×

Lindblad approximation and spin relaxation in quantum electrodynamics. (English) Zbl 1519.81495

Summary: This article is concerned with the time evolution of spin observables for generalized spin boson models. This applies in particular to a model of nuclear magnetic resonance, namely a \(\frac{1}{2}\)-spin particle in a constant external magnetic field and in interaction with the quantized electromagnetic field (photons). We derive a Lindblad (or GKLS) type approximation of the spin dynamics initially in a photon vacuum state together with a precise control of the error coming from this approximation. The error term is bounded by \(g^2\) where \(g\) is the coupling constant of the spin-photon interaction. The point here is the uniformity in time \(t > 0\) of this error control.

MSC:

81V10 Electromagnetic interaction; quantum electrodynamics
81V73 Bosonic systems in quantum theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alicki R and Lendi K 2007 Quantum Dynamical Semigroups and Applications (Lect. Notes Phys.) vol 717 2nd edn (Berlin: Springer)
[2] Amour L, Jager L and Nourrigat J 2019 Infinite dimensional semiclassical analysis and applications to a model in nuclear magnetic resonance J. Math. Phys.60 071503 · Zbl 1416.81222 · doi:10.1063/1.5094396
[3] Amour L, Jager L and Nourrigat J 2020 Ground state photon number at large dsitance Rep. Math. Phys.85 227-38 · Zbl 1441.81135 · doi:10.1016/s0034-4877(20)30026-4
[4] Amour L, Lascar R and Nourrigat J 2017 Weyl calculus in QED I. The unitary group J. Math. Phys.58 013501 · Zbl 1355.81158 · doi:10.1063/1.4973742
[5] Arai A and Hirokawa M 1997 On the existence and uniqueness of ground states of a generalized spin-boson model J. Funct. Anal.151 455-503 · Zbl 0898.47048 · doi:10.1006/jfan.1997.3140
[6] Bloch F 1946 Nuclear induction Phys. Rev.70 460-73 · doi:10.1103/physrev.70.460
[7] Chruściński D and Pascazio S 2017 A brief history of the GKLS equation Open Syst. Inf. Dyn.24 1740001 · Zbl 1377.81081 · doi:10.1142/s1230161217400017
[8] Cohen-Tannoudji C, Dupont-Roc J and Grynberg G 2001 Processus d’interaction entre photons et atomes(Savoirs actuels Paris: EDP Sciences/CNRS Editions)
[9] Davies E B 1974 Markovian master equations Commun. Math. Phys.39 91-110 · Zbl 0294.60080 · doi:10.1007/bf01608389
[10] Davies E B 1976 Quantum Theory of Open Systems (London, New York: Academic) · Zbl 0388.46044
[11] Dereziński J and Gérard C 1999 Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians Rev. Math. Phys.11 383-450 · Zbl 1044.81556 · doi:10.1142/s0129055x99000155
[12] De Roeck W and Kupiainen A 2013 Approach to ground state and time-independent photon bound for massless spin-boson models Ann. Henri Poincaré14 253-311 · Zbl 1266.81163 · doi:10.1007/s00023-012-0190-z
[13] Falconi M, Faupin J, Fröhlich J and Schubnel B 2017 Scattering theory for Lindblad master equations Commun. Math. Phys.350 1185-218 · Zbl 1365.81077 · doi:10.1007/s00220-016-2737-1
[14] Gorini V, Kossakowski A and Sudarshan E C G 1976 Completely positive dynamical semigroups of N-level systems J. Math. Phys.17 821-5 · Zbl 1446.47009 · doi:10.1063/1.522979
[15] Haake F 1973 Statistical treatment of open systems by generalized master equations Springer Tracts in Modern Physics ed Höhler G (Berlin: Springer) · doi:10.1007/978-3-662-40468-3_2
[16] Hübner M and Spohn H 1995 Radiative decay: nonperturbative approaches Rev. Math. Phys.7 363-87 · Zbl 0843.35068 · doi:10.1142/s0129055x95000165
[17] Jakšić V and Pillet C-A 1996 On a model for quantum friction. III. Ergodic properties of the spin-boson system Commun. Math. Phys.178 627-51 · Zbl 0864.47049 · doi:10.1007/bf02108818
[18] Jeener J and Henin F 2002 A presentation of pulsed nuclear magnetic resonance with full quantization of the radio frequency magnetic field J. Chem. Phys.116 8036-47 · doi:10.1063/1.1467332
[19] Kossakowski A 1972 On quantum statistical mechanics of non-Hamiltonian systems Rep. Math. Phys.3 247-74 · doi:10.1016/0034-4877(72)90010-9
[20] Leggett A J, Chakravarty S, Dorsey A T, Fisher M P A, Garg A and Zwerger W 1987 Dynamics of the dissipative two-state system Rev. Mod. Phys.59 1 · doi:10.1103/revmodphys.59.1
[21] Lieb E and Loss M 2004 A note on polarization vectors in quantum electrodynamics Commun. Math. Phys.252 477 · Zbl 1102.81070 · doi:10.1007/s00220-004-1185-5
[22] Lindblad G 1976 On the generators of quantum dynamical semigroups Commun. Math. Phys.48 119-30 · Zbl 0343.47031 · doi:10.1007/bf01608499
[23] Merkli M 2020 Quantum Markovian master equations: resonance theory shows validity for all time scales Ann. Phys., NY412 167996 · Zbl 1432.81047 · doi:10.1016/j.aop.2019.167996
[24] Nakajima S 1958 Prog. Theor. Phys.20 984 · Zbl 0084.21505 · doi:10.1143/ptp.20.948
[25] Prigogine I and Resibois P 1961 Physics27 629 · doi:10.1016/0031-8914(61)90008-8
[26] Reed M and Simon B 1978 Methods of Modern Mathematical Physics (New York, London: Academic) · Zbl 0401.47001
[27] Reuse F A 2007 Electrodynamique et Optique Quantiques (Lausanne: Presses Polytechniques et Universitaires Romandes) · Zbl 1144.78001
[28] Rivas A and Huelga S 2012 Open Quantum Systems (Springer Briefs in Physics) (Heidelberg: Springer) · Zbl 1246.81006 · doi:10.1007/978-3-642-23354-8
[29] Romero R H and Aucar G A 2002 QED approach to the nuclear spin-spin coupling tensor Phys. Rev. A 65 053411 · doi:10.1103/physreva.65.053411
[30] Spohn H 2004 Dynamics of Charged Particles and Their Radiation Field (Cambridge: Cambridge University Press) · Zbl 1078.81004 · doi:10.1017/CBO9780511535178
[31] Van Hove L 1955 Quantum-mechanical perturbations giving rise to a statistical transport equation Physica21 517-40 · Zbl 0074.22804 · doi:10.1016/s0031-8914(55)92832-9
[32] Van Hove L 1957 The approach to equilibrium in quantum statistics. A perturbation treatment to general order Physica23 441-80 · Zbl 0079.19405 · doi:10.1016/s0031-8914(57)92891-4
[33] Zwanzig R 1960 Ensemble method in the theory of irreversibility J. Chem. Phys.33 1338 · doi:10.1063/1.1731409
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.