# A Comparative Study on COVID-19 Fake News Detection Using Different
Transformer Based Models

Sajib Kumar Saha Joy
Dibyo Fabian Dofadar, Riyo Hayat Khan, Md. Sabbir Ahmed, Rafeed Rahman
Department of Computer Science and Engineering
Ahsanullah University of Science and Technology
Dhaka, Bangladesh
<EMAIL_ADDRESS>Department of Computer Science and Engineering
Brac University
Dhaka, Bangladesh
{dibyo.fabian.dofadar, riyo.hayat.khan<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

The rapid advancement of social networks and the convenience of internet
availability have accelerated the rampant spread of false news and rumors on
social media sites. Amid the COVID-19 epidemic, this misleading information
has aggravated the situation by putting people’s mental and physical lives in
danger. To limit the spread of such inaccuracies, identifying the fake news
from online platforms could be the first and foremost step. In this research,
the authors have conducted a comparative analysis by implementing five
transformer-based models such as BERT, BERT without LSTM, ALBERT, RoBERTa, and
a Hybrid of BERT & ALBERT in order to detect the fraudulent news of COVID-19
from the internet. COVID-19 Fake News Dataset has been used for training and
testing the models. Among all these models, the RoBERTa model has performed
better than other models by obtaining an F1 score of 0.98 in both real and
fake classes.

###### Index Terms:

COVID-19, Transformer, Fake News Detection, BERT, RoBERTa

##  I Introduction

One of the most difficult tasks in the field of Natural Language Processing is
to detect misleading news, especially because of the vast expansion of social
networks and easy accessibility of the internet. The massive online resources
surely do include a lot of useful information, but at the same time, they
consist of a lot of fraudulent news and rumors. Hence, it has become crucial
to stop the spreading of misinformation, especially amid the COVID-19 crisis.
COVID-19 is undoubtedly one of the biggest catastrophes that have ever
occurred in history. It is referred to as a global pandemic by the World
Health Organization (WHO). While the whole world has been suffering within
this chaos, people are forced to fight against “infodemic”. In this era of
technological advancement, social media is playing a negative role by
spreading fake news and rumors, resulting in creating panic among people.
Since COVID-19 has involved the lives of people greatly, public attention has
been focused on COVID-19-based information. Misusing this chance, many
adversarial agents had intentionally spread bogus news for economic,
political, or other purposes. A wave of myths, frauds, and misconceptions
about the disease’s etiology, effects, prevention, and treatment [11] had been
widespread across the world of the internet. Rumors like “spraying chlorine on
your own body for preventing the coronavirus”, “drinking alcohol can kill the
virus”, etc. [7] had been viral on various social media platforms. Believing
this misleading information had put people in danger, in some cases, costing
their lives. There had been a study by [7] in which it was estimated that
almost 5800 people were hospitalized and 800 people died because of following
the rumors on social media [10]. This widespread misinformation had not only
damaged the people’s lives physically but also mentally since it causes terror
and anxiety [13].
As a solution to this problem, one must detect misinformation from internet
platforms. However, it is very complicated for humans to distinguish between
false and true news since the process of evidence collection can be very
tedious and also very time-consuming. Keeping all these in mind, a practical
solution is to build models that could identify all the false information. In
this research, the authors have implemented five transformer-based models such
as BERT, BERT without LSTM, ALBERT, RoBERTa, and a Hybrid of BERT & ALBERT for
identifying fake news regarding COVID-19, and have performed a comparative
analysis of those models.
Several pieces of research have been conducted to detect fraudulent news
regarding COVID-19. In [9], the authors had shown the use of Decision Tree,
Logistic Regression, Gradient Boost, and Support Vector Machine (SVM) to
detect fake news related to COVID-19. For this purpose, they prepared and
annotated a dataset of more than ten thousand posts from social media and
reports related to true and fake news about COVID-19. They had shown a binary
classification between fake and real news. Among all the models that they
used, the Support Vector Machine showed the highest value of 93.32% in terms
of F1 score when tested with the testing data.
In [3], the automatic detection of whether a particular tweet related to COVID
is fake or real was shown. The authors applied an ensemble method with pre-
trained models for this objective. Also based upon username and link domains,
they incorporated a unique heuristic algorithm. Genuine news items were
obtained for the dataset from credible sources that supplied crucial COVID-19
information, whereas ”fake” ones were gathered from tweets, posts, and
articles that made false COVID-19 assertions.
A dataset of Contraint@AAAI 2021 COVID-19 Fake news detection was used in
[12]. The models that they have incorporated for fake news detection were CNN,
LSTM, and BERT.
The authors of [2] have incorporated language models which are fine-tuned
transformers with resilient loss functions and used influence calculation to
remove harmful training instances. They evaluated the model on the COVID-19
misinformation test set (Tweets-19). They used Fake-News COVID-19
(FakeNews-19) which was the dataset made available for the collaborative
effort of the CONSTRAINT 2021 workshop. There has been another research [6] in
which the authors showed their best approach using BERT, ALBERT, and XLNET
while detecting fake news related to COVID-19.

##  II Dataset

###  II-A Data Description

The dataset used in this research was referred to [9]. It contains 10,700
social media items and articles that have been manually annotated. This
dataset was given two labels - ’real’ and ’fake.’ Out of the 10,700 instances,
6420 (60%) social media posts and articles were separated for training and
2140 (20%) social media posts and articles were chosen for testing and
validation each. The total number of distinct words in the sample was 37,505
with 5141 words appearing in both actual and fraudulent news articles and
posts. Furthermore, 52.34 percent of the samples were from credible news,
while the remaining 47.66 percent were from fake news.

###  II-B Data Preprocessing and Exploratory Data Analysis

At first, for convenience, the samples for ‘real’ labels were changed to 0 and
the ‘fake’ labels were relabelled to 1. Then the major preprocessing steps
were performed which included different text normalization and cleaning
processes like stop words removal, link removal, lowercase transformation,
contraction expansion, special characters removal, ignoration of ASCII values,
translation table construction, etc.

After cleaning the data, it was checked whether there is any missing value in
the dataset and it was found that there was no data loss in the text cleaning
process. Figure 1 depicts the distribution of the number of words per post in
train data. It can be seen that there are a very small number of posts whose
length is large. So, to get the suitable sequence length, the number of words
in the training data was quantiled. After a few tests, it was found that
quantiling by 98% and 99% gives 52 and 56 length sequences respectively. This
means if the maximum sequence length is set to 52, then no info will be lost
in 98% data and for 56, no info will be lost in 99% data. For implementation
purposes, as the models are computationally expensive, it was decided to take
56 as the maximum sequence length.

Figure 1: Distribution of the number of words per post in train data

Lastly, the distribution of labels can be visualized in Figure 2. It can be
seen that data is nearly balanced across the training, test, and validation
data.

Figure 2: Distribution of labels in train, test and validation data Figure 3:
Block diagram of several implemented models

##  III Methodology

Transformer based approaches used in this research are BERT and other BERT
inspired NLP architectures. The five different models implemented for this
research are as follows:

  * •

BERT based model: BERT[15] is currently one of the most widely used attention-
based linguistic interpretation models. BERT is built to pre-train deep
bidirectional representations from unlabeled data by configuring both the left
and right settings in all the layers. Consequently, with only one extra output
layer, the pre-trained BERT model may be fine-tuned to develop cutting-edge
models for a wide variety of tasks without requiring significant task-specific
engineering changes. In the implementation, some extra preprocessing steps
like tokenization and padding were applied before creating the model. The
token inputs and mask inputs were jointly considered as the input for the
model. BERT then generated the feature vectors of the text and this BERT
output was passed to a 128-unit Long short-term memory (LSTM) [16] layer
followed by a dropout layer with a 20% dropout rate. Finally, the model
generated the final output using the attributes after a fully connected layer
with the sigmoid activation function. Figure 3(a) shows the simple block
diagram of the architecture of this model.

  * •

BERT without LSTM based model: In this model, after word embedding, the
outputs of the BERT model were not followed by an LSTM layer. Instead, an
extra 128-unit fully connected layer and a dropout layer with a 30% dropout
rate were connected after BERT output. Furthermore, another fully connected
layer with 64 units was connected followed by another dropout layer with a 20%
dropout rate. Similar to the previous model, the final output was found after
a fully connected layer with the sigmoid activation function. The simple
architecture behind this model can be visualized in Figure 3(b).

  * •

ALBERT based model: The ALBERT model[17] is a simplified BERT-based model for
self-supervised language representation learning. It describes two parameter
reduction strategies for lowering memory utilization and increasing BERT
training speed. It divides the embedding matrix into two sub-matrices. It also
uses repeated layers to separate between bunches. Architecturally it is almost
similar to the first model. The implementation of this model for this research
can be observed in Figure 3(c).

TABLE I: Precision, Recall, F1-score and Support of Real and Fake class for BERT, BERT without LSTM, ALBERT, RoBERTa and Hybrid (BERT+ALBERT) models | Fake Class | Real Class
---|---|---
Model Name | Precision | Recall | F1 Score | Support | Precision | Recall | F1 Score | Support
BERT | 0.87 | 0.88 | 0.88 | 1020 | 0.89 | 0.88 | 0.89 | 1120
BERT without LSTM | 0.77 | 0.71 | 0.74 |  | 0.75 | 0.81 | 0.78 |
ALBERT | 0.97 | 0.87 | 0.92 |  | 0.90 | 0.98 | 0.93 |
RoBERTa | 0.99 | 0.96 | 0.98 |  | 0.97 | 0.99 | 0.98 |
Hybrid (BERT+ALBERT) | 0.94 | 0.96 | 0.95 |  | 0.96 | 0.94 | 0.95 |
TABLE II: Comparison with Other Literature Approaches Applied on the Same Dataset Author’s work | Methods | Precision | Recall | F1-score
---|---|---|---|---
[9] | SVM | 0.9333 | 0.9332 | 0.9332
[5] | XLNet with Topic Distributions | 0.968 | 0.967 | 0.967
[8] | Cross-SEAN | 0.946 | 0.961 | 0.953
[1] | DistilBERT | 0.923 | 0.949 | 0.936
[4] | ERNIE 2.0 | 0.976 | 0.976 | 0.976
This work | RoBERTa | 0.99 | 0.96 | 0.98
  * •

RoBERTa based model: RoBERTa [18] is also a BERT-inspired model in which the
structure is constructed on BERT and then the model modifies key
hyperparameters, removing the next-sentence pretraining purpose and preparing
with significantly larger mini-batches and learning rates. This model has two
major preferences over other transformer-based models. The expansive batches
move forward with perplexity on masked dialect displaying objective and well
as end-task accuracy. Also, huge batches are moreover less demanding to
parallelize through distributed parallel training. In the implementation of
this research, the structure of the model is similar to the first and third
models in general. This model can be observed with a simple block diagram from
Figure 3(d).

  * •

Hybrid (BERT + ALBERT) model: The last model was created by merging the
outputs of the BERT and ALBERT models. In this model, no LSTM was used during
implementation. Another major difference with other models was, that this
model did not consider individual word’s feature vectors, but took feature
vectors for the overall sentence. Thus a big feature vector was created by
concatenating the embedding of BERT and ALBERT. This feature vector was fed
into a 128-unit fully connected layer and a dropout layer with a 30% dropout
rate. Furthermore, another fully connected layer with 64 units was connected
followed by another dropout layer with a 20% dropout layer. Similar to the
other models, the final output layer was created after a fully connected layer
with the sigmoid activation function. Figure 3(e) represents the simple block
diagram of this model architecture.

##  IV Hyper parameter tuning

Tensorflow API is used to build the models. For all the models, the hyper
parameter was set under manual observation. A small portion of the data (40%
of training data and 30% of validation data) was utilized for finding the best
hyper parameters for the models. It was searched for an acceptable balance
between bias and variance to get well suited hyper parameter. Adam optimizer
[14] was applied on 32-sized batches and the learning rate was set at 0.00002
for every single model.

##  V Results

As discussed earlier, a total of 5 models were used for the mentioned dataset
namely BERT, BERT without LSTM, ALBERT, RoBERTa, and Hybrid (BERT+ALBERT).
From Table 1, it can be observed that the BERT model had achieved 87%
Precision, 88% Recall, and F1 score of 88% for the Fake class and 89%
Precision, 88% Recall, and F1 score of 89% for the Real class. For the BERT
without LSTM the Precision, Recall, and F1 scores of the Fake class are 77%,
71%, and 74% whereas for the Real class those are 75%, 81% and 78%
respectively. ALBERT shows an increase in all the metrics for both the Real
and Fake classes. This model showed a Precision of 97% along with 87% Recall
and 92% F1 score for the Fake class and 90% Precision, 98% Recall, and 93% F1
score for the Real class. A Hybrid model was also implemented for this work.
This model was projected to outperform all other state-of-the-art models.
Unfortunately, this model’s performance was not the best among the other
models; instead, this model gave a decent result. For the Fake class, the
Precision was 94%, Recall was 96% and the F1 score for this model was 95%. For
the Real class, the precision was 96%, the recall was 94% and the F1 score was
95%. Among all models, the RoBERTa model was well suited for the dataset. It
has the highest score for every metric. For the Real class, this model got 97%
Precision, 99% Recall, and 98% F1 scores. For the Fake class, the model
achieved 99% Precision, 96% Recall, and 98% F1 scores. The reason why RoBERTa
performed better compared to other models was the progress of the expansive
batches on masked dialect displaying objective and end-task accuracy.
Moreover, these expansive batches are less challenging for parallelizing while
using distributed parallel training.

Figure 4: Confusion matrix of RoBERTa model

In Figure 4, it can be seen that 2090 news was correctly classified for
RoBERTa model. Here, 1106 news was real news and the rest 984 news were fake
news were classified accurately. On the other end, 14 instances of real news
were mislabeled as false news, while 36 instances of fake news were mislabeled
as real news.
Table 2 compares the strategy to other authors in the literature that were
used on the same dataset. According to the table, the suggested RoBERTa model
outperforms existing techniques from similar literature implementations.

##  VI Conclusion

Widespread misinformation has a negative impact on people’s lives, not just
physically but also mentally, especially when it is linked to a global
epidemic like COVID-19. As a result, detecting such misinformation becomes a
crucial challenge in order to minimize the pandemonium as much as possible.
Manually detecting fake news can be a time-consuming and daunting task. In
this research, the authors suggested five different models for detecting both
real and fake news in order to obtain more reliable, accurate, and simple
detection of these fake news. In this research, RoBERTa has been the best fit
for detecting both real and fake news whereas all other models have shown
decent results. The authors have also proposed a hybrid model that
incorporates BERT & ALBERT models and demonstrated comparable outcomes to the
state-of-the-art models. This paper is an attempt to use different approaches
to recognize COVID-19 related false news to a significant extent. The future
work of this research will include testing additional models as well as
improving the hybrid model to detect COVID-19 related fake news and also
handle similar cases better than existing methods.

## References

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# Thermodynamic perspective on quantum fluctuations

Akira Sone<EMAIL_ADDRESS>Kanu Sinha<EMAIL_ADDRESS>Sebastian Deffner
<EMAIL_ADDRESS>
###### Abstract

What is the major difference between large and small systems? At small length-
scales the dynamics is dominated by fluctuations, whereas at large scales
fluctuations are irrelevant. Therefore, any thermodynamically consistent
description of quantum systems necessitates a thorough understanding of the
nature and consequences of fluctuations. In this chapter, we outline two
closely related fields of research that are commonly considered separately –
fluctuation forces and fluctuation theorems. Focusing on the main gist of
these exciting and vivid fields of modern research, we seek to provide a
instructive entry point for both communities of researchers interested in
learning about the other.

###### keywords:

Quantum fluctuations , quantum thermodynamics , Casimir effect, quantum
fluctuations theorems

††journal: Encyclopedia of Mathematical Physics 2nd edition

[inst1]Department of Physics, University of Massachusetts,
Boston,,city=Boston, postcode=MA 02125, country=USA [inst2]Wyant College of
Optical Sciences, University of Arizona,city=Tucson, postcode=AZ 85721,
country=USA [inst3]Department of Physics, University of Maryland, Baltimore
County,,city=Baltimore, postcode=MD 21250, country=USA

##  1 Introduction

Thermodynamics is one of the, if not the most powerful theory in theoretical
physics. Built upon macroscopic observation and empirical laws, thermodynamics
statements remain true – independent of the microscopic details of the system
under study. Therefore, to apply and utilize thermodynamics it is always
instrumental to identify the good macroscopic observables, which are
measurable quantities immune to fluctuations in time and space, over the time-
and length-scales of external observation [1].

Focusing on macroscopic variables and quasi-static processes, thermodynamics
permits a tremendous reduction of mathematical complexity in the description
of many-body systems. However, all real-life processes occur at finite-rates
and it is frequently desirable to describe also the microscopic details of
fluctuations around the macroscopically observable mean values. This becomes
particularly important for quantum systems, for which fluctuations are
comprised of both, thermal noise and quantum uncertainty [2].

Curiously, the study of fluctuations in quantum systems has been developed in
(at least) two very different directions, that are commonly presented without
much overlap. Quantum stochastic thermodynamics [2, 3, 4] focuses on
formulations of the laws of thermodynamics for quantum systems, whereas
Casimir physics [5, 6, 7, 8, 9] studies the mechanically measurable
consequences of fluctuations of quantum electromagnetic (EM) fields.

Since this two directions of research are obviously closely related, or at
least they should be conceptually close, it appears desirable to have a
coherent presentation. In this entry to the _Encyclopedia of Mathematical
Physics_ we provide, therefore, a concise outline of the main notions, which
may serve as an entry point into the large body of literature on the two
topics.

##  2 Fluctuations in quantum fields and their corresponding forces

In Quantum Field Theory (QFT), empty space is not quite empty but jostling
with fluctuations, both thermal and quantum in origin. Quantum fluctuations
are an inevitable and fascinating feature of QFT, persisting in the absence of
all matter and radiation, even at zero temperature. Phenomena arising from
quantum fluctuations span a range of scales: from spontaneous emission and
Lamb shifts of atoms [10, 11], intermolecular forces responsible for the
stability of colloidal suspensions [12], adhesive properties of gecko feet
[13] and stiction in nano- and micro-mechanical systems [14], to potentially,
density perturbations in the cosmic microwave background [15]. Such quantum
and thermal fluctuations of the EM field can lead to forces between neutral
objects, often addressed by different names depending on the geometry,
separation and material properties of the interacting objects, such as van der
Waals [16], London [17], Casimir-Polder [18], Casimir-Lifshitz [19], or more
generally Casimir [20] forces, arise as a result of the interaction mediated
between the fluctuating dipole moments that constitute two neutral bodies via
the fluctuations of the EM field.

Figure 1: Quantum fluctuation phenomena at various scales: Stability of
colloids [12], friction in micro-electromechanical systems (MEMS) [14], van
der Waals adhesion in gecko feet [13], density perturbations in the Cosmic
Microwave Background (CMB), and Casimir-Polder attraction of atoms near
surfaces. Image credits: [21, 22].

Fluctuation forces are typically studied between bodies that are classical.
The development of nanoscale quantum systems opens an exciting new paradigm
for exploring fluctuation forces between quantum systems that can be
externally driven, prepared in coherent superpositions or entangled states
[23, 24, 25, 26, 27, 28, 29, 30]. This necessitates an Open Quantum System
description of fluctuation forces, that we will review in the following. Such
an approach enables one to treat fluctuation-induced forces as well as the
dissipation and decoherence effects in the same framework. To this end, we
derive a master equation that describes fluctuation-induced phenomena –
forces, dissipation and decoherence – in a system of $N$ two-level atoms
interacting with the quantum and thermal fluctuations of the EM field in the
vicinity of a medium. While such an approach has been previously discussed in
the context of analyzing quantum fluctuation-induced forces [28], here we take
both the quantum and thermal fluctuations of the EM field into consideration
as a generalization of previous analyses.

###  2.1 An Open Quantum System approach to Fluctuation Forces

####  2.1.1 Model and Hamiltonian

Consider a collection of $N$ identical two-level systems or atoms interacting
with the fluctuations of the EM field, as shown in Fig. 2 (a). The Hamiltonian
describing the total system is,

$H_{\mathrm{tot}}=H_{S}+H_{B}+H_{\mathrm{int}}\,,$ (1)

where $H_{S}$ refers to the Hamiltonian for the atoms,

$H_{S}=\sum_{n=1}^{N}\hbar\omega_{0}\,\sigma_{n}^{+}\sigma_{n}^{-}$ (2)

with $\sigma_{n}^{+}$ and $\sigma_{n}^{-}$ being the raising and lowering
operators for the $n$th atom. Moreover, $\omega_{0}$ is the resonance
transition frequency.

The collection of atoms interacts with the medium-assisted EM field that forms
the bath, described by the Hamiltonian,

$H_{B}=\int_{0}^{\infty}\mathrm{d}\omega\int\mathrm{d}^{3}{\mathbf{r}}\sum_{\lambda=e,m}{\mathbf{f}}^{\dagger}_{\lambda}({\mathbf{r}},\omega){\mathbf{f}}_{\lambda}({\mathbf{r}},\omega).$
(3)

The operators ${\mathbf{f}}^{(\dagger)}_{\lambda}({\mathbf{r}},\omega)$ refer
to the bosonic operators for the field in the presence of a medium as
described in the macroscopic Quantum Electrodynamics (QED) formalism [7, 8,
31, 32]. Physically one can understand the operators
${\mathbf{f}}^{(\dagger)}_{\lambda}({\mathbf{r}},\omega)$ to represent the
creation and annihilation operators for the polaritonic excitations of the
field+matter system in the presence of media at frequency $\omega$, position
${\mathbf{r}}$, with $\lambda=e,m$ characterizing the quantum noise
polarization $(e)$ or magnetization $(m)$. These operators obey the canonical
commutation relations,

$\left[{\mathbf{f}}_{\lambda}\left({\mathbf{r}},\omega\right),{\mathbf{f}}_{\lambda^{\prime}}^{\dagger}\left({\mathbf{r}}^{\prime},\omega^{\prime}\right)\right]=\delta(\omega-\omega^{\prime})\delta\left({\mathbf{r}}-{\mathbf{r}}^{\prime}\right)\delta_{\lambda\lambda^{\prime}}\,.$
(4)

Figure 2: (a) Schematic representation of $N$ two-level atoms interacting with
the quantum and thermal fluctuations of the EM field near a medium. Each atom
has a resonant transition frequency $\omega_{0}$, with $\left|g\right\rangle$
and $\left|e\right\rangle$ as the ground and excited levels. (b) Open quantum
system representation of the system in consideration. The collection of atoms
is the system of interest, EM field is the bath, with its density of modes
modified by the presence of the medium.

The system-bath interaction Hamiltonian is given by the electric-dipole
Hamiltonian [33],

$H_{\mathrm{int}}=\sum_{n}{\mathbf{d}}_{n}(\sigma_{n}^{+}+\sigma_{n}^{-})\cdot{\mathbf{E}}({\mathbf{r}}_{n}),$
(5)

where ${\mathbf{d}}_{n}$ is the dipole moment corresponding to the $n$th atom
and ${\mathbf{E}}\left({\mathbf{r}}_{n}\right)$ is the quantized electric
field at the position of the $n$th atom ${\mathbf{r}}_{n}$, given by:

${\mathbf{E}}\left({\mathbf{r}}_{n}\right)=\int_{0}^{\infty}\mathrm{d}\omega\int\mathrm{d}^{3}{\mathbf{r}}\sum_{\lambda}{\mkern
1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}_{\lambda}\left({\mathbf{r}}_{n},{\mathbf{r}},\omega\right)\cdot{\mathbf{f}}_{\lambda}\left({\mathbf{r}},\omega\right)+{\mathbf{f}}^{\dagger}_{\lambda}\left({\mathbf{r}},\omega\right)\cdot{\mkern
1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}^{\dagger}_{\lambda}\left({\mathbf{r}}_{n},{\mathbf{r}},\omega\right).$
(6)

The coefficients ${\mkern 1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}_{\lambda}({\mathbf{r}},{\mathbf{r}}^{\prime},\omega)$ are proportional
to the Green’s tensor ${\mkern 1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}\left({\mathbf{r}},{\mathbf{r}}^{\prime},\omega\right)$ of the EM field,
accounting for the presence of any macroscopic media near the atoms, such that
[7, 31, 32]:

$\sum_{\lambda=e,m}\int\mathrm{d}^{3}{\mathbf{r}}^{\prime}\,{\mkern
1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}_{\lambda}\left({\mathbf{r}}_{1},{\mathbf{r}}^{\prime},\omega\right)\cdot{\mkern
1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}^{\dagger}_{\lambda}\left({\mathbf{r}}_{2},{\mathbf{r}}^{\prime},\omega\right)=\frac{\hbar\mu_{0}}{\pi}\omega^{2}\,\mathfrak{Im}\left({\mkern
1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}\left({\mathbf{r}}_{1},{\mathbf{r}}_{2},\omega\right)\right),$ (7)

where $\mu_{0}$ is the vacuum magnetic permeability.

The Green’s tensor ${\mkern 1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}\left({\mathbf{r}}_{1},{\mathbf{r}}_{2},\omega\right)$ in the above
equation represents the propagator for the EM field in the presence of media,
accounting for how a photon at frequency $\omega$ propagates between positions
${\mathbf{r}}_{1}$ and ${\mathbf{r}}_{2}$. The Green’s tensor can be found as
the solution of the Helmholtz equation [34],

$\left(\nabla_{1}\times\frac{1}{\mu({\mathbf{r}}_{1},\omega)}\nabla_{1}\times-\frac{\omega^{2}}{c^{2}}\epsilon\left({\mathbf{r}}_{1},\omega\right)\right){\mkern
1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}({\mathbf{r}}_{1},{\mathbf{r}}_{2},\omega)={\mathbf{\delta}}\left({\mathbf{r}}_{1}-{\mathbf{r}_{2}}\right),$
(8)

where $\epsilon({\mathbf{r}},\omega)$ and $\mu({\mathbf{r}},\omega)$ represent
the spatially dependent dielectric permittivity and magnetic permeability in
the presence of a medium. The Green’s tensor encompasses the effects of
surface geometry, optical and material properties of the medium, thereby
modifying the density of states of the EM environment.

####  2.1.2 Master equation

We now continue to derive the open system dynamics of the atomic density
matrix $\rho_{S}$, by tracing out the medium-assisted quantized EM field. To
this end, we first move to the interaction picture with respect to the free
Hamiltonian of the system and the bath $H_{0}=H_{S}+H_{B}$. As usual, we
define the interaction Hamiltonian in the interaction picture as
$\tilde{H}_{\mathrm{int}}\equiv\exp{\left(-iH_{0}t/\hbar\right)}\,H_{\mathrm{int}}\,\exp{\left(iH_{0}t/\hbar\right)}$,

$\begin{split}\tilde{H}_{\mathrm{int}}\left(t\right)&=\sum_{n}\int_{0}^{\infty}\mathrm{d}\omega\int\mathrm{d}^{3}{\mathbf{r}}\sum_{\lambda}\left(\sigma_{n}^{+}e^{i\omega_{0}t}+\sigma_{n}^{-}e^{-i\omega_{0}t}\right)\\\
&\times\left[{\mathbf{d}}_{n}\cdot{\mkern
1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}_{\lambda}\left({\mathbf{r}}_{n},{\mathbf{r}},\omega\right)\cdot{\mathbf{f}}_{\lambda}\left({\mathbf{r}},\omega\right)e^{-i\omega
t}+{\mathbf{f}}^{\dagger}_{\lambda}\left({\mathbf{r}},\omega\right)\cdot{\mkern
1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}^{\dagger}_{\lambda}\left({\mathbf{r}}_{n},{\mathbf{r}},\omega\right)\cdot{\mathbf{d}}_{n}e^{i\omega
t}\right].\end{split}$ (9)

The second-order Born-Markov master equation that describes the evolution of
the atomic density matrix is thus given by [35],

$\begin{split}\frac{{d}\rho_{S}}{{d}t}&=-\frac{1}{\hbar^{2}}\,\mathrm{tr_{B}}\left\\{\int_{0}^{\infty}\mathrm{d}\tau\,\left[\tilde{H}_{\mathrm{int}}(t),\left[\tilde{H}_{\mathrm{int}}(t-\tau),\rho_{S}\otimes\rho_{B}\right]\right]\right\\}\\\
&=\underbrace{-\frac{1}{\hbar^{2}}\,\mathrm{tr_{B}}\left\\{\int_{0}^{\infty}\mathrm{d}\tau\,\tilde{H}_{\mathrm{int}}\left(t\right)\tilde{H}_{\mathrm{int}}\left(t-\tau\right)\rho_{S}\otimes\rho_{B}\right\\}}_{\mathrm{(I)}}\\\
&\qquad\underbrace{-\frac{1}{\hbar^{2}}\mathrm{tr_{B}}\left\\{\int_{0}^{\infty}\mathrm{d}\tau\,\rho_{S}\otimes\rho_{B}\tilde{H}_{\mathrm{int}}\left(t-\tau\right)\tilde{H}_{\mathrm{int}}\left(t\right)\right\\}}_{\mathrm{(II)}}\\\
&\qquad\underbrace{+\frac{1}{\hbar^{2}}\,\mathrm{tr_{B}}\left\\{\int_{0}^{\infty}\mathrm{d}\tau\,\tilde{H}_{\mathrm{int}}\left(t\right)\rho_{S}\otimes\rho_{B}\tilde{H}_{\mathrm{int}}\left(t-\tau\right)\right\\}}_{\mathrm{(III)}}\\\
&\qquad\underbrace{+\frac{1}{\hbar^{2}}\,\mathrm{tr_{B}}\left\\{\int_{0}^{\infty}\mathrm{d}\tau\,\tilde{H}_{\mathrm{int}}\left(t-\tau\right)\rho_{S}\otimes\rho_{B}\tilde{H}_{\mathrm{int}}\left(t\right)\right\\}}_{\mathrm{(IV)}}\end{split}$
(10)

In making the Born approximation in the above master equation we assume that
the state of the EM field bath is always in thermal equilibrium at some given
temperature $T$, such that the bath equilibration time scale is much faster
than the system evolution. The thermal density matrix of the bath is given by
$\rho_{B}=\exp{\left(-H_{B}/(k_{B}T)\right)}/Z$, with $Z$ as the partition
function.

We can simplify each of the terms in the above master equation (10), starting
with (I),

$\begin{split}\mathrm{(I)}&=-\frac{1}{\hbar^{2}}\,\mathrm{tr_{B}}\left\\{\int_{0}^{\infty}\mathrm{d}\tau\,\tilde{H}_{\mathrm{int}}\left(t\right)\tilde{H}_{\mathrm{int}}\left(t-\tau\right)\rho_{S}\otimes\rho_{B}\right\\}\\\
&=-\frac{1}{\hbar^{2}}\,\mathrm{tr}_{\mathrm{B}}\bigg{\\{}\int_{0}^{\infty}\mathrm{d}\tau\sum_{n,m}\int_{0}^{\infty}\mathrm{d}\omega\mathrm{d}\omega^{\prime}\int\mathrm{d}^{3}{\mathbf{r}}\mathrm{d}^{3}{\mathbf{r}}^{\prime}\sum_{\lambda,\lambda^{\prime}}\left(\sigma_{n}^{+}e^{i\omega_{0}t}+\sigma_{n}^{-}e^{-i\omega_{0}t}\right)\\\
&\quad\times\left[{\mathbf{d}}_{n}\cdot{\mkern
1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}_{\lambda}\cdot{\mathbf{f}}_{\lambda}\left({\mathbf{r}},\omega\right)e^{-i\omega
t}+{\mathbf{f}}_{\lambda}^{\dagger}\left({\mathbf{r}},\omega\right)\cdot{\mkern
1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}^{\dagger}_{\lambda}\cdot{\mathbf{d}}_{n}e^{i\omega
t}\right]\left(\sigma_{m}^{+}e^{i\omega_{0}(t-\tau)}+\sigma_{m}^{-}e^{-i\omega_{0}(t-\tau)}\right)\\\
&\quad\times\left[{\mathbf{d}}_{m}\cdot{\mkern
1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}_{\lambda^{\prime}}\cdot{\mathbf{f}}_{\lambda^{\prime}}\left({\mathbf{r}}^{\prime},\omega^{\prime}\right)e^{-i\omega^{\prime}(t-\tau)}+{\mathbf{f}}^{\dagger}_{\lambda^{\prime}}\left({\mathbf{r}}^{\prime},\omega^{\prime}\right)\cdot{\mkern
1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}^{\dagger}_{\lambda^{\prime}}\cdot{\mathbf{d}}_{m}e^{i\omega^{\prime}(t-\tau)}\right]\rho_{S}\otimes\rho_{B}\bigg{\\}},\end{split}$

where we have defined the shorthand notation ${\mkern
1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}_{\lambda}^{(\dagger)}\equiv{\mkern
1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}^{(\dagger)}_{\lambda}\left({\mathbf{r}_{n}},{\mathbf{r}},\omega\right)$
and ${\mkern 1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}_{\lambda^{\prime}}^{(\dagger)}\equiv{\mkern
1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}^{(\dagger)}_{\lambda^{\prime}}\left({\mathbf{r}_{m}},{\mathbf{r}}^{\prime},\omega^{\prime}\right)$.
To take the trace over the field, we note that,

$\begin{split}\mathrm{tr_{B}}\left\\{{\mathbf{f}}_{1}^{\dagger}{\mathbf{f}}_{2}\rho_{B}\right\\}=&n_{\mathrm{th}}(\omega_{1})\delta(\omega_{1}-\omega_{2})\delta({\mathbf{r}}_{1}-{\mathbf{r}}_{2})\delta_{\lambda\lambda^{\prime}}\\\
\mathrm{tr_{B}}\left\\{{\mathbf{f}}_{1}{\mathbf{f}}_{2}^{\dagger}\rho_{B}\right\\}=&\left(n_{\mathrm{th}}(\omega_{1})+1\right)\delta(\omega_{1}-\omega_{2})\delta({\mathbf{r}}_{1}-{\mathbf{r}}_{2})\delta_{\lambda\lambda^{\prime}}\\\
\mathrm{tr_{B}}\left\\{{\mathbf{f}}_{1}{\mathbf{f}}_{2}\rho_{B}\right\\}=&\mathrm{tr_{B}}\left\\{{\mathbf{f}}_{1}^{\dagger}{\mathbf{f}}_{2}^{\dagger}\rho_{B}\right\\}=0\end{split}$
(11)

where $n_{\mathrm{th}}(\omega)=1/(\exp{\left(\hbar\omega/(k_{B}T)\right)}-1)$
is the average thermal occupation number in mode $\omega$. Tracing over the
bath and using the fluctuation-dissipation relation for the Green’s tensor
(7), Eq. (2.1.2) can be thus simplified to,

$\displaystyle\mathrm{(I)}=-\frac{\mu_{0}}{\pi\hbar}\int_{0}^{\infty}\mathrm{d}\tau\sum_{n,m}\int_{0}^{\infty}\mathrm{d}\omega~{}\omega^{2}{\mathbf{d}}_{n}\cdot\mathrm{Im}{\mkern
1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}({\mathbf{r}}_{n},{\mathbf{r}}_{m},\omega)\cdot{\mathbf{d}}_{m}$
$\displaystyle\left[\left(n_{\mathrm{th}}(\omega)+1\right)e^{-i\omega\tau}+n_{\mathrm{th}}(\omega)e^{i\omega\tau}\right]\left(\hat{\sigma}_{n}^{+}\hat{\sigma}_{m}^{-}e^{i\omega_{0}\tau}+\hat{\sigma}_{n}^{-}\hat{\sigma}_{m}^{+}e^{-i\omega_{0}\tau}\right)\rho_{S}.$
(12)

Performing the integral over $\tau$ and $\omega$, we get111The integration
over $\tau$ involves the use of the indentiy $\int_{0}^{\infty}\mathrm{d}\tau
e^{iX\tau}=\pi\delta(X)+i\mathcal{P}\frac{1}{X}$. The dirac-delta function
contributes to the dissipative coefficients $\Gamma_{nm}$ and the principal
value part to the dispersive energy shifts $\Delta\omega_{nm}$. The dispersive
shifts are then obtained via performing contour integrals over $\omega$, the
details of the derivation can be found in [8].:

$\displaystyle\mathrm{(I)}=\sum_{m,n}$
$\displaystyle\left[\left\\{-\frac{\Gamma_{nm}}{2}\left(n_{\mathrm{th}}(\omega_{0})+1\right)+i\left(\Delta\omega_{nm}^{\mathrm{OR}}-\Delta\omega_{nm}^{\mathrm{R}}\left(n_{\mathrm{th}}(\omega_{0})+1\right)\right)\right\\}\hat{\sigma}_{n}^{+}\hat{\sigma}_{m}^{-}\right.$
$\displaystyle\left.+\left\\{-\frac{\Gamma_{nm}}{2}n_{\mathrm{th}}(\omega_{0})-i\left(\Delta\omega_{nm}^{\mathrm{OR}}+\Delta\omega_{nm}^{\mathrm{R}}n_{\mathrm{th}}(\omega_{0})\right)\right\\}\hat{\sigma}_{n}^{-}\hat{\sigma}_{m}^{+}\right]\rho_{S},$
(13)

where we have defined

$\displaystyle\Delta\omega_{nm}^{\mathrm{R}}\equiv$
$\displaystyle\frac{\mu_{0}}{\hbar}\omega_{0}^{2}{\mathbf{d}}_{n}\cdot\mathfrak{Re}{\mkern
1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}\left({\mathbf{r}}_{n},{\mathbf{r}}_{m},\omega_{0}\right)\cdot{\mathbf{d}}_{m}$
(14) $\displaystyle\Gamma_{nm}\equiv$
$\displaystyle\frac{2\mu_{0}\omega_{0}^{2}}{\hbar}{\mathbf{d}}_{n}\cdot\mathfrak{Im}{\mkern
1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}\left({\mathbf{r}}_{n},{\mathbf{r}}_{m},\omega_{0}\right)\cdot{\mathbf{d}}_{m}$
(15) $\displaystyle\Delta\omega_{nm}^{\mathrm{OR}}\equiv$
$\displaystyle\frac{2\mu_{0}\omega_{0}k_{B}T}{\hbar^{2}}{\sum_{p=0}^{\infty}}^{\prime}\frac{\omega_{p}^{2}}{\omega_{p}^{2}+\omega_{0}^{2}}{\mathbf{d}}_{n}\cdot{\mkern
1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}\left({\mathbf{r}}_{n},{\mathbf{r}}_{m},i\omega_{p}\right)\cdot{\mathbf{d}}_{m},$
(16)

Here $\Delta\omega_{nm}^{\mathrm{R}}$ represents the resonant dipole-dipole
interaction between atoms $n$ and $m$ mediated by fluctuations of the EM field
in the presence of a medium. Similarly, $\Gamma_{nm}$ denote the dissipative
interaction coefficients between atoms $n$ and $m$, induced by thermal and
quantum fluctuations. We note that the resonant contribution to the total
Casimir-Polder potential $\Delta\omega_{nm}^{\mathrm{R}}$, as well as its
dissipative counterpart $\Gamma_{nm}$, depend on the real and imaginary part
of the response function of the environment at the resonant frequency of the
atoms ($\mathrm{Re}{\mkern 1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}\left({\mathbf{r}}_{n},{\mathbf{r}}_{m},\omega_{0}\right)$ and
$\mathrm{Im}{\mkern 1.5mu\overline{\mkern-1.5mu\hbox{$\mkern
1.5mu\overline{\mkern-1.5muG\mkern-1.5mu}\mkern 1.5mu$}\mkern-1.5mu}\mkern
1.5mu}\left({\mathbf{r}}_{n},{\mathbf{r}}_{m},\omega_{0}\right)$). The off-
resonant interaction coefficient $\Delta\omega_{nm}^{\mathrm{OR}}$ comprises
contribution from the fluctuations at all frequencies, with
$\omega_{p}=p\frac{2\pi k_{B}T}{\hbar}$ $(p\in\mathbb{Z}^{+})$, representing
the Matsubara frequencies [36, 8]. The prime on the summation indicates that
the $p=0$ term comes with a factor of $1/2$.

We can now simplify the remaining terms in the master equation to obtain:

$\displaystyle\mathrm{(II)}=$
$\displaystyle\sum_{m,n}\rho_{S}\left[\left\\{-\frac{\Gamma_{nm}}{2}\left(n_{\mathrm{th}}(\omega_{0})+1\right)-i\left(\Delta\omega_{nm}^{\mathrm{OR}}-\Delta\omega_{nm}^{\mathrm{R}}\left(n_{\mathrm{th}}(\omega_{0})+1\right)\right)\right\\}\hat{\sigma}_{n}^{+}\hat{\sigma}_{m}^{-}\right.$
$\displaystyle\left.+\left\\{-\frac{\Gamma_{nm}}{2}n_{\mathrm{th}}\left(\omega_{0}\right)+i\left(\Delta\omega_{nm}^{\mathrm{OR}}+\Delta\omega_{nm}^{\mathrm{R}}n_{\mathrm{th}}\left(\omega_{0}\right)\right)\right\\}\hat{\sigma}_{n}^{-}\hat{\sigma}_{m}^{+}\right]$
(17) $\displaystyle\mathrm{(III)}=$
$\displaystyle\mathrm{(IV)}=\sum_{m,n}\frac{\Gamma_{nm}}{2}\left(n_{\mathrm{th}}(\omega_{0})+1\right)\hat{\sigma}_{m}^{-}\rho_{S}\hat{\sigma}_{n}^{+}+\frac{\Gamma_{nm}}{2}n_{\mathrm{th}}(\omega_{0})\hat{\sigma}_{n}^{-}\rho_{S}\hat{\sigma}_{m}^{+}.$
(18)

Adding together Eqs. (2.1.2), (2.1.2) and (18), we obtain the full master
equation as:

$\displaystyle\frac{{d}\rho_{S}}{{d}t}=-\frac{i}{\hbar}\left[H_{\mathrm{eff}},\rho_{S}\right]+\mathcal{L}_{\mathrm{eff}}\left[\rho_{S}\right]$
(19)

where $H_{\mathrm{eff}}$ is the medium-assisted effective Hamiltonian that
describes the energy shifts or fluctuation forces on the system, given by:

$\displaystyle H_{\mathrm{eff}}=$
$\displaystyle\sum_{n=1}^{N}\hbar\left[\Delta\omega_{nn}^{\mathrm{OR}}-\Delta\omega_{nn}^{\mathrm{R}}\left(n_{\mathrm{th}}\left(\omega_{0}\right)+1\right)\right]\hat{\sigma}^{+}_{n}\hat{\sigma}^{-}_{n}-\hbar\left[\Delta\omega_{nn}^{\mathrm{OR}}\right.$
$\displaystyle\left.+\Delta\omega_{nn}^{\mathrm{R}}n_{\mathrm{th}}\left(\omega_{0}\right)\right]\hat{\sigma}^{-}_{n}\hat{\sigma}^{+}_{n}-\sum_{n\neq
m}\hbar\left(\Delta\omega_{nm}^{\mathrm{R}}\left(2n_{\mathrm{th}}\left(\omega_{0}\right)+1\right)\right)\hat{\sigma}^{+}_{n}\hat{\sigma}^{-}_{m}$
(20)

The medium-assisted Liouvillian $\mathcal{L}_{\mathrm{eff}}$ accounts for the
dissipation and decoherence of the system, and is given by the Lindblad
superoperator:

$\displaystyle\mathcal{L}_{\mathrm{eff}}\left[\rho_{S}\right]=\sum_{n,m=1}^{N}$
$\displaystyle-\frac{\Gamma_{nm}}{2}\left(n_{\mathrm{th}}(\omega_{0})+1\right)\left[\hat{\sigma}^{+}_{n}\hat{\sigma}^{-}_{m}\rho_{S}+\rho_{S}\hat{\sigma}^{+}_{n}\hat{\sigma}^{-}_{m}-2\hat{\sigma}^{-}_{n}\rho_{S}\hat{\sigma}^{+}_{m}\right]$
$\displaystyle-\frac{\Gamma_{nm}}{2}n_{\mathrm{th}}(\omega_{0})\left[\hat{\sigma}^{-}_{n}\hat{\sigma}^{+}_{m}\rho_{S}+\rho_{S}\hat{\sigma}^{-}_{n}\hat{\sigma}^{+}_{m}-2\hat{\sigma}^{+}_{n}\rho_{S}\hat{\sigma}^{-}_{m}\right],$
(21)

where the first term denotes dissipation and decoherence due to both thermal
and quantum fluctuations. The second term represents absorption of thermal
radiation by the atomic system, and vanishes at $T\rightarrow 0$.

The Hamiltonian $H_{\mathrm{eff}}$ accounts for the fluctuation-induced energy
shifts or forces in the presence of external media for a collection of atoms
in arbitrary states. The fluctuation-induced force between the atoms and the
medium can be found as the spatial derivative of the medium-modified energy of
the atomic system,

$F=\frac{{d}}{{d}z}\,\mathrm{tr_{S}}\left\\{\rho_{S}H_{\mathrm{eff}}\right\\},$
(22)

where $z$ is the coordinate along which the atoms and medium are separated
from each other.

Figure 3: Fluctuation-induced force on a system of two atoms near a silica
surface. The atoms are initially prepared in excited states
$\left|ee\right\rangle$, as can be seen from the green dash-dotted curve in
(b). As the atomic system interacts with the EM field bath, the atoms get de-
excited, eventually going to the ground state as depicted by the blue dotted
curve in (b). The total fluctuation force on the atoms is represented by the
black solid curve in (a). There is an enhancement in the force as the atomic
population in the superradiant state increases.

####  2.1.3 Instructive example: two atoms near a surface

As an example, we can analyze the fluctuation-forces on two atoms prepared in
collective states near a surface, as shown in Fig. 3. It can be seen that
starting with the initial state $\rho_{S}=\left|ee\right\rangle\left\langle
ee\right|$, the dynamics of the two atoms is such that their joint density
matrix can be written as

$\rho_{S}=\rho_{ee}\left|ee\right\rangle\left\langle
ee\right|+\rho_{\mathrm{sup,sup}}\left|\psi_{+}\right\rangle\left\langle\psi_{+}\right|+\rho_{\mathrm{sub,sub}}\left|\psi_{-}\right\rangle\left\langle\psi_{-}\right|+\rho_{gg}\left|gg\right\rangle\left\langle
gg\right|,$ (23)

where
$\left|\psi_{\pm}\right\rangle=\left(\left|eg\right\rangle\pm\left|ge\right\rangle\right)/\sqrt{2}$
refer to the Dicke super- and subradiant states, that are known to exhibit
enhancement and suppression in spontaneous emission [37, 38]. The blue solid
curve $(\tilde{F}_{\mathrm{sup}})$ and the red dashed curve
$(\tilde{F}_{\mathrm{sub}})$ in Fig. 3(a) show that these states exhibit a
similar enhancement and suppression in the fluctuation-induced force [28, 27,
29]. The master equation approach thus allows one to monitor the non-
equilibrium dynamics of fluctuation-induced forces on the collective atomic
density matrix in the presence of correlations.

###  2.2 Remarks

We have introduced a master equation description of quantum and thermal
fluctuation-induced forces on a system of $N$ two-level atoms near a medium.
The master equation (19) is sufficiently general to describe the non-
equilibrium dynamics of the system prepared in general quantum state near an
arbitrary macroscopic body and includes all fluctuation phenomena –
forces/energy shifts, dissipation and decoherence. Our approach combines the
macroscopic QED formalism [7, 8, 31, 32] with an Open Quantum System framework
[35], allowing one to analyze fluctuation-induced forces and dissipative
dynamics in quantum systems on the same footing.

There is an interesting parallel between the QED and thermodynamic
perspectives on fluctuation-induced forces. In the QED framework, fluctuation-
induced forces arise from the sum over interactions between the fluctuating
dipole moments that constitute matter, mediated via virtual and thermal
photons of the quantized EM field. Equivalently, these forces can be seen as
originating via the difference in the free energy of a photon gas with and
without the presence of external boundaries [39, 40, 41, 42, 43, 44, 45]. In
addition to fluctuation-induced forces, the framework of fluctuational QED has
been successfully applied to study radiative heat transfer between bodies in
close proximity [46, 47, 48, 49, 50, 51, 52, 53]. Moreover, the concepts of
non-equilibrium thermodynamics have been used to demonstrate quantum friction
in atoms moving near surfaces [54, 55, 56].

The natural question arises whether thermodynamic considerations can further
constrain and/or characterize the properties of quantum and thermal
fluctuation-induced phenomena.

##  3 Second law of thermodynamics for fluctuations

Some of the most important cornerstones of quantum thermodynamics are
statements of the second law that are collectively known as _fluctuation
theorems_. The first versions were derived for purely classical dynamics [57,
58], which are already regarded as the most significant results in the modern
thermodynamics [59]. Many important principles naturally follow, such as the
arrow of time [60] and linear response theory [61, 62]. To better explore the
laws of thermodynamics at the nanoscale, reformulating the second law of
thermodynamics from the fundamentals of quantum physics is, thus, of
particular significance.

###  3.1 Two-Time Measurement Scheme

In the energy representation, the two central quantities of thermodynamics are
heat and work [1]. In contrast to state functions, though, these two
quantities are described by non-exact differentials. In other words, heat and
work depend on the specific path a thermodynamics system takes in state space.
This is consistent with classical mechanics, where work is defined as a
functional of a force along a trajectory.

Due to Heisenberg’s uncertainty relations, classical trajectories are somewhat
ill-defined in quantum dynamics. Hence, how to define work and heat in quantum
systems becomes significantly more involved [2]. In complete analogy to
classical thermodynamics, “meaningful” notions of quantum thermodynamic work
are motivated by experimental accessibility.

Arguably, the most prominent approach has been called two-time measurement
(TTM) (or two-point measurement) scheme [63, 64, 65, 66, 67, 68, 69, 70, 71].
In this paradigm, thermodynamic work is identified as the difference of the
outcomes of energy measurements preformed at the beginning and the end of a
process. If the system is isolated, i.e., if it evolves under unitary dynamics

$\partial_{t}U_{t}=-i/\hbar\,H_{t}U_{t}\quad\text{with}\quad
U_{0}=\mathbb{I}\,,$ (24)

any difference in energy has to be identified with work. In other words,
isolated quantum systems do not exchange energy with the environment.

Schematically the TTM scheme is illustrated in Fig. 4.

Figure 4: In the TTM scheme, the quantum system is initially prepared in a
Gibbs state $\rho_{0}^{\text{eq}}=\exp{\left(-\beta H_{0}\right)}/Z_{0}$. A
first energy measurement is performed, which projects the state into an
eigenstate of $H_{0}$, $\left|E_{i}\right\rangle$ with the measurement outcome
$E_{i}$. Then, the system is let to evolve under the unitary $U_{\tau}$,
before a the second energy measurement $H_{\tau}$ is performed.

Consider a system initially prepared in a Gibbs state with inverse temperature
$\beta$ and initial Hamiltonian $H_{0}$,
$\rho_{0}^{\text{eq}}=\exp{\left(-\beta H_{0}\right)}/Z_{0}$, where as before
$Z_{0}\equiv\mathrm{tr}\left\\{\exp{\left(-\beta H_{0}\right)}\right\\}$ is
the standard partition function. At $t=0$ a first projective measurement of
the energy is performed, which projects the system into an energy eigenstate
$\\{E_{i},\left|E_{i}\right\rangle\\}$ of $H_{0}$. The corresponding
eigenvalue $E_{i}$ is recorded, and the system is let to evolve under the
unitary $U_{t}$ from $t=0$ to $t=\tau$. At $t=\tau$ a second measurement of
the energy is performed, which now projects the system into an
eigenstate$\\{E_{j}^{\prime},\left|E_{j}^{\prime}\right\rangle\\}$ of
$H_{\tau}$.

The first law of thermodynamics then dictates that the energy difference is
the work is the thermodynamic work,

$W_{i\to j}=E_{j}^{\prime}-E_{i}\,.$ (25)

The corresponding probability density function characterizing the fluctuations
of the work reads

$P_{F}(W)\equiv\sum_{ij}\delta(W-E_{j}^{\prime}+E_{i})\,\left|\langle
E_{j}^{\prime}\hskip 1.0pt|U_{\tau}|\hskip
1.0ptE_{i}\rangle\right|^{2}\,\frac{e^{-\beta E_{i}}}{Z_{0}}\,,$ (26)

where $\left|\langle E_{j}^{\prime}\hskip 1.0pt|U_{\tau}|\hskip
1.0ptE_{i}\rangle\right|^{2}$ is the transition probability from
$\left|E_{i}\right\rangle$ to $\left|E_{j}^{\prime}\right\rangle$. Note that
the such defined work distribution is consistent with thermodynamics, which we
observe from

$\begin{split}\langle W\rangle_{P}&=\int dWP_{F}(W)W=\sum_{ij}\frac{e^{-\beta
E_{i}}}{Z_{0}}\left|\langle E_{j}^{\prime}\hskip 1.0pt|U_{\tau}|\hskip
1.0ptE_{i}\rangle\right|^{2}(E_{j}^{\prime}-E_{i})\\\
&=\mathrm{tr}\left\\{H_{\tau}U_{\tau}\rho_{0}^{\text{eq}}U_{\tau}^{\dagger}\right\\}-\mathrm{tr}\left\\{H_{0}\rho_{0}^{\text{eq}}\right\\}=\left\langle
H_{\tau}\right\rangle-\left\langle H_{0}\right\rangle\equiv\langle
W\rangle\,.\end{split}$ (27)

In standard thermodynamics [1], reversible processes are ideal in the sense
that they dissipate the least amount of energy into the environment. Hence, it
is often also interesting to consider the “reverse” quantum process. To this
end, consider the same quantum system, but now initially prepared in a Gibbs
state with respect to the final Hamiltonian, $H_{\tau}$, and which evolves
under the “backward” dynamics $U_{t}^{\dagger}$. The corresponding work
distribution reads

$P_{B}(-W)\equiv\sum_{ij}\delta(-W-E_{i}+E_{j}^{\prime})\,\left|\langle
E_{i}\hskip 1.0pt|U_{\tau}^{\dagger}|\hskip
1.0ptE_{j}^{\prime}\rangle\right|^{2}\frac{e^{-\beta
E_{j}^{\prime}}}{Z_{\tau}}\,,$ (28)

where $Z_{\tau}\equiv\mathrm{tr}\left\\{\exp{\left(-\beta
H_{\tau}\right)}\right\\}$ is the standard partition function of $H_{\tau}$.

Interestingly, the work distributions for forward and backward processes are
closely related. We have,

$\begin{split}P_{F}(W)&=\sum_{ij}\frac{e^{-\beta E_{i}}}{Z_{0}}\left|\langle
E_{j}^{\prime}\hskip 1.0pt|U_{\tau}|\hskip
1.0ptE_{i}\rangle\right|^{2}\delta(W-E_{j}^{\prime}+E_{i})\\\
&=\frac{Z_{\tau}}{Z_{0}}\sum_{ij}\frac{e^{-\beta(E_{j}^{\prime}-W)}}{Z_{\tau}}\left|\langle
E_{j}^{\prime}\hskip 1.0pt|U_{\tau}|\hskip
1.0ptE_{i}\rangle\right|^{2}\delta(W-E_{j}^{\prime}+E_{i})\\\
&=\frac{Z_{\tau}}{Z_{0}}\left(\sum_{ij}\frac{e^{-\beta
E_{j}^{\prime}}}{Z_{\tau}}\left|\langle E_{i}\hskip
1.0pt|U_{\tau}^{\dagger}|\hskip
1.0ptE_{j}^{\prime}\rangle\right|^{2}\delta(-W-E_{i}+E_{j}^{\prime})\right)e^{\beta
W}\,.\end{split}$ (29)

Hence, we obtain the Quantum Crooks Fluctuation Theorem [63, 64]

$\frac{P_{F}(W)}{P_{B}(-W)}=e^{\beta(W-\Delta F)}\,,$ (30)

where $\Delta F\equiv-\beta^{-1}\ln(Z_{\tau}/Z_{0})$ is the equilibrium free
energy difference.

The Quantum Crooks Fluctuation Theorem (30) is a so-called detailed
fluctuation theorem. The corresponding integral version, the Quantum Jarzynski
Equality [69, 72] is simply obtained from

$\langle e^{-\beta W}\rangle\equiv\int dWe^{-\beta W}P_{F}(W)=e^{-\beta\Delta
F}\,.$ (31)

It is important to recognize that the fluctuation theorems permit to determine
the free energy difference from an average over nonequilibrium fluctuations of
the thermodynamic work. This is a stronger statement then the standard maximum
work theorem, which provides only an inequality. In fact, employing Jensen’s
inequality, the Jarzynski equality directly gives

$\langle W\rangle\geqslant\Delta F\,.$ (32)

Finally, it has been noted that the TTM scheme has a good correspondence with
the classical scenario [73, 74], which makes it a conceptually sound starting
point for quantum thermodynamic considerations.

####  3.1.1 Symmetry relation for the characteristic function

The fluctuation theorems can be written in a more convenient form by
considering the characteristic functions of the work distributions [69]. Let
$C_{F}(u)$ and $C_{B}(u)$ be the characteristic functions of $P_{F}(W)$ and
$P_{B}(-W)$, respectively

$\begin{split}C_{F}(u)\equiv\int dWP_{F}(W)e^{iuW}\quad\text{and}\quad
C_{B}(u)\equiv\int dWP_{B}(-W)e^{-iuW}\,,\end{split}$ (33)

which can be written as

$\begin{split}C_{F}(u)&=\sum_{ij}\frac{e^{iuE_{j}^{\prime}+i(-u+i\beta)E_{i}}}{Z_{0}}\left|\langle
E_{j}^{\prime}\hskip 1.0pt|U_{\tau}|\hskip 1.0ptE_{i}\rangle\right|^{2}\\\
C_{B}(u)&=\sum_{ij}\frac{e^{i(-u+i\beta)E_{j}^{\prime}+iuE_{i}}}{Z_{\tau}}\left|\langle
E_{i}\hskip 1.0pt|U_{\tau}^{\dagger}|\hskip
1.0ptE_{j}^{\prime}\rangle\right|^{2}\,.\end{split}$ (34)

Therefore, we immediately have

$\frac{C_{F}(u)}{C_{B}(-u+i\beta)}=e^{-\beta\Delta F}\,,$ (35)

which expresses the Quantum Crooks Fluctuation theorem (30) as a symmetry
relation for the characteristic functions. In the following, Sec. 3.2, we will
demonstrate that $C_{F}(u)$ and $C_{B}(-u+i\beta)$ can be computed by using
the single-qubit interferometry algorithm on a quantum computer.

###  3.2 Quantum circuits for verifying quantum fluctuation theorem

The quantum fluctuation theorem based on the TTM scheme has been
experimentally studied using various platforms, such as trapped ions [75, 76,
77], nuclear magnetic resonance [78], defect spins [79, 80], superconducting
qubit system [81, 82], and even the D-Wave machine [83].

In the remainder of the present exposition, we focus on how to implement the
symmetry relation (35) on quantum computers. To this end, we re-write the
expressions for the characteristic function as

$C_{F}(u)=\mathrm{tr}\left\\{U^{\dagger}_{\tau}e^{iuH_{\tau}}U_{\tau}e^{-iuH_{0}}\rho_{0}^{\text{eq}}\right\\}\,.$
(36)

and

$C_{B}(u)=\mathrm{tr}\left\\{U_{\tau}e^{iuH_{0}}U^{\dagger}_{\tau}e^{-iuH_{\tau}}\rho_{\tau}^{\text{eq}}\right\\}\,,$
(37)

where $\rho_{\tau}^{\text{eq}}=\exp{\left(-\beta H_{\tau}\right)}/Z_{\tau}$
denotes the Gibbs state defined by the final Hamiltonian $H_{\tau}$. We will
now see, that the expressions of Eqs. (36) and (37) are particularly useful as
they can be computed using single-qubit interferometry, which was
experimentally implemented in Refs. [70, 71].

####  3.2.1 General Idea

@C=1em @R=1.2em |0⟩& H 0 [1] $\sigma_{x}$ and $\sigma_{y}$
ρ U

Figure 5: Quantum circuit of the single-qubit interferometry

Consider the single-qubit interferometry quantum circuit, depicted in Fig. 5.
Here, $\mathbf{H}$ is the Hadamard gate, which can be expressed in the logical
basis as

$\mathbf{H}\equiv\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\ 1&-1\end{pmatrix}$
(38)

and $U$ is a unitary operator. We further define
$\left|0\right\rangle\equiv\begin{pmatrix}1&0\end{pmatrix}^{T}$ and
$\left|1\right\rangle\equiv\begin{pmatrix}0&1\end{pmatrix}^{T}$. Choosing the
input state to be

$\rho_{\text{in}}=\left|0\right\rangle\\!\left\langle
0\right|\otimes\rho=\begin{pmatrix}\rho&0\\\ 0&0\end{pmatrix}\,,$ (39)

it is easy to see that the output state is given by

$\rho_{\text{out}}=\frac{1}{2}\begin{pmatrix}\rho&\rho U^{\dagger}\\\
U\rho&U\rho U^{\dagger}\end{pmatrix}\,.$ (40)

As before, $\sigma_{x},\sigma_{y},\sigma_{z}$ denote the Pauli’s matrices

$\displaystyle\sigma_{x}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},~{}\sigma_{y}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},~{}\quad\text{and}\quad\sigma_{z}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}.$ (41)

Thus, we have

$\langle\sigma_{x}\rangle=\mathrm{tr}\left\\{(\sigma_{x}\otimes\mathbb{I})\rho_{\text{out}}\right\\}=\mathfrak{Re}\left(\mathrm{tr}\left\\{\rho
U\right\\}\right)\,,$ (42)

and

$\langle\sigma_{y}\rangle=\mathrm{tr}\left\\{(\sigma_{y}\otimes\mathbb{I})\rho_{\text{out}}\right\\}=\mathfrak{Im}\left(\mathrm{tr}\left\\{\rho
U\right\\}\right)\,.$ (43)

Consequently, we obtain $\mathrm{tr}\left\\{\rho U\right\\}$ from the quantum
circuit in Fig. 5 as

$\displaystyle\mathrm{tr}\left\\{\rho
U\right\\}=\langle\sigma_{x}\rangle+i\langle\sigma_{y}\rangle\,.$ (44)

####  3.2.2 Quantum circuits for computing characteristic functions

We will now see that the characteristic functions can be computed by employing
the same single-qubit interferometry. The quantum circuit for computing
$C_{F}(u)$ is depicted in Fig. 6, and Fig. 7 shows the the quantum circuit for
computing $C_{B}(u)$.

@C=1em @R=1.2em |0⟩& H 0 [1] 0 [1] $\sigma_{x}$ and $\sigma_{y}$
ρ_0^eq e^-i u H_0 U_τ e^iu H_τ

Figure 6: Quantum circuit for computing $C_{F}(u)$

@C=1em @R=1.2em |0⟩& H 0 [1] 0 [1] $\sigma_{x}$ and $\sigma_{y}$
ρ_τ^eq e^-i u H_τ U_τ^† e^iu H_0

Figure 7: Quantum circuit for computing $C_{B}(u)$.

For the symmetry relation (35), we are interested in
$C_{B}(-u+i\beta)=\mathrm{tr}\left\\{U_{\tau}e^{-\beta
H_{0}}e^{-iuH_{0}}U_{\tau}^{\dagger}e^{\beta
H_{\tau}}e^{iuH_{\tau}}\rho_{\tau}^{\text{eq}}\right\\}$. A minor technical
difficulty arises from the fact that $\exp{\left(\beta H_{\tau}\right)}$ and
$\exp{\left(-\beta H_{0}\right)}$, are not unitary operators, which means that
they cannot be represented by a quantum gate. Hence, to compute
$C_{B}(-u+i\beta)$, we need employ the method developed in Ref. [84]. First,
$\exp{\left(\beta H_{\tau}\right)}$ and $\exp{\left(-\beta H_{0}\right)}$ are
decomposed into a Pauli sequence $\\{\sigma_{\ell}\\}_{\ell=1}^{d^{2}}$, where
$\sigma_{1}\equiv\mathbb{I}$ 222For example, for a two-qubit system $(d=4)$,
the Pauli sequence is the set of operators
$\\{\mathbb{I},\sigma_{x}\otimes\mathbb{I},\sigma_{y}\otimes\mathbb{I},\sigma_{z}\otimes\mathbb{I},\mathbb{I}\otimes\sigma_{x},\mathbb{I}\otimes\sigma_{y},\mathbb{I}\otimes\sigma_{z},\sigma_{x}\otimes\sigma_{x},\sigma_{x}\otimes\sigma_{y},\sigma_{x}\otimes\sigma_{z},\sigma_{y}\otimes\sigma_{x},\sigma_{y}\otimes\sigma_{y},\sigma_{y}\otimes\sigma_{z},\sigma_{z}\otimes\sigma_{x},\sigma_{z}\otimes\sigma_{y},\sigma_{z}\otimes\sigma_{z}\\}$,
which has $4^{2}=16$ elements, and these elements are the bases constructing
any $4\times 4$ matrix. Usually, for a $d\times d$ matrix, the Pauli sequence
has $d^{2}$ elements.. Then, we can write

$e^{-\beta
H_{0}}=\sum_{\ell=1}^{d^{2}}\alpha_{\ell}^{(0)}\sigma_{\ell}\quad\text{and}\quad
e^{\beta
H_{\tau}}=\sum_{\ell=1}^{d^{2}}\alpha_{\ell}^{(\tau)}\sigma_{\ell}\,,$ (45)

where we have

$\alpha_{\ell}^{(0)}=\frac{1}{d}\,\mathrm{tr}\left\\{e^{-\beta
H_{0}}\sigma_{\ell}\right\\}\quad\text{and}\quad\alpha_{\ell}^{(\tau)}=\frac{1}{d}\,\mathrm{tr}\left\\{e^{\beta
H_{\tau}}\sigma_{\ell}\right\\}\,.$ (46)

These parameters can be regarded as known quantities since we have full
knowledge of $H_{0}$ and $H_{\tau}$.

With this, we can write

$\displaystyle
C_{B}(-u+i\beta)=\sum_{k\ell}\alpha_{k}^{(0)}\alpha_{\ell}^{(\tau)}\mathrm{tr}\left\\{U_{\tau}\sigma_{k}e^{-iuH_{0}}U_{\tau}^{\dagger}\sigma_{\ell}e^{iuH_{\tau}}\rho_{\tau}^{\text{eq}}\right\\}.$
(47)

Defining

$\displaystyle
F_{k\ell}\equiv\mathrm{tr}\left\\{U_{\tau}\sigma_{k}e^{-iuH_{0}}U_{\tau}^{\dagger}\sigma_{\ell}e^{iuH_{\tau}}\rho_{\tau}^{\text{eq}}\right\\}\,,$
(48)

we can construct the quantum circuit depicted in Fig. 8

@C=1em @R=1.2em |0⟩& H 0 [1] 0 [1] 0 [1] 0 [1] $\sigma_{x}$ and $\sigma_{y}$
ρ_τ^eq e^iu H_τσ_ℓ U_τ^† e^-iuH_0 σ_k

Figure 8: Quantum circuit for computing $F_{k\ell}$ to construct
$C_{B}(-u+i\beta)$.

to compute $F_{k\ell}$. In conclusion, the characteristic functions can be
directly determined with the depicted quantum circuits.

However, we would be remiss if we forwent a few remarks on the shortcomings of
the TTM scheme. It has even been argued that TTM scheme is thermodynamically
inconsistent since the projective energy measurements inevitably destroy
quantum coherences [85]. To overcome this issue, recent works proposed
alternative paradigms, such as dynamic Bayesian networks [86], the Maggenau-
Hill quasiprobability [87], and the one-time measurement (OTM) scheme [88, 89,
90, 91, 92]. Particularly, in the OTM scheme, the distribution of changes in
internal energy is constructed by considering the expectation value of the
energy conditioned on the initial energy measurement outcomes. This formalism
avoids the second projective measurement and, therefore, the thermodynamic
contribution of quantum coherence or the correlations generated by the
dynamics are naturally contained in the formalism [93]. However, we have shown
that the OTM scheme is nothing but the TTM scheme for non-demolition
measurements, which can also be implemented with quantum circuits [94].

###  3.3 Fluctuation theorems for open quantum dynamics

The natural question arises how the fluctuation theorem can be generalized to
open system dynamics described by master equations such as (19) in an
experimentally meaningful way. While this is still a somewhat open question,
it has been suggested that also for open systems TTM is a conceptually sound
approach [95, 96, 97, 98, 99, 100, 101]. However, also other approaches such
as OTM [89] or alternative notions in phase space [102, 103] have been
considered.

##  4 Concluding remarks

It is a core tenet of statistical mechanics that in macroscopic systems
fluctuations are irrelevant. Mathematically, this is underpinned by the law of
large numbers. However, at the nanoscale fluctuations dominate and hence any
thermodynamically consistent description will have to focus on the nature,
characteristics, and consequences of fluctuations. In this chapter of the
_Encyclopedia of Mathematical Physics_ we have outlined two conceptually
related, but typically separately considered areas of current research:
fluctuation forces and fluctuation theorems. The confluence of these two
topics holds significant potential in terms of building a non-equilibrium
quantum thermodynamics-based description of fluctuation phenomena in QED, with
possible applications in near-field radiative heat transfer, design of
nanoscale quantum systems and fundamental connections between thermodynamics
and fluctuational QED that remain to be elucidated.

Acknowledgements

A.S is supported by the startup package from University of Massachusetts
Boston. This work was supported by the National Science Foundation under Grant
No. PHY-2309341 (K.S.) and No. PHY-1748958 (K.S.), and by the John Templeton
Foundation under Award No. 62422 (K.S. and S.D.).

##

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