Balanced Allocation on Graphs: A Random Walk Approach

In this paper we propose algorithms for allocating $n$ sequential balls into $n$ bins that are interconnected as a $d$-regular $n$-vertex graph $G$, where $d\ge3$ can be any integer.Let $l$ be a given positive integer. In each round $t$, $1\le t\le n$, ball $t$ picks a node of $G$ uniformly at random and performs a non-backtracking random walk of length $l$ from the chosen node.Then it allocates itself on one of the visited nodes with minimum load (ties are broken uniformly at random). Suppose that $G$ has a sufficiently large girth and $d=\omega(\log n)$. Then we establish an upper bound for the maximum number of balls at any bin after allocating $n$ balls by the algorithm, called {\it maximum load}, in terms of $l$ with high probability. We also show that the upper bound is at most an $O(\log\log n)$ factor above the lower bound that is proved for the algorithm. In particular, we show that if we set $l=\lfloor(\log n)^{\frac{1+\epsilon}{2}}\rfloor$, for every constant $\epsilon\in (0, 1)$, and $G$ has girth at least $\omega(l)$, then the maximum load attained by the algorithm is bounded by $O(1/\epsilon)$ with high probability.Finally, we slightly modify the algorithm to have similar results for balanced allocation on $d$-regular graph with $d\in[3, O(\log n)]$ and sufficiently large girth

Loop structure of the lowest Bloch band for a Bose-Einstein condensate

We investigate analytically and numerically Bloch waves for a Bose--Einstein condensate in a sinusoidal external potential. At low densities the dependence of the energy on the quasimomentum is similar to that for a single particle, but at densities greater than a critical one the lowest band becomes triple-valued near the boundary of the first Brillouin zone and develops the structure characteristic of the swallow-tail catastrophe. We comment on the experimental consequences of this behavior.Comment: 4 pages, 7 figure

Gravity-induced Wannier-Stark ladder in an optical lattice

We discuss the dynamics of ultracold atoms in an optical potential accelerated by gravity. The positions and widths of the Wannier-Stark ladder of resonances are obtained as metastable states. The metastable Wannier-Bloch states oscillate in a single band with the Bloch period. The width of the resonance gives the rate transition to the continuum.Comment: 5 pages + 8 eps figures, submitted to Phys. Rev.

A mapping approach to synchronization in the "Zajfman trap": stability conditions and the synchronization mechanism

We present a two particle model to explain the mechanism that stabilizes a bunch of positively charged ions in an "ion trap resonator" [Pedersen etal, Phys. Rev. Lett. 87 (2001) 055001]. The model decomposes the motion of the two ions into two mappings for the free motion in different parts of the trap and one for a compressing momentum kick. The ions' interaction is modelled by a time delay, which then changes the balance between adjacent momentum kicks. Through these mappings we identify the microscopic process that is responsible for synchronization and give the conditions for that regime.Comment: 12 pages, 9 figures; submitted to Phys Rev

Scaling property of the critical hopping parameters for the Bose-Hubbard model

Recently precise results for the boundary between the Mott insulator phase and the superfluid phase of the homogeneous Bose-Hubbard model have become available for arbitrary integer filling factor g and any lattice dimension d > 1. We use these data for demonstrating that the critical hopping parameters obey a scaling relationship which allows one to map results for different g onto each other. Unexpectedly, the mean-field result captures the dependence of the exact critical parameters on the filling factor almost fully. We also present an approximation formula which describes the critical parameters for d > 1 and any g with high accuracy.Comment: 5 pages, 5 figures. to appear in EPJ

Random Scattering by Atomic Density Fluctuations in Optical Lattices

We investigate hitherto unexplored regimes of probe scattering by atoms trapped in optical lattices: weak scattering by effectively random atomic density distributions and multiple scattering by arbitrary atomic distributions. Both regimes are predicted to exhibit a universal semicircular scattering lineshape for large density fluctuations, which depend on temperature and quantum statistics.Comment: 4 pages, 2 figure

GP attitudes and self-reported behaviour in primary care consultations for low back pain

Background. The implementation of guideline recommendations in primary care has become widespread. The treatment of low back pain (LBP) has followed suite. Research shows that the use of LBP guidelines is influenced by the believability of the underlying evidence, the GPs consultation style and uncertainties surrounding diagnosis and treatment

Superfluid Dynamics of a Bose-Einstein Condensate in a Periodic Potential

We investigate the superfluid properties of a Bose-Einstein condensate (BEC) trapped in a one dimensional periodic potential. We study, both analytically (in the tight binding limit) and numerically, the Bloch chemical potential, the Bloch energy and the Bogoliubov dispersion relation, and we introduce {\it two} different, density dependent, effective masses and group velocities. The Bogoliubov spectrum predicts the existence of sound waves, and the arising of energetic and dynamical instabilities at critical values of the BEC quasi-momentum which dramatically affect its coherence properties. We investigate the dependence of the dipole and Bloch oscillation frequencies in terms of an effective mass averaged over the density of the condensate. We illustrate our results with several animations obtained solving numerically the time-dependent Gross-Pitaevskii equation.Comment: 13 pages, 7 figures, movies and published paper available at http://www.iop.org/EJ/abstract/1367-2630/5/1/11

Periodically-dressed Bose-Einstein condensates: a superfluid with an anisotropic and variable critical velocity

Two intersecting laser beams can produce a spatially-periodic coupling between two components of an atomic gas and thereby modify the dispersion relation of the gas according to a dressed-state formalism. Properties of a Bose-Einstein condensate of such a gas are strongly affected by this modification. A Bogoliubov transformation is presented which accounts for interparticle interactions to obtain the quasiparticle excitation spectrum in such a condensate. The Landau critical velocity is found to be anisotropic and can be widely tuned by varying properties of the dressing laser beams.Comment: 5 pages, 4 figure

Bloch oscillations and mean-field effects of Bose-Einstein condensates in 1-D optical lattices

We have loaded Bose-Einstein condensates into one-dimensional, off-resonant optical lattices and accelerated them by chirping the frequency difference between the two lattice beams. For small values of the lattice well-depth, Bloch oscillations were observed. Reducing the potential depth further, Landau-Zener tunneling out of the lowest lattice band, leading to a breakdown of the oscillations, was also studied and used as a probe for the effective potential resulting from mean-field interactions as predicted by Choi and Niu [Phys. Rev. Lett. {\bf 82}, 2022 (1999)]. The effective potential was measured for various condensate densities and trap geometries, yielding good qualitative agreement with theoretical calculations.Comment: 5 pages, 3 figures; accepted for publication in Physical Review Letter