# Quantum Mechanical Engine for the Quantum Rabi Model

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## Abstract

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## 1. Introduction

#### 1.1. Quantum Rabi Model

#### 1.2. First Law of Thermodynamic

#### 1.3. Cycle of Operation

## 2. Quantum Rabi Model as a Working Substance

#### 2.1. Case of $\xi \equiv g$

#### 2.2. Case of $\xi \equiv \omega $

#### 2.3. Case of $\xi \equiv \mathsf{\Omega}$

## 3. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Simulation of Isoenergetic Process

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**Figure 1.**Two lowest energy levels of the quantum Rabi model as a function of (

**a**) the coupling strength g, with $\omega =\mathsf{\Omega}$; (

**b**) the resonator frequency $\omega $, with $g=\mathsf{\Omega}$; and (

**c**) the TLS frequency $\mathsf{\Omega}$ with $g=\omega $. The Solid line denotes the exact diagonalization of Equation (1) and dashed line denotes the approximation given by Equation (2).

**Figure 2.**Diagram of the Isoenergetic cycle for (

**a**) $\xi \equiv g$; (

**b**) $\xi \equiv \omega $; and (

**c**) $\xi \equiv \mathsf{\Omega}$. Stages $1\to 2$ and $3\to 4$ correspond to isoenergetic processes, while stages $2\to 3$ and $4\to 1$ correspond to adiabatic processes.

**Figure 3.**(

**a**) ${g}_{2}$ as a function of ${g}_{1}$, given by the isoenergetic condition ${E}_{\mathbf{0}}\left({g}_{1}\right)={E}_{\mathbf{1}}\left({g}_{2}\right)$; (

**b**) ${g}_{4}$ as a function of ${g}_{1}$, where ${g}_{4}$ is obtained from the isoenergetic condition ${E}_{\mathbf{1}}\left({g}_{3}\right)={E}_{\mathbf{0}}\left({g}_{4}\right)$, and ${g}_{3}={\alpha}^{\left(g\right)}{g}_{2}$. We have chosen ${\alpha}^{\left(g\right)}=1.2$ (blue), ${\alpha}^{\left(g\right)}=1.4$ (red), ${\alpha}^{\left(g\right)}=1.6$ (yellow), ${\alpha}^{\left(g\right)}=1.8$ (purple), and ${\alpha}^{\left(g\right)}=2$ (green). The dots in Figure 3b indicate the threshold ${g}_{3}=2\phantom{\rule{0.166667em}{0ex}}\mathsf{\Omega}$ [57]. Solid lines denote the numerical calculation, and dashed lines are calculated with the approximated energy levels.

**Figure 4.**(

**a**) Energy exchange ${Q}_{\mathrm{in}}$ and (

**b**) total work extracted, ${W}_{\mathrm{total}}$, as a function of ${g}_{1}$, for ${\alpha}^{\left(g\right)}=1.2$ (blue), ${\alpha}^{\left(g\right)}=1.4$ (red), ${\alpha}^{\left(g\right)}=1.6$ (yellow), ${\alpha}^{\left(g\right)}=1.8$ (purple), and ${\alpha}^{\left(g\right)}=2$ (green). The dots in Figure 4b indicate the threshold ${g}_{3}=2\phantom{\rule{0.166667em}{0ex}}\mathsf{\Omega}$ [57]. Solid lines denote the numerical calculation, and dashed lines are calculated with the approximated energy levels.

**Figure 5.**Efficiency $\eta $ as function ${g}_{1}$ for ${\alpha}^{\left(g\right)}=1.2$ (blue), ${\alpha}^{\left(g\right)}=1.4$ (red), ${\alpha}^{\left(g\right)}=1.6$ (yellow), ${\alpha}^{\left(g\right)}=1.8$ (purple), and ${\alpha}^{\left(g\right)}=2$ (green). The dots indicate the threshold ${g}_{3}=2\phantom{\rule{0.166667em}{0ex}}\mathsf{\Omega}$ [57]. In both figures solid line denotes the exact numerical calculation, and dashed line is calculated with the approximated energy levels.

**Figure 6.**(

**a**) ${\omega}_{2}^{-1}$ as a function of ${\omega}_{1}^{-1}$ given by the isoenergetic condition ${E}_{\mathbf{0}}\left({\omega}_{1}\right)={E}_{\mathbf{1}}\left({\omega}_{2}\right)$; (

**b**) ${\omega}_{4}^{-1}$ as a function of ${\omega}_{1}^{-1}$, where ${\omega}_{4}$ is obtained from the isoenergetic condition ${E}_{\mathbf{1}}\left({\omega}_{3}\right)={E}_{\mathbf{0}}\left({\omega}_{4}\right)$, and ${\omega}_{3}={\alpha}^{\left(\omega \right)}{\omega}_{2}$. We have chosen ${\alpha}^{\left(\omega \right)}=0.75$ (blue), ${\alpha}^{\left(\omega \right)}=0.80$ (red), ${\alpha}^{\left(\omega \right)}=0.85$ (yellow), ${\alpha}^{\left(\omega \right)}=0.90$ (purple), and ${\alpha}^{\left(\omega \right)}=0.95$ (green). Solid lines denote the numerical calculation, and dashed lines are calculated with the approximated energy levels.

**Figure 7.**(

**a**) Energy exchange ${Q}_{\mathrm{in}}$ and (

**b**) total work extracted $\left({W}_{\mathrm{total}}\right)$ as a function of ${\omega}_{1}^{-1}$ for ${\alpha}^{\left(\omega \right)}=0.75$ (blue), ${\alpha}^{\left(\omega \right)}=0.8$ (red), ${\alpha}^{\left(\omega \right)}=0.85$ (yellow), ${\alpha}^{\left(\omega \right)}=0.90$ (purple), and ${\alpha}^{\left(\omega \right)}=0.95$ (green). Solid lines denote the numerical calculation, and dashed lines are calculated with the approximated energy levels.

**Figure 8.**Efficiency as function ${\omega}_{1}^{-1}$ for ${\alpha}^{\left(\omega \right)}=0.75$ (blue), ${\alpha}^{\left(\omega \right)}=0.8$ (red), ${\alpha}^{\left(\omega \right)}=0.85$ (yellow), ${\alpha}^{\left(\omega \right)}=0.90$ (purple), and ${\alpha}^{\left(\omega \right)}=0.95$ (green). Solid lines denote the numerical calculation, and dashed lines are calculated with the approximated energy levels.

**Figure 9.**(

**a**) shows ${\mathsf{\Omega}}_{2}^{-1}$ as a function of ${\mathsf{\Omega}}_{1}^{-1}$ given by the isoenergetic condition ${E}_{\mathbf{0}}\left({\mathsf{\Omega}}_{1}\right)={E}_{\mathbf{1}}\left({\mathsf{\Omega}}_{2}\right)$; (

**b**) shows ${\mathsf{\Omega}}_{4}^{-1}$ as a function of ${\mathsf{\Omega}}_{1}^{-1}$ where ${\mathsf{\Omega}}_{4}$ is obtained from the isoenergetic condition ${E}_{\mathbf{1}}\left({\mathsf{\Omega}}_{3}\right)={E}_{\mathbf{0}}\left({\mathsf{\Omega}}_{4}\right)$, and ${\mathsf{\Omega}}_{3}={\alpha}^{\left(\mathsf{\Omega}\right)}{\mathsf{\Omega}}_{2}$. We have chosen ${\alpha}^{\left(\mathsf{\Omega}\right)}=0.75$ (blue), ${\alpha}^{\left(\mathsf{\Omega}\right)}=0.8$ (red), ${\alpha}^{\left(\mathsf{\Omega}\right)}=0.85$ (yellow), ${\alpha}^{\left(\mathsf{\Omega}\right)}=0.90$ (purple), and ${\alpha}^{\left(\mathsf{\Omega}\right)}=0.95$ (green). In this case we have only considered the exact numerical calculation.

**Figure 10.**(

**a**) Energy exchange ${Q}_{\mathrm{in}}$ and (

**b**) total work extracted $\left({W}_{\mathrm{total}}\right)$ as a function of ${\mathsf{\Omega}}_{1}^{-1}$ for ${\alpha}^{\left(\mathsf{\Omega}\right)}=0.75$ (blue), ${\alpha}^{\left(\mathsf{\Omega}\right)}=0.8$ (red), ${\alpha}^{\left(\mathsf{\Omega}\right)}=0.85$ (yellow), ${\alpha}^{\left(\mathsf{\Omega}\right)}=0.90$ (purple), and ${\alpha}^{\left(\mathsf{\Omega}\right)}=0.95$ (green).

**Figure 11.**Efficiency as a function of ${\mathsf{\Omega}}_{1}^{-1}$ for different values of ${\alpha}^{\left(\mathsf{\Omega}\right)}$ given by ${\alpha}^{\left(\mathsf{\Omega}\right)}=0.75$ (blue), ${\alpha}^{\left(\mathsf{\Omega}\right)}=0.8$ (red), ${\alpha}^{\left(\mathsf{\Omega}\right)}=0.85$ (yellow), ${\alpha}^{\left(\mathsf{\Omega}\right)}=0.90$ (purple), and ${\alpha}^{\left(\mathsf{\Omega}\right)}=0.95$ (green).

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**MDPI and ACS Style**

Alvarado Barrios, G.; Peña, F.J.; Albarrán-Arriagada, F.; Vargas, P.; Retamal, J.C.
Quantum Mechanical Engine for the Quantum Rabi Model. *Entropy* **2018**, *20*, 767.
https://doi.org/10.3390/e20100767

**AMA Style**

Alvarado Barrios G, Peña FJ, Albarrán-Arriagada F, Vargas P, Retamal JC.
Quantum Mechanical Engine for the Quantum Rabi Model. *Entropy*. 2018; 20(10):767.
https://doi.org/10.3390/e20100767

**Chicago/Turabian Style**

Alvarado Barrios, Gabriel, Francisco J. Peña, Francisco Albarrán-Arriagada, Patricio Vargas, and Juan Carlos Retamal.
2018. "Quantum Mechanical Engine for the Quantum Rabi Model" *Entropy* 20, no. 10: 767.
https://doi.org/10.3390/e20100767