# Efficiency of Harmonic Quantum Otto Engines at Maximal Power

## Abstract

**:**

## 1. Introdcution

## 2. Carnot Engine at Maximal Power

## 3. Endoreversible Otto Cycle

#### 3.1. Isentropic Compression

#### 3.2. Isochoric Heating

#### 3.3. Isentropic Expansion

#### 3.4. Isochoric Cooling

## 4. Classical Harmonic Engine

## 5. Quantum Harmonic Engine

## 6. Concluding Remarks

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Callen, H. Thermodynamics and an Introduction to Thermostastistics; Wiley: New York, NY, USA, 1985. [Google Scholar]
- Curzon, F.L.; Ahlborn, B. Efficiency of a Carnot engine at maximum power output. Am. J. Phys.
**1975**, 43, 22–24. [Google Scholar] [CrossRef] - Hoffmann, K.H.; Burzler, J.M.; Schubert, S. Endoreversible thermodynamics. J. Non-Equilib. Thermodyn.
**1997**, 22, 311–355. [Google Scholar] [CrossRef] - Leff, H.S. Thermal efficiency at maximum work output: New results for old heat engines. Am. J. Phys.
**1987**, 55, 602–610. [Google Scholar] [CrossRef] - Erbay, L.B.; Yavuz, H. Analysis of the Stirling heat engine at maximum power conditions. Energy
**1997**, 22, 645–650. [Google Scholar] [CrossRef] - Rezek, Y.; Kosloff, R. Irreversible performance of a quantum harmonic heat engine. New J. Phys.
**2006**, 8, 83. [Google Scholar] [CrossRef] - Abah, O.; Roßnagel, J.; Jacob, G.; Deffner, S.; Schmidt-Kaler, F.; Singer, K.; Lutz, E. Single-Ion Heat Engine at Maximum Power. Phys. Rev. Lett.
**2012**, 109, 203006. [Google Scholar] [CrossRef] [PubMed] - Roßnagel, J.; Abah, O.; Schmidt-Kaler, F.; Singer, K.; Lutz, E. Nanoscale Heat Engine Beyond the Carnot Limit. Phys. Rev. Lett.
**2014**, 112, 030602. [Google Scholar] [CrossRef] [PubMed] - Esposito, M.; Kawai, R.; Lindenberg, K.; Van den Broeck, C. Efficiency at Maximum Power of Low-Dissipation Carnot Engines. Phys. Rev. Lett.
**2010**, 105, 150603. [Google Scholar] [CrossRef] [PubMed] - Bonança, M.V.S. Approaching Carnot efficiency at maximum power in linear response regime. arXiv, 2018; arXiv:1809.09163. [Google Scholar]
- Roßnagel, J.; Dawkins, S.T.; Tolazzi, K.N.; Abah, O.; Lutz, E.; Schmidt-Kaler, F.; Singer, K. A single-atom heat engine. Science
**2016**, 352, 325–329. [Google Scholar] [CrossRef] [PubMed][Green Version] - Klaers, J.; Faelt, S.; Imamoglu, A.; Togan, E. Squeezed Thermal Reservoirs as a Resource for a Nanomechanical Engine beyond the Carnot Limit. Phys. Rev. X
**2017**, 7, 031044. [Google Scholar] [CrossRef] - Scovil, H.E.D.; Schulz-DuBois, E.O. Three-Level Masers as Heat Engines. Phys. Rev. Lett.
**1959**, 2, 262. [Google Scholar] [CrossRef] - Scully, M.O. Quantum Afterburner: Improving the Efficiency of an Ideal Heat Engine. Phys. Rev. Lett.
**2002**, 88, 050602. [Google Scholar] [CrossRef] [PubMed] - Scully, M.O.; Zubairy, M.S.; Agarwal, G.S.; Walther, H. Extracting Work from a Single Heat Bath via Vanishing Quantum Coherence. Science
**2003**, 299, 862–864. [Google Scholar] [CrossRef] [PubMed] - Scully, M.O.; Chapin, K.R.; Dorfman, K.E.; Kim, M.B.; Svidzinsky, A. Quantum heat engine power can be increased by noise-induced coherence. Proc. Natl. Acad. Sci. USA
**2011**, 108, 15097–15100. [Google Scholar] [CrossRef] [PubMed][Green Version] - Zhang, K.; Bariani, F.; Meystre, P. Quantum Optomechanical Heat Engine. Phys. Rev. Lett.
**2014**, 112, 150602. [Google Scholar] [CrossRef] [PubMed] - Gardas, B.; Deffner, S. Thermodynamic universality of quantum Carnot engines. Phys. Rev. E
**2015**, 92, 042126. [Google Scholar] [CrossRef] [PubMed] - Hardal, A.Ü.C.; Müstecaplıoğlu, Ö.E. Superradiant Quantum Heat Engine. Sci. Rep.
**2015**, 5, 12953. [Google Scholar] [CrossRef] [PubMed][Green Version] - Cavina, V.; Mari, A.; Giovannetti, V. Slow Dynamics and Thermodynamics of Open Quantum Systems. Phys. Rev. Lett.
**2017**, 119, 050601. [Google Scholar] [CrossRef] [PubMed][Green Version] - Roulet, A.; Nimmrichter, S.; Taylor, J.M. An autonomous single-piston engine with a quantum rotor. Quantum Sci. Technol.
**2018**, 3, 035008. [Google Scholar] [CrossRef][Green Version] - Cherubim, C.; Brito, F.; Deffner, S. Non-thermal quantum engine in transmon qubits. arXiv, 1810; arXiv:1810.04226. [Google Scholar]
- Niedenzu, W.; Mukherjee, V.; Ghosh, A.; Kofman, A.G.; Kurizki, G. Quantum engine efficiency bound beyond the second law of thermodynamics. Nat. Commun.
**2018**, 9, 165. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ronzani, A.; Karimi, B.; Senior, J.; Chang, Y.C.; Peltonen, J.T.; Chen, C.; Pekola, J.P. Tunable photonic heat transport in a quantum heat valve. Nat. Phys.
**2018**, 14, 991. [Google Scholar] [CrossRef] - Scully, M.O. Quantum Photocell: Using Quantum Coherence to Reduce Radiative Recombination and Increase Efficiency. Phys. Rev. Lett.
**2010**, 104, 207701. [Google Scholar] [CrossRef] [PubMed] - Dorfman, K.E.; Svidzinsky, A.A.; Scully, M.O. Increasing Photovoltaic Power by Noise Induced Coherence Between Intermediate Band States. Coherent Opt. Phenom.
**2013**, 1, 42–49. [Google Scholar] [CrossRef] - Einax, M.; Nitzan, A. Network Analysis of Photovoltaic Energy Conversion. J. Phys. Chem. C
**2014**, 118, 27226–27234. [Google Scholar] [CrossRef][Green Version] - Quan, H.T.; Liu, Y.X.; Sun, C.P.; Nori, F. Quantum thermodynamic cycles and quantum heat engines. Phys. Rev. E
**2007**, 76, 031105. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kosloff, R. Quantum Thermodynamics: A Dynamical Viewpoint. Entropy
**2013**, 15, 2100–2128. [Google Scholar] [CrossRef][Green Version] - Kosloff, R.; Rezek, Y. The Quantum Harmonic Otto Cycle. Entropy
**2017**, 19, 136. [Google Scholar] [CrossRef] - Spohn, H.; Lebowitz, J.L. Irreversible Thermodynamics for Quantum Systems Weakly Coupled to Thermal Reservoirs. Adv. Chem. Phys.
**1978**, 38, 109–142. [Google Scholar] [CrossRef]

**Figure 1.**Efficiency of the endoreversible Otto cycle at maximal power (red, solid line), together with the Curzon–Ahlborn efficiency (purple, dashed line) and the Carnot efficiency (blue, dotted line) in the high temperature limit, $\hslash {\omega}_{2}/{k}_{B}{T}_{c}=0.1$. Other parameters are ${\alpha}_{c}=1$, ${\alpha}_{h}=1$, and $\gamma =1$.

**Figure 2.**Efficiency of the endoreversible Otto cycle at maximal power (red, solid line), together with the Curzon–Ahlborn efficiency (purple, dashed line) and the Carnot efficiency (blue, dotted line) in the deep quantum regime, $\hslash {\omega}_{2}/{k}_{B}{T}_{c}=10$. Other parameters are ${\alpha}_{c}=1$, ${\alpha}_{h}=1$, and $\gamma =1$.

© 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Deffner, S. Efficiency of Harmonic Quantum Otto Engines at Maximal Power. *Entropy* **2018**, *20*, 875.
https://doi.org/10.3390/e20110875

**AMA Style**

Deffner S. Efficiency of Harmonic Quantum Otto Engines at Maximal Power. *Entropy*. 2018; 20(11):875.
https://doi.org/10.3390/e20110875

**Chicago/Turabian Style**

Deffner, Sebastian. 2018. "Efficiency of Harmonic Quantum Otto Engines at Maximal Power" *Entropy* 20, no. 11: 875.
https://doi.org/10.3390/e20110875