Infinite dimensional semiclassical analysis and applications to a model in nuclear magnetic resonance. (English) Zbl 1416.81222
Summary: We are concerned in this paper with the connection between the dynamics of a model related to nuclear magnetic resonance in Quantum Field Theory (QFT) and its classical counterpart known as the Maxwell-Bloch equations. The model in QFT is a model of quantum electrodynamics considering fixed spins interacting with the quantized electromagnetic field in an external constant magnetic field. This model is close to the common spin-boson model. The classical model goes back to F. Bloch [Phys. Rev., II. Ser. 70, 460–473 (1946)]. Our goal is not only to study the derivation of the Maxwell-Bloch equations but also to establish a semiclassical asymptotic expansion of arbitrary high order with control of the error terms of these standard nonlinear classical motion equations. This provides therefore quantum corrections of any order in powers of the semiclassical parameter of the Bloch equations. Besides, the asymptotic expansion for the photon number is also analyzed, and a law describing the photon number time evolution is written down involving the radiation field polarization. Since the quantum photon state Hilbert space (radiation field) is infinite dimensional, we are thus concerned in this article with the issue of semiclassical calculus in an infinite dimensional setting. In this regard, we are studying standard notions as Wick and anti-Wick quantizations, heat operator, Beals characterization theorem, and compositions of symbols in the infinite dimensional context which can have their own interest.
©2019 American Institute of Physics
©2019 American Institute of Physics
MSC:
| 81V35 | Nuclear physics |
| 81V10 | Electromagnetic interaction; quantum electrodynamics |
| 81T10 | Model quantum field theories |
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
| 81V80 | Quantum optics |
Keywords:
Maxwell-Bloch equationsReferences:
| [1] | Ammari, Z.; Falconi, M., Wigner measures approach to the classical limit of the Nelson model: Convergence of dynamics and ground state energy, J. Stat. Phys., 157, 2, 330-362 (2014) · Zbl 1302.82009 · doi:10.1007/s10955-014-1079-7 |
| [2] | Ammari, Z.; Falconi, M., Bohr’s correspondence principle for the renormalized Nelson model, SIAM J. Math. Anal., 49, 6, 5031-5095 (2017) · Zbl 1394.35383 · doi:10.1137/17m1117598 |
| [3] | Ammari, Z.; Nier, F., Mean field limit for bosons and infinite dimensional phase-space analysis, Ann. Henri Poincaré, 9, 8, 1503-1574 (2008) · Zbl 1171.81014 · doi:10.1007/s00023-008-0393-5 |
| [4] | Amour, L.; Jager, L.; Nourrigat, J., On bounded Weyl pseudodifferential operators in Wiener spaces, J. Funct. Anal., 269, 2747-2812 (2015) · Zbl 1328.47052 · doi:10.1016/j.jfa.2015.08.004 |
| [5] | Amour, L.; Khodja, M.; Nourrigat, J., Approximative composition of Wick symbols and applications to the time dependent Hartree-Fock equation, Asymptotic Anal., 85, 3-4, 229-248 (2013) · Zbl 1287.47058 · doi:10.3233/ASY-131184 |
| [6] | Amour, L.; Lascar, R.; Nourrigat, J., Beals characterization of pseudodifferential operators in Wiener spaces, Appl. Math. Res. Express, 2017, 1, 42-270 · Zbl 1456.35248 · doi:10.1093/amrx/abw001 |
| [7] | Amour, L.; Lascar, R.; Nourrigat, J., Weyl calculus in QED. I. The unitary group, J. Math. Phys., 58, 013501 (2017) · Zbl 1355.81158 · doi:10.1063/1.4973742 |
| [8] | Amour, L.; Lascar, R.; Nourrigat, J., Weyl calculus in Wiener spaces and in QED · Zbl 1421.81177 |
| [9] | Amour, L.; Nourrigat, J., Hamiltonian systems and semiclassical dynamics for interacting spins in QED (2015) |
| [10] | Appleby, D. M., Husimi transform of an operator product, J. Phys. A: Math. Gen., 33, 21, 3903-3915 (2000) · Zbl 0991.81054 · doi:10.1088/0305-4470/33/21/304 |
| [11] | Arai, A.; Hirokawa, M., On the existence and uniqueness of ground states of a generalized spin-boson model, J. Funct. Anal., 151, 2, 455-503 (1997) · Zbl 0898.47048 · doi:10.1006/jfan.1997.3140 |
| [12] | Bach, V.; Fröhlich, J.; Sigal, I. M., Quantum electrodynamics on confined nonrelativistic particles, Adv. Math., 137, 2, 299-395 (1998) · Zbl 0923.47040 · doi:10.1006/aima.1998.1734 |
| [13] | Beals, R., Characterization of pseudodifferential operators and applications, Duke Math. J., 44, 1, 45-57 (1977) · Zbl 0353.35088 · doi:10.1215/s0012-7094-77-04402-7 |
| [14] | Berezin, F. A.; Šubin, M. A., The Schrödinger Equation (1991) · Zbl 0749.35001 |
| [15] | Bloch, F., Nuclear induction, Phys. Rev., 70, 460-473 (1946) · doi:10.1103/physrev.70.460 |
| [16] | Cohen-Tannoudji, C.; Dupont-Roc, J.; Grynberg, G., Processus d’Interaction Entre Photons et Atomes (2001) |
| [17] | Combescure, M.; Robert, D., Coherent States and Applications in Mathematical Physics (2012) · Zbl 1243.81004 |
| [18] | Correggi, M.; Falconi, M., Effective potentials generated by field interaction in the quasi- classical limit, Ann. Henri Poincaré, 19, 1, 189-235 (2018) · Zbl 1392.81132 · doi:10.1007/s00023-017-0612-z |
| [19] | Dereziński, J.; Gérard, C., Asymptotic completeness in quantum field theory: Massive Pauli-Fierz Hamiltonians, Rev. Math. Phys., 11, 4, 383-450 (1999) · Zbl 1044.81556 · doi:10.1142/s0129055x99000155 |
| [20] | Engelke, F., Virtual photons in magnetic resonance, Concepts Magn. Reson., Part A, 36A, 5, 266-339 (2010) · doi:10.1002/cmr.a.20166 |
| [21] | Folland, G. B., Harmonic Analysis in Phase Space (1989) · Zbl 0682.43001 |
| [22] | Frank, R. L.; Zhou, G., Derivation of an effective evolution equation for a strongly coupled polaron, Anal. PDE, 10, 2, 379-422 (2017) · Zbl 1365.35130 · doi:10.2140/apde.2017.10.379 |
| [23] | Fröhlich, J., On the infrared problem in a model of scalar electrons and massless scalar bosons, Ann. Inst. H. Poincaré Sect. A (N.S.), 19, 1-103 (1973) · Zbl 1216.81151 |
| [24] | Gérard, C., On the existence of ground states for massless Pauli-Fierz Hamiltonians, Ann. Henri Poincaré, 1, 443-459 (2000) · Zbl 1004.81012 · doi:10.1007/s000230050002 |
| [25] | Ginibre, J.; Nironi, F.; Velo, G., Partially classical limit of the Nelson model, Ann. Henri Poincaré, 7, 1, 21-43 (2006) · Zbl 1094.81058 · doi:10.1007/s00023-005-0240-x |
| [26] | Glauber, R. J., Coherent and incoherent states of the radiation field, Phys.Rev., 131, 6, 2766 (1963) · Zbl 1371.81166 · doi:10.1103/physrev.131.2766 |
| [27] | Glimm, J.; Jaffe, A., Quantum Physics: A Functional Integral Point of View (1987) |
| [28] | Gross, L., Existence and uniqueness of physical ground states, J. Funct. Anal., 10, 52-109 (1972) · Zbl 0237.47012 · doi:10.1016/0022-1236(72)90057-2 |
| [29] | Gross, L., The relativistic polaron without cutoffs, Commun. Math. Phys., 31, 25-73 (1973) · Zbl 1125.81310 · doi:10.1007/bf01645589 |
| [30] | Gross, L., Measurable functions on Hilbert space, Trans. Am. Math. Soc., 105, 372-390 (1962) · Zbl 0178.50001 · doi:10.2307/1993726 |
| [31] | Gross, L., Abstract Wiener spaces, 31-42 (1965) · Zbl 0187.40903 |
| [32] | Gross, L., Abstract Wiener measure and infinite dimensional potential theory, Lectures in Modern Analysis and Applications II, 84-116 (1970) · Zbl 0203.13002 |
| [33] | Gross, L., Potential theory on Hilbert space, J. Funct. Anal., 1, 123-181 (1967) · Zbl 0165.16403 · doi:10.1016/0022-1236(67)90030-4 |
| [34] | Hepp, K., The classical limit for quantum mechanical correlation functions, Commun. Math. Phys., 35, 265-277 (1974) · doi:10.1007/bf01646348 |
| [35] | Hörmander, L., The Analysis of Linear Partial Differential Operators (1985) · Zbl 0601.35001 |
| [36] | Hübner, M.; Spohn, H., Spectral properties of the spin-boson Hamiltonian, Ann. l’I. H. P., Sect. A, tome, 62, 3, 289-323 (1995) · Zbl 0827.47053 |
| [37] | Jager, L., Stochastic extensions of symbols in Wiener spaces and heat operator (2016) |
| [38] | Janson, S., Gaussian Hilbert Spaces (1997) · Zbl 0887.60009 |
| [39] | Jeener, J.; Henin, F., A presentation of pulsed nuclear magnetic resonance with full quantization of the radio frequency magnetic field, J. Chem. Phys., 116, 8036-8047 (2002) · doi:10.1063/1.1467332 |
| [40] | Krée, P.; Ra̧czka, R., Kernels and symbols of operators in quantum field theory, Ann. l’I. H. P. Sect., A, 28, 1, 41-73 (1978) · Zbl 0386.47015 |
| [41] | Kuo, H. H., Gaussian Measures in Banach Spaces (1975) · Zbl 0306.28010 |
| [42] | Lascar, B., Une classe d’opérateurs elliptiques du second ordre sur un espace de Hilbert, J. Funct. Anal., 35, 3, 316-343 (1980) · Zbl 0463.47029 · doi:10.1016/0022-1236(80)90086-5 |
| [43] | Lerner, N., Metrics on the phase space and non-selfadjoint pseudo-differential operators, Pseudo-Differential Operators: Theory and Applications (2010) · Zbl 1186.47001 |
| [44] | Leopold, N.; Pickl, P., Derivation of the Maxwell-Schrödinger equations from the Pauli-Fierz Hamiltonian (2016) |
| [45] | Leopold, N.; Pickl, P., Mean-field limits of particles in interaction with quantized radiation fields (2018) · Zbl 1414.81287 |
| [46] | Lieb, E.; Loss, M., A note on polarization vectors in quantum electrodynamics, Commun. Math. Phys., 252, 1-3, 477-483 (2004) · Zbl 1102.81070 · doi:10.1007/s00220-004-1185-5 |
| [47] | Meyer, P. A., Quantum Probability for Probabilists (1991) |
| [48] | Mizrahi, M. M., On the semiclassical expansion in quantum mechanics for arbitrary Hamiltonians, J. Math. Phys., 18, 34, 786-790 (1977) · doi:10.1063/1.523308 |
| [49] | Reed, M.; Simon, B., Methods of Modern Mathematical Physics (1978) · Zbl 0401.47001 |
| [50] | Reuse, F. A., Electrodynamique et Optique Quantiques (2007) · Zbl 1144.78001 |
| [51] | Robert, D., Autour de l’Approximation Semi-Classique (1987) · Zbl 0621.35001 |
| [52] | Romero, R. H.; Aucar, G. A., QED approach to the nuclear spin-spin coupling tensor, Phys. Rev. A, 65, 053411 (2002) · doi:10.1103/physreva.65.053411 |
| [53] | Simon, B., The P(φ)\(_2\) Euclidean (Quantum) Field Theory (1974) · Zbl 1175.81146 |
| [54] | Spohn, H., Ground state(s) of the spin-boson Hamiltonian, Commun. Math. Phys., 123, 277-304 (1989) · Zbl 0667.60108 · doi:10.1007/bf01238859 |
| [55] | Spohn, H., Dynamics of Charged Particles and Their Radiation Field (2004) · Zbl 1078.81004 |
| [56] | Unterberger, A., Les opérateurs métadifférentiels, Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory, 205-241 (1980) · Zbl 0452.35121 |
| [57] | Zworski, M., Semiclassical Analysis (2012) · Zbl 1252.58001 |
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