Narrow Results

In this paper, we construct metastable states of atoms interacting with the quantized radiation field. These states emerge from the excited bound states of the non-interacting system. We prove that these states obey an exponential time-decay law. In detail, we show that their decay is given by an exponential function in time, predicted by Fermi's Golden Rule, plus a small remainder term. The latter is proportional to the (4+β)th power of the coupling constant and decays algebraically in time. As a result, though it is small, it dominates the decay for large times. A central point of the paper is that our remainder term is significantly smaller than the one previously obtained in [1] and as a result we are able to show that the time interval during which the Fermi's Golden Rule can be observed is significantly longer that the time interval obtained in [1]. This improvement is achieved by incorporating a part of the complex dilatation resonance states into our construction of the metastable states rather than using the unperturbed eigenstates (the excited bound states of the non-interacting system). Thus, the connection to resonance states allows us to introduce metastable states which qualify better in the description of unstable excited states of the interacting system.

An accelerated version of the lattice Boltzmann equation aimed at fast attainment of steady-state flow conditions is presented. Significant computational savings as compared to the standard explicit LBE scheme are obtained for cavity flows in two dimensions.

A method is presented for transparent, energy-dependent boundary conditions for open, non-Hermitian systems, and is illustrated on an example of Stark resonances in a single-particle quantum system. The approach provides an alternative to external complex scaling, and is applicable when asymptotic solutions can be characterized at large distances from the origin. Its main benefit consists in a drastic reduction of the dimesnionality of the underlying eigenvalue problem. Besides application to quantum mechanics, the method can be used in other contexts such as in systems involving unstable optical cavities and lossy waveguides.

Mechanism's effect of transversal magnetic field on the populations of excited states of He atoms has been investigated. Furthermore, the dependencies of complete absorption of He(I) spectral line transition 3³P_0,1,2–2³S₁ on optical plasma density are calculated by numerical methods taking into consideration the fine-structure and Zeeman splitting schemes of the spectral lines. Using the calculated values and experimental results, the dynamics of the populations of He(I) metastable states 2³S₁ of the He nanosecond discharge are discussed.

Consider a piecewise smooth expanding map of the interval possessing several invariant subintervals and the same number of ergodic absolutely continuous invariant probability measures (ACIMs). After this system is perturbed to make the subintervals lose their invariance in such a way that there is a unique ACIM, we show how to approximate the diffusion coefficient for an observable of bounded variation by the diffusion coefficient of a related continuous time Markov chain.

The coarsening of an array of vortex ripples prepared in an unstable state is discussed within the framework of a simple mass transfer model first introduced by K. H. Andersen et al. (see Ref. 1). Two scenarios for the selection of the final pattern are identified. When the initial state is homogeneous with uniform random perturbations, a unique final state is reached which depends only on the shape of the interaction function f(λ). A potential formulation of the dynamics suggests that the final wavelength is determined by a Maxwell construction applied to f(λ), but comparison with numerical simulations shows that this yields only an upper bound. In contrast, the evolution from a perfectly homogeneous state with a localized perturbation proceeds through the propagation of wavelength doubling fronts. The front speed can be predicted by standard marginal stability theory. In this case the final wavelength depends on the initial wavelength in a complicated manner which involves multiplication by powers of two and rational ratios such as 4/3.

We present a concise review of the physics of ultra-cold dipolar gases, based mainly on the theoretical developments in our own group. First, we discuss shortly weakly interacting ultra-cold trapped dipolar gases. Dipolar Bose–Einstein condensates exhibit non-standard instabilities and the physics of both Bose and Fermi dipolar gases depends on the trap geometry. We focus then the second part of the paper on strongly correlated dipolar gases and discuss ultra-cold dipolar gases in optical lattices. Such gases exhibit a spectacular richness of quantum phases and metastable states, which may perhaps be used as quantum memories. We comment shortly on the possibility of superchemistry aiming at the creation of dipolar heteronuclear molecules in lattices. Finally, we turn to ultra-cold dipolar gases in artificial magnetic fields, and consider rotating dipolar gases, that provide in our opinion the best option towards the realization of the fractional quantum Hall effect and quantum Wigner crystals.

The coarsening of an array of vortex ripples prepared in an unstable state is discussed within the framework of a simple mass transfer model first introduced by K. H. Andersen et al. (see Ref. 1). Two scenarios for the selection of the final pattern are identified. When the initial state is homogeneous with uniform random perturbations, a unique final state is reached which depends only on the shape of the interaction function f(λ). A potential formulation of the dynamics suggests that the final wavelength is determined by a Maxwell construction applied to f(λ), but comparison with numerical simulations shows that this yields only an upper bound. In contrast, the evolution from a perfectly homogeneous state with a localized perturbation proceeds through the propagation of wavelength doubling fronts. The front speed can be predicted by standard marginal stability theory. In this case the final wavelength depends on the initial wavelength in a complicated manner which involves multiplication by powers of two and rational ratios such as 4/3.