# Quantum Thermodynamics with Degenerate Eigenstate Coherences

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## Abstract

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## 1. Introduction

## 2. General Formalism

#### 2.1. Microscopic Derivation of Lindblad Master Equations

#### 2.2. Average Thermodynamics

#### 2.3. Fluctuating Thermodynamics

#### 2.3.1. Counting Statistics

#### 2.3.2. Finite-Time Fluctuation Theorem

## 3. Degenerate Single Quantum Dot Circuit

#### 3.1. Model

#### 3.2. Model Thermodynamics

#### 3.3. Statistics and Fluctuation Theorem

## 4. Summary

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. KMS Condition with Chemical Potentials

## Appendix B. Positivity of Entropy Production

## Appendix C. Details for the Specific Model

#### Appendix C.1. Reservoir Correlation Functions

#### Appendix C.2. Liouvillian

#### Appendix C.3. Wide-Band Limit

#### Appendix C.4. Current Suppression Point

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**Figure 1.**(

**a**) Illustration of the double quantum dot system with degenerate on-site energies ϵ and Coulomb-interaction U (dashed). The leads are described by Fermi functions ${f}_{L/R}(\omega )$ that depend on lead temperatures and chemical potentials. The peculiar feature of the system is that it is possible to tunnel directly into a superposition of the singly-charged states, described by the rate $\gamma =\sqrt{{\mathrm{\Gamma}}_{A}{\mathrm{\Gamma}}_{B}}$. Mainly for simplicity, we consider in this paper a tunnel-coupling configuration with only two different tunneling rates (bold solid and thin dotted). To avoid a bistable regime, we note that we require ${\mathrm{\Gamma}}_{A}\ne {\mathrm{\Gamma}}_{B}$; (

**b**) Graph associated with the master Equation (41). Solid arrows correspond to conventional transition rates obeying an LDB relation for each reservoir $\nu \in \{L,R\}$; they are proportional to ${\mathrm{\Gamma}}_{\nu t/\nu b}$ as indicated. Dashed arrows connect populations with the coherences; they do not correspond to traditional rates, but vanish as ${\gamma}_{\nu}\to 0$, thus effectively decoupling populations and coherences in the local basis.

**Figure 2.**Plot of the matter current entering the system from the left junction for different times (legend) versus dimensionless bias $\Delta \mu ={\mu}_{L}-{\mu}_{R}$ for the initially completely mixed state. For small times, the current is not point-symmetric, since the system content dominates the dynamics. For intermediate times, the current first approaches the steady-state RWA limit (dotted). For larger times, the coherences induce a large valley of current suppression, with a minimum at $\Delta {\mu}^{*}=\pm (2\u03f5+U)$. The energy currents (upper left inset, same color coding) behave similarly. The bottom right inset compares the time-dependent average matter current obtained within the BMS and the RWA approximation, for two different values of the potential difference $\Delta \mu $ indicated by vertical lines in the main plot. Parameters were chosen as $\beta {\mathrm{\Gamma}}_{A}=0.1$, $\beta {\mathrm{\Gamma}}_{B}=0.225$, $\beta \u03f5=10$ and $\beta U=50$.

**Figure 3.**Plot of the von Neumann (solid) and Shannon (dashed) entropies for different times (legend) versus dimensionless bias for the initially completely mixed state. Initially (grey), both entropies are constant and coincide with the maximum value of $\mathrm{ln}(4)$ (dimension of the Hilbert space). As coherences build up, they start to differ until they reach different steady states. Consistent with the pure delocalized steady state at the current suppression point (see Appendix C.4), the steady-state von Neumann entropy vanishes (solid black), whereas the Shannon entropy does not (dashed black). The dotted curve shows the steady-state entropy (Shannon) for the RWA rate equation. The time evolution of the von Neumann system entropy is illustrated in the bottom left inset for two values of the potential bias taken at the corresponding vertical lines of the main plot. The parameters were chosen as in Figure 2.

**Figure 4.**Top: Plot of the (positive) dimensionless entropy production rate for different times (solid curves) versus dimensionless bias $\beta \Delta \mu $ for the initially completely mixed state. For small times, the entropy production rate does not vanish anywhere since the system is not equilibrated. For large times, the steady-state entropy production rate (bold) is approached, which inherits the minima from the energy and matter currents (bottom). In contrast, the RWA version (dotted black) does not exhibit the coherence-induced dips. Bottom: For orientation, we also plot the dimensionless matter (red) and energy (green) currents and the rescaled absolute value of the coherences (thin dotted magenta). Parameters were chosen as in Figure 2.

**Figure 5.**Comparison of efficiency (black), matter current from left to right (red) and generated power (green) for the Born–Markov and secular (BMS) master equation (solid) and the rotating wave approximation (RWA) rate equation (thin dashed) versus dimensionless bias voltage. The thin dotted line denotes the absolute value of the coherence. The region of finite efficiency is marked by a non-dominating role of coherences in which the RWA and BMS efficiencies are similar. We notice that the quantum (BMS) efficiency is below the classical (RWA) efficiency. Parameters were chosen as in Figure 2.

**Figure 6.**Probability distributions of the number of particles flowing out of the left reservoir during four different time intervals τ as obtained from the dressed quantum master Equation (27) applied to our model. The initial condition on the system density matrix is the grand canonical equilibrium distribution with respect to the right reservoir (45). The long tail of the long-term distribution (blue) results from telegraph-noise averaging over a δ-peak at $\Delta n=1$ (trapped dark state) and a distribution conventionally propagating to the right. Chemical potentials were chosen as $\beta {\mu}_{L}=-\beta {\mu}_{R}=30.$ Other parameters where chosen as in Figure 2.

**Figure 7.**Probability distributions of the number of particles flowing out of the left reservoir during four different time intervals τ obtained within the rotating wave approximation, that is neglecting the influence of quantum coherences ${\rho}_{tb}$ and ${\rho}_{bt}$ on the statistics. The initial condition on the system density matrix is the grand canonical equilibrium distribution with respect to the right reservoir (45). Chemical potentials were chosen as $\beta {\mu}_{L}=-\beta {\mu}_{R}=30.$ Other parameters were chosen as in Figure 2.

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Bulnes Cuetara, G.; Esposito, M.; Schaller, G. Quantum Thermodynamics with Degenerate Eigenstate Coherences. *Entropy* **2016**, *18*, 447.
https://doi.org/10.3390/e18120447

**AMA Style**

Bulnes Cuetara G, Esposito M, Schaller G. Quantum Thermodynamics with Degenerate Eigenstate Coherences. *Entropy*. 2016; 18(12):447.
https://doi.org/10.3390/e18120447

**Chicago/Turabian Style**

Bulnes Cuetara, Gregory, Massimiliano Esposito, and Gernot Schaller. 2016. "Quantum Thermodynamics with Degenerate Eigenstate Coherences" *Entropy* 18, no. 12: 447.
https://doi.org/10.3390/e18120447