Rigorous results on approach to thermal equilibrium, entanglement, and nonclassicality of an optical quantum field mode scattering from the elements of a non-equilibrium quantum reservoir

Stephan De Bievre1, Marco Merkli2, and Paul E. Parris3

1Univ. Lille, CNRS, Inria, UMR 8524, Laboratoire P. Painlevé, F-59000 Lille, France
2Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL, A1C 5S7, Canada
3Missouri University of Science and Technology, Rolla, Missouri, 65409, USA

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Rigorous derivations of the approach of individual elements of large isolated systems to a state of thermal equilibrium, starting from arbitrary initial states, are exceedingly rare. This is particularly true for quantum mechanical systems. We demonstrate here how, through a mechanism of repeated scattering, an approach to equilibrium of this type actually occurs in a specific quantum system, one that can be viewed as a natural quantum analog of several previously studied classical models. In particular, we consider an optical mode passing through a reservoir composed of a large number of sequentially-encountered modes of the same frequency, each of which it interacts with through a beam splitter. We then analyze the dependence of the asymptotic state of this mode on the assumed stationary common initial state $\sigma$ of the reservoir modes and on the transmittance $\tau=\cos\lambda$ of the beam splitters. These results allow us to establish that at small $\lambda$ such a mode will, starting from an arbitrary initial system state $\rho$, approach a state of thermal equilibrium even when the reservoir modes are not themselves initially thermalized. We show in addition that, when the initial states are pure, the asymptotic state of the optical mode is maximally entangled with the reservoir and exhibits less nonclassicality than the state of the reservoir modes.

It is a tenet of thermodynamics that a macroscopic system, starting from almost any initial state, will inevitably evolve towards a well-defined state of thermodynamic equilibrium. Perhaps surprisingly, rigorous mathematical derivations of how, exactly, this approach to equilibrium comes about as a consequence of the underlying dynamics of the individual elements making up the system are exceedingly rare. This rarity is particularly acute for large quantum mechanical systems. In this paper, the authors demonstrate through a rigourous mathematical analysis, how an approach to equilibrium of this type actually occurs in a specific quantum system that can be experimentally realized using an array of simple optical elements. In particular, they consider an optical (e.g. laser) mode of fixed frequency (the system mode) that sequentially passes through a linear sequence of beam splitters all with the same transmittance. Each beam splitter is assumed to be illuminated through a second input port by identically prepared “reservoir” modes of the same frequency. The beam splitters thus allow the system mode to interact and exchange energy with the reservoir modes. The authors’ analysis shows that when the transmittance of the beam splitters is high, the initial optical system mode will, after passing through many beam splitters, approach a state of thermal equilibrium even when the reservoir modes are not themselves initially thermalized.

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[1] Joshua M. Deutsch. Eigenstate thermalization hypothesis. Rep. Progr. Phys., 81(8):082001, 16, 2018. URL: https:/​/​doi.org/​10.1088/​1361-6633/​aac9f1, doi:10.1088/​1361-6633/​aac9f1.

[2] Joel L. Lebowitz. Boltzmann's Entropy and Time's Arrow. Physics Today, 46(9):32–38, 09 1993. doi:10.1063/​1.881363.

[3] Sheldon Goldstein, Takashi Hara, and Hal Tasaki. Time scales in the approach to equilibrium of macroscopic quantum systems. Phys. Rev. Lett., 111:140401, Oct 2013. doi:10.1103/​PhysRevLett.111.140401.

[4] Cédric Villani. (Ir)reversibility and entropy. In Time, volume 63 of Prog. Math. Phys., pages 19–79. Birkhäuser/​Springer Basel AG, Basel, 2013. doi:10.1007/​978-3-0348-0359-5_2.

[5] Hal Tasaki. Typicality of thermal equilibrium and thermalization in isolated macroscopic quantum systems. J. Stat. Phys., 163(5):937–997, 2016. URL: https:/​/​doi.org/​10.1007/​s10955-016-1511-2, doi:10.1007/​s10955-016-1511-2.

[6] Takashi Mori, Tatsuhiko N Ikeda, Eriko Kaminishi, and Masahito Ueda. Thermalization and prethermalization in isolated quantum systems: a theoretical overview. Journal of Physics B: Atomic, Molecular and Optical Physics, 51(11):112001, may 2018. doi:10.1088/​1361-6455/​aabcdf.

[7] Stephan De Bièvre and Paul E. Parris. A rigourous demonstration of the validity of Boltzmann's scenario for the spatial homogenization of a freely expanding gas and the equilibration of the Kac ring. J. Stat. Phys., 168(4):772–793, 2017. URL: https:/​/​doi.org/​10.1007/​s10955-017-1834-7, doi:10.1007/​s10955-017-1834-7.

[8] Subhadip Chakraborti, Abhishek Dhar, Sheldon Goldstein, Anupam Kundu, and Joel L. Lebowitz. Entropy growth during free expansion of an ideal gas. J. Phys. A, 55(39):Paper No. 394002, 30, 2022. URL: https:/​/​doi.org/​10.1088/​1751-8121/​ac8a7e, doi:10.1088/​1751-8121/​ac8a7e.

[9] Saurav Pandey, Junaid Majeed Bhat, Abhishek Dhar, Sheldon Goldstein, David A. Huse, Manas Kulkarni, Anupam Kundu, and Joel L. Lebowitz. Boltzmann entropy of a freely expanding quantum ideal gas. J. Stat. Phys., 190(8):Paper No. 142, 29, 2023. URL: https:/​/​doi.org/​10.1007/​s10955-023-03154-y, doi:10.1007/​s10955-023-03154-y.

[10] Stephan De Bièvre and Paul E. Parris. Equilibration, generalized equipartition, and diffusion in dynamical Lorentz gases. J. Stat. Phys., 142(2):356–385, 2011. URL: https:/​/​doi.org/​10.1007/​s10955-010-0109-3, doi:10.1007/​s10955-010-0109-3.

[11] Stephan De Bièvre, Carlos Mejía-Monasterio, and Paul E. Parris. Dynamical mechanisms leading to equilibration in two-component gases. Phys. Rev. E, 93:050103, May 2016. doi:10.1103/​PhysRevE.93.050103.

[12] Francesco Ciccarello, Salvatore Lorenzo, Vittorio Giovannetti, and G. Massimo Palma. Quantum collision models: Open system dynamics from repeated interactions. Physics Reports, 954:1–70, 2022. Quantum collision models: Open system dynamics from repeated interactions. doi:https:/​/​doi.org/​10.1016/​j.physrep.2022.01.001.

[13] Laurent Bruneau, Alain Joye, and Marco Merkli. Repeated interactions in open quantum systems. J. Math. Phys., 55(7):075204, 67, 2014. URL: https:/​/​doi.org/​10.1063/​1.4879240, doi:10.1063/​1.4879240.

[14] Laurent Bruneau, Alain Joye, and Marco Merkli. Asymptotics of repeated interaction quantum systems. J. Funct. Anal., 239(1):310–344, 2006. URL: https:/​/​doi.org/​10.1016/​j.jfa.2006.02.006, doi:10.1016/​j.jfa.2006.02.006.

[15] Laurent Bruneau, Alain Joye, and Marco Merkli. Random repeated interaction quantum systems. Comm. Math. Phys., 284(2):553–581, 2008. URL: https:/​/​doi.org/​10.1007/​s00220-008-0580-8, doi:10.1007/​s00220-008-0580-8.

[16] Laurent Bruneau, Alain Joye, and Marco Merkli. Repeated and continuous interactions in open quantum systems. Ann. Henri Poincaré, 10(7):1251–1284, 2010. URL: https:/​/​doi.org/​10.1007/​s00023-009-0017-8, doi:10.1007/​s00023-009-0017-8.

[17] Valerio Scarani, Mário Ziman, Peter Štelmachovič, Nicolas Gisin, and Vladimír Bužek. Thermalizing quantum machines: Dissipation and entanglement. Phys. Rev. Lett., 88:097905, Feb 2002. doi:10.1103/​PhysRevLett.88.097905.

[18] Vojkan Jakšić and Claude-Alain Pillet. Non-equilibrium steady states of finite quantum systems coupled to thermal reservoirs. Comm. Math. Phys., 226(1):131–162, 2002. URL: https:/​/​doi.org/​10.1007/​s002200200602, doi:10.1007/​s002200200602.

[19] Marco Merkli, Matthias Mück, and Israel Michael Sigal. Theory of non-equilibrium stationary states as a theory of resonances. Ann. Henri Poincaré, 8(8):1539–1593, 2007. URL: https:/​/​doi.org/​10.1007/​s00023-007-0346-4, doi:10.1007/​s00023-007-0346-4.

[20] Dragi Karevski and Thierry Platini. Quantum nonequilibrium steady states induced by repeated interactions. Phys. Rev. Lett., 102:207207, May 2009. doi:10.1103/​PhysRevLett.102.207207.

[21] Serge Haroche and Jean-Michel Raimond. Exploring the Quantum: Atoms, Cavities, and Photons. Oxford University Press, 08 2006. URL: https:/​/​doi.org/​10.1093/​acprof:oso/​9780198509141.001.0001, doi:10.1093/​acprof:oso/​9780198509141.001.0001.

[22] Christian Weedbrook, Stefano Pirandola, Raúl García-Patrón, Nicolas J. Cerf, Timothy C. Ralph, Jeffrey H. Shapiro, and Seth Lloyd. Gaussian quantum information. Reviews of Modern Physics, 84:621–669, May 2012. doi:10.1103/​RevModPhys.84.621.

[23] Vojkan Jakšić and Claude-Alain Pillet. On a model for quantum friction. II. Fermi's golden rule and dynamics at positive temperature. Comm. Math. Phys., 176(3):619–644, 1996. URL: http:/​/​projecteuclid.org/​euclid.cmp/​1104286117.

[24] Volker Bach, Jürg Fröhlich, and Israel Michael Sigal. Return to equilibrium. J. Math. Phys., 41(6):3985–4060, 2000. URL: https:/​/​doi.org/​10.1063/​1.533334, doi:10.1063/​1.533334.

[25] Jürg Fröhlich and Marco Merkli. Another return of ``return to equilibrium''. Comm. Math. Phys., 251(2):235–262, 2004. URL: https:/​/​doi.org/​10.1007/​s00220-004-1176-6, doi:10.1007/​s00220-004-1176-6.

[26] Marco Merkli. Quantum Markovian master equations: resonance theory shows validity for all time scales. Ann. Physics, 412:167996, 29, 2020. URL: https:/​/​doi.org/​10.1016/​j.aop.2019.167996, doi:10.1016/​j.aop.2019.167996.

[27] Marco Merkli. Dynamics of open quantum systems ii, markovian approximation. Quantum, 6:616, January 2022. doi:10.22331/​q-2022-01-03-616.

[28] C. D. Cushen and R. L. Hudson. A quantum-mechanical central limit theorem. J. Appl. Probability, 8:454–469, 1971. URL: https:/​/​doi.org/​10.2307/​3212170, doi:10.2307/​3212170.

[29] Simon Becker, Nilanjana Datta, Ludovico Lami, and Cambyse Rouzé. Convergence rates for the quantum central limit theorem. Comm. Math. Phys., 383(1):223–279, 2021. URL: https:/​/​doi.org/​10.1007/​s00220-021-03988-1, doi:10.1007/​s00220-021-03988-1.

[30] Alessio Serafini. Quantum Continuous Variables: A Primer of Theoretical Methods (1st ed.). CRC Press., Boca Raton, 2017. URL: https:/​/​doi.org/​10.1201/​9781315118727, doi:10.1201/​9781315118727.

[31] Michael M. Wolf, Geza Giedke, and J. Ignacio Cirac. Extremality of gaussian quantum states. Physical Review Letters, 96(8), March 2006. doi:10.1103/​physrevlett.96.080502.

[32] U. M. Titulaer and R. J. Glauber. Correlation Functions for Coherent Fields. Physical Review, 140(3B):B676–B682, November 1965. doi:10.1103/​PhysRev.140.B676.

[33] Anatole Kenfack and Karol Zyczkowski. Negativity of the Wigner function as an indicator of non-classicality. Journal of Optics B: Quantum and Semiclassical Optics, 6(10):396–404, October 2004. doi:10.1088/​1464-4266/​6/​10/​003.

[34] Stephan De Bièvre, Dmitri B. Horoshko, Giuseppe Patera, and Mikhail I. Kolobov. Measuring nonclassicality of bosonic field quantum states via operator ordering sensitivity. Phys. Rev. Lett., 122:080402, Feb 2019. doi:10.1103/​PhysRevLett.122.080402.

[35] Anaelle Hertz and Stephan De Bièvre. Quadrature coherence scale driven fast decoherence of bosonic quantum field states. Physical Review Letters, 124:090402, March 2020. doi:10.1103/​PhysRevLett.124.090402.

[36] Dmitri B. Horoshko, Stephan De Bièvre, Giuseppe Patera, and Mikhail I. Kolobov. Thermal-difference states of light: Quantum states of heralded photons. Physical Review A, 100:053831, November 2019. doi:10.1103/​PhysRevA.100.053831.

[37] Anaelle Hertz and Stephan De Bièvre. Decoherence and nonclassicality of photon-added and photon-subtracted multimode Gaussian states. Physical Review A, 107(4):043713, 2023. doi:10.1103/​PhysRevA.107.043713.

[38] A. Z. Goldberg, G. S. Thekkadath, and K. Heshami. Measuring the quadrature coherence scale on a cloud quantum computer. Physical Review A, 107(4):042610, 2023. doi:10.1103/​PhysRevA.107.042610.

[39] Célia Griffet, Matthieu Arnhem, Stephan De Bièvre, and Nicolas J. Cerf. Interferometric measurement of the quadrature coherence scale using two replicas of a quantum optical state. Phys. Rev. A, 108:023730, August 2023. doi:10.1103/​PhysRevA.108.023730.

[40] Anaelle Hertz, Nicolas J. Cerf, and Stephan De Bièvre. Relating the Entanglement and Optical Nonclassicality of Multimode States of a Bosonic Quantum Field. Physical Review A, 102(3):032413, September 2020. doi:10.1103/​PhysRevA.102.032413.

[41] Ludovico Lami, Krishna Kumar Sabapathy, and Andreas Winter. All phase-space linear bosonic channels are approximately gaussian dilatable. New J. Phys., 20:113012, 2018. doi:https:/​/​doi.org/​10.1088/​1367-2630/​aae738.

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