Fluctuation relations for heat exchange in the generalized Gibbs ensemble

Abstract

In this work, we investigate the heat exchange between two quantum systems whose initial equilibrium states are described by the generalized Gibbs ensemble. First, we generalize the fluctuation relations for heat exchange discovered by Jarzynski and Wójcik to quantum systems prepared in the equilibrium states described by the generalized Gibbs ensemble at various generalized temperatures. Secondly, we extend the connections between heat exchange and the Rényi divergences to quantum systems under generic initial conditions. These relations are applicable for quantum systems with conserved quantities and universally valid for quantum systems in the integrable and chaotic regimes.

Keywords

exchange fluctuation relation / generalized Gibbs ensemble

References

[1] C. Jarzynski and D. K. Wójcik, Classical and quantum fluctuation theorems for heat exchange, Phys. Rev. Lett. 92(23), 230602 (2004)

[2] B. B. Wei, Relations between heat exchange and the Rényi divergences, Phys. Rev. E 97(4), 042107 (2018)

[3] A. Rényi, in: Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, University of California Press Press, 1961, pp 547–561

[4] T. van Erven and P. Harremoes, Rényi divergence and Kullback–Leibler divergence, IEEE Trans. Inf. Theory 60(7), 3797 (2014)

[5] S. Beigi, Sandwiched Rényi divergence satisfies data processing inequality, J. Math. Phys. 54(12), 122202 (2013)

[6] M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel, On quantum Rényi entropies: A new generalization and some properties, J. Math. Phys. 54(12), 122203 (2013)

[7] M. Esposito, U. Harbola, and S. Mukamel, Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems, Rev. Mod. Phys. 81(4), 1665 (2009)

[8] C. Jarzynski, Equalities and inequalities: Irreversibility and the second law of thermodynamics at the nanoscale, Annu. Rev. Condens. Matter Phys. 2(1), 329 (2011)

[9] M. Campisi, P. Hanggi, and P. Talkner, Quantum fluctuation relations: Foundations and applications, Rev. Mod. Phys. 83(3), 771 (2011)

[10] T. Kinoshita, T. Wenger, and D. S. Weiss, A quantum Newton’s cradle, Nature 440(7086), 900 (2006)

[11] M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Relaxation in a completely integrable many-body quantum system: An ab initio study of the dynamics of the highly excited states of 1D lattice hard-core bosons, Phys. Rev. Lett. 98(5), 050405 (2007)

[12] P. Calabrese, F. H. L. Essler, and M. Fagotti, Quantum quench in the transverse field Ising chain, Phys. Rev. Lett. 106(22), 227203 (2011)

[13] M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl, I. Mazets, D. A. Smith, E. Demler, and J. Schmiedmayer, Relaxation and prethermalization in an isolated quantum system, Science 337(6100), 1318 (2012)

[14] J. P. Ronzheimer, M. Schreiber, S. Braun, S. S. Hodgman, S. Langer, I. P. McCulloch, F. Heidrich-Meisner, I. Bloch, and U. Schneider, Expansion dynamics of interacting bosons in homogeneous lattices in one and two dimensions, Phys. Rev. Lett. 110(20), 205301 (2013)

[15] J. S. Caux and F. H. Essler, Time evolution of local observables after quenching to an integrable model, Phys. Rev. Lett. 110(25), 257203 (2013)

[16] L. Vidmar, J. P. Ronzheimer, M. Schreiber, S. Braun, S. S. Hodgman, S. Langer, F. Heidrich-Meisner, I. Bloch, and U. Schneider, Dynamical quasicondensation of hard-core bosons at finite momenta, Phys. Rev. Lett. 115(17), 175301 (2015)

[17] L. Vidmar, D. Iyer, and M. Rigol, Emergent eigenstate solution to quantum dynamics far from equilibrium, Phys. Rev. X 7(2), 021012 (2017)

[18] C. Gogolin and J. Eisert, Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems, Rep. Prog. Phys. 79(5), 056001 (2016)

[19] L. Vidmar and M. Rigol, Generalized Gibbs ensemble in integrable lattice models, J. Stat. Mech. 2016(6), 064007 (2016)

[20] T. Langen, S. Erne, R. Geiger, B. Rauer, T. Schweigler, M. Kuhnert, W. Rohringer, I. E. Mazets, T. Gasenzer, and J. Schmiedmayer, Experimental observation of a generalized Gibbs ensemble, Science 348(6231), 207 (2015)

[21] L. E. Reichl, A Modern Course in Statistical Physics, Edward Arnold, Austin, TX, 1987

[22] E. T. Jaynes, Information theory and statistical mechanics, Phys. Rev. 106(4), 620 (1957)

[23] E. T. Jaynes, Information theory and statistical mechanics (II), Phys. Rev. 108(2), 171 (1957)

[24] W. Yang, W. L. Ma, and R. B. Liu, Quantum manybody theory for electron spin decoherence in nanoscale nuclear spin baths, Rep. Prog. Phys. 80(1), 016001 (2017)

[25] B. B. Wei, and R. B. Liu, Lee-Yang zeros and critical times in decoherence of a probe spin coupled to a bath, Phys. Rev. Lett. 109(18), 185701 (2012)

[26] B. B. Wei, S. W. Chen, H. C. Po, and R. B. Liu, Phase transitions in the complex plane of a physical parameter, Sci. Rep. 4(1), 5202 (2015)

[27] B. B. Wei, Z. F. Jiang, and R. B. Liu, Thermodynamic holography, Sci. Rep. 5(1), 15077 (2015)

[28] X. H. Peng, H. Zhou, B. B. Wei, J. Y. Cui, J. F. Du, and R. B. Liu, Experimental observation of Lee–Yang zeros, Phys. Rev. Lett. 114(1), 010601 (2015)

[29] B. B. Wei, Probing Yang–Lee edge singularity by central spin decoherence, New J. Phys. 19(8), 083009 (2017)

[30] B. B. Wei, Probing conformal invariant of non-unitary two dimensional system by central spin decoherence, Sci. Rep. 8(1), 3080 (2018)

[31] H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P. Sun, Decay of Loschmidt echo enhanced by quantum criticality, Phys. Rev. Lett. 96(14), 140604 (2006)

[32] J. Zhang, X. Peng, N. Rajendran, and D. Suter, Detection of quantum critical points by a probe qubit, Phys. Rev. Lett. 100(10), 100501 (2008)

[33] S. W. Chen, Z. F. Jiang, and R. B. Liu, Quantum criticality at high temperature revealed by spin echo, New J. Phys. 15(4), 043032 (2013)

[34] B. B. Wei and M. B. Plenio, Relations between dissipated work in non-equilibrium process and the family of Rényi divergences, New J. Phys. 19(2), 023002 (2017)

[35] B. B. Wei, Links between dissipation and Rényi divergences in the PT-symmetric quantum mechanics, Phys. Rev. A 97(1), 012105 (2018)

[36] B. B. Wei, Relations between dissipated work and Rényi divergences in the generalized Gibbs ensemble, Phys. Rev. A 97(4), 042132 (2018)

[37] X. Y. Guo, , Demonstration of irreversibility and dissipation relation of thermodynamics with a superconducting qubit, arXiv: 1710.10234 (2017)

[38] A. Wehrl, General properties of entropy, Rev. Mod. Phys. 50(2), 221 (1978)

[39] J. Goold, U. Poschinger, and K. Modi, Measuring heat exchange of a quantum process, Phys. Rev. E 90(2), 020101 (2014)

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