Quantum Otto engine driven by quantum fields

We consider a quantum Otto engine using an Unruh-DeWitt particle detector model which interacts with a quantum scalar field in curved spacetime. We express a generic condition for extracting positive work in terms of the effective temperature of the detector. This condition reduces to the well-known positive work condition in the literature under the circumstances where the detector reaches thermal equilibrium with the field. We then evaluate the amount of work extracted by the detector in two scenarios: an inertial detector in a thermal bath and a circulating detector in the Minkowski vacuum, which is inspired by the Unruh quantum Otto engine.


Introduction
Quantum thermodynamics is a study of thermodynamic phenomena from the perspective of quantum mechanics.Within this field quantum heat engines, where the working substance is quantum matter that interacts with thermal baths, have emerged as a crucial area of research [1].Notable among quantum heat engines are the quantum Carnot engine and the quantum Otto engine (QOE).The quantum Carnot engine operates through two isothermal processes and two quantum adiabatic processes.The QOE, on the other hand, is composed of two isochoric thermalization processes and two quantum adiabatic processes [1][2][3][4][5][6][7].For a comprehensive review, see [8].
Our interest lies in the QOE using a two-level quantum system in the realm of relativistic quantum information (RQI), where relativity is taken into account.In particular, we consider a specific model for a qubit known as the Unruh-DeWitt (UDW) particle detector model [9,10], which interacts with a quantum field in (curved) spacetime.The UDW detector model exhibits the essential features of the light-atom interaction as long as angular momenta are not exchanged [11,12].A UDW detector reveals interesting phenomena such as the Unruh effect [9], where a uniformly accelerating UDW detector perceives a thermal bath at the Unruh temperature T U = a/2π (where a is the magnitude of the acceleration), whilst an observer at rest sees a vacuum.
Investigations have been carried out previously on the QOE employing the UDW detector model.In [13], it was shown that a linearly accelerating UDW detector can be utilized to extract work in a QOE.This process is facilitated by the Unruh effect, which generates hot and cold thermal baths by varying the magnitude of acceleration a.Such an engine is commonly referred to as the Unruh QOE.The amount of work extracted is determined by a response function of the UDW detector, which is the probability of a transition from the ground state to the excited state, and vice versa.Subsequent studies have further expanded on the Unruh QOE in various scenarios, including a fermionic field [14], a degenerate detector [15], entangled detectors [16,17], and in the presence of a reflecting boundary [18].However, these studies focus on a specific acceleration trajectory, the state of a field, and a switching function.A comprehensive investigation of the QOE in a more general setting still remains largely unexplored.
In this paper, we explore the QOE within the framework of RQI in a generalized setting.We consider a pointlike UDW detector following an arbitrary trajectory in curved spacetime, interacting with a quantum scalar field in a quasi-free state.Thus, the thermalization of the detector is not necessarily required.By employing a perturbative method, we derive general expressions (3.15) and (3.20) for the work extracted through the quantum Otto cycle, which are written in terms of the response function of the detector.In addition, we identify conditions for extracting positive work in a generalized setting, given by (3.16) and (3.22).These conditions are expressed in terms of the effective temperature, an estimator for temperature, as perceived by the detector.Since thermality is not a prerequisite, the expressions for work extracted and the positive work condition are applicable to any situation.Notably, when the field is in a thermal state, known as the Kubo-Martin-Schwinger (KMS) state, and the interaction duration of a rapidly decreasing switching function is sufficiently long, the effective temperature becomes the KMS temperature.In such a case, the condition for positive work (3.22) becomes identical to the original condition found in [2,3].
Our derived expressions are demonstrated concretely within two scenarios: an inertial detector immersed in a KMS state of a quantum field in Minkowski spacetime and a detector in circular motion in the Minkowski vacuum.The first example is the most basic instance of thermality within QFT, for which we employ a Gaussian switching function and explore the effects of interaction duration.
The second example is inspired by the Unruh QOE previously examined in the literature, which has some challenges.A linearly accelerating detector requires an immense amount of acceleration to reach a temperature of 1 Kelvin.This can be seen from the expression of the Unruh temperature in SI units, T U = ℏa/2πk B c, where ℏ, k B , and c are respectively the reduced Planck constant, the Boltzmann constant, and the speed of light.It is well known that an acceleration of a ∼ 10 20 m/s 2 is required to achieve T U ∼ 1 K.This implies that significant work must be performed on the detector to extract a small amount of work from the Unruh QOE.Moreover, the duration of interaction is almost instantaneous.Typically, the Unruh QOE assumes that the detector initially travels at a constant speed v and then accelerates until it comes back to the same speed in the opposite direction.The time interval in this process (in terms of the detector's proper time τ ) is given by ∆τ = (2c/a)arctanh(v/c), and inserting a = 10 20 m/s 2 , it requires v ≈ c to yield ∆τ ∼ 1 s.Otherwise, if we demand a slower speed, say v = c/2, then the interaction duration amounts to ∆τ ∼ 10 −12 seconds, which is an extremely short period of time.The detector certainly cannot thermalize within such a time scale.Finally, executing such a protocol demands considerable space.The procedure in the Unruh QOE consists of linear acceleration and uniform motion at speed v.This means that it requires the detector to consistently move in one direction, which cannot be done in a confined laboratory.
Instead, one can consider a UDW detector in circular motion, a concept that is motivated by [19][20][21], where an experimental setup is proposed to measure the circular Unruh effect.See also [22][23][24][25].A circulating UDW detector overcomes the shortcomings described above, though the temperature induced by the acceleration is no longer T U .In fact, the detector cannot be thermalized.Instead, one should define the effective temperature T circ as observed by the circulating detector [24,26,27].This is where our proposed framework excels, as our expressions remain valid even in the absence of thermality.
The outline of this paper is as follows.After we introduce the basic aspects of the UDW detector and thermality in quantum field theory in 2, we review the mechanism of QOE in 3.1.Our main results are shown in 3.2, followed by examples in 4.1 and 4.2.

Unruh-DeWitt detector model
Consider a pointlike UDW detector, which is a two-level quantum system with an energy gap Ω between ground |g⟩ and excited states |e⟩ locally interacting with the quantum Klein-Gordon field φ(x).Here, the background spacetime is not restricted to a specific geometry but is assumed to be a (curved) spacetime that accommodates a well-behaved Klein-Gordon equation.
In the Schrödinger picture, the total Hamiltonian ĤS,tot reads ĤS,tot = ĤS,D + ĤS,ϕ + ĤS,int , ( where ĤS,D and ĤS,ϕ are the free Hamiltonians for the detector and the field, respectively, and ĤS,int is the interaction Hamiltonian.These are explicitly given by where λ is the coupling constant, τ is the proper time of the detector, and χ(τ /σ) is the switching function (with σ being the typical time scale of interaction), which determines the time dependence of coupling.In this paper, we assume that the L 2 norm of χ(τ ) is unity: Let Ĥτ I (τ ) be the interaction Hamiltonian in the interaction picture.Here, the Hamiltonian is the generator of time-translation with respect to the proper time τ , which is indicated by the superscript.The explicit form is given by where μ(τ ) is the monopole moment describing the internal dynamics of the detector: (2.5) Let us obtain the final state of the detector.The time-evolution operator in the interaction picture is given by where T τ is a time-ordering symbol with respect to the proper time τ .Carrying out the perturbative method, this operator can be expanded as Consider the initial state of the total system, where ρ ϕ is the initial state of the field and ρ D,0 is the detector's initial state given by The final total state ρ tot can be obtained as and by tracing out the field degree of freedom we obtain the final state of the detector, ρ D .In general, ρ D contains n-point correlation functions ⟨ φ(x 1 )... φ(x n )⟩ ρ ϕ , where ⟨•⟩ ρ ϕ is the expectation value with respect to ρ ϕ .In this paper, we are particularly interested in a quantum field in a quasi-free state,1 in which the one-point correlator ⟨ φ(x)⟩ ρ ϕ vanishes and every correlation function can be written in terms of two-point correlators W (x, x ′ ) := ⟨ φ(x) φ(x ′ )⟩ ρ ϕ , also known as the Wightman function.Examples of quasi-free states include the vacuum state |0⟩ and the Kubo-Martin-Schwinger state, which we will describe in the next subsection 2.2.Assuming that ρ ϕ is a quasi-free state, the post-interaction state of the detector to the leading order in λ can be calculated as follows. where with being the response function, i.e. the probability to excite the ground state |g⟩ to |e⟩.Similarly, F(−Ω) is the probability for the transition |e⟩ → |g⟩.The quantity W (x(τ ), x(τ ′ )) := ⟨ φ(x(τ )) φ(x(τ ′ ))⟩ ρ ϕ is the pullback of the Wightman function along the detector's trajectory x(τ ).Note that the odd-powers in λ vanish since the one-point correlators ⟨ φ(x)⟩ ρ ϕ are zero.We remark that ρ D is the state of the detector after interacting with the field once, while in QOE, the detector interacts with the field twice.Fortunately, Eq. (2.11) tells us that the density matrix ρ D stays diagonal after the interaction, meaning the final density matrix also remains diagonal after the second interaction.Nevertheless, careful attention must be paid to the switching timings of these two interactions.

KMS condition
In quantum mechanics, a thermal state is given by the Gibbs state ρ = e −β Ĥ /Z, where β is the inverse temperature, Ĥ is the Hamiltonian, and Z := Tr[e −β Ĥ ] is the partition function.In quantum field theory, the notion of thermality is rigorously defined by the Kubo-Martin-Schwinger (KMS) condition [28][29][30].The argument given in this section is summarized in Fig. 1.
Consider a Wightman function in a field state ρ ϕ : Note that the vacuum Wightman function ⟨0| φ(x(τ )) φ(x(τ ′ ))|0⟩ is a special case corresponding to ρ ϕ = |0⟩ ⟨0|.The Wightman function satisfies the KMS condition with respect to time τ at the KMS temperature In addition, if the KMS condition is satisfied, then the Wightman function is stationary, i.e., it only depends on the difference in time: W (τ, τ ′ ) = W (∆τ ).We identify the KMS temperature T KMS as the temperature of the quantum field.
The KMS condition is related to the so-called detailed balance condition.For any function f (ω), the detailed balance condition at temperature β −1 is given by (2.16) It is known [31] that, assuming the Wightman function is stationary, the KMS condition with respect to τ at temperature β −1 KMS is satisfied if and only if the Fourier transformed Wightman function, satisfies the detailed balance condition (2.16).
The detailed balance condition can be implemented into the response function of a UDW detector.Suppose the Wightman function satisfies the KMS condition and the switching function χ(τ /σ) has a typical interaction duration time scale σ.The response function (2.13) can be written as where ω is a dimensionless variable and χ(ω) is the Fourier transform of the switching function χ(τ /σ).One can show that if χ(ω) decays sufficiently fast as ω increases (such as a Gaussian switching), then the response function satisfies the detailed balance condition in the long interaction limit [31,32]: Inspired by the detailed balance condition for the response function (2.19), one may define the effective temperature T eff : The effective temperature is an estimator for the KMS temperature of the field.In particular, if the Wightman function obeys the KMS condition and the long interaction limit (σ → ∞) is taken, then the effective temperature becomes the KMS temperature due to (2.19).

Review: quantum Otto engine
In this subsection, we review the QOE following the paper by Kieu [3].Note that the whole system is composed of a qubit and thermal baths in the Gibbs state.In the next subsection, we will take the following protocol and apply it to the system composed of a UDW detector interacting with a quantum scalar field in a quasi-free state, namely, we will omit the assumption that the baths are in a thermal state.
1. Prepare a qubit with energy gap Ω 1 in an initial state where p ∈ [0, 1], as shown in Fig. 2.
3. Let the qubit interact with a bath at temperature T H .The qubit's Hamiltonian is time-independent Ĥ = Ω 2 |e⟩ ⟨e|, and so the work done on the qubit is zero: ⟨W 2 ⟩ = 0.After the interaction, the state becomes The first law of thermodynamics gives the dissipated heat: This is an isochoric thermalization process.
4. After isolating the qubit from the bath, perform an adiabatic contraction of the energy gap from Ω 2 to Ω 1 , while the state remains to be ρ 1 .From the same calculation as in step-2, we have The work is extracted from the qubit in this process.
5. Next another isochoric thermalization process is carried out: the qubit interacts with a bath at a colder temperature T C (< T H ). At the end of the process, the state becomes The total extracted work and dissipated heat read Assuming Ω 2 > Ω 1 , the extracted work is positive if δp H > 0.
It is known that the positive work condition reads [2,3]

Work extracted by a UDW detector
Now let us employ a UDW detector as the qubit in this process.As mentioned before, we will not assume that the baths are thermal.Instead, we only assume that the quantum field is in a quasi-free state, i.e., the one-point correlators ⟨ φ(x)⟩ ρ ϕ vanish.Nevertheless, we will keep using subscripts such as δp H and δp C for convenience.
In this subsection, we consider a general scenario in which a UDW detector following an arbitrary path in spacetime interacts with a quantum field in a quasi-free state.Working perturbatively, we derive the extracted work and the condition for it to be positive.

Relativistic quantum Otto engine
We first describe our QOE in the relativistic setting as depicted in Fig. 3.
• (step-1) Before the interaction, the detector with the energy gap Ω 1 is prepared in ρ D,0 in (2.9).

• (Step-2) One performs a quantum adiabatic expansion Ω
During this process, the detector does not interact with the field. ) ) (3.9c) While we identified the switching function as a compactly supported function, it is also possible to employ rapidly decreasing functions such as Gaussian or Lorentzian functions.These are not compactly supported, but the results remain unchanged when the exponentially decreasing tails of these functions are disregarded.We will make use of these functions later in this subsection.
• (Step-4) Adiabatic compression Ω 2 → Ω 1 .Again, the detector does not interact during this process and so the state remains as ρ D,1 .
• (Step-5) Implement another isochoric process using χ C [(τ − τ C )/σ C ] for the switching function.To ensure that the interaction does not overlap with the previous isochoric process, we make sure that the supports of the two switching functions are disjoint by implementing The state of the detector changes as Notice that δp C contains λ 4 terms since δp H is of order λ 2 .As in the traditional QOE, the work extracted reads ⟨W ext ⟩ = δp H ∆Ω.
Since ⟨W ext ⟩ only depends on δp H , it may appear that the cyclicity condition does not affect this value.However there is a subtlety.If we assume that an experimenter can freely choose a value of p ∈ [0, 1] at the beginning of the cycle, this means that the experimenter has to adjust the response functions F H (±Ω 2 ) and F C (±Ω 1 ) in such a way that the cyclicity condition is satisfied.On the other hand, if we allow the response functions to take any value, then the population p has to be adjusted accordingly.

When p is free to choose
We first require the cyclicity condition, δp H + δp C = 0, to close the thermodynamic cycle.Since δp C contains terms of order O(λ 4 ), we omit these by imposing 1 ≫ δp H /p for a given p(̸ = 0).This reduces to In particular, if the long-interaction limit σ H → ∞ is taken, we obtain Thus, in the long interaction duration, T eff H /Ω 2 ≪ 1 is required.Moreover, as a special case of a detector with a rapidly decreasing switching function interacting with the quantum field in the KMS state, this reads Nevertheless, if the interaction duration is finite, then λ ≪ 1 is sufficient for our perturbative analysis. 2nder this restriction, the cyclicity condition for any p ∈ (0, 1] leads to That is, one has to tune the response functions so that this equality is satisfied. We then consider the positive work condition in the case where the value of p is chosen freely by an experimenter.The work extracted (3.7) is and the positive work condition is (1 . We note that F H (−Ω 2 )/F H (Ω 2 ) > 1 since the de-excitation probability is higher than the excitation probability, thereby implying p ∈ (0, 1/2).This is consistent with previous work [13], where the Unruh QOE with a free p is able to extract work only when 0 < p < 1/2.The condition can be written in terms of the effective temperature (2.20): Recall that in the long interaction duration, T eff H /Ω 2 ≪ 1 is required, which means that (3.16) tells us that p ≪ 1 needs to be chosen if we wish to extract positive work in the long-interaction limit within the perturbation theory.
Notice that if positive work is extracted while the cyclicity condition is satisfied, then F C (±Ω 1 ) has to obey (1 − p)/p < F C (−Ω 1 )/F C (Ω 1 ) since the response functions are nonnegative and the positive work condition requires (1−p)/p > F H (−Ω 2 )/F H (Ω 2 ).Therefore, within the perturbation theory, The opposite is not always true since the inequality does not necessarily imply the cyclicity condition (3.14).In addition, the statement (3.17) does not hold without the cyclicity condition.Nevertheless, by contraposition, if T eff H /Ω 2 > T eff C /Ω 1 is violated then positive work cannot be extracted or the cycle is not closed.
The inequality T eff H /Ω 2 > T eff C /Ω 1 can be thought of as the generalization of the positive work condition found in [2,3] to the effective temperature.In fact, in the long-interaction limit, the effective temperatures become the KMS temperatures.In this particular scenario, the work extracted becomes (3.18)

When p is adjusted
Let us focus on the case where p is initially adjusted and derive the work extracted.We again assume that p ̸ = 0 and p ≫ δp H so that the terms higher than λ 4 can be omitted in δp C .From the cyclicity condition, we can solve for p, which reads By inserting p into (3.7)we obtain The extract work is positive if and only if We find that the positive work condition (3.21) can be written in terms of the effective temperature defined by (2.20) as Again, we are not necessarily assuming that the field is in the KMS state, and so the effective temperature is not necessarily the KMS temperature.If the field is in the KMS state and the detector with a rapidly decreasing switching function interacts with the field for a long time (σ H , σ C → ∞), the effective temperature becomes the KMS temperature, leading to which is the condition found in [2,3].The extracted work in this case is Therefore, (3.23) is a special case of (3.21) [and equivalently (3.22)] when the response function obeys the detailed balance condition (2.19).In other words, the condition (3.21) (or alternatively (3.22)) is applicable to any scenario, even when the detector is not thermalized.
In summary, we showed that within the perturbation theory, the positive work condition is associated with If p is not fixed and the cycle is closed by adjusting the response functions, T eff C /Ω 1 < T eff H /Ω 2 is a necessary condition to extract positive work.On the other hand, if we instead adjust p to close the cycle while the response functions are not fixed, then this condition is necessary and sufficient.This recovers the traditional QOE positive work condition [2,3] as a special case.Note that our results hold not only for a quantum field in the KMS state but also for any quasi-free state in curved spacetime.The use of the effective temperature is similar to other papers such as [33].

Examples
We now demonstrate our results in (3 + 1)-dimensional Minkowski spacetime.Throughout this section, we use a Gaussian switching function where τ j is the center of the Gaussian and σ > 0 is the characteristic Gaussian width, which has the units of time.

Inertial detector in thermal quantum field
Consider (3 + 1)-dimensional Minkowski spacetime.We wish to examine the extracted work (3.20) by coupling an inertial UDW detector to a KMS state of the field to explore the basic properties of the work extraction.
The Wightman function in a thermal state of a massless field can be written as [34] W where W M is the Wightman function in the Minkowski vacuum and W th is the contribution from the thermal state: Here, ϵ is the UV cutoff and β = T −1 is the inverse KMS temperature of the field.Assuming the Gaussian switching (4.1), the response function F β (Ω) of an inertial detector in the thermal bath reads [34] where are respectively the response functions for an inertial detector in the Minkowski vacuum and the thermal contribution.Note that it does not depend on the center of the Gaussian switching since the Wightman function is time-translation invariant.Below, we assume that the interaction durations are the same: σ H = σ C ≡ σ.

Case: free choice of p
We first consider the case where p is freely chosen as discussed in Sec.3.2.2.We plot 3, p = 0.315, and Ω 1 = 1 in Fig. 4(a-i).The dashed line represents the asymptotic value, lim σ→∞ ⟨W ext ⟩ /λ 2 σ, given by (3.18).We see that the work can be extracted when the interaction duration is short, but as σ increases extraction becomes exponentially difficult.This behavior can be explained by examining the positive work condition (3.16) in Fig. 4(a-ii).The figure shows that the positive work condition is satisfied when σ is small enough.When σ → ∞ is taken, the effective temperature becomes the KMS temperature: Moreover, based on the discussion in Sec.3.2.2, the perturbative analysis is valid in this limit when T H /Ω 2 ≪ 1, which is not the case in our scenario.Nevertheless, the perturbative analysis is valid in the figure as long as λ ≪ 1.
Although ⟨W ext ⟩ /λ 2 σ > 0 for some σ, the thermodynamic cycle is not closed.The cyclicity condition (3.14) when σ H = σ C reads We plot each side of this equation in Fig. 4(a-iii), fixing T C = 1 and other parameters (except σ), so as to find an optimal interaction duration σ such that the two curves cross.This point in the figure represents the scenario when the cyclicity condition is satisfied.
As one can see, the positive work condition in 4(a-ii) and the cyclicity condition in 4(a-iii) are not satisfied at the same time; hence the work extracted is not coming from a genuine QOE.In fact, Fig. 4(a-ii) tells us that T eff C /Ω 1 > T eff H /Ω 2 , and the statement (3.17) implies that either work cannot be extracted or that the thermodynamic cycle is not closed.
Meanwhile, we can find a set of parameters that enables us to extract positive work from a closed QOE.This scenario is demonstrated in Fig. 4(b), where T H = 4 is chosen.Figure 4(b-i) shows that the extracted work is positive for all σ, which is consistent with the positive work condition examined in 4(b-ii).This time, from Fig. 4(b-iii), there exists a value of σ that satisfies the cyclicity condition while ⟨W ext ⟩ /λ 2 σ > 0.Moreover, Figs.4(b-ii) and (b-iii) exemplify the statement (3.17): if positive work is extracted from a genuine QOE, then the effective temperatures satisfy but the opposite is not always true.

Case: adjusted p
We now consider the QOE described in Sec.3.2.3.We first note that an inertial detector in the Minkowski vacuum (i.e., zero temperature: T H = T C = 0) cannot extract work from vacuum fluctuations since the condition (3.21) cannot be satisfied no matter what Ω and σ are.We then consider the KMS temperatures T H , T C ̸ = 0 and compute the extracted work ⟨W ext ⟩ /λ 2 σ.The interaction duration σ is divided so that the perturbative analysis is valid-otherwise, the extracted work ⟨W ext ⟩ /λ 2 monotonically increases with σ without an upper bound.In Fig. 5(a), we plot the extracted work ⟨W ext ⟩ /λ 2 σ as a function of σ with various T H . Here, quantities are in units of the energy gap Ω 1 at the beginning of the cycle, and we fix Ω 2 /Ω 1 = 3 and T C /Ω 1 = 1.We observe that the detector cannot extract positive work when the interaction duration is too short (i.e., σΩ 1 ≪ 1), while ⟨W ext ⟩ /λ 2 σ asymptotes to a number at σΩ 1 ≫ 1.This is the opposite behavior of the result shown in Fig. 4, and can be attributed to the difference between the positive work conditions given by (3.16) and (3.22).In the case of free p, as illustrated in Fig. 4(b), H /Ω 2 is compared to a constant value.In contrast, for Fig. 5, T eff H /Ω 2 is compared to another effective temperature T eff C /Ω 1 as depicted in 5(b).The asymptotic value for each curve is depicted by the dashed line in Fig. 5(a), which is evaluated from (3.24).This confirms that, at the long interaction duration, the detailed balance condition for the response function (2.19) is satisfied.In addition, we observe that the work extracted when (T H /Ω 1 , Ω 2 /Ω 1 ) = (2, 3) and (3,3) is always non-positive, whereas it is possible to extract positive work from the system with (T H /Ω 1 , Ω 2 /Ω 1 ) = (4, 3).
This can be understood from the condition (3.23); when the detailed balance condition (2.19) is satisfied (σΩ 1 ≫ 1 in this case), the detector can extract positive work when the temperature of the baths and the energy gap obey (3.23).

UDW detector in circular motion
The Unruh QOE utilizes the thermality caused by the Unruh effect at the temperatures T H = a H /2π and T C = a C /2π with a H > a C in the protocol described in Sec. 3 [13].
However, as we pointed out in the Introduction, a linearly accelerating UDW detector is not ideal for extracting thermodynamic work since it requires a tremendous amount of acceleration a as well as a huge space to let the detector travel long distances.Instead, we wish to employ a UDW detector in circular motion, which not only allows us to confine the detector in a compact space, but also less work is required to acquire some temperatures compared to the linear acceleration case.Moreover, the Wightman function along a circulating detector's trajectory does not satisfy the KMS condition.Our general expression for the extracted work (3.15) and (3.20) as well as the positive work condition (3.16) and (3.22) can still be applied to such a scenario.Consider the circular trajectory of the UDW detector, which is given by Here, R(> 0) is the radius of the circle, ω is the angular velocity of the detector, and One can introduce the proper acceleration of the detector, whose magnitude a is given by a = Rω 2 γ 2 , as well as the speed of the detector v = Rω(≤ 1).Using these relations, one can write γ, ω, v in terms of a and R as follows.
In order to calculate the response function (2.13), one needs to have an expression of the pullback of the Wightman function along the trajectory (4.7).Consider a minimally coupled, massless quantum scalar field in (3 + 1)-dimensional Minkowski spacetime.The Wightman function in the Minkowski vacuum is given by (4.3a).Inserting (4.7), we obtain the pullback of the Wightman function: where ∆τ := τ −τ ′ .Since the Wightman function is time-translation invariant, the response functions do not depend on the switching time.

Extracted work
From the Wightman function above, we can numerically calculate the response functions and the work extracted for the detector in circular motion.In Fig. 6, we plot ⟨W ext ⟩ /λ 2 σ as a function of σ in units of the energy gap Ω 1 in step-1.Figures 6(a .However, we find in the latter case that, unlike in Sec.2.2, the extracted work does not reach a maximum at large σ.Instead, there exists a maximum at σΩ 1 ≈ 1.Nevertheless, we confirm that this behavior is contingent on the choice of parameters.We then plot ⟨W ext ⟩ /λ 2 σ as a function of the radius of the circle, log 10 R, in units of Ω 1 in Fig. 7. Here, we set Ω 2 = 2, σ = 1, a C = 1, and each figure includes curves for a H = 3, 4, and 5.As before, Fig. 7(a) corresponds to QOE with p = 1/20, while 7(b) depicts the result with a tuned p.Both figures exhibit a similar pattern; a detector orbiting with a small radius R has difficulty extracting work, and ⟨W ext ⟩ /λ 2 σ is maximized as R increases, eventually asymptoting to some value at R → ∞.Thus, the radius R of the orbit must be optimized to extract the maximum amount of work.

Fixed acceleration
Although the analytic expression for the effective temperature for a circulating detector, T circ , cannot be obtained in general, one can impose an additional assumption to obtain the exact form of T circ .Suppose that the detector is circulating at high speed v ≈ 1 (i.e., ultrarelativistic regime), which can be realized by taking a large radius R. The effective temperature in this case is known to be [24,27] While the effective temperature T circ increases with a, it is also possible to enhance the temperature by enlarging the energy gap Ω with fixed acceleration.That is, for a given a, We then ask if the circulating detector at high speed with a fixed acceleration a can extract positive work by manipulating the effective temperature by adjusting the energy gap.
The answer to this inquiry is negative.To see this, it is easy to check that, for a given

Conclusion
Inspired by the Unruh quantum Otto engines, we considered the quantum Otto engine (QOE) in a general setting utilizing an Unruh-DeWitt (UDW) detector interacting with a quantum scalar field.While the original QOE considered a qubit thermalized in thermal baths, we assumed (i) the field is in a quasi-free state, not necessarily in a Kubo-Martin-Schwinger (KMS) thermal state, and (ii) the detector follows an arbitrary trajectory in curved spacetime.By carrying out a perturbative method, we managed to derive the extracted work (3.15) and (3.20) in terms of the response function of the UDW detector.In addition, we found that the condition for extracting positive work can be expressed in terms of the effective temperature perceived by the detector, (3.16) and (3.22).These formulae are applicable to various scenarios outside (thermal) equilibrium.As a particular case, if the field is in the KMS state and the detector with a rapidly decreasing switching function interacts with it for a long time, the response function is known to obey the detailed balance condition, which makes the effective temperature identical to the KMS temperature.The condition (3.22) then becomes the positive work condition (3.23) originally found by [2,3].Using the formula for the extracted work, we explored two concrete examples: an inertial detector in the KMS state and a circulating detector in the Minkowski vacuum.
In the scenario of an inertial detector in thermal baths, we numerically examined the effect of the interaction duration.The relationship between the extracted work and the interaction duration is drastically changed by the choice of the detector's initial state.Nevertheless, the extracted work asymptotes to the value that corresponds to thermal equilibrium as the interaction duration increases.
Another example involves a circulating detector in the Minkowski vacuum.The motivation for this scenario comes from the previously proposed Unruh QOE [13], in which the thermal baths are generated by the Unruh effect.However, utilizing a linearly accelerating UDW detector in the Unruh QOE is not practical for several reasons such as the work necessary for acceleration and the requirement for a large space.The circulating detector, on the other hand, circumvents these issues.Moreover, the QOE with a circulating detector can be adequately examined only when our general framework is employed; The Wightman function pulled back along the circular trajectory does not satisfy the KMS condition, which suggests that the use of the effective temperature is necessary.
The work extracted by the circulating detector showed behavior similar to that of the inertial detector in a thermal bath.However, we found that there exist optimal values for the interaction duration and radius of the orbit to extract maximum work.Furthermore, we showed that the detector is unable to extract work with a fixed acceleration due to the violation of the positive work condition.
An interesting extension to our work would involve exploring quantum heat engines using non quasi-free states.In particular, coherent and squeezed states-which are not quasi-free-are known to be particular interest in a variety of settings.It would be interesting to see how a non-vanishing one-point correlator contributes to the cycle.

Figure 1 .
Figure 1.The relationship between the KMS state and the response function.

Figure 3 .
Figure 3.Each step in a relativistic QOE.The red and blue stripes indicate the isochoric process, i.e., the interaction between the detector and the field.

Figure 4 .
Figure 4.The extracted work and conditions for a UDW detector in a KMS state of a thermal quantum field for (a) T H = 2 and (b) T H = 4. Here, Ω 2 = 3, T C = 1, and p = 0.315 in units of Ω 1 .(a-i) and (b-i) depict ⟨W ext ⟩ /λ 2 σ as a function of the interaction duration σ in units of Ω 1 .The dashed line indicates the asymptotic value.The positive work condition is examined in (a-ii) and (b-ii), and the cyclicity condition is investigated in (a-iii) and (b-iii).

5 .
(a) The extracted work ⟨W ext ⟩ /λ 2 σ as a function of the interaction duration σ in units of Ω 1 .Here, T C = 1.The dashed lines are the work extracted (3.24) after the detailed balance condition is satisfied.(b) The effective temperature divided by the energy gap in the case of T H = 4 and Ω 2 = 3 in (a).The positive work condition is satisfied when

Figure 6 .
Figure 6.The extracted work ⟨W ext ⟩ /λ 2 σ by a detector in circular motion as a function of the interaction duration σ in units of Ω 1 .Here, Ω 2 = 1.01 and a C = 1.(a) The case where p is adjustable and p = 1/20, and (b) the case where p is tuned to complete the cycle.Each figure contains a H = 2 and a H = 3.
) and 6(b) are respectively the cases where p is freely chosen and tuned.In both figures, we set Ω 2 /Ω 1 = 1.01,RΩ 1 = 1, and a C /Ω 1 = 1, and each figure includes the curves for a H /Ω 1 = 2 and 3.As in the case of the inertial detector in a thermal bath, QOE with a free p (here p = 1/20) extracts more work when the interaction duration is short [Fig.6(a)],whereas if p is tuned to close the cycle, the opposite behavior holds [Fig.6(b)]

Figure 7 .
Figure 7. Work extracted by a circulating UDW detector as a function of the radius log 10 R when (a) p is adjustable (chosen to be p = 1/20), and (b) p is tuned to complete the cycle.Here, Ω 2 = 2, σ = 1, and a C = 1 in units of Ω 1 .Each figure contains three cases of a H : a H = 3, 4, and 5.
which violates the necessary condition for positive work,(3.16)and(3.22).Thus, we find that, for any p ∈ (0, 1), it is impossible to extract positive work from a UDW detector in a circular motion at v ≈ 1 by fixing a.This is one example of the feature of quantum heat engines: T C < T H does not necessarily guarantee the extraction of positive work in quantum heat engines, but rather, one should consider T C /Ω 1 < T H /Ω 2 , where Ω 1 < Ω 2 .