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A novel method is presented which allows a fast computation of complex energy resonance states in Stark systems, i.e. systems in a homogeneous field. The technique is based on the truncation of a shift-operator in momentum space. Numerical results for space periodic and non-periodic systems illustrate the extreme simplicity of the method.

The paper studies metastable states of a Bloch electron in the presence of external ac and dc fields. Provided resonance condition between period of the driving frequency and the Bloch period, the complex quasienergies are numerically calculated for two qualitatively different regimes (quasiregular and chaotic) of the system dynamics. For the chaotic regime an effect of quantum stabilization, which suppresses the classical decay mechanism, is found. This effect is demonstrated to be a kind of quantum interference phenomenon sensitive to the resonance condition.

The Wannier-Bloch resonance states are metastable states of a quantum particle in a space-periodic potential plus a homogeneous field. Here we analyze the states of quantum particle in space- and time-periodic potential. In this case the dynamics of the classical counterpart of the quantum system is either quasiregular or chaotic depending on the driving frequency. It is shown that both the quasiregular and the chaotic motion can also support quantum resonances. The relevance of the obtained result to the problem a of crystal electron under simultaneous influence of d.c. and a.c. electric fields is briefly discussed. PACS: 73.20Dx, 73.40Gk, 05.45.+b

We study the statistics of the Wigner delay time and resonance width for a Bloch particle in ac and dc fields in the regime of quantum chaos. It is shown that after appropriate rescaling the distributions of these quantities have universal character predicted by the random matrix theory of chaotic scattering.

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.

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.

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 Filter-Diagonalization Method is applied to time periodic Hamiltonians and used to find selectively the regular and chaotic quasienergies of a driven 2D rotor. The use of N cross-correlation probability amplitudes enables a selective calculation of the quasienergies from short time propagation to the time T (N). Compared to the propagation time T (1) which is required for resolving the quasienergy spectrum with the same accuracy from auto-correlation calculations, the cross-correlation time T (N) is shorter by the factor N , that is T (1) = N T (N).

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.

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 global dynamical properties of a quantum system can be conveniently visualized in phase space by means of a quantum phase space entropy in analogy to a Poincare section in classical dynamics for two-dimensional time independent systems. Numerical results for the Pullen Edmonds systems demonstrate the properties of the method for systems with mixed chaotic and regular dynamics.

For periodically driven systems, quantum tunneling between classical resonant stability islands in phase space separated by invariant KAM curves or chaotic regions manifests itself by oscillatory motion of wave packets centered on such an island, by multiplet splittings of the quasienergy spectrum, and by phase space localisation of the quasienergy states on symmetry related ,ux tubes. Qualitatively di,erent types of classical resonant island formation | due to discrete symmetries of the system | and their quantum implications are analysed by a (uniform) semiclassical theory. The results are illustrated by a numerical study of a driven non-harmonic oscillator.

The Filter-Diagonalization Method is used to ,nd the broad and even overlapping resonances of a 1D Hamiltonian used before as a test model for new resonance theories and computational methods. It is found that the use of several complex-scaled cross-correlation probability amplitudes from short time propagation enables the calculation of broad overlapping resonances, which can not be resolved from the amplitude of a single complex-scaled autocorrelation calculation.

The quasienergy spectrum of a periodically driven quantum system is constructed from classical dynamics by means of the semiclassical initial value representation using coherent states. For the first time, this method is applied to explicitly time dependent systems. For an anharmonic oscillator system with mixed chaotic and regular classical dynamics, the entire quantum spectrum (both regular and chaotic states) is reproduced semiclassically with surprising accuracy. In particular, the method is capable to account for the very small tunneling splittings.

Quantum Chaos
(1999)

The study of dynamical quantum systems, which are classically chaotic, and the search for quantum manifestations of classical chaos, require large scale numerical computations. Special numerical techniques developed and applied in such studies are discussed: The numerical solution of the time-dependent Schr-odinger equation, the construction of quantum phase space densities, quantum dynamics in phase space, the use of phase space entropies for characterizing localization phenomena, etc. As an illustration, the dynamics of a driven one-dimensional anharmonic oscillator is studied, both classically and quantum mechanically. In addition, spectral properties and chaotic tunneling are addressed.

We present an entropy concept measuring quantum localization in dynamical systems based on time averaged probability densities. The suggested entropy concept is a generalization of a recently introduced [PRL 75, 326 (1995)] phase-space entropy to any representation chosen according to the system and the physical question under consideration. In this paper we inspect the main characteristics of the entropy and the relation to other measures of localization. In particular the classical correspondence is discussed and the statistical properties are evaluated within the framework of random vector theory. In this way we show that the suggested entropy is a suitable method to detect quantum localization phenomena in dynamical systems.

A formalism is developed for calculating the quasienergy states and spectrum for time-periodic quantum systems when a time-periodic dynamical invariant operator with a nondegenerate spectrum is known. The method, which circumvents the integration of the Schr-odinger equation, is applied to an integrable class of systems, where the global invariant operator is constructed. Furthermore, a local integrable approximation for more general non-integrable systems is developed. Numerical results are presented for the doubleresonance model.