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In this work, we discuss the resonance states of a quantum particle in a periodic potential plus static force. Originally this problem was formulated for a crystalline electron subject to the static electric field and is known nowadays as the Wannier-Stark problem. We describe a novel approach to the Wannier-Stark problem developed in recent years. This approach allows to compute the complex energy spectrum of a Wannier-Stark system as the poles of a rigorously constructed scattering matrix and, in this sense, solves the Wannier-Stark problem without any approximation. The suggested method is very efficient from the numerical point of view and has proven to be a powerful analytic tool for Wannier-Stark resonances appearing in different physical systems like optical or semiconductor superlattices.

The paper studies the effect of a weak periodic driving on metastable Wannier-Stark states. The decay rate of the ground Wannier-Stark states as a continuous function of the driving frequency is calculated numerically. The theoretical results are compared with experimental data of Wilkinson et at. [Phys.Rev.Lett.76, 4512 (1996)] obtained for cold sodium atoms in an accelerated optical lattice.

The statistics of the resonance widths and the behavior of the survival probability is studied in a particular model of quantum chaotic scattering (a particle in a periodic potential subject to static and time-periodic forces) introduced earlier in Ref. [5,6]. The coarse-grained distribution of the resonance widths is shown to be in good agreement with the prediction of Random Matrix Theory (RMT). The behavior of the survival probability shows, however, some deviation from RMT.

We study the transitions between the ground and excited Wannier states induced by a weak ac field. Because the upper Wannier states are several order of magnitude less stable than the ground states, these transitions decrease the global stability of the system characterized by the rate of probability leakage or decay rate. Using nonhermitian resonant perturbation theory we obtain an analytical expression for this induced decay rate. The analytical results are compared with exact numerical calculations of the system decay rate.

The quasienergy spectrum of a Bloch electron affected by dc-ac fields is known to have a fractal structure as function of the so-called electric matching ratio, which is the ratio of the ac field frequency and the Bloch frequency. This paper studies a manifestation of the fractal nature of the spectrum in the system "atom in a standing laser wave", which is a quantum optical realization of a Bloch electron. It is shown that for an appropriate choice of the system parameters the atomic survival probability (a quantity measured in laboratory experiments) also develops a fractal structure as a function of the electric matching ratio. Numerical simulations under classically chaotic scattering conditions show good agreement with theoretical predictions based on random matrix theory.

The paper studies quantum states of a Bloch particle in presence of external ac and dc fields. Provided the period of the ac field and the Bloch period are commensurate, an effective scattering matrix is introduced, the complex poles of which are the system quasienergy spectrum. The statistics of the resonance width and the Wigner delay time shows a close relation of the problem to random matrix theory of chaotic scattering.