# Quantum Thermodynamics in Strong Coupling: Heat Transport and Refrigeration

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## Abstract

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## 1. Introduction

## 2. Basic Construction

## 3. Thermodynamical Aspects of the Surrogate Hamiltonian

#### Transport Dynamics

## 4. Heat Pump Operation

## 5. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

SSH | Stochastic Surrogate Hamiltonian |

L-GKS | Lindblad–Goirini–Kossakowski–Sudarshan |

COP | Coefficient of Performance |

## References

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**Figure 1.**The heat transport setup. In the centre is the system composed of an asymmetric double well with harmonic frequencies ${\omega}_{L}$ and ${\omega}_{R}$. Two primary baths are locally coupled to the left and right wells. Secondary baths with identical frequencies to the primary baths impose a temperature by random swap operations.

**Figure 2.**The convergence of the current as a function of the number of realizations. ${J}_{c}$ is shown for 2, 4, 6, 8, 10, and 12 realizations. ${\omega}_{L}=0.1$, ${\omega}_{R}=0.2$, ${T}_{L}=10$ K, ${T}_{R}=25$ K. Number of bath modes is eight in each bath. The fluctuations become smaller when the steady state is approached.

**Figure 3.**The heat current J as a function of the temperature ${T}_{L}$ for fixed ${T}_{R}=15$ K. Three cases are shown; blue ${\omega}_{L}>{\omega}_{R}$, black ${\omega}_{L}<{\omega}_{R}$, and red ${\omega}_{L}={\omega}_{R}$.

**Figure 4.**The rectifier effect: $\frac{{\mathcal{J}}_{\to}}{{\mathcal{J}}_{\leftarrow}}$ as a function of the frequency ratio ${\omega}_{L}/{\omega}_{R}$. $\frac{{\mathcal{J}}_{\to}}{{\mathcal{J}}_{\leftarrow}}$ is the ratio in the heat current from hot to cold when the left and right baths are swithched. ${T}_{c}=5$ K and ${T}_{h}=25$ K.

**Figure 5.**The Heat current J as a function of time for different temperature ${T}_{L}$ for fixed ${T}_{R}=15$ K. The timescale of approach to equilibrium depends on the temperature difference. The number of realizations is nine.

**Figure 6.**The heat pump: (

**a**) Cooling current as a function of external driving amplitude ϵ; (

**b**) Cooling current as a function of the coefficient of performance (COP). The bath temperatures are ${T}_{c}=10$ K and ${T}_{h}=25$ K, and $\Gamma =0.5$. A minimum threshold value of driving ϵ is required to overcome the heat leak (Red region). Increasing the driving strength leads to a maximum in cooling, which is due to a non-local system state. The cooling decreases when the system localizes on the hot bath.

**Figure 7.**The probability density $p(R)=Tr\{{\widehat{\rho}}_{s}\widehat{\mathbf{R}}\}$ of the steady state of the system for different driving amplitudes ϵ superimposed on the potential (black background). For zero driving the system localizes on the cold bath side. The optimum cooling is obtained for delocalized density $\u03f5=0.02$ and $\u03f5=0.05$.

**Figure 8.**The cooling current as a function of the system bath coupling constant Γ. Notice a maximum in dissipation at $\Gamma \sim 0.25$ and a maximum in coolling for $\Gamma \sim 1$. The bath temperatures are ${T}_{c}=10$ K and ${T}_{h}=25$ K and the driving amplitude is $\u03f5=0.25$.

**Figure 9.**The expectation value of the position $\langle R\rangle $ as a function of time. The system is allowed to relax to steady state reached after 1500 fsec. At $t=3500$ fsec, the coupling to the bath and the driving field is turned off. The coherence in steady state manifests itself by complex dynamics.

Potential Parameters | Values | Units |

${\omega}_{L}$ | $1\times {10}^{-3}$ | $a.u.$ |

${\omega}_{R}$ | 0.2–2$\times {10}^{-3}$ | $a.u.$ |

${R}_{L}$ | 0 | $bohr$ |

${R}_{R}$ | 1.5 | $bohr$ |

A | 0.5 | $a.u.$ |

σ | 0.5 | $bohr$ |

Grid Parameters | Typical Values | Units |

Grid spacing, $\Delta r$ | 0.0273 | $bohr$ |

Number of grid points, ${N}_{r}$ | 128 | |

Time steps, $\Delta t$ | 0.12 | fsec |

Order of Chebychev polynomials | 128 | |

Reduced mass, μ | 1836 | $a.u.$ |

Hot and Cold Bath Parameters | Typical Values | Units |

Number of bath modes (h/c) | 8 | |

Cutoff frequency, ${\omega}_{c}$ | 2.0 | eV |

System–bath coupling, Γ | 0.5 | $a.u.$ |

System–bath coupling range, γ | 0.5 | $a.u.$ |

Swap rate, ζ | 1.05 | $a.u.$ |

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Katz, G.; Kosloff, R. Quantum Thermodynamics in Strong Coupling: Heat Transport and Refrigeration. *Entropy* **2016**, *18*, 186.
https://doi.org/10.3390/e18050186

**AMA Style**

Katz G, Kosloff R. Quantum Thermodynamics in Strong Coupling: Heat Transport and Refrigeration. *Entropy*. 2016; 18(5):186.
https://doi.org/10.3390/e18050186

**Chicago/Turabian Style**

Katz, Gil, and Ronnie Kosloff. 2016. "Quantum Thermodynamics in Strong Coupling: Heat Transport and Refrigeration" *Entropy* 18, no. 5: 186.
https://doi.org/10.3390/e18050186