# Quantum Thermodynamics in the Refined Weak Coupling Limit

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## Abstract

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## 1. Introduction

## 2. Weak Coupling and Refined Weak Coupling Limit

#### 2.1. Weak Coupling Limit

#### 2.2. Refined Weak Coupling Limit

#### 2.3. Refined Weak Coupling Limit under Slowly-Varying Time-Dependent Hamiltonians

## 3. Standard Thermodynamics of Open Quantum Systems

#### 3.1. The First Law

#### 3.2. The Second Law

#### 3.3. Difficulties Beyond the Weak Coupling Limit

## 4. Thermodynamics in the Refined Weak Coupling Limit

#### 4.1. Time-Independent System Hamiltonian ${H}_{\mathcal{S}}$

- ${E}^{\mathrm{R}}\left(0\right)=E\left(0\right)$. Thus, the deviation of ${E}^{\mathrm{R}}\left(t\right)$ from $E\left(t\right)$ at finite times is unambiguously caused by the interaction term ${V}_{\mathcal{SE}}$ in the Hamiltonian.
- ${E}^{\mathrm{R}}\left(t\right)$ approaches $E\left(t\right)$ for t large. In such a case ${\Lambda}_{t}^{\mathrm{R}}$ approaches Davies’ semigroup (16), and then, ${H}_{\mathcal{S}}^{\mathrm{R}}\left(t\right)$ approaches ${H}_{\mathcal{S}}$.

#### 4.2. Time-Dependent System Hamiltonian ${H}_{\mathcal{S}}\left(t\right)$

## 5. Example: Spin-Boson Model in the Refined Weak Coupling Limit

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Internal Energy in the Weak Coupling Limit

## Appendix B. Calculation of the Time-Ordered Exponential for the Driven Spin-Boson Model in the Refined Weak Coupling

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**Figure 1.**Results for the entropy production (left column) and internal energy (right column) for the spin-boson model under the refined weak coupling limit (solid lines) and the Davies semigroup dynamics of the weak coupling limit (same color, dashed lines). These are calculated under three different system initial conditions ${\rho}_{\mathcal{S}}\left(0\right)=|e\rangle \langle e|$, ${\rho}_{\mathcal{S}}\left(0\right)=|g\rangle \langle g|$, and ${\rho}_{\mathcal{S}}\left(0\right)=|+\rangle {}_{y}\langle +|$, which are depicted in the first, second, and third row, respectively. The bath is assumed to have an Ohmic spectral density $J\left(\omega \right)=\alpha \omega exp(-\omega /{\omega}_{c})$ with $\alpha =0.05$ and ${\omega}_{c}=5$, in units of ${\omega}_{0}$. The different bath temperatures are highlighted by different colors. As expected, convergence for large time is obtained.

**Figure 2.**Results for the spin-boson model with diagonal driving ${H}_{\mathcal{S}}\left(t\right)={\textstyle \frac{{\omega}_{0}\left(t\right)}{2}}{\sigma}_{z}$ under the adiabatically-deformed refined and Davies weak coupling limit (same color, dashed lines). The entropy production (top left column), power (bottom left column), and internal energy (top right column) are plotted for three different bath temperatures. The internal energy for different values of the modulation frequency at $T=1$ is also depicted (bottom right column) showing convergence to the refined weak coupling result for constant ${H}_{\mathcal{S}}$ ($\nu =0$, blue dotted line). These results are calculated for the system initially prepared in the ground state ${\rho}_{\mathcal{S}}\left(0\right)=|g\rangle \langle g|$. As in Figure 2, the bath is assumed to have an Ohmic spectral density $J\left(\omega \right)=\alpha \omega exp(-\omega /{\omega}_{c})$ with $\alpha =0.05$ and ${\omega}_{c}=5$, in units of ${\omega}_{0}$.

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Rivas, Á. Quantum Thermodynamics in the Refined Weak Coupling Limit. *Entropy* **2019**, *21*, 725.
https://doi.org/10.3390/e21080725

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Rivas Á. Quantum Thermodynamics in the Refined Weak Coupling Limit. *Entropy*. 2019; 21(8):725.
https://doi.org/10.3390/e21080725

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Rivas, Ángel. 2019. "Quantum Thermodynamics in the Refined Weak Coupling Limit" *Entropy* 21, no. 8: 725.
https://doi.org/10.3390/e21080725