Derivation of the Redfield Quantum Master Equation and Corrections to It by the Bogoliubov Method

A. S. Trushechkin

Informatics and Automation, Mathematics of Quantum Technologies, Tome 313 (2021), pp. 263-274.

Résumé

Following the ideas N. N. Bogoliubov used to derive the classical and quantum nonlinear kinetic equations, we derive the Redfield quantum linear master equation, which is widely used in the theory of open quantum systems. This method allows one to calculate the joint state of a system and a reservoir at every instant of time, as well as to obtain quantum master equations as autonomous differential equations in an arbitrary order of perturbation theory. We prove that under certain conditions the expressions for the corrections of all orders are well defined. We also discuss the question of whether the quantum dynamics described by the derived quantum master equations is Markovian.

Bibliographie

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

[2] R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics, J. Wiley Sons, New York, 1975 | MR | Zbl

[3] N. N. Bogoliubov, Problems of a Dynamical Theory in Statistical Physics, North-Holland, Amsterdam, 1962 | MR | MR

[4] Bogolyubov N.N., Gurov K.P., “Kineticheskie uravneniya v kvantovoi mekhanike”, ZhETF, 17:7 (1947), 614–628

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

[6] Chruściński D., Pascazio S., “A brief history of the GKLS equation”, Open Syst. Inf. Dyn., 24:3 (2017), 1740001 | DOI | MR | Zbl

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

[8] Dümcke R., “Convergence of multitime correlation functions in the weak and singular coupling limits”, J. Math. Phys., 24:2 (1983), 311–315 | DOI | MR

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

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

[11] A. S. Holevo, Quantum Systems, Channels, Information: A Mathematical Introduction, de Gruyter Stud. Math. Phys., 16, de Gruyter, Berlin, 2012 | MR | Zbl

[12] Jang S., Cao J., Silbey R.J., “Fourth-order quantum master equation and its Markovian bath limit”, J. Chem. Phys., 116:7 (2002), 2705–2717 | DOI

[13] Lax M., “Formal theory of quantum fluctuations from a driven state”, Phys. Rev., 129:5 (1963), 2342–2348 | DOI | MR

[14] 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

[15] Lo Gullo N., Sinayskiy I., Busch Th., Petruccione F., Non-Markovianity criteria for open system dynamics, E-print, 2014, arXiv: 1401.1126 [quant-ph]

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

[17] Pechen A.N., Volovich I.V., “Quantum multipole noise and generalized quantum stochastic equations”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 5:4 (2002), 441–464 | DOI | MR | Zbl

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

[19] Richter M., Mukamel S., “Relaxation processes in systems strongly coupled to a harmonic bath”, J. Mod. Opt., 57:19 (2010), 2004–2008 | DOI | Zbl

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

[21] Spohn H., “Kinetic equations from Hamiltonian dynamics: Markovian limits”, Rev. Mod. Phys., 52:3 (1980), 569–615 | DOI | MR

[22] Spohn H., Large scale dynamics of interacting particles, Springer, Berlin, 1991 | Zbl

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

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

[25] Teretenkov A.E., “Non-Markovian evolution of multi-level system interacting with several reservoirs. Exact and approximate”, Lobachevskii J. Math., 40:10 (2019), 1587–1605 | DOI | MR | Zbl

[26] Teretenkov A.E., Non-perturbative effects in corrections to quantum master equations arising in Bogolubov–van Hove limit, E-print, 2020, arXiv: 2008.02820 [quant-ph]

[27] A. S. Trushechkin, “Dynamics of reservoir observables within the Zwanzig projection operator method in the theory of open quantum systems”, Proc. Steklov Inst. Math., 306 (2019), 257–270 | DOI | MR | Zbl

[28] Trushechkin A., “Calculation of coherences in Förster and modified Redfield theories of excitation energy transfer”, J. Chem. Phys., 151:7 (2019), 074101 | DOI

[29] Trushechkin A., Unified GKLS quantum master equation beyond the secular approximation, E-print, 2021, arXiv: 2103.12042 [quant-ph] | Zbl

[30] Zwanzig R., “Ensemble method in the theory of irreversibility”, J. Chem. Phys., 33:5 (1960), 1338–1341 | DOI | MR