Dynamics of Open Quantum Systems II, Markovian Approximation

Marco Merkli

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

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A finite-dimensional quantum system is coupled to a bath of oscillators in thermal equilibrium at temperature $T \gt 0$. We show that for fixed, small values of the coupling constant $\lambda$, the true reduced dynamics of the system is approximated by the completely positive, trace preserving Markovian semigroup generated by the Davies-Lindblad generator. The difference between the true and the Markovian dynamics is $O(|\lambda|^{1/4})$ for all times, meaning that the solution of the Gorini-Kossakowski-Sudarshan-Lindblad master equation is approximating the true dynamics to accuracy $O(|\lambda|^{1/4})$ for all times. Our method is based on a recently obtained expansion of the full system-bath propagator. It applies to reservoirs with correlation functions decaying in time as $1/t^{4}$ or faster, which is a significant improvement relative to the previously required exponential decay.

We consider a finite-dimensional quantum system interacting with a thermal Bose field (a reservoir consisting of non-interacting oscillators). The total dynamics is given by a Hamiltonian containing a system, a reservoir plus an interaction term. We give a rigorous argument showing that the reduced dynamics of the system (after tracing out the reservoir degrees of freedom) is approximated by the solution of the markovian master equation. The difference between the approximate (markovian) and the true system density matrix is small in the interaction between the system and reservoir, equally so for all times $t\ge0$. We show this result for reservoirs with polynomially decaying correlation functions (previously, exponentially quick decay was needed).

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[1] R. Alicki, K. Lendi: Lect. Notes Phys., vol. 717, Springer Verlag, 2007.

[2] L. Amour, J. Nourrigat: Lindblad approximation and spin relaxation in quantum electrodynamics, J. Phys. A 53, 245204-245222 (2020).

[3] 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).

[4] V. Bach, J. Fröhlich, I.M. Sigal: Return to equilibrium, J. Math. Phys. 41(6), 3985-4060 (2000).

[5] F. Benatti, R. Floreanini: Open Quantum Dynamics: Complete Positivity and Entanglement, Int. J. Mod. Phys. B 19, 3063 (2005).

[6] O. Bratteli, D.W. Robinson: Operator Algebras and Quantum Statistical Mechanics 1,2, Texts and Monographs in Physics, Springer Verlag 2002.

[7] H.-P. Breuer, F. Petruccione: The Theory of Open Quantum Systems, Oxford University Press, 2002.

[8] D. Chruściński, S. Pascazio: A Brief History of the GKLS Equation, Open Syst. Inf. Dyn. 24, No. 03, 1740001 (2017).

[9] D. Chruściński: On the hybrid Davies like generator for quantum dissipation, Chaos 31, 023110 (2021).

[10] E.B. Davies: Markovian Master Equations, Commun. Math. Phys. 39, 9-110 (1974).

[11] E.B. Davies: Markovian Master Equations, II, Math. Ann. 219, 147-158 (1976).

[12] J. Dereziński, V. Jaksic: Spectral Theory of Pauli-Fierz Operators, J. Funct. Analysis 180, 243-327 (2001).

[13] 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).

[14] R. Dümke, H. Spohn: The Proper Form of the Generator in the Weak Coupling Limit, Z. Physik B, 34, 419-422 (1979).

[15] P. Facchi, S. Pascazio: Deviations from exponential law and Van Hove's “$\lambda^2 t$” limit, Physica A 271, 133-146 (1999).

[16] J. Fröhlich, M. Merkli: Another return of ``return to equilibrium'', Comm. Math. Phys. 251(2), 235-262 (2004).

[17] V. Jaksic, C.-A. Pillet: A Note on Eigenvalues of Liouvilleans , J. Stat. Phys. 105, Nos. 5/​6 (2001).

[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).

[19] M. Könenberg, M. Merkli, H. Song: Ergodicity of the Spin-Boson Model for Arbitrary Coupling Strength, Commun. Math. Phys. 336, 261-285 (2015).

[20] M. Könenberg, M. Merkli: On the irreversible dynamics emerging from quantum resonances, J. Math. Phys. 57, 033302 (2016).

[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.

[22] J.-G. Li, J. Zou, B. Shao: Non-Markovianity of the damped Jaynes-Cummings model with detuning, Phys. Rev. A 81, 062124 (2010).

[23] D. A. Lidar: Lecture notes on the theory of open quantum systems (2019).

[24] D.A. Lidar, Z. Bihary, K.B. Whaley: From completely positive maps to the quantum Markovian semigroup master equation, Chem. Phys. 268 35-53 (2001).

[25] C. Majenz, T. Albash, H.-P. Breuer, and D. A. Lidar: Coarse graining can beat the rotating-wave approximation in quantum markovian master equations, Phys. Rev. A 88, 012103 (2013).

[26] M. Merkli, G.P. Berman, A. Redondo: Application of resonance perturbation theory to dynamics of magnetization in spin systems interacting with local and collective bosonic reservoirs, J. Phys. A Math. Theor. 44, 305306 (2011).

[27] M. Merkli: Positive Commutators in Non-Equilibrium Statistical Mechanics, Comm. Math. Phys. 223, 327-362 (2001).

[28] M. Merkli: Level shift operators for open quantum systems, J. Math. Anal. Appl. 327, Issue 1, 376-399 (2007).

[29] M. Merkli: Quantum Markovian master equations: Resonance theory shows validity for all time scales, Ann. Phys. 412 167996 (2020).

[30] M. Merkli: Dynamics of Open Quantum Systems I, Oscillation and Decay, Quantum 6 615 (2022).

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

[32] M. Merkli: The ideal quantum gas, Lecture Notes in Mathematics, 1880, 183-233 (2006).

[33] M. Mohseni, Y. Omar, G.S. Engel, and M.B. Plenio (Eds): Quantum Effects in Biology, Cambridge University Press, 2014.

[34] E. Mozgunov, D. Lidar: Completely positive master equation for arbitrary driving and small level spacing, Quantum 4 227 (2020).

[35] A.S. Trushechkin, M. Merkli, J.D. Cresser, J. Anders: Open quantum system dynamics and the mean force Gibbs state.

[36] L. Van Hove: Quantum-mechanical perturbations giving rise to a statistical transport equation, Physica 21 (1-5), 517-540 (1955).

[37] Á. Rivas: Refined weak-coupling limit: Coherence, entanglement, and non-Markovianity, Phys. Rev. A 95, 042104 (2017).

[38] Á. Rivas, S.F. Huelga: Open Quantum Systems, An Introduction, in: Springer Briefs in Physics 2012.

[39] 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).

[40] V.A. Zagrebnov: Gibbs Semigroups, in: Operator Theory: Advances and Applications 273, Birkhäuser 2019.

Cited by

[1] A. S. Trushechkin, M. Merkli, J. D. Cresser, and J. Anders, "Open quantum system dynamics and the mean force Gibbs state", AVS Quantum Science 4 1, 012301 (2022).

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