Abstract
For multi-level time-dependent quantum systems one can construct superadiabatic representations in which the coupling between separated levels is exponentially small in the adiabatic limit. Based on results from [BeTe1] for special Hamiltonians we explicitly determine the asymptotic behavior of the exponentially small coupling term for generic two-state systems with real-symmetric Hamiltonian. The superadiabatic coupling term takes a universal form and depends only on the location and the strength of the complex singularities of the adiabatic coupling function. Our proof is based on a new norm which allows to rigorously implement Darboux' principle, a heuristic guideline widely used in asymptotic analysis.
As shown in [BeTe1], first order perturbation theory in the superadiabatic representation then allows to describe the time-development of exponentially small adiabatic transitions and thus to rigorously confirm Michael Berry's [Be] predictions on the universal form of adiabatic transition histories.
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References
Abramowitz, M., Stegun, I.A.(Eds.): Handbook of Mathematical Functions. 9th printing, New York: Dover, 1972
Berry, M.V.: Histories of adiabatic quantum transitions. Proc. R. Soc. Lond. A 429, 61–72 (1990)
Berry, M.V., Lim, R.: Universal transition prefactors derived by superadiabatic renormalization. J. Phys. A 26, 4737–4747 (1993)
Betz, V., Teufel, S.: Precise coupling terms in adiabatic quantum evolution. Annales Henri Poincaré 16, 217–246 (2005)
Betz, V., Teufel, S.: Precise coupling terms in adiabatic quantum evolution: extensions and examples. In preparation
Betz, V., Teufel, S.: Adiabatic transition histories for Born-Oppenheimer type models. In preparation
Born, M., Fock, V.: Beweis des Adiabatensatzes. Zeits. für Phys. 51, 165–169 (1928)
Boyd, J.P.: The Devil's Invention: Asymptotics, Superasymptotic and Hyperasymptotic Series. Acta Appl. 56, 1–98 (1999)
Dingle, R.B.: Asymptotic Expansions: Their Derivation and Interpretation. New York-London: Academic Press, 1973
Hagedorn, G., Joye, A.: Time development of exponentially small non-adiabatic transitions. Commun. Math. Phys. 250, 393–413 (2004)
Henrici, P.: Applied and computational analysis, Vol. 2. New York: Wiley, 1977
Joye, A.: Non-trivial prefactors in adiabatic transition probabilities induced by high order complex degeneracies. J. Phys. A 26, 6517–6540 (1993)
Joye, A., Kunz, H., Pfister, C.-E.: Exponential decay and geometric aspect of transition probabilities in the adiabatic limit. Ann. Phys. 208, 299 (1991)
Joye, A., Pfister, C.-E.: Superadiabatic evolution and adiabatic transition probability between two nondegenerate levels isolated in the spectrum. J. Math. Phys. 34, 454–479 (1993)
Martinez, A.: Precise exponential estimates in adiabatic theory. J. Math. Phys. 35, 3889–3915 (1994)
Lim, R., Berry, M.V.: Superadiabatic tracking of quantum evolution. J. Phys. A 24, 3255–3264 (1991)
Nenciu, G.: Linear adiabatic theory. Exponential estimates. Commun. Math. Phys. 152, 479–496 (1993)
Panati, G., Spohn, H., Teufel, S.: Space-adiabatic perturbation theory. Adv. Theor. Math. Phys. 7, 145–204 (2003)
Sjöstrand, J.: Projecteurs adiabatiques du point de vue pseudodifférentiel. C. R. Acad. Sci. Paris Sér. I Math. 317, 217–220 (1993)
Teufel, S.: Adiabatic perturbation theory in quantum dynamics. Springer Lecture Notes in Mathematics 1821, Berlin-Heidelberg-New York: Springer, 2003
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Communicated by B. Simon
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Betz, V., Teufel, S. Precise Coupling Terms in Adiabatic Quantum Evolution: The Generic Case. Commun. Math. Phys. 260, 481–509 (2005). https://doi.org/10.1007/s00220-005-1419-1
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DOI: https://doi.org/10.1007/s00220-005-1419-1