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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).

11B85 Automata sequences
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
Full Text: DOI
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