# Non-Thermal Quantum Engine in Transmon Qubits

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## Abstract

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## 1. Introduction

## 2. System Description

## 3. Non-Equilibrium Thermodynamics

#### 3.1. Non-Thermal Equilibrium States

#### 3.2. The Cycle

## 4. Work, Heat and Efficiency

## 5. Conclusions and Final Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Non-Thermal Equilibrium States

**Figure A1.**Stationary state’s elements ${\rho}_{\mathrm{T}}^{ee}$ and $|{\rho}_{\mathrm{T}}^{eg}|$ for different values of $({\omega}_{\mathrm{T}},{E}_{d})$. Important amounts of population and quantum coherence changes can be reached during the engine operation.

## Appendix B. Thermodynamic Quantities along Each Stroke

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**Figure 1.**Sketch of the quantum engine with a transmon qubit as working substance interacting with an externally pumped (E(t)) transmission line (cavity). Both systems are embedded in the same cryogenic environment, which plays the role of a standard thermal bath of temperature T. Such a setup gives a dynamics of a working substance in the presence of a controllable non-thermal environment.

**Figure 2.**Sketch of the thermodynamic cycle obtained by varying the tunable parameters ${\omega}_{\mathrm{T}}$ and ${E}_{d}$. Each one of the strokes are obtained by keeping one of the variables constant while quasi-statically varying the other one.

**Figure 3.**Stationary state’s von Neumann entropy in the regime of operation of the thermal engine. Any thermodynamic cycle must be contained on this surface.

**Figure 4.**Efficiency $\eta $ as a function of the upper values $({\omega}_{1},{E}_{1})$ for the cycle depicted in Figure 2. The observed highest efficiency of about $47\%$ was attained when $({\omega}_{1},{E}_{1})=({\omega}_{1,\mathrm{max}},{E}_{1,\mathrm{max}})$, with ${\omega}_{1,\mathrm{max}}/2\pi =1000$ MHz and ${E}_{1,\mathrm{max}}/2\pi \hslash =2$ MHz.

Parameter | Value |
---|---|

${\omega}_{\mathrm{CPW}}/2\pi $ | 4.94 GHz |

$\omega /2\pi $ | 4.94 GHz |

$g/2\pi \hslash $ | 120 MHz |

T | 30 mK |

$\Gamma /2\pi $ | 2 MHz |

${\kappa}_{\mathrm{CPW}}/2\pi $ | 1 MHz |

${\omega}_{0}/2\pi $ | 100 MHz |

${\omega}_{1,\mathrm{max}}/2\pi $ | 1000 MHz |

${E}_{0}/2\pi \hslash $ | 0.2 MHz |

${E}_{1,\mathrm{max}}/2\pi \hslash $ | 2 MHz |

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Cherubim, C.; Brito, F.; Deffner, S. Non-Thermal Quantum Engine in Transmon Qubits. *Entropy* **2019**, *21*, 545.
https://doi.org/10.3390/e21060545

**AMA Style**

Cherubim C, Brito F, Deffner S. Non-Thermal Quantum Engine in Transmon Qubits. *Entropy*. 2019; 21(6):545.
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**Chicago/Turabian Style**

Cherubim, Cleverson, Frederico Brito, and Sebastian Deffner. 2019. "Non-Thermal Quantum Engine in Transmon Qubits" *Entropy* 21, no. 6: 545.
https://doi.org/10.3390/e21060545