# Quantum Stirling heat engine with squeezed thermal reservoir

Nikolaos Papadatos<EMAIL_ADDRESS>Department of Physics, University
of Patras, 26500 Greece

###### Abstract

We analyze the performance of a quantum Stirling heat engine (QSHE), using a
two level system and the harmonic oscillator as the working medium, that
contacts with a squeezed thermal reservoir and a cold reservoir. First, we
derive closed-form expressions for the produced work and efficiency which
strongly depends on the squeezing parameter $r_{h}$. Then, we prove that the
effect of squeezing heats the working medium to a higher effective temperature
which leads to better overall performance. In particular, the efficiency
increases with the degree of squeezing surpassing the standard Carnot limit,
when the ratio of temperatures of hot and cold reservoir is small.
Furthermore, we derive the analytical expressions for the efficiency at
maximum work and the maximum produced work in the high and low temperature
regime and we find that at extreme temperatures the squeezing parameter
$r_{h}$ does not affect the performance of the QSHE. Finally, the performance
of the QSHE depends on the nature of the working medium.

##  I Introduction

###  I.1 Motivation

Over the past few years, the emerging field of quantum thermodynamics has
attracted an increasing attention to the scientific community, trying to find
the quantum origin of the laws of thermodynamics Anders . Fundamental
thermodynamic variables such as temperature $T$, internal energy $U$, etc.,
have been investigated extensively in terms of classical, statistical, quantum
and relativistic quantum thermodynamics Bimalendu -Naka . Furthermore, the
comprehension of the laws of classical thermodynamics is attainable, in the
context of quantum heat engines (QHEs), comparing thermodynamic quantities
such as work and efficiency in terms of quantum variables.
The QHE was first studied by Scovil and Schulz-DuBois in 1959 using a 3-level
maser Scovil . Thenceforward, the study of performance of QHEs Alickiengine
has paved the way to new fields of research. In particular, the work and
efficiency of QHEs Alicki , Eitan Geva and Ronnie Kosloff  have been analyzed
with respect to the role of coherence, Scully1 -Hammam , the impact of
entanglement Zhangentangled -He  the effect of a many-particle working medium
Beau -Jussiau , the effect of the degeneracy factor in the energy levels Pena
Barrios , fast Otto-cycle Erdmanfast , endoreversible Otto engine with
harmonic oscillator as working medium Deffnerendore , endoreversible engines
Deffnerendore , Smith , Myersndore , an engine based on many body localization
as a working fluid Halpern .

A new era of quantum thermodynamics is the relativistic quantum
thermodynamics, especially in the context of QHEs. In particular, the
construction of a QHE that includes relativistic effects. The following
literature is indicative of the need for more research. The performance of
QHEs has been analyzed with a relativistic particle confined in the one-
dimensional potential well as working substance Pritam , with the
thermodynamical properties of relativistic and quantum fieldsBruschi , with
non-relativistic working medium interacting with a relativistic bath nikos1 ,
with QHEs based on Unruh effect UnruhXu , UnruhGray .

One of the goals of studying quantum machines is to find ways to improve its
overall performance Quantum Afterburner , Abah0 . Studies have emphasized that
Reinforcement-Learning Erdman  and nonequilibrium quantum reservoirs Abah1
such as coherent Quancohere -Hardalcohere  and correlated
Dillenschneidercorrel  can improve the performance of a quantum heat engine.
Additionally, the concept of squeezed thermal reservoir in quantum
thermodynamics has started to create an increasing interest. The quantum Otto
heat engine whose working medium is in contact with squeezed reservoirs has
been studied extensively and presents some very interesting features. In
particular, the efficiency at maximum power increases with the degree of
squeezing, surpassing the standard Carnot limit HuangSQUE -Manzano . In the
real world, the efficiency at maximum power is more important quantity instead
of maximum efficiency, at which power is zero Leff . The efficiency at maximum
power was first studied by Curzon and Ahlborn (CA) Curzon  and they found that
$\eta_{CA}=1-\sqrt{\frac{T_{c}}{T_{h}}}$, where $T_{c}$ and $T_{h}$ is the
temperature of the cold and hot reservoir respectively.

The question arises as to what impact the squeezed thermal bath will have to
another thermodynamic cycle, such as the Stirling thermodynamic cycle that has
received less attention. Stirling thermodynamic cycle has two isothermal and
two isochoric processes Quan  and produces more work than that of the Otto
cycle under the same parameters or equal efficiency conditions and operates a
wider range of coupling strengths akmak .

The aim of this paper is to study the performance of a QHE in which the
working medium interacts with a pure cold reservoir and a squeezed thermal
reservoir Scullyquantumpro -Dung2 . In particular, we analyze the quantum
equivalent of the classical Stirling cycle Chenstirling -Purkait  in which the
hot reservoir is replaced by a thermal squeezed reservoir. We observe that the
squeezing parameter $r_{h}$ improves the performance of QSHE, especially in
the small ratio between hot and cold reservoir which means that the engine has
less energetic cost. On the other hand, the generation of squeezing has a
thermodynamic cost which depends on the specific configuration employed Galve
, Zagoskin . Generally speaking, when the cost of generating a squeezed state
is included in the computation of efficiency of the engine, the squeezed
reservoirs are costly Singh . However, we focus on studying the performance of
QSHE without including the energy losses from the generation of squeezing.

###  I.2 Analysis and results

Our QHE consists of a working medium that is weakly coupled to pure cold
reservoir and a squeezed thermal reservoir on alternate pace according to the
Stirling thermodynamic cycle. We analyse two cases as far as the working
medium is concerned, a two level system and a harmonic oscillator. Our results
are the following:

  1. (i)

We derive closed-form expressions for the produced work and efficiency which
strongly depends on the squeezing parameter $r_{h}$.

  2. (ii)

In both cases of working mediums, the produced work and the efficiency of the
QSHE increases with the degree of squeezing, see Figs. 1-9.

  3. (iii)

The performance of QSHE, i.e., the efficiency and the total produced work
depends on the nature of the working medium, see Figs. 1-9.

  4. (iv)

The effective temperature due to squeezing is always higher than the
temperature of the thermal reservoir.

  5. (v)

We derived the analytic expression for the efficiency at maximum work and the
maximum produced work in the high and low temperature regime.

  6. (vi)

In the low temperature regime, the efficiency of the QSHE with a two level
system as a working medium coincides with the efficiency of quantum Otto heat
engine.

  7. (vii)

At extreme temperatures the squeezing parameter $r_{h}$ does not affect the
performance of the QSHE.

The structure of this paper is the following. In Sec.II we examine the first
case of QSHE with a two level system as a working medium. In Sec.III, we
examine the second case of QSHE with harmonic oscillator as a working medium.
The conclusions are presented in Sec. IV.

##  II Quantum Stirling heat engine - Two level atom

In this section, we use a two level system (TLS) of frequency $\omega$ as the
working medium. The Hamiltonian is
$H=\frac{1}{2}{\leavevmode\hbox{$\hbar$}}\omega\sigma^{z}$. The TLS interacts
with the squeezed thermal reservoir at temperature $T_{h}$, squeeze parameter
$r_{h}$ and the phase $\phi$. The density matrix equation is BrePe07

$\displaystyle\frac{\partial\hat{\rho}}{\partial\tau}=$
$\displaystyle\Gamma[N+1]\left(\hat{\sigma}_{-}\hat{\rho}\hat{\sigma}_{+}-\frac{1}{2}\hat{\sigma}_{+}\hat{\sigma}_{-}\hat{\rho}-\frac{1}{2}\hat{\rho}\hat{\sigma}_{+}\hat{\sigma}_{-}\right)+\Gamma
N\left(\hat{\sigma}_{+}\hat{\rho}\hat{\sigma}_{-}-\frac{1}{2}\hat{\sigma}_{-}\hat{\sigma}_{+}\hat{\rho}-\frac{1}{2}\hat{\rho}\hat{\sigma}_{-}\hat{\sigma}_{+}\right)-$
$\displaystyle-\Gamma M\hat{\sigma}_{+}\hat{\rho}\hat{\sigma}_{+}-\Gamma
M^{*}\hat{\sigma}_{-}\hat{\rho}\hat{\sigma}_{-},$ (1)

where

$\displaystyle N=n\cosh(2r_{h})+\sinh^{2}(r_{h}),$ (2)

and

$\displaystyle M=-\cosh(r_{h})\sinh(r_{h})e^{i\phi}(2n+1),$ (3)

and

$\displaystyle
n=\frac{1}{e^{\frac{{\leavevmode\hbox{$\hbar$}}\omega}{k_{B}T}}-1},$ (4)

is the mean number of quanta. We notice that the asymptotic state does not
depend on the squeezed phase $\phi$.

$\displaystyle\rho=\frac{1}{(2n+1)\cosh(2r)}\bordermatrix{&&\cr&n\cosh(2r)+\sinh^{2}(r)&0\cr&0&n\cosh(2r)+\cosh^{2}(r)\cr},$
(5)

equivalently, we have

$\displaystyle\rho=\frac{1}{(2N+1)}\bordermatrix{&&\cr&N&0\cr&0&N+1\cr}.$ (6)

The expectation value of energy is given by

$\displaystyle U=\langle\hat{E}\rangle=\operatorname{Tr}(\hat{\rho}\hat{H}),$
(7)

thus,

$\displaystyle U=-\frac{{\leavevmode\hbox{$\hbar$}}\omega}{2[2N+1]}.$ (8)

The quantum Stirling heat engine is the quantum counterpart of classical
Stirling thermodynamic cycle. It consists of two isothermal prosesses and two
isochoric processes. In the first case, as the working medium, we have a two
level system that interacts with a hot squeezed reservoir and a cold bath on
alternate pace so as to perform a quantum Stirling thermodynamic cycle.
The general quantum Stirling cycle has the following four repeated cyclically
steps:

  1. (i)

Isothermal compression: $A(\omega_{2},T_{h})\rightarrow B(\omega_{1},T_{h})$.
Initially, the TLS contacts with the hot squeezed reservoir at a constant
temperature $T_{h}$ that can be described by a unitary squeezed operator with
squeezed parameters $r_{h}$ and $\phi$. During this process the mean number of
quanta and the energy gaps must change simultaneously in order for the system
to remain in an equilibrium state. The heat absorbed by the working medium is
$Q_{AB}=T_{h}\Delta S_{AB}$.

  2. (ii)

Isochoric compression: $B(\omega_{1},T_{h})\rightarrow C(\omega_{1},T_{c})$.
In the second stroke, the TLS is disconnected from the hot squeezed bath and
is connected with the cold bath at temperature $T_{c}$ with fixed frequency.

  3. (iii)

Isothermal expansion: $C(\omega_{1},T_{c})\rightarrow D(\omega_{2},T_{c})$. In
the third stroke, the TLS is still coupled with the cold bath and the
frequency increases from $\omega_{1}\rightarrow\omega_{2}$. The heat released
is $Q_{CD}=T_{c}\Delta S_{CD}$.

  4. (iv)

Isochoric expansion: $D(\omega_{2},T_{c})\rightarrow A(\omega_{2},T_{h})$.
Finally, at constant frequency the TLS is disconnected from the cold bath and
is connected with the hot squeezed bath at temperature $T_{h}$ with fixed
frequency.

The entropy for a TLS at thermal equilibrium is

$\displaystyle
S[\hat{\rho}]=-k_{B}\operatorname{Tr}(\hat{\rho}\log\hat{\rho}),$ (9)

thus

$\displaystyle\begin{split}S=&-\frac{1}{(2n+1)\cosh
2r_{h}}k_{B}\Big{(}(\cosh^{2}r_{h}+n\cosh 2r_{h})\log\frac{2n+1+(\cosh
2r_{h})^{-1}}{4n+2}+\\\ +&(n\cosh
2r_{h}+\sinh^{2}r_{h})\log\frac{n+\sinh^{2}r_{h}\operatorname{sech}^{2}2r_{h}}{2n+1}\Big{)}.\end{split}$
(10)

We compute the entropy change in process $AB$

$\displaystyle\begin{split}&S_{AB}=S(B)-S(A)=\\\
=&k_{B}\log\Big{[}\frac{\cosh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{h}}\Big{)}}{\cosh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{h}}\Big{)}}\Big{]}+F_{r_{h}}(\omega_{2})\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2T_{h}}\tanh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{h}}\Big{)}-F_{r_{h}}(\omega_{1})\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2T_{h}}\tanh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{h}}\Big{)}+\\\
&+\frac{1}{2}k_{B}\log\Bigg{[}\frac{1+(1-S^{2}_{r_{h}})\sinh^{2}\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{h}}\Big{)}}{1+(1-S^{2}_{r_{h}})\sinh^{2}\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{h}}\Big{)}}\Bigg{]},\end{split}$
(11)

where

$\displaystyle
F_{r_{h}}(\omega_{i})=S_{r_{h}}\Bigg{\\{}1+\frac{k_{B}T_{h}}{{\leavevmode\hbox{$\hbar$}}\omega_{i}}\log\Bigg{[}\frac{1+S_{r_{h}}+e^{-\frac{{\leavevmode\hbox{$\hbar$}}\omega_{i}}{k_{B}T_{h}}}(1-S_{r_{h}})}{1+S_{r_{h}}+e^{\frac{{\leavevmode\hbox{$\hbar$}}\omega_{i}}{k_{B}T_{h}}}(1-S_{r_{h}})}\Bigg{]}\Bigg{\\}},$
(12)

and

$\displaystyle S_{r_{h}}=\operatorname{sech}(2r_{h}).$ (13)

Consequently, the heat from this process is,

$\displaystyle Q_{AB}=T_{h}\Big{[}S(B)-S(A)\Big{]},$ (14)

from (11) we obtain

$\displaystyle\begin{split}Q_{AB}&=k_{B}T_{h}\log\Big{[}\frac{\cosh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{h}}\Big{)}}{\cosh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{h}}\Big{)}}\Big{]}+F_{r_{h}}(\omega_{2})\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2}\tanh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{h}}\Big{)}-F_{r_{h}}(\omega_{1})\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2}\tanh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{h}}\Big{)}+\\\
&+\frac{1}{2}k_{B}T_{h}\log\Bigg{[}\frac{1+(1-S^{2}_{r_{h}})\sinh^{2}\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{h}}\Big{)}}{1+(1-S^{2}_{r_{h}})\sinh^{2}\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{h}}\Big{)}}\Bigg{]}.\end{split}$
(15)

The heat absorbed in the isothermal process AB using a thermal squeezed
reservoir (15), has the usual form but with coefficients that contains the
squeezing parameter $r_{h}$.
In the limit $r_{h}\rightarrow 0$, Eq 15 takes the form,

$\displaystyle
Q_{AB}=k_{B}T_{h}\log\Big{[}\frac{\cosh\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2T_{h}}}{\cosh\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{h}}}\Big{]}+\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2}\tanh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{h}}\Big{)}-\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2}\tanh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{h}}\Big{)}$
(16)

The work output per cycle according to the first law of thermodynamics
AlickiHistory , Kieu

$\displaystyle Q_{AB}=\Delta U_{AB}+W_{AB}.$ (17)

The expectation value of energy from (8) is:

$\displaystyle\begin{split}\langle\hat{H}\rangle_{A}&=-\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2}S(r_{h})\tanh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{h}}\Big{)},\hskip
8.53581pt\langle\hat{H}\rangle_{B}=-\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2}S(r_{h})\tanh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{h}}\Big{)},\\\
\langle\hat{H}\rangle_{C}&=-\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2}\tanh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{c}}\Big{)},\hskip
36.98866pt\langle\hat{H}\rangle_{D}=-\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2}\tanh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{c}}\Big{)}.\end{split}$
(18)

The expectation value of work is:

$\displaystyle\begin{split}W_{AB}&=k_{B}T_{h}\log\Bigg{[}\frac{\cosh\big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{h}}\big{)}}{\cosh\big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{h}}\big{)}}\Bigg{]}+\frac{1}{2}k_{B}T_{h}\log\Bigg{[}\frac{1+(1-S^{2}_{r_{h}})\sinh^{2}\big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{h}}\big{)}}{1+(1-S^{2}_{r_{h}})\sinh^{2}\big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{h}}\big{)}}\Bigg{]}+\\\
&+G(\omega_{2})\tanh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{h}}\Big{)}-G(\omega_{1})\tanh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{h}}\Big{)},\end{split}$
(19)

where

$\displaystyle
G(\omega_{i})=S_{r_{h}}\frac{k_{B}T_{h}}{{\leavevmode\hbox{$\hbar$}}\omega_{i}}\log\Bigg{[}\frac{1+S_{r_{h}}+e^{-\frac{{\leavevmode\hbox{$\hbar$}}\omega_{i}}{k_{B}T_{h}}}(1-S_{r_{h}})}{1+S_{r_{h}}+e^{\frac{{\leavevmode\hbox{$\hbar$}}\omega_{i}}{k_{B}T_{h}}}(1-S_{r_{h}})}\Bigg{]}$
(20)

Following the same procedure as before we compute the heat and work in the
third stroke

$\displaystyle
Q_{CD}=T_{c}\Big{[}S(D)-S(C)\Big{]}=k_{B}T_{c}\log\Bigg{[}\frac{\cosh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{c}}\Big{)}}{\cosh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{c}}\Big{)}}\Bigg{]}+\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2}\tanh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{c}}\Big{)}-\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2}\tanh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{c}}\Big{)},$
(21)

and

$\displaystyle
W_{CD}=k_{B}T_{c}\log\Bigg{[}\frac{\cosh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{c}}\Big{)}}{\cosh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{c}}\Big{)}}\Bigg{]}.$
(22)

In the quantum isochoric process, no work is done, thus the absorbed or
released heat is:

$\displaystyle
Q_{BC}=U(C)-U(B)=\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2}\Big{[}\operatorname{sech}(2r_{h})\tanh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{h}}\Big{)}-\tanh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{c}}\Big{)}\Big{]},$
(23)

$\displaystyle
Q_{DA}=U(A)-U(D)=\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2}\Big{[}\tanh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{c}}\Big{)}-\operatorname{sech}(2r_{h})\tanh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{h}}\Big{)}\Big{]}.$
(24)

The total work is computed from (19) and (22), $W=W_{AB}+W_{CD}$, thus:

$\displaystyle\begin{split}W=&k_{B}T_{h}\log\Bigg{[}\frac{\cosh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{h}}\Big{)}}{\cosh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{h}}\Big{)}}\Bigg{]}+k_{B}T_{c}\log\Bigg{[}\frac{\cosh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{c}}\Big{)}}{\cosh\Big{(}\frac{\omega_{1}}{2k_{B}T_{c}}\Big{)}}\Bigg{]}+\frac{k_{B}T_{h}}{2}\log\Bigg{[}\frac{1+(1-S^{2}_{r_{h}})\sinh^{2}\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{h}}\Big{)}}{1+(1-S^{2}_{r_{h}})\sinh^{2}\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{h}}\Big{)}}\Bigg{]}+\\\
&+G(\omega_{2})\tanh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{h}}\Big{)}-G(\omega_{1})\tanh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{h}}\Big{)}.\end{split}$
(25)

In Figs. 1 and 2 we plot the produced work $W/k_{B}T_{c}$ of the QSHE as a
function of the ratio of frequencies $\omega_{2}/\omega_{1}$ and temperatures
$T_{h}/T_{C}$ respectively, with different values of the squeezing parameter
$r_{h}$. We notice that the squeezing parameter $r_{h}$ plays a pivotal role
in the work production due to the fact that the work is maximum even if the
ratio of hot and cold temperature is small, meaning less energetic cost.

Figure 1: Total work output $W/k_{B}T_{c}$ as a function of the frequency
modulation $\omega_{2}/\omega_{1}$, for different values of the squeezed
parameter $r_{h}$. Figure 2: Total work output $W/k_{B}T_{c}$ as a function of
the temperature $T_{h}/T_{c}$, for different values of the squeezed parameter
$r_{h}$.

The system absords heat only during the first isothermal stroke and the final
isochoric stroke, thus, the total heat absorbed during a complete cycle

$\displaystyle Q_{H}=Q_{AB}+Q_{DA}.$ (26)

The average heat $\langle\hat{Q}\rangle_{H}\geq 0$ absorbed from the hot
squeezed reservoir and the average heat $\langle\hat{Q}\rangle_{C}\leq 0$
absorbed from the cold thermal reservoir. The mean work
$\langle\hat{W}\rangle_{ab}\geq 0$ expresses the consumed work by the system
and the $\langle\hat{W}\rangle_{cd}\leq 0$ means that the system produces
work.

The efficiency of the QHE is the proportion of the total work per cycle with
respect to the heat absorbed from the hot thermal reservoir:

$\displaystyle\eta_{S}$
$\displaystyle=\frac{\langle\hat{W}\rangle_{total}}{\langle\hat{Q}\rangle_{H}}=1+\frac{Q_{BC}+Q_{CD}}{Q_{AB}+Q_{DA}}.$
(27)

In Fig. 3, we plot the efficiency $\eta_{S}$ of our cycle as a function of the
squeezing parameter $r_{h}$ and compare it with Carnot and Curzon-Ahlborn
efficiency Curzon . We notice that the squeezing parameter $r_{h}$ increases
the efficiency, especially in the small ratio of temperatures, and the
efficiency tends to Curzon-Ahlborn efficiency $\eta_{CA}$, which is the
efficiency at maximum work Broeck . On the other hand, when the ratio of
temperatures $T_{h}/T_{C}$ increases the squeezing parameter can not improve
the performance of QSHE so as to surpass the Carnot efficiency $\eta_{C}$.

Figure 3: Efficiency of the heat engine as a function of the squeezing
parameter $r_{h}$ in comparison to Carnot $\eta_{C}$ and Curzon Ahlborn
$\eta_{CA}$ efficiency. Others parameters are $\omega_{2}=5\omega_{1}$. Figure
4: Efficiency of the heat engine as a function of the temperatures
$T_{c}/T_{h}$ for different ratio of frequencies.

As indicated in Fig. 5 the efficiency of the QSHE increases with the increase
of the ratio $\omega_{2}/\omega_{1}$ and the $r_{h}$.

Figure 5: Efficiency of the heat engine as a function of
$\omega_{2}/\omega_{1}$ and the squeezing parameter $r_{h}$ for
$T_{h}=2T_{C}$.

We highlight that the efficiency of the QSHE with squeezed thermal reservoir
surpasses the standard Carnot limit when the ratio of hot and cold bath is
small. This interesting result is caused when the squeezing parameter
$r_{h}\rightarrow 1$, and its mechanisms can be explained as follows. At the
end of the quantum isochoric process, the asymptotic state of the two level
system is given from (6). When $r_{h}\rightarrow 0$ the asymptotic state of
the two level system is given from,

$\displaystyle\rho=\frac{1}{(2n+1)}\bordermatrix{&&\cr&n&0\cr&0&n+1\cr},$ (28)

where $n$ is given from (4). On the contrary, when $r_{h}>0$, the asymptotic
state of the two level system is given again from (28) but with an effective
temperature that contains the squeezing parameter $r_{h}$ and the temperature
T,

$\displaystyle
T^{eff}=\frac{{\leavevmode\hbox{$\hbar$}}\omega}{2k_{B}\operatorname{arctanh}\Big{[}\frac{\tanh\big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega}{2k_{B}T}\big{)}}{\cosh(2r_{h})}\Big{]}},$
(29)

and $T^{eff}$ is always higher than T due to the squeezed parameters $r_{h}$
HuangEffects .

Let us now, proceed to examine the high temperature limit:
$\frac{\omega}{T_{h}}\rightarrow 0$. The total produced work in the first
order approximation is:

$\displaystyle
W_{ht}=\frac{{\leavevmode\hbox{$\hbar$}}^{2}(\omega_{2}^{2}-\omega_{1}^{2})}{8k_{B}}\Big{[}S_{r_{h}}(S_{r_{h}}-2)\frac{1}{T_{h}}+\frac{1}{T_{C}}\Big{]}.$
(30)

We use the second derivative test with respect to $\omega_{2}$ Abah  in order
to find the maximum value of the produced work. In this case the produced work
does not have a local maximum value.

Next, we compute the efficiency in the high temperature regime in the second
order approximation:

$\displaystyle\eta_{mw}^{ht}=1+\frac{2\frac{\omega_{1}^{2}}{\omega_{2}^{2}}S_{r_{h}}(1-\eta_{C})-1)-(1-\frac{\omega_{1}^{2}}{\omega_{2}^{2}})}{2-(1-\eta_{C}){\leavevmode\hbox{$\hbar$}}S_{r_{h}}\Big{[}2-{\leavevmode\hbox{$\hbar$}}S_{r_{h}}(1-\eta_{C})(1-\frac{\omega_{1}^{2}}{\omega_{2}^{2}})\Big{]}}.$
(31)

Let us now, proceed to examine the low temperature limit:
$\frac{\omega}{T_{C}}\rightarrow\infty$. The total produced work is:

$\displaystyle
W_{lt}=\frac{1}{2}{\leavevmode\hbox{$\hbar$}}(\omega_{2}-\omega_{1})+\frac{{\leavevmode\hbox{$\hbar$}}^{2}(\omega_{2}^{2}-\omega_{1}^{2})}{8k_{B}T_{h}}\Big{[}S_{r_{h}}(S_{r_{h}}-2)\Big{]},$
(32)

where the first and the second term is the first and second order
approximation respectively. The maximum work is when

$\displaystyle\omega_{2}=\frac{2k_{B}T_{h}}{{\leavevmode\hbox{$\hbar$}}}\frac{1}{4-3S_{r_{h}}^{2}}.$
(33)

The efficiency in the low temperature regime is

$\displaystyle\eta_{mw}^{lt}=1-\frac{\omega_{1}}{\omega_{2}},$ (34)

which coincides with the efficiency of quantum Otto heat engine Assis , Assis1
and is independent of the squeezing parameter $r_{h}$ and depends on the ratio
of the frequencies only. The efficiency at maximum work is taken from (33) and
(34).

The second order approximation of the efficiency, which depends on squeezing
parameter $r_{h}$, is:

$\displaystyle\eta_{mw}^{lt}=1-\frac{\omega_{1}^{2}}{\omega_{2}^{2}}\frac{\frac{4k_{B}T_{h}}{\omega_{1}}-2{\leavevmode\hbox{$\hbar$}}S_{r_{h}}}{\frac{4k_{B}T_{h}}{\omega_{2}}-{\leavevmode\hbox{$\hbar$}}S_{r_{h}}\Big{[}2-S_{r_{h}}(1-\frac{\omega_{1}^{2}}{\omega_{2}^{2}})\Big{]}}$
(35)

##  III Quantum Stirling heat engine - Harmonic oscillator

In this section we examine the harmonic oscillator as the working medium of
the QSHE. At first, we refer the thermal squeezed state of the harmonic
oscillator. For a harmonic oscillator of mass $m$ and frequency $\omega$, the
squeezed thermal state results from the application of the squeezing operator
to the thermal equilibrium state. The density matrix is as follows:

$\displaystyle\rho_{sq}=\hat{S}(\xi)\rho_{G}\hat{S}^{\dagger}(\xi)=\hat{S}(\xi)\frac{e^{-\beta\hat{H}}}{Z(\beta)}\hat{S}^{\dagger}(\xi)$
(36)

where $\hat{\rho}_{G}$ is the Gibbs state of the harmonic oscillator, and
$\hat{S}(\xi)$ is the squeezed operator,

$\hat{S}(\xi)=e^{\frac{1}{2}(\xi^{*\alpha^{2}}-\xi\alpha^{\dagger^{2}})},$
(37)

and

$\displaystyle\xi=re^{i\theta},$

is a complex number with $r\geq 0$ called the squeezing parameter with
$\theta\in[0,2\pi]$.
The internal energy of the harmonic oscillator in a squeezed thermal state is
BEI

$\displaystyle\langle\hat{H}\rangle$
$\displaystyle={\leavevmode\hbox{$\hbar$}}\omega\Big{(}n+\frac{1}{2}\Big{)}\cosh(2r_{h}),$
(38)

where

$\displaystyle n$
$\displaystyle=\frac{1}{e^{\frac{{\leavevmode\hbox{$\hbar$}}\omega}{k_{B}T}}-1}.$
(39)

In an analogous way, as in the first part of this work, we construct the four
strokes of the QSHE with the harmonic oscillator as the working medium Yair
Rezek and Ronnie Kosloff , and then we compute the expectation value of energy
of each process, thus:

$\displaystyle\begin{split}\langle\hat{H}\rangle_{A}&=\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2S(r_{h})}\coth\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{h}}\Big{)},\hskip
8.53581pt\langle\hat{H}\rangle_{B}=\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2S(r_{h})}\coth\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{h}}\Big{)},\\\
\langle\hat{H}\rangle_{C}&=\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2}\coth\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{c}}\Big{)},\hskip
24.18501pt\langle\hat{H}\rangle_{D}=\frac{\omega_{2}}{2}\coth\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{c}}\Big{)}.\end{split}$
(40)

Consequently, the heat in the isothermal process $AB$ is:

$\displaystyle\begin{split}Q_{AB}&=k_{B}T_{h}\sum_{i=1}^{2}(-1)^{i+1}\Big{\\{}F_{r_{h}}(\omega_{i})\log[F_{r_{h}}(\omega_{i})]-(F_{r_{h}}(\omega_{i})-1)\log[F_{r_{h}}(\omega_{i})-1]\Big{\\}},\end{split}$
(41)

where

$\displaystyle
F_{r_{h}}(\omega)=\frac{1}{2}\Bigg{[}1+S^{-1}_{r_{h}}\coth\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega}{2k_{B}T_{h}}\Big{)}\Bigg{]}.$
(42)

The expectation value of work in this process is obtained from the first law
of thermodynamics:

$\displaystyle\begin{split}W_{AB}&=k_{B}T_{h}\sum_{i=1}^{2}(-1)^{i+1}\Big{\\{}F_{r_{h}}(\omega_{i})\log[F_{r_{h}}(\omega_{i})]-(F_{r_{h}}(\omega_{i})-1)\log[F_{r_{h}}(\omega_{i})-1]\Big{\\}}+\\\
&+\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2S(r_{h})}\coth\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{h}}\Big{)}-\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2S(r_{h})}\coth\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{h}}\Big{)}.\end{split}$
(43)

In the quantum isochoric processes, no work is done, thus the absorbed or
released heat is:

$\displaystyle
Q_{BC}=U(C)-U(B)=\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2}\Bigg{[}\coth\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{c}}\Big{)}-\frac{1}{S(r_{h})}\coth\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{h}}\Big{)}\Bigg{]}$
(44)

$\displaystyle
Q_{DA}=U(A)-U(D)=\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2}\Bigg{[}\frac{1}{S(r_{h})}\coth\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{h}}\Big{)}-\coth\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{c}}\Big{)}\Bigg{]}.$
(45)

Following the same procedure as before we compute the heat and work in the
third stroke

$\displaystyle
Q_{CD}=T_{c}\Big{[}S(D)-S(C)\Big{]}=k_{B}T_{c}\log\Bigg{[}\frac{\sinh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{c}}\Big{)}}{\sinh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{c}}\Big{)}}\Bigg{]}+\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2}\coth\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{c}}\Big{)}-\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2}\coth\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{c}}\Big{)}.$
(46)

Consequently, the total produced work is:

$\displaystyle\begin{split}W_{total}&=k_{B}T_{h}\sum_{i=1}^{2}(-1)^{i+1}\Big{\\{}F_{r_{h}}(\omega_{i})\log[F_{r_{h}}(\omega_{i})]-(F_{r_{h}}(\omega_{i})-1)\log[F_{r_{h}}(\omega_{i})-1]\Big{\\}}+\\\
+&k_{B}T_{c}\log\Bigg{[}\frac{\sinh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}}{2k_{B}T_{c}}\Big{)}}{\sinh\Big{(}\frac{{\leavevmode\hbox{$\hbar$}}\omega_{2}}{2k_{B}T_{c}}\Big{)}}\Bigg{]}.\end{split}$
(47)

In Figs. 6 and 7, we show that the total produced work increases monotonically
from zero as a function of the frequency ration respectively with the increase
of the squeezing parameter $r_{h}$.

Figure 6: Total work output $W/k_{B}T_{c}$ as a function of the frequency
modulation, $\omega_{2}/\omega_{1}$, for different values of the squeezed
parameter $r_{h}$ and temperature ratio. Figure 7: Total work output,
$W/k_{B}T_{c}$ as a function of temperatures ratio, $T_{h}/T_{C}$, for
different values of the squeezed parameter $r_{h}$ and frequency ratio.

Furthermore, we compute the efficiency of the QSHE

$\displaystyle\eta$
$\displaystyle=\frac{\langle\hat{W}\rangle_{total}}{\langle\hat{Q}\rangle_{H}}=1+\frac{Q_{BC}+Q_{CD}}{Q_{AB}+Q_{DA}}.$
(48)

Due to the fact that the analytical expression of the efficiency of the QSHE
is complicated, we plot the numerical result in Figs. 8 and 9. In Fig. 8 we
plot the efficiency as a function to squeezing parameter $r_{h}$ and compare
it with the Curzon Ahlborn $\eta_{CA}$ efficiency and the efficiency of the
classical Stirling heat engine, which is the same with the Carnot efficiency
$\eta_{C}$. We show that the increase of $r_{h}$ can improve the efficiency of
the QSHE so as to approach and surpass the Carnot limit.

Figure 8: Efficiency $\eta$ as a function of the squeezed parameter $r_{h}$
for different frequencies of the oscillator
$\omega_{1}=k_{B}T_{c}/{\leavevmode\hbox{$\hbar$}}$ in comparison to Carnot
$\eta_{C}$ and Curzon Ahlborn $\eta_{CA}$ efficiency. Other parameters are
$T_{h}=2T_{c}$

In Fig. 9 we plot the efficiency of the QSHE as a function of the squeezing
parameter $r_{h}$ and the ratio of frequencies $\omega_{2}/\omega_{1}$. We
show that the efficiency increases with the increase of
$\omega_{1}=k_{B}T_{c}/{\leavevmode\hbox{$\hbar$}}$ and also with $r_{h}$.

Figure 9: Efficiency of QSHE with respect to $\omega_{2}/\omega_{1}$ and the
squeezing parameter $r_{h}$. The oscillator ’s frequency is
$\omega_{1}=k_{B}T_{c}/{\leavevmode\hbox{$\hbar$}}$.

Let us now, proceed to examine the high temperature limit:
$\frac{\omega}{T_{h}}\rightarrow 0$. The total produced work is:

$\displaystyle
W_{ht}=k_{B}(T_{h}-T_{C})\log\Big{(}\frac{\omega_{2}}{\omega_{1}}\Big{)}+\frac{{\leavevmode\hbox{$\hbar$}}}{12k_{B}T_{h}S_{r_{h}}}(\omega_{2}^{2}-\omega_{1}^{2})+\frac{{\leavevmode\hbox{$\hbar$}}^{2}}{24}(\omega_{1}^{2}-\omega_{2}^{2})\Big{(}\frac{S_{r_{h}}}{k_{B}T_{h}S_{2r_{h}}}+\frac{1}{k_{B}T_{C}}\Big{)}$
(49)

where the first term is the first order approximation and the rest terms are
derived from the second-order approximation. Following the same procedure as
in the first part of this work, we find that the maximum work is when

$\displaystyle\omega_{2}=\frac{2\sqrt{3k_{B}T_{C}\eta_{C}}}{{\leavevmode\hbox{$\hbar$}}\sqrt{1+S_{2r_{h}}^{-1}S_{r_{h}}^{2}(1-\eta_{C})}}.$
(50)

Consequently, the efficiency at maximum work in the high temperature regime
is:

$\displaystyle\eta_{mw}^{ht}=\frac{\eta_{C}}{\eta_{C}-1+S_{r_{h}}^{-1}+\log\Big{(}\frac{\omega_{2}}{\omega_{1}}\Big{)}}\log\Big{(}\frac{\omega_{2}}{\omega_{1}}\Big{)}.$
(51)

Let us now, proceed to examine the low temperature limit:
$\frac{\omega}{T_{C}}\rightarrow\infty$. The total produced work is:

$\displaystyle
W_{lt}=k_{B}T_{h}\log\Big{(}\frac{\omega_{2}}{\omega_{1}}\Big{)}+\frac{1}{2}{\leavevmode\hbox{$\hbar$}}(\omega_{1}-\omega_{2})+\frac{{\leavevmode\hbox{$\hbar$}}}{12}(\omega_{1}^{2}-\omega_{2}^{2})\Big{(}\frac{1}{S_{r_{h}}}-\frac{S_{r_{h}}}{2S_{2r_{h}}}\frac{{\leavevmode\hbox{$\hbar$}}}{k_{B}T_{h}}\Big{)}$
(52)

where the first two terms are the first order approximation.

The maximum work is when

$\displaystyle\omega_{2}=\frac{1+S_{2r_{h}}}{2}\frac{k_{B}T_{h}}{{\leavevmode\hbox{$\hbar$}}}\Big{[}-3+\sqrt{3(11-4S_{r_{h}}^{2})}\Big{]}$
(53)

Consequently, the efficiency at maximum work in the low temperature regime is:

$\displaystyle\eta_{mw}^{lt}=1+\frac{{\leavevmode\hbox{$\hbar$}}\omega_{1}-2k_{B}T_{h}S^{-1}_{r_{h}}}{-{\leavevmode\hbox{$\hbar$}}\omega_{2}+2k_{B}T_{h}[S^{-1}_{r_{h}}+\log\big{(}\frac{\omega_{2}}{\omega_{1}}\big{)}]}$
(54)

##  IV Conclusions

In this paper, we investigated a quantum Stirling heat engine (QSHE) with a
two level system and a harmonic oscillator as the working medium that contacts
with a squeezed thermal reservoir and a cold reservoir. We obtained closed-
form expressions for the produced work and efficiency which depends strongly
on the squeezing parameter $r_{h}$. Then we proved that the effect of
squeezing heats the working medium to a higher effective temperature leading
to a better overall performance. In particular, the efficiency increases with
the degree of squeezing surpassing the standard Carnot limit, when the ratio
of temperatures of hot and cold reservoir is small, which means that the
engine has less energetic cost. Additionally, we derived the analytic
expression for the efficiency at maximum work and the maximum produced work in
the high and low temperature regime. We found that the efficiency in the low
temperature regime coincides with the efficiency of quantum Otto heat engine
and is independent of the squeezing parameter $r_{h}$, depending only on the
ratio of frequencies. We noticed that in the first order approximation the
efficiency at maximum work and the maximum produced work is independent of the
squeezing parameter $r_{h}$. On the contrary, the aforementioned result in the
second order approximation depends on the squeezing parameter $r_{h}$. This
means that at extreme temperatures the squeezing parameter $r_{h}$ does not
affect the performance of the QSHE. Finally, the performance of the QSHE
depends on the nature of the working medium.

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