Kinetic State and Emergence of Markovian Dynamics in Exactly Solvable Models of Open Quantum Systems

A. S. Trushechkin

Informatics and Automation, Noncommutative Analysis and Quantum Information Theory, Tome 324 (2024), pp. 198-224.

Résumé

Typically, the theory of open quantum systems studies the dynamics of the reduced state (density operator) of the system. However, in the early stages of evolution, it is impossible to separate the reservoir dynamics from the system dynamics. Among the consequences of this fact is the violation of the positivity of solutions of some quantum master equations for the reduced density operator. In this paper we study the joint dynamics of the system and reservoir at an early stage of evolution and the pre-relaxation of the joint state to a so-called kinetic state. A kinetic state of the system and reservoir is characterized by the fact that it is completely determined by the reduced density operator of the system alone. Only after the formation of a kinetic state, it becomes possible to describe the evolution of the reduced density operator of the system in terms of a semigroup.

Détail

Keywords: open quantum systems, quantum master equations, Bogolyubov–van Hove limit, pre-relaxation.

Bibliographie

[1] Accardi L., Frigerio A., Lu Y.G., “On the relation between the singular and the weak coupling limits”, Acta appl. math., 26:3 (1992), 197–208 | DOI | MR | Zbl

[2] Accardi L., Lu Y.G., Volovich I., Quantum theory and its stochastic limit, Springer, Berlin, 2002 | DOI | MR | Zbl

[3] Accardi L., Pechen A.N., Volovich I.V., “Quantum stochastic equation for the low density limit”, J. Phys. A: Math. Gen., 35:23 (2002), 4889–4902 | DOI | MR | Zbl

[4] Alicki R., “Master equations for a damped nonlinear oscillator and the validity of the Markovian approximation”, Phys. Rev. A, 40:7 (1989), 4077–4081 | DOI

[5] Alicki R., “Comment on “Reduced dynamics need not be completely positive””, Phys. Rev. Lett., 75:16 (1995), 3020 | DOI | MR

[6] Alicki R., “Pure decoherence in quantum systems”, Open Syst. Inf. Dyn., 11:1 (2004), 53–61 | DOI | MR | Zbl

[7] Alicki R., Lendi K., Quantum dynamical semigroups and applications, Lect. Notes Phys., 717, Springer, Berlin, 2007 | DOI | MR | Zbl

[8] Arai A., Hirokawa M., “On the existence and uniqueness of ground states of a generalized spin-boson model”, J. Funct. Anal., 151:2 (1997), 455–503 | DOI | MR | Zbl

[9] Attal S., Pautrat Y., “From repeated to continuous quantum interactions”, Ann. Henri Poincaré, 7:1 (2006), 59–104 | DOI | MR | Zbl

[10] Becker T., Wu L.-N., Eckardt A., “Lindbladian approximation beyond ultraweak coupling”, Phys. Rev. E, 104:1 (2021), 014110 | DOI | MR

[11] Bloch F., “Generalized theory of relaxation”, Phys. Rev., 105:4 (1957), 1206–1222 | DOI | MR | Zbl

[12] N. N. Bogolyubov, “Problems of a dynamical theory in statistical physics”, Studies in Statistical Mechanics, v. 1, North-Holland, Amsterdam, 1962, 1–118 https://bogolubov.jinr.ru | MR | MR

[13] Bogolyubov N.N., Kineticheskie uravneniya i funktsii Grina v statisticheskoi mekhanike, Preprint No57, In-t fiziki AN AzSSR, Baku, 1977 | MR

[14] Bratteli O., Robinson D.W., Operator algebras and quantum statistical mechanics, v. 1, $C^*$- and $W^*$-algebras. Symmetry groups. Decomposition of states, Springer, Berlin, 1987 ; v. 2, Equilibrium states. Models in quantum statistical mechanics, Springer, Berlin, 1997 | DOI | MR | Zbl | DOI | Zbl

[15] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford Univ. Press, Oxford, 2002 | MR | Zbl

[16] Cattaneo M., Giorgi G.L., Zambrini R., Maniscalco S., “A brief journey through collision models for multipartite open quantum dynamics”, Open Syst. Inf. Dyn., 29:3 (2022), 2250015 | DOI | MR | Zbl

[17] Cheng Y.C., Silbey R.J., “Markovian approximation in the relaxation of open quantum systems”, J. Phys. Chem. B, 109:45 (2005), 21399–21405 | DOI

[18] Ciccarello F., Lorenzo S., Giovannetti V., Palma G.M., “Quantum collision models: Open system dynamics from repeated interactions”, Phys. Rep., 954 (2022), 1–70 | DOI | MR | Zbl

[19] Cresser J.D., Facer C., Coarse-graining in the derivation of Markovian master equations and its significance in quantum thermodynamics, E-print, 2017, arXiv: 1710.09939 [quant-ph] | DOI

[20] D'Abbruzzo A., Cavina V., Giovannetti V., “A time-dependent regularization of the Redfield equation”, SciPost Phys., 15:3 (2023), 117 | DOI | MR

[21] Davidović D., “Completely positive, simple, and possibly highly accurate approximation of the Redfield equation”, Quantum, 4 (2020), 326 | DOI

[22] Davidović D., “Geometric–arithmetic master equation in large and fast open quantum systems”, J. Phys. A: Math. Theor., 55:45 (2022), 455301 | DOI | MR | Zbl

[23] Davies E.B., “Dynamics of a multilevel Wigner–Weisskopf atom”, J. Math. Phys., 15:12 (1974), 2036–2041 | DOI | MR

[24] Davies E.B., “Markovian master equations”, Commun. Math. Phys., 39:2 (1974), 91–110 | DOI | MR | Zbl

[25] Davies E.B., “Markovian master equations. II”, Math. Ann., 219:2 (1976), 147–158 | DOI | MR | Zbl

[26] Dümcke R., “The low density limit for an $N$-level system interacting with a free Bose or Fermi gas”, Commun. Math. Phys., 97:3 (1985), 331–359 | DOI | MR | Zbl

[27] Facchi P., Ligabó M., Lonigro D., “Spectral properties of the singular Friedrichs–Lee Hamiltonian”, J. Math. Phys., 62:3 (2021), 032102 | DOI | MR | Zbl

[28] Farina D., Giovannetti V., “Open-quantum-system dynamics: Recovering positivity of the Redfield equation via the partial secular approximation”, Phys. Rev. A, 100:1 (2019), 012107 | DOI | MR

[29] Filippov S.N., Semin G.N., Pechen A.N., “Quantum master equations for a system interacting with a quantum gas in the low-density limit and for the semiclassical collision model”, Phys. Rev. A, 101:1 (2020), 012114 | DOI | MR

[30] Garraway B.M., “Nonperturbative decay of an atomic system in a cavity”, Phys. Rev. A, 55:3 (1997), 2290–2303 | DOI | MR

[31] Garraway B.M., “Decay of an atom coupled strongly to a reservoir”, Phys. Rev. A, 55:6 (1997), 4636–4639 | DOI

[32] Gorini V., Frigerio A., Verri M., Kossakowski A., Sudarshan E.C.G., “Properties of quantum Markovian master equations”, Rep. Math. Phys., 13:2 (1978), 149–173 | DOI | MR | Zbl

[33] Hartmann R., Strunz W.T., “Accuracy assessment of perturbative master equations: Embracing nonpositivity”, Phys. Rev. A, 101:1 (2020), 012103 | DOI

[34] A. S. Holevo, “Stochastic representations of quantum dynamical semi-groups”, Proc. Steklov Inst. Math., 191 (1992), 145–154 | MR | Zbl

[35] A. S. Holevo, “Quantum stochastic calculus”, J. Sov. Math., 56:5 (1991), 2609–2624 | DOI | MR | Zbl

[36] A. S. Holevo, Statistical Structure of Quantum Theory, Springer, Berlin, 2001 | MR | Zbl

[37] Jeske J., Ing D.J., Plenio M.B., Huelga S.F., Cole J.H., “Bloch–Redfield equations for modeling light-harvesting complexes”, J. Chem. Phys., 142:6 (2015), 064104 | DOI

[38] Jordan T.F., Shaji A., Sudarshan E.C.G., “Dynamics of initially entangled open quantum systems”, Phys. Rev. A, 70:5 (2004), 052110 | DOI | MR | Zbl

[39] Kiršanskas G., Franckié M., Wacker A., “Phenomenological position and energy resolving Lindblad approach to quantum kinetics”, Phys. Rev. B, 97:3 (2018), 035432 | DOI

[40] V. V. Kozlov, “Statistical irreversibility of the Kac reversible circular model”, Regul. Chaotic Dyn., 16:5 (2011), 536–549 | DOI | MR | MR | Zbl

[41] Latune C.L., Sinayskiy I., Petruccione F., “Negative contributions to entropy production induced by quantum coherences”, Phys. Rev. A, 102:4 (2020), 042220 | DOI | MR

[42] Li L., Hall M.J.W., Wiseman H.M., “Concepts of quantum non-Markovianity: a hierarchy”, Phys. Rep., 759 (2018), 1–51 | DOI | MR | Zbl

[43] Lill S., Lonigro D., Self-adjointness and domain of generalized spin–boson models with mild ultraviolet divergences, E-print, 2023, arXiv: 2307.14727 [math-ph] | DOI | MR

[44] Łobejko M., Winczewski M., Suárez G., Alicki R., Horodecki M., Towards reconciliation of completely positive open system dynamics with the equilibration postulate, E-print, 2022, arXiv: 2204.00643 [quant-ph] | DOI

[45] Lonigro D., “Generalized spin–boson models with non-normalizable form factors”, J. Math. Phys., 63:7 (2022), 072105 | DOI | MR | Zbl

[46] Lonigro D., “Renormalization of spin–boson interactions mediated by singular form factors”, Quantum mathematics II: INdAM 2022, Springer INdAM Ser., 58, Springer, Singapore, 2023, 103–122 | DOI | MR

[47] Lonigro D., “Self-adjointness of a class of multi-spin–boson models with ultraviolet divergences”, Math. Phys. Anal. Geom., 26:2 (2023), 15 | DOI | MR | Zbl

[48] Lonigro D., Chruściński D., “Quantum regression in dephasing phenomena”, J. Phys. A: Math. Theor., 55:22 (2022), 225308 | DOI | MR | Zbl

[49] Majenz C., Albash T., Breuer H.-P., Lidar D.A., “Coarse graining can beat the rotating-wave approximation in quantum Markovian master equations”, Phys. Rev. A, 88:1 (2013), 012103 | DOI

[50] McCauley G., Cruikshank B., Bondar D.I., Jacobs K., “Accurate Lindblad-form master equation for weakly damped quantum systems across all regimes”, npj Quantum Inf., 6 (2020), 74 | DOI

[51] Merkli M., “Quantum Markovian master equations: Resonance theory shows validity for all time scales”, Ann. Phys., 412 (2020), 167996 | DOI | MR | Zbl

[52] Merkli M., “Correlation decay and Markovianity in open systems”, Ann. Henri Poincaré, 24 (2023), 751–782 | DOI | MR | Zbl

[53] Mozgunov E., Lidar D., “Completely positive master equation for arbitrary driving and small level spacing”, Quantum, 4 (2020), 227 | DOI

[54] Nathan F., Rudner M.S., “Universal Lindblad equation for open quantum systems”, Phys. Rev. B, 102:11 (2020), 115109 | DOI

[55] Ohya M., Volovich I., Mathematical foundations of quantum information and computation and its applications to nano- and bio-systems, Springer, Dordrecht, 2011 | DOI | MR | Zbl

[56] Open quantum systems I: The Hamiltonian approach, Lect. Notes Math., 1880, ed. by S. Attal, A. Joye, C.-A. Pillet, Springer, Berlin, 2006 | DOI | MR | Zbl

[57] Palma G.M., Suominen K.-A., Ekert A.K., “Quantum computers and dissipation”, Proc. R. Soc. London A, 452 (1996), 567–584 | DOI | MR | Zbl

[58] Palmer P.F., “The singular coupling and weak coupling limits”, J. Math. Phys., 18:3 (1977), 527–529 | DOI | MR

[59] Pechukas P., “Reduced dynamics need not be completely positive”, Phys. Rev. Lett., 73:8 (1994), 1060–1062 | DOI | MR | Zbl

[60] Potts P.P., Kalaee A.A.S., Wacker A., “A thermodynamically consistent Markovian master equation beyond the secular approximation”, New J. Phys., 23 (2021), 123013 | DOI | MR

[61] Ptaszyński K., Esposito M., “Thermodynamics of quantum information flows”, Phys. Rev. Lett., 122:15 (2019), 150603 | DOI | MR

[62] Ptaszyński K., Esposito M., “Post-thermalization via information spreading in open quantum systems”, Phys. Rev. E, 106:1 (2022), 014122 | DOI | MR

[63] Redfield A.G., “On the theory of relaxation processes”, IBM J. Res. Dev., 1:1 (1957), 19–31 | DOI | MR

[64] Redfield A.G., “The theory of relaxation processes”, Adv. Magn. Opt. Reson., 1 (1965), 1–32 | DOI

[65] M. Reed and B. Simon, Methods of Modern Mathematical Physics, v. 2, Fourier Analysis, Self-adjointness, Academic Press, New York, 1975 | MR | MR | Zbl

[66] Rivas Á., “Refined weak-coupling limit: Coherence, entanglement, and non-Markovianity”, Phys. Rev. A, 95:4 (2017), 042104 | DOI

[67] Rivas Á., “Quantum thermodynamics in the refined weak coupling limit”, Entropy, 21:8 (2019), 725 | DOI | MR

[68] Rivas Á., Huelga S.F., Open quantum systems: An introduction, Springer, Berlin, 2012 | DOI | MR | Zbl

[69] Rodríguez-Rosario C.A., Modi K., Aspuru-Guzik A., “Linear assignment maps for correlated system-environment states”, Phys. Rev. A, 81:1 (2010), 012313 | DOI

[70] Schaller G., Brandes T., “Preservation of positivity by dynamical coarse graining”, Phys. Rev. A, 78:2 (2008), 022106 | DOI | MR

[71] Soret A., Cavina V., Esposito M., “Thermodynamic consistency of quantum master equations”, Phys. Rev. A, 106:6 (2022), 062209 | DOI | MR

[72] Suárez A., Silbey R., Oppenheim I., “Memory effects in the relaxation of quantum open systems”, J. Chem. Phys., 97:7 (1992), 5101–5107 | DOI

[73] A. E. Teretenkov, “Pseudomode approach and vibronic non-Markovian phenomena in light-harvesting complexes”, Proc. Steklov Inst. Math., 306 (2019), 242–256 | DOI | DOI | MR | Zbl

[74] Teretenkov A.E., “Non-perturbative effects in corrections to quantum master equations arising in Bogolubov–van Hove limit”, J. Phys. A: Math. Theor., 54:26 (2021), 265302 | DOI | MR | Zbl

[75] Trushechkin A.S., “Vyvod kvantovogo kineticheskogo uravneniya Redfilda i popravok k nemu po metodu Bogolyubova”, Tr. MIAN, 313 (2021), 263–274 | DOI | MR | Zbl

[76] Trushechkin A., “Unified Gorini–Kossakowski–Lindblad–Sudarshan quantum master equation beyond the secular approximation”, Phys. Rev. A, 103:6 (2021), 062226 | DOI | MR

[77] Trushechkin A., “Quantum master equations and steady states for the ultrastrong-coupling limit and the strong-decoherence limit”, Phys. Rev. A, 106:4 (2022), 042209 | DOI | MR

[78] Trushechkin A.S., “Long-term behaviour in an exactly solvable model of pure decoherence and the problem of Markovian embedding”, Mathematics, 12:1 (2024), 1 | DOI

[79] Tupkary D., Dhar A., Kulkarni M., Purkayastha A., “Fundamental limitations in Lindblad descriptions of systems weakly coupled to baths”, Phys. Rev. A, 105:3 (2022), 032208 | DOI | MR

[80] Wangsness R.K., Bloch F., “The dynamical theory of nuclear induction”, Phys. Rev., 89:4 (1953), 728–739 | DOI | Zbl