# 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

## References

- Fujisawa, T.; Hayashi, T.; Tomita, R.; Hirayama, Y. Bidirectional counting of single electrons. Science
**2006**, 312, 1634–1636. [Google Scholar] [CrossRef] [PubMed] - Gustavsson, S.; Leturcq, R.; Studer, M.; Shorubalko, I.; Ihn, T.; Ensslin, K.; Driscoll, D.C.; Gossard, A.C. Electron counting in quantum dots. Surf. Sci. Rep.
**2009**, 64, 191–232. [Google Scholar] [CrossRef] - Küng, B.; Rössler, C.; Beck, M.; Marthaler, M.; Golubev, D.S.; Utsumi, Y.; Ihn, T.; Ensslin, K. Irreversibility on the level of single-electron tunneling. Phys. Rev. X
**2012**, 2, 011001. [Google Scholar] [CrossRef] - Saira, O.-P.; Yoon, Y.; Tanttu, T.; Möttönen, M.; Averin, D.V.; Pekola, J.P. Test of the Jarzynski and Crooks fluctuation relations in an electronic system. Phys. Rev. Lett.
**2012**, 109, 180601. [Google Scholar] [CrossRef] [PubMed] - Liebisch, T.C.; Reinhard, A.; Berman, P.R.; Raithel, G. Atom counting statistics in ensembles of interacting Rydberg atoms. Phys. Rev. Lett.
**2005**, 95, 253002. [Google Scholar] [CrossRef] [PubMed] - Malossi, N.; Valado, M.M.; Scotto, S.; Huillery, P.; Pillet, P.; Ciampini, D.; Arimondo, E.; Morsch, O. Full counting statistics and phase diagram of a dissipative Rydberg gas. Phys. Rev. Lett.
**2014**, 113, 023006. [Google Scholar] [CrossRef] [PubMed] - Krinner, S.; Stadler, D.; Husmann, D.; Brantut, J.-P.; Esslinger, T. Observation of quantized conductance in neutral matter. Nature
**2015**, 517, 64–67. [Google Scholar] [CrossRef] [PubMed] - Pekola, J.P. Towards quantum thermodynamics in electronic circuits. Nat. Phys.
**2015**, 11, 118–123. [Google Scholar] [CrossRef] - Esposito, M.; Harbola, U.; Mukamel, S. Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems. Rev. Mod. Phys.
**2009**, 81, 1665–1702. [Google Scholar] [CrossRef] - Campisi, M.; Hänggi, P.; Talkner, P. Colloquium: Quantum fluctuation relations: Foundations and applications. Rev. Mod. Phys.
**2011**, 83, 771–791. [Google Scholar] [CrossRef] - Spohn, H.; Lebowitz, J.L. Irreversible Thermodynamics for Quantum Systems Weakly Coupled to Thermal Reservoirs; John Wiley & Sons: New York, NY, USA, 2007; pp. 109–142. [Google Scholar]
- Kosloff, R. Quantum thermodynamics: A dynamical viewpoint. Entropy
**2013**, 15, 2100–2128. [Google Scholar] [CrossRef] - Gelbwaser-Klimovsky, D.; Niedenzu, W.; Kurizki, G. Thermodynamics of quantum systems under dynamical control. Adv. Atom. Mol. Opt. Phys.
**2015**, 64, 329. [Google Scholar] - Spohn, H. Kinetic equations from Hamiltonian dynamics: Markovian limits. Rev. Mod. Phys.
**1980**, 52, 569. [Google Scholar] [CrossRef] - Breuer, H.-P.; Petruccione, F. The Theory of Open Quantum Systems; Oxford University Press: Oxford, UK, 2002. [Google Scholar]
- Gardiner, C.; Zoller, P. Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics; Springer: Berlin/Heidelberg, Germany, 2004; Volume 56. [Google Scholar]
- Lindblad, G. On the generators of quantum dynamical semigroups. Commun. Math. Phys.
**1976**, 48, 119–130. [Google Scholar] [CrossRef] - Harbola, U.; Esposito, M.; Mukamel, S. Quantum master equation for electron transport through quantum dots and single molecules. Phys. Rev. B
**2006**, 74, 235309. [Google Scholar] [CrossRef] - Seifert, U. Stochastic thermodynamics, fluctuation theorems and molecular machines. Rep. Prog. Phys.
**2012**, 75, 126001. [Google Scholar] [CrossRef] [PubMed] - Van den Broeck, C.; Esposito, M. Ensemble and trajectory thermodynamics: A brief introduction. Physica A
**2015**, 418, 6–16. [Google Scholar] [CrossRef] - Esposito, M. Stochastic thermodynamics under coarse graining. Phys. Rev. E
**2012**, 85, 041125. [Google Scholar] [CrossRef] [PubMed] - Esposito, M.; Harbola, U.; Mukamel, S. Entropy fluctuation theorems in driven open systems: Application to electron counting statistics. Phys. Rev. E
**2007**, 76, 031132. [Google Scholar] [CrossRef] [PubMed] - Harbola, U.; Esposito, M.; Mukamel, S. Statistics and fluctuation theorem for boson and fermion transport through mesoscopic junctions. Phys. Rev. B
**2007**, 76, 085408. [Google Scholar] [CrossRef] - Cuetara, G.B.; Esposito, M.; Gaspard, P. Fluctuation theorems for capacitively coupled electronic currents. Phys. Rev. B
**2011**, 84, 165114. [Google Scholar] [CrossRef] - Correa, L.A.; Palao, J.P.; Adesso, G.; Alonso, D. Performance bound for quantum absorption refrigerators. Phys. Rev. E
**2013**, 87, 042131. [Google Scholar] [CrossRef] [PubMed] - Krause, T.; Brandes, T.; Esposito, M.; Schaller, G. Thermodynamics of the polaron master equation at finite bias. J. Chem. Phys.
**2015**, 142, 134106. [Google Scholar] [CrossRef] [PubMed] - Cuetara, G.B.; Esposito, M. Double quantum dot coupled to a quantum point contact: A stochastic thermodynamics approach. New J. Phys.
**2015**, 17, 095005. [Google Scholar] [CrossRef] - Kosloff, R. A quantum mechanical open system as a model of a heat engine. J. Chem. Phys.
**1984**, 80, 1625. [Google Scholar] [CrossRef] - Cuetara, G.B.; Engel, A.; Esposito, M. Quantum thermodynamics of rapidly driven systems. New J. Phys.
**2015**, 17, 055002. [Google Scholar] [CrossRef] - Uzdin, R.; Levy, A.; Kosloff, R. Equivalence of quantum heat machines, and quantum-thermodynamic signatures. Phys. Rev. X
**2015**, 5, 031044. [Google Scholar] [CrossRef] - Strasberg, P.; Schaller, G.; Brandes, T.; Esposito, M. Thermodynamics of quantum-jump-conditioned feedback control. Phys. Rev. E
**2013**, 88, 062107. [Google Scholar] [CrossRef] [PubMed] - Gelbwaser-Klimovsky, D.; Niedenzu, W.; Brumer, P.; Kurizki, G. Power enhancement of heat engines via correlated thermalization in a three-level “working fluid”. Sci. Rep.
**2015**, 5. [Google Scholar] [CrossRef] [PubMed] - Niedenzu, W.; Gelbwaser-Klimovsky, D.; Kurizki, G. Performance limits of multilevel and multipartite quantum heat machines. Phys. Rev. E
**2015**, 92, 042123. [Google Scholar] [CrossRef] [PubMed] - Cuetara, G.B.; Esposito, M.; Imparato, A. Exact fluctuation theorem without ensemble quantities. Phys. Rev. E
**2014**, 89, 052119. [Google Scholar] [CrossRef] [PubMed] - Braun, M.; König, J.; Martinek, J. Theory of transport through quantum-dot spin valves in the weak-coupling regime. Phys. Rev. B
**2004**, 70, 195345. [Google Scholar] [CrossRef] - Darau, D.; Begemann, G.; Donarini, A.; Grifoni, M. Interference effects on the transport characteristics of a benzene single-electron transistor. Phys. Rev. B
**2009**, 79, 235404. [Google Scholar] [CrossRef] - Schultz, M.G.; von Oppen, F. Quantum transport through nanostructures in the singular-coupling limit. Phys. Rev. B
**2009**, 80, 033302. [Google Scholar] [CrossRef] - Schaller, G.; Kießlich, G.; Brandes, T. Transport statistics of interacting double dot systems: Coherent and non-Markovian effects. Phys. Rev. B
**2009**, 80, 245107. [Google Scholar] [CrossRef] - Schultz, M.G. Quantum transport through single-molecule junctions with orbital degeneracies. Phys. Rev. B
**2010**, 82, 155408. [Google Scholar] [CrossRef] - Nilsson, H.; Karlström, O.; Larsson, M.; Caroff, P.; Pedersen, J.; Samuelson, L.; Wacker, A.; Wernersson, L.-E.; Xu, H. Correlation-induced conductance suppression at level degeneracy in a quantum dot. Phys. Rev. Lett.
**2010**, 104, 186804. [Google Scholar] [CrossRef] [PubMed] - Karlström, O.; Pedersen, J.; Samuelsson, P.; Wacker, A. Canyon of current suppression in an interacting two-level quantum dot. Phys. Rev. B
**2011**, 83, 205412. [Google Scholar] [CrossRef] - Lidar, D.A.; Bihary, Z.; Whaley, K.B. From completely positive maps to the quantum Markovian semigroup master equation. Chem. Phys.
**2001**, 268, 35–53. [Google Scholar] [CrossRef] - Schaller, G.; Brandes, T. Preservation of positivity by dynamical coarse-graining. Phys. Rev. A
**2008**, 78, 022106. [Google Scholar] [CrossRef] - Schaller, G. Open Quantum Systems Far from Equilibrium; Lecture Notes in Physics; Springer: Berlin/Heidelberg, Germany, 2014; Volume 881. [Google Scholar]
- Schaller, G. Quantum equilibration under constraints and transport balance. Phys. Rev. E
**2011**, 83, 031111. [Google Scholar] [CrossRef] [PubMed] - Spohn, H. Entropy production for quantum dynamical semigroups. J. Math. Phys.
**1978**, 19, 1227–1230. [Google Scholar] [CrossRef] - Silaev, M.; Heikkilä, T.T.; Virtanen, P. Lindblad-equation approach for the full counting statistics of work and heat in driven quantum systems. Phys. Rev. E
**2014**, 90, 022103. [Google Scholar] [CrossRef] [PubMed] - Braig, S.; Brouwer, P. Rate equations for coulomb blockade with ferromagnetic leads. Phys. Rev. B
**2005**, 71, 195324. [Google Scholar] [CrossRef] - Jordan, A.N.; Sukhorukov, E.V. Transport statistics of bistable systems. Phys. Rev. Lett.
**2004**, 93, 260604. [Google Scholar] [CrossRef] [PubMed] - Schaller, G.; Kießlich, G.; Brandes, T. Counting statistics in multistable systems. Phys. Rev. B
**2010**, 81, 205305. [Google Scholar] [CrossRef] - Lindblad, G. Completely positive maps and entropy inequalities. Commun. Math. Phys.
**1975**, 40, 147–151. [Google Scholar] [CrossRef]

**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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**MDPI and ACS Style**

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