknowledge elements extraction by
our framework. Meanwhile, detailed examples are given to demonstrate what
exactly metaknowledge is and how to generate it. At the end of this paper, we
propose and analyze the task flow of metaknowledge applications and the
associations between knowledge and metaknowledge.

###### Index Terms:

Metaknowledge, Multi-Modal, Document Layout Analysis, Knowledge Graph.

##  1 Introduction

Recently, widely used large-scale knowledge bases, such as DBpedia [1],
FreeBase [2], and YAGO [3], are all based on semantic entities and relations,
focusing on specific concepts and things. These widely used large-scale
knowledge bases have achieved good performance in information retrieval and
question answering tasks. However, the very knowledge that exists in large-
scale knowledge bases is based on entity-relation triples, which is not
exactly as same as the knowledge in human beings’ perception. Knowledge in
human minds is a complex of hierarchical, structured, and systematized
elements which has strongly logical or topological associations, especially
presented in structure or sequence.

For matching human’s natural intuition of knowledge, it is necessary to find a
brand-new representation that makes knowledge in the computer structured and
hierarchical. Thus, to address the significant issue that current knowledge
bases have difficulty in forming knowledge hierarchy, this work introduces the
concept of metaknowledge into knowledge engineering research. Ref. [4] studies
metaknowledge in scientific articles and holds that ”… metaknowledge results
from the critical scrutiny of what is known, how, and by whom. It can now be
obtained on large scales, enabled by a concurrent informatics revolution.”
Inspired by their thoughts, we believe that metaknowledge is knowledge about
knowledge. It describes the patterns of how human beings get, analyze and
organize knowledge. Similar to the metadata, metaknowledge is the structural
representation of knowledge and knowledge with fine-grained and hierarchical
characteristics.

Analogous to scientific articles in Ref. [4], documents such as textbooks,
papers, theses, governmental documents, laws, and regulations are the most
commonly used materials in getting knowledge. Those materials are normally
formed based on the author’s logic and intentions which people can easily
understand the backgrounds, motivations, themes only from structural patterns
such as titles, section titles, keywords, etc. They are good data sources to
acquire metaknowledge. Nevertheless, these structural patterns about getting
knowledge cannot be well described by triple-based knowledge. This work holds
that the structural elements such as titles, section titles, abstract,
keywords and so on, potentially reflect the structural logical associations
between knowledge. Therefore, in our study, the elements above are defined as
Metaknowledge Elements.

The process of people reading can be divided into three steps: see the words,
know the words, and understand the words. When the computer imitates human
beings’ process, the three steps become object detection, optical character
recognition (OCR), and natural language processing (NLP). Object detection
makes the computer classify and locate the layout elements of documents. OCR
makes the computer know the exact textual content. NLP makes the computer
analyze and understand the textual content. Documents of three modalities are
involved in these signs of process: image modal, layout modal, and text modal.
Therefore, a multi-modal method should be designed to extract metaknowledge
elements from documents.

To extract metaknowledge elements from multi-modal documents, our work
proposes MEF, a multi-modal Metaknowledge Extraction Framework. Through
experiments on multi-modal Chinese governmental documents dataset GovDoc-CN,
MEF performs better than single-modal models. Furthermore, this work refines
and explains the concept, the functions, the applications task flow of
metaknowledge, and proposes that the metaknowledge applications are in a
higher level compared with knowledge applications. Metaknowledge applications
are similar to the knowledge applications partly but play a more significant
role in managing and organizing knowledge.

##  2 Related Works

Visually-Rich Document Analysis (VRDA) is a task aiming to analyzing visually-
rich documents (VRDs) which are scanned or digital-born, such as page images
or PDFs, it plays a crucial role in governmental and commercial applications.
VRDA basically relies on three modalities: the text modal, the layout modal,
and the image modal. Ref. [5] presents an end-to-end and multi-modal fully
convolutional networks to extract document semantic elements. Ref. [6]
introduces a graph convolutional model that combines textual and visual
information in VRDs. Ref. [7] combines large pre-trained language models and
graph neural networks to encode both textual and visual information in VRDs.
Ref. [8] presents an approach based on graph neural network to build a rich
representation of text fields on a webpage and the relationships between them.
The works above have shown significant improvement on VRDA, whereas they do
not comprehensively consider the combination of all three modalities (text,
layout, and image). LayoutLM [9] and its improved model [10] are the pre-
trained and fine-tuned multi-modal framework to analyze VRDs, it unites text
modal, layout modal, and image modal to better extract semantic elements.
Nonetheless, the textual information in LayoutLMs is extracted from the image
region it corresponds to by OCR, that is, the frameworks consider textual
information of a document as independent semantic units instead of a coherent
passage.

Hierarchical Graph is initially proposed in the document reading
comprehension. Focusing on natural questions answering, Ref. [11] presents
multi-grained machine reading comprehension framework for modeling documents
with their hierarchical natures. The framework in Ref. [11] divides a document
into four levels of granularity: document, paragraphs, sentences, and tokens,
then utilizes graph attention networks (GATs) to obtain a multi-grained
representation of the document. Meanwhile, the recent work [12] presents
hierarchical graph network (HGN) for the multi-hop question answering task.
Different from the one-hop question answering that answers originated from a
single paragraph, the multi-hop question answering focuses on acquire answers
from multiple paragraphs or the whole passage. HGN is built by constructing
nodes on different levels of granularity including questions, paragraphs,
sentences and entities, it aggregates clues from texts across multiple
paragraphs. In creating HGN, the initial node representations are updated
through graph propagation and traverses through the graph edges of subsequent
sub-tasks for multi-hop reasoning. The works above present exquisite models
for modeling documents, but restricted in a single modal of text.

To make the computer effectively organizes and utilizes documents described by
natural language, the first task is to convert nonstructural documents in
natural language into structural data, where Natural Language to SQL (NL-to-
SQL) is one of such the task. Approaches (Seq2SQL [13], SQLNet [14], RE-SQL
[15], etc.) and datasets (SPLASH [16], ACL-SQL [17], etc.) all aim to
transform natural language to a structural data format, which generally is the
Structured Query Language (SQL). The natural language materials these
approaches and datasets processed are commonly tokens and sentences, while
whereby the passage-level information is not referred to. Recently, Ref. [18]
designs a weakly-supervised text-to-graph model Doc2Graph to bridge the gap
between concept map construction and neural networks, it is able to translate
passage-level documents into graph data.

Thanks to the inspirational works above, in this work, we consider getting
through the key links between how to extract the semantic structure of
documents (where VRDA focuses on), how to model structured documents (where
HGN focuses on), and how to translated documents in natural language into data
that the computer can understand (where NL2SQL and Doc2Graph focus on).

##  3 Metaknowledge Extraction Framework

In this work, a multi-modal framework is designed to extract metaknowledge
elements from documents in the text modal and the image modal, which is
described as Fig. 1. There are three parts in the framework: (1) metaknowledge
elements extraction modules, which extract metaknowledge elements form
documents in both text modal and image modal, such as titles, subtitles,
authors’ information, etc.; (2) verification and alignment module, which
aligns metaknowledge elements from text modal and image modal; (3)
metaknowledge generating module, which organizes metaknowledge elements by
document topology and hierarchy, then generates metaknowledge through entity
recognition and relation extraction.

Figure 1: The metaknowledge extraction framework (MEF), including: (1)
Metaknowledge elements extraction modules (from both text modal and image
modal); (2) Verification and alignment module; (3) Metaknowledge generating
module. The text modal extraction uses BERT+BiLSTM+CRF [19] framework, the
image modal extraction uses YOLOv4 [20] \+ PaddleOCR [21] framework.

###  3.1 Text Modal Metaknowledge Elements Extraction

BERT [22] is a pre-trained transformer model. It designs two pre-training
tasks: Masked Language Model (MLM) and Next Sentence Prediction (NSP). In MEF,
BERT is applied to vectorize the text in the text modal. When text modal
document $\boldsymbol{D}_{t}$ is input, BERT transform it into vector
$\boldsymbol{v}_{t}$.

The BiLSTM structure adds a layer of reverse LSTM network based on a
unidirectional LSTM network to use the context information more effectively.
The BiLSTM network can better capture the preceding and subsequent messages at
a certain time. It calculates two different hidden layer representations for
each sentence using the sequential and reverse order local methods and then
obtains the final hidden layer representation by splicing vectors.

In the task of NER, BiLSTM can predict the probability of each input word
corresponding to different output labels. For these directly obtained
probabilities, we can judge the label with the highest probability of the
current token. However, only using the BiLSTM network will make the output
ignore the correlation between labels and only pay attention to the
association between input characters and labels, which will lead to great
deviation in recognition results. To solve the problem, a conditional random
field (CRF) is added in BiLSTM. The BiLSTM+CRF model [23] uses the CRF layer
to add constraints, which effectively reduces the false prediction results of
BiLSTM.

Figure 2: An example of BERT+BiLSTM+CRF model: The input Chinese text sequence
means “Notice on epidemic situation (line break) University Office”, in which
“Notice on epidemic situation” is the title and “University Office” is the
addressee. All the tokens in the sequence are annotated in BIO format.

Inspired by the above works, in this paper, we use BERT+BiLSTM+CRF [19] (Fig.
2) to extract metaknowledge elements from text modal documents in a way
similar to NER. The word vector output from the BERT is input into the BiLSTM
network. The output of BiLSTM is spliced and integrated, and the CRF layer is
superimposed behind the BiLSTM to increase the utilization of label
information. The CRF layer uses the transfer feature to consider the
correlation between labels, combines BiLSTM and CRF to obtain the global
optimal output sequence so that the prediction label is a sequence considering
the label correlation.

###  3.2 Image Modal Metaknowledge Elements Extraction

In the image modal, the metaknowledge elements extraction is similar to the
document layout analysis task, which is a combination of optical character
recognition (OCR) and object detection. Specifically, it is to find the
regions of the metaknowledge elements through the object detection and then
recognize the text content of the metaknowledge elements from regions they
belong to.

The image modal metaknowledge elements extraction module is designed based on
the YOLOv4 [20] framework and PaddleOCR [21]. As is shown in Fig. 1, the
extraction module includes an object detection part and an OCR part. The
object detection part uses YOLOv4 framework to detect metaknowledge elements
in document image. Each metaknowledge element corresponds to a vector
$\boldsymbol{v}_{i}=(l_{i},p_{i},x_{i},y_{i},w_{i},d_{i})$, where $l_{i}$
represents the label of the object, $p_{i}$ indicates the probability that the
object belongs to this type of element, $(x_{i},y_{i},w_{i},d_{i})$ is the
bounding box coordinate of the location of the target, representing the
location origin $(x_{i},y_{i})$, width $w_{i}$ and height $d_{i}$.

Due to the accuracy of OCR, the text recognition result of the detected image
region can not be guaranteed to be accurate, but can be corrected by using the
corresponding text modal data. In this work, the Levenshtein distance (Eq. 1)
between the OCR extraction results and the original texts in the text modal is
calculated for correcting the OCR errors.

In the OCR post-process above, for each element of image recognition, the text
element with the shortest Levenshtein distance corresponding to it will be
found. Considering the possible errors in the optical character recognition
module for the extraction of image modal information, this work holds that for
general elements, when the Levenshtein distance between OCR recognition
results and corresponding named entity recognition results is not more than 3
of OCR recognition results, that is, the difference is not more than 3 of the
total sentence length of image recognition, the two are consistent.

${lev_{a,b}(i,j)=\left\\{\begin{array}[]{l}\max(i,j),if\min(i,j)=0;\\\
else\min\left\\{\begin{array}[]{l}le{v_{a,b}}(i-1,j)+1\\\
le{v_{a,b}}(i,j-1)+1\\\
le{v_{a,b}}(i-1,j-1)+1\;({a_{i}}\neq{b_{j}})\end{array}\right.\end{array}\right.}$
(1)

###  3.3 Multi-Modal Metaknowledge Elements Alignment

The single-modal approaches have different performances when extracting
various metaknowledge elements. Therefore, it is necessary to consider the two
modalities’ extraction results at the same time to fix the fault-tolerance
problem of single-modal extraction to improve the extraction quality.

For metaknowledge element $i$, let the extraction result of text modal be the
One-Hot vector $\boldsymbol{v}_{i\\_text}$, the extraction result of image
modal be the One-Hot vector $\boldsymbol{v}_{i\\_image}$. If the total number
of metaknowledge elements is $n$ and the total number of element categories is
$k$, then the total number of all possible combinations of the multi-modal
extraction results is $k^{2}$. Assume the $k^{2}$ matrix $\boldsymbol{W}$ is
the decision matrix, $\boldsymbol{W}=[w_{j1},w_{j2}]_{k^{2}\times 2}$. Each
row in $\boldsymbol{W}$ represents a possible combination of multi-modal
extraction results. Suppose the j-th row in $\boldsymbol{W}$ represents the
case that ”the extraction result of the element $i$ in the text modal is the
$k_{1}$ type, and the extraction result of the same element $i$ in the image
modal is the $k_{2}$ type”, where $j=k_{1}\times k_{2}$. If the extraction
result of the text modal is correct but the extraction result of the image
modal is incorrect, set $w_{j1}=1,w_{j2}=0$, otherwise, set
$w_{j1}=0,w_{j2}=1$. If the extraction results of the two modalities are both
correct, then set $w_{j1}=w_{j2}=0.5$. Therefore, for the metaknowledge
element $i$, the final extraction result is
$\boldsymbol{v}_{i\\_text}=w_{j1}\boldsymbol{v}_{i\\_text}+w_{j2}\boldsymbol{v}_{i\\_image}$.
When $\boldsymbol{W}$ is trained with enough samples, the extraction results
in the two modalities ($k_{1}$ and $k_{2}$) only need to look up the
$k_{1}\times k_{2}$ row of matrix $\boldsymbol{W}$, and the weighted summation
is the final extraction result.

###  3.4 Organizing Metaknowledge Elements

The framework proposed above realizes the classification of various types of
metaknowledge elements. Still, it does not clarify the juxtaposition and
inclusive relationships between the elements, especially the sections at
different levels, and does not form a hierarchical document structure.

From the perspective of people’s writing and reading habits, to determine the
relationship between sections at different levels, it is only necessary to
consider the order in which sections appear in the whole document. Among the
sections of different levels belonging to the ”inclusive” relationship, the
level of the section that appears first must be higher than that of the later;
among the sections of the same level belonging to the ”juxtaposition”
relationship, the order in which they appear in the document can also reflect
the relationship. Generally speaking, it is to rank sections with the
”inclusive” relationship and to sequence sections with the ”juxtaposition”
relationship, through which different levels of sections appear in the
document.

In MEF, through segmenting paragraphs and sentences, the document is divided
into a set
$T_{Document}=\\{(x_{0},y_{0}),(x_{1},y_{1}),\cdots,(x_{n},y_{n})\\}$, where
the coordinate $(x,y)$ is the Paragraph Index (PI) and Sentence Index (SI) of
each sentence in the document. For instance, assume the coordinate of sentence
$i$ is $(x_{i},y_{i})$, where $x_{i}$ represents that sentence $i$ belongs to
the $x_{i}$-th paragraph in the whole document, $y_{i}$ represents that the
sentence $i$ is the $y_{i}$-th sentence in the $x_{i}$-th paragraph. The
smaller the PI, the paragraph in which the sentence is in the front; the
smaller the SI in the paragraph, the more forward the sentence in the
paragraph.

Exact Subgraph Enumeration Tree [24] (ESU-Tree) is a structural model designed
for network motif recognition. This model is used to search for subgraphs of a
specified scale in the network. Since the structure of ESU-Tree can better
reflect the hierarchical and structural relationship, based on ESU-Tree, we
propose a special structure for the hierarchical representation of documents,
which is called ”Document Structure Tree (DST)” (Fig. 3).

Figure 3: Document Structure Tree

DEFINITION 1 The Document Structure Tree (DST) is a directed rooted tree that:

  * •

Each of its child vertices points to its parent vertex;

  * •

Its root vertex is located at level 0, the number of full tree levels is 4,
the depth is 4, and the height is 5;

  * •

Its fourth layer is all leaf vertices;

  * •

Its vertices have weights, but edges have no weights, and the vertex weights
are composed of two parts front weight and back weight. When comparing
weights, the front weights is compared first, and the back weights is compared
when the two front weights is equal;

  * •

Its left vertex has a smaller weight than its right vertex, and its parent
vertex has a smaller weight than its child vertices.

DEFINITION 2 If the vertex $RP$ is the nearest right vertex of a $Child$
vertex’s parent, define $RP$ as the Right Parent of $Child$. Similarly, if the
vertex $LP$ is the nearest left vertex of a $Child$ vertex’s parent, define
$LP$ as the Left Parent of $Child$.

If $P$ is the Parent, $C$ is the Child, $LP$ is the Left Parent, $RP$ is the
Right Parent, $ST$ is the subtree, $DST$ is the entire document structure
tree, $RST$ is the right subtree, and $L$ is the section level. Use
”$\leftarrow$” represents ”assignment”; obviously, by analyzing the
characteristics of the document structure tree, the following three basic
properties can be summarized:

PROPERTY 1 In any $ST$ of $DST$, there is the following weight relationship:

${weight(P)<weight(C)<weight(P)}$ (2)

PROPERTY 2 In any $ST$ of $DST$, there is the following weight relationship:

$\begin{split}&\forall\ vertex\in DST,\exists\ ST\subseteq DT,\\\ &p\leftarrow
Root(ST),rp\leftarrow Root(RST),\\\ &\textbf{IF}\
weight(p)<weight(vertex)<weight(rp),\\\ &\textbf{THEN}\ vertex\subseteq
ST.\end{split}$ (3)

PROPERTY 3 For any vertex in $DST$, its hierarchical attribution satisfies:

$\begin{split}&\forall\ vertex\in DST,\exists\ L_{1},L_{2},\ \textbf{and}\
L_{2}=L_{1}+1,\\\ &\textbf{IF}\
\min\\{weight(L_{1})\\}<weight(vertex)<\min\\{weight(L_{2})\\},\\\
&\textbf{THEN}\ vertex\subseteq L_{1}.\end{split}$ (4)

The establishment order and traversal order of the document structure tree are
consistent with the Chinese reading order, basically in the order that
”root$\rightarrow$left node$\rightarrow$right”. The establishment problem can
be abstracted into the following representation:

Known: (1) the sections levels of vertices; (2) the weight of each vertex.

Solve: (1) the parent-child relationship of each vertex; (2) the hierarchy of
vertices.

To make the computer able to organize metaknowledge elements, it is necessary
to consider designing the data structure of the above-mentioned document
structure tree model. The main problem of the design is to realize the
relationship judgment of ”parent$<$child$<$right parent” on the computer. To
accomplish this task, it is needed to access each vertex from left to right
and from top to bottom to judge the juxtaposition and inclusive relationship
between left and right, superior and subordinate vertices (subtrees).

For the inequality ”parent$<$child$<$right parent”, the condition is
incomplete. Because the parent node is traversed from top to bottom, the
parent vertex must be accessed before the child vertex, that is, the left end
of the inequality must be true. Therefore, it is only necessary to consider
the case where the right end condition is incomplete, that is, the right
parent vertex (right subtree) does not exist.

Obviously, if the classification discussion method is adopted, it is more
expensive to add supplementary rules to where the right parent does not exist.
Therefore, consider constructing the conditions that make the right part of
the inequality always hold to adapt to the original rules, rather than
establishing new rules. For this reason, the concept of ”absolute right
subtree” is introduced.

DEFINITION 3 The Absolute Right Subtree (ARS) is a document structure tree in
which the root vertex weight is sufficiently large, and the child is empty. It
is actually a fully weighted vertex at the rightmost end of the level. It only
participates in weight comparisons but cannot be accessed.

The 4th layer belongs to the 4th section level, all are leaf vertices. Their
subtrees are empty, so it is only necessary to establish an ARS in the 1st
layer, the 2nd layer, and the 3rd layer. Moreover, by setting the traversal
condition, the ARS can participate in the weight comparisons without being
accessed, which solves the situation that the right parent vertex does not
exist.

Figure 4: Absolute Right Subtree: the vertices weights are part of the DST
established by the extracted metaknowledge elements of the _Report on the work
of the Government, Year 2019, The Central People’s Government, P. R. China_.

As is shown in Fig. 4, obviously, for all the child vertices $e$ to $n$ of
vertex $c$, there is no right parent, and Eq. 2 is no longer valid. To ensure
that Eq. 2 is always true, $weight(ARS)$ should be a sufficiently large
number. In this work, 0x3F3F3F3F is set to the sufficiently large number,
which not only avoids data overflow but also has the same order of magnitude
as the maximum 0x7FFFFFFF of 32-bit integer data. Due to the introduction of
the ARS $d$, the right parent of the child vertices $e$ to $n$ becomes $d$,
and then the inequality $38<weight(subnodes)<0x3F3F3F3F$ holds.

Therefore, the document structure tree’s minimum data unit is a structure
containing the attributes of the root node and all the child vertices, which
can be defined recursively to realize the document structure tree’s
construction.

⬇

class DST:

def __init__(self):

self.content = [] # Root

self.subtree = [] # Child

##  4 Experiments and Analysis

To evaluate the metaknowledge elements extraction framework in this work,
experiments are designed to verify the feasibility of extracting metaknowledge
elements from multi-modal documents. Here we build governmental documents
dataset GovDoc-CN to evaluate our framework.

###  4.1 Dataset and Setups

In our previous work, we released a multi-modal governmental documents dataset
GovDoc-CN111https://github.com/RuilinXu/GovDoc-CN, which obtains 6 816
government document pages from the Central People’s Government of China (CPGC)
and its subordinate ministries and commissions from the policy document
database of the CPGC.These documents are automatically typeset by LaTeX and
manually annotated in text modal and image modal.

Table 1 shows the statistical information of GovDoc-CN, in which there are 10
kinds of metaknowledge elements, including the sign of issuing authority, the
document number, the title, the addressee, the 1st level section, the 2nd
level section, the 3rd level section, the main text, the issuing authority and
the date of writing.

TABLE I: The Statistical Information Of GovDoc-CN Types | # Elementsa
---|---
Sign of issuing authority | 1 347
Document number | 1 344
Title | 1 359
Addressee | 1 065
1st level section | 5 184
2nd level section | 4 697
3rd level section | 1 415
Issuing authority | 2 073
Date of writing | 1 178
Paragraph | 10 280
aThe number of these elements comes from the text modal and image modal, and
the annotations of the two modalities correspond to each other.

This work uses the same train, evaluation, and test set to train both the text
modal model (BERTBASE+BiLSTM+CRF) and the image modal model
(YOLOv4+PaddleOCR). Using 3 672 document pages for training, 918 pages for
evaluation and 511 pages for testing. The setups are as follows:

  * $\bullet$

_Text Modal._ Using pre-trained Chinese BERTBASE for fine-tuning. Batch size
is set as 4, learning rate is 2e-5, max length is 512. Train 30 epochs.

  * $\bullet$

_Image Modal._ The YOLOv4 is trained with batch size 64, learning rate 1e-5,
and is trained 20 000 iterations.

###  4.2 Metaknowledge Elements Extraction

TABLE II: Reuslts on GovDoc-CN Metaknowledge Elements |  | BERT
---
+BiLSTM
+CRF
| YOLOv4
---
+PaddleOCRa
MEF
Sign of issuing authority | 0.9870 | 0.8308 | 0.9999
Document number | 0.8977 | 0.8588 | 0.8977
Title | 0.9290 | 0.9211 | 0.9299
Addressee | 0.9608 | 0.8090 | 0.9905
1st level section | 0.9608 | 0.7453 | 0.8845
2nd level section | 0.7770 | 0.6384 | 0.8214
3rd level section | 0.7348 | 0.6194 | 0.8235
Issuing authority | 0.8646 | 0.5342 | 0.9347
Date of writing | 0.8462 | 0.8095 | 0.9510
Main text | 0.7097 | 0.6452 | 0.8146
Average | 0.8668 | 0.7412 | 0.9048
aConsidering OCR mismatch.
Figure 5: Examples of original document pages, annotated ground truth pages,
and outputs of the MEF.

The metaknowledge elements extraction results on GovDoc-CN has been shown in
Table 2. Micro-F1 Score are used for evaluating our framework. Compared with
the single-modal approaches BERT+BiLSTM+CRF and YOLOv4+PaddleOCR, our
framework MEF reaches better performance in average F1-Score ( +3.80%
BERT+BiLSTM+CRF, +16.36% YOLOv4+PaddleOCR ). Table 2 also shows that the two
single-modal approaches of text and image complement each other in extracting
various metaknowledge elements. The example outputs are shown in Fig. 5.

As the layout structure of official documents is relatively simple, to verify
the generalization ability of MEF to extract metaknowledge elements of complex
documents, we originally planned to use large-scale multi-modal document
layout analysis dataset DocBank [25] for evaluation.

Figure 6: The metaknowledge elements extraction results on DocBank_50K.

However, when using the dataset, we find that the computer annotating approach
has a terrible impact on the data quality, and there is a lot of dirty data
mixed in the bounding boxes of the image modal annotations. In the text modal
annotations, the dataset only gives the label of each token but does not
provide the starting and ending position of paragraphs, making it very
difficult to convert these annotated tokens into BIO format data that commonly
used in NLP. Limited by the computing capability, we have to run our
experiments on a subset of 50 000 samples of DocBank. We find that the the
text modal results are deplorable according to either the text classification
or the NER approaches. In the image modal, after a large amount of complex
bounding boxes merging work that has transformed the token-level bounding
boxes into the component-level, we finally get SOTA (Table 3).

As shown in Table 3 and Fig. 6, MEF’s extraction performance of 7 types of
elements exceeds that of LayoutLM trained with 10 times more samples. Due to
issues of the DocBank, MEF’s performance in the text modal is not as good as
pleasant. Nevertheless, considering its performance on GovDoc-CN, we believe
that switching to a better multi-modal document dataset will achieve more
ideal results. Unfortunately, there is currently a lack of relevant datasets.
The layout analysis datasets represented by PubLayNet [26] have only a single
modal of image. Therefore, we will work harder to strengthen the weak links in
the next work and expand GovDoc-CN into a multilingual and multi-modal
document dataset with more document types.

TABLE III: Reuslts on DocBank_50K | Metaknowledge
---
Elements
| MEF
---
(Image Modal)
| Pre-trained
---
LayoutLMa
Abstract | 0.9693 | 0.9537
Author | 0.9778 | 0.9423
Caption | 0.9912 | 0.9788
Equation | 0.9858 | 0.9346
Figure | 0.9934 | 0.9941
Footer | 0.9874 | 0.9826
List | 0.9481 | 0.8699
Paragraph | 0.9665 | 0.9849
Reference | 0.8461 | 0.9643
Section | 0.9977 | 0.9694
Table | 0.6764 | 0.8175
Title | 0.9834 | 0.9999
a Results come from Ref. [25], the model was trained and evaluated on the
complete DocBank.

###  4.3 Generating Metaknowldge

The previous processes have acquired the hierarchical structure of documents,
that is, the typologies of documents metaknowledge. However, the semantic
information has not been linked to document metaknowledge yet. Therefore, this
work picks Ref. [27] from DocBank_50K as an example to demonstrate the whole
process of how to generate metaknowledge from documents.

The metaknowledge elements are extracted through MEF and organized by the DST
model. Furthermore, entity-relation triples, or the Structure Contextual
Triples called in Ref. [28], are extracted from the paragraphs and linked to
the sections or subsections they belong to, and then we get the metaknowledge
of Ref. [27]. The metaknowledge is saved in the Neo4j database, which is shown
in Fig. 7(a).

(a) Metaknowledge of Ref. [27] which is displayed in Neo4j

(b) The hierarchical structure of Ref. [27]’s metaknowledge

Figure 7: An example: metaknowledge of Ref. [27]

We explain the metaknowledge of Ref. [27] in Fig. 7(b). Metaknowledge is a
graph structure with hierarchical characteristics, which is basically
consistent with the directory structure of documents. In Fig. 7(b), the
hierarchical characteristics are represented by three layers: the 1st layer,
the title; the 2nd layer, the sections; and the 3rd layer, the subsections. In
the 1st layer, metaknowledge elements describe the attributes of documents
such as the authors, the institutes, and the supporters. These metaknowledge
elements are represented as a form similar to triple-based knowledge. In the
2nd and 3rd layers, we extract entities and relations from paragraphs, which
consists of triple-based knowledge, and then linked to their nearest section
or subsection they belong to.

This work recognizes that the citation network is also well-organized in
metaknowledge. As shown in Fig. 8, all the references of Ref. [27] are
extracted and organized by hierarchical structure. Compared with the current
citation network analysis research, the citation network based on
metaknowledge no longer takes the authors as the main research target, but
pays more attention to the knowledge structure and knowledge itself. Each
reference is a document that can generate metaknowledge. Therefore, a large-
scale metaknowledge network can be constructed by converting a large number of
references into metaknowledge.

Figure 8: The reference network of Ref. [27] constructed by metaknowledge.
Figure 9: The task flow of metaknowledge applications. The horizontal axis
represents the task flow of knowledge and metaknowledge, and the vertical axis
represents the low-to-high levels of knowledge and metaknowledge.

##  5 Task Flow Of Metaknowledge Applications

Metaknowledge is the knowledge about knowledge, in other words, metaknowledge
realizes the organization and management of specific knowledge. Under the
metaknowledge empowerment, the knowledge can be effectively used at a higher
level. We analyze the whole task flow of current knowledge engineering
applications, inspired by that, we propose a novel task flow of metaknowledge
applications (Fig. 9).

Fig. 9 shows the task flow of knowledge and metaknowledge from two different
levels. From the perspective of tasks, the knowledge application task flow can
be divided into four parts: (1) Knowledge Extraction, including named entity
recognition, relation extraction, attribute extraction, etc. (2) Knowledge
Representation. Generally, it has two types: one is based on entity-relation
triples, where the approaches include TransX [29, 30, 31], Word2Vec [32],
Sen2Vec [33], Para2Vec [34]; the other type is based on knowledge networks,
where the approaches include Node2Vec [35], DeepWalk [36], Graph Neural
Networks (GNNs) [37], etc. (3) Knowledge Base Construction, including entity
alignment, entity disambiguation, ontology construction, knowledge reasoning,
etc. (4) Knowledge Utilization, including intelligent retrieval, knowledge
recommendation, etc.

The metaknowledge application task flow can also be divided into four parts:
(1) Metaknowledge Extraction, the data generally comes from multiple modes,
but the approaches basically include document layout analysis, document
structuring, structure contextual triples extraction, etc. (2) Metaknowledge
Representation. Considering metaknowledge is weighted and directional graph
data, theoretically the approaches such as graph convolutional network (GCN)
[38, 39] and graph attention network (GAT) [40] could be well-performed on
this task. (3) Metaknowledge Networks Construction. Its main task is building
a network from large amounts of metaknowledge, using approaches such as graph
fusion, graph linking, etc. (4) Metaknowledge Utilization, which includes
Metaknowledge-based Question Answering (MbQA) and Metaknowledge-based Document
Understanding (MbDU), among which MbQA seems to be a combination of knowledge
base question answering (KBQA) [41, 42] and document-based question answering
(DBQA) [43, 44], it considers both semantic logic of the knowledge and
hierarchical structure of the document.

From the perspective of levels (Fig. 9), metaknowledge applications is higher,
more holistic and more macroscopic than knowledge applications. The
information elements of knowledge are entities and the relations (entity-
relation triples), while the information elements of metaknowledge should be
knowledge and the relations between knowledge (structure contextual triples).
Specifically, metaknowledge is the weighted and directional graph data, which
could have a better performance than the triple-based knowledge bases on tasks
such as question answering, machine reading comprehension, etc. Knowledge
applications realize the management of information, similarly, metaknowledge
applications will realize the management and organization of knowledge.

##  6 Conclusion

In this work, we introduce the concept of metaknowledge to knowledge
engineering researches, propose a framework MEF to extract metaknowledge
elements from multi-modal documents, and the Document Structure Tree model to
organize metaknowledge with document topological and hierarchical features.
Experiments and analysis demonstrate the effectiveness of MEF, an
metaknowledge example of Ref. [27] is given to analyze what metaknowledge
actually is and how is it organized. Finally, our work proposes the task flow
of metaknowledge applications, analyzes the tasks from upstream to downstream
that metaknowledge is utilized. Future work will focus on metaknowledge
representation learning, metaknowledge network construction, and how to verify
the functionality of metaknowledge.

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# Manifestation of superfluidity in atom-number-imbalanced two-component Bose-
Einstein condensates

S. M. Al-Marzoug1,2, B. B. Baizakov3,4, U. Al Khawaja5 and H. Bahlouli1,2 1
Interdisciplinary Research Center for Intelligent Secure Systems, King Fahd
University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia,
2 Physics Department, King Fahd University of Petroleum and Minerals, Dhahran
31261, Saudi Arabia,
3 Physical-Technical Institute, Uzbek Academy of Sciences, 100084, Tashkent,
Uzbekistan,
4 Institute of Theoretical Physics, National University of Uzbekistan,
Tashkent 100174, Uzbekistan,
5 Department of Physics, School of Science, The University of Jordan, Amman,
11942, Jordan.

###### Abstract

Superfluid and dissipative regimes in the dynamics of a two-component quasi-
one-dimensional Bose-Einstein condensate (BEC) with unequal atom numbers in
the two components have been explored. The system supports localized waves of
the symbiotic type owing to the same-species repulsion and cross-species
attraction. The minority BEC component moves through the majority component
and creates excitations. To quantify the emerging excitations we introduce a
time dependent function called disturbance. Through numerical simulations of
the coupled Gross-Pitaevskii equations with periodic boundary conditions, we
have identified a critical velocity of the localized wave, above which a
transition from superfluid to dissipative regime occurs, as evidenced by a
sharp increase in the disturbance function. The factors responsible for the
discrepancy between the actual critical velocity and the speed of sound,
expected from theoretical arguments, have been discussed.

###### pacs:

67.85.De, 03.75.Kk, 05.30.Jp

##  I Introduction

Superfluidity is the ability of a liquid to flow through tight tubes or narrow
slits without dissipation. The experimental discovery kapitza1938 ; allen1938
and theoretical explanation landau1941  of this macroscopic quantum phenomenon
were among the most important achievements of 20th-century physics. While
initial studies were restricted to liquid helium, superfluid properties were
recently observed and explored in a dilute gas of bosons, called Bose-Einstein
condensates (BEC) pitaevskii-book ; barenghi-book . The advantage of ultracold
gases in studies of superfluidity over liquid helium is that they are
analytically tractable, weakly interacting systems, and free of surface
tension effects.

The phenomenon of superfluidity in quantum liquids and gases has been studied
in different contexts. Among them, the breakdown of superfluidity when the
flow velocity reaches some critical value is particularly intriguing. The
physics behind the suppression of superfluidity and transition to a usual
dissipative regime above the critical velocity is linked to the generation of
excitations in the system such as phonons, rotons, and vortices when the
liquid flows at velocities exceeding some critical value $v_{c}$, which was
predicted by Landau landau1941 ; lifshitz-book  $v_{c}={\rm
min}\left(\epsilon(p)/p\right)$, with $\epsilon(p)$ being the energy of the
excitation with momentum $p$. Measurements of the critical velocity in BEC by
laser stirring have confirmed that the superfluidity indeed breaks down if the
laser beam moves faster than a certain finite speed raman1999 ; onofrio2000 ;
engels2007 . However, the measured values of the critical velocity ($\sim$ 1.6
mm/s) in the experiment with sodium BEC raman1999  appeared to be smaller than
the speed of sound ($\sim$ 6.2 mm/s), expected from theoretical arguments. The
diameter of the laser spot in the experiment was macroscopic ($\sim
13\,\mu{\rm m}$), exceeding the intrinsic length scale of the condensate, so
called healing length $\xi=1/\sqrt{8\pi a_{s}n}\sim 0.3\,\mu{\rm m}$, where
$a_{s}=52\,a_{B}$ \- is the two-body $s$ \- wave scattering length in units of
Bohr radius, $n=1.5\times 10^{14}$ cm-3 \- is the peak density of the
condensate. Under these conditions, vortex shedding by the moving object
kwon2015 ; kokubo2024 , rather than the emission of phonons, was a more
realistic phenomenon. Later investigations have revealed several factors which
could be responsible for the observed discrepancy, such as the size and
strength of the probe object desbuquois2012 ; kiehn2022 ; kwak2023 , the
intrinsic nonlinearity of BEC hakim1997 ; pavloff2002 ; leszczyszyn2009 ;
abdullaev2012 , the system’s dimensionality astrakharchik2004 , etc. All
arguments mentioned indicate the need for further research on the origin of
the critical velocity leading to the breakdown of superfluidity in quantum
gases. This is necessary to gain a better understanding of this macroscopic
quantum phenomenon.

This work aims to explore the manifestation of superfluid properties in the
dynamics of quasi-1D binary quantum gases. In our setting, the mixture
condensate can support stable localized waves of the symbiotic type
adhikari2005 ; perezgarcia2005  due to same-species repulsion and cross-
species attraction. We consider a significant disparity in atom numbers
between the components and a toroidal trap potential for the binary
condensate. Under these conditions, the smaller localized component appears
immersed in the larger extended component, distributed over the whole
integration domain. The minority component is propelled by applying a
potential that only affects this component. The majority component is
subjected to a uniform (flat) potential, while the minority component
experiences a harmonic potential. This causes the minority component to
oscillate, similar to a pendulum, with zero velocity at the turning points and
maximum velocity at the center. By adjusting the amplitude of these
oscillations, we can observe different scenarios where the majority component
is either in a superfluid state or not. In cases where the majority component
is not in a superfluid state, a sound wave is generated, and the energy of the
oscillating “pendulum” is transformed into sound. In fact, suppression of
superfluidity and transition to a dissipative regime occurs when the velocity
of the probe object (localized smaller component) reaches a critical value,
leading to the emergence of density waves superimposed on the uniform
background of the larger component. We introduce a measure called disturbance
to quantify the emerging excitations in the majority component, whose
superfluidity is being tested.

##  II Model and governing equations

We use the following coupled dimensionless Gross-Pitaevskii equations (GPE) to
describe the dynamics of a two-component BEC in a quasi-1D trap potential

$i\frac{\partial\psi_{i}}{\partial
t}+\frac{1}{2}\frac{\partial^{2}\psi_{i}}{\partial
x^{2}}+(g_{i}|\psi_{i}|^{2}+g_{ij}|\psi_{j}|^{2})\psi_{i}=V_{i}\psi_{i},\qquad
i,j=1,2,\qquad i\not=j,$ (1)

where $\psi_{i}(x,t)$ are the mean field wave functions of the condensate
components, $g_{i},g_{ij}$ are the same-species and cross-species atomic
interactions coefficients respectively, $V_{i}$ is external potential for the
$i$-th component. Specifically, we use a uniform potential for the majority
component and a harmonic trap for the minority component. Such a component-
selective potential was employed in studies of two-vortex collisions in
particle-imbalanced heteronuclear BEC mixtures richaud2023 . The dimensionless
quantities in these equations are scaled using the frequency of the radial
confinement $\omega_{\bot}$, atomic mass $m$ and radial harmonic oscillator
length $l_{\bot}=\sqrt{\hbar/m\omega_{\bot}}$ as follows: time $t\rightarrow
t\omega_{\bot}$, space $x\rightarrow x/l_{\bot}$, wave function
$\psi_{i}\rightarrow\sqrt{2|a_{s}|}\,\psi_{i}$, with $a_{s}$ being the atomic
$s$-wave scattering length. When considering imbalanced settings with
different number of atoms in the components
$N_{1,2}=\int|\psi_{1,2}(x)|^{2}dx$, we shall denote $\psi_{1}(x,t)$ and
$\psi_{2}(x,t)$ as minority and majority components, respectively. The system
of coupled GPE (1) is suitable for the description of homonuclear BEC
mixtures, such as two internal states of 87Rb egorov2013  or 39K semeghini2018
. In these mixtures, all the atoms belong to the same element but can occupy
two different internal spin states. We use the periodic boundary conditions
for the GPE (1) implying the condensate is held in a toroidal trap potential
similar to that reported in ryu2007 ; ramanathan2011 .

##  III Numerical simulations

Numerical simulations start with the creation of a localized state in the
particle imbalanced ($N_{1}\not=N_{2}$) binary condensate with repulsive
intra-component ($g_{1}<0,g_{2}<0$) and attractive inter-component
($g_{12}>0$) interactions, as shown in Fig. 1a. The density of the majority
component $|\psi_{2}|^{2}$ beyond the space occupied by the localized wave
$|\psi_{1}|^{2}$ is relatively small and uniform. Here, we are interested in
the superfluid and dissipative behaviors exhibited by the majority component.

a) b)

Figure 1: a) The initial state of the binary condensate with same-species
repulsion ($g_{1}=g_{2}=-1$) and cross-species attraction
($g_{12}=g_{21}=1.05$) obtained by numerical solution of the GPE (1) with
periodic boundary conditions. The smaller component $\psi_{1}$ forms a
localized wave while the larger component $\psi_{2}$ spreads over the whole
integration domain. b) The center-of-mass position of the minority component
$\psi_{1}$, initially shifted by $\delta x$ from the minimum of the harmonic
trap $V_{1}(x)$ and released, performs oscillations near the origin. At
moderate displacement ($\delta x=\pi$) it shows fully conservative/superfluid
dynamics (red), while strong displacement ($\delta x=3\pi$) leads to damped
oscillations (purple) evidencing the absence of superfluidity. Parameter
values: $N_{1}=80$, $N_{2}=100$, $V_{1}(x)=0.001\,x^{2},V_{2}(x)=0$.

In the next step, we displace the minority component by $\delta x$ from its
initial position at $x=0$ and let it oscillate in the harmonic potential
$V_{1}(x)=\beta x^{2}$, as illustrated in Fig. 1b. The center-of-mass position
of the minority component was evaluated according to its definition

$\eta(t)=\frac{1}{N_{1}}\int_{-L/2}^{L/2}x|\psi_{1}(x,t)|^{2}dx,$ (2)

where $\psi_{1}(x,t)$ is the solution of GPE (1) at the given time instance.
By analyzing the oscillation dynamics of the minority component for different
initial displacements we have revealed a critical value $\delta x_{cr}\sim
2.5\,\pi$ below which the dynamics are fully conservative, while larger
displacements demonstrate damped oscillations, as shown in Fig. 1b
respectively for $\delta x=\pi$, and $\delta x=3\pi$. These numerical
experiments emulate the shuttle motion of a probe object or laser beam in a
superfluid. It is natural to expect that at sufficiently large oscillation
amplitudes, the probe object acquires a velocity greater than superfluid
critical velocity ($v>v_{c}$) near the minimum of the harmonic trap at $x=0$,
producing excitations in the condensate. Table 1 presents the velocity of the
localized minority component when it passes through the minimum of the
harmonic trap for different initial displacements

Initial shift, $\delta x$ | $\pi$ | $1.5\,\pi$ | $2\,\pi$ | $2.5\,\pi$ | $3\,\pi$
---|---|---|---|---|---
Max. velocity, $v$ | 0.10 | 0.15 | 0.19 | 0.24 | 0.29
Table 1: The maximal velocity $v$ acquired by the minority component in the
harmonic trap for different initial shifts $\delta x$, according to numerical
solution of the GPE (1).

We introduce a time-dependent function called disturbance to quantify the
generation and growth of excitations in the background condensate of the
majority component paris-mandoki2017

$D(t)=\int_{C}(|\psi_{2}(x,t)|^{2}-|\psi_{2}(x,0)|^{2})^{2}\,dx,$ (3)

where the integration is performed over the spatial domain with vanishing
amplitude of the smaller component $C\in\left[|\psi_{1}(x,t)|\simeq 0\right]$.
The latter condition ensures we consider only the excitations on top of the
uniform part of the repulsive condensate $\psi_{2}$. Figure 2 shows the growth
rate of excitations for various initial displacements of the minority
component in relation to the minimum of the harmonic trap.

a) b)

Figure 2: a) Growth of excitations in the majority component ($\psi_{2}$)
during oscillations of the minority component ($\psi_{1}$) in the harmonic
trap for different initial displacements $\delta x$ according to Eq. (3). A
sharp transition from superfluid to dissipative regime is observed near the
critical shift $\delta x_{c}\simeq 2.5\,\pi$. b) The integral disturbance
${\cal D}=\int\limits_{0}^{t_{f}}D(t)dt$ as a function of maximal velocity $v$
attained by the minority component during oscillations in the harmonic trap
($t_{f}=800$). Data for the velocity (red points) are taken from Table 1,
while the blue curve represents the exponential model ${\cal D}(v)=\alpha
e^{\gamma\,v}$ with $\alpha=0.009,\gamma=32.4$.

As can be seen from Fig. 2 a, the movement of the minority component doesn’t
produce excitations in the majority component, thus the disturbance function
Eq. (3) does not grow with time, if its initial position is not sufficiently
far from the origin ($\delta x<2.5\,\pi$). In contrast, at a greater initial
shift ($\delta x=3\,\pi$), the disturbance function rapidly grows with time,
evidencing the vigorous generation of excitations. These excitations in the
majority component appear as density modulations (sound waves) on the
initially uniform domain of $\psi_{2}$. It is reasonable to suggest that the
generation rate of quasiparticles or density waves starts to grow
exponentially once the velocity of the probe object exceeds the critical
value. Figure 2b shows the behavior of the integral disturbance ${\cal D}$ for
different maximal velocities of the minority component moving through the
majority component. As can be seen from this figure, numerically obtained data
(red points) nicely fit with the exponential model ${\cal D}(v)=\alpha
e^{\gamma\,v}$.

Different dynamical regimes can be explored by altering either the initial
position or the initial velocity of the probe object in the harmonic trap. The
simulation results presented in Fig. 3 demonstrate the transition from the
superfluid regime to the dissipative regime when the initial displacement of
the minority component exceeds the critical value. The generation of
excitations occurs during time intervals when the localized minority component
repeatedly passes the minimum point of the harmonic potential with
supercritical velocity. It is evident that the density modulations on the
background condensate $|\psi_{2}(x,t)|^{2}$ do not emerge when the minority
component undergoes small-amplitude oscillations and reaches subcritical
velocity near the bottom of the parabolic trap, as shown in Fig. 3 a,c.
However, during large-amplitude oscillations, the probe object reaches
supercritical velocity, causing the density modulations on
$|\psi_{2}(x,t)|^{2}$ to strongly amplify (see Fig. 3 b,d).

a) b)

c) d)

Figure 3: The density modulations on the background condensate $\psi_{2}(x,t)$
do not appear when the minority component performs small amplitude
oscillations ($\delta x=\pi$), attaining subcritical velocity $v=0.1$ near the
bottom of the parabolic trap (a, c). At large amplitude oscillations ($\delta
x=3\pi$) the probe object attains supercritical velocity $v=0.29$, thus the
density modulations strongly amplify (b, d). To highlight the density waves in
the condensate, only the low-intensity part $|\psi_{2}(x,t)|^{2}<2$ has been
shown (a, b).

The existence of a definite superfluid critical velocity $v_{c}$ can be
demonstrated by examining the long-time evolution of the oscillating minority
component whose initial condition was in the dissipative domain. This implies
that it is far shifted from the minimum of the harmonic trap and initially
performs large amplitude oscillations, thus attaining a supercritical velocity
($v>v_{c}$) at the origin. It is reasonable to expect, that after gradually
losing its kinetic energy due to generation of excitations in the larger
component, the probe object slows down attaining the subcritical velocity
($v<v_{c}$), and the superfluid regime is reestablished. To illustrate this
idea in Fig. 4 we show the long-time evolution of the center-of-mass position
and velocity of the minority component for two initial displacements,
corresponding to the superfluid and dissipative regimes. As can be seen from
this figure, at a smaller displacement ($\delta x=\pi$), the localized wave
undergoes free oscillations with a constant amplitude, indicating the presence
of the superfluid regime (blue dashed line). On the other hand, a larger
displacement ($\delta x=3\pi$) results in damped oscillations, suggesting the
onset of the dissipative regime (red solid line). Subsequently, after
evolution time $t>3000$ the superfluid regime is reestablished, albeit with
some reduction in energy of the minority component due to the excitations
created in the majority BEC component.

a) b)

Figure 4: Oscillations of the center-of-mass position $\eta(t)$ (a) and
velocity $d\eta/dt$ (b) of the minority component, which was initially shifted
from the minimum of the harmonic trap, according to Eq. (2). At a smaller
displacement ($\delta x=\pi$), the localized wave undergoes free oscillations
with a constant amplitude, indicating the presence of the superfluid regime
(blue dashed line). In comparison, a larger displacement ($\delta x=3\pi$)
leads to damped oscillations, indicating the onset of the dissipative regime
(red solid line). Later, the superfluid regime is reestablished, with some
reduction in energy of the minority component due to the quasiparticles
created in the majority component.

Numerical simulations were performed using the methods of split-step fast
Fourier transform agrawal-book ; numrecipes  and a Runge-Kutta in interaction
picture hult2007 . The integration domain of length $L\in[-6\pi,6\pi]$ has
been employed to accommodate 1024 Fourier modes with a corresponding space
step $\Delta x=0.036$. The time step was $\Delta t=0.0005$. The periodic
boundary conditions $\psi_{i}(-L/2,t)=\psi_{i}(L/2,t)$, $i=1,2$, were adopted
to emulate a toroidal trap configuration. The stationary states of the mixture
condensate described by GPE (1) were constructed using the Pitaevskii damping
procedure choi1998 . The stability of ground-state solutions was confirmed by
observing the long-term evolution of weakly perturbed initial wave profiles.
Different initial conditions for the governing GPE were prepared by shifting
the position of the minority component in the harmonic trap relative to its
minimum at $x=0$.

Now we will estimate the parameter values used for numerical simulations in
physical units and compare them with previously reported experimental data.
Let us consider a BEC of 87Rb atoms prepared in two internal states
$|F=1,m_{F}=-1\rangle$ and $|F=2,m_{F}=1\rangle$. The $s$-wave scattering
lengths of rubidium atoms in these ground hyperfine states are almost equal
$a_{1}\simeq a_{2}\simeq 100\,a_{B}$, while the inter-component coefficient
$a_{12}$ can be independently tuned. The transfer of atoms from one state to
the other can be induced by coherent electromagnetic radiation until the
desired populations $N_{1}$, $N_{2}$ in corresponding states $\psi_{1}(x,t)$
and $\psi_{2}(x,t)$ are achieved. The mixture condensate is supposed to be
held in a toroidal trap with a transverse confinement frequency
$\omega_{\bot}=2\pi\times 100$ Hz. Then the units of time and space are given
by $\tau=1/\omega_{\bot}\simeq 1.6$ ms,
$a_{\bot}=\sqrt{\hbar/m\omega_{\bot}}\simeq 1\,\mu$m, respectively. The length
of the integration domain (circumference of the toroidal trap) is equal to
$L=12\,\pi a_{\bot}\simeq 41\,\mu$m. The density of the uniformly distributed
condensate’s majority component consisting of ${\cal N}_{1}=10^{4}$ rubidium
atoms, required for calculation of the sound velocity and healing length is
$n_{1}={\cal N}_{1}/La_{\bot}^{2}\simeq 2.1\times 10^{20}$ m-3. Using these
data one can estimate the speed of sound
$c_{s}=(4\pi\hbar^{2}|a_{s}|n_{1}/m^{2})^{1/2}\simeq 2.7$ mm/s and the healing
length $\xi=1/\sqrt{8\pi|a_{s}|n_{1}}\simeq 0.19\,\mu{\rm m}$.

It is important to note that the above presented values are associated with
uniformly distributed BEC components, taking place for $g_{12}=0$. A localized
state of symbiotic type emerges when there is attraction between the
components ($g_{12}=-1.05$), leading to a decrease in the density of the
majority component away from the minority one. For a particular case shown in
Fig. 1a the density is reduced by approximately three times. Thus we obtain
the corrected values $c_{s}\simeq 1.6$ mm/s and $\xi\simeq 0.32\,\mu{\rm m}$.

It is evident from Fig. 2b that the transition from a superfluid to a
dissipative regime occurs near the critical velocity $\sim 0.1$, which in
physical units corresponds to $v_{c}\simeq 0.07$ mm/s. Therefore the critical
velocity leading to suppression of superfluidity in BEC appears to be
significantly smaller than the sound velocity, their ratio being
$v_{c}/c_{s}\simeq 0.04$. Similar ratios were reported in the experiments with
sodium onofrio2000  $v_{c}/c_{s}\simeq(0.07\pm 0.02)$ and rubidium condensates
raman1999  $v_{c}/c_{s}\simeq 0.25$.

The reasons why the theoretical prediction overestimates the actual critical
velocity were addressed in many previous works (see e.g. kiehn2022 ). Among
them are the inhomogeneity of the medium, the influence of nonlinearity,
macroscopic size of the moving object and dimensionality of the system. All
quoted factors are inherent to our model. In particular, the size of the
localized wave (moving minority component) calculated for the symmetric
setting ismailov2024  ($N=N_{1}=N_{2},\ g=g_{1}=g_{2}$)
$a_{0}=\sqrt{2\pi}/[(g+g_{12})N]\simeq 1.2\,\mu$m was greater than the healing
length $\xi\simeq 0.3\,\mu{\rm m}$. Thus the probe object in our model is
macroscopic. For now, uncovering the physical mechanisms behind the onset of
the dissipative regime in a BEC superfluid and defining the associated
critical velocity is still a challenge.

##  IV Conclusions

Through numerical simulations, we have studied the superfluid and dissipative
regimes in the dynamics of a binary Bose-Einstein condensate where the
localized minority component, acting as a single entity (probe object), moves
through the majority component and hence creating excitations. We have
designed a configuration in which the minority component is surrounded by the
majority one and experiences regular back-and-forth movements under the action
of a harmonic trap. Its velocity varies from zero at classical turning points
to a maximum value at the lowest point of the harmonic trap. When the velocity
of motion exceeds some critical value, a strong dissipation occurs,
accompanied by a production of quasiparticles in the majority component. After
multiple oscillations, the minority component experiences energy losses and
slowing down, which eventually cause the return to a non-dissipative mode. It
was found that the actual value of the critical velocity leading to the
suppression of superfluidity in BEC is smaller than the speed of sound
expected from theoretical arguments. The possible reasons for the observed
discrepancy were analyzed.

## Acknowledgements

The authors gratefully acknowledge the support provided by the
Interdisciplinary Research Center for Intelligent Secure Systems (IRC-ISS) at
King Fahd University of Petroleum and Minerals (KFUPM) in funding this work
through project No. INSS2302.

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