# Exploring the role of criticality in the quantum Otto cycle fueled by the
anisotropic quantum Rabi-Stark 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  Norton G. de Almeida
Instituto de Física, Universidade Federal de Goiás, 74.001-970, Goiânia \- GO,
Brazil  G. D. de Moraes Neto<EMAIL_ADDRESS>Department of Physics,
Zhejiang Normal University, Jinhua 321004, Zhejiang, P. R. China

###### Abstract

Quantum heat machines, encompassing heat engines, refrigerators, heaters, and
accelerators, represent the forefront of quantum thermodynamics, offering a
novel paradigm for converting heat energy into useful mechanical work.
Leveraging quantum mechanical principles, these machines promise superior
efficiency and performance compared to classical counterparts, with potential
applications in renewable energy and quantum computing. This paper
investigates a quantum Otto engine operating in both ideal and finite-time
scenarios, employing a two-level system interacting with a harmonic oscillator
within the framework of the anisotropic quantum Rabi-Stark model (AQRSM) as
the working medium. This model is notable for exhibiting both first-order and
continuous quantum phase transitions. By focusing on quantum heat engines, our
study reveals that these phase transitions critically modulate the efficiency
and power of AQRSM-based engines, outperforming quantum engines fueled by
working medium with harmonic spectrum. Additionally, we explore the impacts of
quantum friction and conduct limit cycle analysis in finite-time operations,
providing insights into optimizing quantum heat engines for practical
implementation.

###### pacs:

42.50.Ar, 03.65.Yz, 42.50.Pq

##  I Introduction

Quantum thermodynamics stands as a rapidly advancing domain aimed at
comprehending thermodynamic phenomena at the quantum scale [1, 2, 3, 4, 5, 6,
7]. This field attracts significant attention from the scientific community
not only due to its role in bridging quantum mechanics and thermodynamics but
also for its potential in advancing devices that take advantage of the unique
properties of quantum systems as their working medium (WM)[8, 9, 10, 11, 12,
13].

Quantum heat engines (QHEs) play a pivotal role in probing the thermodynamic
characteristics of quantum systems and harnessing work from heat flow between
hot and cold reservoirs [14, 15, 16, 17, 18, 19, 20, 21, 22]. They have
attracted considerable attention from scientists and engineers due to the
growing demand for efficient energy utilization, storage, and transfer through
quantum technologies. Experimentally, these engines have been implemented on
various platforms, including trapped ion systems [23, 24, 25, 26],
optomechanics [27, 28], ultracold atoms [29, 30], and superconducting circuits
[31, 32, 33, 34, 35].

Critical systems, such as those undergoing quantum phase transitions, are
particularly fascinating as WM in quantum heat engines due to their unique
properties at criticality, which have been extensively studied in recent
literature [36, 37, 38, 39, 40, 41, 42, 43, 44, 45]. Near critical points,
these systems exhibit heightened fluctuations and correlations, offering
opportunities to enhance work extraction or gain better control over
thermodynamic processes. This critical behavior introduces intricate dynamics
into the system, resulting in novel scaling laws for work output and
efficiency. For instance, the divergence in correlation length and time near a
quantum critical point profoundly impacts the engine’s efficiency. Moreover,
investigating critical systems as WMs facilitates the exploration of universal
properties of quantum heat engines. Universality suggests that different
systems near criticality may demonstrate similar scaling behaviors, thereby
broadening the applicability of the results. This concept implies that the
performance of quantum heat engines can be predicted based on the critical
exponents of the involved phase transitions [41].

The cornerstone model for light-matter interaction is the quantum Rabi model
(QRM), which involves a two-level atom (qubit) coupled through dipole
interaction to a quantized single-mode cavity field [46, 47, 48, 49]. This
model has been extensively explored across various domains, including quantum
optics [50], entanglement generation [51], quantum thermodynamics [52, 53, 54,
55], and the study of quantum phase transitions [56, 57, 58, 59, 60]. When
different coupling constants are applied to the rotating-wave (RW) and
counter-rotating-wave (CRW) interaction terms, the QRM transforms into the
anisotropic quantum Rabi model (AQRM)[61, 62, 63]. Additionally, omitting the
CRW terms results in the Jaynes-Cummings model (JCM) [64]. Notably, both the
AQRM and the JCM exhibit first-order quantum phase transitions (QPTs)[62, 65].
These QPTs significantly impact the quantum thermalization dynamics and the
persistence of quantum correlations at thermal equilibrium [66, 67, 68].

Moreover, the Quantum Rabi-Stark model (QRSM) is an extension of the QRM that
incorporates nonlinear coupling terms[69, 70, 71]. This model was inspired by
recent experimental advancements[72] and has been extensively studied in
theoretical works[73, 74, 75, 72]. The QRSM has been analyzed using the
Bargmann space approach [71, 73] and solved using the Bogoliubov operator
approach (BOA)[74]. These studies have revealed many exotic properties of the
QRSM, such as first-order QPT and spectral collapse [74]. Furthermore, the
AQRM with Stark coupling terms, also known as the anisotropic quantum Rabi-
Stark model (AQRSM), has been found to exhibit both first-order and continuous
QPTs [76].

In this paper, we perform a comprehensive examination of a thermal engine
utilizing the AQRSM as the working substance. Our study begins by thoroughly
analyzing the ideal scenario of infinite operation time and zero power
delivered, focusing on the intricate relationship between phase transitions
and their influence on enhancing efficiency, work throughout the quantum Otto
cycle. Additionally, we construct a detailed phase diagram to provide insights
into the operational regimes, shedding light on the behavior of the coupled
system compared to engines operating with a qubit or a quantum harmonic
oscillator.

Expanding our investigation beyond the ideal scenario, we explore finite-time
operations, where the trade-off between power and efficiency becomes
prominent, along with the effects of quantum friction. Here, we highlight the
crucial role played by the spectral characteristics of the working medium,
including level crossings, degeneracies, and quasi-degenerate points, in
shaping the performance of quantum engines. Furthermore, we incorporate a
limit cycle analysis to deepen our understanding of the system behavior under
finite-time conditions.

Acknowledgments to previous investigations on light-matter heat engines[52,
53, 54, 55], which considered the QRM as the WM, are due. However, these
studies did not encompass first-order QPTs, which are absent in the QRM, and
focused on the ideal scenario of zero power delivery and extremely low
temperatures. Notably, only Ref.[55] addressed the second-order QPTs occurring
at an infinite frequency ratio for the Stirling cycle.

The paper is organized as follows: In Section II A, we introduce the
Anisotropic Quantum Rabi-Stark model, describe the Quantum Otto Cycle, and
present the calculations for work and heat, as well as the operation in finite
time. In Section III, we present and discuss the results, focusing on the
working operational regimes for the ideal adiabatic process, efficiency, and
work. Additionally, for the finite-time cycle, we focus on the Heat engine and
perform an analysis of entropy production and how the power and efficiency are
affected by the light-matter coupling. In Section IV, we present our
conclusions.

##  II THEORETICAL FRAMEWORK

###  II.1 Anisotropic quantum Rabi-Stark model

The Anisotropic Quantum Rabi-Stark Model (AQRSM) describes the coupling
between a two-level system (qubit) and a single-mode bosonic field,
experiencing both the Rotating-Wave (RW) and Counter-Rotating-Wave (CRW)
interactions asymmetrically. Additionally, it incorporates a nonlinear
coupling term between the atomic inversion and the cavity photon number. The
Hamiltonian of the AQRSM[76] is given by ($\hbar=1$)

$\begin{split}\hat{H}&=\omega\hat{a}^{\dagger}\hat{a}+\frac{\Delta}{2}\hat{\sigma}_{z}+U\hat{a}^{\dagger}\hat{a}\hat{\sigma}_{z}\\\
&+\lambda_{1}(\hat{a}\hat{\sigma}^{+}+\hat{a}^{\dagger}\hat{\sigma}^{-})+\lambda_{2}(\hat{a}^{\dagger}\hat{\sigma}^{+}+\hat{a}\hat{\sigma}^{-}).\end{split}$
(1)

In the above, $\omega$ is the frequency of the bosonic field described by
annihilation (creation) operator $\hat{a}$ ($\hat{a}^{\dagger}$), $\Delta$ is
the transition frequency of qubit described by Pauli matrices
$\hat{\sigma}_{x,y,z},\hat{\sigma}^{\pm}=(\hat{\sigma}_{x}\pm
i\hat{\sigma}_{y})/2$ . $U$ denotes the nonlinear Stark coupling strength,
while $\lambda_{1}$ and $\lambda_{2}$ account for the qubit-boson coupling
strengths for the RW and CRW interactions, respectively. One key aspect of the
AQRSM in Eq.(1) is that it preserves the parity symmetry $\mathbb{Z}_{2}$.
This fact can be evidenced by considering the total number of excitations
operator $\hat{N}=\hat{a}^{\dagger}\hat{a}+\hat{\sigma}^{+}\hat{\sigma}^{-}$
and the parity operator $\hat{\pi}=\mathrm{exp}\left(i\pi\hat{N}\right)$. The
conserved parity $\mathbb{Z}_{2}$ possesses two eigenvalues,
$\left\langle\hat{\pi}\right\rangle=\pm 1$, depending on whether the total
number of excitations are even or odd.

The AQRSM is notable for its manifestation of both first-order and continuous
quantum phase transitions (QPT)[76]. First-order QPTs are marked by abrupt
changes in ground-state properties during the transition, especially in the
infinite-volume (thermodynamic) limit. In contrast, continuous QPTs exhibit
smooth changes in ground-state features, including the emergence of divergent
length scales in quantum correlation functions and a gapless energy spectrum
in the infinite-volume limit.

The first-order QPT arises in the AQRSM at the energy crossing between the
ground state and the first excited state, occurring when $\omega=1$ at

$\lambda_{1c}=\sqrt{\frac{\Delta(1-U^{2})}{U(1+r^{2})+1-r^{2}}},$ (2)

where $r=\lambda_{2}/\lambda_{1}$ represents the degree of anisotropy. This
transition signifies a substantial shift in the system’s behavior and is
crucial for understanding its quantum properties. Continuous QPT, on the other
hand, occurs in the QRM at an infinite frequency ratio of $\Delta/\omega$[56].
At a critical coupling, the system undergoes a second-order quantum phase
transition into a superradiant phase, which has been observed in trapped-ion
settings[77], a phenomenon that naturally manifest itself in the AQRSM as
well[58].

Interestingly, AQRSM exhibits another form of continuous QPT, in a less
restrictive physical realization, at finite frequency ratios $\Delta/\omega$
for the special Stark coupling $|U|=1$. When $U=\pm 1$, AQRSM can be mapped
into an effective quantum oscillator. The low energy spectrum separates into
two branches: the upper one with
$E>\mp\frac{\Delta}{2}-2\kappa^{2}\alpha^{2}$, and the lower one with
$E<E_{c}^{\pm}=\mp\frac{\Delta}{2}-2\alpha^{2}$, where
$\alpha=(\lambda_{1}+\lambda_{2})/2$ and $\kappa=(1-r)/(1+r)\leq 1$. The real
lower spectra only exist before the critical coupling
$\alpha_{c}^{\pm}=\sqrt{\frac{1\mp\Delta\pm\kappa}{2}}$.

At the critical points $\alpha_{c}^{\pm}$, the phenomenon known as spectral
collapse occurs, where the discrete low energy levels converge to a shared
energy value $E_{c}^{\pm}$, signaling the closure of the energy gap within the
system at the critical coupling $\alpha_{c}^{\pm}$. It is therefore suggested
that gapless Goldstone mode excitations appear for $\alpha_{c}^{\pm}$.
Furthermore, beyond the critical point, the system experiences the breaking of
parity symmetry due to the infinite degeneracy of all states, characteristic
of a second-order QPT. We note that the presence of QPT in the AQRSM model
belongs to a distinct universality class compared to models that exhibit phase
transitions in the infinite frequency limit[78].

The spectral characteristics of the working medium, such as level crossings,
degeneracies, and quasi-degenerate points, play a crucial role in determining
the performance of a quantum engine. These features can either enhance or
suppress specific thermodynamic processes, thereby influencing the engine’s
efficiency and overall performance. In Fig.1, we present the energy spectrum
of the AQRSM in relation to the ground state energy, considering various
parameters such as the light-matter coupling strengths $\lambda_{1}$ and
$\lambda_{2}$, and the nonlinear Stark interaction $U$. Notably, the spectrum
reveals level crossings characterized by changes in ground state parity,
leading to infinite first-order quantum phase transitions as the coupling
strength increases. Moreover, the presence of non-degenerate and near-
degenerate points further impacts the operation and thermodynamic behavior of
the engine. Understanding the spectrum of the working medium is vital not only
for uncovering the underlying physics behind the emergence of thermal
correlations but also for determining the characteristics of non-equilibrium
states accessible during the finite-time cycle of the quantum engine.

Figure 1: Energy spectrum diagram for the anisotropic quantum Rabi-Stark
model. The plots show differences in energy levels as functions of the light-
matter coupling strengths $\lambda_{1}$ (RW) and $\lambda_{2}$ (CRW) for
various anisotropy ratios $\lambda_{2}/\lambda_{1}$ and
$\lambda_{1}/\lambda_{2}$. Energy levels are computed over a range of Stark
interaction strengths $U$ from -0.9 to 0.9, in increments of 0.3. The vertical
dash-dot black lines indicate the position of the first-order phase transition
according to Eq.(2), providing insight into the critical behavior of the
system. Additional system parameters are set as $\Delta=1$ and $\omega=1$

###  II.2 Quantum Otto Cycle

A Quantum Otto engine operates by cyclically manipulating the work substance
parameters to perform work, consisting of four main stages: compression,
isochoric heating, expansion, and isochoric cooling[8]. The thermodynamic
cycle we consider is shown in Fig.2 and proceeds as follows:

1\. Quantum Isochoric Heating Process:

The working substance, characterized by the Hamiltonian $H^{h}$ and frequency
$\omega_{h}=\omega=\Delta$, comes into contact with a hot reservoir at
temperature $T_{h}$. It’s important to note that the AQRSM spectrum (see
Fig.1) exhibits both non-degenerate and degenerate eigenvalues. These are
adequately modeled by the secular approximation when deriving the Markovian
master equation, except in cases of quasi-degeneracy. Following the approach
outlined in Ref.[79], we can derive a thermodynamically valid master equation
for the entire spectrum of the Hamiltonian, Eq.(1), up to the spectrum
collapse at $|U|=1$. This derivation holds true under conditions of weak
damping, a high bath cut-off frequency, and a flat spectral density, ensuring
positivity and Markovianity. Under these conditions, the system evolves
Markovianly according to the quantum master equation:

$\displaystyle\frac{d}{dt}\hat{\rho}$ $\displaystyle=$
$\displaystyle-i[\hat{H},\hat{\rho}]+\sum_{\begin{subarray}{c}u=a,\sigma^{-}\\\
k<j\end{subarray}}\left\\{\Gamma^{jk}_{u}n_{u}(\Delta_{jk})\mathcal{D}[|\phi_{j}\rangle\langle\phi_{k}|,\hat{\rho}]\right.$
(3)
$\displaystyle\left.+\Gamma^{jk}_{u}[1+n_{u}(\Delta_{jk})]\mathcal{D}[|\phi_{k}\rangle\langle\phi_{j}|,\hat{\rho}]\right\\},$

where
$\mathcal{D}[\hat{O},\hat{\rho}]=\frac{1}{2}\left[2\hat{O}\hat{\rho}\hat{O}^{{\dagger}}-\hat{\rho}\hat{O}^{{\dagger}}\hat{O}-\hat{O}^{{\dagger}}\hat{O}\hat{\rho}\right]$.

The eigenvalues $E_{k}$ and eigenvectors $|\phi_{k}\rangle$ of the AQRSM
Hamiltonian $\hat{H}|\phi_{k}\rangle=E_{k}|\phi_{k}\rangle$ determine the
system’s dynamics. Dissipation rates
$\Gamma^{jk}_{u}=\gamma_{u}(\Delta_{jk})|S^{jk}_{u}|^{2}$ depend on the
spectral function $\gamma_{u}(\Delta_{jk})$ And the transition coefficients:
${S}^{jk}_{a}={\langle}\phi_{j}|(\hat{a}^{{\dagger}}+\hat{a})|\phi_{k}{\rangle}$,
${S}^{jk}_{\sigma^{-}}={\langle}\phi_{j}|(\hat{\sigma}_{+}+\hat{\sigma}_{-})|\phi_{k}{\rangle}$.
For the Ohmic case,
$\gamma_{u}(\Delta_{jk})=\pi\alpha\Delta_{jk}\exp(-|\Delta_{jk}|/\omega_{c})$,
where $\alpha$ is the coupling strength and $\omega_{c}$ is the cutoff
frequency, throughout all numerical simulations performed, we consider
$\alpha=0.001\omega$ and $\omega_{c}=10\omega$. The Bose-Einstein distribution
$n_{u}(\Delta_{jk},T_{u})=1/[\exp(\Delta_{jk}/T_{u})-1]$ accounts for thermal
effects. The steady-state solution of Eq.(3) yields the density matrix of the
canonical ensemble, as confirmed by straightforward numerical simulations,
given by:

$\displaystyle\hat{\rho}_{\mathrm{ss}}=\sum_{n}\frac{e^{-E_{n}/T}}{\mathcal{Z}}|\phi_{n}{\rangle}{\langle}\phi_{n}|,$
(4)

where $\mathcal{Z}=\sum_{n}e^{-E_{n}/T}$ is the partition function, and the
steady-state population is given by
$P_{n}^{ss}=\frac{e^{-E_{n}/T}}{\mathcal{Z}}$. At steady state, the population
dynamics reach
$\rho_{1}=\rho_{ss}(T_{h})=\sum_{n}P_{n}^{ss}(T_{h})|E_{n}^{h}\rangle\langle
E_{n}^{h}|$, absorbing heat $Q_{h}$ without performing work.

2\. Quantum Adiabatic Expansion Process: The system, isolated from the hot
reservoir, changes energy levels from $E_{n}^{h}$ to $E_{n}^{c}$ by varying
the frequency from $\omega_{h}$ to $\omega_{c}$ ($\omega_{h}>\omega_{c}$). The
populations $P_{n}^{ss}(T_{h})$ remain unchanged. The state becomes
$\rho_{2}=\sum_{n}P_{n}^{ss}(T_{h})|E_{n}^{c}\rangle\langle E_{n}^{c}|$. Only
work is performed.

3\. Quantum Isochoric Cooling Process: The substance, with frequency
$\omega_{c}$, contacts a cold reservoir at temperature $T_{c}<T_{h}$ until
equilibrium is reached. The steady-state population changes to
$P_{n}^{ss}(T_{c})$, and the state becomes
$\rho_{3}=\sum_{n}P_{n}^{ss}(T_{c})|E_{n}^{c}\rangle\langle E_{n}^{c}|$. Heat
$Q_{c}$ is released, with no work done.

4\. Quantum Adiabatic Compression Process: Isolated from the cold reservoir,
the system’s energy levels revert from $E_{n}^{c}$ to $E_{n}^{h}$ by changing
the frequency from $\omega_{c}$ to $\omega_{h}$. The state becomes
$\rho_{4}=\sum_{n}P_{n}^{ss}(T_{c})|E_{n}^{h}\rangle\langle E_{n}^{h}|$. Only
work is performed.

Figure 2: Schematic representation of a Quantum Otto engine fueled by the
anisotropic quantum Rabi-Stark model: (1) The working substance, described by
Hamiltonian $H^{h}$ and frequency $\omega_{h}$, interacts with a hot reservoir
at temperature $T_{h}$. (2) Isolated from the hot reservoir, energy levels
change from $E_{n}^{h}$ to $E_{n}^{c}$ by adjusting the frequency from
$\omega_{h}$ to $\omega_{c}$ ($\omega_{h}>\omega_{c}$). (3) With frequency
$\omega_{c}$, the substance contacts a cold reservoir at temperature
$T_{c}<T_{h}$. (4) Isolated from the cold reservoir, the system’s energy
levels revert from $E_{n}^{c}$ to $E_{n}^{h}$ by changing the frequency from
$\omega_{c}$ to $\omega_{h}$, with work being performed.

###  II.3 Work and Heat Calculations

Using the first law of thermodynamics for a quantum system with discrete
energy levels:

$dU=\delta Q+\delta W=\sum_{n}(E_{n}dP_{n}^{ss}+P_{n}^{ss}dE_{n}),$ (5)

where $E_{n}$ are the energy levels and $P_{n}^{ss}$ are the steady-state
occupation probabilities. The heat $Q_{h}$ and $Q_{c}$ exchanged with the hot
and cold reservoirs, respectively, and the net work $W$ are given by:

$\displaystyle Q_{h}$ $\displaystyle=$
$\displaystyle\sum_{n}E_{n}^{h}[P_{n}^{ss}(T_{h})-P_{n}^{ss}(T_{c})],$ (6)

$\displaystyle Q_{c}$ $\displaystyle=$
$\displaystyle\sum_{n}E_{n}^{c}[P_{n}^{ss}(T_{c})-P_{n}^{ss}(T_{h})],$ (7)
$\displaystyle W$ $\displaystyle=$ $\displaystyle
Q_{h}+Q_{c}=\sum_{n}(E_{n}^{h}-E_{n}^{c})[P_{n}^{ss}(T_{h})-P_{n}^{ss}(T_{c})].$
(8)

Positive values of $Q$ indicate heat absorption from a reservoir, while
negative values indicate heat release. Similarly, positive values of $W$
indicate work done by the engine, and negative values indicate work done on
the engine. The allowed working regimes under the Clausius inequality and the
first law of thermodynamics are:

  1. 1.

Heat engine (E): $Q_{h}>0$, $Q_{c}<0$, $W>0$;

  2. 2.

Refrigerator (R): $Q_{c}>0$, $Q_{h}<0$, $W<0$;

  3. 3.

Heater (H): $Q_{c}<0$, $Q_{h}<0$, $W<0$;

  4. 4.

Accelerator (A): $Q_{c}<0$, $Q_{h}>0$, $W<0$.

Among these four operation modes, the heat engine (E) and the refrigerator (R)
are of the most practical interest. The heat engine operates by absorbing heat
from a hot reservoir, converting part of this energy into work, and releasing
the remaining heat to a cold reservoir. The efficiency of the heat engine,
$\eta$, is a key figure of merit and is defined as the ratio of the work
output to the heat absorbed from the hot reservoir $\eta=\frac{W}{Q_{h}}$.

In contrast, a refrigerator absorbs heat from a cold reservoir and releases it
to a hot reservoir by consuming work. The performance of a refrigerator is
measured by its coefficient of performance (COP), $\xi$, which is the ratio of
the heat removed from the cold reservoir to the work input
$\xi=\frac{Q_{c}}{|W|}.$

###  II.4 Results Relevant to Our Analysis.

We review key findings from recent studies that provide a foundation for our
discussion. These results play a crucial role in comprehending the mechanisms
and behaviors observed in our work and help contextualize our findings within
the broader field of quantum thermodynamics. To aid our analysis, it is useful
to recall the analytical results for the quantum Otto cycle (QOC) when the
working medium is a special multi-level system with all energy gaps changing
by the same ratio in the quantum adiabatic process, i.e.,
$E_{n}^{h}-E_{m}^{h}=\alpha(E_{n}^{l}-E_{m}^{l})$, for $n=0,1,2,\ldots$ and
$\alpha>1$ since $E_{n}^{h}>E_{n}^{l}$. Substituting into the equations
Eqs.(6) and (8), we obtain the efficiency
$\eta_{\alpha}=1-\frac{E_{n}^{l}-E_{m}^{l}}{E_{n}^{h}-E_{m}^{h}}=1-\frac{1}{\alpha}.$
In these instances, the quantum concept of adiabatic evolution aligns with the
thermodynamic definition. This result, which stands for a special multi-level
working fluid, generalizes the qubit engine and the quantum harmonic
oscillator (QHO). For such systems, an adiabatic stroke maps an initial
thermal state to another at a different temperature[8]. In the limit of zero
coupling for the AQRSM model, $\alpha=\omega_{h}/\omega_{c}$, the efficiency
is given by $\eta_{\lambda=U=0}=1-\omega_{c}/\omega_{h}$, and the coefficient
of performance (COP) is
$\xi_{\lambda=U=0}=\omega_{c}/(\omega_{h}-\omega_{c})$, corresponding to a
qubit and a non-interacting bosonic mode. Another important result is the
positive-work condition (PWC) derived from the special multi-level harmonic
spectrum given by $T_{h}>\frac{\omega_{h}}{\omega_{c}}T_{c}.$ Applying Eq.(8)
to the qubit engine yields
$W_{\mathrm{qubit}}=(\omega_{h}-\omega_{c})\left[\frac{1}{1+\exp(\beta_{h}\omega_{h})}-\frac{1}{1+\exp(\beta_{c}\omega_{c})}\right]$,
where $\beta_{h}=\frac{1}{k_{B}T_{h}}$ and $\beta_{c}=\frac{1}{k_{B}T_{c}}$
represent the inverse temperatures of the hot and cold reservoirs,
respectively, with $k_{B}$ being the Boltzmann constant. For a bosonic mode,
the total work done during the QOC is
$W_{\mathrm{QHO}}=(\omega_{h}-\omega_{c})\left[\frac{1}{\exp(\beta_{h}\omega_{h})-1}-\frac{1}{\exp(\beta_{c}\omega_{c})-1}\right]$.

These expressions illustrate how the specific quantum nature of the working
substance affects thermodynamic figures of merit, such as the engine’s
efficiency, by considering the respective energy gaps and temperature
dependencies of the systems. However, in general, a quantum adiabatic stroke
will drive a medium out of equilibrium, as occurs for AQRSM, although it will
still remain in a passive energy-diagonal state.

The spectrum of the AQRSM displays a significant degree of anharmonicity and
criticality, which are essential for understanding the factors influencing the
efficiency and power of quantum Otto engines[80]. This complexity makes it
impractical to derive closed-form expressions for the thermodynamic quantities
of the cycle, necessitating a numerical approach. Furthermore, examining the
expressions for the decoupled system becomes crucial to assess the potential
benefits arising from the interaction between radiation and matter in
enhancing the thermal engine’s performance.

Criticality is an active area of exploration in both classical and quantum
condensed matter physics, driven by the divergence of various parameters near
critical points, which leads to universal behaviors in observed phenomena. In
a first-order phase transition, molar entropy and volume display discontinuity
with temperature, while heat capacity diverges due to latent heat. Conversely,
second-order phase transitions are characterized by a divergent susceptibility
[81]. It has been theorized that near a classical phase transition, the
divergence of equilibrium fluctuations in a second-order QPT, as indicated by
the fluctuation-dissipation theorem, may enhance heat engine performance[36].
At zero temperature, a system can undergo a quantum phase transition, leading
to changes in non-analyticities in its ground state. Evidence supporting this
quantum critical enhancement in engine operation has been found in several
systems undergoing continuous phase transitions[37, 38, 39, 40, 41, 42, 43,
44, 45]. Additionally, such singular behaviors may be reflected in stochastic
or quantum thermodynamic quantities, as the singular behavior of entropy
production indicates a second-order transition[82, 83, 84]. However, the
implications of first-order QPT in heat engines are not fully understood.

First-order quantum phase transitions are intriguing due to potential
differences in the characteristic relaxation frequency spectrum of
fluctuations and alterations in the nature of quantum coherence near the
transition point[85]. These aspects imply that the realm of first-order
quantum phase transitions offers ample opportunities for both theoretical
exploration and experimental investigation in the domain of quantum
thermodynamics. It’s noteworthy to mention that in a quenched system
experiencing a first-order QPT, the average work exhibits discontinuity[82].

Finally, we should mention that the excited-state quantum phase transition
(ESQPT) was thoroughly studied for the QRM [59] and AQRM [60] in the limit of
infinite frequency ratios ($\Delta/\omega\to\infty$). Excited-state quantum
phase transitions (ESQPTs) refer to non-analytic changes in the properties of
excited states of a quantum system as a control parameter is varied. Unlike
ground-state quantum phase transitions, which occur at zero temperature,
ESQPTs manifest at finite energy densities and are characterized by
singularities in the density of states. These transitions have significant
implications for thermodynamic quantities such as entropy, specific heat, and
work. For instance, near an ESQPT, the system’s specific heat may exhibit
anomalous peaks, and the response functions can show critical scaling
behavior[86, 87]. This criticality can influence the efficiency and
performance of quantum heat engines, as the presence of ESQPTs may enhance the
engine’s work output and alter the cycle’s thermodynamic properties.
Understanding ESQPTs thus provides deeper insights into the non-equilibrium
dynamics and the thermodynamic behavior of quantum systems, potentially
leading to optimized designs for quantum thermal engines[88]. However, to the
best of our knowledge, there have been no studies on ESQPTs in the AQRSM at
finite frequency ratios. Investigating ESQPTs in the AQRSM could provide
valuable insights into the behavior of quantum engines operating under these
conditions, potentially revealing new avenues for enhancing engine performance
and efficiency in practical applications.

###  II.5 Operation in Finite Time

In our investigations thus far, we have primarily focused on analyzing the
engine’s performance in the adiabatic limit, where the power output tends
towards zero due to the extended duration of the strokes. Although reducing
the duration of the strokes can increase power, it inevitably introduces
inefficiencies due to the generation of irreversibility and internal
friction[89, 90]. The emergence of quantum friction presents a significant
challenge in finite-time quantum processes. Quantum friction arises from non-
adiabatic excitations during rapid driving of the working substance, leading
to energy dissipation and reduced efficiency. Ongoing efforts are directed
towards comprehending and mitigating quantum friction, with the aim of
enhancing the performance of quantum heat engines operating within finite time
frames[91, 37].

To comprehend the trade-off between power and efficiency in the engine, our
attention is directed towards the heat engine E, which begins with the system
in a thermal state at temperature $T_{h}$. Let us consider a dynamic system
with a generic time-dependent Hamiltonian $H(t)$. The average energy, $U(t)$,
for state $\rho(t)$ is $U(t)=\langle H(t)\rangle=\text{Tr}\\{\rho(t)H(t)\\}$,
with the rate of change of system energy
$\frac{dU}{dt}=\frac{dW}{dt}+\frac{dQ}{dt}$. Here, $Q$ and $W$ represent heat
and work, respectively, and are given by:

$Q(t)=\int_{0}^{t}dt^{\prime}\text{Tr}\\{H(t^{\prime})\dot{\rho}(t^{\prime})\\},$
(9)
$W(t)=\int_{0}^{t}dt^{\prime}\text{Tr}\\{\dot{H}(t^{\prime})\rho(t^{\prime})\\}.$
(10)

For the finite time quantum Otto engine cycle process, we set the working
substance begins in the hot Gibbs state $\rho_{\tau_{0}}$. During an isochoric
process, see Fig.2 strokes $\mathbf{1}$ and $\mathbf{3}$, where $H(t)=H$ and
$W(t)=0$, both thermalization times are set equal, denoted as
$\tau_{2}=\tau_{4}$. Adiabatic processes, see Fig.2 strokes $\mathbf{2}$ and
$\mathbf{4}$, where $Q(t)=0$, are implemented by linearly varying the working
substance frequency as $\omega(t)=\omega_{j}-\tau_{j}t$, with $t$ ranging from
$0$ to $\frac{\omega_{f}-\omega_{i}}{\tau_{n}}$, where $j=h,c$ for $n=1,3$,
and $\tau_{1}=\tau_{3}$. For the ideal quantum Otto engine cycle,
$\tau_{1}=\tau_{3}\rightarrow\infty$, $\tau_{2}=\tau_{4}\rightarrow\infty$,
and $\rho_{\tau_{4}}=\rho_{\tau_{0}}$. Throughout the four stages of the
quantum Otto cycle, energy exchange occurs solely through work or heat. The
total extracted work $W(t)$ is the sum of $Q_{1}(t)$ and $Q_{3}(t)$, yielding
the efficiency $\eta(t)=\frac{W(t)}{Q(t)}$. The power $P(t)$ is the derivative
of work with respect to time, calculated as
$\frac{dW}{dt}=\text{Tr}\\{\dot{H}(t)\rho(t)\\}$.

We investigate the quantum heat engine (QHE) through successive iterations of
the cycle. The isochoric stages entail Markovian evolution, as illustrated in
Fig.3, we observe that the system converges to a limit cycle irrespective of
the initial state. We begin each cycle with the hot Gibbs state
$\rho_{\tau_{0}}$ as our initial condition. Upon completing the cycle, the
system settles into the final state $\rho_{\tau_{4}}$. This final state then
serves as the starting point $\rho_{\tau_{0}}$ for the subsequent cycle,
continuing this iterative process. As a result, multiple iterations of the
cycle are performed. In Fig. 3, we illustrate the fidelity
$F_{h}=F(\rho^{(N-1)}(\tau_{0}),\rho^{(N)}(\tau_{0}))$, which quantifies the
similarity between the state $\rho(\tau_{0})$ across consecutive iterations.
The fidelity $F$ equals 1 when the states are identical. Notably, for each
cycle, when $\tau_{4}>800$, the fidelity $F_{h}\rightarrow 1$. After numerous
iterations, the system’s state exhibits minimal variation during the cycle,
with $F_{h}\rightarrow 1$. This observation indicates that during very short
cycle times, the system’s state remains nearly unchanged (see Fig. 3).

For short time of $\tau_{4}$, the evolution through one full cycle changes the
state of the system, leading to a decrease in the fidelity between consecutive
cycles. This decrease indicates that the system’s state is evolving and not
remaining in a steady state during these short but non-negligible cycle times.

As we consider increasing values of $\tau_{4}$, the interaction with the hot
reservoir drives $\rho(\tau_{4})$ closer to the stationary thermal state. This
thermalization process increases the fidelity because the system spends more
time in contact with the reservoir, allowing it to reach equilibrium. For
times comparable with the relaxation time, $F_{h}\rightarrow 1$, indicating
that the engine operates in the limit cycle from the first iteration. This
behavior is expected, as longer interaction times with the reservoir allow the
system to reach and maintain thermal equilibrium more efficiently.

Interestingly, the limit cycle can also be achieved with a nonequilibrium
steady state. In such cases, the system reaches a steady-state that is not in
thermal equilibrium but remains constant over time under continuous operation.
This steady state can be maintained by balancing the energy exchanges with the
reservoirs, ensuring the system’s stability and consistent performance.

Throughout this work, we characterize the thermodynamic figures of merit in
the limit cycle as defined above. The limit cycle behavior is crucial for
understanding the steady-state performance of the quantum Otto engine,
ensuring consistent and reliable operation.

To assess the detrimental effects of finite-time operation, we consider the
total entropy produced in a cycle, denoted as
$\left\langle\Sigma_{\text{total}}\right\rangle$. This measure encompasses
both the finite-time effects of the engine’s dynamics and the thermalization
process, directly reflecting the irreversibility within the cycle. The total
entropy production can be expressed as

$\begin{split}\langle\Sigma_{\text{total}}\rangle&=-\beta_{h}[\text{Tr}(\rho_{\tau_{4}}H_{h})-\text{Tr}(\rho_{\tau_{3}}H_{h})]\\\
&-\beta_{c}[\text{Tr}(\rho_{\tau_{2}}H_{c})-\text{Tr}(\rho_{\tau_{1}}H_{c})].\end{split}$
(11)

As a complementary measure of the detrimental effects of finite-time
operations, we quantify the work due to internal friction using the Kullback-
Leibler divergence

$W_{\text{fric}}=\frac{1}{\beta_{h(c)}}D(\rho_{1(3)}||\rho_{1(3)}^{\text{qe}}),$
(12)

where
$D(\rho||\rho^{\text{ref}})=\text{Tr}[\rho(\ln\rho-\ln\rho^{\text{ref}})]$.
Here, $\rho_{1(3)}^{\text{qe}}$ denotes the quasistatically evolved state
corresponding to $\rho_{1(3)}$. The total friction work sums contributions
from compression and expansion strokes
$W^{\text{fric}}=W_{\text{comp}}^{\text{fric}}+W_{\text{exp}}^{\text{fric}}$.

An important result that warrants attention is that the efficiency of the
quantum engine can be related to the total entropy produced in a cycle as [92]

$\eta=\eta_{\text{Carnot}}-\frac{\left\langle\Sigma_{\text{total}}\right\rangle}{\beta_{\text{c}}\left\langle
Q_{\text{h}}\right\rangle},$ (13)

where $\eta_{\text{Carnot}}=1-\beta_{\text{h}}/\beta_{\text{c}}$ is the Carnot
efficiency. The quantity
$\mathcal{L}_{\text{therm}}=\left\langle\Sigma_{\text{total}}\right\rangle/\beta_{\text{c}}\left\langle
Q_{\text{h}}\right\rangle$ quantifies the departure of the engine’s efficiency
from the Carnot efficiency. Since
$\left\langle\Sigma_{\text{total}}\right\rangle$ is nonnegative, the quantum
engine’s efficiency is always upper-bounded by the Carnot efficiency.

Furthermore, total entropy production
thisthis$\langle\Sigma_{\text{total}}\rangle$ plays a crucial role in
quantifying the relative fluctuations of integrated currents, such as heat or
energy, through thermodynamic uncertainty relations[93, 94, 95]. These
relations establish a trade-off between optimizing relative fluctuations and
entropy production, providing constraints that are universally applicable
across different scenarios. Consequently, a continuous heat engine operating
within a non-equilibrium steady state must adhere to a trade-off relationship
involving its efficiency, output power, and power fluctuations. This
relationship can be expressed through the integrated current
$\mathcal{Q}_{i}$, where $\mathcal{Q}_{i}$ represents quantities like work
($W$), efficiency ($\eta$), or power ($P$)[95]:

$\frac{\mathrm{Var}(\mathcal{Q}_{i})}{\langle\mathcal{Q}_{i}\rangle^{2}}\geq
f(\langle\Sigma_{total}\rangle),$ (14)

where $f(x)=\text{csch}^{2}(g(x/2))$. Here, $\text{csch}(x)$ represents the
hyperbolic cosecant function, and $g(x)$ is the inverse function of
$x\tanh(x)$.

Figure 3: Fidelity betweem $\rho^{(N)}(\tau_{0})$ and $\rho^{(N-1)}(\tau_{0})$
as a function of the cycle time $\tau_{4}$ where $N=2$ (dash dot blue), $N=3$
(red dotted line), $N=4$ (green dashed line). The other system parameters are
given by $T_{h}=0.5$, $T_{c}=0.1$, $\lambda_{1}=\lambda_{2}=0.5$, $U=0$. All
quantities above are in units of $\omega$.  Figure 4:  Phase diagram
illustrating the operating regimes of the quantum Otto machine. The diagram
shows how the hot reservoir temperature ($T_{h}$) and the qubit-boson coupling
strengths ($\lambda_{1},\lambda_{2}$) influence the machine’s behavior, with a
fixed ratio of $\lambda_{2}/\lambda_{1}$ (or $\lambda_{1}/\lambda_{2}$). The
cold reservoir temperatures are set at $T_{c}=0.1$ (panels 1-5), $T_{c}=0.5$
(panels 6-10), and $T_{c}=2$ (panels 11-15). The color coding represents
different operating modes: heat engine (deep blue), refrigerator (green),
heater (yellow), and accelerator (red). All quantities are in units of
$\omega$, with parameters $U=0$, $\omega_{h}=2\omega$, and
$\omega_{c}=\omega$. Figure 5: Operating regimes of the quantum Otto machine
with varying nonlinear Stark coupling strength $U$ and qubit-boson coupling
strengths $\lambda_{1}(\lambda_{2})$, both in units of $\omega$. Fixed
coupling ratios are: (a),(e) $\lambda_{2}/\lambda_{1}=0$, (b),(f)
$\lambda_{2}/\lambda_{1}=0.5$, (c),(g) $\lambda_{1}/\lambda_{2}=0.5$, and
(d),(h) $\lambda_{1}/\lambda_{2}=1$. Color codes indicate different
operational modes: heat engine (deep blue), refrigerator (green), heater
(red), and accelerator (yellow). Parameters: $T_{h}=0.5$, $T_{c}=0.1$ (first
row); $T_{h}=2$, $T_{c}=0.5$ (second row); $\omega_{h}=2\omega$,
$\omega_{c}=\omega$.

Figure 6: 2D heatmaps depicting the Work (W), the normalized work
$\overline{W}$, and Efficiency ($\eta$) of the Quantum Otto Cycle powered by
the anisotropic quantum Rabi model. Panels (a) and (d) show the net Work and
Panels (b) and (e) show the normalized work, while panels (c) and (f) display
the Efficiency as functions of the coupling strengths $\lambda_{1}$ (RW) and
$\lambda_{2}$ (CRW). The left column presents results for a cold reservoir
temperature $T_{c}=0.1$ and a hot reservoir temperature $T_{h}=0.5$. The right
column shows results for $T_{c}=0.5$ and $T_{h}=2$. The color bars indicate
the magnitude of the Work and Efficiency. Cycle parameters include
$\omega_{h}=2\omega$ and $\omega_{c}=\omega$, with all quantities measured in
units of $\omega$. Gray areas represent regimes where the system does not
operate as a heat engine.

Figure 7: 2D heatmaps depicting the Work (W), the normalized work
$\overline{W}$, and Efficiency ($\eta$) of the Quantum Otto cycle powered by
the quantum Rabi model. The heatmaps show the dependence of Work (a, d), the
normalized work (b, e) and Efficiency (c, f) on the coupling strength
$\lambda_{1}=\lambda_{2}=\lambda$. The left column presents the results for a
cold reservoir with fixed temperature $T_{c}=0.1$. The right column shows the
results for a hot reservoir with fixed temperature $T_{h}=4$. Color bars
indicate the magnitude of the work, normalized work, and Efficiency. The cycle
parameters include $\omega_{h}=2\omega$ and $\omega_{c}=\omega$, with all
quantities measured in units of $\omega$. Gray areas represent regimes where
the system does not operate as a heat engine.

Figure 8: 2D heatmaps depicting the Work W (a,d), the normalized work
$\overline{W}$ (b,e), and Efficiency ($\eta$) (c,f) as functions of the Stark
interaction $U$ and the coupling $\mathbf{\lambda}$ for a Quantum Otto Cycle
powered by the quantum Rabi-Stark model. The left column presents the results
for cold and hot temperatures $T_{c}=0.1$ and $T_{h}=0.5$, while the right
column displays the results for $T_{c}=0.5$ and $T_{h}=2$. Color bars indicate
the magnitude of the work, normalized work, and Efficiency. For reference, the
qubit and quantum harmonic oscillator engine achieve a constant efficiency of
$\eta=0.5$, with work $W_{\mathrm{qubit}}\approx 0.018(0.15)$ and
$W_{\mathrm{QHO}}\approx 0.019(0.43)$, respectively, for the parameters in the
first (second) column. Other cycle parameters are $\omega_{h}=2\omega$ and
$\omega_{c}=\omega$, with all quantities in units of $\omega$. Gray areas
represent regimes where the system does not operate as a heat engine. Figure
9: (a)-(e) Work W extracted from the quantum heat engine as a function of the
qubit-boson coupling strength ratio $\lambda_{1}/\lambda_{2}$ for different
nonlinear Stark coupling strength $U=0$ (solid blue line), $U=0.1$ (dashed red
line), $U=0.3$ (dotted magenta line), $U=0.6$ (dash dot green line), and
$U=0.9$ (solid black line) with fixed coupling ratio
(a)$\lambda_{2}/\lambda_{1}=0$,(b)$\lambda_{1}/\lambda_{2}=0$,(c)$\lambda_{2}/\lambda_{1}=0.5$,
(d)$\lambda_{1}/\lambda_{2}=0.5$ and (e)$\lambda_{2}/\lambda_{1}=1$. (f)-(j)
Work extracted from the quantum heat engine as a function of the qubit-boson
coupling strength $\lambda_{1}$($\lambda_{2}$) in units of $\omega$ for
different nonlinear Stark coupling strength $U=-0.1$ (dashed red line),
$U=-0.3$ (dotted magenta line), $U=-0.6$ (dash dot green line), $U=-0.9$
(solid black line) with fixed coupling ratio (f)
$\lambda_{2}/\lambda_{1}=0$,(g) $\lambda_{1}/\lambda_{2}=0$,(h)
$\lambda_{2}/\lambda_{1}=0.5$, (i) $\lambda_{1}/\lambda_{2}=0.5$ and (j
)$\lambda_{2}/\lambda_{1}=1$. 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 10: Efficiency $\eta$ of
the quantum heat engine as a function of the qubit-boson coupling strength
ratio $\lambda_{1}/\lambda_{2}$ for nonlinear Stark coupling strengths:
(a)-(e) $U=0$ (solid blue line), $U=0.1$ (dashed red line), $U=0.3$ (dotted
magenta line), $U=0.6$ (dash dot green line), and $U=0.9$ (solid black line)
fixing (a) $\lambda_{2}/\lambda_{1}=0$,(b) $\lambda_{1}/\lambda_{2}=0$,(c)
$\lambda_{2}/\lambda_{1}=0.5$, (d) $\lambda_{1}/\lambda_{2}=0.5$ and (e)
$\lambda_{2}/\lambda_{1}=1$; (f)-(j) $U=-0.1$ (dashed red line), $U=-0.3$
(dotted magenta line), $U=-0.6$ (dash dot green line), $U=-0.9$ (solid black
line) with fixed coupling ratio (f) $\lambda_{2}/\lambda_{1}=0$, (g)
$\lambda_{1}/\lambda_{2}=0$, (h) $\lambda_{2}/\lambda_{1}=0.5$, (i)
$\lambda_{1}/\lambda_{2}=0.5$ and (j) $\lambda_{2}/\lambda_{1}=1$. 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
11: Work W and Efficiency $\eta$ as a function of the qubit-boson coupling
strength ratio $\lambda_{1}/\lambda_{2}$ for several nonlinear Stark coupling
strengths $U=-0.95$ (dotted red line), $U=-0.98$ (dash dot blue line), and
$U=-0.99$ (solid black line). (a)-(e) Fixing (a) $\lambda_{2}/\lambda_{1}=0$,
(b) $\lambda_{1}/\lambda_{2}=0$, (c) $\lambda_{2}/\lambda_{1}=0.5$, (d)
$\lambda_{1}/\lambda_{2}=0.5$ and (e) $\lambda_{2}/\lambda_{1}=1$. (f)-(j)
Fixing
(f)$\lambda_{2}/\lambda_{1}=0$,(g)$\lambda_{1}/\lambda_{2}=0$,(h)$\lambda_{2}/\lambda_{1}=0.5$,
(i)$\lambda_{1}/\lambda_{2}=0.5$ and (j)$\lambda_{2}/\lambda_{1}=1$. 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
12: (a)-(e) Panels (1) to (12) illustrate the Work (W) and Efficiency ($\eta$)
as functions of the qubit-boson coupling strength $\lambda_{1}$ (or
$\lambda_{2}$) for different nonlinear Stark coupling strengths (U) and
detuning ($\delta$). The top row (panels 1 to 6) shows the Work (W) and the
bottom row (panels 7 to 12) shows the Efficiency ($\eta$). Each panel
represents different values of U and coupling ratios
($\lambda_{2}/\lambda_{1}$ or $\lambda_{1}/\lambda_{2}$), as indicated in the
top of each panel. The curves within each panel represent different detuning
values ($\delta=0,2,4,10,20,50$) as indicated in the legend. The temperatures
are fixed at $T_{c}=0.1$ and $T_{h}=0.5$. Note that efficiency peaks with
increasing detuning $\delta$ and decreases for negative values of $U$. All
quantities are measured in units of $\omega$.

##  III Results and Discussions

###  III.1 Operational Modes for the Ideal Infinite Time Cycle.

The study of the ideal adiabatic regime (zero power) may lack direct practical
relevance, yet it plays a crucial role in establishing the upper limits of
thermodynamic figures of merit and offering a qualitative understanding of the
different permissible operational regimes—Heat engine (E), Refrigerator (R),
Heater (H), and Accelerator (A)—governed by Clausius’s inequality and the
first law of thermodynamics.

We begin by examining the anisotropic model ($U=0$, as illustrated in the
operational phase diagram Fig.4 with $\omega_{h}/\omega_{c}=2$, as a function
of the coupling strength $\lambda_{1}(\lambda_{2})$ and the hot temperature of
the reservoir $T_{h}$, for representative anisotropies values
$r=0,1/2,1,2,\infty$ and cold temperatures of the reservoir $T_{c}=0.1,0.5,2$.
By manipulating the parameter $\lambda$ within two given operating
temperatures of the Otto cycle, we can achieve four distinct machine types.
Notably, at low temperatures, Fig.4(1)-(5), all four operational regions are
present, predominantly occupied by the refrigerator (green area) and the heat
engine (deep blue area).

As anticipated, within the regions of the spectrum depicted in Fig.1 that
maintain harmonicity, all energy gaps undergo uniform changes during the
quantum adiabatic process. Consequently, the positive working condition (PWC)
mirrors the behavior observed in quantum harmonic oscillators or qubits, as
described by $T_{h}>\frac{\omega_{h}}{\omega_{c}}T_{c}$ and confirmed in
Fig.4. As seen from the figure, the operational modes without practical
applicability, such as the heater (yellow) and accelerator (red), emerge in
regions where quasi-degenerate excited states are more pronounced, gradually
diminishing as the temperature of the cold bath is increased.

Figures 5(a)-(h) depict the operational behaviors of the quantum Otto engine
based on the nonlinear Stark coupling strength $U$ and qubit-boson coupling
strength $\lambda_{1}$, with $r$ increasing from left to right across panels.
Near critical points, abrupt transitions occur among heat engine,
refrigerator, heater, and accelerator modes, where even small changes in
parameters can significantly affect the engine’s performance. The sensitivity
to $\lambda_{1}$ and the range $-1\leq U\leq 1$ emphasizes the profound impact
of $U$ on these operational shifts. Lower cold reservoir temperatures
stabilize the heat engine and refrigerator modes, whereas higher temperatures
enable a broader range of operational modes. This highlights the complex
dependence of the engine on coupling parameters and temperature, which is
crucial for optimizing its performance.

The operational phase diagrams, Fig. 4 and Fig. 5, illustrate the occurrence
of unwanted heater and accelerator cycles around quasi-degenerate energy
levels. This phenomenon is significant in both true first-order QPTs, as given
by $\lambda_{1c}=\sqrt{\frac{\Delta(1-U^{2})}{U(1+r^{2})+1-r^{2}}}$, and in
level degeneracies, such as those observed in the QRM. In quantum engines,
heater cycles involve energy absorption, while accelerator cycles involve
rapid energy transfer, both of which are of no practical interest. It is
noteworthy that close to the continuous QPT, the only region where the PWC is
satisfied, leading to the Heat engine cycle, occurs when the Stark interaction
$U$ is negative ($U\approx-1$), $\lambda_{1}<1$, and the system operates in a
low-temperature regime ($T_{c}=0.1$). For all other parameter configurations
close to the second-order QPT, the engine can only function as a refrigerator.

###  III.2 Ideal Infinite Time Cycle: Work and Efficiency.

To conduct a more in-depth quantitative study of the model in question, in
this work we focus on the heat engine regime of the AQRM working medium.
Figure 6 presents computations of the net work $W$, Figs.6(a,d), the
normalized work $\overline{W}=\frac{W}{W_{\mathrm{qubit}}+W_{\mathrm{QHO}}}$,
Figs.6(b,e), and efficiency $\eta$, Figs.6(c,f), as functions of the coupling
strengths $\lambda_{1}$ (RW) and $\lambda_{2}$ (CRW). The left column presents
outcomes for temperatures $T_{c}=0.1$ and $T_{h}=0.5$, while the right column
elucidates results for $T_{c}=0.5$ and $T_{h}=2$.

For reference, the qubit and quantum harmonic oscillator engine maintain a
constant efficiency of $\eta=0.5$, with work $W_{\mathrm{qubit}}\approx 0.018$
($0.15$) and $W_{\mathrm{QHO}}\approx 0.019$ ($0.43$), corresponding to the
parameters in the first (second) column of Fig.6. The primary advantage of
AQRM over the uncoupled system in generating net work is evident in the low-
temperature cycle regime of $T_{c}=0.1$ and $T_{h}=0.5$, as depicted in
Fig.6(a). This advantage is particularly notable within the coupling region
bounded by $1.2\leq\lambda_{1}\leq 1.7$ and $0\leq\lambda_{2}\leq 0.5$. In
this region, the maximum value of $W_{\mathrm{max}}=0.073$ is achieved for
$\lambda_{1}=1.41$ and $\lambda_{2}=0.22$. Remarkably, $W_{\mathrm{max}}$ is
twice the amount of work delivered by a decoupled qubit plus QHO heat engine.
It’s worth noting that in the deep strong coupling regime at low temperature,
Fig.6(b), and for all coupling regimes at higher temperatures, Fig.6(e), it
becomes apparent that the decoupled system can deliver more work during the
engine cycle. As observed in Figs.6(c),(f), for a broad range of couplings,
$\eta$ surpasses the quantum Otto efficiency of a working medium with a
harmonic spectrum, $\eta_{\alpha}=1-\omega_{c}/\omega_{h}=0.5$. Specifically,
$\eta_{\mathrm{max}}=0.75$ ($0.64$) for the low (high) temperature regime,
very close to the Carnot efficiency, $\eta_{\text{Carnot}}=0.8$ ($0.75$).

We now turn our attention to evaluating the unique effect of the temperature
gradient on engine performance. To do so, we focus on the Rabi model ($r=1$)
by computing the figures of merit, $W$, $\overline{W}$, and $\eta$, for fixed
cold (hot) reservoir temperatures $T_{c}=0.1$ ($T_{h}=4$), varying the light-
matter coupling $\lambda$ and the hot (cold) bath $T_{h}$ ($T_{c}$)
temperature as depicted in Fig.7, left (right) columns. In the scenario of a
low-temperature bath at $T_{c}=0.1$, it’s evident from $\overline{W}$ that the
coupled system can surpass the total net work generated by the engine driven
by the qubit and QHO working in parallel by more than $600\%$ in the
temperature gradient region where $T_{h}<0.5$, and the coupling strength falls
within $0.1<\lambda<1$. Additionally, within this parameter regime,
$\eta_{\mathrm{max}}$ is approximately equal to $\eta_{\text{Carnot}}$. As
depicted in Fig.7(e), for nearly all parameters, the net work remains the same
as that of the decoupled system, except in the deep strong coupling regime,
$\lambda>2$, and at higher cold bath temperatures, $T_{c}>1.8$. Here, the PWC
for the decoupled system, $T_{h}>\frac{\omega_{h}}{\omega_{c}}T_{c}$, is
nearly saturated, resulting in almost negligible total work
($W_{\mathrm{qubit}}\approx W_{\mathrm{QHO}}\approx 0$), indicating a looser
PWC for the QRM in this region. Surprisingly, as shown in Fig.7(f), the
coupled-system WM operates with higher efficiency across a wide range of
parameters in high-temperature engine operation, where quantum correlations
are absent[68]. Examining Figs. 8(a)-(f), we observe that the performance of a
quantum Otto engine based on the ARSM varies significantly with temperature
conditions, the coupling parameter $\lambda$, and the nonlinear Stark coupling
$U$. Under conditions where $T_{c}=0.1$ and $T_{h}=0.5$, both the work $W$ and
efficiency $\eta$ are maximized within specific ranges of $\lambda$ and $U$,
approaching the Carnot efficiency $\eta_{Carnot}=0.8$. The normalized work
$\overline{W}>1$ suggests that under these conditions, the engine can
outperform the combined capabilities of qubit and quantum harmonic oscillator
engines. Conversely, for $T_{c}=0.5$ and $T_{h}=2$, both the work and
efficiency decrease as expected with higher temperatures. These observations
underscore the critical role of temperature and coupling parameters in
optimizing the engine’s performance, emphasizing the need to explore different
coupling regimes to maximize efficiency and work output. Analyzing Figs.
9(a)-(f), we find that the work extracted from the quantum heat engine depends
on the qubit-boson coupling strengths $\lambda_{1}$ ($\lambda_{2}$) and the
nonlinear Stark coupling strength $U$. Positive values of $U$ lead to an
increase in work up to an optimal point before decreasing, while negative $U$
results in a consistent decrease in work. The work output is also markedly
influenced by variations in $\lambda_{1}$ and $\lambda_{2}$, with the
anisotropy ratio $\lambda_{2}/\lambda_{1}$ exhibiting diverse behaviors and
indicating optimal operational points for the engine. Achieving optimal
performance in the engine requires precise adjustments of $U$, $\lambda_{1}$,
and $\lambda_{2}$. Positive $U$ proves advantageous up to a certain limit, and
the ratio between $\lambda_{2}$ and $\lambda_{1}$ plays a crucial role in
maximizing performance. These findings underscore the complexity of optimizing
quantum heat engines and highlight the importance of parameter exploration in
achieving desired efficiencies.

To investigate the ideal quasi-static conditions and comprehend the variations
in efficiency $\eta$ of a quantum heat engine as influenced by the coupling
parameters ($\lambda_{1}$, $\lambda_{2}$) and the nonlinear Stark coupling
strength $U$, Figs. 10(a)–(j) illustrate how the efficiency approaches the
Carnot limit and is impacted by proximity to critical points. The efficiency
maximizes when $\lambda_{1}$ and $\lambda_{2}$ lie between 1.0 and 1.5, and
for positive $U$ values ranging from 0.5 to 0.9, with a noticeable decline
outside these ranges. Close to critical points connected to quantum phase
transitions, an initial efficiency increase is triggered by level structure
effects, followed by sudden decreases, rendering engine operation
impracticable. Figs. 10(a)–(e) indicate higher efficiencies with positive $U$,
peaking near 0.75, whereas Figs. 10(f)–(j) show lower efficiencies with
negative $U$, often nearing Carnot efficiency for certain parameter sets. This
study highlights the potential to enhance quantum heat engine efficiency by
methodically choosing parameters, particularly near critical points.

Similarly, Figs. 11(a)–(e) and (f)–(j) display the work $W$ and efficiency
$\eta$ produced by the quantum heat engine as functions of the coupling
parameters $\lambda_{1}$, $\lambda_{2}$, and the nonlinear Stark coupling
strength $U$. These figures highlight the crucial impact of the coupling
strengths and the Stark parameter on the work output near the second-order
quantum phase transition. Due to the spectrum collapse, achieving convergence
in numerical simulations near the critical point requires considerable
computational resources. It is notable that the model is well-defined for all
parameter values as the Hamiltonian operator remains self-adjoint, although it
lacks a ground state above the critical point [96]. Thus, our analysis focuses
on three illustrative Stark coupling strengths.

Significant peaks in the work are observed, with the maximum work output
declining as the system nears the critical point. Notably, Figs. 11(a)–(e)
show that around the critical point, work extraction is more sensitive to the
anisotropy $r$ than in other regions, as seen in Fig. 9, indicating the
difficulty of tuning parameters for optimal engine performance. Conversely,
Figs. 11(f)–(j) indicate that the engine can achieve efficiencies very close
to the Carnot limit for negative $U$, emphasizing the sensitivity of the
engine to both the sign and magnitude of $U$. This thorough analysis
highlights the essential role of coupling parameters and nonlinear
interactions in optimizing the performance of quantum heat engines.

Additionally, analysis of Figs. 12(1)–(12) suggests that under detuning
conditions, the efficiency $\eta$ of the quantum heat engine can approach
Carnot efficiency. This is depicted in Fig. 12, where the efficiency peaks
notably, particularly with increasing detuning $\delta$. Generally, negative
values of $U$ decrease the efficiency. Achieving optimal efficiency requires
fine-tuning parameters such as detuning $\delta$, the nonlinear Stark coupling
strength $U$, and the coupling ratios $\lambda_{2}/\lambda_{1}$ and
$\lambda_{1}/\lambda_{2}$. Therefore, under ideal conditions, the efficiency
of the quantum heat engine can nearly reach Carnot efficiency at the cost of
lower work output and deep-strong light-matter couplings, as seen in the
figures. Crucially, controlling light-matter detuning offers a valuable
mechanism to enhance the performance of quantum heat engines across various
operational scenarios.

Figure 13: Panels (1)-(15) illustrate, from left to right, the total work $W$,
efficiency $\eta$, total entropy production
$\langle\Sigma_{\text{total}}\rangle$, friction work $W^{\text{fric}}$, and
power $P$ as functions of the adiabatic driving time $\tau_{3}=\tau_{1}$ and
thermalization stroke time $\tau_{2}=\tau_{4}$, for various Rabi couplings
$\lambda_{1}=\lambda_{2}$ and Stark strength $U$, as indicated in the
rightmost column of all figures. The temperatures considered are $T_{h}=0.5$
and $T_{c}=0.1$. The subplots on total entropy production (3),(8) and (13)
indicate the lower bound of the thermodynamic uncertainty relation with
respect to $\langle\Sigma_{\text{total}}\rangle$. The subplots on total
friction work (4) and (9) are zoomed-in views of the corresponding larger
figures. Additional cycle parameters include $\omega_{h}=2\omega$ and
$\omega_{c}=\omega$, with all quantities expressed in units of $\omega$.

###  III.3 Finite Time Cycle.

Finite-time quantum heat engines hold significant promise for practical
applications in quantum technologies, particularly in quantum information
processing where rapid and efficient thermal cycling is essential. These
engines also provide valuable insights into the fundamental limits of
thermodynamic processes in the quantum regime, enhancing our understanding of
how quantum effects impact thermodynamic performance. In light of the findings
from previous sections, we conduct an analysis focusing on three optimal
parameter regimes near the critical points of the first- and second-order QPT
as depicted in Fig. 13, which displays the time-dependent thermodynamic
quantifiers: work $W_{(t)}$, efficiency $\eta_{(t)}$ and power $P_{(t)}$.
Operating under finite-time constraints, where cycle lengths are finite rather
than infinitely long, poses several challenges that directly influence engine
performance and efficiency. We evaluated the detrimental effects of finite-
time processes in Fig. 13 through the total entropy
$\left\langle\Sigma_{\text{total}}\right\rangle$ as defined in Eq. 11 and the
work due to internal friction $W^{\text{fric}}$ calculated from Eq. 12. We
note that all values are calculated after the system reaches the limit cycle.
The necessary number of cycles depends on the total cycle length and the
coupling parameters. For this study, three unique sets of coupling parameters
are chosen to thoroughly cover both quantitative and qualitative results
across the whole parameter space. Additionally, the important topic of finite-
time optimization will be reserved for further research As illustrated in the
first column of Fig. 13, the total amount of work extracted diminishes for
nonequilibrium states reached with short thermalization stroke times
$\tau_{2}=\tau_{4}$ and brief adiabatic stroke times $\tau_{1}=\tau_{3}$.
Interestingly, for couplings near the first-order critical point, as evident
in Fig. 13(1) and (6), a thermalization time of $\tau_{2}\approx 2000$ and
adiabatic transitions with $\tau_{1}>10$ are sufficient to recover the maximum
work realized in the ideal case. On the other hand, close to the continuous
QPT, achieving the same work as in the infinite time cycle is difficult and
not achievable within the simulation times considered, as demonstrated in Fig.
13(11). Similarly, the efficiency of the system also decreases for
nonequilibrium states with short thermalization and adiabatic stroke times, as
shown in Fig. 13(2),(7) and (12). Particularly around the first-order critical
point, the efficiency reaches the ideal maximum value with a thermalization
time of $\tau_{2}\approx 800$ and adiabatic changes with $\tau_{1}>5$,
matching the behavior of the extracted work. Nonetheless, near the continuous
QPT, achieving the same level of efficiency as in the infinite time cycle is
equally challenging and impossible within the simulation times considered. The
performance of quantum engines is critically dependent on power output, with
practical applications often necessitating substantial power delivery within
short durations. This significance is demonstrated in the final panel of Fig.
13. The findings indicate that, unlike work and efficiency, power is less
sensitive to the thermalization time, resulting in comparable power outputs
for nonequilibrium states with brief thermalization periods near the first-
order critical point, as depicted in Fig. 13(5) and (10). Notably, near the
continuous QPT, achieving peak power output is as challenging as attaining
optimal work and efficiency, rendering it unattainable within the simulated
times, as illustrated in Fig. 13(15), where the generated power is
considerably lower by an order of magnitude compared to other cases.

The importance of power in finite-time quantum engines extends to various
applications, ensuring optimal system performance within finite operational
durations, thereby enhancing overall practicality and effectiveness. Moreover,
in the development of nanoscale heat engines, where achieving high efficiency
and power in brief cycles is challenging, understanding the trade-offs between
stroke times and performance metrics is crucial for advancing technology in
energy conversion and management at the quantum level. Therefore,
investigating the conditions under which maximum power can be achieved,
particularly near critical points, provides valuable insights for designing
more efficient and powerful quantum devices.

To deepen our understanding of the finite-time thermodynamic quantifiers
mentioned earlier, we evaluate the adverse effects of finite-time operation
through total entropy $\left\langle\Sigma_{\text{total}}\right\rangle$, as
illustrated in Figs. 13(3), (8), and (13). The subplots show the computations
of the lower bound of thermodynamic uncertainty relations as defined in Eq.
14, and the impact of quantum nonadiabaticity in evolution through friction
work ($W^{\text{fric}}$), shown in Figs. 13(4), (9), and (14).

From Figs. 13(3),(8),(13) we observe that the total entropy changes with
thermalization and adiabatic times are too small to be noticeable on the scale
where the entropy is changing for all graphs. The uncertainty relation shows
that for a small total entropy reached in the longer time limit, the power or
efficiency must tend to zero, or the variance in this quantity will go to
infinity. Conversely, when the thermal quantizer is maximized for short
adiabatic times, the AQRSM around the first-order QPT can lead to small
fluctuations, while the second-order point results in large signal-to-noise
ratios. Let us recall that Eq. 14 provides the best case for the efficiency
and power of the stochastic cycle, and a detailed study of the quantum
fluctuations of the AQRSM as a working fluid is in demand and will be explored
in a future work.

Furthermore, for the friction work plots, Figs. 13(4),(9),(14) we observe that
although the total power around the first-order QPT shows a minor dependence
on the thermalization time, the amount of friction changes by three orders of
magnitude between the fast and slow thermalization strokes. As illustrated in
the figure, the friction work demonstrates a strong dependence on the
adiabatic and thermalization times around the second-order QPT point, leading
to increased wasted work due to friction in this scenario.

We note that to mitigate the effects of quantum friction and irreversibility,
several optimization strategies can be employed. One such approach is
shortcut-to-adiabaticity (STA) protocols, which are designed to mimic finite-
time adiabatic transformations [97, 98]. These protocols use contradiabatic
driving or other techniques to suppress nonadiabatic excitations, thereby
reducing quantum friction and increasing efficiency [99]. Another strategy
focuses on fine-tuning control parameters and cycle timings to balance both
power output and efficiency [100]. This form of optimization necessitates a
thorough comprehension of the system’s dynamics and the interaction between
various dissipation mechanisms [101]. These strategies are essential in
crafting quantum thermodynamic cycles, aiming to achieve high efficiency while
controlling dissipation [102]. We anticipate that incorporating these
techniques in subsequent research will enhance the efficiency of quantum heat
engines operating within finite-time frameworks near critical points.

##  IV Conclusion

In this study, we thoroughly examined the functionality of a quantum Otto heat
engine using the anisotropic quantum Rabi-Stark model (AQRSM) as the working
medium. Our research highlighted that quantum phase transitions, which
influence the spectral characteristics of the medium—such as level crossings,
degeneracies, and quasi-degenerate points—play a crucial role in optimizing
the total work output and efficiency of the quantum Otto cycle.

In particular, the observed first-order and continuous phase transitions in
the AQRSM provided significant insights into leveraging critical properties to
enhance the performance of quantum heat engines. In the theoretical scenario
of infinite operation time with zero power output, we constructed a detailed
phase diagram delineating the operational states of the machine, specifying
the conditions under which it functions as a heat engine, refrigerator,
heater, or accelerator.

Furthermore, by considering finite-time operations, we investigated the effect
of quantum friction and conducted a limit cycle analysis, underscoring the
importance of the spectral properties of the working substance in determining
the optimal operating conditions for the quantum Otto engine. Our study
indicates that quantum heat engines based on coupled critical systems have the
potential to significantly surpass the performance of their isolated
counterparts working independently. This finding opens promising avenues for
advancements in renewable energy technologies and quantum computing,
particularly in cryogenically cooled environments that require efficient
energy and thermal management.

We advocate for future research to focus on the practical implementation of
the AQRSM-based thermal engine in accessible physical systems and to examine
the statistical outcomes derived from stochastic thermodynamic processes.

##  V 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.

## References

  * [1] Kosloff and Ronnie.  Quantum thermodynamics: A dynamical viewpoint.  Entropy, 6(6), 2013.
  * [2] R. Kosloff and A. Levy.  Quantum heat engines and refrigerators: continuous devices.  Annual Review of Physical Chemistry, 65(1):365, 2013.
  * [3] John Goold, Marcus Huber, Arnau Riera, L. d. Rio, and P. Skrzypczyk.  The role of quantum information in thermodynamics—a topical review.  J. Phys. A, 49:143001, 2016.
  * [4] Sai, Vinjanampathy, Janet, and Anders.  Quantum thermodynamics.  Contemporary Physics, 57(4), 2016.
  * [5] J. Millen and A. Xuereb.  Perspective: Quantum thermodynamics.  New Journal of Physics, 18(1):2703–2711, 2015.
  * [6] K. Ronnie and R. Yair.  The quantum harmonic otto cycle.  Entropy, 19(4):136–, 2017.
  * [7] Sebastian Deffner and Steve Campbell.  Quantum Thermodynamics: An introduction to the thermodynamics of quantum information.  Morgan & Claypool Publishers, 2019.
  * [8] Hai-Tao Quan, Yu-xi Liu, Chang-Pu Sun, and Franco Nori.  Quantum thermodynamic cycles and quantum heat engines.  Physical Review E, 76(3):031105, 2007.
  * [9] Haitao T Quan.  Quantum thermodynamic cycles and quantum heat engines. ii.  Physical Review E, 79(4):041129, 2009.
  * [10] Nicolas Brunner, Noah Linden, Sandu Popescu, and Paul Skrzypczyk.  Virtual qubits, virtual temperatures, and the foundations of thermodynamics.  Physical Review E, 85(5):051117, 2012.
  * [11] Victor Mukherjee and Uma Divakaran.  Many-body quantum thermal machines.  Journal of Physics: Condensed Matter, 33(45):454001, aug 2021.
  * [12] Leonardo da Silva Souza, Gonzalo Manzano, Rosario Fazio, and Fernando Iemini.  Collective effects on the performance and stability of quantum heat engines.  Physical Review E, 106(1):014143, 2022.
  * [13] Patrice A Camati, Jonas FG Santos, and Roberto M Serra.  Coherence effects in the performance of the quantum otto heat engine.  Physical Review A, 99(6):062103, 2019.
  * [14] Koji Maruyama, Franco Nori, and Vlatko Vedral.  Colloquium: The physics of maxwell’s demon and information.  Reviews of Modern Physics, 81(1):1, 2009.
  * [15] Marlan O Scully.  Quantum photocell: Using quantum coherence to reduce radiative recombination and increase efficiency.  Physical review letters, 104(20):207701, 2010.
  * [16] XL Huang, Tao Wang, XX Yi, et al.  Effects of reservoir squeezing on quantum systems and work extraction.  Physical Review E, 86(5):051105, 2012.
  * [17] Friedemann Tonner and Günter Mahler.  Autonomous quantum thermodynamic machines.  Physical Review E, 72(6):066118, 2005.
  * [18] W. Hao, S. Liu, and J. He.  Thermal entanglement in two-atom cavity qed and the entangled quantum otto engine.  Phys Rev E Stat Nonlin Soft Matter Phys, 79(4 Pt 1):041113, 2009\.
  * [19] Himangshu Prabal Goswami and Upendra Harbola.  Thermodynamics of quantum heat engines.  Physical Review A, 88(1):013842, 2013.
  * [20] Tien D Kieu.  The second law, maxwell’s demon, and work derivable from quantum heat engines.  Physical review letters, 93(14):140403, 2004.
  * [21] AE Allahverdyan, R Serral Gracia, and Th M Nieuwenhuizen.  Work extraction in the spin-boson model.  Physical Review E, 71(4):046106, 2005.
  * [22] Jianhui Wang, Zhaoqi Wu, and Jizhou He.  Quantum otto engine of a two-level atom with single-mode fields.  Phys Rev E Stat Nonlin Soft Matter Phys, 85(4):041148, 2012.
  * [23] Obinna Abah, Johannes Rossnagel, Georg Jacob, Sebastian Deffner, Ferdinand Schmidt-Kaler, Kilian Singer, and Eric Lutz.  Single-ion heat engine at maximum power.  Physical review letters, 109(20):203006, 2012.
  * [24] Johannes Roßnagel, Obinna Abah, Ferdinand Schmidt-Kaler, Kilian Singer, and Eric Lutz.  Nanoscale heat engine beyond the carnot limit.  Physical review letters, 112(3):030602, 2014.
  * [25] Johannes Roßnagel, Samuel T Dawkins, Karl N Tolazzi, Obinna Abah, Eric Lutz, Ferdinand Schmidt-Kaler, and Kilian Singer.  A single-atom heat engine.  Science, 352(6283):325–329, 2016.
  * [26] Gleb Maslennikov, Shiqian Ding, Roland Hablützel, Jaren Gan, Alexandre Roulet, Stefan Nimmrichter, Jibo Dai, Valerio Scarani, and Dzmitry Matsukevich.  Quantum absorption refrigerator with trapped ions.  Nature communications, 10(1):1–8, 2019.
  * [27] Keye Zhang, Francesco Bariani, and Pierre Meystre.  Quantum optomechanical heat engine.  Physical review letters, 112(15):150602, 2014.
  * [28] K. Zhang, F. Bariani, and P. Meystre.  Theory of an optomechanical quantum heat engine.  Physical Review A, 90(2):023819, 2014.
  * [29] O. Fialko and D. Hallwood.  Isolated quantum heat engine.  Physical Review Letters, 108(8):085303, 2012.
  * [30] Jean-Philippe Brantut, Charles Grenier, Jakob Meineke, David Stadler, Sebastian Krinner, Corinna Kollath, Tilman Esslinger, and Antoine Georges.  A thermoelectric heat engine with ultracold atoms.  Science, 342(6159):713–715, 2013.
  * [31] HT Quan, YD Wang, Yu-xi Liu, CP Sun, and Franco Nori.  Maxwell’s demon assisted thermodynamic cycle in superconducting quantum circuits.  Physical review letters, 97(18):180402, 2006.
  * [32] AO Niskanen, Y Nakamura, and Jukka P Pekola.  Information entropic superconducting microcooler.  Physical Review B, 76(17):174523, 2007.
  * [33] Jukka P Pekola, V Brosco, Mikko Möttönen, Paolo Solinas, and A Shnirman.  Decoherence in adiabatic quantum evolution: Application to cooper pair pumping.  Physical review letters, 105(3):030401, 2010.
  * [34] Jonne V Koski, Aki Kutvonen, Ivan M Khaymovich, Tapio Ala-Nissila, and Jukka P Pekola.  On-chip maxwell’s demon as an information-powered refrigerator.  Physical review letters, 115(26):260602, 2015.
  * [35] JP Pekola, DS Golubev, and DV Averin.  Maxwell’s demon based on a single qubit.  Physical Review B, 93(2):024501, 2016.
  * [36] M. Campisi and R. Fazio.  The power of a critical heat engine.  Nat. Commun., 7:11895, 2016.
  * [37] Thomás Fogarty and Thomas Busch.  A many-body heat engine at criticality.  Quantum Science and Technology, 6(1):015003, nov 2020.
  * [38] Giulia Piccitto, Michele Campisi, and Davide Rossini.  The ising critical quantum otto engine.  New Journal of Physics, 2022.
  * [39] Yu-Han Ma, Shan-He Su, and Chang-Pu Sun.  Quantum thermodynamic cycle with quantum phase transition.  Phys. Rev. E, 96:022143, Aug 2017.
  * [40] YY Chen, G Watanabe, YC Yu, XW Guan, and A del Campo.  An interaction-driven many-particle quantum heat engine and its universal behavior.  npj Quantum Inf, 5, 2019.
  * [41] Revathy B.S, Victor Mukherjee, Uma Divakaran, and Adolfo del Campo.  Universal finite-time thermodynamics of many-body quantum machines from kibble-zurek scaling.  Phys. Rev. Res., 2:043247, 2020.
  * [42] Revathy B.S, Victor Mukherjee, and Uma Divakaran.  Bath engineering enhanced quantum critical engines.  Entropy, 24(10), 2022.
  * [43] Revathy B.S, Harsh Sharma, and Uma Divakaran.  Improving performance of quantum heat engines by free evolution.  arXiv preprint arXiv:2302.07003, 2023.
  * [44] He-Guang Xu, Jiasen Jin, GDM Neto, and Norton G de Almeida.  Universal quantum otto heat machine based on the dicke model.  Physical Review E, 109(1):014122, 2024.
  * [45] Victor Mukherjee, Uma Divakaran, et al.  Quantum critical engine at finite temperatures.  arXiv preprint arXiv:2403.20172, 2024.
  * [46] II Rabi.  On the process of space quantization.  Physical Review, 49(4):324, 1936.
  * [47] Isidor Isaac Rabi.  Space quantization in a gyrating magnetic field.  Physical Review, 51(8):652, 1937.
  * [48] Daniel Braak.  Integrability of the rabi model.  Physical Review Letters, 107(10):100401, 2011.
  * [49] Daniel Braak, Qing-Hu Chen, Murray T Batchelor, and Enrique Solano.  Semi-classical and quantum rabi models: in celebration of 80 years.  Journal of Physics A: Mathematical and Theoretical, 49(30):300301, 2016.
  * [50] MO Scully and MS Zubairy.  Quantum optics cambridge university press cambridge.  1997\.
  * [51] Qing-Hu Chen, Yuan Yang, Tao Liu, and Ke-Lin Wang.  Entanglement dynamics of two independent jaynes-cummings atoms without the rotating-wave approximation.  Physical Review A, 82(5):052306, 2010.
  * [52] Ferdi Altintas, Ali ÜC Hardal, and Özgür E Müstecaplıoğlu.  Rabi model as a quantum coherent heat engine: From quantum biology to superconducting circuits.  Physical Review A, 91(2):023816, 2015.
  * [53] G Alvarado Barrios, F Albarrán-Arriagada, FA Cárdenas-López, G Romero, and JC Retamal.  Role of quantum correlations in light-matter quantum heat engines.  Physical Review A, 96(5):052119, 2017.
  * [54] Gabriel Alvarado Barrios, Francisco J Peña, Francisco Albarrán-Arriagada, Patricio Vargas, and Juan Carlos Retamal.  Quantum mechanical engine for the quantum rabi model.  Entropy, 20(10):767, 2018.
  * [55] Yuan-Sheng Wang, Man-Hong Yung, Dazhi Xu, Maoxin Liu, and Xiaosong Chen.  Critical behavior of the quantum stirling heat engine.  Physical Review A, 109(2):022208, 2024.
  * [56] Myung-Joong Hwang, Ricardo Puebla, and Martin B Plenio.  Quantum phase transition and universal dynamics in the rabi model.  Physical review letters, 115(18):180404, 2015.
  * [57] Myung-Joong Hwang, MS Kim, and Mahn-Soo Choi.  Recurrent delocalization and quasiequilibration of photons in coupled systems in circuit quantum electrodynamics.  Physical review letters, 116(15):153601, 2016.
  * [58] Maoxin Liu, Stefano Chesi, Zu-Jian Ying, Xiaosong Chen, Hong-Gang Luo, and Hai-Qing Lin.  Universal scaling and critical exponents of the anisotropic quantum rabi model.  Physical Review Letters, 119(22):220601, 2017.
  * [59] Ricardo Puebla, Myung-Joong Hwang, and Martin B Plenio.  Excited-state quantum phase transition in the rabi model.  Physical Review A, 94(2):023835, 2016.
  * [60] Gaoke Hu, Wen-Long You, Maoxin Liu, and Haiqing Lin.  Excited-state quantum phase transition in light-matter systems with discrete and continuous symmetries.  Physical Review A, 108(3):033710, 2023.
  * [61] YX Yu, J Ye, and WM Liu.  Goldstone and higgs modes of photons inside a cavity and their detections sci, 2013.
  * [62] Qiong-Tao Xie, Shuai Cui, Jun-Peng Cao, Luigi Amico, and Heng Fan.  Anisotropic rabi model.  Physical Review X, 4(2):021046, 2014.
  * [63] Michael Tomka, Omar El Araby, Mikhail Pletyukhov, and Vladimir Gritsev.  Exceptional and regular spectra of a generalized rabi model.  Physical Review A, 90(6):063839, 2014.
  * [64] Edwin T Jaynes and Frederick W Cummings.  Comparison of quantum and semiclassical radiation theories with application to the beam maser.  Proceedings of the IEEE, 51(1):89–109, 1963.
  * [65] Chuanzhou Zhu, Lin Dong, and Han Pu.  Effects of spin-orbit coupling on jaynes-cummings and tavis-cummings models.  Physical Review A, 94(5):053621, 2016.
  * [66] Li-Bao Fan, Yue-Hui Zhou, Fen Zou, Huan Guo, Jin-Feng Huang, and Jie-Qiao Liao.  Quantum thermalization and vanishing thermal entanglement in the open jaynes–cummings model.  Annalen der Physik, 532(8):2000134, 2020.
  * [67] Wang-Yan Liu, Li-Bao Fan, Ye-Xiong Zeng, Jin-Feng Huang, and Jie-Qiao Liao.  Quantum thermalization and thermal entanglement in the open quantum rabi model.  Annalen der Physik, 535(2):2200502, 2023.
  * [68] He-Guang Xu, V Montenegro, Gao Xianlong, Jiasen Jin, and GD de Moraes Neto.  Persisting quantum effects in the anisotropic rabi model at thermal equilibrium.  Physical Review Research, 6(1):013001, 2024.
  * [69] Arne L Grimsmo and Scott Parkins.  Cavity-qed simulation of qubit-oscillator dynamics in the ultrastrong-coupling regime.  Physical Review A, 87(3):033814, 2013.
  * [70] Arne L Grimsmo and Scott Parkins.  Open rabi model with ultrastrong coupling plus large dispersive-type nonlinearity: nonclassical light via a tailored degeneracy.  Physical Review A, 89(3):033802, 2014.
  * [71] Hans-Peter Eckle and Henrik Johannesson.  A generalization of the quantum rabi model: exact solution and spectral structure.  Journal of Physics A: Mathematical and Theoretical, 50(29):294004, 2017.
  * [72] Lei Cong, Simone Felicetti, Jorge Casanova, L Lamata, Enrique Solano, and Iñigo Arrazola.  Selective interactions in the quantum rabi model.  Physical Review A, 101(3):032350, 2020.
  * [73] Andrzej J Maciejewski, Maria Przybylska, and Tomasz Stachowiak.  An exactly solvable system from quantum optics.  Physics Letters A, 379(24-25):1503–1509, 2015.
  * [74] You-Fei Xie, Liwei Duan, and Qing-Hu Chen.  Quantum rabi–stark model: solutions and exotic energy spectra.  Journal of Physics A: Mathematical and Theoretical, 52(24):245304, 2019.
  * [75] You-Fei Xie and Qing-Hu Chen.  Exact solutions to the quantum rabi-stark model within tunable coherent states.  Communications in Theoretical Physics, 71(5):623, 2019.
  * [76] You-Fei Xie, Xiang-You Chen, Xiao-Fei Dong, and Qing-Hu Chen.  First-order and continuous quantum phase transitions in the anisotropic quantum rabi-stark model.  Physical Review A, 101(5):053803, 2020.
  * [77] M-L Cai, Z-D Liu, W-D Zhao, Y-K Wu, Q-X Mei, Y Jiang, L He, X Zhang, Z-C Zhou, and L-M Duan.  Observation of a quantum phase transition in the quantum rabi model with a single trapped ion.  Nature communications, 12(1):1126, 2021.
  * [78] Xiang-You Chen, You-Fei Xie, and Qing-Hu Chen.  Quantum criticality of the rabi-stark model at finite frequency ratios.  Physical Review A, 102(6):063721, 2020.
  * [79] Gavin McCauley, Benjamin Cruikshank, Denys I Bondar, and Kurt Jacobs.  Accurate lindblad-form master equation for weakly damped quantum systems across all regimes.  npj Quantum Information, 6(1):74, 2020.
  * [80] David Gelbwaser-Klimovsky, Alexei Bylinskii, Dorian Gangloff, Rajibul Islam, Alán Aspuru-Guzik, and Vladan Vuletic.  Single-atom heat machines enabled by energy quantization.  Physical review letters, 120(17):170601, 2018.
  * [81] P. M. Chaikin and T. C. Lubensky.  Principles of Condensed Matter Physics.  Cambridge University Press, Cambridge, UK, 1995.
  * [82] Eduardo Mascarenhas, Helena Bragança, Ross Dorner, M França Santos, Vlatko Vedral, Kavan Modi, and John Goold.  Work and quantum phase transitions: Quantum latency.  Physical Review E, 89(6):062103, 2014.
  * [83] Ross Dorner, John Goold, Cecilia Cormick, Mauro Paternostro, and Vlatko Vedral.  Emergent thermodynamics in a quenched quantum many-body system.  Physical Review Letters, 109(16):160601, 2012.
  * [84] Krissia Zawadzki, Roberto M Serra, and Irene D’Amico.  Work-distribution quantumness and irreversibility when crossing a quantum phase transition in finite time.  Physical Review Research, 2(3):033167, 2020.
  * [85] C Pfleiderer.  Why first order quantum phase transitions are interesting.  Journal of Physics: Condensed Matter, 17(11):S987, 2005.
  * [86] Pavel Cejnar, Stefan Heinze, and Michal Macek.  Monodromy and excited-state quantum phase transitions in integrable systems: collective vibrations of nuclei.  Physical review letters, 99(10):100601, 2006.
  * [87] Armando Relano, J Dukelsky, A Escuderos, and JM Arias.  Quantum phase transitions in the one-dimensional hubbard model: Excited-state quantum phase transitions.  Physical Review A, 78(6):060102, 2008.
  * [88] Pavel Cejnar, Pavel Stránský, Michal Macek, and Michal Kloc.  Excited-state quantum phase transitions.  Journal of Physics A: Mathematical and Theoretical, 54(13):133001, 2021.
  * [89] Ronnie Kosloff and Yair Rezek.  The quantum harmonic otto cycle.  Entropy, 19(4), 2017.
  * [90] Naoto Shiraishi, Keiji Saito, and Hal Tasaki.  Universal trade-off relation between power and efficiency for heat engines.  Physical review letters, 117(19):190601, 2016.
  * [91] Yair Rezek and Ronnie Kosloff.  Irreversible performance of a quantum harmonic heat engine.  New Journal of Physics, 8(5):83, 2006.
  * [92] John P. S. Peterson, Tiago B. Batalhão, Marcela Herrera, Alexandre M. Souza, Roberto S. Sarthour, Ivan S. Oliveira, and Roberto M. Serra.  Experimental characterization of a spin quantum heat engine.  Phys. Rev. Lett., 123:240601, Dec 2019.
  * [93] Patrick Pietzonka and Udo Seifert.  Universal trade-off between power, efficiency, and constancy in steady-state heat engines.  Physical review letters, 120(19):190602, 2018.
  * [94] Gianmaria Falasco, Massimiliano Esposito, and Jean-Charles Delvenne.  Unifying thermodynamic uncertainty relations.  New Journal of Physics, 22(5):053046, 2020.
  * [95] André M Timpanaro, Giacomo Guarnieri, John Goold, and Gabriel T Landi.  Thermodynamic uncertainty relations from exchange fluctuation theorems.  Physical review letters, 123(9):090604, 2019.
  * [96] Daniel Braak, Lei Cong, Hans-Peter Eckle, Henrik Johannesson, and Elinor K Twyeffort.  Spectral continuum in the rabi–stark model.  JOSA B, 41(8):C97–C111, 2024.
  * [97] E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, G. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga.  Shortcuts to adiabaticity.  Advances in Atomic, Molecular, and Optical Physics, 62:117–169, 2013.
  * [98] Sebastian Deffner and Steve Campbell.  Quantum speed limits: From heisenberg’s uncertainty principle to optimal quantum control.  Journal of Physics A: Mathematical and Theoretical, 47(49):49FT01, 2014.
  * [99] Adolfo del Campo.  Shortcuts to adiabaticity by counterdiabatic driving.  Physical Review Letters, 111(10):100502, 2013.
  * [100] Obinna Abah, Mauro Paternostro, and Eric Lutz.  Efficiency and fluctuations of quantum heat engines.  Physics Reports, 789:1–51, 2018.
  * [101] Wolfgang Niedenzu and Karl H. Hoffmann.  Dissipation in quantum thermodynamics.  Entropy, 21(8):809, 2019.
  * [102] David Gelbwaser-Klimovsky and Patricio Campos.  Quantum thermodynamics: A dynamical viewpoint.  Contemporary Physics, 57(4):511–529, 2016.