# Carnot Cycles in a Harmonically Confined Ultracold Gas across Bose–Einstein Condensation

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## Abstract

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## 1. Introduction

## 2. Cycles

## Author Contributions

## Funding

## Institutional Review Board Statement

## Conflicts of Interest

## References

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**Figure 1.**Carnot Cycles in a $\Pi $-$\mathcal{V}$ diagram for a ${}^{87}$Rb gas that is harmonically trapped.

**Figure 2.**Typical dependence of values for global pressure and fitting using Equation (10). While above the critical temperature the system is a linear curve, below this, it is close to a fourth-order dependence with T. ($N=4.2\times {10}^{5}$, $\mathcal{V}=7.8\times {10}^{-9}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{3}$, ${T}_{c}=2.5\times {10}^{-7}\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$).

**Figure 3.**Calculated efficiency $\eta =\frac{W}{{Q}_{H}}$ versus Carnot efficiency $\eta =1-\frac{{T}_{L}}{{T}_{H}}$ for a series of cycles above and below the critical temperature; namely, those with no transition during the cycle. Despite having a condensed phase, the efficiency of the cycle maintains the expected efficiency for a Carnot cycle.

**Figure 4.**Calculated efficiency $\eta =\frac{W}{{Q}_{H}}$ versus Carnot efficiency $\eta =1-\frac{{T}_{L}}{{T}_{H}}$ for a series of cycles that cross the transition line.

**Figure 5.**Global isothermal compressibility ${\mathcal{K}}_{T}=-\frac{1}{\mathcal{V}}{\left(\frac{\partial \mathcal{V}}{\partial \Pi}\right)}_{N,T}$ as a function of $\mathcal{V}$ along a Carnot cycle, whose isothermal processes cross the transition line. These crosses are observed as discontinuities in the compressibility. In the inset, the compressibility is shown for a cycle that does not cross the transition.

**Figure 6.**An example of Ideal, Otto and Carnot Cycles in a $\Pi $-$\mathcal{V}$ diagram, within the same extreme temperatures, marked with large dots, ${T}_{H}$ and ${T}_{L}$. These are the same hot and cold temperatures that occur for the Carnot cycle. The Ideal cycle is composed of two isobaric curves and two isochores, while the Otto cycle has two adiabatic processes and two isochoric ones.

**Figure 7.**Ratio of the calculated efficiency to the Carnot one for a series of ideal, Otto and Carnot cycles, as a function of experimental efficiency. The ideal cycle efficiency is approximately $0.2$; the Carnot efficiency and the Otto cycle efficiency are $0.3$.

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**MDPI and ACS Style**

Reyes-Ayala, I.; Miotti, M.; Hemmerling, M.; Dubessy, R.; Perrin, H.; Romero-Rochin, V.; Bagnato, V.S.
Carnot Cycles in a Harmonically Confined Ultracold Gas across Bose–Einstein Condensation. *Entropy* **2023**, *25*, 311.
https://doi.org/10.3390/e25020311

**AMA Style**

Reyes-Ayala I, Miotti M, Hemmerling M, Dubessy R, Perrin H, Romero-Rochin V, Bagnato VS.
Carnot Cycles in a Harmonically Confined Ultracold Gas across Bose–Einstein Condensation. *Entropy*. 2023; 25(2):311.
https://doi.org/10.3390/e25020311

**Chicago/Turabian Style**

Reyes-Ayala, Ignacio, Marcos Miotti, Michal Hemmerling, Romain Dubessy, Hélène Perrin, Victor Romero-Rochin, and Vanderlei Salvador Bagnato.
2023. "Carnot Cycles in a Harmonically Confined Ultracold Gas across Bose–Einstein Condensation" *Entropy* 25, no. 2: 311.
https://doi.org/10.3390/e25020311