# Magnetically-Driven Quantum Heat Engines: The Quasi-Static Limit of Their Efficiency

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## Abstract

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## 1. Introduction

## 2. General Theory

#### 2.1. Relaxation to Equilibrium

#### 2.2. Quasi-Static Evolution

## 3. Magnetically-Driven Quantum Engine on a Quantum Dot Array

#### 3.1. The Single-Particle Spectrum in a Cylindrical Quantum Dot under an External Magnetic Field

#### 3.2. The Iso-Energetic Cycle

#### 3.3. The Quantum Carnot Cycle

## 4. A Magneto-Strain-Driven Quantum Engine on a Graphene Layer

#### 4.1. The Single-Particle Spectrum

#### 4.2. The Quantum Engine Cycle

## 5. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References and Notes

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**Figure 2.**(Color online) The single-particle spectrum for a cylindrical quantum dot in a constant magnetic field. Depicted are the branches relevant to the quantum heat engine (QHEN).

**Figure 3.**(Color online) The effective two-level system that allows the construction of the iso-energetic cycle.

**Figure 4.**(Color online) Pictorial description of the Magnetization versus an external magnetic field for the iso-energetic cycle.

**Figure 5.**(Color online) The efficiency of the iso-energetic cycle, calculated from Equation (50), is represented as a function of the expansion parameter $\alpha >1$. Different values of the initial magnetic field in the cycle ${B}_{1}$, expressed in terms of the number of flux quanta ${N}_{{\Phi}_{1}}$, are compared. We find that the asymptotic limit represented by Equation (51) (red dashed line in the figure) is achieved in practice for ${N}_{{\Phi}_{1}}>30$.

**Figure 6.**(Color online) The quantum Carnot cycle discussed in this section is pictorially represented. The isothermal trajectories are achieved by bringing the system into contact with macroscopic thermal reservoirs at temperatures ${T}_{H}>{T}_{C}$, respectively.

**Figure 7.**(Color online) The magnetization of the system changes as a function of the applied external magnetic field, along two isothermal and two iso-entropic trajectories of the cycle. The isothermal trajectories are achieved by bringing the system into contact with macroscopic thermal reservoirs at temperatures ${T}_{H}>{T}_{C}$, respectively.

**Figure 8.**(Color online) The phonon spectrum of single-layer graphene along the symmetry directions of the Brillouin zone. Calculated from a force constant model using the elastic parameters in [47].

**Figure 9.**(Color online) The deformation field that induces a uniform pseudo-magnetic field ${\mathbf{B}}_{S}$.

**Figure 10.**(Color online) The cycle is pictorially represented in the entropy (S) versus external magnetic field (B) coordinates. The cycle is composed of two iso-entropic trajectories and two trajectories at constant external magnetic field. The cold reservoir is at ${T}_{1}={T}_{C}$, whereas the hot reservoir is at ${T}_{3}={T}_{H}$.

**Figure 11.**(Color online) The efficiency of the cycle, as a function of the compression ratio $r\left({B}_{2}\right)$, for the case $\overline{\gamma}=0$ (red, dash-dotted line) compared to the case $\overline{\gamma}=1.7{\mu}_{B}$ (blue, solid line). Here ${B}_{1}=4\phantom{\rule{0.166667em}{0ex}}\mathrm{T}$, ${B}_{S}=20\phantom{\rule{0.166667em}{0ex}}\mathrm{T}$, and the temperatures at the reservoirs ${T}_{H}=100\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$, ${T}_{C}=30\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$, respectively.

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**MDPI and ACS Style**

Muñoz, E.; Peña, F.J.; González, A.
Magnetically-Driven Quantum Heat Engines: The Quasi-Static Limit of Their Efficiency. *Entropy* **2016**, *18*, 173.
https://doi.org/10.3390/e18050173

**AMA Style**

Muñoz E, Peña FJ, González A.
Magnetically-Driven Quantum Heat Engines: The Quasi-Static Limit of Their Efficiency. *Entropy*. 2016; 18(5):173.
https://doi.org/10.3390/e18050173

**Chicago/Turabian Style**

Muñoz, Enrique, Francisco J. Peña, and Alejandro González.
2016. "Magnetically-Driven Quantum Heat Engines: The Quasi-Static Limit of Their Efficiency" *Entropy* 18, no. 5: 173.
https://doi.org/10.3390/e18050173