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# Evaluating Tag Recommendations for E-Book Annotation Using a Semantic
Similarity Metric

Emanuel Lacic  Know-Center GmbHGrazAustria<EMAIL_ADDRESS>,  Dominik
Kowald  Know-Center GmbHGrazAustria<EMAIL_ADDRESS>,  Dieter Theiler
Know-Center GmbHGrazAustria<EMAIL_ADDRESS>,  Matthias Traub  Know-
Center GmbHGrazAustria<EMAIL_ADDRESS>,  Lucky Kuffer  HGV
GmbHMunichGermany<EMAIL_ADDRESS>,  Stefanie Lindstaedt  Know-
Center GmbHGrazAustria<EMAIL_ADDRESS>and  Elisabeth Lex  Graz
University of TechnologyGrazAustria<EMAIL_ADDRESS>
(2019)

###### Abstract.

In this paper, we present our work to support publishers and editors in
finding descriptive tags for e-books through tag recommendations. We propose a
hybrid tag recommendation system for e-books, which leverages search query
terms from Amazon users and e-book metadata, which is assigned by publishers
and editors. Our idea is to mimic the vocabulary of users in Amazon, who
search for and review e-books, and to combine these search terms with editor
tags in a hybrid tag recommendation approach. In total, we evaluate 19 tag
recommendation algorithms on the review content of Amazon users, which
reflects the readers’ vocabulary. Our results show that we can improve the
performance of tag recommender systems for e-books both concerning tag
recommendation accuracy, diversity as well as a novel semantic similarity
metric, which we also propose in this paper.

Tag Recommendation; E-Book Annotation; Hybrid Recommendation; Amazon Search
Query Terms; Diversity; Semantic Similarity

††journalyear: 2019††copyright: acmcopyright††conference: REVEAL@RecSys’2019;
September 16-20, 2019; Copenhagen, Denmark††price: 15.00††doi:
xx.xxx/xxxxxx.xxxxxx††isbn: xxx-x-xxxx-xxxx-x/xx/xx

##  1\. Introduction

When people shop for books online in e-book stores such as, e.g., the Amazon
Kindle store, they enter search terms with the goal to find e-books that meet
their preferences. Such e-books have a variety of metadata such as, e.g.,
title, author or keywords, which can be used to retrieve e-books that are
relevant to the query. As a consequence, from the perspective of e-book
publishers and editors, annotating e-books with tags that best describe the
content and which meet the vocabulary of users (e.g., when searching and
reviewing e-books) is an essential task (zubiaga2011tags, ).

Problem and aim of this work. Annotating e-books with suitable tags is,
however, a complex task as users’ vocabulary may differ from the one of
editors. Such a vocabulary mismatch yet hinders effective organization and
retrieval (zhao2010termnecessity, ) of e-books. For example, while editors
mostly annotate e-books with descriptive tags that reflect the book’s content,
Amazon users often search for parts of the book title. In the data we use for
the present study (see Section 2), we find that around 30% of the Amazon
search terms contain parts of e-book titles.

In this paper, we present our work to support editors in the e-book annotation
process with tag recommendations (jaschke2007tag, ; kowald2017tagrec, ). Our
idea is to exploit user-generated search query terms in Amazon to mimic the
vocabulary of users in Amazon, who search for e-books. We combine these search
terms with tags assigned by editors in a hybrid tag recommendation approach.
Thus, our aim is to show that we can improve the performance of tag
recommender systems for e-books both concerning recommendation accuracy as
well as semantic similarity and tag recommendation diversity.

Related work. In tag recommender systems, mostly content-based algorithms
(e.g., (lu2009content, ; lops2013content, )) are used to recommend tags to
annotate resources such as e-books. In our work, we incorporate both content
features of e-books (i.e., title and description text) as well as Amazon
search terms to account for the vocabulary of e-book readers.

Concerning the evaluation of tag recommendation systems, most studies focus on
measuring the accuracy of tag recommendations (e.g., (jaschke2007tag, )).
However, the authors of (belem2013topic, ) suggest also to use beyond-accuracy
metrics such as diversity to evaluate the quality of tag recommendations. In
our work, we measure recommendation diversity in addition to recommendation
accuracy and propose a novel metric termed semantic similarity to validate
semantic matches of tag recommendations.

Approach and findings. We exploit editor tags and user-generated search terms
as input for tag recommendation approaches. Our evaluation comprises of a rich
set of 19 different algorithms to recommend tags for e-books, which we group
into (i) popularity-based, (ii) similarity-based (i.e., using content
information), and (iii) hybrid approaches. We evaluate our approaches in terms
of accuracy, semantic similarity and diversity on the review content of Amazon
users, which reflects the readers’ vocabulary. With semantic similarity, we
measure how semantically similar (based on learned Doc2Vec (le2014distributed,
) embeddings) the list of recommended tags is to the list of relevant tags. We
use this additional metric to measure not only exact “hits” of our
recommendations but also semantic matches.

Our evaluation results show that combining both data sources enhances the
quality of tag recommendations for annotating e-books. Furthermore, approaches
that solely train on Amazon search terms provide poor performance in terms of
accuracy but deliver good results in terms of semantic similarity and
recommendation diversity.

##  2\. Method

In this section, we describe our dataset as well as our tag recommendation
approaches we propose to annotate e-books.

###  2.1. Dataset

Our dataset contains two sources of data, one to generate tag recommendations
and another one to evaluate tag recommendations. HGV GmbH has collected all
data sources111Currently the data used in this study cannot be made publicly
available because of copyright issues, but we will try to provide a public
version of it soon in the future. and we provide the dataset statistics in
Table 1.

Data used to generate recommendations. We employ two sources of e-book
annotation data: (i) editor tags, and (ii) Amazon search terms. For editor
tags, we collect data of 48,705 e-books from 13 publishers, namely Kunstmann,
Delius-Klasnig, VUR, HJR, Diogenes, Campus, Kiwi, Beltz, Chbeck, Rowohlt,
Droemer, Fischer and Neopubli. Apart from the editor tags, this data contains
metadata fields of e-books such as the ISBN, the title, a description text,
the author and a list of BISACs, which are identifiers for book categories.

For the Amazon search terms, we collect search query logs of 21,243 e-books
for 12 months (i.e., November 2017 to October 2018). Apart from the search
terms, this data contains the e-books’ ISBNs, titles and description texts.

Table 1 shows that the overlap of e-books that have editor tags and Amazon
search terms is small (i.e., only 497). Furthermore, author and BISAC (i.e.,
the book category identifier) information are primarily available for e-books
that contain editor tags. Consequently, both data sources provide
complementary information, which underpins the intention of this work, i.e.,
to evaluate tag recommendation approaches using annotation sources from
different contexts.

Data used to evaluate recommendations. For evaluation, we use a third set of
e-book annotations, namely Amazon review keywords. These review keywords are
extracted from the Amazon review texts and are typically provided in the
review section of books on Amazon. Our idea is to not favor one or the other
data source (i.e., editor tags and Amazon search terms) when evaluating our
approaches against expected tags. At the same time, we consider Amazon review
keywords to be a good mixture of editor tags and search terms as they describe
both the content and the users’ opinions on the e-books (i.e., the readers’
vocabulary). As shown in Table 1, we collect Amazon review keywords for 2,896
e-books (publishers: Kiwi, Rowohlt, Fischer, and Droemer), which leads to
33,663 distinct review keywords and on average 30 keyword assignments per
e-book.

Data used to generate recommendations | #
---|---
Number of e-books | 69,451
thereof with editor tags | 48,705
thereof with Amazon search terms | 21,243
thereof with editor tags and Amazon search terms | 497
Number of distinct authors | 25,086
Number of distinct BISACs (= category IDs) | 1,448
Number of distinct editor tags | 114,707
Number of distinct Amazon search terms | 8,240
Data used to evaluate recommendations | #
Number of e-books with Amazon review keywords | 2,896
Number of distinct Amazon review keywords | 33,663
Avg. number of distinct Amazon review keywords per e-book | 30
Table 1. Statistics of our e-book annotation dataset used to generate and
evaluate tag recommendations.

###  2.2. Tag Recommendation Approaches

We implement three types of tag recommendation approaches, i.e., (i)
popularity-based, (ii) similarity-based (i.e., using content information), and
(iii) hybrid approaches. Due to the lack of personalized tags (i.e., we do not
know which user has assigned a tag), we do not implement other types of
algorithms such as collaborative filtering (marinho2008collaborative, ). In
total, we evaluate 19 different algorithms to recommend tags for annotating
e-books.

Popularity-based approaches. We recommend the most frequently used tags in the
dataset, which is a common strategy for tag recommendations
(kowald2016influence, ). That is, a most popular $MP_{Editor}$ approach for
editor tags and a most popular $MP_{Amazon}$ approach for Amazon search terms.
For e-books, for which we also have author (= $MP^{Author}_{Editor}$ and
$MP^{Author}_{Amazon}$) or BISAC (=$MP^{BISAC}_{Editor}$ and
$MP^{BISAC}_{Amazon}$) information, we use these features to further filter
the recommended tags, i.e., to only recommend tags that were used to annotate
e-books of a specific author or a specific BISAC.

We combine both data sources (i.e., editor tags and Amazon search terms) using
a round-robin combination strategy, which ensures an equal weight for both
sources. This gives us three additional popularity-based algorithms (=
$MP_{Combined}$, $MP^{Author}_{Combined}$ and $MP^{BISAC}_{Combined}$).

Similarity-based approaches. We exploit the textual content of e-books (i.e.,
description or title) to recommend relevant tags (cantador2010content, ). For
this, we first employ a content-based filtering approach (balabanovic1997fab,
) based on TF-IDF (ramos2003using, ) to find top-$N$ similar e-books222In our
experiments, we set $N=20$, the minimum document frequency to $10$ and the
minimum word length to $5$.. For each of the similar e-books, we then either
extract the assigned editor tags (= $SIM^{Description}_{Editor}$ and
$SIM^{Title}_{Editor}$) or the Amazon search terms (=
$SIM^{Description}_{Amazon}$ and $SIM^{Title}_{Amazon}$). To combine the tags
of the top-$N$ similar e-books, we use the cross-source algorithm
(bostandjiev2012tasteweights, ), which favors tags that were used to annotate
more than one similar e-book (i.e., tags that come from multiple
recommendation sources). The final tag relevancy is calculated as:

(1) $\displaystyle W_{t_{i}}=|S_{t_{i}}|\cdot\sum\limits_{s_{t_{i}}\in
S}{W_{s_{t_{i}}}}$

where $|S_{t_{i}}|$ denotes the number of distinct e-books, which yielded the
recommendation of tag $t_{i}$, to favor tags that come from multiple sources
and $W_{s_{t_{i}}}$ is the similarity score of the corresponding e-book. We
again use a round-robin strategy to combine both data sources (=
$SIM^{Description}_{Combined}$ and $SIM^{Title}_{Combined}$).

Hybrid approaches. We use the previously mentioned cross-source algorithm
(bostandjiev2012tasteweights, ) to construct four hybrid recommendation
approaches. In this case, tags are favored that are recommended by more than
one algorithm.

Hence, to create a popularity-based hybrid (= $HYB^{MP}$), we combine the best
three performing popularity-based approaches from the ones (i) without any
contextual signal, (ii) with the author as context, and (iii) with BISAC as
context. In the case of the similarity-based hybrid (= $HYB^{SIM}$), we
utilize the two best performing similarity-based approaches from the ones (i)
which use the title, and (ii) which use the description text. We further
define $HYB^{All}$, a hybrid approach that combines the three popularity-based
methods of $HYB^{MP}$ and the two similarity-based approaches of $HYB^{SIM}$.
Finally, we define $HYB^{Best}$ as a hybrid approach that uses the best
performing popularity-based and the best performing similarity-based approach
(see Figure 1 in Section 4 for more details about the particular algorithm
combinations).

##  3\. Experimental Setup

In this section, we describe our evaluation protocol as well as the measures
we use to evaluate and compare our tag recommendation approaches.

###  3.1. Evaluation Protocol

For evaluation, we use the third set of e-book annotations, namely Amazon
review keywords. As described in Section 2.1, these review keywords are
extracted from the Amazon review texts and thus, reflect the users’
vocabulary. We evaluate our approaches for the 2,896 e-books, for whom we got
review keywords. To follow common practice for tag recommendation evaluation
(kowald2015evaluating, ), we predict the assigned review keywords (= our test
set) for respective e-books.

###  3.2. Evaluation Metrics

In this work, we measure (i) recommendation accuracy, (ii) semantic
similarity, and (iii) recommendation diversity to evaluate the quality of our
approaches from different perspectives.

Recommendation accuracy. We use Normalized Discounted Cumulative Gain (nDCG)
(ParraSahebi, ) to measure the accuracy of the tag recommendation approaches.
The nDCG measure is a standard ranking-dependent metric that not only measures
how many tags can be correctly predicted but also takes into account their
position in the recommendation list with length of $k$. It is based on the
Discounted Cummulative Gain, which is given by:

(2) $\displaystyle
DCG@k=\sum\limits_{k=1}\limits^{|r_{b}^{k}|}(\frac{2^{T(k)}-1}{log_{2}(1+k)})$

where $T(k)$ is a function that returns 1 if the recommended tag at position
$i$ in the recommended list is relevant. We then calculate DCG@$k$ for every
evaluated e-book by dividing DCG@$k$ with the ideal DCG value iDCG@$k$, which
is the highest possible DCG value that can be achieved if all the relevant
tags would be recommended in the correct order. It is given by the following
formula (ParraSahebi, ):

(3) $\displaystyle nDCG@k=\frac{1}{|B|}\sum\limits_{b\in
B}{(\frac{DCG@k}{iDCG@k})}$

Semantic similarity. One precondition of standard recommendation accuracy
measures is that to generate a “hit”, the recommended tag needs to be an exact
syntactical match to the one from the test set. When tags are recommended from
one data source and compared to tags from another source, this can be
problematic. For example, if we recommend the tag “victim” but expect the tag
“prey”, we would mark this as a mismatch, therefore being a bad
recommendation. But if we know that the corresponding e-book is a crime novel,
the recommended tag would be (semantically) descriptive to reflect the book’s
content. Hence, in this paper, we propose to additionally measure the semantic
similarity between recommended tags and tags from the test set (i.e., the
Amazon review keywords).

(a) Popularity-based approaches

(b) Similarity-based approaches

(c) Hybrid approaches

Figure 1. Accuracy results with respect to $nDCG$ for (a) popularity-based,
(b) similarity-based and (c) hybrid tag recommendation approaches. All results
are reported for different numbers of recommended tags (i.e., $k\in[1,10]$).

Over the last four years, there have been several notable publications in the
area of applying deep learning to uncover semantic relationships between
textual content (e.g., by learning word embeddings with Word2Vec
(mikolov2013distributed, ; kenter2015short, )). Based on this, we propose an
alternative measure of recommendation quality by learning the semantic
relationships from both vocabularies and then using it to compare how
semantically similar the recommended tags are to the expected review keywords.
For this, we first extract the textual content in the form of the description
text, title, editor tags and Amazon search terms of e-books from our
dataset333We also pre-process the extracted text by removing bad characters,
stop words and changing all words to lowercase.. We then train a Doc2Vec
(le2014distributed, ) model444We use the DBOW approach with a size of $50$ for
the latent vector representation, negative sampling of $10$, a learning rate
of $0.025$ and train it for $10$ epochs. on the content. Then, we use the
model to infer the latent representation for both the complete list of
recommended tags as well as the list of expected tags from the test set.
Finally, we use the cosine similarity measure to calculate how semantically
similar these two lists are.

Recommendation diversity. As defined in (SmythMcClave01, ), we calculate
recommendation diversity as the average dissimilarity of all pairs of tags in
the list of recommended tags. Thus, given a distance function $d(t_{i},t_{j})$
that corresponds to the dissimilarity between two tags $t_{i}$ and $t_{j}$ in
the list of recommended tags, $D$ is given as the average dissimilarity of all
pairs of tags:

(4) $\displaystyle D@k=\frac{1}{|B|}\sum\limits_{b\in
B}{(\frac{1}{k\cdot(k-1)}\sum\limits_{i\in R}{\sum\limits_{j\in
r_{b}^{k},j\neq i}{}{d(t_{i},t_{j})}})}$

where $|B|$ is the number of evaluated e-books and the dissimilarity function
is defined as $d(t_{i},t_{j})=1-sim(t_{i},t_{j})$. In our experiments, we use
the previously trained Doc2Vec model to extract the latent representation of a
specific tag. The similarity of two tags $sim(t_{i},t_{j})$ is then calculated
with the Cosine similarity measure using the latent vector representations of
respective tags $t_{i}$ and $t_{j}$.

##  4\. Results

Concerning tag recommendation accuracy, in this section, we report results for
different values of $k$ (i.e., number of recommended tags). For the beyond-
accuracy experiment, we use the full list of recommended tags (i.e., $k=10$).

###  4.1. Recommendation Accuracy Evaluation

Figure 1 shows the results of the accuracy experiment for the (i) popularity-
based, (ii) similarity-based, and (iii) hybrid tag recommendation approaches.

Popularity-based approaches. In Figure 1a, we see that popularity-based
approaches based on editor tags tend to perform better than if trained on
Amazon search terms. If we take into account contextual information like BISAC
or author, we can further improve accuracy in terms of $nDCG$. That is, we
find that using popular tags from e-books of a specific author leads to the
best accuracy of the popularity-based approaches. This suggests that editors
and readers do seem to reuse tags for e-books of same authors. If we use both
editor tags and Amazon search terms, we can further increase accuracy,
especially for higher values of $k$ like in the case of $MP_{Combined}$. This
is, however, not the case for $MP^{BISAC}_{Combined}$ as the accuracy of the
integrated $MP^{BISAC}_{Amazon}$ approach is low. The reason for this is the
limited amount of e-books from within the Amazon search query logs that have
BISAC information (i.e., only $2.38\%$).

Similarity-based approaches. We further improve accuracy if we first find
similar e-books and then extract their top-$k$ tags in a cross-source manner
as described in Section 2.2.

As shown in Figure 1b, using the description text to find similar e-books
results in more accurate tag recommendations than using the title (i.e.,
<EMAIL_ADDRESS>for $SIM^{Description}_{Editor}$). This is somehow expected
as the description text consists of a bigger corpus of words (i.e., multiple
sentences) than the title. Concerning the collected Amazon search query logs,
extracting and then recommending tags from this source results in a much lower
accuracy performance. Thus, these results also suggest to investigate beyond-
accuracy metrics as done in Section 4.2.

(a) Semantic similarity

(b) Recommendation diversity

Figure 2. Beyond-accuracy evaluation results of our tag recommendation
approaches. We use the list of $10$ recommended tags to calculate the (a)
semantic similarity, and (b) recommendation diversity. We provide the boxplots
and the mean values for the approaches.

Hybrid approaches. Figure 1c shows the accuracy results of the four hybrid
approaches. By combining the best three popularity-based approaches, we
outperform all of the initially evaluated popularity algorithms (i.e.,
<EMAIL_ADDRESS>for $HYB^{MP}$). On the contrary, the combination of the two
best performing similarity-based approaches $SIM^{Description}_{Editor}$ and
$SIM^{Title}_{Editor}$ does not yield better accuracy. The negative impact of
using a lower-performing approach such as $SIM^{Title}_{Editor}$ within a
hybrid combination can also be observed in $HYB^{All}$ for lower values of
$k$. Overall, this confirms our initial intuition that combining the best
performing popularity-based approach with the best similarity-based approach
should result in the highest accuracy (i.e<EMAIL_ADDRESS>for
$HYB^{Best}$). Moreover, our goal, namely to exploit editor tags in
combination with search terms used by readers to increase the metadata quality
of e-books, is shown to be best supported by applying hybrid approaches as
they provide the best prediction results.

###  4.2. Beyond-Accuracy Evaluation

Figure 2 illustrates the results of the experiments, which measure the
recommendation impact beyond-accuracy.

Semantic similarity. Figure 2a illustrates the results of our proposed
semantic similarity measure. To compare our proposed measure to standard
accuracy measures such as $nDCG$, we use Kendall’s Tau rank correlation
(kendall1938new, ) as suggested by (lapata2006automatic, ) for automatic
evaluation of information-ordering tasks. From that, we rank our
recommendation approaches according to both accuracy and semantic similarity
and calculate the relation between both rankings. This results in $\tau=0.743$
with a p-value ¡ $0.00001$, which suggests a high correlation between the
semantic similarity and the standard accuracy measure.

Therefore, the semantic similarity measure helps us interpret the
recommendation quality. For instance, we achieve the lowest $nDCG$ values with
the similarity-based approaches that recommend Amazon search terms (i.e.,
$SIM^{Description}_{Amazon}$ and $SIM^{Title}_{Amazon}$). When comparing these
results with others from Figure 1b, a conclusion could be quickly drawn that
the recommended tags are merely unusable. However, by looking at Figure 2a, we
see that, although these approaches do not provide the highest recommendation
accuracy, they still result in tag recommendations that are semantically
related at a high degree555A semantic similarity of $1.0$ would denote a
semantically (and syntactically) perfect fit of tag recommendations to the
test set. to the expected annotations from the test set. Overall, this
suggests that approaches, which provide a poor accuracy performance concerning
$nDCG$ but provide a good performance regarding semantic similarity could
still be helpful for annotating e-books.

Recommendation diversity. Figure 2b shows the diversity of the tag
recommendation approaches. We achieve the highest diversity with the
similarity-based approaches, which extract Amazon search terms. Their accuracy
is, however, very low. Thus, the combination of the two vocabularies can
provide a good trade-off between recommendation accuracy and diversity.

##  5\. Conclusion and Future Work

In this paper, we present our work to support editors in the e-book annotation
process. Specifically, we aim to provide tag recommendations that incorporate
both the vocabulary of the editors and e-book readers. Therefore, we train
various configurations of tag recommender approaches on editors’ tags and
Amazon search terms and evaluate them on a dataset containing Amazon review
keywords. We find that combining both data sources enhances the quality of tag
recommendations for annotating e-books. Furthermore, while approaches that
train only on Amazon search terms provide poor performance concerning
recommendation accuracy, we show that they still offer helpful annotations
concerning recommendation diversity as well as our novel semantic similarity
metric.

Future work. For future work, we plan to validate our findings using another
dataset, e.g., by recommending tags for scientific articles and books in
BibSonomy. With this, we aim to demonstrate the usefulness of the proposed
approach in a similar domain and to enhance the reproducibility of our results
by using an open dataset.

Moreover, we plan to evaluate our tag recommendation approaches in a study
with domain users. Also, we want to improve our similarity-based approaches by
integrating novel embedding approaches (mikolov2013distributed, ;
kenter2015short, ) as we did, for example, with our proposed semantic
similarity evaluation metric. Finally, we aim to incorporate explanations for
recommended tags so that editors of e-book annotations receive additional
support in annotating e-books (vig2009tagsplanations, ). By making the
underlying (semantic) reasoning visible to the editor who is in charge of
tailoring annotations, we aim to support two goals: (i) allowing readers to
discover e-books more efficiently, and (ii) enabling publishers to leverage
semi-automatic categorization processes for e-books. In turn, providing
explanations fosters control over which vocabulary to choose when tagging
e-books for different application contexts.

Acknowledgments. The authors would like to thank Peter Langs, Jan-Philipp Wolf
and Alyona Schraa from HGV GmbH for providing the e-book annotation data. This
work was funded by the Know-Center GmbH (FFG COMET Program), the FFG Data
Market Austria project and the AI4EU project (EU grant 825619). The Know-
Center GmbH is funded within the Austrian COMET Program - Competence Centers
for Excellent Technologies - under the auspices of the Austrian Ministry of
Transport, Innovation and Technology, the Austrian Ministry of Economics and
Labor and by the State of Styria. COMET is managed by the Austrian Research
Promotion Agency (FFG).

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# Synthetic Hall ladder with tunable magnetic flux

Jeong Ho Han<EMAIL_ADDRESS>Korea Research Institute of Standards
and Science, Daejeon 34113, Korea  Dalmin Bae  Department of Physics and
Astronomy, Seoul National University, Seoul 08826, Korea  Center for
Correlated Electron Systems, Institute for Basic Science, Seoul 08826, Korea
Yong-il Shin<EMAIL_ADDRESS>Department of Physics and Astronomy, Seoul
National University, Seoul 08826, Korea  Center for Correlated Electron
Systems, Institute for Basic Science, Seoul 08826, Korea  Institute of Applied
Physics, Seoul National University, Seoul 08826, Korea

###### Abstract

We describe a synthetic three-leg Hall ladder system with a tunable magnetic
flux for neutral 173Yb atoms in a one-dimensional optical lattice. The ladder
legs are formed by three hyperfine ground spin states of the atoms, and the
complex interleg links are generated through Raman couplings between the spin
states using multiple laser beams. The effective magnetic flux through a
ladder plaquette, $\phi$, is controlled by the angles of the Raman laser beams
with the lattice axis. We investigate the quench dynamics of the Hall ladder
system for $\phi\approx\frac{\pi}{3},\frac{\pi}{2},$ and $\frac{2\pi}{3}$
after a sudden application of the Raman coupling in various interleg link
configurations. The semi-classical trajectory of the atoms in the plane of the
spin composition and lattice position exhibits the characteristic motion for
the effective magnetic field. In a tube configuration with the three legs
cyclically linked, the quench evolution was observed to be substantially
damped, which is attributed to the random flux threading the Hall tube.

###### pacs:

XX.XX.XX

##  I Introduction

Understanding topological states of matter is among the major challenges of
condensed matter physics, with relevance for new classes of materials beyond
the conventional symmetry-breaking paradigm Hasan2010 . Owing to the
capabilities to engineer tunneling amplitudes and address individual lattice
sites, ultracold atoms in optical lattices represent a promising platform for
exploring such topological phases in a defect-free environment Goldman2016 ;
Cooper2019 . One of the prominent examples is the quantum simulation of a
Harper-Hofstadter (HH) Hamiltonian, which is the paradigmatic model for
quantum Hall physics. The HH Hamiltonian was first achieved in two-dimensional
(2D) optical lattice systems using laser-assisted tunneling Aidelsburger2013 ;
Miyake2013  and further realized in novel forms based on so-called synthetic
dimensional lattices Celi2014 ; Mancini2015 ; Stuhl2015 . In the synthetic
dimension framework, a set of atomic internal states is employed as virtual
lattice sites in an extra dimension and an effective gauge potential is
generated by engineering the couplings between the states. Various Hall
ladders Livi2016 ; An2017-1 ; Kolkowitz2017  and topological lattice systems
An2018 ; Meier2016 ; Kang2020 ; Kanungo2021  were successfully demonstrated.
Further, this synthetic dimension approach enables Hall effects to be studied
even under hypothetical situations such as in tube geometries Han2019 ;
Chalopin2020 ; Liang2021  and in four dimensions Lohse2018 .

In our recent work Han2019 , we constructed a synthetic three-leg Hall ladder
for fermionic 173Yb atoms in a one-dimensional (1D) optical lattice, where
three hyperfine ground spin states of atoms were utilized as the indices of
the three legs and coupled to each other through two-photon Raman transitions.
The linking structure of the ladder was altered by controlling the number of
activated Raman couplings, including a tube configuration where the three spin
states were cyclically coupled. In this study, a Hall ladder with an effective
magnetic flux per unit plaquette of $\phi=2\pi/3$ was investigated as a
representative case, featuring the existence of a symmetry-protected 1D
topological phase Barbarino2018 . Such a Hall tube system is known to belong
to a 1D quasicrystal, exhibiting a fractal energy spectrum such as
Hofstadter’s butterfly for varying $\phi$ and carrying a number of
topologically nontrivial states. Therefore, a natural extension for the Hall
ladder system is to attain magnetic flux tunability for a full study of the
quantum Hall physics.

In this paper, we present a detailed description of the synthetic three-leg
Hall ladder system and experimentally demonstrate the flux tunability of the
system. We investigate the quench evolution of the system after a sudden
application of the Raman couplings for different ladder structures and
effective magnetic fluxes. The semi-classical trajectories of the atoms are
reconstructed in the synthetic space composed of real and synthetic
dimensions, and the effect of the tunable magnetic flux is demonstrated by
comparing the measured dynamics with the numerical simulation results
calculated from the corresponding Bloch equations. For the periodic boundary
ladder, we observe a large damping in the spin evolution. In the tube
geometry, there is an additional magnetic flux, which pierces the tube cross-
section and affects the quasi-momentum along the circumference of the tube
Laughlin1981 ; Luo2020 ; Liang2021 . We attribute the large damping to the
random flux through the tube owing to our Raman beam configuration, which is
also compared with numerical simulation results. We discuss various options
for controlling the longitudinal flux in the experiment.

The remainder of this paper is structured as follows. In Sec. II, we introduce
a model Hamiltonian governing the Hall ladder system for open and periodic
boundary conditions. In Sec. III, we describe the experimental protocol to
generate the synthetic Hall ladder system using neutral atoms in a 1D optical
lattice. In Sec. IV, we present the quench experiments and discuss their
results. Finally, some concluding remarks and outlooks are provided in Sec. V.

##  II Theoretical model

Figure 1: Hall ladders with a uniform magnetic flux $\phi$ per plaquette. The
ladder consists of a finite number of legs, indexed by $s$ with (a) open
boundaries on its lateral sides or (b) a periodic boundary to form a
cylindrical geometry. $j$ labels the lattice site along the leg direction.

A Hall ladder is a 2D square lattice system in the presence of a gauge
potential, having a finite width for one dimension. We consider non-
interacting spinless fermions in a Hall ladder, whose sites are indexed by
$(j,s)$ with $j\in\mathbb{Z}$ and $s\in\\{1,\cdots,\rho\\}$ (Fig. 1). In the
tight-binding approximation, the system’s Hamiltonian is given by

$\displaystyle\hat{H}/\hbar$
$\displaystyle=\sum_{j,s}\left[-t_{x}\hat{a}^{\dagger}_{j+1,s}\hat{a}_{j,s}+\frac{\Omega_{s}}{2}e^{i\phi
j}\hat{a}^{\dagger}_{j,s+1}\hat{a}_{j,s}+\mathrm{h.c.}\right],$ (1)

where $\hat{a}_{j,s}$ ($\hat{a}^{\dagger}_{j,s}$) is the annihilation
(creation) operator for the localized state on site $(j,s)$. In addition,
$t_{x}$ and $\Omega_{s}$ are tunneling amplitudes between adjacent lattice
sites, where we assume a real $t_{x}$ and complex-valued $\Omega_{s}$ in
general. The tunneling along the $s$ direction has a $j$-dependent Peierls
phase, resulting in a net phase $\phi$ for a loop around a unit plaquette,
which corresponds to a magnetic flux per plaquette,
$\Phi_{B}=(\phi/2\pi)\Phi_{0}$ with $\Phi_{0}$ being the magnetic flux
quantum. Taking the transformation of
$\hat{b}^{\dagger}_{q,s}=L^{-1/2}\sum_{j}\hat{a}^{\dagger}_{j,s}e^{i[q+(s-1)\phi]j}$
(where $L$ is the length of the ladder), $\hat{H}$ is expressed as

$\displaystyle\hat{H}/\hbar=\sum_{q,s}\big{[}$ $\displaystyle
h_{q}(\phi,s)\hat{b}^{\dagger}_{q,s}\hat{b}_{q,s}$ (2)
$\displaystyle+\frac{\Omega_{s}}{2}\hat{b}^{\dagger}_{q,s+1}\hat{b}_{q,s}+\frac{\bar{\Omega}_{s-1}}{2}\hat{b}^{\dagger}_{q,s-1}\hat{b}_{q,s}\big{]},$

where $q$ is the quasimomentum along $j$, and
$h_{q}(\phi,s)=-2t_{x}\cos[q+(s-1)\phi]$ represents the energy dispersion of
leg $s$ with momentum shifted by $(s-1)\phi$. The off-diagonal terms infer
that the Hall ladder can be viewed as a coupled $\rho$-leg system.

In the case of $\Omega_{\rho}=0$, the Hall ladder has an open boundary with
edges at $s=1$ and $\rho$ [Fig. 1(a)]. The Bloch Hamiltonian is given in a
$\rho\times\rho$ matrix form by

$\displaystyle\hat{H}_{q}/\hbar$ (3)
$\displaystyle=\begin{pmatrix}h_{q}(\phi,1)&\frac{\Omega_{1}}{2}&0&\cdots&0\\\
\frac{\bar{\Omega}_{1}}{2}&h_{q}(\phi,2)&\frac{\Omega_{2}}{2}&\cdots&0\\\
0&\frac{\bar{\Omega}_{2}}{2}&\ddots&&\vdots\\\
\vdots&\vdots&&\ddots&\frac{\Omega_{\rho-1}}{2}\\\
0&0&\cdots&\frac{\bar{\Omega}_{\rho-1}}{2}&h_{q}(\phi,\rho)\end{pmatrix}.$

In Fig. 2, we display the density of states of an equilateral ladder, i.e.,
$\Omega_{s}=\Omega$, as a function of $\phi$ for various $\Omega/t_{x}$. As
the interleg coupling strength increases, energy gaps are opened, indicating
strong avoided crossings. All bands in the open boundary Hall ladder have a
vanishing 1D winding number. The topological bands can be formed when
additional hopping terms are introduced, such as next nearest neighbor
tunneling, i.e., diagonal links between the legs Hugel2014 .

Figure 2: Density of states (DoS) of open 3-leg Hall ladders as functions of
magnetic flux $\phi$ for (a) $\Omega/t_{x}=0.1$, (b) 1, and (c) 10. (d)
Evolution of DoS as a function of $\Omega/t_{x}$ for $\phi=\pi$.

When $\Omega_{\rho}\neq 0$, the system has a periodic boundary, forming a tube
geometry [Fig. 1(b)]. The periodic condition is achieved only if
$\rho\phi/(2\pi)=P/Q$, where $P$ and $Q$ are co-prime integers. This
requirement corresponds to the wavefunction of a particle being single-valued
after $Q$ loops around the tube. With $\phi/(2\pi)=P^{\prime}/Q^{\prime}$
(where $P^{\prime}$ and $Q^{\prime}$ are co-prime integers), we have $\rho
P^{\prime}/Q^{\prime}=P/Q$ and $Q=\mathcal{D}/\rho$ with
$\mathcal{D}=\mathrm{lcm}(\rho,Q^{\prime})$. The Bloch Hamiltonian of the
periodic system is given by

$\displaystyle\hat{H}_{q}/\hbar$ (4)
$\displaystyle=\begin{pmatrix}h_{q}(\phi,1)&\frac{\Omega_{1}}{2}&0&\cdots&\frac{\bar{\Omega}_{\mathcal{D}}}{2}\\\
\frac{\bar{\Omega}_{1}}{2}&h_{q}(\phi,2)&\frac{\Omega_{2}}{2}&\cdots&0\\\
0&\frac{\bar{\Omega}_{2}}{2}&\ddots&&\vdots\\\
\vdots&\vdots&&\ddots&\frac{\Omega_{\mathcal{D}-1}}{2}\\\
\frac{\Omega_{\mathcal{D}}}{2}&0&\cdots&\frac{\bar{\Omega}_{\mathcal{D}-1}}{2}&h_{q}(\phi,\mathcal{D})\end{pmatrix},$

where periodic labels along $s$ are adopted, i.e.,
$h_{q}(\phi,s+\rho)=h_{q}(\phi,s)$ and $\Omega_{s+\rho}=\Omega_{s}$. In
general, the system exhibits a $\mathcal{D}$-band energy spectrum.

In a tube geometry, we need to consider an additional magnetic flux that
threads the longitudinal direction of the tube, which is defined as
$\phi_{\mathrm{L}}=\mathrm{arg}(\Omega_{1}\cdots\Omega_{\rho})$. For the
equilateral case with $\Omega_{s}=\Omega=|\Omega|e^{i\phi_{\mathrm{L}}/\rho}$,
the physical meaning of $\phi_{\mathrm{L}}$ is clear. The translational
symmetry along the tube circumference defines the quasi-momentum
$p=0,\frac{2\pi}{\rho},\cdots,\frac{2\pi}{\rho}(\rho-1)$ and the wavefunction
fulfills the Bloch theorem. Following a substitution
$\hat{b}_{q,s}=e^{ips}\hat{c}_{q,s}$, $\hat{H}$ is expressed as

$\displaystyle\hat{H}/\hbar=\sum_{q,s}\big{[}$ $\displaystyle
h_{q}(\phi,s)\hat{c}^{\dagger}_{q,s}\hat{c}_{q,s}$ (5)
$\displaystyle+\frac{\Omega}{2}e^{-ip}\hat{c}^{\dagger}_{q,s+1}\hat{c}_{q,s}+\frac{\bar{\Omega}}{2}e^{ip}\hat{c}^{\dagger}_{q,s-1}\hat{c}_{q,s}\big{]},$

where the form of the off-diagonal terms shows that the effect of the
longitudinal flux $\phi_{\mathrm{L}}$ is a shift of the momentum $p$ to
$p^{\prime}=p-(\phi_{\mathrm{L}}/\rho)$, analogous to the Aharonov-Bohm effect
for a charged particle. According to Laughlin’s argument, an adiabatic change
in $\phi_{\mathrm{L}}$ results in topological charge pumping along the leg
direction in the system, as demonstrated in recent experiments Fabre2021 .
Note that the total magnetic flux piercing the cross section of the tube at
$j$ is $\Phi_{j}=\rho\phi j+\phi_{\mathrm{L}}$.

In Fig. 3, we display the energy spectra of the equilateral Hall ladder with
$\rho=3$ as a function of $\phi$ for various $\Omega/t_{x}$ and
$\phi_{\mathrm{L}}=0$. Because Eq. (4) satisfies the recursive Harper
equation, the spectra exhibit a self-similarity like Hofstadter’s butterfly,
which is a hallmark of quasi-crystals Rajagopal2019 ; Dean2013 . Figure 3(b)
shows an anisotropic case for $\Omega/t_{x}=11.7$, which is close to the value
for the experiments that follow. In Ref. Barbarino2018 , it was shown that
$\hat{H}_{q}$ for $\phi=2\pi/\rho$ ($\rho$ is odd) and $\phi_{\mathrm{L}}=0$
embeds a 1D topologically insulating phase with nontrivial winding number
protected by generalized inversion symmetry. From Zak phase calculations, we
found that there are more topologically nontrivial bands for different $\phi$,
which are indicated as red lines in Fig. 3. When the system is project onto
the Brillouin zone spanned by $q$ and $p$, it is also characterized by a 2D
topological invariant, i.e., a Chern number Abanin2013 .

Figure 3: Hofstadter’s butterfly emergent in periodic 3-leg Hall ladder.
Energy spectra of the system for (a) the isotropic tunneling strength
$\Omega/t_{x}=1$ and (b) an anisotropic case $\Omega/t_{x}=11.7$. The
topologically nontrivial energy bands with nonzero Zak phase are indicated by
red lines.

##  III Experimental setup

###  III.1 Sample preparation

Figure 4: (a) Sketch of the experimental setup. Ultracold fermionic atoms of
173Yb are trapped in 1D optical lattice. A set of Raman beams drive the
complex interleg tunnelings between the synthetic lattice sites, generating
the effective magnetic flux $\phi(\theta)$ perpendicular to the $xs$-plane.
(b) The linking structure of the synthetic Hall ladder is determined by the
Raman coupling schemes in the three lowest atomic spin states. (c) Exemplary
images of the atoms obtained after optical Stern-Gerlach detection and (d)
adiabatic band mapping, where the samples are quenched for (c) 0.1 and (d)
0.15 ms, respectively.

We begin by preparing a degenerate Fermi gas of 173Yb spin-polarized in a
ground state $|F,m_{F}\rangle=|5/2,-5/2\rangle$ confined in a three-
dimensional optical lattice potential, as described in our previous experiment
Han2019 . A typical sample contains $\approx 1.2\times 10^{5}$ atoms with a
residual spin fraction ($m_{F}\neq-5/2$) of less than 3% and has a temperature
$\approx 0.35T_{F}$, where $T_{F}$ is the Fermi temperature of the trapped
sample. The optical lattice potential is formed by three pairs of interfering
laser beams at a wavelength $\lambda_{L}=532$ nm, resulting in lattice
constants $d_{x,z}=\lambda_{L}/2$ and $d_{y}=\lambda_{L}/\sqrt{3}$. The final
lattice depths are $(V_{x},V_{y},V_{z})=(4.5,20,20)E_{L,\alpha}$, where
$E_{L,\alpha}=h^{2}/8md^{2}_{\alpha}$, $\alpha\in\\{x,y,z\\}$ are the
corresponding recoil energies, $h$ is the Planck constant, and $m$ is the
atomic mass. Within our experiment timescale (a few milliseconds), the high
lattice depths $V_{y}$ and $V_{z}$ strongly suppress the atomic motion along
$y$ and $z$ between the adjacent individual planes, generating effective 1D
lattice with a tunneling rate of $t_{x}=2\pi\times 298$ Hz. The trapping
frequencies of the overall harmonic confinement are estimated to be
$(\omega_{x},\omega_{y},\omega_{z})=2\pi\times(57,40,130)$ Hz at the end of
the preparation sequence.

###  III.2 Realization of synthetic Hall ladders

We realize a 2D Hall ladder by hybridizing the 1D optical lattice with
synthetic tunneling between the internal atomic spin states. The sites of the
synthetic dimension are provided by a subset of 1S0 ground spin states, which
we denote as $|1\rangle\equiv|m_{F}=-5/2\rangle$,
$|2\rangle\equiv|m_{F}=-3/2\rangle$, and $|3\rangle\equiv|m_{F}=-1/2\rangle$,
constituting the legs of the ladder system. The ladder rungs are generated by
two-photon Raman transitions between these spin states. As shown in Figure
4(a), three Raman beams $R_{1,2,3}$ propagating in the $xy$ plane are
irradiated on the sample with wavevectors
$\vec{k}_{r1}=k_{R}(\cos{\theta}\hat{x}+\sin{\theta}\hat{y})$ and
$\vec{k}_{r2}=\vec{k}_{r3}=k_{R}\hat{x}$, respectively. The polarization
directions of the Raman beams are horizontal for $R_{1,3}$ (responsible for
$\sigma^{\pm}$) and vertical for $R_{2}$ (responsible for $\pi$) toward the
$xy$ plane. To lift the ground state degeneracy, an external magnetic field of
$B=153~{}$G is applied along $z$. In this configuration, a Raman pair
$R_{2}R_{1}$($\pi\sigma$) drives the couplings
$|1\rangle$$\leftrightarrow$$|2\rangle$ and
$|2\rangle$$\leftrightarrow$$|3\rangle$ changing the quantum number by $\Delta
m_{F}=1$, whereas $R_{1}R_{3}$($\sigma\sigma$) generates the coupling between
$|1\rangle$$\leftrightarrow$$|3\rangle$ with $\Delta m_{F}=2$.

In a synthetic dimensional frame, the Raman transition between $|s\rangle$ and
$|s^{\prime}\rangle$ can be interpreted as a interleg tunneling with complex
amplitude $\Omega_{ss^{\prime}}e^{i\phi j}/2$, where $\Omega_{ss^{\prime}}$ is
the Rabi frequency and $j$ is the site index for the real lattice. We adjust
the Raman beam intensities to satisfy $\Omega_{12}=\Omega_{31}=11.7t_{x}$; we
emphasize that $\Omega_{12}/\Omega_{23}$ equals almost unity (within 2%)
because both transitions originate from the same Raman beam pair,
$R_{2}R_{1}$. The momentum transfer owing to the Raman photons imprints the
spatial phase modulation on the synthetic tunneling amplitude,

$\phi(\theta)=[(\vec{k}_{r2,r3}-\vec{k}_{r1})\cdot\hat{x}]d_{x}=k_{R}d_{x}(1-\cos{\theta}),$
(6)

which is experimentally tunable by varying $\theta$. The atoms hopping around
the unit plaquette acquire a uniform phase $\phi$. In this manner, we generate
a synthetic Hall ladder system governed by Eq. (1).

The essence of producing a ladder geometry is the precise manipulation of
atomic energy levels $\nu_{s}$, because we employ an energy mismatch to
isolate the ladder manifold from the residual spin states. In Ref. Stuhl2015 ,
unwanted Raman transitions are suppressed by the second-order Zeeman shift in
$\nu_{s}$; in Ref. An2017-1 ; An2018 ; Meier2016 , the separation originates
from the nonlinearity of the free particle energy spectrum. In this
experiment, we exploit the state-dependent light shift by dressing an
additional $\sigma^{-}$ polarized laser beam on the atoms to detach the
superfluous link, particularly between $|3\rangle$ and $|4\rangle$. This laser
beam has an intensity of 8.5 mW cm-2 and is detuned $-70$ MHz from the
$|^{1}S_{0},F=5/2\rangle$$\rightarrow$$|^{3}P_{1},F^{\prime}=7/2\rangle$
resonance, adding a positive (negative) light shift for $\nu_{1}$, $\nu_{2}$,
and $\nu_{3}$ ($\nu_{4}$, $\nu_{5}$, and $\nu_{6}$), respectively, which
drives the two-photon transition between $|3\rangle$ and $|4\rangle$ off-
resonant. Along with the light shifts from $R_{1,2,3}$, the final experimental
condition is
$\left(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5}\right)$$\approx$$\left(0,-0.28,0,-2.52,2.25\right)\Omega_{12}$,
where $\xi_{s}=\left(\nu_{s}-\nu_{1}\right)-(s-1)\delta\omega$ is the detuning
of $|s\rangle$ from the two-photon resonance, and $\delta\omega=2\pi\times
30.4$ kHz. In this scheme, $R_{1,2,3}$ are dominated by $\sigma^{\pm}$ beams,
which generate light shifts in atomic energy levels with a small asymmetry in
$\xi_{2}$. This gives rise to a minor leakage channel between $|2\rangle$ and
$|4\rangle$, causing a typical atom loss of less than 8$\%$ after a $1$-ms
evolution.

Figure 5: Time dependence of the spin composition $n_{s}$ and total lattice
momentum $\langle k\rangle$ of open boundary synthetic Hall ladders for
different values of an effective magnetic flux. (a) Schematic, (b) $n_{s}$,
(c) $\langle k\rangle$ of the 2-leg ladder, (d) schematic, (e) $n_{s}$, and
(f) $\langle k\rangle$ of the 3-leg ladder are shown. The dashed lines are
fitted curves using a damped oscillator model [Eq. (9)] and the solid lines
indicate the results from the numerical simulation obtained by solving the
Bloch equations. Each data point comprises five independent measurements of
the same experiment and the error bar is their standard deviation.

The structure of the Hall ladder system is determined by the connectivity
between the legs of the synthetic lattices. As depicted in Figure 4(b), we
vary the number of legs by controlling the number of resonant Raman
transitions employed in the system. To create a periodic boundary, we place
the laser frequencies of $R_{1,2,3}$ at $\omega_{1}=\omega$,
$\omega_{2}=\omega+\delta\omega$, and $\omega_{3}=\omega-2\delta\omega$, where
$\omega$ is the laser frequency blue-detuned by $1.97$ GHz from the
$|^{1}S_{0},F=5/2\rangle$$\rightarrow$$|^{3}P_{1},F^{\prime}=7/2\rangle$
transition. The two(three)-leg ladder under an open boundary condition can be
realized by shifting $\omega_{2}$($\omega_{3}$) by $2\pi\times 400$($-400$)
kHz from the Raman resonance, deactivating the corresponding interleg links
$R_{2}R_{1}$($R_{1}R_{3}$), respectively. For a large detuning, the associated
interleg couplings are effectively switched off but the light shifts for the
Hall ladder formation are nearly unaffected.

##  IV Results and Discussions

###  IV.1 Quench dynamics for open boundary ladders

Quench dynamics refers to the evolution of a system after a sudden change in
the system parameters and it has been used as an essential tool in the study
of topological lattice systems Eisert2015 . Examples include reconstructing
the band structure of a spin-orbit coupled lattice through Fourier
spectroscopy VCuriel2017 , the observation of dynamical vortices in a momentum
space Flaschner2018 , spin relaxation after quenching across the symmetry-
protected topological phases Song2018 , and the determination of topological
invariant based on a post-quench evolution Sun2018 . The quench dynamics of a
system can also be exploited to extract macroscopic topological properties
Unal2016 .

We conducted a quench experiment of the synthetic Hall ladders for various
effective magnetic fluxes. Our experimental sequence is directed as follows.
After preparing the atoms in $|1\rangle$ in a 1D optical lattice, we suddenly
allow the interleg tunneling by illuminating the Raman beams on the samples
for a variable time. The time dependence of the spin composition is measured
through optical Stern-Gerlach detection OSG . Figure 4(c) displays an image of
a spin-separated sample containing the atoms transferred from $|1\rangle$ to
$|2\rangle$ and $|3\rangle$. Because the optical Stern-Gerlach process
destroys the momentum profile of the atoms, we separately measure the lattice
momentum distribution $n(k)$ using adiabatic band mapping in the samples under
the same conditions [Fig. 4(d)]. In the band mapping, slowly switching off the
trapping lattice potential transforms the lattice momentum state $|q\rangle$
into the corresponding free-space momentum states $|k_{s}\rangle$, where the
momentum $k_{s}$ of the spin $|s\rangle$ is related to $q$ as
$k_{s}d_{x}=[q+(s-1)\phi]$ modulo $2\pi$ and $-k_{L}<k_{s}\leq k_{L}$ with
$k_{L}=\pi/d_{x}$. To investigate the effective magnetic flux dependence of
the quench dynamics, we carried out the experiment for three different fluxes,
$\phi=\pi/3,\pi/2,$ and $2\pi/3$ by setting the angle of incidence of $R_{1}$
to $\theta=50^{\circ},60^{\circ},$ and $72^{\circ}$, respectively.

We first investigate the quench dynamics in an open 2-leg Hall ladder, which
is realized by deactivating the $|1\rangle$-$|2\rangle$ and
$|2\rangle$-$|3\rangle$ interleg couplings. In Fig. 5(b), we plot the time
evolution of the spin compositions for $\Omega_{12}/t_{x}=11.7$. During the
evolution, the ladder demonstrates two-level Rabi dynamics between $|1\rangle$
and $|3\rangle$, whereas the population of $|2\rangle$ remains at a low
fraction. From the measured $n(k)$, we obtain the total average momentum
$\langle
k\rangle=\int^{k_{\mathrm{L}}}_{-k_{\mathrm{L}}}kn(k)dk/\int^{k_{\mathrm{L}}}_{-k_{\mathrm{L}}}n(k)dk$,
the time evolution of which is shown for a different $\phi$ in Fig. 5(c). We
observe that $\langle k\rangle$ oscillates in the negative direction and its
amplitude gradually increases over time. This originates from the interplay
between the coupling structure and the initial momentum distribution. The
$|k,1\rangle$ state is coupled to the $|k-\phi/d_{x},3\rangle$ for
$k\in[-k_{L}+\phi/d_{x},k_{L}]$, which drives the negative momentum recoil,
whereas the $|k,1\rangle$ state is coupled to $|k+2k_{L}-\phi/d_{x},3\rangle$
for $k\in[-k_{L},-k_{L}+\phi/d_{x})$, which gives rise to a positive shift in
lattice momentum owing to the lattice periodicity [Fig. 5(a)]. Under our
initial condition, the $s=1$ band is not completely filled and is more
populated at near $k=0$, causing $\langle k\rangle$ to start oscillating in
the negative direction. The gradual increase in the oscillation amplitude
arises, because the average oscillation period for $k$ giving a positive shift
differs from that for the other momentum range owing to different coupling
strengths. For a higher $\phi$, the momentum range giving a positive shift in
$\langle k\rangle$ is larger, and thus the growth in amplitude increases in
speed [Fig. 5(c)].

Figure 6: Semi-classical trajectories of the atoms in the plane of the spin
and real lattice positions for the quench dynamics of (a)-(c) the open
boundary 2-leg and (d)-(f) 3-leg ladder systems for different values of
effective magnetic flux. The initial positions are denoted by the red marks.

To validate our results, we perform a numerical simulation of the time
evolution of the system by solving the Bloch equation,

$i\hbar\frac{\partial}{\partial t}\Psi(q,t)=\hat{H}_{q}\Psi(q,t),$ (7)

where $\Psi(q,t)=[c_{1},c_{2},c_{3}]^{T}$ is the wavefunction of the fermions
with probability amplitude $c_{s}(q,t)$ for spin $|s\rangle$. The atomic
density for spin $s$ and momentum $k_{s}$ is given by
$n_{s}(k_{s},t)=|c_{s}(q,t)|^{2}$, and $n(k,t)=\sum_{s}n_{s}(k_{s},t)$. We use
the initial condition $\Psi(q,0)=[\sqrt{n_{1}(k_{1},0)},0,0]$, where
$n_{1}(k_{1},0)$ is obtained by averaging the experimentally measured lattice
momentum distributions of the spin-polarized atoms at $t=0$. In Fig. 5(c), the
simulation results are displayed as solid lines, which correctly anticipate
the time evolution in $\langle k\rangle$ for a different $\phi$. In the
simulation, we also observe that the increasing slope of the oscillation
envelope is reversed after $t>4$ ms.

Next, we investigate the quench dynamics in an open 3-leg Hall ladder, which
is realized by deactivating the $|1\rangle$-$|3\rangle$ interleg coupling. The
time evolutions of $n_{s}$ and $\langle k\rangle$ for the open 3-leg Hall
ladder are displayed in Fig. 5(e) and 5(f), respectively. In comparison with
the 2-leg case, a $\phi$-dependence of the $\langle k\rangle$ oscillations
dramatically appears with a large change in amplitude. For $\phi=\pi/3$, the
dynamics of $\langle k\rangle$ evolves mostly on $\langle k\rangle>0$, whereas
a change in sign of $\langle k\rangle$ occurs for $\phi=\pi/2$ and $2\pi/3$
[Fig. 5(f)]. This behavior originates from the multiple Raman processes
involved in the 3-leg case and the fact that the large magnetic flux causes
the atoms to be reflected at the Brillouin zone boundary by the Bloch
oscillation Han2019 . For example, the atoms initially located at $\langle
k\rangle=0$ experience two consecutive Raman transitions, which exerts a
momentum recoil exceeding $k_{L}$ if $\phi\geq\pi/2$, forcing the atoms to
shift forward $\langle k\rangle<0$ states at near $t\approx 0.25$ ms. We found
that numerical calculations including a phenomenological damping effect
damping  are in good agreement with the experiment results.

###  IV.2 Semi-classical trajectories for open boundary ladders

In the presence of a magnetic field, a charged particle shows a characteristic
cyclotron motion owing to the Lorentz force perpendicular to the velocity of a
particle. For the particle in a Hall ladder system, a similar but more
complicated motion is expected, because of the reflection from the lattice
potential as well as the lateral edges. From the measurements, we
reconstructed the trajectory of the average position of the atoms in the $xs$
plane, following the analysis described in Ref. Mancini2015 . In brief, the
atom in $|k\rangle$ travels with the group velocity
$v_{k}={\partial\epsilon(k)}/{\hbar\partial k}$, where
$\epsilon(k)=2t_{x}[1-\cos(kd_{x})]$ is the lattice dispersion of the ground
band. We neglected the effect of the trapping potential, whose timescale is an
order of magnitude slower than the experimental timescale $\approx 1$ ms. The
average velocity of the atoms, $\langle v\rangle$, is obtained based on the
density-weighted sum of the group velocity and the average position of the
atoms, $\langle x\rangle$, is calculated by integrating the average velocity
over time, i.e.,

$\langle x(t)\rangle=\int_{0}^{t}\langle v(t)\rangle
d\tau=\int_{0}^{t}\int_{-k_{L}}^{k_{L}}n(k,\tau)v_{k}~{}dkd\tau,$ (8)

whereas the average position in $s$ is determined as $\langle
s(t)\rangle=n_{3}(t)-n_{1}(t)$.

In Fig. 6, we plot the reconstructed quench trajectory of the system in the
$xs$ plane. The 2-leg ladder case [Fig. 6(a)-6(c)] shows a damped cyclotron
motion truncated by the ladder edge. Noticeably, as $\phi$ increases, the
cyclotron motion tilts toward the negative $x$-direction with growing
overlapping regions. This is compatible with the observations in Figs. 5(c)
and 5(d), where the time dependence of momentum shows a rather increasing
envelope, whereas the amplitude of the spin composition oscillations is slowly
damped.

The 3-leg ladder case reveals more intriguing trajectories [Fig. 6(d)-6(f)].
In the $\phi=\pi/3$ ladder, the atoms are shifted toward positive
$x$-direction, because their $\langle k\rangle$ does not change its sign. In
the $\phi=\pi/2$ and $2\pi/3$ cases, the atoms display bouncing motions,
because of the Bloch oscillations during the cyclotron orbits. We emphasize
that the trajectory is almost retraced back for $\phi=\pi/2$ at the first
bounce, indicating the symmetric reflection at $k=k_{L}$ (corresponding to
$2\phi=\pi$).

To further understand the observed behaviors, we numerically simulate the
semi-classical trajectories of the atoms in the $xs$ plane and study their
initial condition dependence (Fig. 7). We examined four different lattice
momentum distributions, including $n_{1}(k_{1},0)$ measured from the
experiments, the case of a completely filled band, and the hypothetical case
of a partially filled band with a complete depletion in $[-k_{L},-0.85k_{L}]$.

Figure 7: Numerical simulation of the atomic trajectories for various magnetic
flux after the quench at $\Omega_{31}/t_{x}=11.7$. The trajectories for (a)
2-leg ladder with measured initial momentum profile (black), (b) with
partially occupied band (red) and completely filled band (blue), (c) 3-leg
ladder with measured initial momentum profile (black), (d) with partially
occupied band (red) and completely filled band (blue) are shown, respectively.
Insets display the corresponding initial momentum profiles.

The simulation for the 2-leg ladders demonstrates that under all three initial
conditions the trajectories of the atoms gradually tilt toward the negative
$x$-direction as $\phi$ increases, which is consistent with the experiment
results. In Fig. 7(a), the case using the measured $n_{1}(k_{1},0)$ shows a
cyclotron motion similar to the trajectories from the experiment data, whereas
in Fig. 7(b), the case of the completely filled band has $\langle
v(0)\rangle=0$ and exhibits a trajectory confined along $\langle s\rangle$ at
the initial times. For the case of the partially filled band, the trajectories
proceed toward the negative $x$-direction from the beginning, because the
depleted population leads to finite $\langle v(0)\rangle\neq 0$.

The simulation for the 3-leg ladders also agrees with the experiment results
in terms of the $\phi$ dependence in the winding directions of the
trajectories in the $xs$ plane. As $\phi$ increases, the atomic trajectory
bends counter-clockwise, regardless of the initial momentum distributions
[Fig. 7(c)]. Similar to the 2-leg case, the completely filled band shows a
motion localized within a small region owing to $\langle v(0)\rangle=0$,
whereas the partially filled bands with an asymmetric occupation leads to
spiral trajectories [Fig. 7(d)]. In both the simulation and experiment, the
semi-classical trajectories for open boundary Hall ladders demonstrate strong
spin-momentum locking, which implies that the atoms experience a force
perpendicular to their velocity, similar to the Lorentz force acting on a
charged particle in a magnetic field.

###  IV.3 Characterization of damping effect in periodic boundary ladders

Finally, we repeat the quench experiment for the synthetic 3-leg Hall ladders
under periodic boundary conditions. As seen in the trajectory change from the
2-leg to 3-leg ladder, the quench evolution in the periodic boundary Hall
ladder will be much more complicated because of the circling motion of the
atoms around the tube. In Fig. 8(a)-8(c), we display the time dependencies of
$n_{s}(t)$ obtained for different magnetic fluxes. The oscillation is quickly
damped with the time constant $t_{d}<300~{}\mu$s. To characterize the damping,
we fit the data of the spin populations in $|1\rangle$ and $|3\rangle$ to a
damped oscillator model,

$n_{s}(t)=n_{s,0}+[n_{s}(0)-n_{s,0}]e^{-t/t_{d}}\cos(2\pi ft),$ (9)

where $n_{s,0}$ is the equilibrium value after a long time, and $f$ is the
oscillation frequency. Table 1 summarizes the fitting results. In comparison
with the open boundary Hall ladders, the periodic boundary systems feature a
strong suppression of the oscillations with over a 6-times shorter damping
time. In our experiment, the magnitude of the damping effect in the synthetic
Hall ladder system is enhanced as $\phi$ increases.

Figure 8: Time dependence of the spin composition of periodic boundary
synthetic Hall ladders for different values of effective magnetic flux (a)
$\phi=\pi/3$, (b) $\pi/2$, and (c) $2\pi/3$. The dash-dotted line indicates
fitted curve for corresponding spin compositions using Eq. (9). (d)-(f)
Numerical simulations accounting the effect of random flux $\Delta\varphi$ are
shown. The solid lines represent the spin compositions averaged over a hundred
1D chains and the gray dots are evolutions of $n_{1}(t)$ for individual
chains. Each data point comprises five independent measurements of the same
experiment and the error bar is their standard deviation.

The intense damping observed in the synthetic Hall ladders under periodic
boundary condition can be attributed to fluctuations of the longitudinal
magnetic flux $\phi_{\mathrm{L}}$ threading the Hall tube Luo2020 . When an
atom encircles the Hall tube at site $j$, i.e., a cyclic transition of
$|j,1\rangle\rightarrow|j,2\rangle\rightarrow|j,3\rangle\rightarrow|j,1\rangle$
is made, a net phase of $\Phi_{j}=\Delta\vec{k}\cdot\vec{r}_{j}+\Delta\varphi$
with
$\Delta\vec{k}=2\vec{k}_{r2}+\vec{k}_{r3}-3\vec{k}_{r1}=3k_{R}[(1-\cos\theta)\hat{x}-\sin\theta\hat{y}]$
and $\Delta\varphi=2\varphi_{2}+\varphi_{3}-3\varphi_{1}$ is acquired, where
$\varphi_{s}$ is the relative phase of Raman laser beam $s$ and $\vec{r}_{j}$
is the position vector of site $j$. From the relation of $\Phi_{j}=\rho\phi
j+\phi_{\mathrm{L}}$ described in Sec. II, the relative phases of the Raman
laser beams directly determine the threading magnetic flux,
$\phi_{\mathrm{L}}=\Delta\varphi$. In our experiment, the relative phases
between the Raman beams were not actively stabilized, and thus large
fluctuations of $\Delta\varphi$ might lead to a damping effect in the
experiment results, which involve averaging many independent measurements.
Here, we remark that in the open boundary ladders, the relative phases between
the Raman laser beams can be gauged away and have no active roles in the
system dynamics.

Table 1: Measured $t_{d}$ (in ms) obtained from spin compositions $|1\rangle$ and $|3\rangle$ in synthetic Hall ladders for different boundary conditions and effective magnetic flux $\phi$. Numbers in brackets represent the 95% confidence intervals from the fits. | $\phi$ | $\pi/3$ | $\pi/2$ | $2\pi/3$
---|---|---|---|---
Exp. $n_{1}$ | open 2-leg | 1692.1(265.2) | 1234.0(215.3) | 1480.2(309.3)
periodic | 141.3(30.6) | 145.3(33.3) | 102.5(22.0)
Exp. $n_{3}$ | open 2-leg | 1873.1(280.7) | 1316.9(253.2) | 1461.4(305.3)
periodic | 226.6(50.0) | 210.7(46.1) | 145.2(37.7)
Sim. $n_{1}$ | periodic | 443.3(93.2) | 246.2(71.4) | 96.2(23.6)

In Fig. 8(d)-8(f), we present the numerical simulation results for the quench
dynamics with the periodic boundary conditions, including the random phase
effect by averaging over 100 1D chains, while assuming evenly distributed
$\phi_{\mathrm{L}}$. Our simulation explains well the damping behavior
observed in the experiment. For a quantitative comparison with the experiment
results, we determine the damping time by fitting the model curve of Eq. (9)
to the calculated $n_{1}$ at early times $t<0.5$ ms. The fitting results,
listed in Table 1, are found to be in reasonable agreement with the experiment
data. As an example, a damping time of $t_{d}=96.2(23.6)$ ms for $\phi=2\pi/3$
is obtained, consistent with the experimentally measured value of
$t_{d}=102.5(22.0)$ ms.

In Ref. Luo2020 , Luo et al. pointed out that in our Raman beam setup, 1D
tubes with different $y$ positions experience a different $\phi_{\mathrm{L}}$.
Indeed, in the expression of $\Phi_{j}$, the wavevector $\Delta\vec{k}$ has a
nonzero $y$-directional component, and thus the longitudinal magnetic flux
$\phi_{\mathrm{L}}$ changes by $\Delta\vec{k}\cdot(d_{y}\hat{y})$ in the next
tube for the $y$ direction. This means that the measurement in our experiment
is intrinsically ensemble averaged even with $\Delta\varphi$ being perfectly
controlled. In Fig. 8(d)-8(f), we plot the time evolution of $n_{1}$ of
individual chains for the random distribution of $\phi_{\textrm{L}}$ as grey
curves. Meanwhile, our experiment results shown in Fig. 8(a)-8(c) indicate
that the variance of the measured value is much smaller than the expected
value for a single tube experiment with a random $\phi_{\mathrm{L}}$, implying
the averaging aspect of the system.

##  V Conclusion

We presented a synthetic Hall ladder system for atoms in a 1D optical lattice
and demonstrated the control of the effective magnetic flux by investigating
its quench dynamics for various ladder structures. For the open boundary
ladders, we observed the characteristic spiral motions of the atoms, which
were well explained by our numerical simulations based on the Bloch equations,
including the effective damping. For the cylindrical Hall ladder, we observed
a significantly strong damping in the quench evolution. We attributed this to
fluctuations of the longitudinal magnetic flux $\phi_{\mathrm{L}}$ threading
the Hall cylinder Luo2020  and verified it through the numerical simulations
of the experiment. The control of $\phi_{\mathrm{L}}$ could be improved in our
setup by employing a 2D sample in the $xz$ plane or modifying the geometric
configuration of the Raman beams. For example, we may rotate the irradiation
directions of all Raman beams by $(\pi-\theta)/2$ in the $xy$ plane to make
$\Delta\vec{k}$ aligned to the leg direction, resulting in a uniform
$\phi_{\mathrm{L}}$ over the entire sample, which will be conducted as an
immediate extension of this work. Combined with tunable atomic interactions
based on the orbital Feshbach resonance Pagano2015 ; Hofer2015 , we expect
that the Hall ladder system provides interesting opportunities for
investigating the correlation effects, such as the fractional charge behavior
Zeng2015 . Finally, we point out that when tunneling is allowed for a
direction orthogonal to the leg, the 1D ladder system is transformed into a 2D
multi-layer Hall system with an in-plane magnetic field Hou2020 .

##  VI Acknowledgments

We thank Jin Hyoun Kang for the fruitful discussion. This work was supported
by the National Research Foundation of Korea (2014-H1A8A1021987,
NRF-2018R1A2B3003373, NRF-2019M3E4A1080399) and the Institute for Basic
Science in Korea (IBS-R009-D1).

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