Abstract
We consider a discrete-time non-Hamiltonian dynamics of a quantum system consisting of a finite sample locally coupled to several bi-infinite reservoirs of fermions with a translation symmetry. In this setup, we compute the asymptotic state, mean fluxes of fermions into the different reservoirs, as well as the mean entropy production rate of the dynamics. Formulas are explicitly expanded to leading order in the strength of the coupling to the reservoirs.
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Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: Quantum walks on graphs. In Proceedings of the thirty-third annual ACM symposium on Theory of computing, pp. 50–59. ACM (2001)
Andréys, S.: Repeated interaction processes in the continuous-time limit, applied to quadratic fermionic systems. Ann. Henri Poincaré 21(1), 115–154 (2020)
Asch, J., Bourget, O., Joye, A.: Spectral stability of unitary network models. Rev. Math. Phys. 27(07), 1530004 (2015)
Aschbacher, W.H., Pillet, C.-A.: Non-equilibrium steady states of the XY chain. J. Stat. Phys. 112, 1153–1175 (2003)
Aschbacher, W., Jakšić, V., Pautrat, Y., Pillet, C.-A.: Topics in non-equilibrium quantum statistical mechanics. In: Open Quantum Systems. III, pp. 1–66. Springer, Berlin (2006)
Aschbacher, W., Jakšić, V., Pautrat, Y., Pillet, C.-A.: Transport properties of quasi-free fermions. J. Math. Phys. 48(3), 032101 (2007)
Attal, S., Joye, A., Pillet, C.-A. (eds.): Open quantum systems. III Recent developments, volume 1882 of Lecture Notes in Mathematics. Springer, 2006. Lecture notes from the Summer School held in Grenoble, June 16–July 4, 2003
Attal, S., Petruccione, F., Sabot, C., Sinayskiy, I.: Open quantum random walks. J. Stat. Phys. 147(4), 832–852 (2012)
Benoist, T., Jakšić, V., Pautrat, Y., Pillet, C.-A.: On entropy production of repeated quantum measurements I. General theory. Commun. Math. Phys. 357(1), 77–123 (2018)
Bougron, J.-F. , Bruneau, L.: Linear response theory and entropic fluctuations in repeated interaction quantum systems. J. Stat. Phys. 181(5), 1636–1677 (2020)
Bruneau, L., Joye, A., Merkli, M.: Asymptotics of repeated interaction quantum systems. J. Funct. Anal. 239(1), 310–344 (2006)
Bruneau, L., Joye, A., Merkli, M.: Repeated interactions in open quantum systems. J. Math. Phys 55(7), 075204 (2014)
Chalker, J.T., Coddington, P.D.: Percolation, quantum tunnelling and the integer Hall effect. J. Phys. C Solid State Phys. 21(14), 2665 (1988)
Dereziński, J., de Roeck, W., Maes, C.: Fluctuations of quantum currents and unravelings of master equations. J. Stat. Phys. 131(2), 341–356 (2008)
Dierckx, B., Fannes, M., Pogorzelska, M.: Fermionic quasifree states and maps in information theory. J. Math. Phys. 49(3), 032109 (2008)
Gudder, S.: Quantum Markov chains. J. Math. Phys. 49(7), 072105 (2008)
Hamza, E., Joye, A.: Thermalization of fermionic quantum walkers. J. Stat. Phys. 166(6), 1365–1392 (2017)
Hanson, E.P., Joye, A., Pautrat, Y., Raquépas, R.: Landauer’s principle in repeated interaction systems. Commun. Math. Phys. 349(1), 285–327 (2017)
Hanson, E.P., Joye, A., Pautrat, Y., Raquépas, R.: Landauer’s principle for trajectories of repeated interaction systems. Ann. Henri Poincaré 19(7), 1939–1991 (2018)
Jakšić, V., Pillet, C.-A.: On entropy production in quantum statistical mechanics. Commun. Math. Phys. 217(2), 285–293 (2001)
Jakšić, V., Pillet, C.-A.: Mathematical theory of non-equilibrium quantum statistical mechanics. J. Stat. Phys. 108(5–6), 787–829 (2002)
Jakšić, V., Ogata, Y., Pautrat, Y., Pillet,C.-A.: Entropic fluctuations in quantum statistical mechanics an introduction. In J. Fröhlich, M. Salmhofer, V. Mastropietro, W. De Roeck, and L. F. Cugliandolo, editors, Quantum Theory from Small to Large Scales, volume 95 of Lecture Notes of the Les Houches Summer School, pp. 213–410. Oxford University Press (2011)
Jakšić, V., Landon, B., Pillet, C.-A.: Entropic fluctuations in xy chains and reflectionless jacobi matrices. Ann. Henri Poincaré 14(7), 1775–1800 (2013)
Jakšić, V., Pillet, C.-A., Westrich, M.: Entropic fluctuations of quantum dynamical semigroups. J. Stat. Phys. 154(1–2), 153–187 (2014)
Karski, M., Förster, L., Choi, J.-M., Steffen, A., Alt, W., Meschede, D., Widera, A.: Quantum walk in position space with single optically trapped atoms. Science 325(5937), 174–177 (2009)
Kassal, I., Aspuru-Guzik, A.: Environment-assisted quantum transport in ordered systems. New J. Phys. 14(5), 053041 (2012)
Kato, T.: Perturbation Theory for Linear Operators, Volume 132 of Grundlehren der mathematischen Wissenschaften, 2nd edn. Springer, New York (1995)
Kempe, J.: Quantum random walks: an introductory overview. Contemp. Phys. 44(4), 307–327 (2003)
Konno, N.: Quantum walks. In Quantum Potential Theory, pp. 309–452. Springer, New York (2008)
Mareš, J., Novotnỳ, J., Štefaňák, M., Jex, I.: Counterintuitive role of geometry in transport by quantum walks. Phys. Rev. A 101(3), 032113 (2020)
Merkli, M., Mück, M., Sigal, I.: Instability of equilibrium states for coupled heat reservoirs at different temperatures. J. Funct. Anal. 243(1), 87–120 (2007)
Merkli, M., Mück, M., Sigal, I.M.: Theory of non-equilibrium stationary states as a theory of resonances. Ann. Henri Poincaré 8(8), 1539–1593 (2007)
Portugal, R.: Quantum Walks and Search Algorithms. Springer, New York (2013)
Raquépas, R.: On fermionic walkers interacting with a correlated structured environment. Lett. Math. Phys. 110(1), 121–145 (2020)
Ruelle, D.: Natural nonequilibrium states in quantum statistical mechanics. J. Stat. Phys. 98(1–2), 57–75 (2000)
Ruelle, D.: Entropy production in quantum spin systems. Commun. Math. Phys. 224(1), 3–16 (2001)
Sajid, M., Asbóth, J.K., Meschede, D., Werner, R.F., Alberti, A.: Creating anomalous floquet chern insulators with magnetic quantum walks. Phys. Rev. B 99, 214303 (2019)
Santha, M.: Quantum walk based search algorithms. In International Conference on Theory and Applications of Models of Computation, pp. 31–46. Springer (2008)
Spagnolo, N., Vitelli, C., Aparo, L., Mataloni, P., Sciarrino, F., Crespi, A., Ramponi, R., Osellame, R.: Three-photon bosonic coalescence in an integrated tritter. Nat. Commun. 4(1), 1–6 (2013)
Spohn, H.: Entropy production for quantum dynamical semigroups. J. Math. Phys. 19(5), 1227–1230 (1978)
Spohn, H., Lebowitz, J.L.: Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs. Adv. Chem. Phys. 38, 109–142 (1978)
Tamura, M., Mukaiyama, T., Toyoda, K.: Quantum walks of a phonon in trapped ions. Phys. Rev. Lett. 124, 200501 (2020)
Venegas-Andraca, S.E.: Quantum walks: a comprehensive review. Quantum Inf. Process. 11(5), 1015–1106 (2012)
Wang, J., Manouchehri, K.: Physical Implementation of Quantum Walks. Springer, New York (2013)
Zähringer, F., Kirchmair, G., Gerritsma, R., Solano, E., Blatt, R., Roos, C.: Realization of a quantum walk with one and two trapped ions. Phys. Rev. Lett. 104(10), 100503 (2010)
Acknowledgements
The research of the authors is partially supported by the French National Agency through the Grant NonStops (ANR-17-CE40-0006). The research of S. A. is supported by the French National Research Agency in the framework of the “Investissements d’avenir” program (ANR-15-IDEX-02). The research of R. R. is partially funded by the National Sciences and Engineering Research Council of Canada. The authors would like to thank the anonymous referees for their comments, which improved the quality of the presentation.
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Communicated by Yoshiko Ogata.
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Andréys, S., Joye, A. & Raquépas, R. Fermionic Walkers Driven Out of Equilibrium. J Stat Phys 184, 14 (2021). https://doi.org/10.1007/s10955-021-02795-1
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DOI: https://doi.org/10.1007/s10955-021-02795-1