,

# Exploring Fine-Tuned Embeddings that
Model Intensifiers for Emotion Analysis

Laura Bostan  Roman Klinger
Institut für Maschinelle Sprachverarbeitung
University of Stuttgart
Pfaffenwaldring 5b, 70569 Stuttgart, Germany
<EMAIL_ADDRESS>
###### Abstract

Adjective phrases like “a little bit surprised”, “completely shocked”, or “not
stunned at all” are not handled properly by currently published state-of-the-
art emotion classification and intensity prediction systems which use
predominantly non-contextualized word embeddings as input. Based on this
finding, we analyze differences between embeddings used by these systems in
regard to their capability of handling such cases. Furthermore, we argue that
intensifiers in context of emotion words need special treatment, as is
established for sentiment polarity classification, but not for more fine-
grained emotion prediction. To resolve this issue, we analyze different
aspects of a post-processing pipeline which enriches the word representations
of such phrases. This includes expansion of semantic spaces at the phrase
level and sub-word level followed by retrofitting to emotion lexica. We
evaluate the impact of these steps with À La Carte and Bag-of-Substrings
extensions based on pretrained GloVe, Word2vec, and fastText embeddings
against a crowd-sourced corpus of intensity annotations for tweets containing
our focus phrases. We show that the fastText-based models do not gain from
handling these specific phrases under inspection. For Word2vec embeddings, we
show that our post-processing pipeline improves the results by up to 8% on a
novel dataset densely populated with intensifiers.

##  1 Introduction

Emotion detection in text includes tasks of mapping words, sentences, and
documents to a discrete set of emotions following a psychological model such
as those proposed by Ekman (1992) and Plutchik (1980), or to intensity scores
or continuous values of valence–arousal–dominance Posner et al. (2005). The
shared task on intensity prediction for discrete classes proposed to combine
both Mohammad et al. (2018); Mohammad and Bravo-Marquez (2017a). In this task
a tweet and an emotion are given and the goal is to determine an intensity
score between 0 and 1.

Especially, but not only in social media, users use degree adverbs (also
called intensifiers Quirk, 1985), for instance in “I am kinda happy” vs. “I am
very happy.” to express different levels of emotion intensity. This is a
relevant task: 10% of tweets containing an emotion word are modified with such
an adverb in the corpus we describe in Section 3.1. In this paper, we
challenge the assumption that models developed for intensity prediction
perform well on tweets containing such phrases and analyze which of the
established embedding methods Word2vec Mikolov et al. (2013), GloVe Pennington
et al. (2014), and fastText embeddings Bojanowski et al. (2017) performs well
when predicting intensities for tweets containing such phrases. We will see
that the performance of the popular and fast-to-train Word2vec method can be
increased with a simple postprocessing pipeline which we present in this
paper.

As a motivating example, the DeepMoji model Felbo et al. (2017) predicts anger
for both the example sentences “I am not angry.” and “I am
angry.”111https://deepmoji.mit.edu. Using the model by Wu et al. (2018) (one
of the state-of-the-art intensity prediction models from Mohammad et al.
(2018), building their model on top of Word2vec embeddings) we also obtain
anger as having the highest intensity for both examples. We argue that the
models should be more sensitive to the difference between negations,
downtoners and amplifiers.

With this paper, we contribute to alleviate this situation in three aspects.
Firstly, we provide an analysis of the distribution of degree adverbs
(including negations) with emotion words and show that not all such
combinations are equally common. Secondly, we perform a crowdsourcing
experiment in which we collect scores for different combinations of degree
adverbs and emotion adjectives. We use these data, which we make publicly
available, as an additional challenging test set for the task of intensity
prediction for English. Thirdly, we use a state-of-the-art intensity
prediction model Wu et al. (2018) on this test set and evaluate two methods to
improve these predictions, namely the inclusion Zhao et al. (2018) and
$n$-gram embeddings via À La Carte of additional subword information with Bag-
of-Substrings Khodak et al. (2018). We evaluate based on Word2vec, GloVe and
fastText embeddings and show that particularly the first two benefit from
these changes, but to different extents.

##  2 Related Work

###  2.1 Degree Adverbs in Linguistics

Adverbs that express intensity are named degree adverbs, degree modifiers or
intensifiers.222In this paper, we will use these terms interchangeably. The
entities they intensify are located on an abstract scale of intensity Quirk
(1985). The intensifiers that scale upward are named amplifiers and are
further categorised as maximizers, such as “completely” and “totally” or
boosters, such as “really” or “truly”. Those that scale downward are called
downtoners and are further classified as approximators, such as “almost” or
“kind of”, compromisers, such as “fairly”, “pretty” and “quite”, diminishers,
such as “slightly” and “a bit”, and minimizers (Quirk, 1985; Paradis, 1997;
Nevalainen and Rissanen, 2002, i. a.). Further distinction of degree modifiers
is concerned with the fact that there are intensifiers that imply boundaries,
such as “totally”, “fully”, and “completely” and those that do not, such as
“very”, “utterly”, “pretty” Paradis (1997, 2001, 2000). Finally, in the
context of discourse, there is the property of expressing focus, which is
present in the so-called focus modifiers, such as “only” and “just”, which are
also further classified in additives, such as “also” and “too” and
restrictives, such as “only” and “merely” Quirk (1985); Athanasiadou (2007).

English degree modifiers have also long history of research in English studies
and more generally in Language Studies. Most English studies focus on the
incidence and distribution of these adverbs in different corpora, e.g. Peters
(1994) study letters from Early Modern English and shows the how the
distributions of boosters change across time. Nevalainen (2008) study the
social variation in intensifier use, with a focus on the suffix -ly. More
recently, Napoli and Ravetto (2017) collect a volume of papers that explore
the process of intensification following a corpus-based, cross-linguistic and
contrastive approach. The volume contains various works on the variation in
the distribution and incidence of the intensifiers based on sociolinguistic
features and in a diachronic fashion. The work brings in attention
intensification in ancient languages as well as modern languages.

A more recent work investigates the differences in the use of intensifiers and
considers English speech of adults and teenagers as corpus. It explores two
maximizers in-depth, namely “absolutely” and “totally” and shows that those
prove to be more “flexible“ in the language used by teenagers Pertejo and
Martínez (2014).

###  2.2 Modifiers in the context of Sentiment and Emotion Analysis

In the context of sentiment analysis the discussion of intensifiers and
negations has gained quite some attention, since those are primarily markers
of subjectivity Athanasiadou (2007).

Negations, and in particular negation cue detection (with the goal of scope
recognition) have been the research interest of Councill et al. (2010) and
Reitan et al. (2015), who use a lexicon for negation cue detection and a
linear-chain conditional random field for scope recognition. In the area of
distributional semantics, the investigation of word vectors with a focus on
negated adjectives Aina et al. (2018) is complementary to our work with
regards to negation in terms of the methods and data used. Following this
approach, one could build a distributional semantic model whose vocabulary
includes the modified phrases. In practice, each occurrence of a modified
adjective by a degree adverb could be treated as a single token (e. g. “not
happy” would be represented as “not_happy”). For a general overview of
modality and negation in computational linguistics we refer the interested
reader to the work by Morante and Sporleder (2012).

Furthermore, Zhu et al. (2014) study the effect of negation words on sentiment
and evaluate a neural composition model. Kiritchenko and Mohammad (2016a)
create a sentiment lexicon of phrases that include modifiers such as negators,
modals, and degree adverbs. The phrases and their constituent words are
annotated manually with the same annotation procedure we will discuss in
detail. We follow this work closely and apply the same procedures in the
context of emotion analysis.

Dragut and Fellbaum (2014) study the effect of intensifiers on the sentiment
ratings and shows that the degree adverbs do not carry an inherent sentiment
polarity but alter the degree of the polarity of the constituents they modify.

We argue that there is not enough work on transferring the methods used in
sentiment analysis to the more fine-grained analysis of emotions, except for
Strohm and Klinger (2018), who limit themselves to analysis and do not apply
state-of-the-art prediction models for handling degree adverbs, and Carrillo-
de Albornoz and Plaza (2013) who consider modified emotions but predict
sentiment.

##  3 Methods

In the following, we explain how we create the data sets for our analysis
(Section 3.1) and then how we set up the experiments to measure the impact of
À La Carte and Bag-of-Substrings on the modified phrases (Section 3.2).

###  3.1 Data Collection and Annotation

As a basis of our work, we create a compositional emotion lexicon for English
Twitter and retrieve crowdsourced ratings using Best-Worst Scaling Louviere et
al. (2015); Kiritchenko and Mohammad (2016b). We show later that these ratings
are by and large independent of context and can therefore be interpreted as a
labeled emotion lexicon of compositional phrases.333Our data is available at
https://www.ims.uni-stuttgart.de/data/modifieremotion.

####  3.1.1 Data Collection

Each query we use to retrieve tweets consists of a pair of an adjective with
one or a combination of several degree adverbs (intensifiers (including
amplifiers and downtoners) and negations), for instance “not at all surprised”
or “not very happy”. We first generate a comprehensive list by mapping each of
Ekman’s fundamental emotions Ekman (1992) to their corresponding adjective
sad, happy, disgusted, afraid, surprised, angry and augment this list to $333$
emotion adjectives and their synonyms from the Oxford Dictionary of English
Ehrlich (1980), the New Oxford American Dictionary Stevenson and Lindberg
(2010) and Macmillan Online English
Dictionary444http://www.macmillandictionaries.com/dictionary-online/ and
further filter this list to 43 entries which are intersubjectively agreeable.
This filter step is performed via crowdsourcing on
Prolific555https://prolific.ac, in which we asked native speakers of English
which emotion is the closest to each synonym. We only keep those synonyms
where all annotators agreed. The inter-annotator agreement is $\kappa=0.63$
(Fleiss’ $\kappa$ over 9 annotators).

The list of degree modifiers is a combination of Quirk (1985); Paradis (1997);
Strohm and Klinger (2018). From the cartesian product of degree modifiers with
emotion adjectives, we keep those which we find at least 10 times in the
general Twitter corpus we discuss below. That leads to $266$ phrases.

We base our analysis on a set of 32 million tweets obtained from Twitter with
the official API between March 2006 and October 2018, using a combination of
diverse search terms corresponding to isolated emotion word synonyms, those in
combination with degree adverbs, and frequent hashtags. We filter out retweets
and full quotes, tweets with more than 140 characters and those with less than
10 tokens, as well as those consisting of more than 30% hashtags, links, or
usernames, which we replace by generic respective tokens otherwise. Tweets
with more than 30% of non-ASCII characters are also removed.

####  3.1.2 Annotation Procedure

For each tweet ($t$) and emotion ($e$) we obtain emotion intensity scores
$s_{t,e}\in[-1,1]$ via Best-Worst Scaling (BWS, Louviere et al., 2015). In
general with BWS, the annotators are shown a subset of a number of items from
a list and are asked to select the best and worst items (or most and least
some given property of interest). Within our study, we show four items at once
to the annotators. In a first setting, we show them four tweets that contain
the queries we want to have scores assigned for. In a second setting, we show
them only the queries without the context (the tweet) in which they were
found. In both scenarios, the annotators need to select the tweet or the query
with the highest and lowest intensity of each emotion.

Figure 1: An example of what contributors see on the Figure Eight Platform.
The 4 sentences shown are an example of a group of four tweets the
contributors have to annotate. The questions Q1 to Q4 that follow below are a
subset of the questionnaire.

These groups of tweets are sampled under following constraints that have been
empirically proven to lead to reliable scores Kiritchenko and Mohammad
(2016c), resulting in $532$ samples (twice the amount of queries): (1) no two
samples have the same four queries (in any order), (2) no two queries within a
sample are identical, (3) each query occurs in 8 ($\pm 1$) different samples,
(4) each pair of queries appears in the same number of samples. We perform two
annotation experiments on the crowdsourcing platform Figure
Eight666https://www.figure-eight.com: In Experiment 1, we present the whole
tweet to the annotator, in Experiment 2, we only show the query phrase. This
enables us to evaluate the importance of context, shown in Section 4.2. Each
sample was annotated by three contributors that confirmed to be English native
speakers.

###  3.2 Adaptations of Embeddings

In the following, we discuss the three methods to improve the embeddings and
later to test if these improvements add additional information with respect to
intensifiers for emotion analysis. The evaluation will be on the downstream
task of emotion intensity prediction.

We focus on subword-level information and phrase-level information, as those,
presumably, capture intensity information.

####  3.2.1 À La Carte

With this method we learn a representation of yet unseen phrases within an
embedding space through a linear transformation of the average of the word
embeddings in the feature’s contexts. The method constructs a representation
for a new phrase given a set of contexts where this phrase occurs in.

Given our Twitter corpus $\mathcal{C}_{w}$ consisting of contexts of words $w$
and the pre-trained word embeddings ${\bf v}_{w}\in\mathbb{R}^{d}$, of
dimension $d$, our goal is to construct a representation ${\bf
v}_{q}\in\mathbb{R}^{d}$ of a query $q$ given a set $\mathcal{C}_{q}$ of
contexts it occurs in.

We learn the transform ${\bf A\in\mathbb{R}^{d\times d}}$ that can recover
existing word vectors ${\bf v}_{w}$ via linear regression by summing their
context embeddings

${\bf v}_{w}\approx{\bf
A}\left(\frac{1}{|\mathcal{C}_{w}|}\sum\limits_{c\in\mathcal{C}_{w}}\sum\limits_{w^{\prime}\in
c}{\bf v}_{w}^{\prime}\right)\,.$ (1)

Using the learned transformation matrix ${\bf A}$ we can embed any new query
${\bf v}_{q}$ in the same semantic space as the pre-trained word embeddings
via

${\bf v}_{q}={\bf
A}\left(\frac{1}{|\mathcal{C}_{q}|}\sum\limits_{c\in\mathcal{C}_{q}}\sum\limits_{w\in
c}{\bf v}_{w}\right)\,.$ (2)

####  3.2.2 Bag-of-Substrings

BoS generalizes pre-trained semantic spaces to unseen words. The established
approach to represent word phrases or sentences is to take a bag of words of
word embeddings.

BoS achieves its goal by first learning a mapping between the subwords present
in each word and its corresponding pre-trained vector. Then, by using this
learned subword transformation, the model is able to generate new
representations for any new word as a set of its character $n$-grams. For us
this is relevant, since we can consider our focus phrases to be character
$n$-grams instead of word $n$-grams.

Formally, the representation for a word ${\bf v}_{w}$ from the lookup table V
(which stores the embeddings of dimension $d$ for each possible substring of
length within a range) is:

${\bf
v}_{w}=\frac{1}{|{\mathcal{S}_{w}}|}\sum\limits_{t\in\mathcal{S}_{w}}{\bf
v}_{t},$ (3)

where $\mathcal{S}_{w}$ is the set of each possible character $n$-grams of
length within a given range over ${w}$ and ${\bf v}_{t}$ is the vector in V
indexed by t.

The model views the vector of a phrase as the average vector of all its
substrings, which are trained by minimizing the overall mean squared loss
between the generated and given vectors for each word:

$\min_{V}\frac{1}{|W|}\sum\limits_{w\in
W}l\left(\frac{1}{|{\mathcal{S}_{w}}|}\sum\limits_{t\in\mathcal{S}_{w}}{\bf
v}_{t},{\bf u}_{w}\right)$ (4)

where ${\bf u}_{w}$ $\in\mathbb{R}^{d\times|W|}$ are the target vectors of the
dimension d over the vocabulary $W$ and $l(\bf v,\bf
u)=\frac{1}{2}\mathinner{\\!\left\lVert\bf{v}-\bf{u}\right\rVert}_{2}^{2}$ .

After training, similarly to the previous method, one can use the learned
space to generate a new word vector ${\bf v}_{q}$ as the average of the
vectors of all of its substrings through Equation 3.

Since BoS produces vectors for unknown words from vectors of substrings of
characters contained in it, this allows to build vectors for misspelled words
and concatenation of words. Particularly on Twitter data, we benefit from
getting a representation for phrases like “sooooexcited:)”, “verrry cheerful”,
“soo unhappy:(”. Relevant for our analysis is that BoS uses special characters
to mark the start and the end of the word and thus helps the model to
distinguish morphemes that occur at different word parts, like prefixes or
suffixes. Through that we learn to distinguish morphemes like “un-”, “-er” and
“-est” that are part of our focus phrases.

Note that this method uses the same idea as in fastText Bojanowski et al.
(2017), but is for our case computationally more efficient, since the BoS
model is trained directly on top of pre-trained vectors, instead of predicting
over text corpora.

####  3.2.3 Retrofitting

We use the method of retrofitting existing embeddings Faruqui et al. (2015) in
order to enrich word vectors using synonymity constraints provided by semantic
lexicons. The algorithm learns the word embedding matrix
$A=\\{{\bf{v}}_{1},{\bf{v}}_{2},\dots,{\bf{v}}_{n}\\}$ with the objective
function:

$\Psi(A)=\sum_{i\in V}\
[\alpha_{i}||{\bf{v}}_{i}-\hat{\bf{v}}_{i}||^{2}+\sum_{(i,j)\in
E}\beta_{ij}||\hat{\bf v}_{i}-\hat{\bf v}_{j}||^{2}]$ (5)

where an original word vector is ${\bf v}_{i}$, its synonym vector is ${\bf
v}_{j}$, and inferred word vector is $\hat{\bf v}_{i}$.

Our lexicon of synonymity constraints was automatically constructed from the
data we collected in Section 3.1.1 by adding an entry for each emotion
adjective with its synonyms crowdsourced as previously described. We also
added entries for the phrases in the lexicon for retrofitting, as follows, for
each emotion phrase according to the intensifiers classification described in
Section 2.1. For instance, “not happy” had as an entry in the lexicon the
phrases “unhappy” and “not happy at all” while “completely cheerful” had in
its entry phrases like “totally cheerful”, “totally happy”, “completely
happy”, among others (since “completely” and “totally” are in the same class).
We apply retrofitting on phrases which origin from the extension of the space
with À La Carte.

Focus phrase |  joy |  sadness |  anger |  fear |  surprise |  disgust
---|---|---|---|---|---|---
so happy | $+.73$ | $-.43$ | $-.50$ | $-.51$ | $-.10$ | $-.66$
not happy | $-.52$ | $+.41$ | $+.02$ | $-.16$ | $-.11$ | $+.17$
kinda happy | $+.53$ | $-.70$ | $-.67$ | $-.55$ | $-.47$ | $-.76$
so sad | $-.50$ | $+.66$ | $+.04$ | $+.13$ | $-.16$ | $+.03$
not sad | $+.55$ | $-.60$ | $-.57$ | $-.55$ | $-.45$ | $-.52$
kinda sad | $-.41$ | $+.62$ | $-.02$ | $+.02$ | $-.18$ | $+.02$
so angry | $-.39$ | $+.26$ | $+.86$ | $+.21$ | $+.02$ | $+.63$
not angry | $+.40$ | $-.36$ | $-.82$ | $-.17$ | $-.27$ | $-.45$
kinda angry | $-.80$ | $+.68$ | $+.84$ | $+.32$ | $+.08$ | $+.64$
so scared | $-.07$ | $+.15$ | $-.21$ | $+.83$ | $+.15$ | $-.13$
not scared | $+.35$ | $-.35$ | $-.28$ | $-.66$ | $-.53$ | $-.33$
kinda scared | $+.03$ | $+.10$ | $-.03$ | $+.71$ | $-.13$ | $-.13$
so surprised | $+.34$ | $-.27$ | $-.09$ | $+.02$ | $+.81$ | $.00$
not surprised | $+.60$ | $-.56$ | $-.50$ | $-.60$ | $-.83$ | $-.60$
kinda surprised | $+.37$ | $-.37$ | $-.17$ | $+.01$ | $+.72$ | $-.20$
so disgusted | $-.11$ | $-.02$ | $+.30$ | $+.16$ | $+.33$ | $+.88$
not disgusted | $+.42$ | $-.39$ | $-.42$ | $-.36$ | $-.36$ | $-.84$
kinda disgusted | $-.16$ | $+.08$ | $+.41$ | $-.01$ | $+.08$ | $+.80$
Table 1: Example queries with their BWS crowdsourced scores for the modifiers
“so”, “kinda” and the negation “not”. For every focus phrase we have an
intensity score between $-1$ and $+1$ for each emotion. The focus phrases are
shown in groups made around the emotion adjectives.

##  4 Results

In the following, we explore the Twitter corpus described previously, the
results of the BWS annotation of the pairs of degree adverbs and adjectives,
and finally we discuss our experimental setting and evaluation on the
downstream task of emotion intensity prediction on the different embedding
adaptation methods.

###  4.1 Corpus Analysis

The Twitter corpus described in Section 3.1 contains 34297941 tweets out of
which 2948397 contain emotion phrases. Most dominant are amplifiers (49%)
followed by downtoners ($24\%$) and negations ($19\%$). Only $8\%$ contain the
emotion adjectives in superlative or comparative.

Figure 2: Relative frequencies of the most common 30 modifiers in the Twitter
Corpus (from dark (infrequent) to yellow (frequent).

Figure 2 shows how often the top 30 modifiers are used with adjectives from
the basic set of emotions. We see that disgust is rarely downtoned and anger,
sadness, and surprise are amplified most often. Joy and fear are relatively
equally amplified, with joy being more negated and fear being more downtoned.
The amplifiers “so” and “really”, as well as the downtoners “just” and “kind
of/kinda” are frequently used. The downtoner “just” is the most frequently
used downtoner and acts at times as an amplifier, which could explain its
frequent use. We hypothesize that this is due to their use as fillers and
their grammaticalization (cf. Tagliamonte, 2006). Most frequently downtoned
emotion is surprise (which is often used in phrases like “a little surprised”,
“quite surprised”, “a bit surprised”).

Figure 3: Amplifiers and downtoners that vary the most in use with regards to
emotion

In Figure 3 and Figure 4 we observe that the use of modifiers with respect to
an emotion vary a lot within the same class of modifiers among both more
frequent and less frequent modifiers. In Figure 3, we observe that the focus
modifier “only” scales downward surprise the least, while all the other “true”
scaling adverbs are more impactful. Sadness is the emotion that is mostly
expressed through the focus adverb “only” in this setting. The figure also
(implicitly) shows that certain modifiers prefer certain adjectives, e.g. the
adjectives that express disgust, such as “disgusted” is mostly modified by
“absolutely”, “truly”, “utterly”, “pretty” and not by “extremely”,
“incredibly” or “only”. This distinction show the “harmony” between adjectives
and degree adverbs Quirk (1985).

Figure 4: Most frequent three amplifiers and downtoners used across all
emotions and their variation with respect to emotion.

Looking in more depth into the most frequent used amplifiers and downtoners in
Figure 4 we see that among the top used amplifiers “so”, “really”, “very” we
find that joy, anger, and disgust prefer “so” over “really” and “very”, the
emotions fear and surprise prefer “very” over “so” and “really” and sadness is
modified rather equally by the three amplifiers. Between the downtoners “kind
of” and “kinda” there is a notable difference in use for sadness, fear and
anger, with “kind of” being prefered over “kinda” in the context of sadness,
with the opposite holding true for fear.

| Spearman’s rank correlation
---|---
Emotion | w/ context | w/o context | between
anger | .84 | .82 | .88
fear | .84 | .73 | .81
joy | .90 | .86 | .91
sadness | .90 | .86 | .88
surprise | .71 | .71 | .81
disgust | .86 | .86 | .88
average | .84 | .80 | .86
Table 2: Split-half reliabilities and Spearman’s rank correlation between
these settings.

###  4.2 Annotation Analysis

Table 1 shows examples of phrases annotated with real-valued scores following
the annotation procedure described in Section 3.1. We see that we have scores
for each phrase in the context of each emotion. For instance, “kinda
surprised” has the score $-.37$ for sadness and $+.17$. We observe that the
negation “not” paired with any emotion adjective, excluding happy obtains a
positive score for joy, and a negative score for every other emotion. The
phrase “not happy” obtains a negative score of only $-.52$. In the complete
annotation results we include as negations also the phrase “not happy at all”,
which in this case gets closer to the lower limit of the potential scores.

We measure the reliability by randomly dividing the sets of 4 responses to
each question into two halves and comparing the Spearman rank correlation
coefficient between the two sets Kiritchenko and Mohammad (2016b). Both with
and without having access to context, the annotators mostly agree regarding
their annotations, as Table 2 shows in the first two columns. Lowest
reliability is achieved for surprise, with .71 Spearman’s rank correlation and
the highest for joy and sadness (.9). The reliability drops most when context
is not available for fear (by 11 percentage points).

Figure 5: Distribution of the aggregated emotion scores obtained by applying
the counting procedure BWS

Figure 5 shows the distribution of the scores assigned through the annotation
per emotion. We observe that disgust is mostly amplified and rarely negated
(only once). The outliers in each boxplot mostly correspond to negated
phrases.

Figure 6: Experimental Setup. The green arrow from Word2vec to the regressor
unit shows the information flow in the baseline. The black solid arrows show
the different experimental settings. The purple dashed arrows at the bottom
show the prediction phase.

###  4.3 Embedding Adaptations

Figure 6 summarizes our experimental setup. We build on top of pretrained
embeddings obtained with Word2vec Mikolov et al. (2013) (300d, negative
sampling, Google News corpus), fastText Bojanowski et al. (2017) (300d, news
corpora), or GloVe Pennington et al. (2014) (300d, Common Crawl). Each
embedding is then optionally augmented with phrase and subword embeddings and
fed into a CNN-LSTM model as proposed by Wu et al. (2018), trained on the
Affect in Tweets Dataset used at Sem Eval 2018 Task 1 Mohammad et al. (2018).
Their system achieved an average Pearson correlation score of 0.722, and
ranked 12/48 in the emotion intensity regression task.

Table 3 shows Spearman’s rank correlation between the predicted intensity
scores and the emotion scores obtained in the annotation of our Twitter corpus
or the EmoInt data Mohammad and Bravo-Marquez (2017b).

The fastText-based models underperform constantly on our Twitter dataset. For
GloVe embeddings, À La Carte (ALC) and Bag-of-Substrings (BoS) lead to a
substantial improvement, of 7pp (see Table 3, G vs. G+ALC) and 8pp (G vs.
G+BoS) over the baseline of using the pretrained embeddings unchanged. On
Word2vec embeddings BoS and ALC show the same improvement of 7pp (W2V vs.
W2V+ALC/BoS).

While on average, ALC and BoS can only substantially contribute based on GloVe
and Word2vec, this is not the case for individual emotions. For Word2vec,
sadness figures to be particularly challenging, leading to an overall
comparably low performance. Most importantly, we observe that our extensions
of the semantic spaces do not negatively affect the results on the EmoInt
dataset.

Unexpectedly, retrofitting does not help in all settings in our post-
processing pipeline except for fastText embeddings. We assume that is a
consequence of using a too small lexicon for retrofitting, and the method
would improve the embeddings if sentiment or emotion lexicons would be used
instead. However, this needs further investigation.

|  joy |  sadness |  anger |  fear |  average
---|---|---|---|---|---
| T EI | T EI | T EI | T EI | T EI
G | .20 .60 | .21 .59 | .24 .60 | .27 .61 | .23 .60
G+ALC | .23 .61 | .31 .63 | .33 .62 | .35 .62 | .30 .63
G+BoS | .24 .58 | .30 .60 | .34 .59 | .36 .57 | .31.59
G+ALC+RF | .19 .60 | .21 .61 | .26 .63 | .28 .61 | .24.61
G+BoS+RF | .19 .62 | .21 .60 | .28 .62 | .25 .61 | .23 .61
W2V | .16 .60 | .12 .59 | .19 .60 | .23 .63 | .18 .62
W2V+ALC | .20 .60 | .24 .64 | .28 .65 | .28 .64 | .25 .63
W2V+BoS | .20 .61 | .23 .64 | .28 .66 | .29 .60 | .25 .64
W2V+ALC+RF | .21 .60 | .25 .54 | .28 .69 | .28 .64 | .26 .62
W2V+BoS+RF | .16 .60 | .12 .61 | .24 .67 | .20 .60 | .18 .63
FT | .16 .58 | .14 .53 | .21 .65 | .22 .60 | .18 .61
FT+ALC | .16 .59 | .14 .52 | .21 .59 | .23 .62 | .19 .59
FT+BoS | .16 .60 | .14 .59 | .22 .63 | .23 .61 | .18 .62
FT+ALC+RF | .18 .54 | .16 .62 | .22 .64 | .25 .59 | .20 .60
FT+ BoS+RF | .16 .60 | .14 .57 | .22 .62 | .21 .57 | .18 .63
Table 3: Evaluation: Spearman’s rank correlation between predicted emotion
intensity scores and annotated scores on our dataset (T) or the EmoInt dataset
(EI). We report results only for the 4 emotions annotated in the EmoInt data.

##  5 Conclusion & Future Work

With this paper, we presented the first analysis of the distribution of degree
adverbs and negations on Twitter in the context of emotions. In addition, we
proposed a pipeline with different modules to expand embeddings particularly
for emotion phrases. Our evaluation shows substantial differences based on the
combination of input embeddings and the postprocessing method. Our pipeline
improves the results obtained while evaluating the downstream task of emotion
intensity prediction on our dataset. Finally, we contribute a novel emotion
phrase lexicon of high precision.

For future work we propose to analyze other baseline approaches, particularly
learning a composition function over pairs of adjectives with degree adverbs.
The modifiers could be considered as functions over adjectives and would be
represented as matrices.

Another further improvement of this work would be to expand this analysis to
verbal and nominal expressions of emotion, which we hypothesize as also being
frequent. In order to obtain meaningful representations for the phrases we
focus on, another natural next step is expanding the postprocessing pipeline
and including a comparison to other adaptation methods such as counterfitting
Mrkšić et al. (2016). Presumably, this will also generate additional insights
into the aspect that we were only able to show a limited improvement based on
retrofitting.

Given the recent advances in representing contextualized word embeddings as
functions computing dynamically the embeddings for words given their context,
we hypothesize and intend to further verify that these embeddings would be a
better choice for input to systems that predict intensity scores. It would be
interesting to compare models such as word embeddings from language models
(Elmo) Peters et al. (2018), bidirectional encoder representations from
transformers (BERT) Devlin et al. (2018), and generative pre-training OpenAI
(GPT) Radford et al. (2019) to the ones we already discussed, since the
contextualized embeddings assign a different vector for a word in each given
context. These approaches presumably produce a different vector for “happy” in
the context of “not” than in the content of “very” or “completely”.

Lastly, we plan to also adjust the lexica created such that it covers more
domains, sources, and languages.

## Acknowledgements

This research has been funded by the German Research Council (DFG), projects
SEAT (Structured Multi-Domain Emotion Analysis from Text, KL 2869/1-1). We
thank Jeremy Barnes, Evgeny Kim, Sean Papay, Sebastian Padó and Enrica Troiano
for fruitful discussions and the reviewers for the helpful comments.

## References

  * Aina et al. (2018) Laura Aina, Raffaella Bernardi, and Raquel Fernández. 2018.  A distributional study of negated adjectives and antonyms.  In _Proceedings of CLiC-it 2018 the 5th Italian Conference on Computational Linguistics_.
  * Carrillo-de Albornoz and Plaza (2013) Jorge Carrillo-de Albornoz and Laura Plaza. 2013.  An emotion-based model of negation, intensifiers, and modality for polarity and intensity classification.  _Journal of the American Society for Information Science and Technology_ , 64(8):1618–1633.
  * Athanasiadou (2007) Angeliki Athanasiadou. 2007.  On the subjectivity of intensifiers.  _Language sciences_ , 29(4):554–565.
  * Bojanowski et al. (2017) Piotr Bojanowski, Edouard Grave, Armand Joulin, and Tomas Mikolov. 2017.  Enriching word vectors with subword information.  _Transactions of the Association for Computational Linguistics_ , 5:135–146.
  * Councill et al. (2010) Isaac Councill, Ryan McDonald, and Leonid Velikovich. 2010.  What’s great and what’s not: learning to classify the scope of negation for improved sentiment analysis.  In _Proceedings of the Workshop on Negation and Speculation in Natural Language Processing_ , pages 51–59. University of Antwerp.
  * Devlin et al. (2018) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2018.  Bert: Pre-training of deep bidirectional transformers for language understanding.  _arXiv preprint arXiv:1810.04805_.
  * Dragut and Fellbaum (2014) Eduard Dragut and Christiane Fellbaum. 2014.  The role of adverbs in sentiment analysis.  In _Proceedings of Frame Semantics in NLP: A Workshop in Honor of Chuck Fillmore (1929-2014)_ , pages 38–41. Association for Computational Linguistics.
  * Ehrlich (1980) E. Ehrlich. 1980.  _Oxford American Dictionary_.  Oxford University Press.
  * Ekman (1992) Paul Ekman. 1992.  An argument for basic emotions.  _Cognition & emotion_, 6(3-4):169–200.
  * Faruqui et al. (2015) Manaal Faruqui, Jesse Dodge, Sujay Kumar Jauhar, Chris Dyer, Eduard Hovy, and Noah A. Smith. 2015.  Retrofitting word vectors to semantic lexicons.  In _Proceedings of the 2015 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 1606–1615. Association for Computational Linguistics.
  * Felbo et al. (2017) Bjarke Felbo, Alan Mislove, Anders Søgaard, Iyad Rahwan, and Sune Lehmann. 2017\.  Using millions of emoji occurrences to learn any-domain representations for detecting sentiment, emotion and sarcasm.  In _Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing_ , pages 1615–1625, Copenhagen, Denmark. Association for Computational Linguistics.
  * Khodak et al. (2018) Mikhail Khodak, Nikunj Saunshi, Yingyu Liang, Tengyu Ma, Brandon Stewart, and Sanjeev Arora. 2018.  A la carte embedding: Cheap but effective induction of semantic feature vectors.  In _Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 12–22. Association for Computational Linguistics.
  * Kiritchenko and Mohammad (2016a) Svetlana Kiritchenko and Saif Mohammad. 2016a.  The effect of negators, modals, and degree adverbs on sentiment composition.  In _Proceedings of the 7th Workshop on Computational Approaches to Subjectivity, Sentiment and Social Media Analysis_ , pages 43–52. Association for Computational Linguistics.
  * Kiritchenko and Mohammad (2016b) Svetlana Kiritchenko and Saif M. Mohammad. 2016b.  Capturing reliable fine-grained sentiment associations by crowdsourcing and best–worst scaling.  In _Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 811–817. Association for Computational Linguistics.
  * Kiritchenko and Mohammad (2016c) Svetlana Kiritchenko and Saif M. Mohammad. 2016c.  Sentiment composition of words with opposing polarities.  In _Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 1102–1108. Association for Computational Linguistics.
  * Louviere et al. (2015) Jordan J. Louviere, Terry N. Flynn, and A. A. J. Marley. 2015.  _Best-Worst Scaling: Theory, Methods and Applications_.  Cambridge University Press.
  * Mikolov et al. (2013) Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, and Jeff Dean. 2013.  Distributed representations of words and phrases and their compositionality.  In C. J. C. Burges, L. Bottou, M. Welling, Z. Ghahramani, and K. Q. Weinberger, editors, _Advances in Neural Information Processing Systems 26_ , pages 3111–3119. Curran Associates, Inc.
  * Mohammad and Bravo-Marquez (2017a) Saif Mohammad and Felipe Bravo-Marquez. 2017a.  Wassa-2017 shared task on emotion intensity.  In _Proceedings of the 8th Workshop on Computational Approaches to Subjectivity, Sentiment and Social Media Analysis_ , pages 34–49. Association for Computational Linguistics.
  * Mohammad and Bravo-Marquez (2017b) Saif Mohammad and Felipe Bravo-Marquez. 2017b.  Wassa-2017 shared task on emotion intensity.  In _Proceedings of the 8th Workshop on Computational Approaches to Subjectivity, Sentiment and Social Media Analysis_ , pages 34–49. Association for Computational Linguistics.
  * Mohammad et al. (2018) Saif Mohammad, Felipe Bravo-Marquez, Mohammad Salameh, and Svetlana Kiritchenko. 2018.  Semeval-2018 task 1: Affect in tweets.  In _Proceedings of The 12th International Workshop on Semantic Evaluation_ , pages 1–17. Association for Computational Linguistics.
  * Morante and Sporleder (2012) Roser Morante and Caroline Sporleder. 2012.  Modality and negation: An introduction to the special issue.  _Computational Linguistics_ , 38(2):223–260.
  * Mrkšić et al. (2016) Nikola Mrkšić, Diarmuid Ó Séaghdha, Blaise Thomson, Milica Gašić, Lina M. Rojas-Barahona, Pei-Hao Su, David Vandyke, Tsung-Hsien Wen, and Steve Young. 2016.  Counter-fitting word vectors to linguistic constraints.  In _Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 142–148. Association for Computational Linguistics.
  * Napoli and Ravetto (2017) Maria Napoli and Miriam Ravetto. 2017.  New insights on intensification and intensifiers.  _Exploring Intensification: Synchronic, diachronic and cross-linguistic perspectives_ , 189:1.
  * Nevalainen (2008) Terttu Nevalainen. 2008.  Social variation in intensifier use: constraint on-ly adverbialization in the past? _English Language & Linguistics_, 12(2):289–315.
  * Nevalainen and Rissanen (2002) Terttu Nevalainen and Matti Rissanen. 2002.  Fairly pretty or pretty fair? on the development and grammaticalization of english downtoners.  _Language Sciences_ , 24(3-4):359–380.
  * Paradis (1997) Carita Paradis. 1997.  _Degree modifiers of adjectives in spoken British English_, volume 92 of _Lund Studies in English_.  Lund University Press.
  * Paradis (2000) Carita Paradis. 2000.  It’s well weird: Degree modifiers of adjectives revisited: The nineties.  _Language and computers_ , 30:147–160.
  * Paradis (2001) Carita Paradis. 2001.  Adjectives and boundedness.
  * Pennington et al. (2014) Jeffrey Pennington, Richard Socher, and Christopher Manning. 2014.  Glove: Global vectors for word representation.  In _Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 1532–1543. Association for Computational Linguistics.
  * Pertejo and Martínez (2014) Paloma Núñez Pertejo and Ignacio M Palacios Martínez. 2014.  That’s absolutely crap, totally rubbish: The use of the intensifiers absolutely and totally in the spoken language of british adults and teenagers.  _Functions of Language_ , 21(2):210–237.
  * Peters (1994) Hans Peters. 1994.  _Degree Adverbs in Early Modern English_, volume 13.  De Gruyter Mouton.
  * Peters et al. (2018) Matthew E. Peters, Mark Neumann, Mohit Iyyer, Matt Gardner, Christopher Clark, Kenton Lee, and Luke Zettlemoyer. 2018.  Deep contextualized word representations.  In _Proc. of NAACL_.
  * Plutchik (1980) Robert Plutchik. 1980.  A general psychoevolutionary theory of emotion.  _Theories of emotion_ , 1(3-31):4.
  * Posner et al. (2005) Jonathan Posner, James A Russell, and Bradley S Peterson. 2005.  The circumplex model of affect: An integrative approach to affective neuroscience, cognitive development, and psychopathology.  _Development and psychopathology_ , 17(3):715–734.
  * Quirk (1985) R. Quirk. 1985.  _A Comprehensive grammar of the English language_.  General Grammar Series. Longman.
  * Radford et al. (2019) Alec Radford, Jeff Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. 2019.  Language models are unsupervised multitask learners.
  * Reitan et al. (2015) Johan Reitan, Jørgen Faret, Björn Gambäck, and Lars Bungum. 2015.  Negation scope detection for twitter sentiment analysis.  In _Proceedings of the 6th Workshop on Computational Approaches to Subjectivity, Sentiment and Social Media Analysis_ , pages 99–108. Association for Computational Linguistics.
  * Stevenson and Lindberg (2010) A. Stevenson and C.A. Lindberg. 2010.  _New Oxford American Dictionary, Third Edition_.  OUP USA.
  * Strohm and Klinger (2018) Florian Strohm and Roman Klinger. 2018.  An empirical analysis of the role of amplifiers, downtoners, and negations in emotion classification in microblogs.  In _The 5th IEEE International Conference on Data Science and Advanced Analytics, Special Track on Sentiment, Emotion, and Credibility of Information in Social Data_ , DSAA, Turin, Italy. IEEE.
  * Tagliamonte (2006) Sali A Tagliamonte. 2006.  “So cool, right?”: Canadian english entering the 21st century.  _Canadian Journal of Linguistics/Revue canadienne de linguistique_ , 51(2-3):309–331.
  * Wu et al. (2018) Chuhan Wu, Fangzhao Wu, Junxin Liu, Zhigang Yuan, Sixing Wu, and Yongfeng Huang. 2018.  Thu_ngn at semeval-2018 task 1: Fine-grained tweet sentiment intensity analysis with attention cnn-lstm.  In _Proceedings of The 12th International Workshop on Semantic Evaluation_ , pages 186–192. Association for Computational Linguistics.
  * Zhao et al. (2018) Jinman Zhao, Sidharth Mudgal, and Yingyu Liang. 2018.  Generalizing word embeddings using bag of subwords.  In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pages 601–606. Association for Computational Linguistics.
  * Zhu et al. (2014) Xiaodan Zhu, Hongyu Guo, Saif Mohammad, and Svetlana Kiritchenko. 2014.  An empirical study on the effect of negation words on sentiment.  In _Proceedings of the 52nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 304–313. Association for Computational Linguistics.


# Phase and Thermal Driven Transport across T-Shaped Double Quantum Dot
Josephson Junction

Bhupendra Kumar<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology, Roorkee, 247667, Uttarakhand, India  Sachin Verma
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology,
Roorkee, 247667, Uttarakhand, India  Ajay<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Technology, Roorkee, 247667, Uttarakhand, India

The phase and thermal driven transport properties of the T-shaped uncorrelated
double quantum dot Josephson junction are analyzed by using Keldysh non-
equilibrium Green’s function equation of motion technique. In this setup, we
have shown that the side-attached quantum dot provides an additional route for
electron transmission which is affecting the transport properties by adjusting
the interdot hopping between the main dot and the side dot. We began with
investigating the impact of interdot hopping on Andreev bound states and
Josephson supercurrent. When a small thermal bias is applied across the
superconducting leads, the system exhibits a finite thermal response which is
primarily due to the, thermally induced, quasi-particle current. The behavior
of the Josephson supercurrent and the quasi-particle current flowing through
the quantum dots is examined for various interdot hopping and thermal biasing.
Finally, the system is considered in an open circuit configuration where the
thermally driven quasi-particle current is compensated by the phase-driven
Josephson supercurrent and the thermophase effect is observed. The effect of
interdot hopping and the position of quantum dot energy level on the
thermophase Seebeck coefficient is investigated.

Keywords: Quantum dots, T-shaped Josephson junction, thermophase Seebeck
effect

##  1 Introduction

A quantum dot (QD)-based Josephson junction is made up of two Bardeen-Cooper-
Schrieffer (BCS) superconducting leads separated by a quantum dot. A DC
Josephson supercurrent can flow across the junction without applying potential
difference, as the Josephson supercurrent largely depends on the phase
difference between the superconductors [1, 2]. Quantum dots have discrete
energy levels and can be controlled by tunning their gate voltage or by
changing the size of quantum dot [3, 4]. Single-electron (quasi-particle)
tunneling and cooper pair tunneling are responsible for charge transport in
quantum dot-based Josephson junctions. Charge transport in these single
quantum dot-based Josephson junctions have been studied extensively both
theoretically [5, 6, 7, 8, 9, 10, 11, 12, 13] as well as experimentally [14,
15, 16, 17, 18, 19, 20, 21]. Using quantum dots allow one to control the
current flowing through Josephson junctions. Further, various authors have
explored the charge transport properties of double quantum dot Josephson
junctions. In such junctions, the double quantum dots are coupled with
superconducting leads in series, parallel, and T-shaped geometry [22, 23, 24,
25, 26, 27, 28, 29]. References [30, 31, 32] provides recent detailed reviews
on the charge transport properties of single and double quantum dot based
Josephson junctions.

On the other hand, due to the limited temperature range, the thermal transport
properties of the ordinary S-I-S Josephson junction and quantum dot-based
junctions have not been widely explored. Despite this limitation, the thermal
transport properties of Josephson junctions are recently attracting great
attention [33, 34, 35, 36, 37, 38, 39]. Recently, very few studies have been
conducted on the thermal transport properties of quantum dot-based Josephson
junctions i.e when both the leads are superconducting [40, 41]. Further, the
thermoelectric transport properties of systems where the quantum dot is
coupled between a normal metal and BCS superconductor (N-QD-S) [42, 43, 44]
and ferromagnet and BCS superconductor (F-QD-S) [45, 46, 47, 48] have been
studied recently. Further, the thermoelectric transport properties of multi-
dot and multi-terminal systems with one superconducting lead are also gaining
attention [49, 50, 51, 52, 53].

Phase and thermal-driven transport properties of quantum dot-based Josephson
junctions can be analyzed through a combination of three currents: quasi-
particle current, interference current, and pair current [33, 40, 54]. A
thermal gradient induces the quasi-particle current to flow across the
junction. Quasi-particle is the only current that contributes to thermal
transport in the S-QD-S system. The interference current, which is due to
coupling between quasi-particle and condensate shows no contribution to
thermal transport and will be ignored in the present study. The pair current
or Josephson supercurrent flows across the junction in absence of voltage
difference or temperature difference between the superconducting leads. This
Josephson current depends on the phase difference between the superconducting
leads. In reference, [40, 41] author demonstrates that quantum dot-based
Josephson junction shows a significant thermal response on applying the
thermal biasing across the superconducting leads. By applying the thermal
biasing across the superconducting leads, there appears to be a phase gradient
across the superconductors. Therefore, a supercurrent will flow across the
junction and it will counterbalance the thermally induced quasi-particle
current. This is the open circuit configuration for S-QD-S system i.e. total
current $I_{C}=0$. The cancellation of quasi-particle current by reverse
supercurrent is the origin of concept of thermophase Seebeck effect in quantum
dot-based Josephson junctions as shown by the schematic diagram in figure 1.

Figure 1: Schematic diagram showing the origin of thermophase Seebeck effect.
The cancellation of quasi-particle current, induced by the temperature
difference, by phase-driven reverse supercurrent is the origin of thermophase.

In the present work, we provide a study of the low-temperature phase and
thermal-driven transport properties of a system where uncorrelated double
quantum dots are coupled with two superconducting leads in T-shaped geometry
(figure 2). In this configuration the main quantum dot ($QD_{1}$) is directly
coupled with the leads and the side quantum dot ($QD_{2}$) is coupled with the
main dot but not with the superconducting leads. To study the thermal
transport properties of a T-shaped double quantum dot Josephson junction, we
have employed Keldysh non-equilibrium Green’s equation of motion technique
[55, 56]. First, we have studied the interdot hopping dependence of Andreev
Bound States (ABS) and supercurrent. Next, total current (which is the
combination of quasi-particle current and Josephson supercurrent) is
calculated for different temperature differences $\Delta T$ and interdot
hopping (t). Finally, the thermophase Seebeck coefficient (TPSC) for the
T-shaped double quantum dot Josephson junction is analyzed. Since, both the
leads are superconductors, so we have taken into account the temperature
dependence of the superconducting gap ($\Delta_{\alpha}$) having a background
temperature always less than the superconducting critical temperature $T_{c}$.

Figure 2: Schematic diagram for double quantum dot in T-shaped geometry
coupled with superconducting leads. Main dot ($QD_{1}$) is directly coupled
with superconducting leads while the side dot ($QD_{2}$) is only coupled with
the main dot.

This paper can be read in the following order: in the preceding section 2, we
provide a detailed description of model Hamiltonian and theoretical formalism.
Section 3, discusses numerical results. Lastly, section 4 concludes the
present work.

##  2 THEORETICAL FORMULATION

To calculate the transport properties of the T-shaped double quantum dot
Josephson junction, we use the generalized Anderson + BCS Hamiltonian in
second quantization formalism.

$\hat{H}=\hat{H}_{leads}+\hat{H}_{QD}+\hat{H}_{tunnel}+\hat{H}_{interdot-
hopping}$ (1)

where

$\hat{H}_{leads}=\sum_{k\sigma,\alpha}\epsilon_{k\alpha}c^{\dagger}_{k\sigma,\alpha}c_{k\sigma,\alpha}-\left(\sum_{k\alpha}\Delta_{\alpha}c^{\dagger}_{k\uparrow,\alpha}c^{\dagger}_{-k\downarrow,\alpha}+h.c\right)$

$\hat{H}_{QD}=\sum_{i=1}^{i=2}\sum_{\sigma}\epsilon_{d_{i\sigma}}d^{\dagger}_{i\sigma}d_{i\sigma}$
$\hat{H}_{tunnel}=\sum_{k\sigma,\alpha}V_{k,\alpha}c^{\dagger}_{k\sigma,\alpha}d_{1\sigma}+h.c$

$\hat{H}_{interdot-
hopping}=\sum_{\sigma}t(d^{\dagger}_{1\sigma}d_{2\sigma}+h.c)$

where h.c stands for Hermitian conjugate.

${\hat{H}}_{leads}$ is the Hamiltonian for left and right superconducting
leads ($\alpha\in L,R$). The first term, describes the free electrons in the
superconducting leads; ${c^{\dagger}}_{k\sigma,\alpha}(c_{k\sigma,\alpha})$ is
the creation (annihilation) operator of electron with spin $\sigma$ and wave
vector $\vec{k}$ and energy $\epsilon_{k,\alpha}$. The second term in
${\hat{H}}_{leads}$ is the BCS term and gives the information about the
interaction between Cooper pair with temperature dependent superconducting
energy gap which is given by [41]

$\Delta_{\alpha}(T_{\alpha})=\Delta_{0}\tanh\left\\{1.74\sqrt{\left(\frac{k_{B}T_{c}}{k_{B}T_{\alpha}}-1\right)}\right\\}$
(2)

where $\Delta_{0}=\lvert\Delta_{0}\rvert e^{i\phi_{\alpha}}$ is the
superconducting gap at absolute zero temperature with $\phi_{\alpha}$ as the
phase of superconducting leads, $T_{c}$ is superconducting critical
temperature, $T_{\alpha}$ is the temperature of superconducting leads and
$k_{B}$ is the Boltzman constant.

$\hat{H}_{QD}$ is the Hamiltonian for main dot ($QD_{1}$) and side dot
($QD_{2}$). $QD_{1}$ (i=1) and $QD_{2}$ (i=2) has energy
$\epsilon_{d_{i\sigma}}$ with ${d^{\dagger}}_{i\sigma}(d_{i\sigma}$) as the
fermionic creation operator (annihilation operator) of electrons with spin
$\sigma$ and $n_{d_{i\sigma}}=d^{\dagger}_{i\sigma}d_{i\sigma}$ is the number
operator. We have neglected the onsite Coulomb interaction $U_{i}$ for
simplification.

$\hat{H}_{tunnel}$ is tunneling Hamiltonian between the energy level of the
main dot and superconducting leads with interaction strength
$V_{k_{1}},\alpha$. Further, we have consider the symmetric coupling strength
of $QD_{1}$ to the left and right leads i.e. $V_{k_{1};L}$=$V_{k_{1};R}$.

The last term $\hat{H}_{interdot-hopping}$ describes the interaction of
electrons of two quantum dots via a hopping like term of strength t. Note
that, there is no direct interaction between superconducting leads and
$QD_{2}$.

Bogoliubov transformation is used to diagonalize the BCS part of the
Hamiltonian. For this, we introduce a new fermionic quasiparticle operator
$\beta$ with coefficient $u_{k}$ and $v_{k}$ which satisfies the normalization
condition $\lvert u_{k}\rvert^{2}$ \+ $\lvert v_{k}\rvert^{2}$ = 1

$c_{k\uparrow}=u_{k}^{*}\beta_{k\uparrow}+v_{k}\beta^{\dagger}_{-k\downarrow}$
(3)
$c^{\dagger}_{-k\downarrow}=u_{k}\beta^{\dagger}_{-k\downarrow}-v^{*}_{k}\beta_{k\uparrow}$
(4)

By replacing the fermionic operator $c_{k\uparrow}$ and
$c^{\dagger}_{-k\downarrow}$ with new quasi-particle operator, we get the
effective Hamiltonian

$\displaystyle\hat{H}=$
$\displaystyle\sum_{k\alpha}E_{k\alpha}(\beta^{\dagger}_{k\uparrow,\alpha}\beta_{k\uparrow,\alpha}+\beta^{\dagger}_{-k\downarrow,\alpha}\beta_{-k\downarrow,\alpha})$
(5)
$\displaystyle+\sum_{k\alpha}(V_{k\alpha}u^{*}_{k}\beta^{\dagger}_{k\uparrow,\alpha}d_{1\uparrow}+V_{k\alpha}u^{*}_{k}\beta^{\dagger}_{-k\downarrow,\alpha}d_{1\downarrow})$
$\displaystyle+\sum_{k\alpha}(V^{*}_{k\alpha}u_{k}d^{\dagger}_{1\uparrow}\beta_{k\uparrow,\alpha}+V^{*}_{k\alpha}u_{k}d^{\dagger}_{1\downarrow}\beta_{-k\downarrow,\alpha})$
$\displaystyle+\sum_{k\alpha}V_{k\alpha}v_{k}(\beta_{-k\downarrow,\alpha}d_{1\uparrow}-\beta_{k\uparrow,\alpha}d_{1\downarrow})$
$\displaystyle+\sum_{k\alpha}V^{*}_{k\alpha}v^{*}_{k}(d^{\dagger}_{1\uparrow}\beta^{\dagger}_{-k\downarrow,\alpha}-d^{\dagger}_{1\downarrow}\beta^{\dagger}_{k\uparrow,\alpha})$
$\displaystyle+\epsilon_{d_{1}}(d^{\dagger}_{1\uparrow}d_{1\uparrow}+d^{\dagger}_{1\downarrow}d_{1\downarrow})$
$\displaystyle+\epsilon_{d_{2}}(d^{\dagger}_{2\uparrow}d_{2\uparrow}+d^{\dagger}_{2\downarrow}d_{2\downarrow})$
$\displaystyle+t(d^{\dagger}_{1\uparrow}d_{2\uparrow}+d^{\dagger}_{1\downarrow}d_{2\downarrow}+d^{\dagger}_{2\uparrow}d_{1\uparrow}+d^{\dagger}_{2\downarrow}d_{1\downarrow})$

where
$E_{k\alpha}=\sqrt{{\epsilon^{2}}_{k\alpha}+\lvert\Delta_{\alpha}\rvert^{2}}$
is excitation quasi-particle energy of the superconducting leads.The
coefficients $u_{k}$ and $v_{k}$ can be expressed as

$\lvert
u_{k}\rvert^{2}=\frac{1}{2}\left(1+\frac{\epsilon_{k,\alpha}}{\sqrt{\epsilon^{2}_{k,\alpha}+\lvert\Delta_{\alpha}\rvert^{2}}}\right)$
(6) $\lvert
v_{k}\rvert^{2}=\frac{1}{2}\left(1-\frac{\epsilon_{k,\alpha}}{\sqrt{\epsilon^{2}_{k,\alpha}+\lvert\Delta_{\alpha}\rvert^{2}}}\right)$
(7)

We have used Green’s equation of motion method (EOM) to solve the above
effective Hamiltonian (Eq. 5). To calculate the spectral and transport
properties of T-shaped double quantum dot Josephson junction system, we need
single-particle retarded Green’s function of main quantum dot $QD_{1}$ which
is defined in Zubarev notation [55, 56, 57]

$\langle\langle
d_{\sigma}(t);d^{\dagger}_{\sigma}(0)\rangle\rangle=-i\theta(t)\langle[d_{\sigma}(t),d^{\dagger}_{\sigma}(0)]_{+}\rangle$

The Fourier transform of the above retarded Green’s function should satisfy
the following equation of motion

$\omega\langle\langle
d_{\sigma}|d^{\dagger}_{\sigma}\rangle\rangle_{\omega}=\langle\\{d_{\sigma},d^{\dagger}_{\sigma}\\}_{+}\rangle+\langle\langle[d_{\sigma},H]_{-}|d^{\dagger}_{\sigma}\rangle\rangle_{\omega}$
(8)

In Nimbu space, Green’s function of the main quantum dot can be represented by
a $2\times 2$ matrix:

$\bm{G^{r}(\omega)}=\begin{pmatrix}{G}^{r}_{11}(\omega)&{G}^{r}_{12}(\omega)\\\
\\\
{G}^{r}_{21}(\omega)&G^{r}_{22}(\omega)\end{pmatrix}=\begin{pmatrix}\langle\langle
d_{1\uparrow}|d^{\dagger}_{1\uparrow}\rangle\rangle&\langle\langle
d_{1\downarrow}|d_{\uparrow}\rangle\rangle\\\ \\\ \langle\langle
d^{\dagger}_{1\downarrow}|d^{\dagger}_{1\uparrow}\rangle\rangle&\langle\langle
d^{\dagger}_{1\downarrow}|d_{1\downarrow}\rangle\rangle\end{pmatrix}$ (9)

By using Green’s function EOM technique (Eq. 8), we obtain the following
coupled equations for $QD_{1}$

$\displaystyle(\omega-\epsilon_{d_{1}})\langle\langle
d_{1\uparrow}|d^{\dagger}_{1\uparrow}\rangle\rangle=$ $\displaystyle
1+\sum_{k\alpha}V_{k\alpha}u^{*}_{k}\langle\langle\beta_{k\uparrow,\alpha}|d^{\dagger}_{1\uparrow}\rangle\rangle$
(10)
$\displaystyle+\sum_{k\alpha}V_{k\alpha}v_{k}\langle\langle\beta^{\dagger}_{-k\downarrow,\alpha}|d^{\dagger}_{1\uparrow}\rangle\rangle$
$\displaystyle+t\langle\langle
d_{2\uparrow}|d^{\dagger}_{1\uparrow}\rangle\rangle$

$\displaystyle(\omega-
E_{k\alpha})\langle\langle\beta_{k\uparrow,\alpha}|d^{\dagger}_{1\uparrow}\rangle\rangle=$
$\displaystyle\sum_{k\alpha}V^{*}_{k\alpha}u_{k}\langle\langle
d_{1\uparrow}|d^{\dagger}_{1\uparrow}\rangle\rangle$ (11)
$\displaystyle+\sum_{k\alpha}V_{k\alpha}v_{k}\langle\langle
d^{\dagger}_{1\downarrow}|d^{\dagger}_{1\uparrow}\rangle\rangle$

$\displaystyle(\omega+E_{k\alpha})\langle\langle\beta^{\dagger}_{-k\downarrow,\alpha}|d^{\dagger}_{1\uparrow}\rangle\rangle=$
$\displaystyle\sum_{k\alpha}V^{*}_{k\alpha}v^{*}_{k}\langle\langle
d_{1\uparrow}|d^{\dagger}_{1\uparrow}\rangle\rangle$ (12)
$\displaystyle-\sum_{k\alpha}V_{k\alpha}u^{*}_{k}\langle\langle
d^{\dagger}_{1\downarrow}|d^{\dagger}_{1\uparrow}\rangle\rangle$

$\displaystyle(\omega+\epsilon_{d_{1}})\langle\langle
d^{\dagger}_{1\downarrow}|d^{\dagger}_{1\uparrow}\rangle\rangle=$
$\displaystyle\sum_{k\alpha}V^{*}_{k\alpha}v^{*}_{k}\langle\langle\beta_{k\uparrow,\alpha}|d^{\dagger}_{1\uparrow}\rangle\rangle$
(13)
$\displaystyle+\sum_{k\alpha}V^{*}_{k\alpha}u_{k}\langle\langle\beta^{\dagger}_{-k\downarrow,\alpha}|d^{\dagger}_{1\uparrow}\rangle\rangle$
$\displaystyle-t\langle\langle
d^{\dagger}_{2\downarrow}|d^{\dagger}_{1\uparrow}\rangle\rangle$

$\displaystyle(\omega-\epsilon_{d_{2}})\langle\langle
d_{2\uparrow}|d^{\dagger}_{1\uparrow}\rangle\rangle=$ $\displaystyle
t\langle\langle d_{1\uparrow}|d^{\dagger}_{1\uparrow}\rangle\rangle$ (14)

$\displaystyle(\omega-\epsilon_{d_{2}})\langle\langle
d^{\dagger}_{2\downarrow}|d^{\dagger}_{1\uparrow}\rangle\rangle=$
$\displaystyle-t\langle\langle
d^{\dagger}_{1\downarrow}|d^{\dagger}_{1\uparrow}\rangle\rangle$ (15)

After solving these closed set of coupled equations (Eq. 10-15) the expression
for single particle retarded Green’s function of the main dot with spin
$\sigma=\uparrow$ can be written as:

${G^{r}}_{11}(\omega)=\langle\langle
d_{1\uparrow}|d^{\dagger}_{1\uparrow}\rangle\rangle=\frac{\omega+\epsilon_{d_{1}}-\frac{t^{2}}{\omega+\epsilon_{d_{2}}}-I_{1}}{(\omega+\epsilon_{d_{1}}-\frac{t^{2}}{\omega+\epsilon_{d_{2}}}-I_{1})(\omega-\epsilon_{d_{1}}-\frac{t^{2}}{\omega-\epsilon_{d_{2}}}-I_{2})-(I_{3})^{2}}$
(16)

In above Green’s function (Eq. 16) $I_{1}$, $I_{2}$ are the diagonal, and
$I_{3}$ is the off-diagonal part of self-energy, which corresponds to the
induced pairing, due to the coupling between the quantum dot and
superconducting leads.
The expressions for $I_{1}$, $I_{2}$ and $I_{3}$ are

$I_{1}=\sum_{k\alpha}\lvert V_{k\alpha}\rvert^{2}\left({\frac{\lvert
u_{k}\rvert^{2}}{\omega+E_{k\alpha}}}+{\frac{\lvert v_{k}\rvert^{2}}{\omega-
E_{k\alpha}}}\right)$ (17) $I_{2}=\sum_{k\alpha}\lvert
V_{k\alpha}\rvert^{2}\left({\frac{\lvert u_{k}\rvert^{2}}{\omega-
E_{k\alpha}}}+{\frac{\lvert v_{k}\rvert^{2}}{\omega+E_{k\alpha}}}\right)$ (18)
$I_{3}=\sum_{k\alpha}\lvert
V_{k\alpha}\rvert^{2}u_{k}v^{*}_{k}\left({\frac{1}{\omega-
E_{k\alpha}}}-{\frac{1}{\omega+E_{k\alpha}}}\right)$ (19)

Transforming the summation into integration and by defining the tunneling rate
$\Gamma_{\alpha}=2\pi\rho_{0}\lvert V_{k\alpha}\rvert^{2}$ where $\rho_{0}$ is
the density of states in normal metallic state, we obtained the following
expressions for $I_{1}$, $I_{2}$ and $I_{3}$:

$I_{1}=I_{2}=-\sum_{\alpha\in
L,R}\left(\frac{\Gamma_{\alpha}\omega}{\sqrt{\Delta_{\alpha}^{2}-\omega^{2}}}\theta(\Delta-\lvert\omega\rvert)+i(\frac{\Gamma_{\alpha}\omega}{\sqrt{\omega^{2}-\Delta_{\alpha}^{2}}}\theta(\lvert\omega\rvert-\Delta)\right)$
(20) $I_{3}=-\sum_{\alpha\in
L,R}\left(\frac{\Gamma_{\alpha}\Delta_{\alpha}}{\sqrt{\Delta_{\alpha}^{2}-\omega^{2}}}\theta(\Delta-\lvert\omega\rvert)+i(\frac{\Gamma_{\alpha}\Delta_{\alpha}}{\sqrt{\omega^{2}-\Delta_{\alpha}^{2}}}\theta(\lvert\omega\rvert-\Delta)\right)$
(21)

Similarly, the off-diagonal Green’s function ${G^{r}}_{21}(\omega)$ can be
calculated with the help of coupled equations (10-15).

$\displaystyle{G^{r}}_{21}(\omega)=\langle\langle
d^{\dagger}_{1\downarrow}|d^{\dagger}_{1\uparrow}\rangle\rangle=\frac{I_{3}}{\big{[}({\omega+\epsilon_{d_{1\downarrow}}-\frac{t^{2}}{\omega+\epsilon_{d_{2\downarrow}}}+I_{1})({\omega-\epsilon_{d_{1\uparrow}}-\frac{t^{2}}{\omega-\epsilon_{d_{2\uparrow}}}+I_{2})-(I_{3})^{2}\big{]}}}}$
(22)

Other Green’s function can be obtained with the help of the following
relations:

$\displaystyle{G}^{r}_{22}(\omega)=-{G}^{r}_{11}(-\omega)^{*}$ (23)
$\displaystyle{G}^{r}_{12}(\omega)={G}^{r}_{21}(-\omega)^{*}$

By equating the denominator of single particle retarded Green’s function
${G}^{r}_{11}$ (Eq. 16) equal to zero, the energies of Andreev Bound States
(ABS) can be analyzed which is discussed in section III.

As discussed in section 1, for quantum dot-based Josephson junction, we can
distribute the charge current in three parts [33, 40]:

$\displaystyle I_{C}=$ $\displaystyle I_{QP}(\epsilon_{di},\phi,T,\Delta
T)+I_{pair-QP}(\epsilon_{di},\phi,T,\Delta T)\cos^{2}\frac{\phi}{2}$ (24)
$\displaystyle+I_{SC}(\epsilon_{di},\phi,T,\Delta T)$

The first term is quasi-particle current and is responsible for thermal
transport in this system. The second term has no contribution to thermal
transport and can be neglected here. The third term in current is due to
cooper pair tunneling and is responsible for the supercurrent in the system.
Here, we present the full derivation of Josephson supercurrent and quasi-
particle current. For the current expression, we follow the formulation given
in reference [9, 55, 58]. The retarded Green’s function $\bm{G^{r}}$ (Eq. 9)
of main quantum dot can also be written as:

$\bm{G^{r}}=[\bm{g}^{r^{-1}}-\bm{\Sigma}^{r}]^{-1}=\frac{1}{A(\omega)}\begin{pmatrix}\bm{g}^{r}_{22}(\omega)-\bm{\Sigma}^{r}_{22}&\bm{\Sigma}^{r}_{12}\\\
\\\
\bm{\Sigma}^{r}_{21}&\bm{g}^{r}_{{11}}(\omega)-\bm{\Sigma}^{r}_{11}\end{pmatrix}$
(25)

Where $\bm{G^{r}}$ is Green’s function of the main quantum dot with leads and
$\bm{g^{r}}$ is Green’s function of the main quantum dot without leads.
$\bm{\Sigma}^{r}=\bm{\Sigma}^{r}_{L}+\bm{\Sigma}^{r}_{R}$ is the retarded
self-energy due to coupling between the main quantum dot and superconducting
leads. $A(\omega)$ is the denominator of Green’s function and is defined as

$A(\omega)=det[\bm{g}^{r^{-1}}-\bm{\Sigma}^{r}]$ (26)

From Eq’s. (9,16,22,23), $\bm{g}^{r}$ for T-shaped double quantum dot
Josephson junction, can be written as

$\bm{g}^{r^{-1}}=\begin{pmatrix}\omega-\epsilon_{d_{1}}-\frac{t^{2}}{\omega-\epsilon_{d_{1}}}+i0^{+}&0\\\
\\\
0&\omega+\epsilon_{d_{1}}-\frac{t^{2}}{\omega+\epsilon_{d_{1}}}+i0^{+}\end{pmatrix}$
(27)

The retarded self-energy ($\bm{{\Sigma}^{r}}_{\alpha}$) of $\alpha$ lead is
defined as

$\bm{{\Sigma}^{r}}_{\alpha}(\omega)=-\frac{i}{2}\Gamma_{\alpha}\rho(\omega)\begin{pmatrix}1&-\frac{\Delta}{\omega}e^{-i\phi_{\alpha}}\\\
\\\ -\frac{\Delta}{\omega}e^{i\phi_{\alpha}}&1\end{pmatrix}$ (28)

For simplicity, we assumed that both superconductors are identical and have a
finite phase difference, $\Gamma_{\alpha}=\Gamma_{L}=\Gamma_{R}=\Gamma$, and
$\Delta_{\alpha}=\Delta_{L}=\Delta_{R}=\Delta$

$\bm{\Sigma^{r}}=\bm{\Sigma^{r}}_{L}+\bm{\Sigma^{r}}_{R}=-i\Gamma\rho(\omega)\begin{pmatrix}1&-\frac{\Delta}{\omega}cos\frac{\phi}{2}\\\
\\\ -\frac{\Delta}{\omega}cos\frac{\phi}{2}&1\end{pmatrix}$ (29)

$\bm{\tilde{\Sigma}^{r}}=\bm{\Sigma^{r}}_{L}+\bm{\Sigma^{r}}_{R}=-i\Gamma\rho(\omega)\begin{pmatrix}0&-\frac{\Delta}{\omega}(-i)sin\frac{\phi}{2}\\\
\\\ -\frac{\Delta}{\omega}(i)sin\frac{\phi}{2}&0\end{pmatrix}$ (30)

where $\Gamma$ is the symmetric coupling strength between the main quantum dot
and superconducting leads. $\rho(\omega)$ is the modified density of state of
the superconductor and can be defined as

$\rho(\omega)=\begin{cases}\frac{\lvert\omega\rvert}{\sqrt{\omega^{2}-\Delta^{2}}},\lvert\omega\rvert>\Delta\\\
\\\
\frac{\lvert\omega\rvert}{i\sqrt{{\Delta^{2}}-\omega^{2}}},\lvert\omega\rvert<\Delta\end{cases}$
(31)

By taking the current conservation condition $I_{L}+I_{R}=0$, the general
formula of Josephson supercurrent for the superconductor-quantum dot systems
can be written as [9]:

$I_{SC}=\frac{1}{2}(I_{L}-I_{R})=\frac{e}{h}\int{d\omega}Re[\bm{G\tilde{\Sigma}}]^{<}_{11-22}$
(32)

Where $\bm{G}$ and $\bm{\Sigma}_{\alpha}$ are $2\times 2$ Fourier transformed
Nambu matrices.

By using the Langreth relation we can write [55];

$Re[\bm{G\tilde{\Sigma}}]^{<}_{11-22}=Re[\bm{{G^{<}}\tilde{\Sigma}^{a}}+\bm{G^{r}\tilde{\Sigma}^{<}}]_{11}-Re[\bm{{G^{<}}\tilde{\Sigma}^{a}}+\bm{G^{r}\tilde{\Sigma}^{<}}]_{22}$
(33)

with $(\bm{G^{r}})^{\dagger}=\bm{G^{a}}$,
$\bm{(\tilde{\Sigma}^{r})}^{\dagger}=\bm{\tilde{\Sigma}^{r}}$

On applying the fluctuation-dissipation theorem, we can define:
$\bm{G^{<}}=f(\omega)[\bm{G}^{a}-\bm{G}^{r}]$ and
$\tilde{\bm{\Sigma}}^{<}=f(\omega)[\tilde{\bm{\Sigma}}^{a}-\tilde{\bm{\Sigma}}^{r}]$,
where $f(\omega)=\frac{1}{e^{{\omega}/{k_{B}T}}+1}$ is Fermi distribution
function and $\bm{G}^{a}$, $\bm{G}^{<}$ are advanced and lesser Green’s
function respectively.

Substituting the above relations in Eq. (33) and using Eq. (25) and Eq. (30)

$Re[\bm{G\tilde{\Sigma}^{r}}]^{<}_{11-22}=f(\omega)Re[(\bm{\tilde{\Sigma}}^{r}\bm{G}^{r})^{\dagger}-\bm{G}^{r}\bm{\tilde{\Sigma}}^{r}]_{11}-f(\omega)Re[(\bm{\tilde{\Sigma}}^{r}\bm{G}^{r})^{\dagger}-\bm{G}^{r}\bm{\tilde{\Sigma}}^{r}]_{22}$
(34)
$(\bm{\tilde{\Sigma}}^{r}\bm{G}^{r})^{\dagger}_{11}=\frac{1}{A(\omega)}\begin{pmatrix}0&-\frac{\Delta}{\omega}(-i)sin\frac{\phi}{2}\\\
\\\
-\frac{\Delta}{\omega}(i)sin\frac{\phi}{2}&0\end{pmatrix}\begin{pmatrix}\bm{g^{r}}_{22}(\omega)-\bm{\Sigma}^{r}_{22}&\bm{\Sigma}^{r}_{12}\\\
\\\
\bm{\Sigma}^{r}_{21}&\bm{g}^{r}_{11}(\omega)-\bm{\Sigma}^{r}_{11}\end{pmatrix}$
(35)

Similarly, we obtain the $(\bm{G^{r}}\bm{\tilde{\Sigma}^{r}})_{11}$ and the
$Re[(\bm{\tilde{\Sigma}}^{r}\bm{G}^{r})^{\dagger}-\bm{G}^{r}\bm{\tilde{\Sigma}}^{r}]_{22}$
in Eq. (34). Finally it can be written as:

$Re[\bm{G\tilde{\Sigma}}]^{<}_{11-22}=\frac{\Gamma^{2}\Delta^{2}}{\omega^{2}-\Delta^{2}}f(\omega)sin{\phi}Im\left[\frac{-1}{A(\omega)}\right]$
(36)

By substituting Eq. (36) in Eq. (32) and taking both spin up and spin down
into consideration, the final expression for Josephson supercurrent can be
written as

$I_{SC}=\frac{2e}{h}\int{d{\omega}}\frac{\Gamma^{2}\Delta^{2}}{\omega^{2}-\Delta^{2}}f(\omega)sin{\phi}Im\left[\frac{-1}{A(\omega)}\right]$
(37)

Similiar to Josephson supercurrent, we can also derive the expression of
quasi-particle current for superconductor-quantum dot systems by following the
formalism given in reference [55, 59, 58].

$I_{\alpha}=\frac{4ie}{h}\int{d\omega}\Gamma_{\alpha}\frac{\lvert\omega\rvert}{\sqrt{\omega^{2}-\Delta^{2}_{\alpha}}}\\{f_{\alpha}(\omega)[G^{r}_{11}(\omega)-G^{a}_{11}(\omega)]+G^{<}_{11}(\omega)\\}$
(38)

By current conservation condition i.e. $I_{L}+I_{R}=0$, the symmetrize form of
current can be written as

$I_{QP}=I_{L}=-I_{R}=\frac{1}{2}(I_{L}-I_{R})$

$\begin{split}I_{QP}=\frac{2ie}{h}\int{d\omega}\\{[\Gamma_{L}\frac{\lvert\omega\rvert}{\sqrt{\omega^{2}-\Delta^{2}_{L}}}f_{L}(\omega)-\Gamma_{R}\frac{\lvert\omega\rvert}{\sqrt{\omega^{2}-\Delta^{2}_{R}}}f_{R}(\omega)][G^{r}_{11}(\omega)-G^{a}_{11}(\omega)]\\\
+[\Gamma_{L}(\omega)-\Gamma_{R}(\omega)]G^{<}_{11}(\omega)\\}\end{split}$ (39)

For the symmetric case ($\Gamma_{\alpha}=\Gamma_{L}=\Gamma_{R}=\Gamma$ and
$\Delta_{\alpha}=\Delta_{L}=\Delta_{R}=\Delta$) the above equation becomes

$\displaystyle I_{QP}=\frac{2ie}{h}\int
d{\omega}{\Gamma}\frac{\lvert\omega\rvert}{\sqrt{\omega^{2}-\Delta^{2}}}\\{f_{L}(\omega)-f_{R}(\omega)\\}[G^{r}_{11}(\omega)-G^{a}_{11}(\omega)]$
(40)

where $f_{L}(\omega)=\frac{1}{e^{{\omega}/{k_{B}(T+\Delta T)}}+1}$,
$f_{R}(\omega)=\frac{1}{e^{{\omega}/{k_{B}T}}+1}$ are the Fermi-distribution
function of left and right superconducting leads and
$[G^{r}_{11}(\omega)-G^{a}_{11}(\omega)]=Im[-G^{r}_{11}(\omega)]$.

The expression of quasi-particle current for linear response regime (thermal
gradient between superconducting leads will be small, $\Delta T\rightarrow 0$)
can be simplified by using power series expansion of the Fermi-distribution
function of left and right superconducting leads:

$I_{QP}=\frac{2e}{h}\int{d\omega}\frac{df(\omega)}{dT}\Gamma
Re({\rho(\omega)})Im[-G^{r}_{{11}}(\omega)]\Delta T$ (41)

In section 1, we have already discussed the origin of the thermophase effect.
Within the linear response regime, the thermophase Seebeck coefficient
$S_{\phi}$ is defined analogous to the thermovoltage Seebeck coefficient and
can be simplified using Eq. (24);

$S_{\phi}=-\left(\frac{\Delta\phi}{\Delta
T}\right)_{I=0}=\frac{dI_{QP}/{d\Delta T}}{I_{SC}}$ (42)

##  3 RESULTS and DISCUSSION

In this section, we present the numerical results and discussion for the
T-shaped double quantum dot Josephson junction. Transport properties are
discussed for uncorrelated quantum dots. The superconducting gap at absolute
zero temperature ($\Delta_{0}$) is considered as the energy unit, where
$\Delta_{0}$ is in meV.

Figure 3: (a) Energy of Andreev bound states and (b) Josephson current as a
function of superconducting phase difference $(\phi)$ for different values of
interdot hopping (t) at absolute zero temperature. The other parameters are
$\Gamma=0.1\Delta_{0}$, $\epsilon_{d_{1}}=0$,
$\epsilon_{d_{2}}=0.5\Delta_{0}$.

In figure 3, we plot the energy of Andreev Bound states (ABS) and Josephson
current as a function of superconducting phase difference ($\phi$) for
different values of interdot hopping (t). First, when $QD_{2}$ is decoupled
from $QD_{1}$ i.e. t=0, the system shows the properties of the usual S-QD-S
Josephson junction. In this case, the supercurrent is discontinuous at
$\phi=\pm\pi$ and upper and lower ABS crosses the Fermi energy ($\omega=0$).
Thus for t=0, the system acts as a perfect transmitting channel. When $QD_{2}$
is coupled with $QD_{1}$ then supercurrent shows sinusoidal behaviour and a
finite gap is generated between lower and upper ABS at $\phi=\pm\pi$. Further
increasing the value of interdot hopping, this supercurrent is suppressed.
This suppression of supercurrent is because coupling $QD_{1}$ with the
$QD_{2}$, the electrons tends to tunnel into $QD_{2}$. This causes
interference destruction between two transport channels and as a result the
supercurrent decreases. Thus for $t>0$, the system does not acts as a perfect
transmitting channel. The suppression of supercurrent can also be explained in
terms of the splitting of QDs energy level due to interdot hopping. When
$t\neq 0$, the equivalent level of QDs splits into two levels i.e.
$\bar{\epsilon_{d_{i}}}=\epsilon_{d_{i}}\pm t$. The equivalent energy level
$\bar{\epsilon_{d_{i}}}$ moves far away from the Fermi level with increasing
interdot hopping and supercurrent decreases.

Figure 4: Total current (supercurrent + quasi-particle current) as a function
of superconducting phase difference $(\phi)$ for different interdot hopping
(t) and thermal biasing $\Delta T$. Insets ($b_{1}$) and ($b_{2}$) show the
separate behaviour of Josephson current and quasi-particle current with $\phi$
and $\Delta T$. Inset in figure (d) shows the variation of quasi-particle
current as a function of interdot hopping. The other parameters are
$\Gamma=0.1\Delta_{0}$, $k_{B}T=0.2\Delta_{0}$,
$\epsilon_{d_{1}}=\epsilon_{d_{2}}=-1.0\Delta_{0}$.

In figure 4 [(a)-(d)], we plot the total current as a function of
superconducting phase difference ($\phi$) for several values of thermal
biasing ($\Delta T$) and interdot hopping. First, by decoupling $QD_{2}$ form
$QD_{1}$ (Fig 4 a) the results of S-QD-S system are reproduced [40]. When both
$QD_{1}$ and $QD_{2}$ are coupled, then there is suppression in the magnitude
of total current with increasing interdpt hopping, which is discussed in the
previous paragraph. In insets 4(b), the individual behavior of Josephson
current and quasi-particle current are shown. It is observed that Josephson
current is almost independent of thermal biasing ($\Delta T$) and largely
depends on superconducting phase difference ($\phi$). On other hand, the
quasi-particle current totally depends on the thermal biasing ($\Delta T$). It
is important to note that in total current, the sinusoidal nature is due to
Josephson current, and the shift in magnitude is due to quasi-particle
current. With an increase in interdot hopping, the amplitude of the
supercurrent and total current vanishes. In inset 4(d), the behaviour of
quasi-particle current is plotted as a function of interdot hopping. It is
observed that quasi-particle current first increases with interdot hopping and
then attain a maximum value for $k_{B}T\sim{t}$, and then decreases with a
further increase in the value of interdot hopping.

Figure 5: The variation of thermophase Seebeck coefficient ($S_{\phi}$) with
$QD_{1}$ energy level $\epsilon_{d_{1}}$ for different interdot hopping (t).
The other parameters are $\Gamma=0.1\Delta_{0}$, $k_{B}T=0.2\Delta_{0}$,
$\epsilon_{d_{2}}=0.5\Delta_{0}$.

As discussed previously, the origin of the thermophase effect is due to the
vanishing total current in open circuit configuration i.e. thermal-driven
quasi-particle current is compensated by the phase-driven Josephson
supercurrent flowing in the reverse direction. In figure 5 and 6, we have
analyzed thermophase Seebeck coefficient $(S_{\phi})$ of S-TDQD-S in linear
response regime for uncorrelated quantum dots.

In figure 5, we have plotted the thermophase Seebeck coefficient (TPSC) as a
function of $QD_{1}$ energy level for different interdot hopping. It is
observed that TPSC ($S_{\phi}$) peaks are highest for t=0 i.e., when $QD_{2}$
is decoupled from $QD_{1}$ [40]. When $QD_{2}$ is coupled with $QD_{1}$, TPSC
peaks start decreasing with increasing interdot hopping (t). To achieve a high
thermophase peak Josephson current should compensate the quasi-particle
current totally. As we have shown that Josephson supercurrent decreases with
increasing interdot hopping, therefore it compensates less quasi-particle
current. Thus, TPSC peaks decrease and produce a shift in peaks with
increasing interdot hopping.

Figure 6: The variation of thermophase Seebeck coefficient ($S_{\phi}$) with
$QD_{1}$ energy level $\epsilon_{d_{1}}$ for different values of $QD_{2}$
energy level $\epsilon_{d_{2}}$ and $t$. The other parameters are
$\Gamma=0.1\Delta_{0}$, $k_{B}T=0.2\Delta_{0}$.

The magnitude of TPSC not only depends on interdot hopping but also depends on
the position of $QD_{2}$ energy levels whether it lies above or below the
Fermi level. In figure 6, we plot the thermophase Seebeck coefficient
($S_{\phi}$) as a function of $QD_{1}$ energy level for different values of
$QD_{2}$ energy level. When $QD_{2}$ is decoupled from $QD_{1}$, then system
reproduces the results of S-QD-S for TPSC ($S_{\phi}$). For finite interdot
hopping, the magnitude of TPSC peaks are enhanced when $QD_{2}$ energy level
lies below the Fermi level. The enhancement of TPSC peaks can be explained as
follows: when $QD_{2}$ energy level lies below the Fermi level, the equivalent
or effective level lies close to the Fermi level which supports the resonant
cooper pair tunneling. Therefore thermally induced quasi-particle is
compensated by Josephson supercurrent completely i.e. large magnitude of TPSC
peaks. These results for different values of $QD_{2}$ energy levels can be
directly compared with the TPSC plots as discussed in figure (5).

##  4 CONCLUSION

We have addressed the phase-driven and thermal-driven transport properties
through a T-shaped double quantum dot Josephson junction. For uncorrelated
quantum dots, the impact of interdot hopping on Andreev bound states (ABS) and
Josephson supercurrent are investigated. For a finite value of interdot
hopping, Josephson supercurrent exhibits sinusoidal nature while ABS shows a
finite gap around the Fermi level. The magnitude of Josephson supercurrent
decreases with increasing interdot hopping because the electrons have a
tendency to tunnel into the side dot with increasing interdot hopping, which
results in interference destruction between two transport channels. Further,
this system exhibits a finite thermal response when a small thermal biasing
($\Delta T$) is applied across the superconducting leads. The quasi-particle
current flows across the junction due to thermal biasing, while the Josephson
current is almost insensitive to thermal biasing. With increasing thermal
biasing, the quasi-particle current produces a finite shift in the magnitude
of the total current. Also, the magnitude of the total current decreases with
increasing interdot hopping because the equivalent level of quantum dots moves
further away from the Fermi level and thus cooper pair tunneling is
suppressed.

Finally, we investigate the influence of interdot hopping and quantum dot
energy levels on the thermophase Seebeck coefficient (TPSC). The magnitude of
TPSC ($S_{\phi}$) decreases with increasing interdot hopping when the energy
levels of the side dot ($QD_{2}$) lie above the Fermi level and increase when
energy levels of the side dot ($QD_{2}$) lie below the Fermi level. In the
later case, when energy levels lie below the Fermi level, the equivalent level
($\epsilon_{d_{i}}\pm t$) of the quantum dot moves towards the Fermi level.
Thus, the cooper pair tunneling increases with interdot hopping, and quasi-
particle current is completely compensated by the Josephson current and the
magnitude of TPSC peaks is enhanced.

We believe that the results presented in this study can be tested
experimentally with the advancement in nano-fabrication techniques. In this
paper, we consider the uncorrelated quantum dots, and the effect of the
Coulomb correlation will be explored in future work. The present study can
also be extended to investigate the thermal transport properties in systems
where double quantum dots are coupled with superconducting leads in series,
and parallel geometry and also for multi-terminal configurations. The concept
of thermophase effect in quantum dot-based Josephson junction can be useful
for future low-temperature thermal applications [60, 61, 62, 63] and need
further investigation.

Acknowledgments

The authors acknowledge the financial support from the research project DST-
SER-1644-PHY 2021-22. Bhupendra Kumar also acknowledges the support from the
Ministry of Education (MoE), India, in the form of a Ph.D. fellowship.

## References

  * [1] B. D. Josephson.  Possible new effects in superconductive tunnelling.  Physics Letters, 1:251–253, 7 1962.
  * [2] Philip W Anderson.  How josephson discovered his effect.  Phys. Today, 23(11):23–29, 1970.
  * [3] Leo P Kouwenhoven, DG Austing, and Seigo Tarucha.  Few-electron quantum dots.  Reports on Progress in Physics, 64(6):701, 2001.
  * [4] Marc A. Kastner.  Artificial atoms.  Physics Today, 46, 1993.
  * [5] AV Rozhkov and Daniel P Arovas.  Josephson coupling through a magnetic impurity.  Physical review letters, 82(13):2788, 1999.
  * [6] E Vecino, A Martín-Rodero, and A Levy Yeyati.  Josephson current through a correlated quantum level: Andreev states and $\pi$ junction behavior.  Physical Review B, 68(3):035105, 2003.
  * [7] Mahn-Soo Choi, Minchul Lee, Kicheon Kang, and Wolfgang Belzig.  Kondo effect and josephson current through a quantum dot between two superconductors.  Physical Review B, 70(2):020502, 2004.
  * [8] Jong Soo Lim and Mahn-Soo Choi.  Andreev bound states in the kondo quantum dots coupled to superconducting leads.  Journal of Physics: Condensed Matter, 20(41):415225, 2008.
  * [9] Yu Zhu, Qing-feng Sun, and Tsung-han Lin.  Andreev bound states and the $\pi$-junction transition in a superconductor/quantum-dot/superconductor system.  Journal of Physics: Condensed Matter, 13(39):8783, 2001.
  * [10] Christoph Karrasch, Akira Oguri, and Volker Meden.  Josephson current through a single anderson impurity coupled to bcs leads.  Physical Review B, 77(2):024517, 2008.
  * [11] DY Vodolazov and FM Peeters.  Superconducting rectifier based on the asymmetric surface barrier effect.  Physical Review B, 72(17):172508, 2005.
  * [12] Yasunari Tanuma, Yukio Tanaka, and Koichi Kusakabe.  Josephson current through a nanoscale quantum dot contacted by conventional superconductors.  Physica E: Low-dimensional Systems and Nanostructures, 40(2):257–260, 2007.
  * [13] A Dhyani, BS Tewari, et al.  Interplay of the single particle and josephson cooper pair tunneling on supercurrent across the superconducting quantum dot junction.  Physica E: Low-dimensional Systems and Nanostructures, 42(2):162–166, 2009.
  * [14] Jorden A Van Dam, Yuli V Nazarov, Erik PAM Bakkers, Silvano De Franceschi, and Leo P Kouwenhoven.  Supercurrent reversal in quantum dots.  Nature, 442(7103):667–670, 2006.
  * [15] K Grove-Rasmussen, H Ingerslev Jørgensen, and PE Lindelof.  Kondo resonance enhanced supercurrent in single wall carbon nanotube josephson junctions.  New Journal of Physics, 9(5):124, 2007.
  * [16] A Eichler, M Weiss, S Oberholzer, C Schönenberger, A Levy Yeyati, JC Cuevas, and A Martín-Rodero.  Even-odd effect in andreev transport through a carbon nanotube quantum dot.  Physical review letters, 99(12):126602, 2007.
  * [17] Yulin Ma, Tianqi Cai, Xiyue Han, Yaowen Hu, Hongyi Zhang, Haiyan Wang, Luyan Sun, Yipu Song, and Luming Duan.  Andreev bound states in a few-electron quantum dot coupled to superconductors.  Physical Review B, 99(3):035413, 2019.
  * [18] J-D Pillet, P Joyez, MF Goffman, et al.  Tunneling spectroscopy of a single quantum dot coupled to a superconductor: From kondo ridge to andreev bound states.  Physical Review B, 88(4):045101, 2013.
  * [19] Eduardo JH Lee, Xiaocheng Jiang, Manuel Houzet, Ramón Aguado, Charles M Lieber, and Silvano De Franceschi.  Spin-resolved andreev levels and parity crossings in hybrid superconductor–semiconductor nanostructures.  Nature nanotechnology, 9(1):79–84, 2014.
  * [20] R Delagrange, R Weil, A Kasumov, M Ferrier, H Bouchiat, and R Deblock.  0-$\pi$ quantum transition in a carbon nanotube josephson junction: Universal phase dependence and orbital degeneracy.  Physical Review B, 93(19):195437, 2016.
  * [21] DB Szombati, S Nadj-Perge, Diane Car, SR Plissard, EPAM Bakkers, and LP Kouwenhoven.  Josephson $\phi$0-junction in nanowire quantum dots.  Nature Physics, 12(6):568–572, 2016.
  * [22] Shu-guang Cheng and Qing-feng Sun.  Josephson current transport through t-shaped double quantum dots.  Journal of Physics: Condensed Matter, 20(50):505202, 2008.
  * [23] Feng Chi and Shu-Shen Li.  Current–voltage characteristics in strongly correlated double quantum dots.  Journal of applied physics, 97(12):123704, 2005.
  * [24] Yu Zhu, Qing-feng Sun, and Tsung-han Lin.  Probing spin states of coupled quantum dots by a dc josephson current.  Physical Review B, 66(8):085306, 2002.
  * [25] Rosa López, Mahn-Soo Choi, and Ramón Aguado.  Josephson current through a kondo molecule.  Physical Review B, 75(4):045132, 2007.
  * [26] Stephanie Droste, Sabine Andergassen, and Janine Splettstoesser.  Josephson current through interacting double quantum dots with spin–orbit coupling.  Journal of Physics: Condensed Matter, 24(41):415301, 2012.
  * [27] Gagan Rajput, Rajendra Kumar, et al.  Tunable josephson effect in hybrid parallel coupled double quantum dot-superconductor tunnel junction.  Superlattices and Microstructures, 73:193–202, 2014.
  * [28] Minchul Lee, Rosa López, Ramón Aguado, Mahn-Soo Choi, et al.  Josephson current in strongly correlated double quantum dots.  Physical review letters, 105(11):116803, 2010.
  * [29] JC Estrada Saldaña, A Vekris, G Steffensen, R Žitko, P Krogstrup, J Paaske, K Grove-Rasmussen, and J Nygård.  Supercurrent in a double quantum dot.  Physical review letters, 121(25):257701, 2018.
  * [30] Silvano De Franceschi, Leo Kouwenhoven, Christian Schönenberger, and Wolfgang Wernsdorfer.  Hybrid superconductor–quantum dot devices.  Nature nanotechnology, 5(10):703–711, 2010.
  * [31] A Martín-Rodero and A Levy Yeyati.  Josephson and andreev transport through quantum dots.  Advances in Physics, 60(6):899–958, 2011.
  * [32] V Meden.  The anderson–josephson quantum dot—a theory perspective.  Journal of Physics: Condensed Matter, 31(16):163001, 2019.
  * [33] Glen D Guttman, Benny Nathanson, Eshel Ben-Jacob, and David J Bergman.  Thermoelectric and thermophase effects in josephson junctions.  Physical Review B, 55(18):12691, 1997.
  * [34] F Giazotto, JWA Robinson, JS Moodera, and FS Bergeret.  Proposal for a phase-coherent thermoelectric transistor.  Applied Physics Letters, 105(6):062602, 2014.
  * [35] Maria José Martínez-Pérez, Antonio Fornieri, and Francesco Giazotto.  Rectification of electronic heat current by a hybrid thermal diode.  Nature nanotechnology, 10(4):303–307, 2015.
  * [36] F Giazotto, TT Heikkilä, and FS Bergeret.  Very large thermophase in ferromagnetic josephson junctions.  Physical review letters, 114(6):067001, 2015.
  * [37] G Marchegiani, P Virtanen, F Giazotto, and M Campisi.  Self-oscillating josephson quantum heat engine.  Physical Review Applied, 6(5):054014, 2016.
  * [38] G Marchegiani, A Braggio, and F Giazotto.  Phase-tunable thermoelectricity in a josephson junction.  Physical Review Research, 2(4):043091, 2020.
  * [39] Alexander G Bauer and Björn Sothmann.  Phase-dependent transport in thermally driven superconducting single-electron transistors.  Physical Review B, 104(19):195418, 2021.
  * [40] Yaakov Kleeorin, Yigal Meir, Francesco Giazotto, and Yonatan Dubi.  Large tunable thermophase in superconductor–quantum dot–superconductor josephson junctions.  Scientific Reports, 6(1):1–7, 2016.
  * [41] Mathias Kamp and Björn Sothmann.  Phase-dependent heat and charge transport through superconductor–quantum dot hybrids.  Physical Review B, 99(4):045428, 2019.
  * [42] M. Krawiec.  Thermoelectric transport through a quantum dot coupled to a normal metal and bcs superconductor.  Acta Physica Polonica A, 114, 2008.
  * [43] Sun-Yong Hwang, Rosa López, and David Sánchez.  Cross thermoelectric coupling in normal-superconductor quantum dots.  Physical Review B, 91(10):104518, 2015.
  * [44] Sachin Verma and Ajay Singh.  Non-equilibrium thermoelectric transport across normal metal–quantum dot–superconductor hybrid system within the coulomb blockade regime.  Journal of Physics: Condensed Matter, 34(15):155601, 2022.
  * [45] Sun-Yong Hwang, David Sánchez, and Rosa López.  A hybrid superconducting quantum dot acting as an efficient charge and spin seebeck diode.  New Journal of Physics, 18(9):093024, 2016.
  * [46] Sun-Yong Hwang, Rosa López, and David Sánchez.  Large thermoelectric power and figure of merit in a ferromagnetic–quantum dot–superconducting device.  Physical Review B, 94(5):054506, 2016.
  * [47] Sun-Yong Hwang, David Sánchez, and Rosa López.  Nonlinear electric and thermoelectric andreev transport through a hybrid quantum dot coupled to ferromagnetic and superconducting leads.  The European Physical Journal B, 90(10):1–7, 2017.
  * [48] Piotr Trocha and Józef Barnaś.  Spin-dependent thermoelectric phenomena in a quantum dot attached to ferromagnetic and superconducting electrodes.  Physical Review B, 95(16):165439, 2017.
  * [49] Wei-Ping Xu, Yu-Ying Zhang, Qiang Wang, Zhi-Jian Li, and Yi-Hang Nie.  Thermoelectric effects in triple quantum dots coupled to a normal and a superconducting leads.  Physics Letters A, 380(7-8):958–964, 2016.
  * [50] Hui Yao, Chao Zhang, Peng-bin Niu, Zhi-Jian Li, and Yi-Hang Nie.  Enhancement of charge and spin seebeck effect in triple quantum dots coupling to ferromagnetic and superconducting electrodes.  Physics Letters A, 382(44):3220–3229, 2018.
  * [51] Karol Izydor Wysokiński.  Thermoelectric transport in the three terminal quantum dot.  Journal of Physics: Condensed Matter, 24(33):335303, 2012.
  * [52] G Michałek, Marcin Urbaniak, BR Bułka, T Domański, and KI Wysokiński.  Local and nonlocal thermopower in three-terminal nanostructures.  Physical Review B, 93(23):235440, 2016.
  * [53] Robert Hussein, Michele Governale, Sigmund Kohler, Wolfgang Belzig, Francesco Giazotto, and Alessandro Braggio.  Nonlocal thermoelectricity in a cooper-pair splitter.  Physical Review B, 99(7):075429, 2019.
  * [54] Glen D Guttman, Benny Nathanson, Eshel Ben-Jacob, and David J Bergman.  Phase-dependent thermal transport in josephson junctions.  Physical Review B, 55(6):3849, 1997.
  * [55] Hartmut Haug, Antti-Pekka Jauho, et al.  Quantum kinetics in transport and optics of semiconductors, volume 2.  Springer, Berlin, 2008.
  * [56] Leonid V Keldysh et al.  Diagram technique for nonequilibrium processes.  Sov. Phys. JETP, 20(4):1018–1026, 1965.
  * [57] Dmitrii Nikolaevich Zubarev.  Double-time green functions in statistical physics.  Soviet Physics Uspekhi, 3(3):320, 1960.
  * [58] Yigal Meir and Ned S Wingreen.  Landauer formula for the current through an interacting electron region.  Physical review letters, 68(16):2512, 1992.
  * [59] Kicheon Kang.  Transport through an interacting quantum dot coupled to two superconducting leads.  Physical Review B, 57(19):11891, 1998.
  * [60] Francesco Giazotto, Tero T Heikkilä, Arttu Luukanen, Alexander M Savin, and Jukka P Pekola.  Opportunities for mesoscopics in thermometry and refrigeration: Physics and applications.  Reviews of Modern Physics, 78(1):217, 2006.
  * [61] HT Quan, YD Wang, Yu-xi Liu, CP Sun, and Franco Nori.  Maxwell’s demon assisted thermodynamic cycle in superconducting quantum circuits.  Physical review letters, 97(18):180402, 2006.
  * [62] MJ Martínez-Pérez, P Solinas, and F Giazotto.  Coherent caloritronics in josephson-based nanocircuits.  Journal of Low Temperature Physics, 175(5):813–837, 2014.
  * [63] Antonio Fornieri, Christophe Blanc, Riccardo Bosisio, Sophie D’ambrosio, and Francesco Giazotto.  Nanoscale phase engineering of thermal transport with a josephson heat modulator.  Nature nanotechnology, 11(3):258–262, 2016.