Random conformal snowflakes

In many problems of classical analysis extremal configurations appear to exhibit complicated fractal structure. This makes it much harder to describe extremals and to attack such problems. Many of these problems are related to the multifractal analysis of harmonic measure. We argue that, searching for extremals in such problems, one should work with random fractals rather than deterministic ones. We introduce a new class of fractals random conformal snowflakes and investigate its properties developing tools to estimate spectra and showing that extremals can be found in this class. As an application we significantly improve known estimates from below on the extremal behaviour of harmonic measure, showing how to constuct a rather simple snowflake, which has a spectrum quite close to the conjectured extremal value

Field-induced decay dynamics in square-lattice antiferromagnet

Dynamical properties of the square-lattice Heisenberg antiferromagnet in applied magnetic field are studied for arbitrary value S of the spin. Above the threshold field for two-particle decays, the standard spin-wave theory yields singular corrections to the excitation spectrum with logarithmic divergences for certain momenta. We develop a self-consistent approximation applicable for S >= 1, which avoids such singularities and provides regularized magnon decay rates. Results for the dynamical structure factor obtained in this approach are presented for S = 1 and S = 5/2.Comment: 12 pages, 11 figures, final versio

Collapse and revival of excitations in Bose-Einstein condensates

We study the energies and decay of elementary excitations in weakly interacting Bose-Einstein condensates within a finite-temperature gapless second-order theory. The energy shifts for the high-lying collective modes turn out to be systematically negative compared with the Hartree-Fock-Bogoliubov-Popov approximation and the decay of the low-lying modes is found to exhibit collapse and revival effects. In addition, perturbation theory is used to qualitatively explain the experimentally observed Beliaev decay process of the scissors mode.Comment: 9 pages, 5 figure

Controlling quasiparticle excitations in a trapped Bose-Einstein condensate

We describe an approach to quantum control of the quasiparticle excitations in a trapped Bose-Einstein condensate based on adiabatic and diabatic changes in the trap anisotropy. We describe our approach in the context of Landau-Zener transition at the avoided crossings in the quasiparticle excitation spectrum. We show that there can be population oscillation between different modes at the specific aspect ratios of the trapping potential at which the mode energies are almost degenerate. These effects may have implications in the expansion of an excited condensate as well as the dynamics of a moving condensate in an atomic wave guide with a varying width

Functional renormalization for Bose-Einstein Condensation

We investigate Bose-Einstein condensation for interacting bosons at zero and nonzero temperature. Functional renormalization provides us with a consistent method to compute the effect of fluctuations beyond the Bogoliubov approximation. For three dimensional dilute gases, we find an upper bound on the scattering length a which is of the order of the microphysical scale - typically the range of the Van der Waals interaction. In contrast to fermions near the unitary bound, no strong interactions occur for bosons with approximately pointlike interactions, thus explaining the high quantitative reliability of perturbation theory for most quantities. For zero temperature we compute the quantum phase diagram for bosonic quasiparticles with a general dispersion relation, corresponding to an inverse microphysical propagator with terms linear and quadratic in the frequency. We compute the temperature dependence of the condensate and particle density n, and find for the critical temperature T_c a deviation from the free theory, Delta T_c/T_c = 2.1 a n^{1/3}. For the sound velocity at zero temperature we find very good agreement with the Bogoliubov result, such that it may be used to determine the particle density accurately.Comment: 21 pages, 16 figures. Reference adde

A Multiscale Guide to Brownian Motion

We revise the Levy's construction of Brownian motion as a simple though still rigorous approach to operate with various Gaussian processes. A Brownian path is explicitly constructed as a linear combination of wavelet-based "geometrical features" at multiple length scales with random weights. Such a wavelet representation gives a closed formula mapping of the unit interval onto the functional space of Brownian paths. This formula elucidates many classical results about Brownian motion (e.g., non-differentiability of its path), providing intuitive feeling for non-mathematicians. The illustrative character of the wavelet representation, along with the simple structure of the underlying probability space, is different from the usual presentation of most classical textbooks. Similar concepts are discussed for fractional Brownian motion, Ornstein-Uhlenbeck process, Gaussian free field, and fractional Gaussian fields. Wavelet representations and dyadic decompositions form the basis of many highly efficient numerical methods to simulate Gaussian processes and fields, including Brownian motion and other diffusive processes in confining domains

Coherence time of a Bose-Einstein condensate

Temporal coherence is a fundamental property of macroscopic quantum systems, such as lasers in optics and Bose-Einstein condensates in atomic gases and it is a crucial issue for interferometry applications with light or matter waves. Whereas the laser is an "open" quantum system, ultracold atomic gases are weakly coupled to the environment and may be considered as isolated. The coherence time of a condensate is then intrinsic to the system and its derivation is out of the frame of laser theory. Using quantum kinetic theory, we predict that the interaction with non-condensed modes gradually smears out the condensate phase, with a variance growing as A t^2+B t+C at long times t, and we give a quantitative prediction for A, B and C. Whereas the coefficient A vanishes for vanishing energy fluctuations in the initial state, the coefficients B and C are remarkably insensitive to these fluctuations. The coefficient B describes a diffusive motion of the condensate phase that sets the ultimate limit to the condensate coherence time. We briefly discuss the possibility to observe the predicted phase spreading, also including the effect of particle losses.Comment: 17 pages, 8 figures; typos correcte

Optically-Induced Polarons in Bose-Einstein Condensates: Monitoring Composite Quasiparticle Decay

Nonresonant light-scattering off atomic Bose-Einstein condensates (BECs) is predicted to give rise to hitherto unexplored composite quasiparticles: unstable polarons, i.e., local ``impurities'' dressed by virtual phonons. Optical monitoring of their spontaneous decay can display either Zeno or anti-Zeno deviations from the Golden Rule, and thereby probe the temporal correlations of elementary excitations in BECs.Comment: 4 pages, 3 figure

Spectral function and quasi-particle damping of interacting bosons in two dimensions

We employ the functional renormalization group to study dynamical properties of the two-dimensional Bose gas. Our approach is free of infrared divergences, which plague the usual diagrammatic approaches, and is consistent with the exact Nepomnyashchy identity, which states that the anomalous self-energy vanishes at zero frequency and momentum. We recover the correct infrared behavior of the propagators and present explicit results for the spectral line-shape, from which we extract the quasi-particle dispersion and damping.Comment: 4 pages, 3 figures, revisited version, to appear as Phys. Rev. Lette

On Littlewood's Constants

In two papers, Littlewood studied seemingly unrelated constants: (i) the best α such that for any polynomial f, of degree n, the areal integral of its spherical derivative is at most ·nα, and (ii) the extremal growth rate rβ of the length of Green's equipotentials for simply connected domains. These two constants are shown to coincide, thus greatly improving known estimates on α. 2000 Mathematics Subject Classification 30C50 (primary), 30C85, 30D35 (secondary