labelinglabel

# Analysis of basic emotions in texts
based on bert vector representation

A. Artemov, A. Veselovskiy, I. Khasenevich, I. Bolokhov
Kognitivnie Sistemi
<EMAIL_ADDRESS>

###### Abstract

In the following paper the authors present a GAN-type model and the most
important stages of its development for the task of emotion recognition in
text. In particular, we propose an approach for generating a synthetic dataset
of all possible emotions combinations based on manually labelled incomplete
data.

Keywords: 7-dimensional emotional model, Pytorch, GAN, deep learning, cosine
similarity, Ekman, BERT, NLP, BRAIN2NLP, emotions, collisions, Big Data

## Introduction

There are several approaches to classification of human emotions. Authors
consider the methodology according to P.Ekman to be the most reasonable [4].
In accordance with it, the following seven human emotions can be
distinguished: fear, sadness, anger, disgust, calm, happiness, surprise. These
emotions were classified according to the analysis of facial expressions of a
person during events of different semantic meaning. The unambiguous perception
of facial expressions with different people and nations was the proof of the
sufficiency and validity of the seven basic emotions.

At the same time, although there is a number of solutions for emotional
recognition, a standard for the number of basic emotions for text data has not
yet been created. For example, in Rodrigo Masaru Ohashi’s work [7] the model
of bidirectional LSTM with a convolutional neural network layer was
successfully trained to identify 4 emotions: joy, fear, sadness and anger.
Chew-Yean Yam’s article “Emotion Detection and Recognition from Text Using
Deep Learning [2]” describes the neural network with 3 hidden layers of 125,
25 and 5 neurons for determining 5 emotions: anger, sadness, fear, happiness,
excitement. Jangwon Park [8] created a Huggingface Transformers based model
that predicts 7 emotions, the F1-measure of the model is 59.81 and 61.48 for
the train and test dataset, respectively. This model is based on the
GoEmotions [3] dataset with 27 emotion classes (see Table 5).

Let us assume, as a hypothesis, that the 7 basic emotions by Ekman are
applicable to texts. That is, the labelling of texts for this set of emotion
classes by a person is complete. Based on the possibility of obtaining such a
dataset, the task is to train a model that can predict the vector of seven
emotions for an arbitrary text. The result of this work is described in the
article.

A positive solution to this problem opens up prospects for combining different
data channels (audio and video) to determine emotions when machines are
communicating with humans. Moreover, of particular interest is the description
of information spread through means of mass communication (social networks),
taking into account the phenomenon of emotions social exchange.

##  1 Dataset vectorization

Initially, it is necessary to form a dataset for training the model. The
dataset should contain data about seven basic emotions for each certain piece
of text - a paragraph or a sentence. The authors couldn’t find a publicly
available dataset with this or similar data. Therefore, it was decided to form
it automatically using data on the emotional reaction of people, expressed in
texts. Such data were emoticons in texts from Twitter, YouTube and other
social networks.

The original texts were divided into sentences, emoticons were extracted from
each sentence. In order to vectorize emotions, a special dictionary was
compiled containing emoticons divided into emotion types according to uniquely
interpreted classes:

The classification of the texts’ emotions was carried out using the created
dictionary. By counting the number of encounters of each emotion in the text,
vectors of 7 emotions were generated for each sentence.

Text vectorization was carried out using our own NLP framework -
BRAIN2NLP111Russian cognitive processor for automated labelling of natural
text. The main functionality of BRAIN2NLP includes language detection, stem
extraction, morphological and syntactic labelling of text, extraction of
nominal entities and definition of a semantic vector of words.. The text
vectorization was based on the BERT language model trained by Google. Thus, at
the output, pairs of the following form were obtained: 512-dimensional vector
BERT to seven-dimensional vector of emotion. In total, more than 100,000 such
pairs were constructed. In the process, it turned out that the resulting
dataset contains a lot of conflicting data (collisions). Collisions are
examples of data, where the same properties of an object lead to different
classes or to a significant number of classes. The automation of the collision
search process required the development of a small script. The algorithm of
its work is given below.

  * 1.

Determine the allowed fraction of the k-number of classes to which objects in
the same cluster can belong.

  * 2.

Classify dataset objects by the vector of their properties (features).

  * 3.

Assign the same ID to dataset objects belonging to the same cluster.

  * 4.

Add up the values by class for objects with the same ID.

  * 5.

Estimate for each ID the mathematical expectation of the vector of values by
class.

  * 6.

Calculate the number of Z values of the class vector that is greater than or
equal to the expected value.

  * 7.

Evaluate each cluster, and where $Z>k$ mark it as a collision.

  * 8.

Assign a collision mark to all objects in the cluster.

Algorithm 1 Finding collisions in a dataset

For the purpose of "human" marking, 10,000 texts were selected, among which
there were only 6913 examples without collisions with a length of no more than
two sentences. The experts were asked to assign no more than 2 emotions to
each of these texts. But the dataset formed in this way also contained minor
examples of collisions, which were also eliminated [1]. As a result, a golden
dataset with 2813 examples was obtained.

##  2 Search for Architecture

BERT-vectors are successfully used for classification tasks of texts [9] and
their fragments [5]. Considering this fact, we have chosen these vectors as a
basic description of the space of text properties, which can be used to
predict the emotion vector.

During the search for the optimal solution for the classification of emotions,
different variants of embedding length BERT (lengths 768 and 512) and
different sizes of datasets were tested. Also, we have tested a certain number
of negative samples, various dictionaries for the extraction of emoticons,
normalization of the dataset, different model architectures, loss functions
and activation functions. As a result of this work, the following hypothesis
was formulated and confirmed in practice: It is possible to improve the
quality of classification by abandoning the attempt to normalize the
unbalanced initial data - the number of examples for a set of classes,
replacing it with uniformly generated BERT vectors for each set of classes.

Among all tested neural network architectures, Generative adversarial network
(GAN) [6] showed the best results - an architecture consisting of a generative
network and a discriminative network configured to work against each other.
Discriminative models learn the boundary between the classes; generative model
generates new data instances. A collision free dataset (Dataset_1) was used
for training the generative network and for final testing of the
discriminative network. In the formulation of the study, the vector of
emotions has 7 dimensions, we denote it as V1. Table 1 shows a list of
combinations of vector V1 values found in Dataset_1.

Table 1: Combinations of emotion vectors Dataset_1 # | fear | sadness | anger | disgust | calm | happiness | surprise
---|---|---|---|---|---|---|---
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1
1 | 0 | 0 | 0 | 0 | 0 | 1 | 0
2 | 0 | 0 | 0 | 0 | 0 | 1 | 1
3 | 0 | 0 | 0 | 0 | 1 | 0 | 0
4 | 0 | 0 | 0 | 0 | 1 | 0 | 1
5 | 0 | 0 | 0 | 0 | 1 | 1 | 0
6 | 0 | 0 | 0 | 0 | 1 | 1 | 1
7 | 0 | 0 | 0 | 1 | 0 | 0 | 0
8 | 0 | 0 | 0 | 1 | 0 | 0 | 1
9 | 0 | 0 | 0 | 1 | 1 | 0 | 0
10 | 0 | 0 | 1 | 0 | 0 | 0 | 0
11 | 0 | 0 | 1 | 0 | 0 | 0 | 1
12 | 0 | 0 | 1 | 0 | 0 | 1 | 0
13 | 0 | 0 | 1 | 0 | 0 | 1 | 1
14 | 0 | 0 | 1 | 0 | 1 | 0 | 0
15 | 0 | 0 | 1 | 0 | 1 | 0 | 1
16 | 0 | 0 | 1 | 0 | 1 | 1 | 0
17 | 0 | 0 | 1 | 1 | 0 | 0 | 0
18 | 0 | 0 | 1 | 1 | 0 | 0 | 1
19 | 0 | 1 | 0 | 0 | 0 | 0 | 0
20 | 0 | 1 | 0 | 0 | 0 | 0 | 1
21 | 0 | 1 | 0 | 0 | 0 | 1 | 0
22 | 0 | 1 | 0 | 0 | 1 | 0 | 0
23 | 0 | 1 | 0 | 0 | 1 | 0 | 1
24 | 0 | 1 | 0 | 0 | 1 | 1 | 0
25 | 0 | 1 | 0 | 1 | 0 | 0 | 0
26 | 0 | 1 | 1 | 0 | 0 | 0 | 0
27 | 0 | 1 | 1 | 0 | 0 | 0 | 1
28 | 0 | 1 | 1 | 1 | 0 | 0 | 0
29 | 1 | 0 | 0 | 0 | 0 | 0 | 0
30 | 1 | 0 | 0 | 0 | 0 | 0 | 1
31 | 1 | 0 | 0 | 0 | 0 | 1 | 1
32 | 1 | 0 | 0 | 0 | 1 | 0 | 0
33 | 1 | 0 | 0 | 0 | 1 | 0 | 1
34 | 1 | 0 | 0 | 1 | 0 | 0 | 0
35 | 1 | 0 | 1 | 0 | 0 | 0 | 1
36 | 1 | 1 | 0 | 0 | 0 | 0 | 0
37 | 1 | 1 | 1 | 0 | 0 | 0 | 0

Since 37 combinations222Not entirely confident in ”Les 36 situations
dramatiques” concept, we decided to generate all possible combinations of
emotions. do not make up a complete list of 128 combinations, the missing ones
had to be generated.

For this purpose, based on Dataset_1, we have trained a generator model for
known combinations of emotions. Then the combinations V1 were initiated, where
the initial - has the values [0, 0, 0, 0, 0, 0, 0], and the final one is [1,
1, 1, 1, 1, 1, 1]. The total number of vectors V1 was 128. Using the
generator, 128 BERT - V2 vectors were obtained corresponding to various V1
combinations.

Thus, a new dataset (Dataset_2) was formed, where each vector V1 corresponded
to the generated vector V2 (Figure 1). Dataset_2 was used for fine-tuning the
Generator model and Discriminator training, predicting the emotion vector from
the BERT vector of the text. Figure 1 shows the process of sequential training
of the Generator, and Figure 2 shows the Discriminator.

Figure 1: The process of generator training with the formation of Dataset_2
Figure 2: The process of joint training of the Generator and the Discriminator

Figure 3 shows graphs of the "descent rate" when training the Generator and
the Discriminator. The large value of the generator error is explained by the
fact that a simple product of vectors is used, and the error is considered by
MSE. The opposite situation is with the discriminator, where not only the
output value is normalized, but also the error is calculated by the cosine
measure.

a)

b)

Figure 3: Gradient descent of a) Generator model; b) Discriminator model

In order to move to a non-negative real number, the output vector of the
discriminator value was normalized as follows

$M_{ij}^{\ast}=\dfrac{M_{ij}-min|M_{ij}|}{\sum_{M_{ij}\in
V}(M_{ij}-min|M_{ij}|)}$ (1)

where V is a vector of forecast values.

##  3 Results

Model training was performed on 70% of the data with testing on the remaining
30%. The evaluation of the quality of the model is illustrated in Figure 4.
Comparative results of the model training process are presented in Table 2.

Table 2: Accuracy rating Emotion | Accuracy |  Figure 4: The matrix of the predictive distribution and the real value of the Discriminator based on Dataset_1
---|---|---
fear | 0.62
sadness | 0.70
anger | 0.63
disgust | 0.76
calm | 0.64
happiness | 0.68
surprise | 0.54
MEAN | 0.65
Table 3: The result of the model training process

Network | Dataset | Architecture | Learn
---|---|---|---
|  Collisions
---
Train |  Test |  |  1-st layer
---
|  2-nd layer
---
|  Weight Initialization
---
|  Activation Function
---
Error |  |  Gradient descent
---
Epochs |  |  Train accuracy
---
|  Test accuracy
---
| Generator
---
Yes | 4855 | 2082 | 7x128 | 128x512 | Random | No | MSE | ADAM | 10 | 0.99 | 0.99
| Discriminator
---
Yes | 89 | 39 | 512x7 | No |  | Metod
---
FM*
| Cosine
---
similarity
| Cosine
---
Loss
ADAM | 50 | 0.75 | 0.81
| Generator
---
No | 1969 | 844 | 7x128 | 128x512 | Random | No | MSE | ADAM | 10 | 0.96 | 0.9
| Discriminator
---
No | 89 | 39 | 512x7 | No |  | Metod
---
FM*
| Cosinus
---
similarity
| Cosine
---
Loss
ADAM | 50 | 0.74 | 0.87

*FM Method (frequency-matrix) is a method for forming a frequency matrix based on pairs of property and class vectors. The original dataset of the emotion vector and BERT vector pairs of text was divided into two matrices of emotion vectors and BERT vectors. Then, by multiplying the obtained matrices, a frequency matrix is formed.

Based on the input data, the model makes a forecast in the form of a
7-dimensional vector of emotions. To be able to compare it with gold dataset,
it was converted into 2-dimensional vector by selecting emotion classes with
maximum values. If at least one of the two emotions coincided with the gold
labelling, then it was considered that the emotion was predicted correctly.
The accuracy of the forecast for such an assessment varies depending on the
class of emotion (see Table 3). The average accuracy of emotion predictions is
0.65.

The results of the model for different words with different emotional
sentiment are shown in Table 4.

Table 4: An example of model’s evaluation the emotional sentiment of a sentence - | Example of sentence |  Expert’s Emotion | Predicted emotion by the Model
---|---|---|---
fear |  sadness |  anger |  disgust |  calm |  happiness |  surprise
1 |  | There is danger all around,
---
it’s scary to go outside.
| fear,
---
sadness
0.49 | 0.12 | 0.17 | 0.13 | 0.00 | 0.00 | 0.09
2 |  | It is very sad that nothing
---
can be done. The city
burned down and everyone
died
| sadness,
---
fear
0.18 | 0.27 | 0.21 | 0.27 | 0.00 | 0.00 | 0.07
3 |  | I am super angry,
---
I’m in a fury!
You’re dead, bastards!
anger | 0.21 | 0.15 | 0.22 | 0.19 | 0.00 | 0.12 | 0.11
4 |  | The soup was terrible,
---
I’ve never tasted such
disgusting meal. I’ve been
sick of it for an hour.
| disgust,
---
anger
0.17 | 0.26 | 0.15 | 0.30 | 0.00 | 0.07 | 0.06
5 |  | Relax and listen to nature, feel this
---
light summer wind. The
silence calms better than a
thousand words.
| calm,
---
happiness
0.03 | 0.20 | 0.19 | 0.14 | 0.14 | 0.30 | 0.00
6 |  | What could be better than
---
watching the sunset together
with close friends!
Just wonderful!
| happiness,
---
calm
0.01 | 0.05 | 0.10 | 0.03 | 0.00 | 0.63 | 0.18
7 |  | John won more than
---
$1 million in the lottery,
that was a pleasant surprise
for him.
| surprise,
---
happiness
0.00 | 0.04 | 0.03 | 0.15 | 0.07 | 0.47 | 0.25

Based on the study of the data on model’s performance, the following
conclusions were made:

  * -

in all cases, the model correctly identified at least one of the dominant
emotions, a little less often – two of the two.

  * -

the model does not define the calm emotion well, in most cases its value is
the lowest.

  * -

Intends to show more emotions than necessary (see examples in the first five
sentences of Table 4).

That is, the model "gives a chance" to emotions that were not defined in the
gold dataset. We consider this to be a special property of the model rather
than a bug, since the model was trained on a larger variety (of generated
data) than those observed by experts. This is a very important consequence
that needs to be taken into account when reconstructing complete data based on
expert assessments.

##  4 Conclusion

Among all the architectures that were tested, GAN performed better than the
others (see Table 5). When compared with the models of emotions of some other
authors, our model showed the best result. The F1-measure value is 0.025
higher than the "goemotions-pytorch" model, and the accuracy value is 0.0053
higher than the "Emotion Detection and Recognition from Text Using Deep
Learning" model, but 0.19 lower than the "Bidirectional LSTM with a
Convolutional Neural Network"model. In the latter case, it is worth saying
that in our model the number of defined emotions is seven against four in
"Bidirectional LSTM with a Convolutional Neural Network".

Table 5: Comparing the quality of different emotion models Model | Emotions | F1 macro | Accuracy
---|---|---|---
| «Bidirectional LSTM with a Convolutional Neural
---
Network» by Rodrigo Masaru Ohashi (2019)
4 | - | 0.84
| «Emotion Detection and Recognition from Text Using
---
Deep Learning» by Chew-Yean Yam (2015)
5 | - | 0.64
«GoEmotions-pytorch» by Jangwon Park (2020) | 7 | 0.61 | -
GAN BERT-EMO model (2020) | 7 | 0.64 | 0.65

The hypothesis about the possibility of improving the quality of
classification by refusing to normalize the class distribution of a real
sample and replacing it with a sample obtained by generating all possible
combinations of classes (BERT vectors) for a finite number of variables was
fully justified. The authors have not encountered such an approach in other
publications. At the same time, it opens up new perspectives for machine
learning with minimizing human involvement by automatically generating "big
data" for training neural networks on "small" incomplete data.

## References

  * [1] A.Artemov, & A. Veselovskiy. CLASSIFICATION-OF-BASIC-EMOTIONS-USING-BERT. 2021. https://github.com/cogsyscompany/datasets
  * [2] Chew-Yean. Emotion Detection and Recognition from Text Using Deep Learning. November 29, 2015. https://devblogs.microsoft.com/cse/2015/11/29/emotion-detection-and-recognition-from-text-using-deep-learning/
  * [3] D. Demszky, D. Movshovitz-Attias, Jeongwoo Ko, A. Cowen, G. Nemade, & Sujith Ravi. GoEmotions: A Dataset of Fine-Grained Emotions. 2020. https://arxiv.org/abs/2005.00547
  * [4] P. E.Ekman, , & R. J.Davidson, (1994). The Nature of Emotion Fundamental Questions. Oxford Oxford University Press.
  * [5] Y. Feng, Y. Wang, & Hang Li. A Sequence-to-Sequence Approach to Dialogue State Tracking. 2020. https://arxiv.org/pdf/2011.09553.pdf
  * [6] Ian J. Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, Yoshua Bengio. Generative Adversarial Networks. 2014. https://arxiv.org/abs/1406.2661
  * [7] R. M. Ohashi. From Sentiment Analysis to Emotion Recognition: A NLP story. 2019. https://medium.com/neuronio/from-sentiment-analysis-to-emotion-recognition-a-nlp-story-bcc9d6ff61ae
  * [8] Jangwon Park. GoEmotions Pytorch. 2020. https://github.com/monologg/GoEmotions-pytorch
  * [9] M. Tan, & J. Jiang. A BERT-based Dual Embedding Model for Chinese Idiom Prediction. 2020. https://arxiv.org/abs/2011.02378


# Universal quantum Otto heat machine based on the Dicke model

He-Guang Xu  School of Physics, Dalian University of Technology, 116024
Dalian, China  Jiasen Jin<EMAIL_ADDRESS>School of Physics, Dalian
University of Technology, 116024 Dalian, China  G.D.M. Neto
<EMAIL_ADDRESS>Department of Physics, Zhejiang Normal University, Jinhua
321004, China  Norton G. de Almeida  Instituto de Física, Universidade Federal
de Goiás, 74.001-970, Goiânia \- GO, Brazil

###### Abstract

In this paper we study a quantum Otto thermal machine where the working
substance is composed of N identical qubits coupled to a single mode of a
bosonic field, where the atoms and the field interact with a reservoir, as
described by the so-called open Dicke model. By controlling the relevant and
experimentally accessible parameters of the model we show that it is possible
to build a universal quantum heat machine (UQHM) that can function as an
engine, refrigerator, heater or accelerator. The heat and work exchanges are
computed taking into account the growth of the number N of atoms as well as
the coupling regimes characteristic of the Dicke model for several ratios of
temperatures of the two thermal reservoirs. The analysis of quantum features
such as entanglement and second-order correlation shows that these quantum
resources do not affect either the efficiency or the performance of the UQHM
based on the open Dicke Model. In addition, we show that the improvement in
both efficiency and coefficient of performance of our UQHM occurs for regions
around the critical value of the phase transition parameter of the model.

##  I Introduction

Quantum thermodynamics [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], which is
described by the laws of quantum mechanics and thermodynamics, plays a
fundamental role in understanding the transitions between various forms of
energy and become a vibrant branch of modern research. A quantum thermal
machine [4, 14, 15, 16, 17, 18, 19, 20, 21, 22] is a quantum device to study
the thermodynamic properties of quantum systems. In recent years, the study of
thermal nanomachines has been driven by the great theoretical and experimental
effort dedicated to the investigation of their properties in the quantum
regime. Nowadays, there are many experimental platforms to explore QHE, such
as trapped ion systems [23, 24, 25, 26], optomechanics [27, 28], ultracold
atoms [29, 30], nuclear magnetic resonance (NMR) [31, 32, 33], superconducting
circuits [34, 35, 36, 37, 38].

Among thermal machines, great interest has been devoted to cyclic thermal
machines, both refrigerators and engines, operating in the quantum regime
where energy exchanges can occur, for example, between the reservoir and just
two levels of a single atom or between levels of a quantum harmonic
oscillator. Several typical quantum cycles have been extensively studied, such
as Carnot, Otto, and Stirling cycles [4, 12, 13, 20, 39, 40, 41, 42, 43, 44,
45, 46, 47]. In this paper, we are only concerned with quantum Otto cycle. The
performance of quantum Otto cycle depend strongly on the choice of the working
substance. For example, recent studies show that with two temperatures fixed,
the Otto cycle performed with fermionic substances can surpass the performance
of the same cycle when performed with bosonic substances [48]. Regarding the
Otto cycle there are several cases being considered, such as single spin
systems [49, 50], two-level atoms [22], coupled spin systems [40, 51, 52],
coupled spin-$3/2$ [53], harmonic oscillators [23, 54], relativistic
oscillators [55], Bose-Einstein condensates [56, 57], and light-matter systems
described by the Jaynes-Cumming [58, 59, 60, 61] and quantum Rabi [62, 63, 64]
models.

Despite the several studies on light-matter systems, there are very little
works devoted to investigate a quantum Otto heat engine operating with multi-
qubits interacting with a single cavity mode in the dressed picture and taking
into account dissipation as well as the number of two-level atoms, as
described by the open Dicke model (ODM). Over the past decades, the Dicke
model has been theoretically studied in several contexts, as for example
quantum phase transition [65, 66, 67, 68, 69, 70, 71, 72], quantum
entanglement [73, 74, 75], chaos [76], lasing [77] and quantum thermodynamics
[78]. According to the qubit-photon coupling ratio $\lambda/\omega$, where
$\lambda$ is the coupling strength and $\omega$ is the frequency of the cavity
mode field, the ODM can be divided into different coupling regimes: weak and
strong coupling regime ($\lambda/\omega<0.1$), ultrastrong coupling regime
(USC) ($0.1\leq\lambda/\omega<1$), which was experimentally realized in a
variety of quantum systems [79, 80, 81, 82, 83, 84, 85, 86, 87], and deep
strong coupling regime (DSC) ($1\leq\lambda/\omega$).

In this work, we study a quantum Otto heat machine (QOHM) operating under two
thermal reservoirs and having as working substance N atoms and one mode of an
electromagnetic field, as modelled by the ODM. We calculate the total work
extracted and the amount of heat exchanged between the system and the
reservoir and both the efficiency of the engine and the coefficient of
performance (COP) of the refrigerator by numerically solving the ODM using the
extended bosonic coherent state approach and the dressed master equation,
which is suitable for any coupling strength regime to describe the ODM
dynamics [88, 89]. As we will show, it is possible, by controlling the ODM
parameters, to build a universal quantum thermal machine [90] that, depending
on the choice of parameters, can work either as an engine, or as a
refrigerator, or as heater, or as an accelerator. Furthermore, our results
indicate that it is not possible, for the model analyzed here, to use quantum
resources to improve the engine efficiency or the refrigerator performance.

This paper is organized as follows. In Sec. II we introduce the open Dicke
model and numerically solve it by using the extended bosonic coherent state
approach. In Sec. III we present our model for a universal QOHM, having as the
working substance N two-level atoms and one mode of the electromagnetic field,
both atoms and field interacting with their respective reservoirs through the
so-called open Dicke model. In Sec. IV, we study the roles of the qubit-mode
coupling strength, the number N of qubits and the temperature ratio between
the cold and hot thermal reservoirs on the amount of work and heat extractable
as well as the impact on the engine efficiency and the refrigerator
performance when varying the system parameters. Finally, in Sec. V we present
our conclusions.

##  II THE MODEL

The Hamiltonian describing the Dicke model consisting of a single bosonic
field interacting with $N$ identical two-level qubits is expressed as
($\hbar=1$) [91, 92]

$~{}\hat{H}_{0}={\omega_{0}}\hat{a}^{\dagger}\hat{a}+\Delta\hat{J}_{z}+\frac{2\lambda}{\sqrt{N}}(\hat{a}^{\dagger}+\hat{a})\hat{J}_{x},$
(1)

where $\omega_{0}$ and $\Delta$ are the frequencies of the single bosonic mode
and qubits, respectively, $\lambda$ is the qubit-boson coupling strength,
$\hat{a}^{\dagger}$($\hat{a}$) denotes the creation (annihilation) operator of
the bosonic field, $\hat{J}_{x}=\frac{1}{2}(\hat{J}_{+}+\hat{J}_{-})$ and
$\hat{J}_{z}$ are the pseudospin operators given by
$\hat{J}_{{\pm}}=\text{$\sum$}_{i}^{N}\hat{\sigma}_{\pm}^{i},\hat{J}_{z}=\sum_{i}^{N}\hat{\sigma}_{z}^{i}$,
with $\hat{\sigma}_{\alpha}~{}(\alpha=x,y,z)$ being the Pauli operators. The
pseudospin operators satisfy the commutation relation
$[\hat{J}_{+},\hat{J}_{-}]=2\hat{J}_{z}$ ,
$[\hat{J}_{z},\hat{J}_{\pm}]=\pm\hat{J}_{\pm}$. In this work, we will consider
the resonance condition $\omega_{0}=\Delta=\omega$.

Dicke model has numerically exact solution by using extended bosonic coherent
state approach [72]. For convenience of numerical solution, we first rotate
the angular momentum operators with $\pi/2$ along the $y$-axis
$\hat{H_{s}}=\exp(i{\pi}\hat{J}_{y}/2)\hat{H}_{0}\exp(-i{\pi}\hat{J}_{y}/2)$,
resulting in

$\displaystyle\hat{H_{s}}=\omega_{0}\hat{a}^{\dagger}\hat{a}-\frac{\Delta}{2}(\hat{J}_{+}+\hat{J}_{-})+\frac{2\lambda}{\sqrt{N}}(\hat{a}^{\dagger}+\hat{a})\hat{J}_{z}.$
(2)

For the two-level qubits, its basis can be spanned by the Dicke state
$\\{|j,m{\rangle},m=-j,-j+1,...,j-1,j$} with $j=N/2$, and the Hilbert space of
the total system can be expressed in terms of the basis
$\\{|\varphi_{m}{\rangle}_{b}\otimes|j,m{\rangle}\\}$. In the Dicke model, the
excitation number $\hat{N}=\hat{a}^{\dagger}{\hat{a}}+\hat{J}_{z}+N/2$ is not
conserved. Therefore, the truncation of the bosonic excitation number
procedure has to be applied in this system, especially in the strong qubit-
boson coupling regime. By considering the displacement transformation
$\hat{A}_{m}=\hat{a}+g_{m}$ with $g_{m}=2\lambda m/\omega\sqrt{N}$ and taking
the total system basis into the Schrödinger equation, we obtain

$\displaystyle-\Delta j_{m}^{+}|\varphi_{m}{\rangle}_{b}|j,m+1{\rangle}-\Delta
j_{m}^{-}|\varphi_{m}{\rangle}_{b}|j,m-1{\rangle}$ (3)
$\displaystyle+\omega_{0}(\hat{A}_{m}^{\dagger}\hat{A}_{m}-g_{m}^{2})|\varphi_{m}{\rangle}_{b}|j,m{\rangle}=E|\varphi_{m}{\rangle}_{b}|j,m{\rangle},$

where $\hat{J}_{\pm}|j,m{\rangle}=j_{m}^{\pm}|j,m\pm 1{\rangle}$, with
$j_{m}^{\pm}=\sqrt{j(j+1)-m(m\pm 1)}$. Next, we multiply Eq. (3) on the left
by $\\{{\langle}n,j|\\}$, which results in

$-\Delta j_{n}^{+}|\varphi_{n+1}{\rangle}_{b}-\Delta
j_{n}^{-}|\varphi_{n-1}{\rangle}_{b}+\omega_{0}(\hat{A}_{n}^{\dagger}\hat{A}_{n}-g_{n}^{2})|\varphi_{n}{\rangle}_{b}=E|\varphi_{n}{\rangle}_{b},$
(4)

where $n=-j,-j+1,...,j$. Furthermore, the bosonic state can be expanded as

$\displaystyle|\varphi_{m}{\rangle}_{b}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\textrm{N}_{\textrm{tr}}}\frac{1}{\sqrt{k!}}c_{m,k}(\hat{A}_{m}^{\dagger})^{k}|0{\rangle}_{A_{m}}$
(5) $\displaystyle=$
$\displaystyle\sum_{k=0}^{\textrm{N}_{\textrm{tr}}}\frac{1}{\sqrt{k!}}c_{m,k}(\hat{a}^{\dagger}+g_{m})^{k}e^{-g_{m}\hat{a}^{\dagger}-g_{m}^{2}/2}|0{\rangle}_{a},$

where $\textrm{N}_{\textrm{tr}}$ is the truncation number of bosonic
excitations. Finally, we obtain the eigen-value equation

$\displaystyle\omega_{0}(l-g_{n}^{2})c_{n,l}-\Delta
j_{n}^{+}\sum_{k=0}^{\textrm{N}_{\textrm{tr}}}c_{n+1,kA_{n}}{\langle}l|k{\rangle}_{A_{n+1}}$
$\displaystyle-\Delta
j_{n}^{-}\sum_{k=0}^{\textrm{N}_{\textrm{tr}}}c_{n-1,kA_{n}}{\langle}l|k{\rangle}_{A_{n-1}}=Ec_{n,l},~{}$
(6)

where the coefficients are
${}_{A_{n}}{\langle}l|k{\rangle}_{A_{n-1}}=(-1)^{l}D_{l,k}$ and
${}_{A_{n}}{\langle}l|k{\rangle}_{A_{n+1}}=(-1)^{k}D_{l,k}$, with

$\displaystyle
D_{l,k}=e^{-G^{2}/2}\sum_{r=0}^{\min[l,k]}\frac{(-1)^{-r}\sqrt{l!k!}G^{l+k-2r}}{(l-r)!(k-r)!r!},G=\frac{2\lambda}{\omega_{0}\sqrt{N}}.$
(7)

In the following work, we select the maximum truncation number
$\textrm{N}_{\textrm{tr}}=50$, which is sufficient to give the convergent
excited state energies with relative error less than $10^{-5}$. As is well
known, in the thermodynamic limit $N\to\infty$ the Dicke model undergoes a
transition from normal (ground-state with zero photonic and atomic
excitations) to superradiant (ground-state with a macroscopic population)
phase when the qubit-boson coupling strength crosses the critical value
$\lambda_{c}=\frac{1}{2}\sqrt{\omega_{0}\Delta\coth(\beta\omega_{0}/2)}$. The
zero and finite temperature transitions belong to different classes of
universality with this difference manifested, for example, in photon-atom
entanglement, which diverges for $T=0$ and remains finite for $T\neq 0$.
Moreover, when $N=1$ the Dicke model is reduced to the seminal quantum Rabi
model [63, 64].

To help clarify the numerical results, we explore two limit regimes of our
model, $(i)$ the thermodynamic limit with $N\rightarrow\infty$ and
$\sqrt{N}\lambda=$_constant_ , and $(ii)$ the deep-strong coupling regime with
fixed $N$ and $\lambda\rightarrow\infty.$ Both regimes allow to derive a
diagonalizable effective Hamiltonian through the Holstein-Primakoff (HP)
representation of the angular momentum operators which maps the total spin
operators $\hat{J}_{\alpha}$ to a bosonic mode $\hat{b}$ .

To the case $(i)$, the quantization axis is
$\mathit{\,}\hat{J}_{z}\mathit{\,}$ and the HP transformation,
$\hat{J}_{z}=(\hat{b}^{\dagger}\hat{b}-\frac{N}{2})$,
$\hat{J}_{+}=\hat{b}^{\dagger}\sqrt{N-\hat{b}^{\dagger}\hat{b}}$,
$\hat{J}_{-}=\sqrt{N-\hat{b}^{\dagger}\hat{b}}$ $\hat{b},$ leading to the
large $N$ limit ( $N\gg\langle\hat{b}^{\dagger}\hat{b}\rangle$)

$~{}\hat{H}_{HP(N)}={\omega_{0}}\hat{a}^{\dagger}\hat{a}+\Delta\hat{b}^{\dagger}\hat{b}+\lambda(\hat{a}^{\dagger}+\hat{a})(\hat{b}^{\dagger}+\hat{b}).$
(8)

The above Hamiltonian can be diagonalized in the normal phase
$\lambda\leq\sqrt{{\omega_{0}}\Delta}/2=\lambda_{c}$ to $\hat{H}_{NP}={\
\varepsilon}_{-}c_{-}^{\dagger}c_{-}+{\varepsilon}_{+}c_{+}^{\dagger}c_{+}\,\
$with the the energies given by

$({\varepsilon}_{\pm})^{2}=\frac{{\omega_{0}^{2}+}\Delta^{2}}{2}\pm\frac{1}{2}\sqrt{({\omega_{0}^{2}-}\Delta^{2})^{2}+16\lambda^{2}{\omega_{0}}\Delta}.\text{
\ }$ (9)

After a suitable displacement of the HP bosons, the super-radiant phase
$\lambda>\lambda_{c}\,$ can be cast in a bilinear form and diagonalized with
the normal mode frequencies

$({\varepsilon}_{\pm})^{2}=\frac{{\omega_{0}^{2}\lambda}^{4}{+}\Delta^{2}\lambda_{c}^{4}}{2\lambda_{c}^{4}}\pm\frac{1}{2\lambda_{c}^{4}}\sqrt{({\omega_{0}^{2}{\lambda}^{4}-}\Delta^{2}\lambda_{c}^{4})^{2}+4{\omega_{0}^{2}}\Delta^{2}\lambda_{c}^{8}}.$
(10)

The proper quantization axis to case $(ii)$ is $\hat{J}_{x}\mathit{\,}$ and
the HP leading Hamiltonian term to large $\lambda$ (
$N\gg\langle\hat{b}^{\dagger}\hat{b}\rangle$) and eigenvalues limit are

$\displaystyle~{}\hat{H}_{HP(\lambda)}$ $\displaystyle=$
$\displaystyle{\omega_{0}}\hat{a}^{\dagger}\hat{a}+\frac{4N\lambda^{2}}{{\omega_{0}}}\hat{b}^{\dagger}\hat{b}-N\lambda(\hat{a}^{\dagger}+\hat{a})$
$\displaystyle E_{mn}$ $\displaystyle=$ $\displaystyle
m\frac{4N\lambda^{2}}{{\omega_{0}}}+n{\omega_{0}-}\frac{N^{2}\lambda^{2}}{{\omega_{0}}}.$
(11)

We note that, the eigenstates of $\hat{H}_{HP(\lambda)}$ are product states of
photons displaced Fock states and atomic states being $x$-polarized.
Furthermore, both limiting cases lead to decoupled quantum harmonic
oscillators that can be used to calculate the average energy analytically for
each stage of the thermodynamic cycle and hence the work and efficiency.

##  III QUANTUM OTTO CYCLE

Figure 1: Schematic representation of the four strokes of an Otto cycle for
the realization of a universal heat machine based on the open Dicke mode, as
detailed in Section III. During the isochoric stroke the frequency of the
working substance, as modelled by the Dicke Hamiltonian, is held fixed while
interacting with a hot (cold) reservoir at temperature $T_{h}$ ($T_{c}$). Only
heat is exchanged during this stroke. In the two quantum adiabatic strokes the
working substance is isolated from the reservoir and has its frequency
shifted, thus producing work. No heat is exchanged during this stroke. By
controlling the parameters $\omega_{0}$, $\Delta$ and $\lambda$ of the model
the machine can work as an engine, refrigerator, heater, or accelerator.

To perform a quantum Qtto cycle, which is composed of two adiabatic and two
isochoric processes [4, 5], we consider N two-level atoms and one
electromagnetic field mode, as described by the Dicke model, as the working
substance. During the isochoric process we left the N atoms and the
electromagnetic field to interact with a hot (cold) reservoir at temperature
$T_{h}$ ($T_{c}$ ). The four strokes of the quantum Otto cycle are described
as follows (Fig. 1).

1\. Quantum isochoric process. The working substance as modelled by the Dicke
Hamiltonian $H_{s}^{h}$ with frequency $\omega=\omega_{h}$ is brought into
contact with a hot reservoir at temperature $T_{h}$. In this process, the
system undergoes a Markovian evolution, which is described by the quantum
dressed master equation [88, 89]

$\displaystyle\frac{d}{dt}{\hat{\rho}}_{s}$ $\displaystyle=$
$\displaystyle-i[{\hat{H}}_{0},{\hat{\rho}}_{s}]+\sum_{u;k<j}\\{\Gamma^{jk}_{u}n_{u}(\Delta_{jk})\mathcal{D}[|\phi_{j}{\rangle}{\langle}\phi_{k}|,{\hat{\rho}}_{s}]$
(12)
$\displaystyle+\Gamma^{jk}_{u}[1+n_{u}(\Delta_{jk})]\mathcal{D}[|\phi_{k}{\rangle}{\langle}\phi_{j}|,{\hat{\rho}}_{s}]\\},$

where $|\phi_{k}{\rangle}$ is the dressed eigenbasis of the Dicke Hamiltonian
$\hat{H}$ as $\hat{H}_{0}|\phi_{k}{\rangle}=E_{k}|\phi_{k}{\rangle}$,
$\mathcal{{D}}[{\hat{O}},{\hat{\rho}}_{s}]=\frac{1}{2}[2{\hat{O}}{\hat{\rho}}_{s}{\hat{O}}^{{\dagger}}-{\hat{\rho}}_{s}{\hat{O}}^{{\dagger}}{\hat{O}}-{\hat{O}}^{{\dagger}}{\hat{O}}{\hat{\rho}}_{s}]$
is the dissipator, $\Gamma^{jk}_{u}=\gamma_{u}(\Delta_{jk})|S^{jk}_{u}|^{2}$
is the rate, with
$S^{jk}_{q}=\frac{1}{\sqrt{N}}{\langle}\phi_{j}|({\hat{J}}_{+}+{\hat{J}}_{-})|\phi_{k}{\rangle}$
and
$S^{jk}_{c}={\langle}\phi_{j}|({\hat{a}}^{{\dagger}}+{\hat{a}})|\phi_{k}{\rangle}$,
where we consider the Ohmic case
$\gamma_{u}(\Delta_{jk})=\pi\alpha\Delta_{jk}){\rm
exp}(-|\Delta_{jk}|/\omega_{co})$, with $\alpha$ being the coupling strength
and $\omega_{co}$ being the cutoff frequency of the thermal baths. In the
eigenbasis, the dynamics of the population
$P_{n}=\langle\phi_{n}|\hat{\rho}_{s}|\phi_{n}\rangle$ is given by

$\displaystyle\frac{d}{dt}P_{n}$ $\displaystyle=$
$\displaystyle\sum_{u,k{\neq}n}\Gamma^{nk}_{u}n_{u}(\Delta_{nk})P_{k}$ (13)
$\displaystyle-\sum_{u,k{\neq}n}\Gamma^{nk}_{u}[1+n_{u}(\Delta_{nk})]P_{n},$

where $\Gamma^{nk}_{u}=-\Gamma^{kn}_{u}$.

After a long enough evolution, the system will reach the only steady state
$\rho_{1}=\text{$\rho_{ss}$}(T_{h})=\sum_{n}P_{n}^{ss}(T_{h})|E_{n}^{h}\rangle\langle
E_{n}^{h}|$ of Eq. (8) with $\frac{d\rho}{dt}=0$, $P_{n}^{ss}(T_{h})$ being
the corresponding population. The system eigenstates $|\phi_{k}^{h}\rangle$
and eigenvalues $E_{k}^{h}$ of $H_{s}^{h}$ were obtained by using the extended
bosonic coherent state approach method [72]. During this process, a heat
amount $Q_{h}$ is absorbed from the hot reservoir, without any work being
done.

2\. Quantum adiabatic expansion process. The system is isolated from the hot
reservoir and the energy levels is changed from $E_{n}^{h}$ to $E_{n}^{c}$ by
varying the frequency from $\omega_{c}$ to $\omega_{h}$ (with
$\omega_{h}>\omega_{c}$). This process must be done slow enough to ensure that
the populations $P_{n}^{ss}$ ($T_{h}$) remain unchanged according to the
quantum adiabatic theorem. At the end of this adiabatic expansion the state
becomes $\rho_{2}=\sum_{n}P_{n}^{ss}(T_{h})|E_{n}^{c}\rangle\langle
E_{n}^{c}|$ . During this process only work is performed, with no heat being
exchanged.

3\. Quantum isochoric process. The working substance with frequency
$\omega=\omega_{c}$ and modelled by the Hamiltonian $H_{s}^{c}$ is now put
into contact with a cold reservoir at temperature $T_{c}<T_{h}$ until they
reach thermal equilibrium. In this case, we have a change in the steady state
population from $P_{n}^{ss}$($T_{h}$) to $P_{n}^{ss}$($T_{c}$), while the
eigenvalues $E_{n}^{c}$ of the system remain unchanged, and the state becomes
$\rho_{3}=\sum_{n}P_{n}^{ss}(T_{c})|E_{n}^{c}\rangle\langle E_{n}^{c}|$.
During this process, only heat is exchanged, an amount of heat $Q_{c}$ is
released to the reservoir, but no work is done.

4\. Quantum adiabatic compression process. The system is isolated from the
cold reservoir and its energy levels is changed back from $E_{n}^{c}$ to
$E_{n}^{h}$ by varying the frequency from $\omega_{h}$ to $\omega_{c}$. At the
end of the process, the populations $P_{n}^{ss}$ ($T_{c}$) remain unchanged,
the state becomes $\rho_{4}=\sum_{n}P_{n}^{ss}(T_{c})|E_{n}^{h}\rangle\langle
E_{n}^{h}|$, and only work is performed on the working substance, but no heat
is exchanged.

Next, let us calculate the work and heat exchanged in each stroke. According
to the first law of thermodynamics, a quantum system with discrete energy
levels can be written as

$\displaystyle dU=\delta Q+\delta
W=\sum_{n}(E_{n}dP_{n}^{ss}+P_{n}^{ss}dE_{n}),$ (14)

where $E_{n}$ are the energy levels and $P_{n}^{ss}$ are the occupation
probabilities at steady state. Accordingly, the heat $Q_{h}$ ($Q_{c}$)
exchanged with the hot (cold) reservoir, and the net work $W$ satisfy the
following relations [20]

$\displaystyle Q_{h}=\sum_{n}E_{n}^{h}[P_{n}^{ss}(T_{h})-P_{n}^{ss}(T_{c})],$
(15) $\displaystyle
Q_{c}=\sum_{n}E_{n}^{c}[P_{n}^{ss}(T_{c})-P_{n}^{ss}(T_{h})],$ (16)
$\displaystyle
W=Q_{h}+Q_{c}=\sum_{n}(E_{n}^{h}-E_{n}^{c})[P_{n}^{ss}(T_{h})-P_{n}^{ss}(T_{c})].$
(17)

In this work we will adopt the following convention: $Q>0$ ($Q<0$) correspond
to absorption (release) of heat from (to) the reservoir while $W>0$ ($W<0$)
correspond to work performed by (on) the quantum heat engine. There are only
four working regimes allowed under not violating the Clausius inequality with
the first law of thermodynamics [93]: (1) Heat engine (E): $Q_{h}>0$,
$Q_{c}<0$, and $W>0$; (2) Refrigerator (R): $Q_{c}>0$, $Q_{h}<0$, and $W<0$;
(3) Heater (H): $Q_{c}<0$, $Q_{h}<0$, and $W<0$; (4) Accelerator (A):
$Q_{c}<0$, $Q_{h}>0$, and $W<0$. In this article we are more concerned with
the heat engine and the refrigerator, which are of most interest for useful
applications and whose figures of merit are the efficiency
$\eta=\frac{W}{Q_{h}}$ and the coefficient of performance (COP)
$\xi=\frac{Q_{c}}{|W|}$, respectively.

##  IV Results and Discussions

Figure 2: The various operating regimes of the quantum Otto machine achieved
by varying the temperatures of the hot thermal reservoir $T_{h}$ and the
qubit-boson coupling strength $\lambda$, both in units of $\omega$, keeping
fixed the temperature of the cold thermal reservoir, also in units of $\omega$
as (a) $T_{c}=0.1$ (b) $T_{c}=0.2$, (c) $T_{c}=0.4$, and (d) $T_{c}=2$. The
color code stands for heat engine (magenta), refrigerator (cyan), heater
(green), and accelerator (yellow). The other system parameters are given by
$N=8$, $\omega_{h}=2\omega$, $\omega_{c}=\omega$. Figure 3: (a) Work output
$W$, (b) heat $Q_{h}$ and (c) heat $Q_{c}$ as a function of the qubit-boson
coupling strength $\lambda$ for different number of qubits $N=2$ (solid blue
line), $N=4$ (solid red line), $N=8$ (solid cyan line), $N=12$ (solid magenta
line), $N=32$ (solid green line), $N=56$ (solid black line). Vertical dotted
lines divide the different operating regimes of our universal quantum Otto
heat machine by the same color used to designate the number N of atoms. The
solid horizontal gray line indicates the zero of each quantity. The other
system parameters are given by $T_{h}=0.5$, $T_{c}=0.1$, $\omega_{h}=2\omega$,
$\omega_{c}=\omega$. All quantities above are in units of $\omega$.  Figure 4:
The efficiency $\eta$ of the quantum heat engine as a function of the qubit-
boson coupling strength $\lambda$ (in units of $\omega$) for different number
of qubits $N=2$ (solid blue line), $N=4$ (dashed red line), $N=8$ (dash dot
cyan line), $N=12$ (dotted magenta line), $N=32$ (dash dot green line), $N=56$
(dotted deep blue line), and infinity N (solid black line), with fixed
$T_{h}=0.5$, $T_{c}=0.1$ for (a) and (b), $T_{h}=6$, $T_{c}=2$ for (c).
Likewise, (d) and (e) and (f) are for the efficiency $\eta$ of the quantum
heat engine as a function of the qubit-boson coupling strength $\lambda$ under
different temperature ratios $T_{c}/T_{h}=1/7$ (solid blue line),
$T_{c}/T_{h}=1/6$ (dashed red line), $T_{c}/T_{h}=1/5$ (dash dot cyan line),
$T_{c}/T_{h}=1/4$ (dotted magenta line), $T_{c}/T_{h}=1/3$ (solid black line),
and fixed $N=8$, with cold reservoir temperatures $T_{c}=0.1$ for (d) and (e),
and $T_{c}=2$ for (f). Here the temperatures are in units $\omega$. The other
system parameters are given by $\omega_{h}=2\omega$, $\omega_{c}=\omega$.
Figure 5: (a) and (b) show the COP $\xi$ as a function of the qubit-boson
coupling strength $\lambda$ (in units of $\omega$) for different $T_{h}$ with
fixed (a) $T_{c}=0.4$, (b) $T_{c}=2$, and $N=8$ qubits. (c) COP $\xi$ as a
function of the qubit-boson coupling strength $\lambda$ (in units of $\omega$)
under different qubit numbers with fixed $T_{c}=2$, $T_{h}=2.2$. The
temperatures are in units of $\omega$.The other parameters of the system are
$\omega_{h}=2\omega$, $\omega_{c}=\omega$.  Figure 6: (a) Work $W$, (b)
efficiency $\eta$, and (c) COP as a function of the frequency ratio
$\omega_{h}/\omega_{c}$ for coupling strength $\lambda=0.05$ (solid blue
line), $\lambda=0.1$ (dashed red line), $\lambda=0.2$ (dash dot cyan line),
$\lambda=0.4$ (dotted magenta line), $\lambda=0.5$ (dash dot green line),
$\lambda=0.7$ (dash dot deep green line) and $\lambda=1.2$ (solid black line).
The temperatures of the cold and hot reservoirs, given in units of $\omega$,
were fixed at $T_{c}=0.1$ and $T_{h}=0.4$, respectively. The number of qubits
is fixed to $N=8$.

###  IV.1 Working regimes for the universal quantum Otto machine based on the
ODM

We can gain some insight of the Otto cycle by making a qualitative description
of the different working regimes for the universal QOHM, as shown in Fig. 2
for $N=8$ two-level atoms and $\omega_{h}/\omega_{c}=2$. Note that given two
operating temperatures of the Otto cycle, by controlling the parameter
$\lambda$ we obtain the four different types of machine. Also, note that the
four regions are all present at low temperatures, Fig.2(a)-(c), and mainly
occupied by refrigerator (cyan area) and heat engine (magenta area). As the
temperature rises, Figs.2(d), the engine and refrigerator operating regions
stand out even more. Fig. 2(a)-(c) show that heat engine and refrigerator
regions are mainly distributed in weak coupling and ultrastrong regime.
Strikingly, up to $\lambda\approx 0.3$ and for $\lambda\gg 2$, due to the
relative harmonicity of the spectrum, the positive-work condition (PWC)
follows the one for the quantum harmonic oscillator or qubit, i.e.,
$T_{h}>\frac{\omega_{h}}{\omega_{c}}T_{c}$.

Next, we analyze the work regimes for the universal QOHM for different numbers
N of qubits when the temperatures of the hot and cold thermal reservoirs are
fixed. Fig. 3 shows the (a) work $W$, (b) heat $Q_{h}$, and (c) heat $Q_{c}$
as a function of the qubit-boson coupling strength $\lambda$ for different
values of the qubits number N, thus evidencing the four working regimes for
different qubits numbers. As mentioned above, heat engine is distributed in
the weak and strong coupling regimes and the deep strong coupling regime,
whereas the PWC $T_{h}>\frac{\omega_{h}}{\omega_{c}}T_{c}$ is satisfied, while
the refrigerator, heater and accelerator are located around the critical
coupling
$\lambda_{c}=\frac{1}{2}\sqrt{\omega_{0}\Delta\coth(\beta\omega_{0}/2)}$,
where the spectrum is highly anharmonic. Note that increasing the number of N
qubits shifts the engine region to the left allowing engines to be built for
smaller values of $\lambda$ as N grows. On the right side, in the deep strong
regime, this behavior is maintained: as N grows, the region corresponding to
the engine also shifts to the left, increasing the area corresponding to the
engine. It is to be noted that for $\lambda\lesssim 0.4$ and for
$\lambda\gtrsim 1.5$ the work extracted by the engine practically remains
constant with the increase of N. This behavior is mimicked by the amounts of
heat $Q_{h}$ and $Q_{c}$ exchanged with the reservoirs. Given that increasing
the number of atoms means that more work and heat can be exchanged, it is
somewhat surprising that for certain values of $\lambda$ increasing N these
quantities remain unchanged. On the other hand, for the other types of
machines, whose regions correspond to the middle region of Fig. 3, the
exchanged work and heat increases with N.

###  IV.2 Efficiency and coefficient of performance

Next, we study the efficiency and COP of the Otto quantum engine and the
refrigerator, which are the heat machines of greatest practical interest. To
help our analysis, it is useful to recall the analytical result of efficiency
$\eta_{\lambda=0}=1-\omega_{c}/\omega_{h}$, and COP
$\xi_{\lambda=0}=\omega_{c}/(\omega_{h}-\omega_{c})$, of an Otto cycle when
$\lambda=0$, which corresponds to $N$ qubits and a non-interacting bosonic
mode. First, we focus on the engine. In Fig. 4(a), (b) and (c) we plot
efficiency as a function of the qubit-boson coupling strength $\lambda$ for
different number of qubits $N=2$ (solid blue), $N=4$ (dashed red), $N=8$ (dash
dot cyan), $N=12$ (dotted magenta), $N=32$ (dash dot green), $N=56$ (dashed
dark blue) and the thermodynamic limit $N\rightarrow\infty$ (solid black) with
fixed $T_{h}=0.5$, $T_{c}=0.1$ for Figs. 4(a)-(b), and $T_{h}=6$, $T_{c}=2$
for Fig. 4(c). Fig. 4(d) and (e) with $T_{c}=0.1$, and Fig. 4(f) with
$T_{c}=2$, show the efficiency for various temperature ratios when varying the
qubit-boson coupling strength $\lambda$ with fixed $N=8$. The drop to zero in
efficiency, Fig. 4(a) and (d), and its growth from zero to a maximum, Fig.
4(b) and (e), occur due to the transition from the engine to the refrigerator
regime, corresponding to the regions shown in Fig. 3. But in any case, it is
notable that the falls to zero and the rises to the maximum occur suddenly
rather than smoothly. Notably, for $\lambda\lesssim 0.3$ the efficiency is
independent of the number of atoms used as working substance. Also, note that
the efficiency drops to zero for smaller values of $\lambda$ as $N$ grows, as
shown in Fig. 4(a) and (d), because of the shift of the engine region to the
left as $N$ grows, as already mentioned when analyzing Fig. 3.

The main advantage of the ODM over the decoupled system as a working substance
happens around the critical region
$\lambda_{c}=\frac{1}{2}\sqrt{\omega_{0}\Delta\coth(\beta\omega_{0}/2)}$,
where $\eta>\eta_{\lambda=0}$ only to $\lambda<\lambda_{c}$, an unexpected
result, that means the normal phase of the Dicke model is more suitable to the
engine operation than the superradiante phase, as depicted in Figs. 4(a)-(f).
Remarkably, for small temperatures as in Fig. 4(a) the number of qubits
saturates quickly to the thermodynamic limit $N\rightarrow\infty$, being
$N\approx 30$ enough to extract the maximum efficiency. Interestingly, for
values of temperature ratios for which there is always the engine condition,
see Figs. 4(c) and (f) and also Fig. 2, there is still a drop below the
$\eta_{\lambda=0}$ around the critical coupling. In the deep-strong coupling
regime the efficiency tends to $\eta=0.5$, what is predicted by the effective
Hamiltonian Eq. (II) and the harmonicity of the spectrum for the considered
temperatures. We point out that, to small number of atoms $N=8$, there is a
decrease in the engine operating region as the temperature gap increases as
shows in Fig. 4(d), the smallest region corresponding to $T_{c}/T_{h}=1/3$
(solid black line). Second, note that the efficiency is smaller the greater
the temperature gap is, a somewhat expected behavior when compared with the
Carnot efficiency $\eta_{Carnot}=1-T_{c}/T_{h}$. As expected, the efficiency
of our universal QOHM based on the open Dicke model never exceeds the Carnot
efficiency. From Fig. 4(a),(b),(d),(e), it would appear that the abrupt drop
and sudden resurgence of efficiency values is a characteristic of engine
efficiency. However, as can be seen from Fig. 2(a)-(d), which shows the region
of the various machines, there are temperature ratios for which there will
always be a condition for the engine to exist. For these temperature ratios,
there will be neither a sudden decrease to zero nor, consequently, an abrupt
resurgence in efficiency. In fact, for $T_{c}>1.0$ in Fig. 2(a), $T_{c}>1.2$
in Fig. 2(b), $T_{c}>1.5$ in Fig. 2(c), and $T_{c}>4.5$ in Fig. 2(d) the
engine condition will always be fulfilled. This behavior is exemplified in
Fig. 4-(c) and (f), where we explored other temperature ratios for fixed $N=8$
and $T_{c}=2$. Note from Fig. 4-(c) that in addition to the abrupt drop in
efficiency, which is a signature of the passage from the region that
determines engine condition to that of refrigerator, for N=2, 4, and 8, there
is also a smooth drop and rise, indicating that despite the increase of
$\lambda$ the engine condition continues to be satisfied.

Next, we focus on the refrigerator regime. In Fig. 5 the COP $\xi$ as function
of the qubit-boson coupling strength is investigated for several ratios of
temperatures with fixed $N=8$ and $T_{c}=0.4$ \- Fig. 5(a), and $T_{c}=2$ \-
Fig. 5(b). In Fig. 5(c) we see the effect of the number of qubits on the COP
$\xi$. Noteworthy, for the normal phase, for all temperatures and number of
qubits, we found $\xi<\xi_{\lambda=0}$. Besides, similar to the heat engine in
the deep-strong coupling regime, the effective Hamiltonian Eq. (II) leads to
an accurate result of $\xi=\xi_{\lambda=0}$. As evidenced from Fig. 5(a)-(c),
the region of coupling $\lambda_{c}<\lambda\leq 3$ is where the COP for
universal QOHM having the Dicke model as working substance surpass that of the
decoupled system used to fuel the quantum refrigerator. In addition, as
observed in Fig. 5(a) and (b), the COP strongly depends on the temperature
ratio, thus differing from
$\xi_{\lambda=0}=\omega_{c}/(\omega_{h}-\omega_{c})$, being higher to small
temperature ratios, as the Carnot COP, keeping the limit
$\xi\ll\xi_{Carnot}=T_{c}/T_{h}-T_{c}$. We note from Fig. 5 that for ratios of
temperature where the universal QOHM would work as heat engine with
$\lambda=0$ (uncoupled case), corresponding to
$\frac{T_{h}}{T_{c}}<\frac{\omega_{h}}{\omega_{c}}$, for some coupling ranges
the universal QOHM works as a refrigerator with COP lower than that of the
uncoupled case.

Lastly, we explore in Fig. 6(a)-(c) the effect of the frequency ratio
$\omega_{h}/\omega_{c}$ on the work exchanged, Fig. 6(a), as well on the
efficiency, Fig. 6(b), and performance, Fig. 6(c), for the universal QOHM. The
coupling strengths are $\lambda=0.05$ (solid blue line), $\lambda=0.1$ (dashed
red line), $\lambda=0.2$ (dash dot cyan line), $\lambda=0.4$ (dotted magenta
line), $\lambda=0.5$ (dash dot green line), $\lambda=0.7$ (dash dot deep green
line) and $\lambda=1.2$ (solid black line). The temperatures were fixed as
$T_{c}=0.1$ to the cold thermal reservoir and $T_{h}=0.4$ to the hot thermal
reservoir. The frequency ratios, in addition to indicating more clearly the
PWC condition, also allow extracting the point that maximizes both the
efficiency and the COP for the universal QOHM. Note the same behavior already
observed in other figures, both for efficiency and for COP. In Fig. 6(b) we
see that after a growth until reaching a maximum, there is an abrupt drop,
precisely at the point where the engine operating condition changes to the
refrigerator condition. In Fig. Fig. 6(c), where there is a sudden appearance
of COP in the refrigerator region, the COP also reaches a maximum and then
decreases smoothly. The value of the maximum frequency ratio can thus be used
to maximize both efficiency and COP, and for the engine this condition is the
so-called efficiency at maximum power [23].

To finish this section, we point out that we also studied two other protocols
to carry out the adiabatic processes, namely (i) keeping the frequencies
constant and changing the coupling strength and (ii) changing the number of
qubits that interact with the quantum mode and fixing both the frequencies and
the coupling strength. As verified by our numerical calculations (not shown
here), in both protocols the efficiency and the coefficient of performance do
not overcome the case in which the working substance is composed of a field
mode decoupled from the qubits.

###  IV.3 Quantum correlations at thermal equilibrium

In this section, we investigate whether quantum correlations are present at
thermal equilibrium and if so, whether they affect the efficiency or COP of
the universal quantum heat machine (UQHM) based on the open Dicke model (ODM).
In accordance with our analytical and numerical studies, shown in Figs.
7(a)-(f), we claim that quantum properties surviving thermalization are not
the reason for the superior performance of efficiency, extractable work and
COP for the UQHM based on the ODM. The improvements we observed both in
efficiency (Fig.5) and COP (Fig.6) are due to the structure of the energy
levels, as evidenced by the validity condition
$N\gg\langle\hat{b}^{\dagger}\hat{b}\rangle$ to derive the effective
Hamiltonian Eq. (8), that is, small temperatures require a smaller number of
qubits to lead to the anharmonicity around the critical point that is present
at all temperatures in the thermodynamic limit.

To calculate quantum correlations, we resort to the following quantities: (i)
the second-order correlation function, which captures the occurrence of sub-
Poissonian statistics, and (ii) the negativity, which quantifies entanglement.
We emphasize that other quantum measures, such as mutual information,
squeezing and quantum discord, were investigated and omitted because they
showed the same general behavior of the considered quantities.

First, note that the conventional definition of the normalized zero-delay
second-order correlation function is [94]

$g^{(2)}(0)=\frac{\langle(\hat{a}^{\dagger})^{2}(\hat{a})^{2}\rangle}{\langle\hat{a}^{\dagger}\hat{a}\rangle^{2}}.$
(18)

This quantity describes the probability of detecting two photons
simultaneously. This definition holds for weak light-matter couplings, where
the intracavity photons, whose annihilation operator is described by
$\hat{a}$, suffice to explain the observed photon correlations. On the other
hand, in the USC regime, where the qubit system strongly dresses the bosonic
field, the second-order correlation function is derived from the input-output
formalism as [95, 96]

$G^{(2)}(0)=\frac{{\langle}(\hat{X}^{-})^{2}(\hat{X}^{+})^{2}{\rangle}}{{\langle}\hat{X}^{-}\hat{X}^{+}{\rangle}^{2}},~{}$
(19)

where

$\displaystyle~{}\hat{X}^{+}=-i\sum_{k>j}\Delta_{kj}X_{jk}|\phi_{j}{\rangle}{\langle}\phi_{k}|,$
(20)

with $\hat{X}^{-}=(\hat{X}^{+})^{{\dagger}}$, $\Delta_{kj}=E_{k}-E_{j}$ is the
energy gap, and
$X_{jk}={\langle}\phi_{j}|(\hat{a}^{\dagger}+\hat{a})|\phi_{k}{\rangle}$.
Here, ${X}_{jk}$ describes the transition from the higher eigenstate
$|\phi_{k}{\rangle}$ to the lower one $|\phi_{j}{\rangle}$. Notice that, in
the weak qubit-photon interaction limit (i.e. $\lambda_{i}\ll 1$), the
operator $\hat{X}^{+}$ is reduced to $\hat{X}^{+}=-i\omega\hat{a}$. Thus, the
correlation function in Eq. (19) simplifies to the conventional case. When
$G^{(2)}(0)<1$ the light presents the non-classical effect of anti-bunching,
and can be taken as an unequivocal indication of quantumness.

As for the negativity $\mathcal{N}(\rho)$ of a subsystem A, it can be defined
in terms of a density matrix $\rho$ as [97, 98]:

$\mathcal{N}(\rho)=\frac{\|\rho^{T_{A}}\|_{1}-1}{2}$ (21)

where $\rho^{T_{A}}$ is the partial transpose of $\rho$ with respect to
subsystem A, and $\|X\|_{1}=Tr|X|=Tr\sqrt{X^{\dagger}X}$ is the trace norm or
the sum of the singular value for the operator $X$. Non-zero negativity values
indicate the presence of quantum correlations in the form of entanglement,
being greater the greater the amount of entanglement present.

Figure 7: Two-photon correlation function $G^{(2)}_{N}(0)$ (a), (c), (e) and
negativity $\mathcal{N}(\rho)$ (b),(d),(f) as a function of the coupling
strength $\lambda$. In (a)-(d) both the two-photon correlation function and
negativity are shown for several qubit number $N$, including the thermodynamic
limit, with the cold thermal reservoir temperature fixed at $T_{c}=0.1$
(a)-(b) and $T_{c}=2$ (c)-(d). In (e)-(f), the qubit number was fixed at $N=8$
and the cold thermal reservoir temperature was chosen as $T_{c}=0.1$ (blue
solid line), $T_{c}=0.4$ (red dash line), $T_{c}=0.2$ (green dot line), and
$T_{c}=5$ (black dash line). The other system parameters are
$\omega_{c}=\omega$. The coupling strength $\lambda$ and temperatures are in
units of $\omega$.

We computed $G^{(2)}(0)$ in Figs. 7(a),(c),(e) and $\mathcal{N}(\rho)$ in
Figs. 7(b),(d), (f) as a function of coupling strength for different number of
qubits. In Figs. 7(e),(f) we set $N=8$ and investigated the effect of
different temperatures on the quantumness of the work substance. Note that the
quantum correlations are degraded by increasing the number $N$ of qubits and
increasing the temperature $T_{h}$, whereas $\eta$ and $\xi$ increase with the
number of qubits for all temperatures. If we compare the Figs. 7(a)-(f)
showing maximum antibunching and maximum entanglement with Figs 4(a)-(f) and
5(a)-(b), which respectively show efficiency and COP, we will see that there
is no correspondence between the maximum of quantum correlations and the
maximum of efficiency and COP. As far as the second-order correlation is
concerned, the efficiency is higher in the deep strong coupling regime
($\lambda>2$), Figs. 4(b),(e), and therefore, far from the region where the
second-order correlation shows the sub-Poissonian effect. The same conclusion
can be drawn from Fig.5(a)-(c) for the COP, whose maxima lie in regions far
from the value of the critical parameter $\lambda$. With regard to negativity,
Figs.7(b), (d), and (f) show that the maximum of negativity, and therefore of
entanglement, does not coincide with the maximum of efficiency and COP. For
example, in Fig. 4(b) for $T_{c}=0.1$ and various values of $N$, the
efficiency is practically constant with the coupling parameter, and therefore
independent of the amount of entanglement, whereas in Fig.7(b) the negativity,
also at $T_{c}=0.1$ and the various values of $N$ presents maximums and
minimums when varying the coupling parameter. The same can be said about the
COP: there is nothing in the analysis of the maximums that indicates the
relevance of negativity for its improvement. Take for example Fig.7(d) for
$T_{c}=2$, where negativity remains zero for a large range of values of
$\lambda$ and then increases monotonically until to $\lambda=2$, with similar
behavior even for different values of $N$. Compare with Fig.5(c), where the
COP has a very different behavior depending on $N$, with no correspondence
with negativity. These same conclusions are supported by additional numerical
calculations that we performed (not shown here). To summarize this Section,
and as previously mentioned, we point out that the improvement in the
efficiency and performance of the UQOHM using the Dicke model as the working
substance cannot be attributed to quantum resources [99, 100], but it is due
to the high anharmonicity of the spectrum around the critical point of the
Dicke model.

##  V Conclusion

In summary, in this work we propose a universal quantum Otto heat machine
(UQOHM) based on the open Dicke model (ODM). The ODM is composed of N atoms of
two levels (qubits) that interact with a mode of the electromagnetic field and
both the mode and the N qubits, which constitute the working substance of the
universal machine, interact with thermal reservoirs. This model presents a
critical point and can be solved analytically in the thermodynamic limit
$N\rightarrow\infty$. By universal thermal machine we mean that it is
possible, by adjusting the atom-field coupling parameter $\lambda$ of the ODM,
to build all types of thermal machines, namely engines, refrigerators,
heaters, and accelerators. Focusing on engines and refrigerators, which are
the machines with the greatest applicability, we show, for a wide temperature
range and a large number of qubits, including in the thermodynamic limit, how
the engine efficiency and the refrigerator coefficient of performance change
with the parameter $\lambda$ of the ODM.

We also conducted a study of the quantum correlations present in the ODM using
the second order correlation function and negativity showing that, for certain
values of the coupling parameter $\lambda$ of the Dicke model, both the
antibunching effect and the entanglement survive thermalization. Next, we show
that it is possible, close to the critical point, to obtain both an efficiency
and a performance for the UQOHM that is greater than the case in which the
system is uncoupled, thus showing the advantage of using the Dicke model as
the working substance. Furthermore, the detailed study of the second-order
correlation function and negativity indicates no correspondence between the
improvement in the efficiency and the coefficient of performance of the UQOHM
and the quantum resources arising from anti-bunching and entanglement.

##  VI Acknowledgement

We acknowledge financial support from the Brazilian agencies CNPq and FAPEG.
This work was performed as part of the Brazilian National Institute of Science
and Technology (INCT) for Quantum Information Grant No. 465469/2014-0. H.-G.
X. and J. J. are supported by National Natural Science Foundation of China
under Grant No. 11975064.

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