Performance of 3D quantum Otto engine with partial thermalization

We investigate the phenomenon of partial thermalization in the context of the efficiency at maximum power (EMP) for a quantum Otto engine. This engine utilizes Bose-Einstein Condensation in a cubic potential. The occurrence of partial thermalization is observed during a finite-time isochoric process, preventing the system from reaching an equilibrium state with the reservoirs and leaving it in a state of residual coherence. The engine’s performance can be evaluated based on its power output and EMP. The cubic potential is employed to induce energy excitation during the expansion and compression phases. The total energy is determined by the work done over a complete cycle. Utilizing Fourier’s law for heat conduction, we have determined that the power output is explicitly influenced by the duration of the heating and cooling strokes as well as the engine’s efficiency. Specifically, a longer stroke time and higher efficiency result in reduced power output. To calculate EMP, we optimize power by varying the compression ratio (κ), and we have found that EMP is also influenced by the isochoric heating and cooling processes. When varying the duration of the isochoric process, EMP shows a slight decrease as isochoric time increases due to entropy production. However, significant improvements in the EMP of the Otto Engine can be achieved by extending the cooling stroke time beyond the heating stroke time.


Introduction
Driven by the exploration of novel paradigms and systems that challenge our understanding of energy conversion at the quantum level, the field of quantum thermodynamics has made remarkable progress in recent years even reaching the practical level [1].The main idea is to discover the practical quantum engine, which is expected to be more efficient in performance and compact in size than the classical heat engine [2][3][4][5][6].Quantum heat engine (QHE) was pioneered by a duo of physicists expert in masers, i.e., Scovil and Schulz-DuBois in 1959 [7], a quantum analogue of the classical heat engine that operates in a cyclic manner to extract work from heat reservoirs while interacting with a quantum working substance [3,[8][9][10][11][12][13].
The quantum Carnot engine, a very ideal model of an engine whose highest efficiency, serves as a benchmark.The ideal Carnot cycle ensures there is no entropy production by operating the engine very slowly, the quasi-static state, and the cycle is reversibly processed.Yet this ideal cycle cannot be realized practically, since its shortfall in producing nearly zero power per cycle, until endoreversible was introduced by Curzon-Alhborn [14].Though the efficiency is a bit lesser than the quasi-static efficiency, the Curzon-Alhborn model produced a non-zero power output.Moreover, in the quantum regime, the application of endoreversible shows respectfully better efficiency than the classical [4,9].This idea can be applied to every thermodynamic cycle, following starting from Carnot cycle [15][16][17][18][19][20], Lenoir cycle [21][22][23], Otto cycle [3,[8][9][10][11][12][13]24], and Stirling cycle [25,26].
One such intriguing system is the quantum Otto engine.The engine operates through a series of thermodynamic processes, including isentropic expansion and compression, and two isochoric interactions with thermal reservoirs [27].By seeing through a microscopical view, the quantum material used as a working substance is believed to be able to boost the performance of the engine instead of using classical material [21,28,29].Bose-Einstein Condensate (BEC), a remarkable quantum state of matter, emerges as an ideal candidate for exploring the Quantum Otto Engine within a confined potential.BEC, composed of ultra-cold bosonic particles, exhibits wave-like properties and macroscopic coherence, making it an intriguing system for studying quantum thermodynamics [30].Its unique properties, such as the ability to occupy the ground state of the trapping potential, offer a way more eligible to achieve better efficiency than fermions due to its symmetrical advantages [31].
The quest for optimizing the performance of quantum heat engines has garnered significant attention in recent years, driven by both the fundamental principles of thermodynamics and the pursuit of efficient energy conversion technologies [1,22,[32][33][34][35][36].Among the various figures of merit used to assess the performance of quantum engines, the Efficiency at Maximum Power (EMP) stands out as a crucial indicator, offering insights into the delicate balance between power output and thermodynamic efficiency [8,9,14,37].The optimization of EMP is of paramount importance in practical applications of quantum engines.
By scrutinizing the EMP as a function of various control parameters, we aim to identify the optimal operating conditions that strike the ideal balance between extracting maximum work from the engine while maintaining a high level of thermodynamic efficiency for the design and optimization of quantum heat engines in future technologies.This manuscript is structured as follows: we begin by providing a theoretical framework for the Quantum Otto Engine trapped in a cubic potential with BEC as the working substance.We present the mathematical formalism and quantum mechanical description of the system, emphasizing the interplay between the trapping potential and the quantum thermodynamic processes.Subsequently, we delve into the thermodynamic performance of the engine, examining its efficiency at maximum power and time stroke under various operating conditions.Our analysis sheds light on the role of quantum statistics and confinement effects in shaping the engine's behavior.

Thermodynamic Properties of BEC in Cubic Potential
In this study, we utilize N -number of ideal Bose gas in the Bose-Einstein Condensate (BEC) phase as the working substance of the quantum Otto engine under the influence of potential as follows with x, y, and z are the coordinate of particle position and ε 0 referred as energy constant.
Here we consider the cubic potential as p = q = l = n → ∞. [38][39][40].In order to find the thermodynamic properties of BEC in cubic potential, we use the formulation of grand canonical potential for the bosonic system formulated as with z = e µ k B T as fugacity and β = 1/k B T as inverse thermal energy.By integrating Eq. ( 2), we can derive other thermodynamic properties; using the relation Ω = U − T S − µN [39].We obtained the entropy of BEC under the cubic potential by deriving Ω with respect to the T , written as with as Zeta function series.Given that BEC occurs when the fugacity, denoted as z, reaches its maximum value of 1, the Bose function at z = 1 can be substituted with the zeta function, ζ(p).The phase transition to BEC happens at a specific temperature known as the critical temperature.This critical temperature for BEC depends on factors such as the number of particles, the nature of the trapping potential, and the mass of the bosons [3,41,42].By differentiating Eq. ( 2) with respect to µ, we can obtain the critical temperature for BEC within a three-dimensional harmonic potential, Here from [30] we use N = 10 8 atom Rb-87 and N/V = 10 11 atom/m 3 .Furthermore, with V = 10 −9 m 3 , and m = 1.419 × 10 −25 kg, we have T c = 8, 33nK.As the system approaches the critical temperature, not all bosons instantaneously occupy the ground state; instead, they progressively populate it as the temperature decreases.The proportion of bosons that remain in an unoccupied state can be characterized using Eq. ( 5) [3], Another important thermodynamic parameter to determine is the internal energy.Substitute equations ( 2), (3), and (6) into the relation Ω = U − T S − µN , we derive the internal energy of a three-dimensional BEC in a cubic potential as follows, All these variables will be utilized in the context of the Otto cycle to determine the power and efficiency of the Otto engine.Internal energy and entropy are essential parameters for finding the amount of work, power, and efficiency of an engine.We will explain this in the subsequent section.

Endoreversible Quantum Otto Engine
The Otto cycle consists of four strokes: isochoric heating (1-4), isentropic expansion (4-3), isochoric cooling (3-2), and isentropic compression (2-1).In the classical Otto engine, expansion and compression are carried out by moving the piston.Unlike in the quantum Otto engine, expansion and compression are done by varying the external field [43], which is the inverse of the volume.During the isochoric stroke, the field is held constant, and the system is enabled to contact reservoirs.Meanwhile, in isentropic stroke the field is varied, but the system is isolated from the reservoirs.The process in each cycle is represented in Figure 1.The transition phase to BEC occurs at a specific critical temperature, which in the previous study [9] is about 8 nK.
Because of this, the temperature of the hot and cold reservoirs is assumed to be around this value in all simulations.
The fundamental difference between quasistatic and endoreversible cycles is in the isochoric stroke.In an endoreversible cycle, the working substance never reaches equilibrium with reservoirs due to finite time.To link the temperature of the working substance and reservoir, we used Fourier's law for conduction [3,44].The amount of work and heat during each process are discussed below.

Isochoric Heating
During isochoric heating, medium contacts with the hot reservoir while the external field is kept constant at V l .Since the medium will never reach the temperature of the reservoir, the temperature of the reservoir and the medium can be derived by using Fourier's conduction law [44] which fulfilled the boundary condition, as follows: at t = 0, the temperature of the medium is T 1 , while at the end of the heating process (after reaching τ h ), the temperature of the medium becomes T 4 .T (τ h ) = T 4 dan T (0) = T 1 have the solutions as T − T h = Ae −α h t .With those two conditions, we obtain: According to the first law of thermodynamics, the amount of heat transferred to the medium is,

Isentropic Expansion
After the heat flows, the medium expands from V l to V h , but the system is isolated from the reservoir, so there are no heat exchanges within this process.The value of work that occurs during the expansion is 10th Asian Physics Symposium (APS 2023) Journal of Physics: Conference Series 2734 (2024) 012031 Since there is no heat exchange within the expansion, then the entropy is also unchanged during the process.By using the Eq. ( 3), we can obtain the relation between volume and temperature, as follows

Isochoric Cooling
After the expansion, the medium is contacted back to the cold reservoir, thus the heat is flowed out from the medium.At the end of the process, the temperature of medium will not be the same as the reservoir, so we can use Fourier's law of conduction once again in this cooling process, By harnessing the boundary conditions at the beginning and the end of the isochoric process T (0) = T 3 and T (τ l ) = T 2 , we obtain

Isentropic Compression
After thermal contact with a cold reservoir, the medium is compressed adiabatically, so the entropy is constant within the process.Similar to the isentropic expansion, the amount of work in this process is defined as As before, we obtain the relation between volume and temperature as The relation of these temperatures and volume is important to eliminate the free variable at work and power formulations.

Power and Efficiency
Power, P is the total work produced during a cycle divided by the total time during the cycle.The total time in a cycle is the summation of time in the isochoric and isentropic processes.As referred to references [43,44], the power produced in a cycle is formulated below, where γ is the time multiplication constant.Using equations ( 11) and ( 15), we get the power during one cycle as below, 3 (e α h τ h +α l τ l − 1)

×
T l e α h τ h (e α l τ l − 1) + T h κ 2 3 (e α h τ h − 1) Efficiency is the fraction of work produced in a cycle to the amount of heat transferred to a working substance.The total work produced in a cycle is the summation of work in expansion and compression.The general formulation of efficiency is defined as [43,44], The higher the efficiency, the more heat can be converted into work.Substitute equations ( 11), (15), and ( 10) to (19), we get the formulation of efficiency Otto engine, where κ = V l /V h is a compression ratio.We see that efficiency only depends on the compression ratio and not on other physical properties of the medium.

Result and Discussion
In accordance with Eq. ( 18) for power, in addition to the reservoir temperature, power also depends on the compression ratio and the duration of the isochoric process.As the denominator indicates, a longer cycle will result in lower power generation.However, achieving high efficiency does not always guarantee high power output; the maximum efficiency is limited by the Carnot limit, specifically when η = 1 − (T l /T h ).Substituting this relationship into Eq.( 18), it becomes obvious that the power output decreases at the Carnot limit, making it unattainable in practical applications.Therefore, the search for the optimum efficiency, which is the efficiency when the power is at its maximum, known as the efficiency at maximum power (EMP), is crucial.EMP is determined by differentiating Eq. ( 18) with respect to κ, as expressed by Here, we kept τ l and τ h constant.The objective is to examine how the EMP evolves as these two parameters are varied.κ max is then substituted back into Eq.( 20) to determine the EMP.The representation of the EMP can be seen in Figure 2. It is important to highlight that the critical temperature of Bose-Einstein condensation (BEC) in a harmonic potential is about 8 nK, as indicated by previous research [3,9].Therefore, the temperatures of the hot and cold reservoirs we used were adjusted to match this value.In Figure 2, where we simultaneously vary τ l and τ h , we observe a slight decrease in EMP as τ increases.This finding is consistent with the results of [9].Due to the fact that power is inversely proportional to isochoric stroke time in Eq. ( 18), power decreases as both τ l and τ h increase, resulting in a decrease in efficiency at maximum power.This decrease in efficiency with increasing isochoric time is attributed to increasing entropy production.Higher entropy production implies a more irreversible cycle, and this irreversibility negatively affects the efficiency [45].However, during the cooling phase, entropy decreases, offsetting the entropy production during heating.This suggests that the decrease in EMP shown in Figure ( 2) is negligible, as the overall change in entropy for a cycle is also insignificant.Furthermore, the increase in entropy during heating does not exactly offset the decrease in entropy during cooling [33] due to the irreversible nature of the process.
The calculation of entropy production during isochoric heating is expressed using the relation ∆S heating = S(T 4 , V l ) − S(T 1 , V l ), which is derived by substituting Eq. ( 9) into Eq.( 3) as shown below, We keep the initial temperature at the start of isochoric heating (T 1 ) constant because we want to investigate how entropy changes as the temperature of the medium increases.Similarly, we use the same approach to determine the entropy production during isochoric cooling by substituting Eq. ( 14) into Eq.( 3), which gives the entropy change during the cooling process as follows, For the purpose of visualization, the temperature at the initial of isochoric cooling (T 3 ) is kept constant.We then plot Eq. ( 22) and ( 23) in Figure 3, where the red line represents the heating process (Figure 3.a) and the blue line represents the cooling process (Figure 3.b).Figure 3 shows that entropy increases as the heating stroke time increases and decreases as the cooling stroke time increases, eventually reaching a saturation point at thermal equilibrium.The initial rapid increase in entropy production during the heating stroke introduces quantum friction, resulting in a decrease in efficiency [33].However, changing the duration of the heating and cooling strokes with different values can have a significant impact on the EMP.In Figure 2.a, we provide a visual representation of EMP with different values of τ l and τ h .As can be seen in Figure 2.a, the EMP decreases significantly with longer heating stroke times, even approaching the Curzon-Ahlborn efficiency, but improves with longer cooling stroke times.Continuously increasing the cooling time does not always lead to a meaningful change in the EMP, as can be seen in Figure 2.a, where the EMP at τ l = 1 is almost identical to that at τ l = 5.This phenomenon is closely related to the entropy generation during the heating and cooling strokes, as shown in Figure 3.The generation of entropy during these strokes serves as a quantum signature of the system and depends on the intrinsic properties of the system.Different media will exhibit different behaviors in entropy production [16].Entropy production can also be related to the coherence effect [32].In the case of partial thermalization, this entropy production enhances the EMP through residual coherence.
In the context of Bose-Einstein condensates (BECs), coherence is associated with the simultaneous collapse of each atom's wave function into the lowest quantum state, forming a single macroscopic wave [46].The number of atoms in the condensed state is temperature dependent, with fewer atoms in the condensed state as the temperature approaches the critical point, and the highest condensate occurring at 0 K. Increasing the heating time excites atoms from the ground state, reducing the fraction of atoms in the condensate and consequently decreasing coherence and EMP.Conversely, increasing the cooling time causes atoms to return to the ground state, increasing coherence and EMP.To optimize efficiency, it is necessary to have different heating and cooling times, which means that the heating time should be slightly shorter than the cooling time.
However, it is important to note that higher efficiency does not necessarily mean higher power output.The highest efficiency is achieved in long and gradual processes where no entropy is generated throughout the cycle.At slow rates of expansion and compression, friction also becomes negligible, making the cycle perfectly ideal and reversible.However, in such cases, the power output drops to zero.To illustrate this, we carry out a simulation of dimensionless power (P * ) as a function of efficiency.We modify Eq. ( 18) by replacing (1 − κ 2 3 ) term from Eq. ( 20), resulting in the formulation of P * , 2 (e α h τ h +α l τ l − 1) In this context, we set α l = α h = 1.As a simplification, we assume the cycle operates with τ h = 5 and τ h = 0.1 namely as "Case A", while the cycle operates with τ h = 0.1 and τ l = 5, namely as "Case B" as shown in Figure 4. Case A achieves its maximum efficiency at a lower level compared to case B. However, the maximum power generated by case A is significantly higher than that of case B. This disparity arises because power generation depends on the fraction of atoms outside the condensate state [3].In the ideal Bose gas scenario, atoms in the condensate state (BEC) cannot contribute or receive work because their compressibility is infinite [39].Consequently, work can only be done by the fraction of atoms that are outside the condensed state, as their compressibility values remain finite.The more atoms that are in an excited state, the greater the potential for work production.As mentioned above, the proportion of excited atoms is temperature dependent, with higher temperatures approaching the critical point leading to a greater number of excited atoms.Therefore, even though the total isochoric times for cases A and B are the same, A will yield a greater power output than B. This result highlights the trade-off between efficiency and power.High efficiency in a motor does not guarantee high power, and engines with lower efficiency often produce higher power.

Conclusion
The findings indicate that the efficiency at maximum power (EMP) in a quantum Otto engine utilizing Bose-Einstein condensate (BEC) remains relatively stable when heating and cooling stroke times are equalized.This stability arises because the entropy generated during the heating phase is counterbalanced by the entropy production during the cooling phase.However, when different values of heating and cooling stroke times are employed, EMP can experience significant improvements or decreases.
Due to the interplay of entropy production and the coherence effect, EMP tends to be lower with long heating stroke times and short cooling stroke times but increases with short heating stroke times and long cooling stroke times.Despite the higher EMP in the latter case, the power output is considerably lower compared to scenarios with long heating stroke times and short cooling stroke times.This discrepancy arises from the fact that power generation relies on the fraction of atoms existing outside the condensate state, and this fraction is temperature-dependent.Higher temperatures lead to more excited atoms, resulting in greater power generation.
In an alternative scenario, EMP relates to the coherence effect which arises from the fraction of atoms within the condensate state.The more atoms present in the condensate, the higher the coherence of the system, leading to increased EMP.However, this interaction also introduces a finite compressibility to BEC, as work can be produced from atoms within the condensate.This aspect requires further investigation.

1 Figure 2 .
Figure 2. EMP as a function of T l in (a) variation of τ h and τ l , and (b) τ h = τ l in a time dimension, together with Curzon-Ahlborn (CA) efficiency as a comparison.The temperature of the hot reservoir is T h = 8nK.

Figure 3 .
Figure 3.The change of entropy during (a) the heating process as a function of τ h and (b) the change of entropy during the cooling process as a function τ l .The temperature of reservoirs are T h = 8 nK and T l = 2 nK.

Figure 4 .
Figure 4. Power as a function of efficiency with different values of isochoric cooling and heating stroke time. 2