Abstract
We consider two-level atoms in a strong external quasi-periodic field with Diophantine frequency vector. We show that if the field is an analytic function with zero average, then for a large set of values of its frequency vector, characterized by imposing infinitely many Diophantine conditions, the spectrum of the quasi-energy operator is pure point, as in the case of nonzero average which was already known in literature.
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References
D. Bambusi and S. Graffi, Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods, Comm. Math. Phys. 219:465–480 (2001).
J. C. A. Barata, On formal quasi-periodic solutions of the Schrödinger equation for a two-level system with a Hamiltonian depending quasi-periodically on time, Rev. Math. Phys. 12:25–64 (2000).
J. C. A. Barata and W. F. Wreszinski, Strong-coupling theory of two-level atoms in periodic fields, Phys. Rev. Lett. 84:2112–2115 (2000).
J. Bellissard, Stability and chaotic behavior of quantum rotators, Stochastic Processes in Classical and Quantum Systems (Ascona, 1985), S. Albeverio, G. Casati, and D. Merlini, eds., Lecture Notes in Physics, Vol. 262 (Springer, Berlin, 1986), pp. 24–38.
P. M. Blekher, H. R. Jauslin, and J. L. Lebowitz, Floquet spectrum for two-level systems in quasiperiodic time-dependent fields, J. Stat. Phys. 68:271–310 (1992).
O. Costin, R. D. Costin, J. L. Lebowitz, and A. Rokhlenko, Evolution of a model quantum system under time periodic forcing: Conditions for complete ionization, Comm. Math. Phys. 221:1–26 (2001).
P. Duclos, O. Lev, P. Š\({{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{t} }}\)ovíček, and M. Vittot, Weakly regular Floquet Hamiltonians with pure point spectrum, Rev. Math. Phys. 14:531–568 (2002).
G. Gentile, Quasi-periodic solutions for two-level systems, Comm. Math. Phys. 242: 221–250 (2003).
H. R. Jauslin and J. L. Lebowitz, Spectral and stability aspects of quantum chaos, Chaos 1:114–121 (1991).
H. M. Nussenzveig, Introduction to Quantum Optics (Gordon & Breach, New York, 1973).
W. F. Wreszinski and S. Casmeridis, Models of two-level atoms in quasiperiodic external fields, J. Stat. Phys. 90:1061–1068 (1998).
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Gentile, G. Pure Point Spectrum for Two-Level Systems in a Strong Quasi-Periodic Field. Journal of Statistical Physics 115, 1605–1620 (2004). https://doi.org/10.1023/B:JOSS.0000028070.11031.0c
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DOI: https://doi.org/10.1023/B:JOSS.0000028070.11031.0c