Dynamics of Open Quantum Systems I, Oscillation and Decay

Marco Merkli

Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, A1C 5S7, Canada

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Abstract

We develop a framework to analyze the dynamics of a finite-dimensional quantum system $\rm S$ in contact with a reservoir $\rm R$. The full, interacting $\rm SR$ dynamics is unitary. The reservoir has a stationary state but otherwise dissipative dynamics. We identify a main part of the full dynamics, which approximates it for small values of the $\rm SR$ coupling constant, uniformly for all times $t\ge 0$. The main part consists of explicit oscillating and decaying parts. We show that the reduced system evolution is Markovian for all times. The technical novelty is a detailed analysis of the link between the dynamics and the spectral properties of the generator of the $\rm SR$ dynamics, based on Mourre theory. We allow for $\rm SR$ interactions with little regularity, meaning that the decay of the reservoir correlation function only needs to be polynomial in time, improving on the previously required exponential decay.
In this work we distill the structural and technical ingredients causing the characteristic features of oscillation and decay of the $\rm SR$ dynamics. In the companion paper [27] we apply the formalism to the concrete case of an $N$-level system linearly coupled to a spatially infinitely extended thermal bath of non-interacting Bosons.

We consider a quantum system in contact with an environment – think of a few spins/qubits interacting with a bath of oscillating degrees of freedom. The environment is very large, has a single stationary (equilibrium) state and is dispersive, meaning that local perturbations created around the system's location travel away from it eventually. We analyze the evolution of the system-environment complex, given by the Schrödinger equation.

We show that the evolution is decomposed into a sum of three parts: One first main part describes the system evolving according to a markovian (memory-less) dynamics while the environment is constantly in its equilibrium (“Born approxiation''). A second main part describes the evolution of initial states away from (orthogonal to) the reservoir equilibrium. A third part is a remainder term, which is small in the system-environment coupling parameter, for all times. The markovian part is composed of explicit (exponentially in time) decaying and oscillating terms and the second main part decays as an inverse power of time.

The paper explains the strategy of the approach and contains the details and estimates leading to the main result. This makes it suitable for researchers who want to learn about the relevant mathematical methods developed in the theory of open quantum systems. In a companion paper, the results are applied to the ubiquitous open system model of an $N$ level system coupled to a bath (field) of free bosonic degrees of freedom. We show there that the markovian approximation, given by the Davies generator, holds for all times, even for reservoirs with slowly (polynomially) decaying correlations.

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[1] R. Alicki, K. Lendi: Lect. Notes Phys., vol. 717, Springer Verlag, 2007.
https:/​/​doi.org/​10.1007/​3-540-70861-8

[2] H. Araki, E.J. Woods: Representation of the canonical commutation relations describing a nonrelativistic infinite free bose gas, J. Math. Phys. 4, 637-662 (1963).
https:/​/​doi.org/​10.1063/​1.1704002

[3] V. Bach, J. Fröhlich, I.M. Sigal: Return to equilibrium, J. Math. Phys. 41(6), 3985-4060 (2000).
https:/​/​doi.org/​10.1063/​1.533334

[4] O. Bratteli, D.W. Robinson: Operator Algebras and Quantum Statistical Mechanics 1,2, Texts and Monographs in Physics, Springer Verlag 2002.
https:/​/​doi.org/​10.1007/​978-3-662-02520-8

[5] H.-P. Breuer, F. Petruccione: The Theory of Open Quantum Systems, Oxford University Press, 2002.
https:/​/​doi.org/​10.1093/​acprof:oso/​9780199213900.001.0001

[6] D. Chruściński, S. Pascazio: A Brief History of the GKLS Equation, Open Syst. Inf. Dyn. 24, No. 03, 1740001 (2017).
https:/​/​doi.org/​10.1142/​S1230161217400017

[7] J. Cesar: Commentarii de Bello Gallico (50 B.C.).

[8] H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon: Schrödinger Operators, Texts and Monographs in Physics, Springer Verlag 1986.
https:/​/​doi.org/​10.1007/​978-3-540-77522-5

[9] E.B. Davies: Markovian Master Equations, Commun. Math. Phys. 39, 9-110 (1974).
https:/​/​doi.org/​10.1007/​BF01608389

[10] E.B. Davies: Markovian Master Equations, II, Math. Ann. 219, 147-158 (1976).
https:/​/​doi.org/​10.1007/​BF01351898

[11] J. Dereziński, V. Jaksic: Spectral Theory of Pauli-Fierz Operators, J. Funct. Analysis 180, 243-327 (2001).
https:/​/​doi.org/​10.1006/​jfan.2000.3681

[12] J. Dereziński, V. Jaksic, C.-A. Pillet: Perturbation Theory of $W^*$-Dynamics, Liouvilleans and KMS-States, Rev. Math. Phys. 15, No. 5, 447-489 (2003).
https:/​/​doi.org/​10.1142/​S0129055X03001679

[13] R. Dümke, H. Spohn: The Proper Form of the Generator in the Weak Coupling Limit, Z. Physik B, 34, 419-422 (1979).
https:/​/​doi.org/​10.1007/​BF01325208

[14] K.-J. Engel, R. Nagel: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics 194, Springer Verlag 2000.
https:/​/​doi.org/​10.1007/​b97696

[15] J. Fröhlich, M. Merkli: Another return of ``return to equilibrium'', Comm. Math. Phys. 251(2), 235-262 (2004).
https:/​/​doi.org/​10.1007/​s00220-004-1176-6

[16] W. Hunziker: Resonances, Metastable States and Exponential Decay Laws in Perturbation Theory, Comm. Math. Phys. 132, 177-188 (1990).
https:/​/​doi.org/​10.1007/​BF02278006

[17] W. Hunziker, I.M. Sigal: Time-dependent scattering theory of $N$-body quantum systems, Rev. Math. Phys. 12, No. 08, 1033-1084 (2000).
https:/​/​doi.org/​10.1142/​S0129055X0000040X

[18] V. Jaksic, C.-A. 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).
https:/​/​doi.org/​10.1007/​BF02099252

[19] T. Kato: Perturbation Theory for Linear Operators. Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Volume 132, Springer Verlag 1966.
https:/​/​doi.org/​10.1007/​978-3-662-12678-3

[20] M. Könenberg, M. Merkli: On the irreversible dynamics emerging from quantum resonances, J. Math. Phys. 57, 033302 (2016).
https:/​/​doi.org/​10.1063/​1.4944614

[21] M. Könenberg, M. Merkli: Completely positive dynamical semigroups and quantum resonance theory, Lett. Math. Phys. 107, Issue 7, 1215-1233 (2017) and Correction to: Completely positive dynamical semigroups and quantum resonance theory.
https:/​/​doi.org/​10.1007/​s11005-019-01177-9

[22] C. Majenz, T. Albash, H.-P. Breuer, D. A. Lidar: Coarse graining can beat the rotating-wave approximation in quantum Markovian master equations, Phys. Rev. A 88, 012103 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.88.012103

[23] J.-G. Li, J. Zou, B. Shao: Non-Markovianity of the damped Jaynes-Cummings model with detuning, Phys. Rev. A 81, 062124 (2010).
https:/​/​doi.org/​10.1103/​PhysRevA.81.062124

[24] M. Merkli, I.M. Sigal, G.P. Berman: Decoherence and Thermalization, Phys. Rev. Lett. 98, 130401 (2007); Resonance Theory of Decoherence and Thermalization, Ann. Phys. 323, 373-412 (2008); Dynamics of Collective Decoherence and Thermalization, Ann. Phys. 323, 3091-3112 (2008).
https:/​/​doi.org/​10.1016/​j.aop.2008.07.004

[25] M. Merkli: Positive Commutators in Non-Equilibrium Statistical Mechanics, Comm. Math. Phys. 223, 327-362 (2001).
https:/​/​doi.org/​10.1007/​s002200100545

[26] M. Merkli: Quantum Markovian master equations: Resonance theory shows validity for all time scales, Ann. Phys. 412 167996 (2020).
https:/​/​doi.org/​10.1016/​j.aop.2019.167996

[27] M. Merkli: Dynamics of Open Quantum Systems II, Markovian Approximation, Quantum 6 616 (2022).
https:/​/​doi.org/​10.22331/​q-2022-01-03-616

[28] M. Merkli: Correlation decay and Markovianity in open systems, submitted (2021).
arXiv:2107.02515

[29] M. Merkli, H. Song and G.P. Berman: Multiscale dynamics of open three-level quantum systems with two quasi-degenerate levels, J. Phys. A: Math. Theor. 48, 275304 (2015).
https:/​/​doi.org/​10.1088/​1751-8113/​48/​27/​275304

[30] M. Merkli, G.P. Berman, A. Saxena: Quantum Electron Transport in Degenerate Donor-Acceptor Systems, J. Math. Phys. 61, 072102 (2020).
https:/​/​doi.org/​10.1063/​1.5138725

[31] M. Merkli: The ideal quantum gas, Lecture Notes in Mathematics, 1880, 183-233 (2006).
https:/​/​doi.org/​10.1007/​3-540-33922-1_5

[32] E. Mozgunov, D. Lidar: Completely positive master equation for arbitrary driving and small level spacing, Quantum 4 227 (2020).
https:/​/​doi.org/​10.22331/​q-2020-02-06-227

[33] F. Nathan, M.S. Rudner: Universal Lindblad equation for open quantum systems, Phys. Rev. B 102, 115109 (2020).
https:/​/​doi.org/​10.1103/​PhysRevB.102.115109

[34] C.-A. Pillet, C. E. Wayne: Invariant Manifolds for a Class of Dispersive, Hamiltonian, Partial Differential Equations, J. Diff. Equ., 141, Issue 2, p. 310-326 (1997).
https:/​/​doi.org/​10.1006/​jdeq.1997.3345

[35] Á. Rivas: Refined weak-coupling limit: Coherence, entanglement, and non-Markovianity, Phys. Rev. A 95, 042104 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.95.042104

[36] Á. Rivas, S.F. Huelga: Open Quantum Systems, An Introduction, in: Springer Briefs in Physics 2012.
https:/​/​doi.org/​10.1007/​978-3-642-23354-8

[37] Á. Rivas, A. D. K. Plato, S. F. Huelga, and M. B. Plenio: Markovian master equations: A critical study, New J. Phys. 12, 113032 (2010).
https:/​/​doi.org/​10.1088/​1367-2630/​12/​11/​113032

[38] B. Simon: Resonances in $n$-body quantum systems with dilation analytic potentials and the foundations of time-dependent perturbation theory, Ann. Math. 97, 247-274 (1973); Resonances and complex scaling: A rigoros overview, Int. J. Quant. Chem. 14, 529-542 (1978).
https:/​/​doi.org/​10.1002/​qua.560140415

[39] A. Soffer, I.M. Weinstein: Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys. 133, Issue 1, pp.119-146 (1990).
https:/​/​doi.org/​10.1007/​BF02096557

[40] A. Trushechkin: Unified Gorini-Kossakowski-Lindblad-Sudarshan quantum master equation beyond the secular approximation, Phys. Rev. A 103, 062226 (2021).
https:/​/​doi.org/​10.1103/​PhysRevA.103.062226

Cited by

[1] Marco Merkli, "Dynamics of Open Quantum Systems II, Markovian Approximation", Quantum 6, 616 (2022).

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