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Recurrent versus diffusive dynamics for a kicked quantum system. (English) Zbl 0746.11014
The author studies the dynamics of a two-level quantum system subject to a time-dependent kicking perturbation modulated along the Prouhet-Thue- Morse sequence: the Hamiltonian she studies is given by: $H(t)=E\sigma_ z + \lambda\sigma_ x \sum^ \infty_{- \infty}\gamma_ n\delta(t-n),$ where $$E$$ and $$\lambda$$ are real parameters, $$\sigma_ x$$ and $$\sigma_ z$$ are the Pauli matrices $$\sigma_ x = \left({0\atop 1}{1\atop 0}\right)$$, $$\sigma_ z = \left({1\atop 0}{0\atop -1}\right)$$, and $$(\gamma_ n)$$ is the two-sided Thue-Morse sequence. The author proves that, for a nontrivial set of parameters $$E$$, $$\lambda$$ (that she conjectures to be a dense set), the quantum dynamics is both recurrent and diffusive. The quantum autocorrelation function (for any initial state) is the Fourier transform of a measure with a pure point part and a singular continuous part (the latter being closely related to the Fourier transform of the Thue-Morse sequence).

MSC:
 11B85 Automata sequences 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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References:
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