# Periodic Energy Transport and Entropy Production in Quantum Electronics

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## Abstract

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## 1. Introduction

## 2. Theoretical Model

## 3. First Law

## 4. Heat Current and Power

- It leads at low frequencies to a correct Joule law valid for all times [52].
- It shows perfect agreement between the Green function approach and the scattering matrix formalism [52].
- It displays parity symmetry upon reversal of the AC frequency even for interacting quantum conductors [79].
- It reduces to the conventional definition in the stationary case since the term ${J}_{c\alpha}^{E}\left(t\right)/2$ vanishes after time averaging [53].

## 5. Entropy Production

## 6. Calculation of the Currents within Green’s Function Approach for Non Interacting Systems

#### 6.1. Time Resolved Charge and Energy Currents Entering the Reservoirs

#### 6.2. Energy Stored in the Contact Regions

#### 6.3. Power Developed by the AC Sources

## 7. Relation to the Scattering Matrix Formalism

## 8. Low Frequency Expansion

## 9. Application

#### 9.1. Adiabatic Regime and Linear Response in the Bias Voltage

#### 9.2. Nonadiabatic Regime

## 10. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Sketch of the system under consideration. A quantum conductor (described by the Hamiltonian ${H}_{S}$), is coupled to two reservoirs (${H}_{L}$ and ${H}_{R}$) kept at the same temperature T, but with different chemical potentials ${\mu}_{L}$ and ${\mu}_{R}$. The conductor is also driven out of equilibrium by the application of AC local power sources, which are all collected in the vector $\mathbf{V}\left(t\right)$. The Hamiltonians representing the left and right contact regions are ${H}_{cL}$ and ${H}_{cR}$, respectively.

**Figure 2.**The total heat flux in the system is produced by the AC and DC driving sources, which is summarized by the relation ${\dot{Q}}_{\mathrm{tot}}\left(t\right)=-{P}_{\mathrm{AC}}\left(t\right)-{P}_{\mathrm{el}}\left(t\right)$. The definition of the heat currents entering the reservoirs, as well as the heat variation in the central part, takes into account half of the energy stored in the contact regions. The heat flux of the central piece contributes purely dynamically, since it is zero when averaged over one period of the AC driving potentials. In the stationary state, charge and energy can be stored only at the reservoirs.

**Figure 3.**Adiabatic regime of a single driven quantum level coupled to two fermionic baths. The different components of the total heat production ${Q}_{\mathrm{tot}}^{\mathrm{diss}}\left(t\right)=-{P}_{\mathrm{tot}}^{\mathrm{diss}}\left(t\right)$ as a function of time. Dashed lines correspond to reservoirs at $T=0$, while solid lines are for ${k}_{B}T=0.02$. Energies are expressed in units of ${\Gamma}_{L}$. The energy of the level evolves in time with ${V}_{AC}=-8$ and $\hslash \omega =1\times {10}^{-3}$. Parameters: $\mu =1$, $\delta \mu =0.004$, ${\epsilon}_{0}=0$ and hybridization widths ${\Gamma}_{L}=1$ and ${\Gamma}_{R}=0.5$.

**Figure 4.**Nonadiabatic regime of a single driven quantum level coupled to two fermionic baths at $T=0$. The different components of the total heat production ${Q}_{\mathrm{tot}}^{\mathrm{diss}}\left(t\right)=-{P}_{\mathrm{tot}}^{\mathrm{diss}}\left(t\right)$ as a function of time. The energy of the level evolves in time with ${V}_{AC}=0.7$ and $\hslash \omega =0.3$. Parameters: $\mu =0.2$, $\delta \mu =0$, ${\epsilon}_{0}=0$, and hybridization widths ${\Gamma}_{L}={\Gamma}_{R}=0.5$. Energies are expressed in units of $\Gamma ={\Gamma}_{L}+{\Gamma}_{R}$. The upper panel shows that the heat flux at the reservoirs may instantaneously attain negative values. The dissipative heat flux at the driven dot ${Q}_{S}^{\mathrm{diss}}\left(t\right)$ is highly fluctuating but the sum of the two contributions satisfies ${Q}_{\mathrm{tot}}^{\mathrm{diss}}\left(t\right)\ge 0$, consistent with the second law.

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**MDPI and ACS Style**

Ludovico, M.F.; Arrachea, L.; Moskalets, M.; Sánchez, D.
Periodic Energy Transport and Entropy Production in Quantum Electronics. *Entropy* **2016**, *18*, 419.
https://doi.org/10.3390/e18110419

**AMA Style**

Ludovico MF, Arrachea L, Moskalets M, Sánchez D.
Periodic Energy Transport and Entropy Production in Quantum Electronics. *Entropy*. 2016; 18(11):419.
https://doi.org/10.3390/e18110419

**Chicago/Turabian Style**

Ludovico, María Florencia, Liliana Arrachea, Michael Moskalets, and David Sánchez.
2016. "Periodic Energy Transport and Entropy Production in Quantum Electronics" *Entropy* 18, no. 11: 419.
https://doi.org/10.3390/e18110419