# Scaling-Up Quantum Heat Engines Efficiently via Shortcuts to Adiabaticity

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## Abstract

**:**

## 1. Introduction

## 2. Modeling a Many-Particle Quantum Heat Engine

#### 2.1. Trapped Quantum Fluids as Working Media

#### 2.2. Quantum Otto Cycle and Fundamental Limits

#### 2.3. Optimization of Power and Efficiency

#### 2.3.1. General Method

#### 2.3.2. Optimizing Adiabatic Output Power

## 3. Superadiabatic Many-Particle Quantum Heat Engines

#### 3.1. First Approach: Finite-Time Optimization and Accidental STA

#### 3.2. Second Approach: Adiabatic Optimization and STA

#### 3.2.1. Reverse Engineering of the Scaling Dynamics

#### 3.2.2. Counterdiabatic Driving

#### 3.2.3. Local Counterdiabatic Driving

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

QHE | Quantum heat engine |

CSM | Calogero–Sutherland model |

STA | Shortcut to adiabaticity |

CD | Counterdiabatic driving |

LCD | Local counterdiabatic driving |

## Appendix A. Nonadiabaticity of the Accidental Protocol

#### Appendix A.1. Derivation of the Scaling Factor b(t)

#### Appendix A.2. Derivation of the Nonadiabatic Factor Q*

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**Figure 1.**Many-particle quantum heat engine. Quantum Otto cycle with a quantum fluid as a working medium confined in a harmonic trap with frequency varying between ω

_{1}and ω

_{2}. States at A and C are thermal, while those at B and D are generally nonequilibrium states.

**Figure 2.**Nonadiabatic coefficient during a shortcut to adiabaticity. (

**a**) Nonadiabatic coefficient during an expansion/compression engineered via accidental shortcut to adiabaticity (STA) (acc), counterdiabatic driving (CD; discussed in Section 3.2.2) and local counterdiabatic driving (LCD; see Section 3.2.3) as a function of the evolution time 0 ≤ t/τ

_{1}≤ 1, where τ

_{1}is the minimum duration for an accidental STA, given in Equation (32) (take n = 1). (

**b**) Variation of the nonadiabatic factor as a function of the duration τ of the protocols, in units of ${\omega}_{1}^{-1}$. In (a) and (b), we take ω

_{1}/ω

_{2}= 0.15.

**Figure 3.**Efficiency of a many-particle quantum heat engine (QHE) run in finite-time at optimal power. Top panels (

**a**–

**c**) display the efficiency of a many-particle QHE at maximum power as a function of β

_{h}/β

_{c}for the accidental protocol (acc) with τ = 1.5/ω

_{1}, a sudden quench (sq) and adiabatic driving (ad). The interaction strength takes values λ = 0, 1, 2 from left to right and N = 500. A similar representation as a function of τ, in units of ${\omega}_{1}^{-1}$, is done in (

**d**–

**f**), where β

_{h}/β

_{c}= 0.3. In both representations, a transition is observed from the sudden-quench to the adiabatic limits that is governed by the nonadiabatic coefficient computed via Equation (31).

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**MDPI and ACS Style**

Beau, M.; Jaramillo, J.; Del Campo, A.
Scaling-Up Quantum Heat Engines Efficiently via Shortcuts to Adiabaticity. *Entropy* **2016**, *18*, 168.
https://doi.org/10.3390/e18050168

**AMA Style**

Beau M, Jaramillo J, Del Campo A.
Scaling-Up Quantum Heat Engines Efficiently via Shortcuts to Adiabaticity. *Entropy*. 2016; 18(5):168.
https://doi.org/10.3390/e18050168

**Chicago/Turabian Style**

Beau, Mathieu, Juan Jaramillo, and Adolfo Del Campo.
2016. "Scaling-Up Quantum Heat Engines Efficiently via Shortcuts to Adiabaticity" *Entropy* 18, no. 5: 168.
https://doi.org/10.3390/e18050168