aainstitutetext: Department of Physics, Kyushu University,
744 Motooka, Nishi-Ku, Fukuoka 819-0395, Japan

# Relativistic quantum Otto engine: Instant work extraction from a quantum
field

Kensuke Gallock-Yoshimura<EMAIL_ADDRESS>
###### Abstract

In this study, we carry out a non-perturbative approach to a quantum Otto
engine, employing an Unruh-DeWitt particle detector to extract work from a
quantum Klein-Gordon field in an arbitrary globally hyperbolic curved
spacetime. We broaden the scope by considering the field in any quasi-free
state, which includes vacuum, thermal, and squeezed states. A key aspect of
our method is the instantaneous interaction between the detector and the
field, which enables a thorough non-perturbative analysis. We demonstrate that
the detector can successfully extract positive work from the quantum Otto
cycle, even when two isochoric processes occur instantaneously, provided the
detector in the second isochoric process receives a signal from the first
interaction. This signaling allows the detector to release heat into the
field, thereby the thermodynamic cycle is completed. As a demonstration, we
consider a detector at rest in flat spacetime and compute the work extracted
from the Minkowski vacuum state.

##  1 Introduction

In the realm of quantum thermodynamics, the exploration of heat engines driven
by quantum systems has gained significant attention over the past few decades.
A particularly important example of quantum heat engines is the quantum Otto
engine (QOE), where a quantum working substance such as a qubit extracts work
from heat baths by operating through two isochoric and adiabatic processes
Scovil.heat.engine.maser ; Feldmann.QOE.2000 ; Kieu.secondlaw.demon.otto ;
Kieu.heat.engine.2006 ; Rostovtsev.Otto.2003 ; Quan.multilevel.engine.2005 ;
Quan.multilevel.heat.engine . The QOE is particularly significant as it
enables separate evaluations of heat and work Bhattacharjee2021 .

While quantum heat engines have been extensively studied within the domain of
quantum mechanics, their exploration in the context of relativistic quantum
mechanics Munoz.QHE.relativistic.dirac.2012 ; Pena.RQM.engine.2016 ;
Purwanto.RQHE.2016 ; YIN201858 ; Chattopadhyay.relativistic.uncertainty.2019 ;
Myers.QOE.relativistic.2021 ; Chattopadhyay.noncommutative.thermo.2021 ;
Sukamto.RQHE.minimal.length.2023  and quantum field theory remains relatively
unexplored. Recent investigations UnruhOttoEngine ; Finn.UnruhOtto  have
delved into QOEs where the working substance is an Unruh-DeWitt (UDW) particle
detector Unruh1979evaporation ; DeWitt1979  interacting with quantum scalar
and spinor fields. In their QOE, the field is assumed to be in the Minkowski
vacuum state, and work is extracted from a thermal bath induced by the Unruh
effect Unruh1979evaporation  experienced by a linearly accelerating detector.
Such a QOE driven by the Unruh effect is particularly called the Unruh QOE.
See also Xu.UnruhOtto.degenerate ; Kane.entangled.Unruh.Otto ;
Barman.entangled.UnruhOtto ; Mukherjee.UnruhOtto.boundary  for subsequent
studies. The Unruh QOE concept extends to scenarios where a UDW detector
follows an arbitrary timelike trajectory in curved spacetime Gallock2023Otto .
We will refer to quantum heat engines that include relativistic effects as
‘relativistic quantum heat engines.’

In both traditional and Unruh QOEs, the working substance is typically assumed
to experience thermality, reflecting the engines’ roots in thermodynamics.
However, it raises the question of what occurs if the baths are in a non-
thermal state Abah.nonequilibrium.2014 . Remarkably, non-thermal baths, such
as quantum coherent Scully.QHE.coherent.2003  and squeezed thermal baths
Huang.squeezing.work ; Rossnagel.Nanoscale.squeezed.QHE.2014 ;
Manzano.squeezed.Otto.2016 ; Niedenzu.squeezed.QOE.2016 ;
Klaers.squeezed.thermal.QOE.2017 , could offer significant advantages.
Therefore, a generalized version of QOE, where the quantum state of reservoirs
is not restricted, is of great interest. In addition, the absence of a
thermality requirement means that a working substance need not interact for a
long time with a bath. As a result, the time dependence of this interaction
can also be arbitrary.

In this context, Ref. Gallock2023Otto  provides a pertinent example. The study
extends the Unruh QOE to a more general relativistic QOE, where the background
geometry, the state of the field (reservoir), the detector’s trajectory, and
the time dependence of the interaction are not specified. Using perturbation
theory, the study has shown that the positive work condition is determined by
the effective temperature observed by the qubit.

Building upon these ideas, this paper examines the relativistic QOE under the
conditions of instantaneous interaction, characterized by delta-coupling. This
approach contrasts with the traditionally assumed prolonged interactions in
QOE literature and allows us to apply a non-perturbative method to our
analysis. Our model assumes an arbitrary globally hyperbolic spacetime and
permits the UDW detector to follow any timelike trajectory. Furthermore, the
state of a quantum scalar field is chosen to be one of the quasi-free states
such as vacuum and thermal states in quantum field theory in curved spacetime
(QFTCS). Since QFTCS is known to have infinitely many unitarily inequivalent
Hilbert space representations, we adopt an algebraic quantum field theoretic
approach to treat all representations on an equal footing.

Our findings reveal that a relativistic QOE with delta-coupling can
successfully extract work from a quantum scalar field, provided the UDW
detector is capable of signal exchange through the quantum field across time.
This means work extraction is contingent on the detector receiving a signal
during the second isochoric process, which was initially sent during the first
interaction. In other words, if such signal exchange is not feasible then it
is impossible to extract work irrespective of the state of the field,
background geometry, or the trajectory of the detector. As an example, we show
that a detector at rest in Minkowski spacetime can extract work from the
Minkowski vacuum due to this signaling effect.

Throughout this paper, we use natural units $\hbar=k_{\text{B}}=c=1$ and the
mostly plus metric convention, and denote a spacetime point by
$\mathsf{x}\equiv(t,{\bm{x}})$.

##  2 Quantum Otto engine: Review

We first review the quantum Otto cycle following the paper by Kieu
Kieu.secondlaw.demon.otto . Here, the working substance is a qubit interacting
with heat baths at temperatures $T_{\text{H}}$ and $T_{\text{C}}$. Figure 1
depicts each step in the cycle. It should be noted that the assumption
regarding the baths being in a thermal state in this context will be replaced
with an arbitrary quasi-free state in Section 5.

Figure 1: A traditional QOE using a qubit interacting with thermal baths at
temperatures $T_{\text{H}}$ and $T_{\text{C}}$.

  1. 1.

Prepare a qubit with energy gap $\Omega_{2}$ in an initial state
$\hat{\rho}_{0}=p\ket{e}\bra{e}+(1-p)\ket{g}\bra{g}$ with $p\in[0,1]$.

  2. 2.

An adiabatic process. The qubit adiabatically (i.e., no heat exchange) expands
the energy gap from $\Omega_{2}$ to $\Omega_{1}$, where
$\Omega_{1}>\Omega_{2}$. The Hamiltonian of the qubit during this process is
$\hat{H}(t)=\Omega(t)\ket{e}\bra{e}$ and so the work done on the qubit is

$\displaystyle\braket{W_{1}}$
$\displaystyle=\int\text{d}t\,\operatorname{Tr}\left[\hat{\rho}_{0}\dfrac{\text{d}\hat{H}(t)}{\text{d}t}\right]=p\Delta\Omega\,,$
(1)

where $\Delta\Omega\coloneqq\Omega_{1}-\Omega_{2}$. The quantum adiabatic
theorem tells us that the state of the qubit, $\hat{\rho}_{0}$, remains the
same during this process. Therefore, the heat flow from the environment to the
qubit is zero:

$\displaystyle\braket{Q_{1}}$
$\displaystyle=\int\text{d}t\,\operatorname{Tr}\left[\dfrac{\text{d}\hat{\rho}_{0}}{\text{d}t}\hat{H}(t)\right]=0\,.$

  3. 3.

An isochoric process. The qubit with a fixed energy gap $\Omega_{1}$ interacts
with a bath at temperature $T_{\text{H}}$. Since the qubit’s Hamiltonian is
time-independent the work done on the qubit is zero: $\braket{W_{2}}=0$.
Meanwhile the state of the qubit changes from $\hat{\rho}_{0}$ to
$\hat{\rho}_{1}=p_{1}\ket{e}\bra{e}+(1-p_{1})\ket{g}\bra{g}$. The first law of
thermodynamics gives the absorbed heat:

$\displaystyle\braket{Q_{2}}$
$\displaystyle=\operatorname{Tr}[\hat{\rho}_{1}\hat{H}]-\operatorname{Tr}[\hat{\rho}_{0}\hat{H}]=\Omega_{1}(p_{1}-p)\,.$
(2)

  4. 4.

Another adiabatic process. Isolate the qubit from the heat bath and contract
the energy gap adiabatically from $\Omega_{1}$ to $\Omega_{2}$. The state
remains to be $\hat{\rho}_{1}$ throughout this process. As in step-2, the heat
and work are

$\displaystyle\braket{Q_{3}}$ $\displaystyle=0\,,$ (3)
$\displaystyle\braket{W_{3}}$ $\displaystyle=-p_{1}\Delta\Omega\,.$ (4)

  5. 5.

Another isochoric process. We put the qubit in a colder bath at temperature
$T_{\text{C}}(<T_{\text{H}})$. At the end of the process, the state becomes
$\hat{\rho}_{2}=p_{2}\ket{e}\bra{e}+(1-p_{2})\ket{g}\bra{g}$, and the heat and
work are

$\displaystyle\braket{Q_{4}}$ $\displaystyle=\Omega_{2}(p_{2}-p_{1})\,,$ (5)
$\displaystyle\braket{W_{4}}$ $\displaystyle=0\,.$ (6)

  6. 6.

Completing a thermodynamic cycle. The final state of the qubit should coincide
with the initial one. Thus we impose $p_{2}=p$. We refer to this as a
cyclicity condition.

From the quantum Otto cycle given above, the total extracted work
$\braket{W_{\text{ext}}}$ and absorbed heat $\braket{Q}$ read

$\displaystyle\braket{W_{\text{ext}}}$
$\displaystyle=-\sum_{j=1}^{4}\braket{W_{j}}=(p_{1}-p)\Delta\Omega(=\braket{Q})\,.$
(7)

Observe that the first law of thermodynamics is satisfied:
$\braket{Q}=\braket{W_{\text{ext}}}$. The positive work condition (assuming
$\Omega_{1}>\Omega_{2}$) reads $p_{1}-p>0$, which turns out to be
$T_{\text{C}}/\Omega_{1}<T_{\text{H}}/\Omega_{2}$ Feldmann.QOE.2000 ;
Kieu.secondlaw.demon.otto . Moreover, the efficiency of the QOE is
$\eta_{\text{O}}\coloneqq\braket{W_{\text{ext}}}/Q_{2}=1-\Omega_{1}/\Omega_{2}$.

We again note that the baths at temperatures $T_{\text{H}}$ and $T_{\text{C}}$
will be replaced with a quantum field in an arbitrary quasi-free state in
Section 5. Moreover, we will employ instantaneous isochoric processes in our
analysis. Thus, our relativistic QOE is characterized as an engine with: (i) a
four-stroke cycle comprising two isochoric and two adiabatic processes, where
the isochoric processes may have arbitrary time-dependence (including
instantaneous interactions) and the adiabatic processes are assumed ideal,
ensuring no heat exchange occurs; (ii) a complete thermodynamic cycle that
returns the qubit to its initial state.

##  3 Quantum field theory in curved spacetime

In this section, we provide a comprehensive review of algebraic quantum field
theory (AQFT) based on Wald:QFTCS ; KayWald1991theorems ;
khavkine2015algebraic ; Hollands.and.Wald2015 ; fewster2020algebraic  as well
as Tjoa.RQC ; Tjoa.nonperturbative.gaussian . Employing AQFT is crucial to
address the inherent challenges of QFTCS, giving rise to many unitarily
inequivalent representations. Unlike the conventional Hilbert space approach
in QFTCS, in which a specific representation is chosen initially, AQFT treats
field observables as elements of an abstract algebra rather than as operators
on a Hilbert space, thereby postponing the selection of a particular
representation. This approach enables us to treat all the unitarily
inequivalent Hilbert spaces on the same footing Wald:QFTCS . Figure 2
schematically depicts the relationships among the concepts in AQFT.

Figure 2: A schematic diagram showing the relationship among mathematical
concepts. A manifold $\mathcal{M}$ is depicted as a spherical object, with a
compact smooth function $f$ indicated in blue. Black arrows represent
mathematical mappings from one space to another, e.g.,
$\phi:C_{0}^{\infty}(\mathcal{M})\to\mathcal{A}(\mathcal{M})$.
$\mathcal{A}^{\star}(\mathcal{M})$ and $\mathcal{L}(\mathcal{D})$ are linear
spaces of algebraic states and linear operators, respectively, where
$\mathcal{D}\subset\mathcal{H}$ is a dense subspace in Hilbert space
$\mathcal{H}$ shown as a dashed box. The correspondence between a
representation $(\mathcal{H},\mathcal{D},\pi)$ and an algebraic state $\omega$
is indicated by green arrows, with one such relationship being derived from
the GNS theorem.

###  3.1 Algebraic quantum field theory

Let $(\mathcal{M},g)$ be an $(n+1)$-dimensional globally hyperbolic spacetime.
In the traditional Hilbert space approach to QFTCS, a free quantum scalar
field $\hat{\phi}$ in this spacetime is considered to be an operator on a
Hilbert space satisfying the Klein-Gordon equation:

$\displaystyle P\hat{\phi}=0\,,\quad P\coloneqq\nabla_{a}\nabla^{a}-m^{2}-\xi
R\,,$ (8)

where $\nabla$ is the Levi-Civita connection with respect to the metric $g$,
$m\geq 0$ is the mass of the field, and $R$ is the Ricci scalar with a
constant $\xi\in\mathbb{R}$. A globally hyperbolic spacetime can be shown to
be foliated by a one-parameter family of Cauchy surfaces, which leads to the
existence of well-defined solutions to the Klein-Gordon equation on
$\mathcal{M}$ Wald:QFTCS .

Let $f\in C_{0}^{\infty}(\mathcal{M})$ be a compactly supported real-valued
smooth function on $\mathcal{M}$. In AQFT, a scalar field is instead regarded
as an $\mathbb{R}$-valued linear map from $C_{0}^{\infty}(\mathcal{M})$ to a
unital $*$-algebra111Here, $*$-algebra $\mathcal{A}$ is an algebra over
$\mathbb{C}$ equipped with an involution $*:\mathcal{A}\to\mathcal{A}$ obeying
(i) anti-linearity: $\forall A,B\in\mathcal{A}$ and
$\forall\lambda_{1},\lambda_{2}\in\mathbb{C}$,
$(\lambda_{1}A+\lambda_{2}B)^{*}=\bar{\lambda}_{1}A^{*}+\bar{\lambda}_{2}B^{*}$,
where $\bar{\lambda}_{i}$ is the complex conjugate of $\lambda_{i}$; (ii)
$\forall A,B\in\mathcal{A}$, $(AB)^{*}=B^{*}A^{*}$; (iii) $\forall
A\in\mathcal{A}$, $(A^{*})^{*}=A$. In particular, a $*$-algebra is called
unital if a multiplicative identity $\mathds{1}\in\mathcal{A}$ is defined.
known as the canonical commutation relation (CCR) algebra:

$\displaystyle\phi:C_{0}^{\infty}(\mathcal{M})\to\mathcal{A}(\mathcal{M})\,,\quad
f\mapsto\phi(f)\,.$ (9)

The CCR algebra $\mathcal{A}(\mathcal{M})$ is generated by smeared field
$\phi(f)$ obeying the following relations khavkine2015algebraic , which we
refer to as the Klein-Gordon relations:

  * •

Linearity: $\phi(af+bf^{\prime})=a\phi(f)+b\phi(f^{\prime})$ for all
$a,b\in\mathbb{R}$ and $f,f^{\prime}\in C_{0}^{\infty}(\mathcal{M})$;

  * •

Hermiticity: $\phi(f)^{*}=\phi(f)$ for all $f\in C_{0}^{\infty}(\mathcal{M})$;

  * •

Klein-Gordon equation: $\phi(Pf)=0$ for all $f\in
C_{0}^{\infty}(\mathcal{M})$;

  * •

Commutation relation:
$[\phi(f),\phi(f^{\prime})]=\mathsf{i}E(f,f^{\prime})\mathds{1}$ for all
$f,f^{\prime}\in C_{0}^{\infty}(\mathcal{M})$;

Here, $E(f,f^{\prime})$ is known as the causal propagator, which can be
defined as follows. Let $E^{+}$ and $E^{-}$ be the retarded and advanced Green
operators on $C_{0}^{\infty}(\mathcal{M})$ associated with the Klein-Gordon
equation, i.e., $E^{\pm}:C_{0}^{\infty}(\mathcal{M})\to
C^{\infty}(\mathcal{M})$ that solve the inhomogeneous Klein-Gordon equation:
$P(E^{\pm}f)=f$. The causal propagator operator is the advanced-minus-retarded
operator $E\coloneqq E^{-}-E^{+}$ on $C_{0}^{\infty}(\mathcal{M})$, which
solves the Klein-Gordon equation:

$\displaystyle
E:C_{0}^{\infty}(\mathcal{M})\to\text{Sol}_{\mathbb{R}}(\mathcal{M})\,,\quad
P(Ef)=0\,,$ (10)

where $\text{Sol}_{\mathbb{R}}(\mathcal{M})$ is a vector space of real
solutions to the Klein-Gordon equation. The causal propagator
$E(f,f^{\prime})\in\mathbb{R}$ can be understood as a smeared real solutions:

$\displaystyle E(f,f^{\prime})$
$\displaystyle\coloneqq\int_{\mathcal{M}}\text{dvol}_{g}\,f(\mathsf{x})(Ef^{\prime})(\mathsf{x})\quad
f,f^{\prime}\in C_{0}^{\infty}(\mathcal{M})\,,$ (11)

where $\text{dvol}_{g}\equiv\sqrt{-\det g}\,\text{d}^{n+1}\mathsf{x}$ is a
volume element.

We note that the field $\phi(f)$ is not an operator that acts on a Hilbert
space, but rather it is an abstract entity. In fact, the aforementioned
definition of $\phi(f)$ is an abstraction of the smeared field operator in
QFTCS:

$\displaystyle\hat{\phi}(f)$
$\displaystyle=\int_{\mathcal{M}}\text{dvol}_{g}\,f(\mathsf{x})\hat{\phi}(\mathsf{x})\,.$
(12)

We will see that $\phi(f)$ can be related to the operator $\hat{\phi}(f)$ on a
Hilbert space when a representation is introduced.

We now define an algebraic state $\omega$, which corresponds to an expectation
value of an observable. Formally, the algebraic state is defined as a linear
functional $\omega:\mathcal{A}(\mathcal{M})\to\mathbb{C}$ satisfying (i) the
normalization condition: $\omega(\mathds{1})=1$; and (ii) the positivity
condition: $\forall A\in\mathcal{A}(\mathcal{M})$, $\omega(A^{*}A)\geq 0$. We
will write a set of algebraic states as $\mathcal{A}^{\star}(\mathcal{M})$.
For any two distinct algebraic states $\omega_{1}$ and $\omega_{2}$
($\omega_{1}\neq\omega_{2}$), one can form a new state $\omega$ as a convex
linear combination of them: $\omega=\lambda\omega_{1}+(1-\lambda)\omega_{2}$
for $\lambda\in(0,1)$. If this is the case, $\omega$ is called mixed.
Otherwise, if an algebraic state $\omega$ cannot be written in this form (thus
it is an extreme point in $\mathcal{A}^{\star}(\mathcal{M})$) then it is
called pure.

The algebraic state $\omega$ is related to the states in Hilbert space. To see
this, we need to introduce a concept of representation of an algebra.

Let us define a $*$-representation of $\mathcal{A}(\mathcal{M})$ as a triple
$(\mathcal{H},\mathcal{D},\pi)$, where $\mathcal{H}$ is Hilbert space,
$\mathcal{D}\subset\mathcal{H}$ is a dense subspace of $\mathcal{H}$, and
$\pi$ (also refer to as a representation) is a linear homomorphism,

$\displaystyle\pi:\mathcal{A}(\mathcal{M})\to\mathcal{L}(\mathcal{D})\,,\quad
A\mapsto\pi(A)\,,$ (13)

where $\mathcal{L}(\mathcal{D})$ is the linear space of linear operators that
acts on $\mathcal{D}$. Here, an involution $*$ is related to a Hermitian
adjoint operation in $\mathcal{L}(\mathcal{D})$ in such a way that

$\displaystyle\pi(A)^{\dagger}|_{\mathcal{D}}=\pi(A^{*})\quad\forall
A\in\mathcal{A}(\mathcal{M})\,.$ (14)

For a unital $*$-algebra, we have $\pi(\mathds{1})=\mathds{1}|_{\mathcal{D}}$.
The field operator on a Hilbert space is then expressed as
$\hat{\phi}(f)\equiv\pi(\phi(f))$.

Given a representation $(\mathcal{H},\mathcal{D},\pi)$, each state vector
$\ket{\psi}\in\mathcal{H}$ can be related to $\omega$ in such a way that
Hollands.and.Wald2015

$\displaystyle\omega_{\psi}(A)$
$\displaystyle=\dfrac{\braket{\psi}{\pi(A)}{\psi}}{\braket{\psi}{\psi}}\,,$
(15)

which can also be applied to
$\omega_{\rho}(A)=\operatorname{Tr}[\hat{\rho}\pi(A)]$ for a given density
matrix $\hat{\rho}$. A collection of such algebraic states associated to each
$\hat{\rho}$,
$S_{\pi}(\mathcal{A}(\mathcal{M}))\equiv\\{\omega_{\rho}\\}\subset\mathcal{A}^{\star}(\mathcal{M})$,
for a given representation $\pi$ is called a folium of $\pi$
hollands2017entanglement ,222Other literatures often use the terminology
‘folium of $\omega$’, which is equivalent to ‘folium of $\pi_{\omega}$ in the
GNS representation of $\omega$’ in our case. and each element
$\omega_{\rho}\in S_{\pi}(\mathcal{A}(\mathcal{M}))$ is referred to as a
normal (algebraic) state in the representation $\pi$.

Conversely, for each algebraic state $\omega$, there exists a corresponding
unique (up to unitary transformation) quadruple
$(\mathcal{H}_{\omega},\mathcal{D}_{\omega},\pi_{\omega},\ket{\Omega_{\omega}})$
satisfying

$\displaystyle\omega(A)$
$\displaystyle=\braket{\Omega_{\omega}}{\pi_{\omega}(A)}{\Omega_{\omega}}\quad\forall
A\in\mathcal{A}(\mathcal{M})\,,$ (16)

as well as
$\pi_{\omega}(\mathcal{A}(\mathcal{M}))\ket{\Omega_{\omega}}=\mathcal{D}_{\omega}$.
The formation of
$(\mathcal{H}_{\omega},\mathcal{D}_{\omega},\pi_{\omega},\ket{\Omega_{\omega}})$
is known as the Gelfand-Naimark-Segal (GNS) construction.

This correspondence between the algebraic states and the state vectors defined
in a Hilbert space tells us that AQFT and the traditional Hilbert space
approach of QFTCS are essentially equivalent. However, as we previously
mentioned, AQFT differs in that it does not necessitate specifying a Hilbert
space, or equivalently, selecting a representation $\pi$, until the last step.

So far, we have introduced a CCR algebra and its $*$-representation. However,
the operators $\pi(A)\in\mathcal{L}(\mathcal{D})$ are generally unbounded,
which can lead to certain challenges such as domain issues. To overcome such
challenges, we use an exponentiated version of $\mathcal{A}(\mathcal{M})$
called the Weyl algebra.

The Weyl algebra $\mathcal{W}(\mathcal{M})$ is a unital $C^{*}$-algebra
generated by $W(Ef)\equiv e^{\mathsf{i}\phi(f)}$ obeying the Weyl relations:

$\displaystyle W(Ef)^{*}=W(-Ef)\,,$ (17a) $\displaystyle
W(Ef_{1})W(Ef_{2})=e^{-\mathsf{i}E(f_{1},f_{2})/2}W(E(f_{1}+f_{2}))\,.$ (17b)

We note that $W(0)=\mathds{1}$ is the identity element of
$\mathcal{W}(\mathcal{M})$. Then, the notion of the algebraic state $\omega$
carries over as $\omega:\mathcal{W}(\mathcal{M})\to\mathbb{C}$ (we denote the
linear space of $\omega$ by $\mathcal{W}^{\star}(\mathcal{M})$). Moreover, a
representation of $C^{*}$-algebra provides a set of bounded linear operators
$\mathcal{B}(\mathcal{H})$ on a Hilbert space $\mathcal{H}$.

###  3.2 Quasi-free states

The Wightman $n$-point function is an essential quantity in QFT. For $f_{j}\in
C_{0}^{\infty}(\mathcal{M})$ and $\phi(f_{j})\in\mathcal{A}(\mathcal{M})$, the
Wightman $n$-point function in an algebraic state $\omega$ is defined as

$\displaystyle\mathsf{W}(f_{1},f_{2},...,f_{n})\coloneqq\omega(\phi(f_{1})\phi(f_{2})...\phi(f_{n}))\,.$
(18)

We now restrict ourselves to a class of physically admissible algebraic states
known as Hadamard states Hollands.and.Wald2015 ; KayWald1991theorems .
Hadamard algebraic states are characterized by their Wightman $n$-point
functions, which, at short distances, exhibit behaviors similar to those in
the Minkowski vacuum state in Minkowski spacetime. A subset of these Hadamard
states allows for the expression of $n$-point functions using only one- and
two-point Wightman functions, which we refer to as Gaussian states. Of
particular interest are the quasi-free states, a subset of Gaussian states,
where the one-point Wightman functions vanish: $\omega(\phi(f))=0$.333In the
literature, quasi-free states are also called Gaussian states. Our use of
’quasi-free’ specifically refers to ’quasi-free states with vanishing one-
point Wightman function.’ KayWald1991theorems  In these quasi-free algebraic
states, every even-point Wightman function can be expressed only in terms of
the two-point Wightman functions $\omega(\phi(f_{1})\phi(f_{2}))$, while all
odd-point functions vanish. For example, while a coherent state is a non-
quasi-free Gaussian state, states like vacuum, squeezed, and thermal Kubo-
Martin-Schwinger (KMS) states fall under the quasi-free type.

The properties above also hold for the Weyl algebra
$\mathcal{W}(\mathcal{M})$. In this case, the two-point Wightman function is
understood as KayWald1991theorems

$\displaystyle\mathsf{W}(f_{1},f_{2})$
$\displaystyle=-\dfrac{\partial^{2}}{\partial s\partial
t}\bigg{|}_{s,t=0}\omega(W(sEf_{1})W(tEf_{2}))\,,$ (19)

and if $\omega:\mathcal{W}(\mathcal{M})\to\mathbb{C}$ is a quasi-free state,
then we have

$\displaystyle\omega(W(Ef))$
$\displaystyle=e^{-\frac{1}{2}\mathsf{W}(f,f)}\,.$ (20)

In what follows, we will first consider a general algebraic state $\omega$ and
then impose (20) at the end to classify $\omega$ as a quasi-free state. It
will be useful to decompose the two-point Wightman function into its real and
imaginary parts as

$\displaystyle\mathsf{W}(f_{1},f_{2})$
$\displaystyle=\text{Re}[\mathsf{W}(f_{1},f_{2})]+\mathsf{i}\,\text{Im}[\mathsf{W}(f_{1},f_{2})]\equiv\mu(Ef_{1},Ef_{2})+\dfrac{\mathsf{i}}{2}E(f_{1},f_{2})\,,$
(21)

where $\mu(Ef_{1},Ef_{2})\in\mathbb{R}$ originates from an inner product of
solutions to the Klein-Gordon equation (see Wald:QFTCS ; KayWald1991theorems
for details), and $E(f_{1},f_{2})\in\mathbb{R}$ is the causal propagator given
in (11).

##  4 Unruh-DeWitt Detector Model

We now introduce our qubit model in the quantum Otto cycle. The qubit model we
employ here is called the Unruh-DeWitt (UDW) particle detector model
Unruh1979evaporation ; DeWitt1979 . A UDW detector has ground $\ket{g}$ and
excited $\ket{e}$ states and their energy gap is denoted by $\Omega$. The free
Hamiltonian $\hat{H}_{\text{D,0}}$ is given by

$\displaystyle\hat{H}_{\text{D,0}}$
$\displaystyle=\dfrac{\Omega}{2}(\hat{\sigma}_{z}+\mathds{1}_{\text{D}})\,,$
(22)

where $\hat{\sigma}_{z}$ is the Pauli-$z$ operator and $\mathds{1}_{\text{D}}$
is the identity operator on the detector’s Hilbert space.

We then let a detector interact with the field $\hat{\phi}$ in a spacetime
region specified by $f\in C_{0}^{\infty}(\mathcal{M})$ in the following
manner. In the interaction picture, the time-evolution unitary operator
$\hat{U}_{\text{I}}$ is given by

$\displaystyle\hat{U}_{\text{I}}$
$\displaystyle=\mathcal{T}_{\tau}\exp\left(-\mathsf{i}\int_{\mathcal{M}}\text{dvol}_{g}\,\hat{h}_{\text{I}}(\mathsf{x})\right)\,,\quad\hat{h}_{\text{I}}(\mathsf{x})\coloneqq
f(\mathsf{x})\hat{\mu}(\tau(\mathsf{x}))\otimes\hat{\phi}(\mathsf{x})$ (23)

where $\mathcal{T}_{\tau}$ is a time-ordering symbol with respect to the
proper time $\tau$ of the detector, and $\hat{\mu}(\tau(\mathsf{x}))$ is a
monopole moment operator given by

$\displaystyle\hat{\mu}(\tau(\mathsf{x}))$
$\displaystyle=e^{\mathsf{i}\Omega\tau(\mathsf{x})}\ket{e}\bra{g}+e^{-\mathsf{i}\Omega\tau(\mathsf{x})}\ket{g}\bra{e}\,.$
(24)

As mentioned earlier, $f\in C_{0}^{\infty}(\mathcal{M})$ is the interaction
region between the detector and the field. Roughly speaking, one can think of
the spatial extension of $f$ on each time slice as the ‘shape’ of the
detector, and its timelike extension as the ‘interaction duration.’ In fact,
it is possible to separate these parts in $f$ by adopting a suitable
coordinate system called the Fermi normal coordinates
$\bar{\mathsf{x}}=(\tau,\bar{\bm{x}})$, which is adapted to the trajectory of
the detector’s center-of-mass. In this coordinate system,
$\bar{\bm{x}}=\bm{0}$ is defined to be the position of the detector’s center-
of-mass, and $\tau$ is its proper time. Then the interaction region $f$ can be
decomposed into

$\displaystyle f(\mathsf{x}(\bar{\mathsf{x}}))$
$\displaystyle=\lambda\chi(\tau/\eta)F(\bar{\bm{x}})\,,$ (25)

where $\lambda\geq 0$ is the coupling constant, $\chi(\tau/\eta)$ is a
switching function (with $\eta>0$ a quantity in units of time) describing the
time-dependence of interaction, and $F(\bar{\bm{x}})$ is a smearing function
representing the shape of the detector.

It is well-known that the existence of the time-ordering symbol in (23)
prevents us from obtaining an exact solution. As a result, one might consider
employing a perturbative analysis for further exploration. In this paper,
however, we will specifically focus on a special class of $\hat{h}_{\text{I}}$
known as delta-coupling Simidzija.Nonperturbative ;
Funai.negative.energy.delta ; Simidzija2018no-go ; Henderson.Bandlimited ;
SahuSabotage ; GallockEntangledDetectors ; Perche.curvature.delta ; Tjoa.RQC ;
Shalabi.locally.UV.delta ; Tjoa.nonperturbative.gaussian ; Jose.tran.delta  to
carry out a non-perturbative analysis. The delta-coupling approach is
characterized by an interaction time scale that is significantly shorter than
the typical time scale relevant to the system under study. The switching
function in the Fermi normal coordinates in the delta-coupling approach is
given by

$\displaystyle\chi((\tau-\tau_{0})/\eta)$
$\displaystyle=\eta\delta(\tau-\tau_{0})\,,$ (26)

where $\delta$ is the Dirac delta distribution, $\tau_{0}\in\mathbb{R}$ is the
time when the detector switches on and off instantaneously, and we used the
identity $\delta(\tau/\eta)=\eta\delta(\tau)$. Since it only deals with an
instant time, the time-ordering will not contribute to the unitary operator in
(23), thereby the resulting $\hat{U}_{\text{I}}$ reads Jose.tran.delta

$\displaystyle\hat{U}_{\text{I}}$
$\displaystyle=\exp\left[-\mathsf{i}\hat{\mu}(\tau_{0})\otimes\hat{\phi}(\mathfrak{f})\right]\,,$
(27)

where

$\displaystyle\hat{\phi}(\mathfrak{f})\coloneqq\int_{\Sigma_{\tau_{0}}}\text{d}\Sigma\,\lambda\eta
F(\bar{\bm{x}})\hat{\phi}(\tau_{0},\bar{\bm{x}})$ (28)

with $\text{d}\Sigma$ being a volume element on a time slice
$\Sigma_{\tau_{0}}$ at $\tau=\tau_{0}$. The function $\mathfrak{f}\in
C_{0}^{\infty}(\Sigma_{\tau_{0}})$ can be thought of as a spatial version of
$f\in C_{0}^{\infty}(\mathcal{M})$. The unitary operator of the form (27) is a
particular example of the so-called Schmidt rank-1 operators.444A Schmidt
rank-$r$ unitary operator $\hat{U}$ on a bipartite system
$\mathcal{H}_{\text{A}}\otimes\mathcal{H}_{\text{B}}$ is defined as
$\hat{U}=\sum_{i=1}^{r}c_{i}\hat{A}_{i}\otimes\hat{B}_{i}$ with $\forall
c_{i}\neq 0$, where $\\{\hat{A}_{i}\\}$ and $\\{\hat{B}_{i}\\}$ are linearly
independent operators. Using a property
$\hat{\mu}(\tau_{0})^{2}=\mathds{1}_{\text{D}}$, Eq. (27) can be expressed as

$\displaystyle\hat{U}_{\text{I}}$
$\displaystyle=\mathfrak{\mathds{1}}_{\text{D}}\otimes\cos(\hat{\phi}(\mathfrak{f}))-\mathsf{i}\hat{\mu}(\tau_{0})\otimes\sin(\hat{\phi}(\mathfrak{f}))\,.$
(29)

In the quantum Otto cycle, the detector interacts with the field twice, as
depicted in Figure 3. In this scenario, the unitary operator becomes

$\displaystyle\hat{U}_{\text{I}}$
$\displaystyle=\mathcal{T}_{\tau}\exp\left(-\mathsf{i}\int_{\mathcal{M}}\text{dvol}_{g}\,\left[\hat{h}_{\text{I,1}}(\mathsf{x})+\hat{h}_{\text{I,2}}(\mathsf{x})\right]\right)\,,$
(30)

where $\hat{h}_{\text{I,$j$}}(\mathsf{x})$ $j\in\\{1,2\\}$ corresponds to the
first ($j=1$) and the second ($j=2$) interaction between the field and the
detector with energy gap $\Omega_{j}$ in the region $f_{j}\in
C_{0}^{\infty}(\mathcal{M})$, respectively. Employing the delta-coupling for
each interaction, Eq. (30) can be written as a product of Schmidt rank-1
operators Jose.tran.delta :

$\displaystyle\hat{U}_{\text{I}}$
$\displaystyle=\hat{U}_{\text{I}}^{(2)}\hat{U}_{\text{I}}^{(1)}\,,$ (31)
$\displaystyle\hat{U}_{\text{I}}^{(j)}$
$\displaystyle\coloneqq\exp\left[-\mathsf{i}\hat{\mu}(\tau_{j})\otimes\hat{\phi}(\mathfrak{f}_{j})\right]=\mathfrak{\mathds{1}}_{\text{D}}\otimes\cos(\hat{\phi}(\mathfrak{f}_{j}))-\mathsf{i}\hat{\mu}(\tau_{j})\otimes\sin(\hat{\phi}(\mathfrak{f}_{j}))\,.$
(32)

Here, we assumed $\tau_{2}>\tau_{1}$ (i.e., the first interaction at
$\tau=\tau_{1}$ is in the causal past of the second interaction at
$\tau=\tau_{2}$). In the following discussion, we will write
$\hat{\mu}_{j}\equiv\hat{\mu}(\tau_{j})$ and
$\hat{\phi}_{j}\equiv\hat{\phi}(\mathfrak{f}_{j})$ for simplicity.

Figure 3: A UDW detector interacting with the field instantaneously.

##  5 Results: Extracted Work from QOE

We now consider a single UDW detector operating within the quantum Otto cycle.
The background is a globally hyperbolic $(n+1)$-dimensional spacetime, in
which we define a quantum Klein-Gordon field in a quasi-free state. The
interaction between the detector and the field is described by the Schmidt
rank-1 unitary operator (31). To maintain generality, we begin our
calculations with a generic algebraic state $\omega$, then later specify it to
a quasi-free state.

###  5.1 General formula of work

####  5.1.1 The first interaction: from step-1 to step-3

As step-1 in the quantum Otto cycle, let the initial state of the total system
be

$\displaystyle\hat{\rho}_{\text{tot,0}}=\hat{\rho}_{\text{D,0}}\otimes\hat{\rho}_{\phi,0}\,,$
(33)

where $\hat{\rho}_{\text{D,0}}$ and $\hat{\rho}_{\phi,0}$ are the initial
state of the detector and the field, respectively, and
$\hat{\rho}_{\text{D,0}}$ is specified to be

$\displaystyle\hat{\rho}_{\text{D,0}}$
$\displaystyle=p\ket{e}\bra{e}+(1-p)\ket{g}\bra{g}\,,\quad p\in[0,1]\,.$ (34)

As depicted in Figure 3, the initial energy gap is $\Omega_{2}$ and then
adiabatically expanded to $\Omega_{1}(>\Omega_{2})$ in step-2.

In step-3 (the isochoric process) of the cycle, the detector interacts with
the field within the region $\mathfrak{f}_{1}$, where the detector’s proper
time is $\tau=\tau_{1}$. The resulting state of the total system is

$\displaystyle\hat{\rho}_{\text{tot,1}}$
$\displaystyle=\hat{U}_{\text{I}}^{(1)}\hat{\rho}_{\text{tot,0}}\hat{U}_{\text{I}}^{(1)\dagger}$
$\displaystyle=\hat{\rho}_{\text{D,0}}\otimes\hat{C}_{1}\hat{\rho}_{\phi,0}\hat{C}_{1}+\hat{\mu}_{1}\hat{\rho}_{\text{D,0}}\hat{\mu}_{1}\otimes\hat{S}_{1}\hat{\rho}_{\phi,0}\hat{S}_{1}-\mathsf{i}\hat{\mu}_{1}\hat{\rho}_{\text{D,0}}\otimes\hat{S}_{1}\hat{\rho}_{\phi,0}\hat{C}_{1}+\mathsf{i}\hat{\rho}_{\text{D,0}}\hat{\mu}_{1}\otimes\hat{C}_{1}\hat{\rho}_{\phi,0}\hat{S}_{1}\,,$
(35)

where we have used (32) and wrote $\hat{S}_{1}\equiv\sin(\hat{\phi}_{1})$ and
$\hat{C}_{1}\equiv\cos(\hat{\phi}_{1})$ for simplicity. We note that
$[\hat{C}_{1},\hat{S}_{1}]=0$ is used. The density matrix of the detector
after the first interaction is obtained by tracing out the field degree of
freedom:

$\displaystyle\hat{\rho}_{\text{D,1}}$
$\displaystyle=\operatorname{Tr}_{\phi}[\hat{U}_{\text{I}}^{(1)}\hat{\rho}_{\text{tot,0}}\hat{U}_{\text{I}}^{(1)\dagger}]$
$\displaystyle=\hat{\rho}_{\text{D,0}}\omega(C_{1}^{2})+\hat{\mu}_{1}\hat{\rho}_{\text{D,0}}\hat{\mu}_{1}\omega(S_{1}^{2})+\mathsf{i}[\hat{\rho}_{\text{D,0}},\hat{\mu}_{1}]\omega(S_{1}C_{1})\,,$
(36)

where $C_{1}$ and $S_{1}$ are the elements of the Weyl algebra corresponding
to $\hat{C}_{1}$ and $\hat{S}_{1}$, respectively, via
$\pi(C_{1})\equiv\hat{C}_{1}$ and $\pi(S_{1})\equiv\hat{S}_{1}$ in a
representation $\pi$ that gives $\hat{\rho}_{\phi,0}$ in its folium. In this
notation, $\omega(A)=\operatorname{Tr}[\hat{\rho}_{\phi,0}\hat{A}]$ is used.

We now calculate each algebraic state in (36). Let us write $C_{1}$ and
$S_{1}$ in terms of the elements of the Weyl algebra
$W_{\mathfrak{f}_{j}}\equiv W(E\mathfrak{f}_{j})$:

$\displaystyle
C_{1}=\dfrac{e^{\mathsf{i}\phi_{1}}+e^{-\mathsf{i}\phi_{1}}}{2}=\dfrac{W_{\mathfrak{f}_{1}}+W_{\mathfrak{f}_{1}}^{*}}{2}\,,$
(37a) $\displaystyle
S_{1}=\dfrac{e^{\mathsf{i}\phi_{1}}-e^{-\mathsf{i}\phi_{1}}}{2\mathsf{i}}=\dfrac{W_{\mathfrak{f}_{1}}-W_{\mathfrak{f}_{1}}^{*}}{2\mathsf{i}}\,.$
(37b)

These can be written in a unified manner as

$\displaystyle\dfrac{W_{\mathfrak{f}_{1}}+(-1)^{u}W_{\mathfrak{f}_{1}}^{*}}{2\mathsf{i}^{u}}\,,\quad
u\in\\{0,1\\}$ (38)

where $u=0$ and $u=1$ correspond to $C_{1}$ and $S_{1}$, respectively. Then
each algebraic state in (36) takes the form

$\displaystyle\omega\left(\dfrac{W_{\mathfrak{f}_{1}}+(-1)^{r}W_{\mathfrak{f}_{1}}^{*}}{2\mathsf{i}^{r}}\dfrac{W_{\mathfrak{f}_{1}}+(-1)^{s}W_{\mathfrak{f}_{1}}^{*}}{2\mathsf{i}^{s}}\right)\,.$
(39)

Using the Weyl relations (17) and the fact that
$\omega:\mathcal{W}(\mathcal{M})\to\mathbb{C}$ is a linear map, we get

$\displaystyle\omega\left(\dfrac{W_{\mathfrak{f}_{1}}+(-1)^{r}W_{\mathfrak{f}_{1}}^{*}}{2\mathsf{i}^{r}}\dfrac{W_{\mathfrak{f}_{1}}+(-1)^{s}W_{\mathfrak{f}_{1}}^{*}}{2\mathsf{i}^{s}}\right)$
$\displaystyle=\dfrac{1}{4\mathsf{i}^{r+s}}\Big{[}\omega(W_{2\mathfrak{f}_{1}})+(-1)^{r}+(-1)^{s}+(-1)^{r+s}\omega(W_{2\mathfrak{f}_{1}}^{*})\Big{]}\,.$
(40)

So far, we have assumed that the state of the field is arbitrary. We now
specify $\hat{\rho}_{\phi,0}$ to be a quasi-free state defined in (20). We
then define

$\displaystyle\nu_{\mathfrak{f}_{1}}\equiv\omega(W_{2\mathfrak{f}_{1}})=e^{-2\mathsf{W}(\mathfrak{f}_{1},\mathfrak{f}_{1})}\in(0,1]\,.$
(41)

In this case, the algebraic state is a real number, which leads to
$\omega(W_{\mathfrak{f}}^{*})=\omega(W_{\mathfrak{f}})$. Then

$\displaystyle\omega\left(\dfrac{W_{\mathfrak{f}_{1}}+(-1)^{r}W_{\mathfrak{f}_{1}}^{*}}{2\mathsf{i}^{r}}\dfrac{W_{\mathfrak{f}_{1}}+(-1)^{s}W_{\mathfrak{f}_{1}}^{*}}{2\mathsf{i}^{s}}\right)$
$\displaystyle=\dfrac{1}{4\mathsf{i}^{r+s}}\Big{[}(1+(-1)^{r+s})\nu_{\mathfrak{f}_{1}}+(-1)^{r}+(-1)^{s}\Big{]}\,.$
(42)

This tells us that $\omega(S_{1}C_{1})=\omega(C_{1}S_{1})=0$ and so the
detector’s state after the first interaction reads

$\displaystyle\hat{\rho}_{\text{D,1}}$
$\displaystyle=\hat{\rho}_{\text{D,0}}\omega(C_{1}^{2})+\hat{\mu}_{1}\hat{\rho}_{\text{D,0}}\hat{\mu}_{1}\omega(S_{1}^{2})$
$\displaystyle=\dfrac{1+\nu_{\mathfrak{f}_{1}}}{2}\hat{\rho}_{\text{D,0}}+\hat{\mu}_{1}\hat{\rho}_{\text{D,0}}\hat{\mu}_{1}\dfrac{1-\nu_{\mathfrak{f}_{1}}}{2}$
(43) $\displaystyle\equiv p_{1}\ket{e}\bra{e}+(1-p_{1})\ket{g}\bra{g}\,,$ (44)

where

$\displaystyle p_{1}$
$\displaystyle=\dfrac{1}{2}+\left(p-\dfrac{1}{2}\right)\nu_{\mathfrak{f}_{1}}\,,$
(45)

when $\hat{\rho}_{\text{D,0}}$ is initially chosen to be (34).

####  5.1.2 The second interaction: from step-4 to step-5

Next, we move to the adiabatic compression process $\Omega_{1}\to\Omega_{2}$
in step-4, which is then followed by a second interaction within the region
$\mathfrak{f}_{2}$ during step-5.

The final state of the total system, $\hat{\rho}_{\text{tot,2}}$, is given by

$\displaystyle\hat{\rho}_{\text{tot,2}}$
$\displaystyle=\hat{U}_{\text{I}}^{(2)}\hat{\rho}_{\text{tot,1}}\hat{U}_{\text{I}}^{(2)\dagger}=\hat{U}_{\text{I}}^{(2)}\hat{U}_{\text{I}}^{(1)}\hat{\rho}_{\text{tot,0}}\hat{U}_{\text{I}}^{(1)\dagger}\hat{U}_{\text{I}}^{(2)\dagger}\,,$
(46)

where $\hat{U}_{\text{I}}^{(j)}$ is given by (32). We will again write
$\hat{S}_{j}\equiv\sin(\hat{\phi}_{j})$ and
$\hat{C}_{j}\equiv\cos(\hat{\phi}_{j})$ and use the properties
$[\hat{C}_{j},\hat{S}_{j}]=0$ and $[\hat{C}_{j},\hat{S}_{k}]\neq 0$ for $j\neq
k$.

After the second interaction in step-5, the final density matrix for the
detector, $\hat{\rho}_{\text{D,2}}$, can be straightforwardly derived as in
the previous case (see Appendix A for the detailed calculations):

$\displaystyle\hat{\rho}_{\text{D,2}}$
$\displaystyle=\hat{\rho}_{\text{D,0}}\omega(C_{1}C_{2}^{2}C_{1})+\hat{\mu}_{2}\hat{\rho}_{\text{D,0}}\hat{\mu}_{2}\omega(C_{1}S_{2}^{2}C_{1})+\hat{\mu}_{1}\hat{\rho}_{\text{D,0}}\hat{\mu}_{1}\omega(S_{1}C_{2}^{2}S_{1})$
$\displaystyle\quad+\hat{\mu}_{2}\hat{\mu}_{1}\hat{\rho}_{\text{D,0}}\hat{\mu}_{1}\hat{\mu}_{2}\omega(S_{1}S_{2}^{2}S_{1})+[\hat{\mu}_{1}\hat{\rho}_{\text{D,0}},\hat{\mu}_{2}]\omega(C_{1}S_{2}C_{2}S_{1})-[\hat{\rho}_{\text{D,0}}\hat{\mu}_{1},\hat{\mu}_{2}]\omega(S_{1}S_{2}C_{2}C_{1})\,,$
(47)

where

$\displaystyle\omega(C_{1}C_{2}^{2}C_{1})=\dfrac{1}{4}\Big{[}1+\nu_{\mathfrak{f}_{1}}+\nu_{\mathfrak{f}_{1}}\nu_{\mathfrak{f}_{2}}\cosh[4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})]+\nu_{\mathfrak{f}_{2}}\cos(2E(\mathfrak{f}_{1},\mathfrak{f}_{2}))\Big{]}\,,$
(48a)
$\displaystyle\omega(C_{1}S_{2}^{2}C_{1})=\dfrac{1}{4}\Big{[}1+\nu_{\mathfrak{f}_{1}}-\nu_{\mathfrak{f}_{1}}\nu_{\mathfrak{f}_{2}}\cosh[4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})]-\nu_{\mathfrak{f}_{2}}\cos(2E(\mathfrak{f}_{1},\mathfrak{f}_{2}))\Big{]}\,,$
(48b)
$\displaystyle\omega(S_{1}C_{2}^{2}S_{1})=\dfrac{1}{4}\Big{[}1-\nu_{\mathfrak{f}_{1}}-\nu_{\mathfrak{f}_{1}}\nu_{\mathfrak{f}_{2}}\cosh[4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})]+\nu_{\mathfrak{f}_{2}}\cos(2E(\mathfrak{f}_{1},\mathfrak{f}_{2}))\Big{]}\,,$
(48c)
$\displaystyle\omega(S_{1}S_{2}^{2}S_{1})=\dfrac{1}{4}\Big{[}1-\nu_{\mathfrak{f}_{1}}+\nu_{\mathfrak{f}_{1}}\nu_{\mathfrak{f}_{2}}\cosh[4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})]-\nu_{\mathfrak{f}_{2}}\cos(2E(\mathfrak{f}_{1},\mathfrak{f}_{2}))\Big{]}\,,$
(48d)
$\displaystyle\omega(C_{1}C_{2}S_{2}S_{1})=\omega(C_{1}S_{2}C_{2}S_{1})=\dfrac{\nu_{\mathfrak{f}_{2}}}{4}\Big{[}\nu_{\mathfrak{f}_{1}}\sinh[4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})]-\mathsf{i}\sin(2E(\mathfrak{f}_{1},\mathfrak{f}_{2}))\Big{]}\,,$
(48e)
$\displaystyle\omega(S_{1}C_{2}S_{2}C_{1})=\omega(S_{1}S_{2}C_{2}C_{1})=\dfrac{\nu_{\mathfrak{f}_{2}}}{4}\Big{[}\nu_{\mathfrak{f}_{1}}\sinh[4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})]+\mathsf{i}\sin(2E(\mathfrak{f}_{1},\mathfrak{f}_{2}))\Big{]}\,.$
(48f)

Here, $\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})\in\mathbb{R}$ is given in
(21). Inserting these expressions, the final density matrix
$\hat{\rho}_{\text{D,2}}$ reads

$\displaystyle\hat{\rho}_{\text{D,2}}$
$\displaystyle=p_{2}\ket{e}\bra{e}+(1-p_{2})\ket{g}\bra{g}\,,$ (49)

where

$\displaystyle p_{2}$
$\displaystyle=\dfrac{1}{2}\left[1+\nu_{\mathfrak{f}_{2}}\sin(2E(\mathfrak{f}_{1},\mathfrak{f}_{2}))\sin\theta+(2p-1)\nu_{\mathfrak{f}_{1}}\nu_{\mathfrak{f}_{2}}\left(e^{4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})}\sin^{2}\dfrac{\theta}{2}+e^{-4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})}\cos^{2}\dfrac{\theta}{2}\right)\right]\,,$
(50)

where $\theta\coloneqq\Omega_{1}\tau_{1}-\Omega_{2}\tau_{2}$. Note that

$\displaystyle
0<\nu_{\mathfrak{f}_{1}}\nu_{\mathfrak{f}_{2}}\left(e^{4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})}\sin^{2}\dfrac{\theta}{2}+e^{-4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})}\cos^{2}\dfrac{\theta}{2}\right)\leq
1\,.$ (51)

####  5.1.3 Closing the cycle and work extracted: step-6

Let us complete the thermodynamic cycle and evaluate the work extracted by the
delta-coupled UDW detector.

We first impose the cyclicity condition in step-6, $p_{2}=p$, which can be
recast into

$\displaystyle p$
$\displaystyle=\dfrac{1}{2}-\dfrac{\dfrac{\nu_{\mathfrak{f}_{2}}}{2}\sin(2E(\mathfrak{f}_{1},\mathfrak{f}_{2}))\sin\theta}{\nu_{\mathfrak{f}_{1}}\nu_{\mathfrak{f}_{2}}\left(e^{4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})}\sin^{2}\dfrac{\theta}{2}+e^{-4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})}\cos^{2}\dfrac{\theta}{2}\right)-1}\,.$
(52)

This means the initial population $p$ in the excited state $\ket{e}$ has to be
adjusted according to (52) to ensure that the thermodynamic cycle is closed.

Under the cyclicity condition (52), let us calculate the work extracted from
the field [see (7)]. From (45), we obtain

$\displaystyle\braket{W_{\text{ext}}}$ $\displaystyle=(p_{1}-p)\Delta\Omega$
$\displaystyle=-\left(p-\dfrac{1}{2}\right)(1-\nu_{\mathfrak{f}_{1}})\Delta\Omega$
(53)
$\displaystyle=\dfrac{\dfrac{\nu_{\mathfrak{f}_{2}}}{2}\sin(2E(\mathfrak{f}_{1},\mathfrak{f}_{2}))\sin\theta(1-\nu_{\mathfrak{f}_{1}})}{\nu_{\mathfrak{f}_{1}}\nu_{\mathfrak{f}_{2}}\left(e^{4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})}\sin^{2}\dfrac{\theta}{2}+e^{-4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})}\cos^{2}\dfrac{\theta}{2}\right)-1}\Delta\Omega\,,$
(54)

where $p$ in (52) is substituted to meet the cyclicity condition. Recalling
that $\nu_{\mathfrak{f}}\in(0,1]$, $\Delta\Omega>0$, and Eq. (51), the
positive work condition, $\braket{W_{\text{ext}}}>0$, reads

$\displaystyle\sin(2E(\mathfrak{f}_{1},\mathfrak{f}_{2}))\sin\theta<0\,.$ (55)

Therefore, we obtained the work extracted (54) and its positive work condition
(55) for a UDW detector, which instantaneously interacts with a quantum scalar
field in a quasi-free state in an arbitrary globally hyperbolic spacetime.

###  5.2 Observations and remarks

Figure 4: Image of self-communication. For a massless scalar field in
$(3+1)$-dimensional Minkowski spacetime, the field propagates in the lightlike
direction. The detector can receive a signal from itself in the past if the
interaction region $\mathfrak{f}_{2}$ intersects with the yellow lightlike
region. Otherwise, $E(\mathfrak{f}_{1},\mathfrak{f}_{2})=0$.

Let us discuss the implication of the final results (54) and (55).

  * •

Work from signaling;

We observe from (55) that a QOE with instantaneous isochoric processes is
capable of extracting positive work, depending on the causal propagator
$E(\mathfrak{f}_{1},\mathfrak{f}_{2})$. The causal propagator
$E(\mathfrak{f}_{1},\mathfrak{f}_{2})$ represents the signaling between the
regions $\mathfrak{f}_{1}$ and $\mathfrak{f}_{2}$ through the quantum field.
The vanishing causal propagator, $E(\mathfrak{f}_{1},\mathfrak{f}_{2})=0$,
implies no signaling between the regions. Since we only have a single UDW
detector, this signaling refers to a self-communication, i.e., the detector in
region $\mathfrak{f}_{2}$ receiving its own signal sent from
$\mathfrak{f}_{1}$ when it first coupled to the field. This point is
visualized in Figure 4.

The positive work condition (55) implies that if the detector is unable to
self-communicate, then $E(\mathfrak{f}_{1},\mathfrak{f}_{2})=0$ leading to
$\braket{W_{\text{ext}}}=0$. Thus, self-communication is necessary to extract
positive work for a delta-coupled UDW detector. It should be noted that merely
receiving a signal from the past does not guarantee work extraction since the
sign in (55) might be positive even when
$E(\mathfrak{f}_{1},\mathfrak{f}_{2})\neq 0$. We will address the mechanism
for the work extraction at the end of this subsection.

  * •

The state of the field;

The causal propagator $E(\mathfrak{f}_{1},\mathfrak{f}_{2})$ is a state-
independent quantity, i.e., its value is the same no matter what the field
state $\omega$ is. Therefore, the positive work condition (55) is state-
independent as well.

  * •

Extracting work from a vacuum state and the second law;

As we will discuss in Section 5.3, it is possible to extract positive work
even when an inertial detector interacts with the vacuum state $\ket{0}$ in
Minkowski spacetime. This can be qualitatively explained as follows. The
second law of thermodynamics states that work cannot be extracted from a
system in equilibrium. However, if the system is brought out of equilibrium,
extracting work becomes feasible. A typical example is a system with two
thermal baths at different temperatures. In our case, the quantum field
becomes an out-of-equilibrium state due to the disturbance caused by the first
interaction. If the detector interacts with this disturbed field, it may be
possible to extract work.

  * •

Comparison to a perturbative analysis;

A perturbative analysis of QOE utilizing a UDW detector was investigated in
Gallock2023Otto  (see UnruhOttoEngine ; Finn.UnruhOtto ;
Xu.UnruhOtto.degenerate ; Kane.entangled.Unruh.Otto ;
Barman.entangled.UnruhOtto ; Mukherjee.UnruhOtto.boundary  for analyses
involving a linearly accelerating UDW detector). In these scenarios, a
pointlike (i.e., zero-size) UDW detector interacts with a quasi-free state
over a finite time duration. The extracted work depends only on the response
function (the probability of transition between
$\ket{g}\leftrightarrow\ket{e}$) and communication is irrelevant in the
leading order of perturbation theory. See also Causality2015Eduardo  for the
communication effect involving two detectors.

Based on the findings presented in this paper, we expect that a weakly-coupled
UDW detector may be able to extract work from a vacuum state if higher-order
terms are accounted for in the perturbation theory.

Before proceeding to the next subsection, let us examine the role of signaling
in our QOE. To understand the work extraction mechanism, let us remind
ourselves the condition for positive work extraction from QOE. As shown in
Section 2, the extracted work $\braket{W_{\text{ext}}}$ after completing the
thermodynamic cycle reads

$\displaystyle\braket{W_{\text{ext}}}$
$\displaystyle=\braket{Q}=\braket{Q_{\mathfrak{f}_{1}}}+\braket{Q_{\mathfrak{f}_{2}}}\,,$
(56)

where $\braket{Q}$ is the total absorbed heat from the baths during the
isochoric processes, and here we additionally defined
$\braket{Q_{\mathfrak{f}_{j}}}$ as heat exchanged during the interaction in
the region $\mathfrak{f}_{j}$. To be clear, $\braket{Q_{\mathfrak{f}_{1}}}$
and $\braket{Q_{\mathfrak{f}_{2}}}$ correspond to the heat exchanged in step-3
and step-5, respectively, which read
$\braket{Q_{\mathfrak{f}_{1}}}=\Omega_{1}(p_{1}-p)$ and
$\braket{Q_{\mathfrak{f}_{2}}}=\Omega_{2}(p_{2}-p_{1})$. Assuming
$\Omega_{1}>\Omega_{2}$, we have seen that $p_{1}-p>0$ needs to be satisfied
to extract positive work. This requirement implies
$\braket{Q_{\mathfrak{f}_{1}}}>0$, meaning the qubit needs to absorb heat from
the field during its first interaction in region $\mathfrak{f}_{1}$.
Meanwhile, the cyclicity condition $p_{2}=p$ in step-6 leads to
$\braket{Q_{\mathfrak{f}_{2}}}=\Omega_{2}(p-p_{1})<0$, implying that heat must
be dissipated into the field during the second interaction in region
$\mathfrak{f}_{2}$. In summary, the qubit must absorb heat and increase the
population $p_{1}$ in the excited state $\ket{e}$ in step-3, followed by an
adiabatic process in step-4 and another isochoric process in step-5 where the
population decreases from $p_{1}$ back to $p_{2}(=p)$, releasing heat in the
process to complete the thermodynamic cycle.

Next, we demonstrate that the delta-coupled UDW detector always absorbs heat
and transitions to the excited state, as long as the population in $\ket{e}$
is less than $1/2$. This can be readily seen from Eqs. (45) and (50). The
population $p_{1}$ in $\ket{e}$ after the first interaction in
$\mathfrak{f}_{1}$ is

$\displaystyle
p_{1}=\dfrac{1}{2}+\left(p-\dfrac{1}{2}\right)\nu_{\mathfrak{f}_{1}}\,\begin{cases}\in[p,1/2)&\text{if}~{}p<\dfrac{1}{2}\vspace{2mm}\\\
\in(1/2,p]&\text{if}~{}p>\dfrac{1}{2}\end{cases}$ (57)

where we have used the fact that $\nu_{\mathfrak{f}_{1}}\in(0,1]$ for a quasi-
free state. In words, if the initial population is $p<1/2$ then an interaction
governed by the delta-coupling always increases the population by absorbing
heat from the field. Conversely, if $p>1/2$ then de-excitation and heat
dissipation occur. For the second interaction in region $\mathfrak{f}_{2}$,
$p_{2}$ is given by (50). In the absence of signaling, where
$E(\mathfrak{f}_{1},\mathfrak{f}_{2})=0$, (50) can be written as

$\displaystyle p_{2}$
$\displaystyle=\dfrac{1}{2}+\left(p-\dfrac{1}{2}\right)\nu_{\mathfrak{f}_{1}}\nu_{\mathfrak{f}_{2}}\alpha\,\begin{cases}\in[p,1/2)&\text{if}~{}p<\dfrac{1}{2}\vspace{2mm}\\\
\in(1/2,p]&\text{if}~{}p>\dfrac{1}{2}\end{cases}$ (58)

where

$\displaystyle\alpha$ $\displaystyle\coloneqq
e^{4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})}\sin^{2}\dfrac{\theta}{2}+e^{-4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})}\cos^{2}\dfrac{\theta}{2}\,.$
(59)

The ranges of $p_{2}$ in (58) is due to the fact that
$\nu_{\mathfrak{f}_{1}}\nu_{\mathfrak{f}_{2}}\alpha\in(0,1]$, as given by
(51).

Eqs. (57) and (58) explain how the UDW detector extracts work from a field.
First, for the positive work condition $p_{1}-p>0$ to be met, it is necessary
that $p<1/2$, as the first interaction (57) solely increases the population.
In this scenario, Eq. (57) implies that $p_{1}$ also remains below $1/2$.
Suppose the second interaction in region $\mathfrak{f}_{2}$ is unaffected by
the first in $\mathfrak{f}_{1}$, i.e., no signal is exchanged. Then, $p_{2}$
is given by (58). However, since $p<1/2$ the detector gets excited by
absorbing heat from the field, conflicting with the condition
$\braket{Q_{\mathfrak{f}_{2}}}<0$. Thus, in a QOE framework, a detector
without signaling cannot extract work from the field because it only absorbs
heat, thereby failing to close the thermodynamic cycle.

The situation changes when the detector communicates with itself from the
past. In this case, $p_{2}$ is given by (50), which is not restricted to
$p_{2}\in[p_{1},1/2)$. If this self-communication results in the detector
dissipating heat, the thermodynamic cycle can be successfully closed. Thus,
communication plays a vital role in completing the cycle.

###  5.3 Example: An inertial detector extracts work from the Minkowski vacuum

As a demonstration, consider an inertial UDW detector in $(3+1)$-dimensional
Minkowski spacetime. We choose a massless minimally coupled scalar field
[i.e., $m=0$ and $\xi=0$ in (8)] in the Minkowski vacuum. In this case, the
unsmeared field operator $\hat{\phi}(\mathsf{x})$ at
$\mathsf{x}\in\mathcal{M}$ can be written as

$\displaystyle\hat{\phi}(\mathsf{x})$
$\displaystyle=\int_{\mathbb{R}^{3}}\dfrac{\text{d}^{3}k}{\sqrt{(2\pi)^{3}2|\bm{k}|}}\left(\hat{a}_{\bm{k}}e^{-\mathsf{i}|\bm{k}|t+\mathsf{i}{\bm{k}}\cdot{\bm{x}}}+\hat{a}_{\bm{k}}^{\dagger}e^{\mathsf{i}|\bm{k}|t-\mathsf{i}{\bm{k}}\cdot{\bm{x}}}\right)\,,$
(60)

with $\hat{a}_{\bm{k}}\ket{0}=0$ for all ${\bm{k}}$.

To evaluate the extracted work (54), we shall compute
$\nu_{\mathfrak{f}},E(\mathfrak{f}_{1},\mathfrak{f}_{2})$, and
$\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})$. As one can see from Eqs. (41) and
(21), the spatially smeared Wightman function
$\mathsf{W}(\mathfrak{f}_{1},\mathfrak{f}_{2})$ is a key ingredient. The
unsmeared vacuum Wightman function from (60) is known to be

$\displaystyle\mathsf{W}(\mathsf{x},\mathsf{x}^{\prime})$
$\displaystyle=\braket{0}{\hat{\phi}(\mathsf{x})\hat{\phi}(\mathsf{x}^{\prime})}{0}=\int_{\mathbb{R}^{3}}\dfrac{\text{d}^{3}k}{(2\pi)^{3}2|\bm{k}|}e^{-\mathsf{i}|\bm{k}|(t-t^{\prime})+\mathsf{i}{\bm{k}}\cdot({\bm{x}}-{\bm{x}}^{\prime})}\,.$
(61)

Therefore, the spatially smeared vacuum Wightman function can be written as

$\displaystyle\mathsf{W}(\mathfrak{f}_{1},\mathfrak{f}_{2})$
$\displaystyle=\int_{\mathbb{R}^{3}}\text{d}^{3}x\int_{\mathbb{R}^{3}}\text{d}^{3}x^{\prime}\,W(\tau_{1},{\bm{x}};\tau_{2},{\bm{x}}^{\prime})\tilde{\lambda}_{1}\tilde{\lambda}_{2}F_{1}({\bm{x}})F_{2}({\bm{x}}^{\prime})$
(62)
$\displaystyle=\dfrac{\tilde{\lambda}_{1}\tilde{\lambda}_{2}}{2}\int_{\mathbb{R}^{3}}\text{d}^{3}k\,\dfrac{e^{-\mathsf{i}|\bm{k}|\Delta\tau}}{|\bm{k}|}\tilde{F}_{1}({\bm{k}})\tilde{F}_{2}(-{\bm{k}})\,,$
(63)

where $\tilde{\lambda}_{j}\coloneqq\lambda_{j}\eta$,
$\Delta\tau\coloneqq\tau_{2}-\tau_{1}$, and

$\displaystyle\tilde{F}_{j}({\bm{k}})$
$\displaystyle\coloneqq\int_{\mathbb{R}^{3}}\dfrac{\text{d}^{3}x}{\sqrt{(2\pi)^{3}}}F_{j}({\bm{x}})e^{\mathsf{i}{\bm{k}}\cdot{\bm{x}}}$
(64)

is the Fourier transform of the smearing function $F_{j}({\bm{x}})$. Hence, we
obtain $\nu_{\mathfrak{f}}$, $E(\mathfrak{f}_{1},\mathfrak{f}_{2})$, and
$\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})$ once an explicit form of the
smearing function $F_{j}({\bm{x}})$ is specified.

Figure 5: The extracted work $\braket{W_{\text{ext}}}\sigma$ in units of the
characteristic width $\sigma$ of the Gaussian smearing function. (a) Work as a
function of the second interaction time $\tau_{2}/\sigma$ when
$\tau_{1}/\sigma=0$, $\Omega_{1}\sigma=1$, $\Omega_{2}\sigma=3$,
$\tilde{\lambda}_{1}/\sigma=100$, and $\tilde{\lambda}_{2}/\sigma=1$. Negative
$\braket{W_{\text{ext}}}\sigma$ corresponds to work done on the detetor. (b)
Work as a function of the coupling constants. Here, we choose
$\tau_{1}/\sigma=0$, $\Omega_{1}\sigma=1$, $\Omega_{2}\sigma=3$, and
$\tau_{2}/\sigma=1.5$.

Let us choose a Gaussian smearing function with a characteristic Gaussian
width $\sigma$:

$\displaystyle F({\bm{x}})\equiv
F_{1}({\bm{x}})=F_{2}({\bm{x}})=\dfrac{1}{(\sqrt{\pi}\sigma)^{3}}e^{-|{\bm{x}}|^{2}/\sigma^{2}}\,,$
(65)

whose Fourier transform reads

$\displaystyle\tilde{F}(|\bm{k}|)$
$\displaystyle=\dfrac{1}{\sqrt{(2\pi)^{3}}}e^{-|\bm{k}|^{2}\sigma^{2}/4}\,.$
(66)

Although the Gaussian function does not have compact support, its ‘tails’ are
exponentially suppressed, allowing us to assume that the detector is
effectively localized.

Inserting (66) into (63), a straightforward calculation gives us
GallockEntangledDetectors ; Tjoa.RQC

$\displaystyle\nu_{\mathfrak{f}_{j}}$
$\displaystyle=e^{-2\mathsf{W}(\mathfrak{f}_{j},\mathfrak{f}_{j})}=\exp\left(-\dfrac{\tilde{\lambda}_{j}^{2}}{2\pi^{2}\sigma^{2}}\right)\,,$
(67a) $\displaystyle E(\mathfrak{f}_{1},\mathfrak{f}_{2})$
$\displaystyle=\dfrac{1}{2}\text{Im}[\mathsf{W}(\mathfrak{f}_{1},\mathfrak{f_{2}})]=\dfrac{\tilde{\lambda}_{1}\tilde{\lambda}_{2}}{2\sqrt{\pi^{3}}\sigma^{2}}\dfrac{\Delta\tau}{\sqrt{2}\sigma}\exp\left[-\left(\dfrac{\Delta\tau}{\sqrt{2}\sigma}\right)^{2}\right]\,,$
(67b) $\displaystyle\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})$
$\displaystyle=\text{Re}[\mathsf{W}(\mathfrak{f}_{1},\mathfrak{f_{2}})]=\dfrac{\tilde{\lambda}_{1}\tilde{\lambda}_{2}}{4\pi^{2}\sigma^{2}}\left[1-2\dfrac{\Delta\tau}{\sqrt{2}\sigma}D^{+}\left(\dfrac{\Delta\tau}{\sqrt{2}\sigma}\right)\right]\,,$
(67c)

where $D^{+}(x)$ is the Dawson function given by

$\displaystyle D^{+}(x)\coloneqq\dfrac{\sqrt{\pi}}{2}e^{-x^{2}}\text{erfi}(x)$
(68)

with the imaginary error function $\text{erfi}(x)$.

After substituting these expressions into the extracted work (54), the results
are shown in Figure 5. In Figure 5(a), we depict the dependence of the second
interaction’s timing, $\tau_{2}/\sigma$, in units of the characteristic
Gaussian width $\sigma$, given $\tau_{1}/\sigma=0$, $\Omega_{1}\sigma=1$,
$\Omega_{2}\sigma=3$, $\tilde{\lambda}_{1}/\sigma=100$, and
$\tilde{\lambda}_{2}/\sigma=1$. As discussed in Section 5.2, the detector can
extract positive work from the vacuum state of the field instantaneously as
long as the first interaction affects the second one. In fact, we have
$\braket{W_{\text{ext}}}\sigma=0$ once the detector cannot receive the signal
from the first interaction (i.e., when $\tau_{2}/\sigma>4$).

Figure 5(b) shows the coupling-dependence of the work
$\braket{W_{\text{ext}}}\sigma$ when $\tau_{1}/\sigma=0$,
$\Omega_{1}\sigma=1$, $\Omega_{2}\sigma=3$, and $\tau_{2}/\sigma=1.5$. We see
that a pair $(\tilde{\lambda}_{1},\tilde{\lambda}_{2})$ must be chosen
carefully to satisfy the positive work condition. In particular, the first
coupling $\tilde{\lambda}_{1}$ is preferred to be strong while the second one
$\tilde{\lambda}_{2}$ should be much weaker. This is due to the fact that
$\braket{W_{\text{ext}}}\to 0$ as $\nu_{\mathfrak{f}_{2}}\to 0$ in (54), which
is the case for $\tilde{\lambda}_{2}/\sigma\gg 1$ in (67a).

##  6 Conclusions

We have examined the relativistic QOE using the UDW particle detector model as
a working substance interacting with a quantum scalar field. Here, the
background spacetime is assumed to be an arbitrary globally hyperbolic
spacetime and the state of the field is considered to be a quasi-free state
such as vacuum and thermal states. By adopting algebraic quantum field
theoretical formulation, we were able to consider such a very general setting.

We were particularly interested in the so-called delta-coupling model of the
UDW detector for two reasons. First, the delta-coupling—which is implemented
by writing the time dependence of the interaction as Dirac’s delta
distribution—enables us to employ a non-perturbative analysis, thereby a
strong coupling can be considered. Secondly, it allows us to examine if a
detector can extract work from a quantum field by means of instantaneous
interactions.

Under these assumptions, we have derived the extracted work (54) and its
positive work condition (55) non-perturbatively and found that it is indeed
possible to extract work instantaneously from a relativistic QOE as long as
the detector receives the signal from its past self. That is, the detector
must receive a signal during its second interaction that was transmitted by
the quantum field from the first interaction. This is because each interaction
without signaling only absorbs heat from the field, thus failing to complete
the thermodynamic cycle. On the other hand, self-communication may assist the
detector in releasing heat back into the field, thereby the completion of the
cycle. As an example, we considered a scenario where the detector is at rest
in flat spacetime. We showed that the detector is able to extract work from
the Minkowski vacuum due to the signaling effect.

Since our interaction model is instantaneous, one might question its
practicality in the context of a quantum heat engine. Indeed, the necessity
for the detector to receive a signal during the second isochoric process
suggests that the time interval between the two isochoric processes must be
exceptionally short, thereby necessitating an immediate adiabatic process in
between. Nevertheless, our results highlight the crucial role of the back-
reaction effect from the reservoir in quantum heat engines. That is, the
influence of the first isochoric process on the second through a quantum field
(serving as the bath in this context) makes it possible for the detector to
extract work.

As an extension to this paper, it would be interesting to include a quantum
field in the Gaussian states with a non-vanishing one-point Wightman function
such as a coherent state.

Note: A related study was recently conducted in kollas2023exactly . In this
study, the authors consider a relativistic QOE utilizing a delta-coupled UDW
detector in Minkowski spacetime to extract work from a quantum field in a
thermal state.

##  Appendix A Second interaction

We provide a detailed calculation for the second interaction. We first assume
the initial field state $\hat{\rho}_{\phi,0}$ to be arbitrary and consider a
quasi-free state at the end.

The total density operator $\hat{\rho}_{\text{tot,2}}$ right after the second
interaction is given by

$\displaystyle\hat{\rho}_{\text{tot,2}}$
$\displaystyle=\hat{U}_{\text{I}}^{(2)}\hat{U}_{\text{I}}^{(1)}\hat{\rho}_{\text{tot,0}}\hat{U}_{\text{I}}^{(1)\dagger}\hat{U}_{\text{I}}^{(2)\dagger}$
$\displaystyle=\hat{\rho}_{\text{D,0}}\otimes\hat{C}_{2}\hat{C}_{1}\hat{\rho}_{\phi,0}\hat{C}_{1}\hat{C}_{2}-\mathsf{i}\hat{\mu}_{2}\hat{\rho}_{\text{D,0}}\otimes\hat{S}_{2}\hat{C}_{1}\hat{\rho}_{\phi,0}\hat{C}_{1}\hat{C}_{2}+\mathsf{i}\hat{\rho}_{\text{D,0}}\hat{\mu}_{2}\otimes\hat{C}_{2}\hat{C}_{1}\hat{\rho}_{\phi,0}\hat{C}_{1}\hat{S}_{2}$
$\displaystyle\quad+\hat{\mu}_{2}\hat{\rho}_{\text{D,0}}\hat{\mu}_{2}\otimes\hat{S}_{2}\hat{C}_{1}\hat{\rho}_{\phi,0}\hat{C}_{1}\hat{S}_{2}+\hat{\mu}_{1}\hat{\rho}_{\text{D,0}}\hat{\mu}_{1}\otimes\hat{C}_{2}\hat{S}_{1}\hat{\rho}_{\phi,0}\hat{S}_{1}\hat{C}_{2}$
$\displaystyle\quad-\mathsf{i}\hat{\mu}_{2}\hat{\mu}_{1}\hat{\rho}_{\text{D,0}}\hat{\mu}_{1}\otimes\hat{S}_{2}\hat{S}_{1}\hat{\rho}_{\phi,0}\hat{S}_{1}\hat{C}_{2}+\mathsf{i}\hat{\mu}_{1}\hat{\rho}_{\text{D,0}}\hat{\mu}_{1}\hat{\mu}_{2}\otimes\hat{C}_{2}\hat{S}_{1}\hat{\rho}_{\phi,0}\hat{S}_{1}\hat{S}_{2}$
$\displaystyle\quad+\hat{\mu}_{2}\hat{\mu}_{1}\hat{\rho}_{\text{D,0}}\hat{\mu}_{1}\hat{\mu}_{2}\otimes\hat{S}_{2}\hat{S}_{1}\hat{\rho}_{\phi,0}\hat{S}_{1}\hat{S}_{2}-\mathsf{i}\hat{\mu}_{1}\hat{\rho}_{\text{D,0}}\otimes\hat{C}_{2}\hat{S}_{1}\hat{\rho}_{\phi,0}\hat{C}_{1}\hat{C}_{2}$
$\displaystyle\quad-\hat{\mu}_{2}\hat{\mu}_{1}\hat{\rho}_{\text{D,0}}\otimes\hat{S}_{2}\hat{S}_{1}\hat{\rho}_{\phi,0}\hat{C}_{1}\hat{C}_{2}+\hat{\mu}_{1}\hat{\rho}_{\text{D,0}}\hat{\mu}_{2}\otimes\hat{C}_{2}\hat{S}_{1}\hat{\rho}_{\phi,0}\hat{C}_{1}\hat{S}_{2}$
$\displaystyle\quad-\mathsf{i}\hat{\mu}_{2}\hat{\mu}_{1}\hat{\rho}_{\text{D,0}}\hat{\mu}_{2}\otimes\hat{S}_{2}\hat{S}_{1}\hat{\rho}_{\phi,0}\hat{C}_{1}\hat{S}_{2}+\mathsf{i}\hat{\rho}_{\text{D,0}}\hat{\mu}_{1}\otimes\hat{C}_{2}\hat{C}_{1}\hat{\rho}_{\phi,0}\hat{S}_{1}\hat{C}_{2}$
$\displaystyle\quad+\hat{\mu}_{2}\hat{\rho}_{\text{D,0}}\hat{\mu}_{1}\otimes\hat{S}_{2}\hat{C}_{1}\hat{\rho}_{\phi,0}\hat{S}_{1}\hat{C}_{2}-\hat{\rho}_{\text{D,0}}\hat{\mu}_{1}\hat{\mu}_{2}\otimes\hat{C}_{2}\hat{C}_{1}\hat{\rho}_{\phi,0}\hat{S}_{1}\hat{S}_{2}$
$\displaystyle\quad+\mathsf{i}\hat{\mu}_{2}\hat{\rho}_{\text{D,0}}\hat{\mu}_{1}\hat{\mu}_{2}\otimes\hat{S}_{2}\hat{C}_{1}\hat{\rho}_{\phi,0}\hat{S}_{1}\hat{S}_{2}\,,$
(69)

and the final density matrix for the detector, $\hat{\rho}_{\text{D,2}}$, can
be obtained by tracing out the field degree of freedom:

$\displaystyle\hat{\rho}_{\text{D,2}}$
$\displaystyle=\operatorname{Tr}_{\phi}[\hat{\rho}_{\text{tot,2}}]$
$\displaystyle=\hat{\rho}_{\text{D,0}}\omega(C_{1}C_{2}^{2}C_{1})+\mathsf{i}[\hat{\rho}_{\text{D,0}},\hat{\mu}_{2}]\omega(C_{1}S_{2}C_{2}C_{1})+\hat{\mu}_{2}\hat{\rho}_{\text{D,0}}\hat{\mu}_{2}\omega(C_{1}S_{2}^{2}C_{1})$
$\displaystyle\quad+\hat{\mu}_{1}\hat{\rho}_{\text{D,0}}\hat{\mu}_{1}\omega(S_{1}C_{2}^{2}S_{1})+\mathsf{i}[\hat{\mu}_{1}\hat{\rho}_{\text{D,0}}\hat{\mu}_{1},\hat{\mu}_{2}]\omega(S_{1}S_{2}C_{2}S_{1})+\hat{\mu}_{2}\hat{\mu}_{1}\hat{\rho}_{\text{D,0}}\hat{\mu}_{1}\hat{\mu}_{2}\omega(S_{1}S_{2}^{2}S_{1})$
$\displaystyle\quad-\mathsf{i}\hat{\mu}_{1}\hat{\rho}_{\text{D,0}}\omega(C_{1}C_{2}^{2}S_{1})+[\hat{\mu}_{1}\hat{\rho}_{\text{D,0}},\hat{\mu}_{2}]\omega(C_{1}S_{2}C_{2}S_{1})-\mathsf{i}\hat{\mu}_{2}\hat{\mu}_{1}\hat{\rho}_{\text{D,0}}\hat{\mu}_{2}\omega(C_{1}S_{2}^{2}S_{1})$
$\displaystyle\quad+\mathsf{i}\hat{\rho}_{\text{D,0}}\hat{\mu}_{1}\omega(S_{1}C_{2}^{2}C_{1})-[\hat{\rho}_{\text{D,0}}\hat{\mu}_{1},\hat{\mu}_{2}]\omega(S_{1}S_{2}C_{2}C_{1})+\mathsf{i}\hat{\mu}_{2}\hat{\rho}_{\text{D,0}}\hat{\mu}_{1}\hat{\mu}_{2}\omega(S_{1}S_{2}^{2}C_{1})\,,$
(70)

where we used $[\hat{C}_{j},\hat{S}_{j}]=0$ at the end.

We now calculate each algebraic state. As we did in Section 5.1.1, we write
$S_{j}$ and $C_{j}$ in terms of the elements of the Weyl algebra, Eq. (37).
Each algebraic state in (70) takes the form

$\displaystyle\omega\left(\dfrac{W_{\mathfrak{f}_{1}}+(-1)^{r}W_{\mathfrak{f}_{1}}^{*}}{2\mathsf{i}^{r}}\dfrac{W_{\mathfrak{f}_{2}}+(-1)^{s}W_{\mathfrak{f}_{2}}^{*}}{2\mathsf{i}^{s}}\dfrac{W_{\mathfrak{f}_{2}}+(-1)^{u}W_{\mathfrak{f}_{2}}^{*}}{2\mathsf{i}^{u}}\dfrac{W_{\mathfrak{f}_{1}}+(-1)^{v}W_{\mathfrak{f}_{1}}^{*}}{2\mathsf{i}^{v}}\right)\,,$
(71)

where $r,s,u,v\in\\{1,\,-1\\}$. A straightforward calculation ends up with

$\displaystyle\omega\left(\dfrac{W_{\mathfrak{f}_{1}}+(-1)^{r}W_{\mathfrak{f}_{1}}^{*}}{2\mathsf{i}^{r}}\dfrac{W_{\mathfrak{f}_{2}}+(-1)^{s}W_{\mathfrak{f}_{2}}^{*}}{2\mathsf{i}^{s}}\dfrac{W_{\mathfrak{f}_{2}}+(-1)^{u}W_{\mathfrak{f}_{2}}^{*}}{2\mathsf{i}^{u}}\dfrac{W_{\mathfrak{f}_{1}}+(-1)^{v}W_{\mathfrak{f}_{1}}^{*}}{2\mathsf{i}^{v}}\right)$
$\displaystyle=\dfrac{1}{16\mathsf{i}^{r+s+u+v}}$
$\displaystyle\times\Big{[}\omega(W_{2\mathfrak{f}_{1}+2\mathfrak{f}_{2}})+(-1)^{r}e^{2\mathsf{i}E(\mathfrak{f}_{1},\mathfrak{f}_{2})}\omega(W_{2\mathfrak{f}_{2}})+(-1)^{v}e^{-2\mathsf{i}E(\mathfrak{f}_{1},\mathfrak{f}_{2})}\omega(W_{2\mathfrak{f}_{2}})+(-1)^{r+v}\omega(W_{2\mathfrak{f}_{1}-2\mathfrak{f}_{2}}^{*})$
$\displaystyle\quad+(-1)^{s}\omega(W_{2\mathfrak{f}_{1}})+(-1)^{u}\omega(W_{2\mathfrak{f}_{1}})+(-1)^{r+s}+(-1)^{r+u}+(-1)^{s+v}+(-1)^{u+v}$
$\displaystyle\quad+(-1)^{r+s+v}\omega(W_{2\mathfrak{f}_{1}}^{*})+(-1)^{r+u+v}\omega(W_{2\mathfrak{f}_{1}}^{*})+(-1)^{s+u}\omega(W_{2\mathfrak{f}_{1}-2\mathfrak{f}_{2}})$
$\displaystyle\quad+(-1)^{r+s+u}e^{-2\mathsf{i}E(\mathfrak{f}_{1},\mathfrak{f}_{2})}\omega(W_{2\mathfrak{f}_{2}}^{*})+(-1)^{s+u+v}e^{2\mathsf{i}E(\mathfrak{f}_{1},\mathfrak{f}_{2})}\omega(W_{2\mathfrak{f}_{2}}^{*})$
$\displaystyle\quad+(-1)^{r+s+u+v}\omega(W_{2\mathfrak{f}_{1}+2\mathfrak{f}_{2}}^{*})\Big{]}\,.$
(72)

Here, we used the fact that $\omega:\mathcal{W}(\mathcal{M})\to\mathbb{C}$ is
a linear map, and utilized an identity

$\displaystyle
W_{a\mathfrak{f}_{1}}W_{b\mathfrak{f}_{2}}W_{c\mathfrak{f}_{2}}W_{d\mathfrak{f}_{1}}=e^{-\mathsf{i}(a-d)(b+c)E(\mathfrak{f}_{1},\mathfrak{f}_{2})/2}W_{(a+d)\mathfrak{f}_{1}+(b+c)\mathfrak{f}_{2}}\,,$
(73)

for $a,b,c,d\in\mathbb{R}$, which can be derived from the Weyl relations (17).

So far, we have assumed that the initial state of the field is arbitrary. We
now specify $\hat{\rho}_{\phi,0}$ to be a quasi-free state (20), and define

$\displaystyle\nu_{f_{j}}\equiv\omega(W_{2\mathfrak{f}_{j}})=e^{-2\mathsf{W}(\mathfrak{f}_{j},\mathfrak{f}_{j})}\in(0,1]\,,$
(74)

which leads to

$\displaystyle\omega\left(\dfrac{W_{\mathfrak{f}_{1}}+(-1)^{r}W_{\mathfrak{f}_{1}}^{*}}{2\mathsf{i}^{r}}\dfrac{W_{\mathfrak{f}_{2}}+(-1)^{s}W_{\mathfrak{f}_{2}}^{*}}{2\mathsf{i}^{s}}\dfrac{W_{\mathfrak{f}_{2}}+(-1)^{u}W_{\mathfrak{f}_{2}}^{*}}{2\mathsf{i}^{u}}\dfrac{W_{\mathfrak{f}_{1}}+(-1)^{v}W_{\mathfrak{f}_{1}}^{*}}{2\mathsf{i}^{v}}\right)$
$\displaystyle=\dfrac{1}{16\mathsf{i}^{r+s+u+v}}$
$\displaystyle\times\Big{(}[1+(-1)^{r+s+u+v}]\nu_{\mathfrak{f}_{1}+\mathfrak{f}_{2}}+[(-1)^{r}+(-1)^{s+u+v}]e^{2\mathsf{i}E(\mathfrak{f}_{1},\mathfrak{f}_{2})}\nu_{\mathfrak{f}_{2}}$
$\displaystyle\quad+[(-1)^{v}+(-1)^{r+s+u}]e^{-2\mathsf{i}E(\mathfrak{f}_{1},\mathfrak{f}_{2})}\nu_{\mathfrak{f}_{2}}+[(-1)^{r+v}+(-1)^{s+u}]\nu_{\mathfrak{f}_{1}-\mathfrak{f}_{2}}$
$\displaystyle\quad+[(-1)^{s}+(-1)^{u}+(-1)^{r+s+v}+(-1)^{r+u+v}]\nu_{\mathfrak{f}_{1}}$
$\displaystyle\quad+(-1)^{r+s}+(-1)^{r+u}+(-1)^{s+v}+(-1)^{u+v}\Big{)}\,.$
(75)

One can directly check that if one of $r,s,u,v$ has a different value than the
rest of the three then the whole term vanishes. This corresponds to, for
instance, $\omega(C_{1}S_{2}C_{2}C_{1})=0$ and so on. Therefore, if the field
is initially in a quasi-free state, the final density operator of the detector
(70) reads

$\displaystyle\hat{\rho}_{\text{D,2}}$
$\displaystyle=\hat{\rho}_{\text{D,0}}\omega(C_{1}C_{2}^{2}C_{1})+\hat{\mu}_{2}\hat{\rho}_{\text{D,0}}\hat{\mu}_{2}\omega(C_{1}S_{2}^{2}C_{1})+\hat{\mu}_{1}\hat{\rho}_{\text{D,0}}\hat{\mu}_{1}\omega(S_{1}C_{2}^{2}S_{1})$
$\displaystyle\quad+\hat{\mu}_{2}\hat{\mu}_{1}\hat{\rho}_{\text{D,0}}\hat{\mu}_{1}\hat{\mu}_{2}\omega(S_{1}S_{2}^{2}S_{1})+[\hat{\mu}_{1}\hat{\rho}_{\text{D,0}},\hat{\mu}_{2}]\omega(C_{1}S_{2}C_{2}S_{1})-[\hat{\rho}_{\text{D,0}}\hat{\mu}_{1},\hat{\mu}_{2}]\omega(S_{1}S_{2}C_{2}C_{1})\,,$
(76)

where each algebraic state is explicitly written as

$\displaystyle\omega(C_{1}C_{2}^{2}C_{1})=\dfrac{1}{16}\Big{[}4+4\nu_{\mathfrak{f}_{1}}+2\nu_{\mathfrak{f}_{1}-\mathfrak{f}_{2}}+2\nu_{\mathfrak{f}_{1}+\mathfrak{f}_{2}}+4\cos(2E(\mathfrak{f}_{1},\mathfrak{f}_{2}))\nu_{\mathfrak{f}_{2}}\Big{]}\,,$
(77a)
$\displaystyle\omega(C_{1}S_{2}^{2}C_{1})=\dfrac{1}{16}\Big{[}4+4\nu_{\mathfrak{f}_{1}}-2\nu_{\mathfrak{f}_{1}-\mathfrak{f}_{2}}-2\nu_{\mathfrak{f}_{1}+\mathfrak{f}_{2}}-4\cos(2E(\mathfrak{f}_{1},\mathfrak{f}_{2}))\nu_{\mathfrak{f}_{2}}\Big{]}\,,$
(77b)
$\displaystyle\omega(S_{1}C_{2}^{2}S_{1})=\dfrac{1}{16}\Big{[}4-4\nu_{\mathfrak{f}_{1}}-2\nu_{\mathfrak{f}_{1}-\mathfrak{f}_{2}}-2\nu_{\mathfrak{f}_{1}+\mathfrak{f}_{2}}+4\cos(2E(\mathfrak{f}_{1},\mathfrak{f}_{2}))\nu_{\mathfrak{f}_{2}}\Big{]}\,,$
(77c)
$\displaystyle\omega(S_{1}S_{2}^{2}S_{1})=\dfrac{1}{16}\Big{[}4-4\nu_{\mathfrak{f}_{1}}+2\nu_{\mathfrak{f}_{1}-\mathfrak{f}_{2}}+2\nu_{\mathfrak{f}_{1}+\mathfrak{f}_{2}}-4\cos(2E(\mathfrak{f}_{1},\mathfrak{f}_{2}))\nu_{\mathfrak{f}_{2}}\Big{]}\,,$
(77d)
$\displaystyle\omega(C_{1}C_{2}S_{2}S_{1})=\omega(C_{1}S_{2}C_{2}S_{1})=\dfrac{1}{16}\Big{[}2\nu_{\mathfrak{f}_{1}-\mathfrak{f}_{2}}-2\nu_{\mathfrak{f}_{1}+\mathfrak{f}_{2}}-4\mathsf{i}\sin(2E(\mathfrak{f}_{1},\mathfrak{f}_{2}))\nu_{\mathfrak{f}_{2}}\Big{]}\,,$
(77e)
$\displaystyle\omega(S_{1}C_{2}S_{2}C_{1})=\omega(S_{1}S_{2}C_{2}C_{1})=\dfrac{1}{16}\Big{[}2\nu_{\mathfrak{f}_{1}-\mathfrak{f}_{2}}-2\nu_{\mathfrak{f}_{1}+\mathfrak{f}_{2}}+4\mathsf{i}\sin(2E(\mathfrak{f}_{1},\mathfrak{f}_{2}))\nu_{\mathfrak{f}_{2}}\Big{]}\,.$
(77f)

One can further simplify these by using the following:

$\displaystyle\nu_{\mathfrak{f}_{1}-\mathfrak{f}_{2}}+\nu_{\mathfrak{f}_{1}+\mathfrak{f}_{2}}$
$\displaystyle=2\nu_{\mathfrak{f}_{1}}\nu_{\mathfrak{f}_{2}}\cosh[4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})]\,,$
(78a)
$\displaystyle\nu_{\mathfrak{f}_{1}-\mathfrak{f}_{2}}-\nu_{\mathfrak{f}_{1}+\mathfrak{f}_{2}}$
$\displaystyle=2\nu_{\mathfrak{f}_{1}}\nu_{\mathfrak{f}_{2}}\sinh[4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})]\,.$
(78b)

Here, $\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})\in\mathbb{R}$ is given in
(21). This can be proven as follows. First, we recall that
$\nu_{\mathfrak{f}}=\omega(W_{2\mathfrak{f}})$. For quasi-free states, we can
write this as

$\displaystyle\nu_{\mathfrak{f}}$
$\displaystyle=\omega_{\mu}(W_{2\mathfrak{f}})=e^{-2\mu(E\mathfrak{f},E\mathfrak{f})}\,.$
(79)

Also, from the linearity of the causal propagator operator,
$E(f+f^{\prime})=Ef+Ef^{\prime}$, we have

$\displaystyle\nu_{\mathfrak{f}_{1}-\mathfrak{f}_{2}}+\nu_{\mathfrak{f}_{1}+\mathfrak{f}_{2}}$
$\displaystyle=e^{-2\mu(E(\mathfrak{f}_{1}-\mathfrak{f}_{2}),E(\mathfrak{f}_{1}-\mathfrak{f}_{2}))}+e^{-2\mu(E(\mathfrak{f}_{1}+\mathfrak{f}_{2}),E(\mathfrak{f}_{1}+\mathfrak{f}_{2}))}$
$\displaystyle=e^{-2\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{1})}e^{2\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})}e^{2\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})}e^{-2\mu(E\mathfrak{f}_{2},E\mathfrak{f}_{2})}$
$\displaystyle\quad+e^{-2\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{1})}e^{-2\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})}e^{-2\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})}e^{-2\mu(E\mathfrak{f}_{2},E\mathfrak{f}_{2})}$
$\displaystyle=\nu_{\mathfrak{f}_{1}}\nu_{\mathfrak{f}_{2}}e^{4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})}+\nu_{\mathfrak{f}_{1}}\nu_{\mathfrak{f}_{2}}e^{-4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})}$
$\displaystyle=2\nu_{\mathfrak{f}_{1}}\nu_{\mathfrak{f}_{2}}\cosh[4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})]\,.$
(80)

Then the aforementioned algebraic states are reduced to

$\displaystyle\omega(C_{1}C_{2}^{2}C_{1})=\dfrac{1}{4}\Big{[}1+\nu_{\mathfrak{f}_{1}}+\nu_{\mathfrak{f}_{1}}\nu_{\mathfrak{f}_{2}}\cosh[4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})]+\nu_{\mathfrak{f}_{2}}\cos(2E(\mathfrak{f}_{1},\mathfrak{f}_{2}))\Big{]}\,,$
(81a)
$\displaystyle\omega(C_{1}S_{2}^{2}C_{1})=\dfrac{1}{4}\Big{[}1+\nu_{\mathfrak{f}_{1}}-\nu_{\mathfrak{f}_{1}}\nu_{\mathfrak{f}_{2}}\cosh[4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})]-\nu_{\mathfrak{f}_{2}}\cos(2E(\mathfrak{f}_{1},\mathfrak{f}_{2}))\Big{]}\,,$
(81b)
$\displaystyle\omega(S_{1}C_{2}^{2}S_{1})=\dfrac{1}{4}\Big{[}1-\nu_{\mathfrak{f}_{1}}-\nu_{\mathfrak{f}_{1}}\nu_{\mathfrak{f}_{2}}\cosh[4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})]+\nu_{\mathfrak{f}_{2}}\cos(2E(\mathfrak{f}_{1},\mathfrak{f}_{2}))\Big{]}\,,$
(81c)
$\displaystyle\omega(S_{1}S_{2}^{2}S_{1})=\dfrac{1}{4}\Big{[}1-\nu_{\mathfrak{f}_{1}}+\nu_{\mathfrak{f}_{1}}\nu_{\mathfrak{f}_{2}}\cosh[4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})]-\nu_{\mathfrak{f}_{2}}\cos(2E(\mathfrak{f}_{1},\mathfrak{f}_{2}))\Big{]}\,,$
(81d)
$\displaystyle\omega(C_{1}C_{2}S_{2}S_{1})=\omega(C_{1}S_{2}C_{2}S_{1})=\dfrac{\nu_{\mathfrak{f}_{2}}}{4}\Big{[}\nu_{\mathfrak{f}_{1}}\sinh[4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})]-\mathsf{i}\sin(2E(\mathfrak{f}_{1},\mathfrak{f}_{2}))\Big{]}\,,$
(81e)
$\displaystyle\omega(S_{1}C_{2}S_{2}C_{1})=\omega(S_{1}S_{2}C_{2}C_{1})=\dfrac{\nu_{\mathfrak{f}_{2}}}{4}\Big{[}\nu_{\mathfrak{f}_{1}}\sinh[4\mu(E\mathfrak{f}_{1},E\mathfrak{f}_{2})]+\mathsf{i}\sin(2E(\mathfrak{f}_{1},\mathfrak{f}_{2}))\Big{]}\,.$
(81f)

###### Acknowledgements.

KGY is thankful to Erickson Tjoa for reviewing this manuscript and Adam
Teixidó-Bonfill for a helpful comment.

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