# Finite-time quantum Otto engine with a squeezed thermal bath: Role of
quantum coherence and squeezing in the performance and fluctuations

Yang Xiao1 Dehua Liu1 Jizhou He1 Wu-Ming Liu2<EMAIL_ADDRESS>Jianhui Wang1,3
<EMAIL_ADDRESS>${}^{1}\,$ Department of Physics, Nanchang University,
Nanchang, 330031, China
${}^{2}\,$ Beijing National Laboratory for Condensed Matter Physics, Institute
of Physics, Chinese Academy of Sciences, Beijing 100190, China
${}^{3}\,$ State Key Laboratory of Surface Physics and Department of Physics,
Fudan University, Shanghai 200433, China

###### Abstract

We consider a finite-time quantum Otto heat engine that consists of two
isochoric (thermal-contact) process, where the system is alternatively coupled
to a hot squeezed and a cold thermal reservoir, and two unitary driven
strokes, where the system is isolated from these two baths and its von Neumann
entropy keeps constant. Both quantum inner friction and coherence are
generated along the driven stroke and coherence cannot be fully erased after
the finite-time hot isochore. Using full counting statistics, we present the
probability distribution functions of heat injection and total work per cycle,
which are dependent on the time duration along each process. With these, we
derive the analytical expressions for the thermodynamic quantities of the two-
level heat engine, such as total work, thermodynamic efficiency, entropy
production, and work fluctuations, in which effects of coherence, squeezing,
inner friction and finite-time heat exchange are included. We then numerically
determine the thermodynamic quantities and the fluctuations using the
parameters employed in the experimental implementation. Our results clarify
the role of coherence and squeezing in the performance and fluctuations in the
quantum Otto engines.

PACS number(s):05.70.Ln

##  I Introduction

Heat engines produce useful work by consuming thermal energy. A great effort
has been devoted to the extension from macroscopic engines to quantum heat
engines Kos14 ; Nie18 ; Scu11 . Practically all heat engines should run far
from the ideal maximum efficiency at which the power becomes vanishing; the
working system proceeding in irreversible processes can not reach Gibbs
thermal state due to finite-time operation. Therefore, quantum heat engines
provide excellent platforms for understanding nonequilibrium thermodynamic and
nonequilibrium statistical implications of open quantum systems.

Towards the ever-smaller scale of thermal machines, quantum effects
increasingly manifest themselves, such as quantum coherenceCam19 ; AW16 ; KM16
; Scu11 ; Bra17 ; Rah12 ; RA15 ; KE18 , entanglementZhang07 ; Wangwu19 ;
Huang09 ; Fun13 , and correlationsJO02 ; LJ11 ; JJ13 , and quantum
measurementsHor14 ; Bra15 ; Kam16 ; Cha21 ; Su21 . Quantum coherence, for
instance, was observed in recent , experimental realizations of quantum heat
engines based on nuclear magnetic resonance RJ19 ; Ser19  and nitrogen-vacancy
centers in diamondJK19 . For quantifying the coherence, several measurements
were recently discussedFra19 ; TM14 ; AG17 ; XY15 ; IM16 ; among them, the so-
called relative entropy of coherence was usually used in quantum thermal
machines as it can give quantitative relations between coherence generation
and irreversibility due to finite-time operation of enginesTM14 ; Fra19 . For
a finite-time unitary driving, the irreversible work comes from not only the
coherence but also the incoherent transitions among the energy levels Fra19 .
A growing interest has been attracted in understanding the thermodynamic
implication of the cyclic quantum thermal machine in which unitary driving
strokes are involved Kos02 ; Pla14 ; Ale15 ; Tur19 ; Jiao21 ; Cam19 ; Abah12 .
These studies showed the finite-time performance of the intrinsically
irreversible quantum heat engines, and also revealed the deep connection
between coherence and quantum friction responsible for the diabatic
transitions Cam19 .

Apart from the standard cyclic engines working with two thermal baths, heat
engines may be fulled by miniaturized, nonthermal baths which drive systems
far away from thermal equilibrium. The nonthermal baths may be quantum
coherentHT06 ; Scu11 , quantum correlatedRD09 ; Ber17 ; And19 ; Per15 ,
quantum-measurement-inducedLB19 ; CE17 ; CE18 ; Su21 , and squeezedAba14 ;
Wang19 ; Nie16 ; Manz18 ; Manz16 ; You18 ; Sin20 ; Kla17 ; Liu15 . Quite
naturally, recent studies have started studying novel finite-time performance
and the fluctuations in quantum heat engines subjected to nonthermal baths.
Adopting squeezed baths are particularly interesting for quantum Otto engines
Kla17 ; Aba14 ; Wang19 ; Assis20  where novel performance beyond the
conventional engines with no violation of thermodynamic principles.

For microscopic systems, heat and work are no longer deterministic Sek10 ;
Sei12 ; Smi18  as is the case for macroscopic systems. As a result, the
efficiency and power for quantum heat engines are stochastic, and both of them
are fluctuating. The power fluctuations, together with the efficiency
fluctuations, as a limiting factor for the practical usefulness in heat
engines, measure the machine stability Bou21 . Ideally, the quantum heat
engine should have high efficiency (small entropy production), large power,
and small fluctuations for these performance measures. Nevertheless, the
machine performance and fluctuations are always governed by the thermodynamic
uncertainty relation associated with tradeoff between relative fluctuations
and irreversible entropy productionMa18 ; Sei18 . In that context, strong
emphasis has been put on the finite-time performance of the quantum heat
engines, and in particular on fluctuations of power and efficiency Ver14 ;
Jiang15 ; Lut20 ; Lut21 . On the other hand, it has been shown that the
efficiency for the standard heat engines can be higher than the Carnot value,
the least likely in the long-time limitVer14 ; Lut21 .

In this paper, we investigate the performance and fluctuations in a finite-
time quantum Otto engine fuelled by a squeezed bath. We determine the
probability distribution functions for heat and work for the engine cycle by
using full counting statistics. We then present general formulae of the
thermodynamic efficiency, power and power fluctuations, in which the
irreversiblities are associated both with quantum friction and coherence
generated along the finite-time unitary driving and with system-bath
interaction involved. By a numerical simulation with a qubit employed in the
experiment, we show that the squeezing suppresses the coherence contributing
to the unwanted irreversible work and thus results in enhancing the power and
thermodynamic efficiency. We specifically study the relative power
fluctuations, which characterize the instability of the quantum engines, and
the large deviation function of efficiency. Our results show that reservoir
squeezing can significantly reduce the relative power fluctuations and quicken
the convergence speed of the efficiency to the typical value.

##  II Quantum Otto cycle in finite time

###  II.1 Machine model

We consider a quantum Otto engine cycle working between a hot squeezed and a
cold thermal bath [see Fig. 1(a)]. This engine cycle consists of two unitary
driving strokes, where the system is isolated from two heat reservoirs, and
two isochoric branches, along each of which the system with constant
Hamiltonian is weakly coupled to the hot squeezed or cold thermal bath. Such
an engine model is described as follows.

(1)Adiabatic compression from time $t=t_{0}$ to $t=t_{1}$. For simplicity, we
assume that the initial time $t_{0}=0$. During this stroke where the system is
isolated from a heat reservoir in time duration $\tau_{ch}$ with
$\tau_{ch}=t_{1}$, the system energy gap is enlarged by the driven Hamiltonian
$H_{ch}(t)=\frac{\hbar\omega(t)}{2}[\cos(\frac{\pi
t}{2\tau_{ch}})\sigma_{x}+\sin(\frac{\pi t}{2\tau_{ch}})\sigma_{z}]$, where
$\omega(t)=\omega_{c}(1-{t}/\tau_{ch})+\omega_{h}({t}/{\tau_{ch}})$ with
$0\leq t\leq\tau_{ch}$, and $\sigma_{x,y,z}$ are the Pauli matrices. As this
driven Hamiltonian does not commute at different times, the quantum coherence
is generated in the energy basis of the system. We use this protocol design
because it was employed in recent experimental realization of the quantum Otto
enginesSer19 ; RJ19 . The system dynamics can be described by a unitary
evolution, provided that the driven time is small enough in order for the
energy exchange between the system and its environment to be neglected. Hence,
the state of the system $\rho_{t}$, with $\rho_{t}\equiv\rho(t)$, is described
in terms of the initial state $\rho_{t_{0}}$,

$\rho_{t}=U_{ch}\rho_{t_{0}}U_{ch}^{{\dagger}},$ (1)

where the time evolution operator
$U_{ch}=\mathcal{T}_{>}\exp\\{-\frac{i}{\hbar}\int_{t_{0}}^{t_{1}}dtH_{ch}(t)\\}$,
with the time-ordering operator $\mathcal{T}_{>}$. In a realistic scenario, we
consider that the interaction between the system and the external filed is
weak and the change of internal energy is equal to the work extracted from the
system. We then use full counting statistics method Sol15 ; Xu18  to obtain
the work distribution as

$\displaystyle p(\mathrm{w}_{ch})$ $\displaystyle=$
$\displaystyle\sum_{n,n^{\prime},m}U_{ch}^{mn}\langle
n(t_{0})|\rho_{t_{0}}|n^{\prime}(t_{0})\rangle U_{ch}^{n^{\prime}m{\dagger}}$
(2) $\displaystyle\times$
$\displaystyle\delta\left[\mathrm{w}_{ch}-\left(\varepsilon_{m}^{h}-\frac{\varepsilon_{n}^{c}+\varepsilon_{n^{\prime}}^{c}}{2}\right)\right]$

where we have used
$H_{ch}(t_{0})|n(t_{0})\rangle=\varepsilon_{n}^{c}|n(t_{0})\rangle$,
$H_{ch}(t_{0})|n^{\prime}(t_{0})\rangle=\varepsilon_{n^{\prime}}^{c}|n^{\prime}(t_{0})\rangle$,
$H_{ch}(t_{1})|m(t_{1})\rangle=\varepsilon_{m}^{h}|m(t_{1})\rangle$,
$U_{ch}^{mn}=\langle m(t_{1})|U_{ch}|n(t_{0})\rangle$, and
$U_{ch}^{n^{\prime}m{\dagger}}=\langle
n^{\prime}(t_{0})|U_{ch}^{{\dagger}}|m(t_{1})\rangle$. Here the term $|\langle
n(t_{0})|U_{ch}|m(t_{1})\rangle|^{2}$ denotes the transition probability
between the system eigenstates $|n(t_{0})\rangle$ and $|m(t_{1})\rangle$RJ19 ,
and $\langle n(t_{0})|\rho_{t_{0}}|n(t_{0})\rangle$ is the probability of the
system being in state $|n(t_{0})\rangle$. For the two-level system under
consideration, we use $\big{|}g\rangle$ and $\big{|}e\rangle$ $(n,m=g,e)$ to
denote the ground and excited eigenstates, respectively. When the driven
stroke is a quantum adiabatic, transition between any two eigenstates will not
happen and therefore $|\langle
n(t_{0})|U_{ch}|m(t_{1})\rangle|^{2}=\delta_{nm}$. However, along the finite-
time driven stroke, the inner friction causes the transitions among the
instantaneous energy eigenstates, resulting in the irreversible work.

Figure 1:  (a) Illustration of the engine cycle driven by a squeezed
reservoir. Beginning with a state $\rho_{t_{0}}$ at time $t=t_{0}$, the
working substance undergoes a unitary compression mediated by a time-dependent
Hamiltonian in time duration $\tau_{ch}$. In the second stroke from time
$t=t_{1}$ to $t=t_{2}$, the working substance with its constant Hamiltonian is
coupled to a squeezed bath with inverse temperature $\beta_{h}$ in time
$\tau_{h}$. The third stroke is a unitary expansion where the system state
evolves from $\rho_{t_{2}}$ to $\rho_{t_{3}}$. In the fourth stroke, where the
system Hamiltonian is again kept constant, the working substance in contact
with the cold thermal bath of inverse temperature $\beta_{c}$ relaxes to the
initial state after time duration $\tau_{c}$. (b) Schematic diagram of the
engine cycle in the ($\omega,\langle n\rangle$) plane. Here $\langle
n_{h}^{{ss}}\rangle=[\exp(\beta_{h}^{\mathrm{eff}})-1)]$ is excitation number
of the system reaching the steady state under reservoir squeezing, and
$\langle n_{c}^{eq}\rangle$ ($\langle n_{h}^{eq}\rangle$) is excitation number
of the system at thermal equilibrium with the cold (hot) thermal bath of
inverse temperature $\beta_{h}(\beta_{c})$. We see that $\langle
n_{t_{2}}\rangle(\langle n_{t_{0}}\rangle)$ deviates from its asymptotic value
$\langle n_{h}^{ss}\rangle$ ($\langle n_{c}^{eq}\rangle$) due to finite time
operation.

(2) Isochoric heating from time $t=t_{1}$ to $t=t_{2}$. The two-level system
is weakly coupled to a hot squeezed heat reservoir at inverse temperature
$\beta_{h}$ during time duration $\tau_{h}$ with $\tau_{h}=t_{2}-t_{1}$, while
its Hamiltonian keeps constant:
$H_{h}(t)=H_{ch}(t_{1})=\frac{\hbar\omega_{h}}{2}\sigma_{z}$. As no work is
produced along the isochoric process, the stochastic heat injection is
equivalent to the increase of the system eigenenergy. The transition
probability from eigenstate $k$ to $l$ along the hot isochore, $\big{|}\langle
k(t_{1})|U_{h}(t)|l(t_{2})\rangle\big{|}^{2}$, with the time evolution
operator $U_{h}(t)$, can be obtained by using
$\rho_{t}=U_{h}\big{|}k(t_{1})\rangle\langle k(t_{1})\big{|}U_{h}^{\dagger}$
to arrive at $\big{|}\langle
k(t_{1})|U_{h}(t)|l(t_{1})\rangle\big{|}^{2}=\langle
l\big{|}\rho_{t}|l\rangle$. Using
$H_{h}(t)\big{|}l(t_{2})=\varepsilon_{l}^{h}|l(t_{2})\rangle$ and
$H_{h}(t)\big{|}k(t_{1})=\varepsilon_{k}^{h}|k(t_{1})\rangle$, it follows
that, the probability distribution for the heat absorbed during this stroke
reads

$p(q_{h}|\mathrm{w}_{ch})=\sum_{k,l}\delta[q_{h}-(\varepsilon_{l}^{h}-\varepsilon_{k}^{h})]\langle
l(t_{2})\big{|}\rho_{t_{2}}\big{|}l(t_{2})\rangle\delta_{km},$ (3)

where $\langle l(t_{2})\big{|}\rho_{t_{2}}\big{|}l(t_{2})\rangle$ is the
probability of finding the system to be in eigenstate $|l(t_{2})\rangle$ after
the second projective measurement at the isochore.

In the hot isochoric process within $t_{1}\leq t\leq t_{2}$, where the system
weakly interacts with a squeezed thermal bath, the system dynamics of density
operator $\rho_{t}$ in interaction picture HF02 ; MM97 ; Sri18  can be
described by

$\displaystyle\frac{d}{dt}\rho_{t}$
$\displaystyle=\gamma_{h}(N_{h}^{ss}+1)(\sigma_{-}\rho_{t}\sigma_{+}-\frac{1}{2}\sigma_{+}\sigma_{-}\rho_{t}$
(4)
$\displaystyle-\frac{1}{2}\rho_{t}\sigma_{+}\sigma_{-})+\gamma_{h}N_{h}^{ss}(\sigma_{+}\rho_{t}\sigma_{-}$
$\displaystyle-\frac{1}{2}\sigma_{-}\sigma_{+}\rho_{t}-\frac{1}{2}\rho_{t}\sigma_{-}\sigma_{+})$
$\displaystyle-\gamma_{h}M\sigma_{+}\rho_{t}\sigma_{+}-\gamma_{h}M^{*}\sigma_{-}\rho_{t}\sigma_{-},$

where

$\displaystyle N_{h}^{ss}=N_{h}^{th}[\cosh^{2}(r)+\sinh^{2}(r)]+\sinh^{2}(r)$
(5)

denotes the excitation number of the system which reaches the steady state
with the squeezed bath, with the mean population at thermal state
$N_{h}^{th}=\frac{1}{e^{\beta_{h}\hbar w_{h}}-1}$,
$M=-\frac{1}{2}\sinh(2*r)e^{i\theta}(2N_{h}^{th}+1)$, with the phase factor
$\theta$, and $\gamma_{h}$ is the vacuum decay rate which indicates the
system-bath interaction strength. Here and hereafter we use
$\sigma_{+}=|1\rangle\langle 0|=\frac{1}{2}(\sigma_{x}+i\sigma_{y})$ and
$\sigma_{-}=|0\rangle\langle 1|=\frac{1}{2}(\sigma_{x}-i\sigma_{y})$.

From Eq. (4), one can find Sri18  that the time-dependent state $\rho(t)$
along the hot isochore can be determined in the Schrödinger picture according
to

$\displaystyle\rho_{t}=\begin{pmatrix}\frac{1+\langle\sigma_{z}(t)\rangle}{2}&\mathcal{X}e^{-i\omega_{h}t}\\\
\mathcal{X}^{*}e^{i\omega_{h}t}&\frac{1-\langle\sigma_{z}(t)\rangle}{2}\end{pmatrix},$
(6)

where
$\mathcal{X}=(1+\frac{e^{\gamma_{h}at}-1}{2})e^{-\frac{\gamma_{h}(2N^{ss}_{h}+1+a)t}{2}}\langle\sigma_{-}(t_{1})\rangle+\sinh(\frac{\gamma_{h}at}{2})e^{i\Phi-\frac{\gamma_{0}(2N^{ss}_{h}+1)t}{2}}\langle\sigma_{+}(t_{1})\rangle,$
with $a=\sinh(2r)(1+2N_{h}^{th})$, and
$\langle\sigma_{z}(t)\rangle=e^{-\gamma_{h}(2N^{ss}_{h}+1)t}\langle\sigma_{z}(t_{1})\rangle-\frac{1-e^{-\gamma_{h}(2N^{ss}_{h}+1)t}}{2N^{ss}_{h}+1}$.

When the isochoric stroke is slow enough such that
$\tau_{h}\gg\tau_{h,relax}$, where $\tau_{h,relax}$ is the relaxation time of
the system with the hot bath, the density matrix (6) reduces to

$\rho_{t_{2}}^{\mathrm{ss}}\Big{|}_{\tau_{h}\gg\tau_{h,relax}}=\begin{pmatrix}p_{t_{2}}^{e,ss}&0\\\
0&p_{t_{2}}^{g,ss}\end{pmatrix}$ (7)

where $p_{t_{2}}^{e,ss}=N_{h}^{ss}/(2N_{h}^{ss}+1)$ and
$p_{t_{2}}^{g,ss}=1-p_{t_{2}}^{e,ss}$. If we introduce the effective inverse
temperature

$\beta_{h}^{\mathrm{eff}}=\frac{1}{\hbar\omega_{h}}\ln\frac{N_{h}^{ss}+1}{N_{h}^{ss}}$
(8)

to write the excitation number as
$N_{h}^{ss}=1/(e^{\beta_{h}^{\mathrm{eff}}\omega_{h}}-1)$, the detailed
balance is restored in the squeezing case owing to the relation $\frac{\langle
e(t_{2})\big{|}\rho_{t_{2}}\big{|}e(\tau_{t_{2}})\rangle}{\langle
g(t_{2})\big{|}\rho_{t_{2}}\big{|}g(t_{2})\rangle}\Big{|}_{\tau_{h}\gg\tau_{h,relax}}=e^{-\beta_{h}^{\mathrm{eff}}\omega_{h}}$.
When $\tau_{h}\gg\tau_{h,relax}$, the system could be fully thermalized along
the hot isochore and all the coherence produced in the compression would be
erased [see Eq. (7)]. However, for incomplete thermalization with
$\tau_{h}\leq\tau_{h,relax},$ a residual amount of the coherence is retained
and thus the parameter $\mathcal{X}$ in Eq. (6) is nonzero. Such coherence
that endures in this finite-time isochoric process will be present in the next
driven stroke.

(3) Adiabatic expansion from time $t=t_{2}$ to $t=t_{3}$. The stroke, during
which the driven Hamiltonian $H_{hc}(t_{3}-t)=H_{ch}(t)$ is realized by
reversing the protocol used in the adiabatic compression, such that the
expansion Hamiltonian takes the values the same as the compression
Hamiltonian, namely, $\tau_{hc}=\tau_{ch}$. Since the evolution is unitary, we
obtain

$\rho_{t_{3}}=U_{hc}\rho_{t_{2}}U_{hc}^{{\dagger}},$ (9)

where
$U_{hc}=\mathcal{T}_{>}\exp\\{-\frac{i}{\hbar}\int_{t_{2}}^{t_{3}}dtH_{hc}(t)\\}$.
Like the first stroke, the internal friction (due to finite time evolution)
will bring coherence and results in the transition between the eigenstates of
the Hamiltonian $H_{hc}(t_{3})$ and $H_{hc}(t_{2})$. The quantum work
distribution along the driven stroke can be expressed as

$\displaystyle p(\mathrm{w}_{hc}|\mathrm{w}_{ch},q_{h})$
$\displaystyle=\sum_{i,i^{\prime},j}U_{hc}^{j,i}\langle
i(t_{2})|\rho_{t_{2}}|i^{\prime}(t_{2})\rangle U_{hc}^{i^{\prime}j{\dagger}}$
(10)
$\displaystyle\times\delta[\mathrm{w}_{hc}-(\varepsilon_{j}^{c}-\frac{\varepsilon_{i}^{h}+\varepsilon_{i\prime}^{h}}{2})]\delta_{i,l},$

where we have used
$H_{hc}(t_{2})|i(t_{2})\rangle=\varepsilon_{i}^{h}|i(t_{2})\rangle$,
$H_{hc}(t_{2})|i^{\prime}(t_{2})\rangle=\varepsilon_{i^{\prime}}^{h}|i^{\prime}(t_{2})\rangle$,
$H_{hc}(t_{3})|j(t_{3})\rangle=\varepsilon_{j}^{c}|j(t_{3})\rangle$,
$U_{hc}^{ji}=\langle j(t_{3})|U_{hc}|i(t_{2})\rangle$, and
$U_{hc}^{i^{\prime}j}=\langle
i^{\prime}(t_{2})|U_{hc}^{{\dagger}}|j(t_{3})\rangle$. The term $|\langle
i(t_{2})|U_{hc}|j(t_{3})\rangle|^{2}$ is the transition probability from state
$|i(t_{2})\rangle$ to $|j(t_{3})\rangle$, and $\langle
i(t_{2})|\rho_{t_{2}}|i^{\prime}(t_{2})\rangle\delta_{i,i^{\prime}}$ denotes
the probability for finding the system in state $|i\rangle$.

(4) Isochoric cooling from $t=t_{3}$ to $t=t_{3}+\tau_{c}$. The system is
weakly coupled with a cold thermal at inverse temperature $\beta_{c}$ in time
period $\tau_{c}$, and its Hamiltonian is kept constant at
$H_{c}(t)$=$H_{ch}(0)=\frac{\omega_{0}}{2}\sigma_{x}$. The dynamics of the
state $\rho_{t}$ along this stroke, where no squeezing is present and $r=0$,
can be given in interaction picture by

$\displaystyle\frac{d}{dt}\rho_{t}$
$\displaystyle=\gamma_{c}(N_{c}^{th}+1)[\sigma_{-}\rho_{t}\sigma_{+}-\frac{1}{2}\sigma_{+}\sigma_{-}\rho_{t}$
(11)
$\displaystyle-\frac{1}{2}\rho_{t}\sigma_{+}\sigma_{-}]+\gamma_{c}N_{c}^{th}[\sigma_{+}\rho_{t}\sigma_{-}$
$\displaystyle-\frac{1}{2}\sigma_{-}\sigma_{+}\rho_{t}-\frac{1}{2}\rho_{t}\sigma_{-}\sigma_{+}],$

where $N_{c}^{th}=\frac{1}{e^{\beta_{c}\hbar w_{c}}-1}$ is the excitation
number of the system at thermal equilibrium with the cold bath, and
$\gamma_{c}$ is the interaction strength between the system and the cold
thermal bath. Using Eq. (11) , we can obtain the density operator $\rho_{t}$
of the system along the cold isochore, and then make a unitary transformation
of the density matrix from the $\sigma_{z}$ basis to the $\sigma_{x}$ basis to
arrive at

$\rho_{t}=\mathcal{U}\begin{pmatrix}\frac{1+\langle\sigma_{z}(t)\rangle}{2}&\mathcal{Y}e^{-i\omega_{c}t}\\\
\mathcal{Y}^{*}e^{i\omega_{c}t}&\frac{1-\langle\sigma_{z}(t)\rangle}{2}\end{pmatrix}\mathcal{U}^{{\dagger}},$
(12)

where the transformation matrix $\mathcal{U}=\begin{pmatrix}\langle
e(t_{1})|e(t_{0})\rangle&\langle g(t_{1})|e(t_{0})\rangle\\\ \langle
e(t_{1})|g(t_{0})\rangle&\langle g(t_{1})|g(t_{0})\rangle\end{pmatrix}$ and
$\mathcal{Y}=e^{-\frac{\gamma_{c}(2N_{c}^{th}+1)t}{2}}\langle\sigma_{-}(t_{3})\rangle$.
We assume that the final density matrix of the stroke is equal to
$\rho_{t_{0}}$ for closing the engine cycle. When the time duration $\tau_{c}$
is much larger than the relaxation time $\tau_{c,relax}$, $\mathcal{Y}$
becomes vanishing and the system reaches the thermal equilibrium at the end of
the cooling stroke, and then the state of the system takes the form (in the
$\sigma_{x}$ basis):

$\rho_{t_{0}}\Big{|}_{\tau_{c}\gg\tau_{c,relax}}=\begin{pmatrix}\frac{1}{2}&\frac{p_{t_{0}}^{e,eq}-p_{t_{0}}^{g,eq}}{2}\\\
\frac{p_{t_{0}}^{e,eq}-p_{t_{0}}^{g,eq}}{2}&\frac{1}{2}\end{pmatrix},$ (13)

where $p_{t_{0}}^{e,eq}=N_{c}^{eq}/(2N_{c}^{eq}+1)$ and
$p_{t_{0}}^{g,eq}=1-p_{t_{0}}^{e,eq}$.

###  II.2 Statistics of machine performance parameters

The stochastic work extracted from the system in a single cycle is
-$\mathrm{w}_{tot}=-(\mathrm{w}_{ch}+\mathrm{w}_{hc})$, and the stochastic
efficiency reads $\eta=-\mathrm{w}_{tot}/q_{h}$. The joint distribution Hol18
for the work output $\mathrm{w}_{ch}$, $\mathrm{w}_{hc}$ and heat $q_{h}$ can
be determined according to Eqs. (2), (3), and (10)

$\displaystyle p(\mathrm{w}_{ch},q_{h},\mathrm{w}_{hc})$ $\displaystyle=$
$\displaystyle
p(\mathrm{w}_{2}|\mathrm{w}_{ch},q_{h})p(q_{h}|\mathrm{w}_{ch})p(\mathrm{w}_{ch})$
(14) $\displaystyle=$
$\displaystyle\sum_{n,n^{\prime}m,i,i^{\prime},j}\delta[\mathrm{w}_{ch}-(\varepsilon_{m}^{h}-\frac{\varepsilon_{n}^{c}+\varepsilon_{n^{\prime}}^{c}}{2})]$
$\displaystyle\times$
$\displaystyle\delta[q_{h}-(\varepsilon^{h}_{i}-\varepsilon^{h}_{m})]$
$\displaystyle\times$
$\displaystyle\delta[\mathrm{w}_{hc}-(\varepsilon_{j}^{c}-\frac{\varepsilon_{i}^{h}+\varepsilon_{i\prime}^{h}}{2})]$
$\displaystyle\times$ $\displaystyle U_{ch}^{mn}\langle
n(t_{0})|\rho_{t_{0}}|n^{\prime}(t_{0})\rangle U_{ch}^{n^{\prime}m}$
$\displaystyle\times$ $\displaystyle U_{hc}^{ji}\langle
i(t_{2})|\rho_{t_{2}}|i^{\prime}(t_{2})\rangle U_{hc}^{i^{\prime}j}.$

This sets the joint distribution of the total work $\mathrm{w}_{tot}$,

$p(q_{h},\mathrm{w}_{tot})=\int
d\mathrm{w}_{ch}d\mathrm{w}_{hc}\delta[\mathrm{w}-(\mathrm{w}_{ch}+\mathrm{w}_{hc})]p(\mathrm{w}_{ch},q_{h},\mathrm{w}_{hc}),$
(15)

for a cycle with quantum heat injection $q_{h}$. Let $\xi\equiv|\langle
n(t_{0})|U_{ch}|m(t_{1})\rangle|^{2}=|\langle
i(t_{2})|U_{hc}|j(t_{3})\rangle|^{2}$ ( $m,n,i,j=e,g$) be the level transition
probability during expansion or compression for the two level system. Using
Eq. (15), we obtain (see Appendix A for details) average work
$\langle\mathrm{w}_{tot}\rangle$ and average injection $\langle q_{h}\rangle$
as

$\displaystyle-\langle\mathrm{w}_{tot}\rangle$ $\displaystyle=$
$\displaystyle\hbar(w_{h}-w_{c})(\langle n_{t_{2}}\rangle-\langle
n_{t_{0}}\rangle)$ (16) $\displaystyle+$ $\displaystyle 2\hbar\xi(w_{c}\langle
n_{t_{2}}\rangle+w_{h}\langle n_{t_{0}}\rangle)$ $\displaystyle-$
$\displaystyle 2\hbar w_{h}\zeta_{ch}-2\hbar w_{c}\zeta_{hc},$
$\displaystyle\langle q_{h}\rangle=\hbar w_{h}[\langle
n_{t_{2}}\rangle+\langle n_{t_{0}}\rangle(2\xi-1)-2\zeta_{ch}],$ (17)

where we have used $2\langle n_{t_{0}}\rangle:=-\langle
g(t_{0})|\rho_{t_{0}}|g(t_{0})\rangle+\langle
e(t_{0})|\rho_{t_{0}}|e(t_{0})\rangle$, $2\langle n_{t_{2}}\rangle:=-\langle
g(t_{2})|\rho_{t_{2}}|g(t_{2})\rangle+\langle
e(t_{2})|\rho_{t_{2}}|e(t_{2})\rangle$, $\zeta_{ch}:=-Re[U_{ch}^{gg}\langle
g(t_{0})|\rho_{t_{0}}|e(t_{0})\rangle U_{ch}^{eg{\dagger}}]$, and
$\zeta_{hc}:=-Re[U_{hc}^{gg}\langle g(t_{2})|\rho_{t_{2}}|e(t_{2})\rangle
U_{hc}^{eg{\dagger}}]$. After a single cycle, the working substance produces
the total work $-\langle\mathrm{w}_{tot}\rangle$ by absorbing average heats
from the hot and cold baths, $\langle q_{h}\rangle$ and $\langle
q_{c}\rangle$, where $\langle
q_{c}\rangle=-\langle\mathrm{w}_{tot}\rangle-\langle q_{h}\rangle$ due to the
energy conservation, as sketched in Fig. 1(b). Here $\langle n_{t_{0}}\rangle$
and $\langle n_{t_{2}}\rangle$ are the average populations of the two-level
system at times $t=t_{0}$ and $t=t_{2}$ [see also Fig. 1 (b)], respectively.

The parameters $\zeta_{hc}$ and $\zeta_{ch}$ in Eq. (16), associated with off-
diagonal elements, refer to the quantum coherence between thee two energy
eigenstates. These two parameters are not independent but are correlated for
finite-time, incomplete thermalizations where coherence is not fully erased.
$\xi_{ch}$ (determined by the off-diagonal elements) affects the state
$\rho_{t_{1}}$ which evaluates to $\rho_{t_{2}}$ after the hot thermal contact
in finite time, and thus $\xi_{hc}$ is dependent on $\xi_{ch}$; similarly,
$\xi_{ch}$ is affected by $\xi_{hc}$ due to finite time $\tau_{c}$. It is
therefor indicated that $\xi_{ch}$ and $\rho_{hc}$ are correlated with each
other. The residual coherence, transferred from the first to the third strokes
due to incomplete thermalization, would interfere with the coherence generated
in the third stroke. To reveal such a dynamical interference effect on the
thermodynamic quantities of the machine, our quantum heat engine is compared
with an alternative cycle, where a full dephasing operation is performed to
completely remove all coherence after the hot isochore with any value of
thermal-contact time $\tau_{h}$. Throughout the paper we use the superscript
``deph'' to describe the quantities corresponding to the dephased engine
cycle.

The average work -$\langle\mathrm{w}_{tot}\rangle$ (16) can be split up into
the two terms:

$-\langle\mathrm{w}_{tot}\rangle=\langle\mathrm{w}_{\mathrm{deph}}\rangle+\langle\mathrm{w}_{\mathrm{coh}}\rangle,$
(18)

where

$\langle\mathrm{w}_{\mathrm{deph}}\rangle=\hbar(w_{h}-w_{c})(\langle
n_{t_{2}}\rangle-\langle
n_{t_{0}}\rangle)+\langle\mathrm{w}_{\mathrm{fri}}\rangle$ (19)

with $\langle\mathrm{w}_{\mathrm{fri}}\rangle=:2\hbar\xi(w_{c}\langle
n_{t_{2}}\rangle+w_{h}\langle n_{t_{0}}\rangle)$, is the average work in the
dephased case, and

$\langle{w}_{\mathrm{coh}}\rangle=-2\hbar w_{h}\zeta_{ch}-2\hbar
w_{c}\zeta_{hc}$ (20)

is the average work associated with quantum coherence. The second term in Eq.
(19), $\langle w_{\mathrm{fri}}\rangle$, represents the additional work due to
overcome the inner friction causing unwanted diabatic transitions in
instantaneous energy eigenstates. The irreversible work can be decomposed into
two contributions, which come from the inner friction associated with the
diabatic transitions and the coherence generated along the unitary driven
strokes. Due to finite-time operation, the system at end of either the hot
(cold) isochore can not reach the stationary state (Gibbs thermal state),
meaning that $\langle n_{t_{2}}\rangle<\langle n_{h}^{ss}\rangle$ and $\langle
n_{t_{0}}\rangle<\langle n_{c}^{eq}\rangle$ [cf. Fig. 1(b)]. The first term in
Eq. (19) should monotonically increase with increase of $\tau_{h}$ and
$\tau_{c}$ to reach a maximum value when $\tau_{h}\gg\tau_{h,relax}$ and
$\tau_{c}\gg\tau_{c,relax}$ and $\langle n_{t_{2}}\rangle\rightarrow\langle
n_{h}^{ss}\rangle$ and $\langle n_{t_{0}}\rangle\rightarrow\langle
n_{c}^{eq}\rangle$. We therefore conclude that the total work for the cyclic
heat engine is affected by the irreversibility originating from both two
driven and two isochoric processes.

With consideration of Eqs. (16) and (17), the thermodynamic efficiency,
$\eta_{th}=-\langle\mathrm{w}_{tot}\rangle/\langle{q_{h}}\rangle$, is then
obtained as

$\displaystyle\eta_{th}$ $\displaystyle=1+\frac{w_{c}}{w_{h}}\frac{\langle
n_{t_{0}}\rangle+\langle n_{t_{2}}\rangle(2\xi-1)-2\zeta_{hc}}{\langle
n_{t_{2}}\rangle+\langle n_{t_{0}}\rangle(2\xi-1)-2\zeta_{ch}}.$ (21)

Since the times taken for the thermal contacts and driven strokes are finite,
the quantum coherence and quantum friction are created, resulting that the
efficiency (21) depends not only on the inner friction but also on the
coherence. If quantum coherence is negligible ($\zeta_{ch,hc}\rightarrow 0$),
the efficiency reduces to the oneJiao21 ,

$\eta_{th}=1+\frac{w_{c}}{w_{h}}\frac{\langle n_{t_{0}}\rangle+\langle
n_{t_{2}}\rangle(2\xi-1)}{\langle n_{t_{2}}\rangle+\langle
n_{t_{0}}\rangle(2\xi-1)},$ (22)

for the cycle in which the incomplete thermalization was approximately
described by local thermal equilibrium, and it further simplifies to the one,
$\eta_{th}=1-{w_{c}}/{w_{h}}$, in the case when inner friction is absent
($\xi\rightarrow 0$).

From Eq. (15), we find that the work fluctuations,
$\delta\mathrm{w}^{2}=\langle\mathrm{w}^{2}\rangle-\langle\mathrm{w}\rangle^{2}$
can by given by (see Appendix A)

$\displaystyle\delta\mathrm{w}^{2}$ $\displaystyle=$
$\displaystyle\hbar^{2}w_{h}^{2}\\{\frac{1}{2}-\langle
n_{t_{2}}\rangle^{2}-[\langle n_{t_{0}}\rangle(1-2\xi)+2\zeta_{ch}]^{2}\\}$
(23) $\displaystyle+$ $\displaystyle\hbar^{2}w_{c}^{2}\\{\frac{1}{2}-\langle
n_{t_{0}}\rangle^{2}-[\langle n_{t_{2}}\rangle(1-2\xi)+2\zeta_{hc}]^{2}\\}$
$\displaystyle+$ $\displaystyle\hbar^{2}w_{c}w_{h}\\{2\langle
n_{t_{0}}\rangle[\langle n_{t_{0}}\rangle(1-2\xi)+2\zeta_{ch}]$
$\displaystyle+$ $\displaystyle 2\langle n_{t_{2}}\rangle[\langle
n_{t_{2}}\rangle(1-2\xi)+2\zeta_{hc}]+2\xi-1\\}.$

If $\zeta_{ch}=\zeta_{hc}=0$, this simplifies to
$\delta\mathrm{w}^{2}=\hbar^{2}w_{h}^{2}[\frac{1}{2}-\langle
n_{t_{2}}\rangle^{2}-\langle
n_{t_{0}}\rangle^{2}(1-2\xi)]+\hbar^{2}w_{c}^{2}[\frac{1}{2}-\langle
n_{t_{0}}\rangle^{2}-\langle
n_{t_{2}}\rangle^{2}(1-2\xi)]+\hbar^{2}w_{c}w_{h}[2\langle
n_{t_{0}}\rangle^{2}(1-2\xi)+2\langle n_{t_{2}}\rangle^{2}(1-2\xi)+2\xi-1]$,
previously derived in Ref. Jiao21  where no quantum coherence was considered.

In contrast to the average work, the average efficiency for the quantum Otto
engine may be ill-defined due to the possible divergence of the stochastic
efficiency. This implies that one could not use the efficiency variance (or
efficiency fluctuations) to investigate the efficiency statistics. We thus
resort to large deviation theory Tou09  associated with the exponential decay
of probabilities of large fluctuations, assuming that the quantum engine
proceeds in long-time limit. We recall that the large deviation function of
the joint distribution $p(q_{h},w)$ and the efficiency distribution
$p_{K}(\eta)$ for a large number of cycles ($K\gg 1$) are governed by the
respective asymptotic forms of $p_{K}(q_{h},\mathrm{w}_{tot})\asymp
e^{-KI(q_{h},\mathrm{w}_{tot})},$ and $p_{K}(\eta)\asymp e^{-KJ(\eta)}.$ The
large deviation functions, $I(q_{h},\mathrm{w}_{tot})$ and $J(\eta)$, describe
the exponentially unlikely deviations of $q_{h}$, $\mathrm{w}_{tot}$ and
$\eta$ from their most probable values. The rate function $J(\eta)$ can be
obtained from $I(q_{h},\mathrm{w}_{tot})$ by the contraction:
$J(\eta)=\underset{q_{h}}{\mathrm{min}}I(q_{h},-\eta q_{h}).$ Let us define
$q_{h}^{(K)}:=\sum^{K}_{j=1}q^{(j)}_{h}/K$ and
$\mathrm{w}^{(K)}:=\sum^{K}_{j=1}\mathrm{w}_{tot}^{(j)}/K$

$\displaystyle\phi(\varphi_{1},\varphi_{2})$ $\displaystyle=$
$\displaystyle\underset{K\rightarrow\infty}{\mathrm{lim}}\frac{1}{K}\ln\langle
e^{K(\varphi_{1}q_{h}^{(K)}+\varphi_{2}\mathrm{w}^{(K)})}\rangle$ (24)
$\displaystyle=$ $\displaystyle\ln\langle
e^{\varphi_{1}q_{h}+\varphi_{2}\mathrm{w}_{tot}}\rangle,$

where we have used $\langle
e^{\varphi_{1}q_{h}+\varphi_{2}\mathrm{w}_{tot}}\rangle=\int\int
dq_{h}d\mathrm{w}_{tot}e^{-\varphi_{1}q_{h}-\varphi_{2}\mathrm{w}_{tot}}p(q_{h},\mathrm{w}_{tot}).$
It follows, using the Legendre-Fenchel transform, that the large deviation
function of quantum efficiency readsLut21 ; Gver14

$J(\eta)=-\underset{\varphi_{2}}{\mathrm{min}}\phi(\varphi_{2}\eta,\varphi_{2}),$
(25)

where function $\phi$ was defined in Eq. (24).

##  III Explicit connection between efficiency and entropy production

The Kullback-Leibler-Umegaki (KL) divergence can be used to describe the
distance between an arbitrary state $\rho$ and a reference state
$\rho^{\mathrm{ref}}$:
$D(\rho||\rho^{\mathrm{ref}})=\mathrm{Tr}(\rho\mathrm{ln}\rho)-\mathrm{Tr}(\rho\mathrm{ln}\rho^{\mathrm{ref}})$Vedral02
; Wilde17 . The KL divergence indicates the degree of nonequilibrium of the
state $\rho$ if the reference state $\rho^{\mathrm{ref}}$ is chosen as the
thermal state $\rho^{eq}$. Using ${\Pi_{n}}:=\sum_{n}|n\rangle\langle
n|\rho|n\rangle\langle n|$ to denote the propagators of $\rho^{\mathrm{ref}}$,
the thermal divergence can be rewritten as TM14
$D(\rho||\rho^{\mathrm{ref}})=D(\mathcal{E}(\rho)||\rho^{\mathrm{ref}})+C(\rho)$,
where $\mathcal{E}(\cdot)=\sum_{n}\Pi_{n}(\cdot)\Pi_{n}$ is the dephasing map,
removing all coherence in the energy basis. Here

$C(\rho)=S[\mathcal{E}(\rho)]-S(\rho)~{}(\mathrm{with}~{}S(\rho)=-\mathrm{Tr}[\rho\mathrm{ln}\rho])$
(26)

is the relative entropy of coherence, which quantifies the amount of
coherence.

The total entropy production for the quantum Otto engine can be related to the
KL divergence along the two thermal contacts through
$\Sigma_{tot}=-[\mathrm{Tr}(\rho_{t_{2}}\ln\rho_{h}^{ss})-\mathrm{Tr}(\rho_{t_{1}}\ln\rho_{h}^{ss}]-[\mathrm{Tr}(\rho_{t_{0}}\ln\rho_{c}^{eq})-\mathrm{Tr}(\rho_{t_{3}}\ln\rho_{c}^{eq}]$
(for details see Appendix B), where $\rho_{t_{2}}$ and $\rho_{t_{1}}$
($\rho_{t_{0}}$ and $\rho_{t_{3}}$) are the final and initial instants of the
hot (cold) isochore, respectively. However, this does not mean that the
entropy production does not depend on the coherences generated in the unitary
expansion and compression whatsoever. The states
$\rho_{t_{0},t_{1},t_{2},t_{3}}$ depend on the coherence generated in the two
driven strokes due to finite time driving, and thus the entropy production
$\Sigma_{tot}$ depends on the protocols of these two processes. In terms of
this entropy production $\Sigma_{tot}$, the machine efficiency can be written
as

$\eta_{th}=\eta_{C}^{\mathrm{gen}}-\frac{\Sigma_{tot}}{\beta_{c}\langle
q_{h}\rangle},$ (27)

where $\eta_{C}^{\mathrm{gen}}=1-{\beta_{h}^{\mathrm{eff}}}/{\beta_{c}}$ is
the generalized Cannot efficiency with $\beta_{h}^{\mathrm{eff}}$ defined in
Eq. (8). In the limit of either high temperature or small squeezing, the
effective temperature (8) becomes
$\beta_{h}^{\mathrm{eff}}=\beta_{h}\sinh(2r)$, leading to
$\eta=\sinh(2r)\eta_{C}-{\Sigma_{tot}}/({\beta_{c}\langle q_{h}\rangle})$. Eq.
(27) shows that reservoir squeezing improves the efficiency but the
irreversibility captured by the entropy production deteriorates the
performance. Since the squeezing contributes significantly to the efficiency,
the upper bound of the efficiency may be beyond the standard Carnot value
$\eta_{C}$ even at positive power where the entropy production is finite.
However, the efficiency must be bounded by the generalized Carnot value
$\eta_{C}^{gen}$ due to positive entropy production $\Sigma_{tot}$ and average
heat injection $\langle q_{h}\rangle$, thereby showing that the second law of
thermodynamics holds. The second term in Eq. (27) quantifies the decrease in
the efficiency due to the irreversible entropy production which is dependent
on the protocol of the finite-time cycle.

##  IV Numerical analysis

The energy scales of the system will be set to be compatible with experiments
based on nuclear magnetic resonance setupsSer19 . We choose the energy gaps of
the system along the cold and hot isochores as $\omega_{c}=4\pi\mathrm{kHz}$
and $\omega_{h}=7.2\pi\mathrm{kHz}$, respectively. The inverse temperatures of
the cold and hot baths are set as $\beta_{c}=2/(\hbar\omega_{c})$ and
$\beta_{h}=1/(2\hbar\omega_{h})$. The system-bath interaction strength along
the hot (cold) isochore is assumed as $\gamma_{c}=\gamma_{h}=1\mathrm{Hz}$,
from which the relaxation times are found to be $\tau_{h,relax}=244.92$ms and
$\tau_{c,relax}=761.59$ms Cam19 .

Figure 2:  Coherence at state $\rho_{t_{3}}$ as a function of driving time
$\tau_{ch}$ for different values of $r$. (b) Coherence at states
$\rho_{t_{2}}$ and $\rho_{t_{3}}$ as a function of squeezing parameter $r$ for
$\tau_{ch}=200\mu$s. The time $\tau_{h}$ is set as $\tau_{h}=0.07515$s.

Figure 3: Relative entropy at time $t=t_{2}$ as a function of (a) driving time
$\tau_{dri}$ and (b) thermal-contact time $\tau_{h}$ for different values of
$r$. In (a) $\tau_{h}=0.07515$s and in (b) $\tau_{dri}=200\mu$s. Figure 4:
Power $P$ in unit of J$\cdot$s$/\hbar$ as a function of (a) driving time
$\tau_{dri}$, (b) squeezing parameter $r$, and (c) thermal-contact time
$\tau_{h}$. In (a) and (b) $\tau_{h}=75.15$ms, and in (b)
$\tau_{dri}=460\mu$s. In (c) the cases of $r=0,1,2$ in the dephased engine
cycle (labelled for``deph'') by red dotted line, blue dashed line, and black
dot-dashed line, respectively.

The time spent on the hot isochore is smaller than or of the same order as the
relaxation time, $\tau_{h}\lesssim\tau_{h,relax}$, indicating that the system
could not reach the steady state even at the end of the stroke. The time spent
along the cold isochore is chosen as $\tau_{c}=5$s$\gg\tau_{c,relax}$ which
indicates that $\rho_{t_{0}}$ is close to $\rho_{c}^{eq}$ . For the machine
under consideration, the thermal contact has the time of second scale, while
the time spent on the driven stroke is always smaller than $1$ ms, thereby
indicating that the cycle period,
$\tau_{cyc}=\tau_{h}+\tau_{c}+\tau_{ch}+\tau_{hc}$, would be dominated by
$\tau_{h}$ and $\tau_{c}$.

Figure 5: Thermodynamic efficiency $\eta_{th}$ as a function of (a) driving
time $\tau_{ch}$, (b) squeezing parameter $r$, and (c) thermal-contact time
$\tau_{h}$. The parameters in (a), (b), and (c) are same as in Figs.4 (a), 4
(b), and 4 (c), respectively.

Figure 6: Root-mean-square relative fluctuation of power, $\sqrt{\delta
P^{2}}/P$, as a function of (a) thermal-contact time $\tau_{h}$ and (b)
driving time $\tau_{dri}$. A logarithmic scale is used in the root-mean-square
relative fluctuation (ordinate axis). The parameters in (a) and (b) are the
same as in Fig. 4(a) and 4(b), respectively.  Figure 7: Large deviation
function of efficiency, $J(\eta)$, as a function of stochastic efficiency
$\eta$. The parameters are $\tau_{dri}=460\mu$s, $\tau_{h}=2.8$s. $J(\eta)$ is
situated between a maximum at the generalized Carnot efficiency
$\eta_{C}^{gen}$ and a minimum at the thermodynamic efficiency $\eta_{th}$,
covering the special case when squeezing was absent Ver14 ; Lut21 . Inset:
Large deviation function $J(\eta)$ in the regime near the macroscopic
efficiency. Both $\eta_{th}$ and $\eta_{C}^{gen}$ in Eq. (27) are increased
with increasing $r$ as they should.

In the limit of long driving time $\tau_{dri}$ , no coherence is generated in
the first and third driven strokes, and thus the coherence $C(\rho_{t_{3}})$
should decay as $\tau_{ch}$ increases, as in Fig. 2(a), where the relative
entropies of coherence for different values of squeezing parameter $r$ are
qualitatively similar. The coherence $C(\rho_{t_{3}})$ is always larger than
$C(\rho_{t_{2}})$ [see Fig. 2(b)], and the difference between them is
equivalent to the amount of coherence generated during the third stroke. Both
Figs. 2(a) and 2(b) show that the coherence is a monotonic decreasing function
of squeezing parameter $r$. This can be understood by the fact that, in Eq.
(6) the off-diagonal density matrix elements (determined by $\mathcal{X}$ and
$\mathcal{X}^{*}$ and related to the quantum coherence) monotonically decrease
as $r$ increases.

Coherence indicates how far the system in contact with the squeezed bath
deviates from the stationary state. The squeezed reservoir reducing the
coherence leads to a decrease in the LK divergence at the end of the hot
isochore, as shown in Fig. 3 (a). The coherence generated in the first stroke
is only partially erased after the incomplete thermalization, and the residual
coherence at the instant is thus decreasing as time $\tau_{dri}$ increases.
Therefore, we observe from Fig. 3(a) that $D(\rho_{t_{2}}||\rho_{h}^{ss})$
decreases as time $\tau_{ch}$. Fig. 3(b) shows that, as expected,
$D(\rho_{t_{2}}||\rho_{h}^{ss})$ monotonically decreases as $\tau_{h}$ for
fixed $r$.

Let us consider the average power which reads
$P={(\langle\mathrm{w}_{\mathrm{deph}}\rangle+\langle\mathrm{w}_{\mathrm{coh}}\rangle)}/\tau_{cyc}$,
where $\langle\mathrm{w}_{\mathrm{deph}}\rangle$ and
$\langle\mathrm{w}_{\mathrm{coh}}\rangle$ were defined in Eqs. (19) and (20),
respectively. In Figs. 4(a), 4(b) and 4(c), we plot the average power $P$ as a
function of driving time $\tau_{dri}$, squeezing parameter $r$, and thermal-
contact time $\tau_{h}$, respectively. The power oscillates as a function of
the driving time $\tau_{dri}$, and very quick driving speed results in poor
power output [Fig. 4(a)]. Under the condition that the total cycle period
$\tau_{cyc}$ is dominated by $\tau_{h,c}$, the major contribution of time
$\tau_{dri}$ associated with driving speed to the power is contributed by
quantum inner friction responsible for irreversible work along the two driven
strokes.

Fig. 4(b) shows that the power $P$ increases monotonically with increasing
squeezing parameter $r$. Increasing $r$ leads to a decrease in the so-called
coherent work (20), and thus the total work (18) increases as $r$ increases.
On the other hand, involving squeezing would shorten the total cycle period
$\tau_{cyc}$ by decreasing time $\tau_{h}$ for fixed divergence
$D(\rho_{t_{2}}||\rho_{h}^{ss})$ [Fig.3 (b)].

The thermal-contact-time variation of the power $P$ is sensitively dependent
on the squeezing parameter $r$ (Fig. 4(c)]. We observe that the power first
increases in small $\tau_{h}$ and then decreases with further increase in
$\tau_{h}$ for $r=0$ or $r=1$, but the power decreases monotonically as
$\tau_{h}$ increases for $r=2$. For small and vanishing squeezing ( $r=1,0$),
fast hot isochoric stroke (relaxation process) suppresses the decoherence of
the system. This implies that quick hot isochoric process requires additional,
coherent work $\langle\mathrm{w}_{\mathrm{coh}}\rangle$ compared to the slow
isochoric evolution. $\langle\mathrm{w}_{\mathrm{coh}}\rangle$ decreases
substantially as $\tau_{h}$ increases; the total work increases faster than
linearly with increasing $\tau_{h}$. Therefore, the power increases with
increasing $\tau_{h}$ to a certain maximum value, at which the effects of
coherence are extremely small, and then decreases gradually. The oscillations
in Fig. 4(c) comes from the effect of the dynamical interference between the
residual coherence after the second stroke and the coherence generated in the
third stroke. Interestingly, in the large squeezing case ($r=2$),
$\mathcal{X}$ in Eq. (6) is always negligible and
$\langle\mathrm{w}_{tot}\rangle\rightarrow\langle\mathrm{w}_{\mathrm{deph}}\rangle$
; hence, zero coherence leads to vanishing interference effect and the total
work (18) becomes equivalent to the work in dephased case.

Another benchmark parameter describing the machine performance is the
thermodynamic efficiency $\eta$, in addition to the power $P$. Efficiency as a
function of driving time $\tau_{dri}$ for different values of squeezing
parameter is shown in Fig. 5(a), where the oscillation of $\eta$ comes from
the inner friction captured by $\xi$. The efficiency increases with increasing
driving time, although not monotonically. Fig. 5(a) also shows that the
oscillation amplitude decreases as $r$ increases, with the amplitude in $r=1$
much less than in $r=0$. It is again shown that the thermodynamic efficiency
$\eta_{th}$ (27) increases with increasing $r$.

As $r$ increases, the irreversible work induced by inner friction decreases.
Therefore, increasing $r$ leads to an increase in efficiency [see also Fig.
5(b)] but a decrease in oscillation magnitude of efficiency. In Fig. 5(c), the
efficiency corresponding to the power in Fig. 4(c) is shown. The shapes of the
efficiency and power curves are similar, except that $\eta$ increases with
$\tau_{h}$ to reach its maximum value consistent with Eq. (22). As emphasized,
the effect of the dynamical interference on the thermodynamic efficiency
behaves similarly to the power output [Fig. 4(c)].

To describe the stability of the heat engine, we consider the root-mean-square
relative fluctuation of power, $\sqrt{\delta P^{2}}/P$, which is equivalent to
the coefficient of variation of the work,
$\sqrt{\delta\mathrm{w}^{2}}/\langle\mathrm{w}\rangle$, and measures the
dispersion of the probability distribution. Fig. 6(a) and 6(b) display the
square root of the relative power fluctuations,
$\sqrt{\delta\mathrm{w}^{2}}/\langle\mathrm{w}_{tot}\rangle$, as a function of
thermal-contact time $\tau_{h}$ and driving time $\tau_{dri},$ respectively.
Fig. 6(a) shows that the oscillation timescale of power fluctuation with
respect to the thermal-contact time $\tau_{h}$ agrees with the corresponding
power [4(c)] and efficiency [5(c)]. However, the power fluctuation
$\sqrt{\delta P^{2}}/P$ decreases monotonically while thermal-contact time
$\tau_{h}$ increases. The larger the time of the system-bath interaction
interval is, the closer the system to the stationary state, so the non-
equilibrium thermal fluctuation of the power is expected to decrease [see Fig.
6(a)]. Increasing $r$ decreases the coherence generated in the unitary driven
strokes, leading to decrease in the the magnitude of fluctuation oscillations.
Note that, for fast driving stroke, the relative power fluctuation is
particularly large in the small and vanishing squeezing case, as demonstrated
in Fig. 6(b). This is because for fast driving speed, large irreversible work
induced by both coherence and quantum friction results in small total work
produced by the quantum engine. However, under very large squeezing, the
effect of coherence is trivial [cf. Fig. 4(c)], leading to small relative
power fluctuations. This implies that, the finite-time quantum Otto engines
run more stably in the presence of squeezing than in the absence of squeezing.

Since the average efficiency, $\langle\eta\rangle=\langle w/q_{h}\rangle$, is
ill-defined due to its divergence when the inner friction is present Lut20 ,
we investigate the efficiency statistics in the long-time trajectories by
determining the large deviation function of efficiency (25). We plot the large
deviation function of efficiency for the quantum Otto engine in Fig. 7, where
the curve has a maximum when $\eta=\eta_{C}^{\mathrm{gen}}$
($\eta_{C}^{\mathrm{gen}}$ reduces to $\eta_{C}$ if $r=0$) and a minimum at
$\eta=\eta_{th}$. We therefore find that the standard thermodynamic efficiency
is the most likely value, and the generalized Carnot efficiency is the least
likely. Furthermore, the rate function $J(\eta$) is strictly larger in
presence of squeezing than the case without squeezing, with the exception of
the point at $\eta=\eta_{th}$. This shows that the convergence of the heat
engine towards to the thermodynamic efficiency is improved by including the
reservoir squeezing.

##  V conclusion

Squeezed baths have been employed to fuel models of choice for investigating
the novel performance of the quantum heat engines as compared to their
macroscopic counterparts. Such machines proceeding finite time have been
focused on the optimization on the average power at high temperature limit,
with special emphasis on the efficiency at maximum power. We have here
extended these studies to include the effects of quantum coherence and inner
friction, two essential quantum features. In terms of squeezing parameter, we
have derived compact expressions of the power, thermodynamic efficiency, and
relative power fluctuations for the quantum Otto engine. We have found that
the quantum Otto engine under squeezing can outperform its non-squeezing
counterparts by enhancing efficiency and power output with higher stability.
We have additionally shown that squeezed reservoir leads to engines with
better stability and faster convergence of efficiency to its most probable
value. In addition, the numerical simulation adopted parameters employed in
the recent experiment, which enable us to experience future experimental
realization of high-performance quantum engines with good stability.

###### Acknowledgements.

This work was supported by the NSFC under Grants No. 11875034 and No.
61835013). W.-M. Liu also acknowledges the financial support by the National
Key R$\\!\And\\!$D Program of China under Grants No. 2021YFA1400900,
No.2021YFA0718300, and NO. 2021YFA1400243.

##  Appendix A The out power and heat of the cycle

In this section, we derive the analytical expressions for averages of work and
heat, and work fluctuations. The joint distribution of quantum work
$\mathrm{w}_{tot}$ and heat $q_{h}$ determines the distribution functions of
work and heat respectively. For the two-level system, the ground and excited
eigenenergies are $\varepsilon_{g}^{\alpha}=-\hbar\omega_{\alpha}$ and
$\varepsilon_{e}^{\alpha}=\hbar\omega_{\alpha}$, where $\alpha=c,h$ correspond
to the cold and hot isochores, respectively. It follows, using Eq.(15), the
probability distribution of work and heat can be calculated as

$\displaystyle p(\mathrm{w}_{tot})$ $\displaystyle=$ $\displaystyle\int
dq_{h}p(\mathrm{w}_{tot},q_{h})$ (A.1) $\displaystyle=$
$\displaystyle[p_{t_{0}}^{e}p_{t_{2}}^{e}+p_{t_{0}}^{g}p_{t_{2}}^{g}-2(p_{t_{0}}^{e}p_{t_{2}}^{e}+p_{t_{0}}^{g}p_{t_{2}}^{g})\xi+\xi^{2}]\delta(\mathrm{w})$
$\displaystyle+$ $\displaystyle 2\zeta_{hc}(\xi-
p_{t_{0}}^{g})\delta(\mathrm{w}+\frac{\hbar
w_{h}}{2})+p_{t_{0}}^{e}p_{t_{2}}^{e}\xi^{2}\delta(\mathrm{w}+\hbar
w_{c}+\hbar w_{h})$ $\displaystyle+$ $\displaystyle
p_{t_{0}}^{g}(1-\xi)\xi\delta(\mathrm{w}-\hbar
w_{c})+2p_{t_{0}}^{g}\zeta_{hc}(1-\xi)\delta(\mathrm{w}-\hbar
w_{c}+\frac{\hbar w_{h}}{2})$ $\displaystyle+$ $\displaystyle
p_{t_{2}}^{e}(1-\xi)\xi\delta(\mathrm{w}+\hbar
w_{h})+p_{t_{0}}^{g}p_{t_{2}}^{e}(1-\xi)^{2}\delta(\mathrm{w}-\hbar
w_{c}+\hbar w_{h})$ $\displaystyle+$ $\displaystyle 2\zeta_{ch}(\xi-
p_{t_{2}}^{g})\delta(\mathrm{w}+\frac{\hbar
w_{c}}{2})+4\zeta_{ch}\zeta_{hc}\delta(\mathrm{w}+\frac{\hbar w_{c}+\hbar
w_{h}}{2})$ $\displaystyle+$ $\displaystyle
2\zeta_{ch}(p_{t_{2}}^{e}-\xi)\delta(\mathrm{w}-\frac{\hbar
w_{c}}{2})-4\zeta_{ch}\zeta_{hc}\delta(\mathrm{w}-\frac{\hbar w_{c}-\hbar
w_{h}}{2})$ $\displaystyle-$ $\displaystyle
2p_{t_{2}}^{e}\zeta_{ch}\xi\delta(\mathrm{w}+\frac{\hbar w_{c}}{2}+\hbar
w_{h})+2p_{t_{2}}^{e}\zeta_{ch}(\xi-1)\delta(\mathrm{w}-\frac{\hbar
w_{c}}{2}+\hbar w_{h})$ $\displaystyle+$ $\displaystyle
p_{t_{2}}^{g}(1-\xi)\xi\delta(\mathrm{w}-\hbar
w_{h})+2\zeta_{hc}(p_{t_{0}}^{e}-\xi)\delta(\mathrm{w}-\frac{\hbar w_{h}}{2})$
$\displaystyle+$ $\displaystyle
p_{t_{0}}^{g}p_{t_{2}}^{g}\xi^{2}\delta(\mathrm{w}-\hbar w_{c}-\hbar
w_{\tau_{h}})+2p_{t_{0}}^{g}\zeta_{hc}\xi\delta(\mathrm{w}-\hbar
w_{c}-\frac{\hbar w_{h}}{2})$ $\displaystyle+$ $\displaystyle
2p_{t_{2}}^{g}\zeta_{ch}(1-\xi)\delta(\mathrm{w}-\hbar w_{h}+\frac{\hbar
w_{c}}{2})-4\zeta_{ch}\zeta_{hc}\delta(\mathrm{w}-\frac{\hbar w_{h}-\hbar
w_{c}}{2})$ $\displaystyle+$ $\displaystyle
2p_{t_{2}}^{g}\zeta_{ch}\xi\delta(\mathrm{w}-\frac{\hbar w_{c}}{2}-\hbar
w_{h})+4\zeta_{ch}\zeta_{hc}\delta(\mathrm{w}-\frac{\hbar w_{c}+\hbar
w_{h}}{2})$ $\displaystyle+$ $\displaystyle
p_{t_{0}}^{e}(1-\xi)\xi\delta(\mathrm{w}+\hbar
w_{c})-2p_{t_{0}}^{e}\zeta_{hc}\xi\delta(\mathrm{w}+\hbar w_{c}+\frac{\hbar
w_{h}}{2})$ $\displaystyle+$ $\displaystyle
p_{t_{0}}^{e}p_{t_{2}}^{g}(1-\xi)^{2}\delta(\mathrm{w}-\hbar w_{h}+\hbar
w_{c})+2p_{t_{0}}^{e}\zeta_{hc}(\xi-1)\delta(\mathrm{w}-\frac{\hbar
w_{h}}{2}+\hbar w_{c}),$

and

$\displaystyle p(q_{h})$ $\displaystyle=$ $\displaystyle\int
d\mathrm{w}p(\mathrm{w},q_{h})$ (A.2) $\displaystyle=$
$\displaystyle\\{[p_{t_{0}}^{g}(1-\xi)+p_{t_{0}}^{e}\xi-2\zeta_{ch}]p_{t_{2}}^{g}+[p_{t_{0}}^{e}(1-\xi)+p_{t_{0}}^{g}\xi+2\zeta_{ch}]p_{t_{2}}^{e}\\}\delta(q_{h})$
$\displaystyle+$
$\displaystyle[p_{t_{0}}^{g}(1-\xi)+p_{t_{0}}^{e}\xi-2\zeta_{ch}]p_{t_{2}}^{e}\delta(q_{h}-\hbar
w_{h})+[p_{1}^{e}(1-\xi)+p_{t_{0}}^{g}\xi+2\zeta_{ch}]p_{t_{2}}^{g}\delta(q_{h}+\hbar
w_{h}).$

Here and hereafter we use $p_{t_{0}}^{g}=\langle
g(t_{0})|\rho_{t_{0}}|g(t_{0})\rangle$, $p_{t_{0}}^{e}=\langle
e(t_{0})|\rho_{t_{0}}|e(t_{0})\rangle$, $p_{t_{2}}^{g}=\langle
g(t_{2})|\rho_{t_{2}}|g(t_{2})\rangle$, $p_{t_{2}}^{e}=\langle
e(t_{2})|\rho_{t_{2}}|e(t_{2})\rangle$, $\zeta_{ch}=-Re[U_{ch}^{gg}\langle
g(t_{0})|\rho_{t_{0}}|e(t_{0})\rangle U_{ch}^{eg{\dagger}}]$,
$\zeta_{hc}=-Re[U_{hc}^{gg}\langle g(t_{2})|\rho_{t_{2}}|e(t_{2})\rangle
U_{hc}^{eg{\dagger}}]$. Integrating over the probability distribution function
Eq. (A.1) allows us to find the expressions for the first two central moments
of quantum work, which are

$\displaystyle-\langle\mathrm{w}\rangle$ $\displaystyle=$ $\displaystyle-\int
d\mathrm{w}p(\mathrm{w})\mathrm{w}$ (A.3) $\displaystyle=$
$\displaystyle\hbar(w_{h}-w_{c})(p_{t_{0}}^{g}p_{t_{2}}^{e}-p_{t_{0}}^{e}p_{t_{2}}^{g})-2\hbar
w_{h}\zeta_{ch}-2\hbar w_{c}\zeta_{hc}$ $\displaystyle-$
$\displaystyle[\hbar(w_{c}-w_{h})(p_{t_{0}}^{e}p_{t_{2}}^{g}-p_{t_{0}}^{g}p_{t_{2}}^{e})+\hbar(w_{0}+w_{h})(p_{t_{0}}^{g}p_{t_{2}}^{g}-p_{t_{0}}^{e}p_{t_{2}}^{e})]\xi$
$\displaystyle=$ $\displaystyle\hbar(w_{h}-w_{c})(\langle
n_{t_{2}}\rangle-\langle n_{t_{0}}\rangle)+2\hbar\xi(w_{c}\langle
n_{t_{2}}\rangle+w_{h}\langle n_{t_{0}}\rangle)$ $\displaystyle-$
$\displaystyle 2\hbar w_{h}\zeta_{ch}-2\hbar w_{c}\zeta_{hc},$

and

$\displaystyle\langle\mathrm{w}^{2}\rangle$ $\displaystyle=$
$\displaystyle\int d\mathrm{w}p(\mathrm{w})\mathrm{w}^{2}$ (A.4)
$\displaystyle=$
$\displaystyle\hbar^{2}w_{c}^{2}[p_{t_{0}}^{g}p_{t_{2}}^{e}+p_{t_{0}}^{e}p_{t_{2}}^{g}+(p_{t_{0}}^{e}-p_{t_{0}}^{g})(p_{t_{2}}^{e}-p_{t_{2}}^{g})\xi-2(p_{t_{0}}^{e}-p_{t_{0}}^{g})\zeta_{hc}]$
$\displaystyle+$
$\displaystyle\hbar^{2}w_{h}^{2}[p_{t_{0}}^{g}p_{t_{2}}^{e}+p_{t_{0}}^{e}p_{t_{2}}^{g}+(p_{t_{0}}^{e}-p_{t_{0}}^{g})(p_{t_{2}}^{e}-p_{t_{2}}^{g})\xi-2(p_{t_{2}}^{e}-p_{t_{2}}^{g})\zeta_{ch}]$
$\displaystyle+$
$\displaystyle\hbar^{2}w_{h}w_{c}[2(p_{t_{0}}^{g}p_{t_{2}}^{e}+p_{t_{0}}^{e}p_{t_{2}}^{g})(2\xi-1)+2(p_{t_{0}}^{e}-p_{t_{0}}^{g})(p_{t_{2}}^{e}-p_{t_{2}}^{g})\xi^{2}$
$\displaystyle+$ $\displaystyle
2\zeta_{hc}(p_{t_{0}}^{e}-p_{t_{0}}^{g})+2\zeta_{ch}(p_{t_{2}}^{e}-p_{t_{2}}^{g})-4\zeta_{hc}\xi(p_{t_{0}}^{e}-p_{t_{0}}^{g})-4\zeta_{ch}\xi(p_{t_{2}}^{e}-p_{t_{2}}^{g})+8\zeta_{ch}\zeta_{hc}]$
$\displaystyle=$ $\displaystyle w_{h}^{2}[1/2-2\langle
n_{t_{2}}\rangle(\langle n_{t_{0}}\rangle+2\zeta_{ch}-2\langle
n_{t_{0}}\rangle\xi)]+w_{c}^{2}[1/2-2\langle n_{t_{0}}\rangle(\langle
n_{t_{2}}\rangle+2\zeta_{hc}-2\langle n_{t_{2}}\rangle\xi)]$ $\displaystyle+$
$\displaystyle w_{c}w_{h}[2\xi-1+4\zeta_{ch}\langle
n_{t_{2}}\rangle(1-2\xi)+4\zeta_{hc}\langle n_{t_{0}}\rangle(1-2\xi)+4\langle
n_{t_{0}}\rangle\langle
n_{t_{2}}\rangle(2\xi^{2}+1-2\xi)+8\zeta_{ch}\zeta_{hc}].$

In deriving Eqs.(A.3) and (A.4), we have used, $\langle
n_{t_{0}}\rangle=({p_{t_{0}}^{e}-p_{t_{0}}^{g}})/2$ and $\langle
n_{t_{2}}\rangle=({p_{t_{2}}^{e}-p_{t_{2}}^{g}})/{2}$, which are average
populations of the system at time $t=t_{0}$ and $t=t_{2}$, respectively. By
combining these two equations, we can obtain the work fluctuations,
$\delta\mathrm{w}^{2}=\langle\mathrm{w}^{2}\rangle-\langle\mathrm{w}\rangle^{2}$
[cf. Eq.(23) in the main text]. By very simple algebra, we use Eq. (A.2) to
obtain the macroscopic heat injection, $\langle q_{h}\rangle=\int
dq_{h}q_{h}p(q_{h})=\hbar
w_{h}[p_{t_{0}}^{g}p_{t_{2}}^{e}-p_{t_{0}}^{e}p_{t_{2}}^{g}+(p_{t_{0}}^{e}-p_{t_{0}}^{g})\xi-2\zeta_{ch}]$,
which, in terms of the populations $n_{t_{0}}$ and $n_{t_{2}}$, can be re-
expressed by Eq. (17).

##  Appendix B Efficiency and Entropy production

When the open quantum system evolves along an isochoric process, the external
field is frozen and the control parameter is fixed, leading to static system
Hamiltonian $H(t)=H$. The dynamics of the system with density operator
$\rho_{t}$ is well described by the master equation of Lindblad form MM97 ;
HF02 ; Kos14 ; Wang19 :

$\frac{{d\rho_{t}}}{dt}=-\frac{i}{\hbar}[{H},{\rho_{t}}]+\mathcal{L}_{D}({\rho_{t}}),$
(B.1)

where
$\mathcal{L}_{D}({\rho})=\sum_{\alpha}{L}_{\alpha}\rho{L}_{\alpha}^{\dagger}-\frac{1}{2}\left[{L}_{\alpha}^{\dagger}{L}_{\alpha},{\rho}\right]_{+}$,
with $[~{}~{}]_{+}$ being the anticommutator. The dissipation term
$\mathcal{L}_{D}(\rho)$ is responsible for driving the quantum system to relax
to a unique, positive-definite, steady state with density matrix
$\rho^{eq}_{t}$, given as the solution of $\mathcal{L}_{D}(\rho^{eq}_{t})=0$.
In the two-level system, the Kraus operators ${L}_{\alpha}$ can be written in
terms of the quantum jump operators, namely, $L_{-}=\sigma_{-}\sqrt{N^{th}+1}$
and $L_{+}=\sigma_{+}\sqrt{N^{th}}$, with $N^{th}=1/(e^{\beta\omega}-1)$, and
then Eq.(B.1) reduces to the form given by Eq. (11) in the main part. The
system under consideration would reach equilibrium with the thermal bath if
the system-bath interaction time is long enough. Such a Gibbs thermal state
for the system can be obtained by determining the steady state solution of
differential equation (B.1) to arrive at

$\rho^{eq}_{t}=\rho^{eq}=\frac{e^{-\beta H}}{\mathrm{Tr}\left(e^{-\beta
H}\right)}.$ (B.2)

Let us consider the case of the two-level system interacting with a squeezed
thermal bath of inverse temperature $\beta$. We introduce the Lindblad
operators Kos14 ; Manz16
$\mathcal{S}_{-}=\sigma_{-}\cosh(r)+\sigma_{+}\sinh(r)e^{i\theta}$ and
$\mathcal{S}_{+}=\sigma_{+}\cosh(r)+\sigma_{-}\sinh(r)e^{-i\theta}$ to write
the Kraus operators as $L_{-}^{sq}=\sqrt{N^{th}+1}\mathcal{S}_{-}$ and
$L_{+}^{sq}=\mathcal{S}_{+}\sqrt{N^{th}}$. Equation (4) in the main text can
then be re-expressed in the Lindblad form as

$\frac{{d\rho_{t}}}{dt}=\mathcal{L}^{sq}_{D}(\rho_{t})=\sum_{\alpha\neq\alpha^{\prime}}{L}_{\alpha}^{sq}\rho_{t}{L}_{\alpha^{\prime}}^{sq}-\frac{1}{2}\left[{L}_{\alpha^{\prime}}^{sq}{L}_{\alpha}^{sq},{\rho_{t}}\right]_{+},$
(B.3)

with $\alpha=-,+$. As we show in the main part, we can introduce the effective
inverse temperature of the squeezed bath,
$\beta^{\mathrm{eff}}=\frac{1}{\hbar\omega}\ln\frac{N^{ss}+1}{N^{ss}}$, where
the excitation number can be written in terms of the squeezing parameter $r$,
$N^{ss}=N^{th}+(2N^{th}+1)\sinh^{2}(r).$ Under squeezing, these Lindblad
operators $L_{\pm}^{sq}$ in Eq. (B.3) satisfy the detailed balance condition,
ensuring that the system with frequency $\omega_{t}=\mathrm{const}$ can evolve
asymptotically in a specific way to the unique, steady state $\rho_{t}^{ss}$,
namely, $\mathcal{L}_{D}^{sq}(\rho_{t}^{ss})=0$. The steady-state solution of
Eq. (B.3) for the open system of Hamiltonian $H(t)$ can be easily obtained,

$\rho^{ss}_{t}=\rho^{ss}=\frac{e^{-\beta^{\mathrm{eff}}H(t)}}{\mathrm{Tr}\left(e^{-\beta^{\mathrm{eff}}H(t)}\right)}.$
(B.4)

For a Markovian open quantum system, the irreversiblity can be well described
by the so-called nonadiabatic entropy production rate Par13 ; Bro10 ; Manz16 ,

$\dot{\Sigma}=-\frac{d}{dt}D(\rho_{t}||\rho^{ss}_{t})=\dot{S}_{t}-\dot{\Psi}_{t},$
(B.5)

where
$D(\rho_{t}||\rho^{ss}_{t})=\mathrm{Tr}[\rho_{t}(\ln\rho_{t}-\ln\rho^{ss}_{t})]$
is the quantum relative entropy. Here $\dot{S}_{t}$ can be given by
$\dot{S}_{t}=-\mathrm{Tr}(\dot{\rho}_{t}\ln\rho_{t}+\dot{\rho}_{t})=-\mathrm{Tr}(\dot{\rho}_{t}\ln\rho_{t})$,
and it indicates the rate of change of the system's von Neumann entropy
$S_{t}=-\mathrm{Tr}(\rho_{t}\ln\rho_{t})$. The second contribution
$\dot{\Psi}$, called the excess entropy production rate, is determined by

$\dot{\Psi}=-\mathrm{Tr}(\dot{\rho}_{t}\ln\rho^{eq}_{t}),$ (B.6)

which defines the effective rate of entropy flow into the system from its
surroundings. Under squeezing, the excess entropy production rate can be
expressed as

$\dot{\Psi}^{sq}=-\mathrm{Tr}(\dot{\rho}_{t}\ln\rho_{t}^{ss}).$ (B.7)

We are now in a position to discuss the entropy production for the quantum
Otto engine sketched in Fig. 1. The machine consists of two isentropic
(unitary) compression, and two isochoric heating stroke where $H(t)=H_{h}$ and
cooling stroke with $H(t)=H_{c}$. The working substance returns to its initial
state after a single cycle and thus its entropy change is zero, which means
$\int_{0}^{\tau_{cyc}}\dot{S}(t)dt=0$,
$\tau_{cyc}=\tau_{h}+\tau_{c}+2\tau_{dri}$. We therefore find that, for the
cyclic engine, the nonadiabatic entropy $\Sigma$ is equivalent to the excess
entropy and it is coming exclusively from the two system-bath interaction
intervals. As emphasized, by incorporating the condition of detailed balance
in the Lindblad master equations [c.f. Eq. (B.3) or (B.1)], the working system
in the long time intervals of system-bath interaction should reach the thermal
state $\rho^{eq}$ in the absence of reservoir squeezing or steady state
$\rho^{ss}$ in the presence of squeezing. For the engine cycle, the total
entropy production turns out to be
$\Sigma_{tot}=\int_{0}^{\tau_{cyc}}\dot{\Sigma}dt=-\int_{t_{1}}^{t_{2}}\dot{\Psi}^{sq}dt-\int_{t_{3}}^{\tau_{cyc}}\dot{\Psi}^{sq}dt$,
where $\dot{\Psi}^{sq}$ was defined by Eq . (B.7).

Hence, the irreversible production $\Sigma_{tot}$ of the cycle can be
calculated as
$\Sigma_{tot}=-[\mathrm{Tr}(\rho_{t_{2}}\ln\rho_{h}^{ss})-\mathrm{Tr}(\rho_{t_{1}}\ln\rho_{h}^{ss}]-[\mathrm{Tr}(\rho_{t_{0}}\ln\rho_{c}^{eq})-\mathrm{Tr}(\rho_{t_{3}}\ln\rho_{c}^{eq}]$,
where $\rho_{h}^{ss}$ can be determined according to Eq. (B.4) by setting
$\beta^{\mathrm{eff}}=\beta_{h}^{\mathrm{{eff}}}$ and $H(t)=H_{h}$, and
$\rho_{c}^{eq}$ according to Eq. (B.2) by using $\beta=\beta_{c}$ and
$H(t)=H_{c}$. It then follows that the total entropy production is given by

$\displaystyle\Sigma_{tot}$ $\displaystyle=$
$\displaystyle-\beta_{h}^{\mathrm{eff}}[\mathrm{Tr}(\rho_{t_{2}}H_{h})]-\mathrm{Tr}(\rho_{t_{1}}H_{h})]$
(B.8) $\displaystyle-$
$\displaystyle\beta_{c}[\mathrm{Tr}(\rho_{t_{0}}H_{c})-\mathrm{Tr}(\rho_{t_{3}}H_{c})]$
$\displaystyle=$ $\displaystyle-\beta_{h}^{\mathrm{eff}}\langle
q_{h}\rangle-\beta_{c}\langle q_{c}\rangle,$

where we have used $\langle
q_{h}\rangle=\mathrm{Tr}(\rho_{t_{2}}H_{h})-\mathrm{Tr}(\rho_{t_{1}}H_{h})$
and $\langle
q_{c}\rangle=\mathrm{Tr}(\rho_{t_{0}}H_{c})-\mathrm{Tr}(\rho_{t_{3}}H_{c})$.
Using Eq.(B.8), we derive the machine efficiency as

$\eta=1+\frac{\langle q_{c}\rangle}{\langle
q_{h}\rangle}=1-\frac{\beta_{h}^{\mathrm{eff}}\langle
q_{h}\rangle+\Sigma_{tot}}{\beta_{c}\langle q_{h}\rangle}.$ (B.9)

Introducing the so-called generalized efficiency
$\eta_{C}^{gen}=1-{\beta_{h}^{\mathrm{eff}}}/{\beta_{c}}$, we then obtain Eq.
(27) in the main text.

## References

  * (1) M. O. Scully, K. R. Chapin, K. E. Dorfman, M. B. Kim, and A. Svidzinsky, Quantum heat engine power can be increased by noise-induced coherence, Proc. Natl. Acad. Sci. USA 108, 15097 (2011).
  * (2) R. Kosloff and Y. Rezek, The quantum harmonic Otto cycle, Entropy 19, 136 (2017).
  * (3) W. Niedenzu, V. Mukherjee, A. Ghosh, A. G.Kofman, and G. kurizki, Quantum engine efficiency bound beyond the second law of thermodynamics, Nat. Commun 9, 165 (2018).
  * (4) P. A. Camati, J. F. G. Santos, and R. M. Serra, Coherence effects in the performance of the quantum Otto heat engine, Phys. Rev. A 99, 062103 (2019).
  * (5) A. Winter and D. Yang, Operational Resource Theory of Coherence, Phys. Rev. Lett. 116, 120404 (2016).
  * (6) K. Korzekwa, M. Lostaglio, J. Oppenheim, and D. Jennings, The extraction of work from coherence, New J. Phys. 18, 023045 (2016).
  * (7) K. Brandner, M. Bauer, and U. Seifert, Universal Coherence Induced Power Losses of Quantum Heat Engines in Linear Response, Phys. Rev. Lett. 119, 170602 (2017).
  * (8) S. Rahav, U. Harbola, and S. Mukamel, Heat fluctuations and coherences in a quantum heat engine, Phys. Rev. A 86, 043843 (2012).
  * (9) R. Uzdin, A. Levy, and R. Kosloff, Equivalence of Quantum Heat Machines, and Quantum-Thermodynamic Signatures, Phys. Rev. X 5, 031044 (2015).
  * (10) K. E. Dorfman, D. Z. Xu, and J. S. Cao, Efficiency at maximum power of a laser quantum heat engine enhanced by noise-induced coherence, Phys. Rev. E 97, 042120 (2018).
  * (11) T. Zhang, W. T. Liu, P. X. Chen, and C. Z. Li, Four-level entangled quantum heat engines, Phys. Rev. A 75, 062102 (2007).
  * (12) K. Funo, Y. Watanabe, and M. Ueda, Thermodynamic work gain from entanglement, Phys. Rev. A 88, 052319 (2013).
  * (13) X. L. Huang, J. L. Guo, and X. X. Yi, Nonequilibrium thermal entanglement in a three-qubit XX model, Phys. Rev. E 80, 054301 (2009).
  * (14) Z. H. Wang, W. Wu, and J. Wang, Steady-state entanglement and coherence of two coupled qubits in equilibrium and nonequilibrium environments, Phys. Rev. E, 99, 042320 (2019).
  * (15) J. Oppenheim, M. Horodecki, P. Horodecki, and R. Horodecki, Thermodynamical approach to quantifying quantum correlations, Phys. Rev. Lett. 89, 180402 (2002).
  * (16) L. delRio, J. Åberg, R. Renner, O. Dahlsten, and V. Vedral, The thermodynamic meaning of negative entropy, Nature (London) 474, 61 (2011).
  * (17) J. J. Park, K. H. Kim, T. Sagawa, and S. W. Kim, Heat engine driven by purely quantum information, Phys. Rev. Lett. 111, 230402 (2013).
  * (18) J. M. Horowitz and K. Jacobs, Quantum effects improve the energy efficiency of feedback control, Phys. Rev. E 89, 042134 (2014).
  * (19) K. Brandner, M. Bauer, M. T. Schmid, and U. Seifert, Coherence-enhanced efficiency of feedback-driven quantum engines, New J. Phys. 17, 065006 (2015).
  * (20) P. Kammerlander and J. Anders, Coherence and measurement in quantum thermodynamics, Sci. Rep. 6, 22174 (2016).
  * (21) S. Chand, S. Dasgupta, and Asoka Biswas, Finite-time performance of a single-ion quantum Otto engine, Phys. Rev. E 103, 032144 (2021).
  * (22) Z. Y. Lin, S. H. Su, J. Y. Chen, J. C. Chen, and J. F. G. Santos, Suppressing coherence effects in quantum-measurement-based engines, Phys. Rev. A 104, 062210 (2021).
  * (23) R. J. de Assis, T. M. de Mendonca, C. J. Villas-Boas, A. M. de Souza, R. S. Sarthour, I. S. Oliveira, and N. G. de Almeida, Efficiency of a Quantum Otto Heat Engine Operating under a Reservoir at Effective Negative Temperatures, Phys. Rev. Lett.122, 240602 (2019).
  * (24) J. P. S. Peterson, T. B. Batalhão, M. Herrera, A. M. Souza, R. S.Sarthour, I. S. Oliveira, and R. M. Serra, Experimental Characterization of a Spin Quantum Heat Engine, Phys. Rev. Lett. 123, 240601 (2019).
  * (25) J. Klatzow, J. N. Becker, P. M. Ledingham, C. Weinzetl, K. T. Kaczmarek, D. J. Saunders, J. Nunn, I. A. Walmsley, R. Uzdin, and E. Poem, Experimental Demonstration of Quantum Effects in the Operation of Microscopic Heat Engines, Phys. Rev. Lett. 122, 110601 (2019).
  * (26) T. Baumgratz, M. Cramer, and M. B. Plenio, Quantifying coherence, Phys. Rev. Lett. 113, 140401 (2014).
  * (27) G. Francica, J. Goold, and F. Plastina, Role of coherence in the nonequilibrium thermodynamics of quantum systems, Phys. Rev. E 99, 042105 (2019).
  * (28) A. Streltsov, G. Adesso, and M. B. Plenio, _Colloquium_ : Quantum coherence as a resource, Rev. Mod. Phys. 89, 041003 (2017).
  * (29) X. Yuan, H. Y. Zhou, Z. Cao, and X. F. Ma, Intrinsic randomness as a measure of quantum coherence, Phys. Rev. A 92, 022124 (2015).
  * (30) I. Marvian and R. W. Spekkens, How to quantify coherence: Distinguishing speakable and unspeakable notions, Phys. Rev. A 94, 052324 (2016).
  * (31) A. Smith, Y. Lu, S. M. An, X. Zhang, J. N. Zhang, Z. P. Gong, H. T. Quan, C. Jarzynski, and K. Kim, Verification of the quantum nonequilibrium work relation in the presence of decoherence, New J. Phys. 20, 013008 (2018).
  * (32) R. Kosloff and T. Feldmann, Discrete four-stroke quantum heat engine exploring the origin of friction, Phys. Rev. E 65, 055102(R) (2002).
  * (33) F. Plastina, A. Alecce, T. J. G. Apollaro, G. Falcone, G. Francica, F. Galve, N. Lo Gullo, and R. Zambrini, Irreversible work and inner friction in quantum thermodynamic processes, Phys. Rev. Lett. 113, 260601 (2014).
  * (34) A. Alecce, F. Galve, N. Lo Gullo, L. Dell’Anna, F. Plastina, and R. Zambrini, Quantum Otto cycle with inner friction: finite-time and disorder effects, New J. Phys. 17, 075007 (2015).
  * (35) D. Türkpence, and F. Altintas, Coupled quantum Otto heat engine and refrigerator with inner friction, Quantum Information Processing, 18, 255 (2019).
  * (36) G. Q. Jiao, S. B. Zhu, J. Z. He, Y. L. Ma, J. H. Wang, Fluctuations in irreversible quantum Otto engines, Phys. Rev. E 103, 032130 (2021).
  * (37) O. Abah, J. Roßnagel, G. Jacob, S. Deffner, F. Schmidt-Kaler, K. Singer, and E. Lutz, Single-ion heat engine at maximum power, Phys. Rev. Lett. 109, 203006 (2012).
  * (38) H. T. Quan, P. Zhang, and C. P. Sun, Quantum-classical transition of photon-Carnot engine induced by quantum decoherence, Phys. Rev. E 73, 036122 (2006).
  * (39) R. Dillenschneider and E. Lutz, Energetics of quantum correlations, Europhys. Lett. 88, 50003 (2009).
  * (40) M. N. Bera, A. Riera, M. Lewenstein, and A. Winter, Generalized laws of thermodynamics in the presence of correlations, Nat. Commun. 8, 2180 (2017).
  * (41) G. M. Andolina, M. Keck, A. Mari, M. Campisi, V. Giovannetti, and M. Polini, Extractable Work, the Role of Correlations, and Asymptotic Freedom in Quantum Batteries, Phys. Rev. Lett. 122, 047702 (2019).
  * (42) M. Perarnau-Llobet, K. V. Hovhannisyan, M. Huber, P. Skrzypczyk, N. Brunner, and A. Aíın, Extractable work from correlations, Phys. Rev. X 5, 041011 (2015).
  * (43) L. Buffoni, A. Solfanelli, P. Verrucchi, A. Cuccoli, and M. Campisi, Quantum Measurement Cooling, Phys. Rev. Lett. 122, 070603 (2019).
  * (44) C. Elouard, D. Herrera-Martí, B. Huard, and A. Auffèves, Extracting Work from Quantum Measurement in Maxwell Demon Engines, Phys. Rev. Lett. 118, 260603 (2017).
  * (45) C. Elouard and A. N. Jordan, Efficient quantum measurement engines, Phys. Rev. Lett. 120, 260601 (2018).
  * (46) J. Roßnagel, O. Abah, F. Schmidt-Kaler, K. Singer, and E. Lutz, Nanoscale heat engine beyond the Carnot limit, Phys. Rev. Lett. 112, 030602 (2014).
  * (47) R. Long and W. Liu, Performance of quantum Otto refrigerators with squeezing, Phys. Rev. E 91, 062137 (2015).
  * (48) G. Manzano, F. Galve, R. Zambrini, and J. M. R. Parrondo, Entropy production and thermodynamic power of the squeezed thermal reservoir, Phys. Rev. E 93, 052120 (2016).
  * (49) J. Klaers, S. Faelt, A. Imamoglu, and E. Togan, Squeezed thermal reservoirs as a resource for a nanomechanical engine beyond the carnot limit, Phys. Rev. X 7, 031044 (2017).
  * (50) V. Singh and Özgür E. Müstecaplıoǧlu, Performance bounds of nonadiabatic quantum harmonic Otto engine and refrigerator under a squeezed thermal reservoir, Phys. Rev. E 102, 062123 (2020).
  * (51) Y. N. You and S. W. Li, Entropy dynamics of a dephasing model in a squeezed thermal bath, Phys. Rev. A 97, 012114 (2018).
  * (52) G. Manzano, Squeezed thermal reservoir as a generalized equilibrium reservoir, Phys. Rev. E 98, 042123 (2018).
  * (53) W. Niedenzu, D. Gelbwaser-Klimovsky, A. G.Kofman, and G. kurizki, On the operation of machines powered by quantum non-thermal baths, New. J. Phys. 18, 083012 (2016).
  * (54) J. H. Wang and J. Z. He, Finite-time performance of a quantum heat engine with a squeezed thermal bath, Phys. Rev. E 100, 052126 (2019).
  * (55) R. J. de Assis, J. S. Sales, J. A. R. da Cunha, and N. G. de Almeida, Universal two-level quantum Otto machine under a squeezed reservoir, Phys. Rev. E, 102, 052131 (2020).
  * (56) G. Manzano, F. Galve, R. Zambrini, and J. M. R. Parrondo, Entropy Production and Thermodynamic Power of the Squeezed Thermal Reservoir, Phys. Rev. E 93, 052120 (2016).
  * (57) U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012).
  * (58) K. Sekimoto, Stochastic Energetics (Berlin:Spinger), 2010.
  * (59) Q. Bouton, J. Nettersheim, S. Burgardt , D. Adam , E. Lutz, and A. Widera, A quantum heat engine driven by atomic collisions, Nat. Commun. 12, 2063 (2021).
  * (60) P. Pietzonka and U. Seifert, Universal Trade-Off between Power, Efficiency and Constancy in Steady-State Heat Engines, Phys. Rev. Lett. 120, 190602 (2018).
  * (61) Y. H. Ma, D. Z. Xu, H. Dong, and C. P. Sun, Universal constraint for efficiency and power of a low-dissipation heat engine, Phys. Rev. E 98, 042112 (2018).
  * (62) G. Verley, M.Esposito, T. Willaert and C. Van den Broeck, The unlikely Carnot efficiency, Nat. Commun. 5, 4721 (2014).
  * (63) J. H. Jiang, B. K. Agarwalla, and D. Segal, Efficiency Statistics and Bounds for Systems with Broken Time-Reversal Symmetry, Phys. Rev. Lett. 115, 040601 (2015).
  * (64) T. Denzler and E. Lutz, Efficiency fluctuations of a quantum heat engine, Phys. Rev. Res. 2, 032062 (2020).
  * (65) T. Denzler, E. Lutz, Efficiency large deviation function of quantum heat engines, New. J. Phys. 23, 075003 (2021).
  * (66) P. Solinas and S. Gasparinetti, Full distribution of work done on a quantum system for arbitrary initial states, Phys. Rev. E 92, 042150 (2015).
  * (67) B. M. Xu, J. Zou, L. S. Guo, and X. M. Kong, Effects of quantum coherence on work statistics, Phys. Rev. E 97, 052122 (2018).
  * (68) H. P. Breuer and F. Petruccione,_The Theory of Open Quantum Systems_ (Oxford University Press, New York, 2002).
  * (69) M. O. Scully and M. S. Zubairy, _Quantum Optics_ (Cambridge University Press, Cambridge, 1997).
  * (70) R. Srikanth and S. Banerjee, Squeezed generalized amplitude damping channel, Phys. Rev. A 77, 012318 (2018).
  * (71) V. Holubec and A. Ryabov, Cycling Tames Power Fluctuations Near Optimum Efficiency, Phys. Rev. Lett. 121, 120601 (2018).
  * (72) H.Touchette, The large deviation approach to statistical mechanics, Phys. Rep 478, 1 (2009).
  * (73) G. Verley, T. Willaert, C. Van den Broeck, and M.Esposito, Universal theory of efficiency fluctuations, Phys. Rev. E 90, 052145 (2014).
  * (74) V. Vedral, The role of relative entropy in quantum information theory, Rev. Mod. Phys. 74, 197 (2002).
  * (75) M. M. Wilde, _Quantum Information Theory_ , 2nd ed. (Cambridge University Press, Cambridge, England, 2017).
  * (76) J. M. Horowitz and J. M. R. Parrondo, Entropy production along nonequilibrium quantum jump trajectories, New J. Phys. 15, 085028 (2013).
  * (77) M. Esposito and C. Van den Broeck, Three faces of the second law. I. Master equation formulation, Phys. Rev. E 82, 011143 (2010).