Dimensional reduction and localization of a Bose-Einstein condensate in a quasi-1D bichromatic optical lattice

We analyze the localization of a Bose-Einstein condensate (BEC) in a one-dimensional bichromatic quasi-periodic optical-lattice potential by numerically solving the 1D Gross-Pitaevskii equation (1D GPE). We first derive the 1D GPE from the dimensional reduction of the 3D quantum field theory of interacting bosons obtaining two coupled differential equations (for axial wavefuction and space-time dependent transverse width) which reduce to the 1D GPE under strict conditions. Then, by using the 1D GPE we report the suppression of localization in the interacting BEC when the repulsive scattering length between bosonic atoms is sufficiently large.Comment: 10 pages, 2 figures, presented at the 7th Workshop on Quantum Chaos and Localisation Phenomena, May 29-31, 2015 - Warsaw, Poland; to be published in a special issue of Acta Physica Polonica

Quantum scattering in one dimension

A self-contained discussion of nonrelativistic quantum scattering is presented in the case of central potentials in one space dimension, which will facilitate the understanding of the more complex scattering theory in two and three dimensions. The present discussion illustrates in a simple way the concept of partial-wave decomposition, phase shift, optical theorem and effective-range expansion.Comment: 8 page

Dynamics of gap solitons in a dipolar Bose-Einstein condensate on a three-dimensional optical lattice

We suggest and study the stable disk- and cigar-shaped gap solitons of a dipolar Bose-Einstein condensate of $^{52}$Cr atoms localized in the lowest band gap by three optical-lattice (OL) potentials along orthogonal directions. The one-dimensional version of these solitons of experimental interest confined by an OL along the dipole moment direction and harmonic traps in transverse directions is also considered. Important dynamics of (i) breathing oscillation of a gap soliton upon perturbation and (ii) dragging of a gap soliton by a moving lattice along axial $z$ direction demonstrates the stability of gap solitons. A movie clip of dragging of three-dimensional gap soliton is included.Comment: To see the dragging movie clip please download sourc

Large-scale anomalies in the cosmic microwave background as signatures of non-Gaussianity

We derive a general expression for the probability of observing deviations from statistical isotropy in the cosmic microwave background (CMB) if the primordial fluctuations are non-Gaussian and extend to superhorizon scales. The primary motivation is to properly characterize the monopole and dipole modulations of the primordial power spectrum that are generated by the coupling between superhorizon and subhorizon perturbations. Unlike previous proposals for generating the hemispherical power asymmetry, we do not assume that the power asymmetry results from a single large superhorizon mode. Instead, we extrapolate the observed power spectrum to superhorizon scales and compute the power asymmetry that would result from a specific realization of non-Gaussian perturbations on scales larger than the observable universe. Our study encompasses many of the scenarios that have been put forward as possible explanations for the CMB hemispherical power asymmetry. We confirm our analytic predictions for the probability of a given power asymmetry by comparing them to numerical realizations of CMB maps. We find that non-local models of non-Gaussianity and scale-dependent local non-Gaussianity produce scale-dependent modulations of the power spectrum, thereby potentially producing both a monopolar and a dipolar power modulation on large scales. We then provide simple examples of finding the posterior distributions for the parameters of the bispectrum from the observed monopole and dipole modulations.Comment: 21 pages, 11 figures; v2: minor changes to match the PRD accepted versio

Using data network metrics, graphics, and topology to explore network characteristics

Yehuda Vardi introduced the term network tomography and was the first to propose and study how statistical inverse methods could be adapted to attack important network problems (Vardi, 1996). More recently, in one of his final papers, Vardi proposed notions of metrics on networks to define and measure distances between a network's links, its paths, and also between different networks (Vardi, 2004). In this paper, we apply Vardi's general approach for network metrics to a real data network by using data obtained from special data network tools and testing procedures presented here. We illustrate how the metrics help explicate interesting features of the traffic characteristics on the network. We also adapt the metrics in order to condition on traffic passing through a portion of the network, such as a router or pair of routers, and show further how this approach helps to discover and explain interesting network characteristics.Comment: Published at http://dx.doi.org/10.1214/074921707000000058 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Effective Nonlinear Schr\"odinger Equations for Cigar-Shaped and Disk-Shaped Fermi Superfluids at Unitarity

In the case of tight transverse confinement (cigar-shaped trap) the three-dimensional (3D) nonlinear Schr\"odinger equation, describing superfluid Fermi atoms at unitarity (infinite scattering length $|a|\to \infty$), is reduced to an effective one-dimensional form by averaging over the transverse coordinates. The resultant effective equation is a 1D nonpolynomial Schrodinger equation, which produces results in good agreement with the original 3D one. In the limit of small and large fermion number $N$ the nonlinearity is of simple power-law type. A similar reduction of the 3D theory to a two-dimensional form is also performed for a tight axial confinement (disk-shaped trap). The resultant effective 2D nonpolynomial equation also produces results in agreement with the original 3D equation and has simple power-law nonlinearity for small and large $N$. For both cigar- and disk-shaped superfluids our nonpolynomial Schr\"odinger equations are quite attractive for phenomenological application.Comment: 22 pages, 5 figure

Simulation of a stationary dark soliton in a trapped zero-temperature Bose-Einstein condensate

We discuss a computational mechanism for the generation of a stationary dark soliton, or black soliton, in a trapped Bose-Einstein condensate using the Gross-Pitaevskii (GP) equation for both attractive and repulsive interaction. It is demonstrated that the black soliton with a "notch" in the probability density with a zero at the minimum is a stationary eigenstate of the GP equation and can be efficiently generated numerically as a nonlinear continuation of the first vibrational excitation of the GP equation in both attractive and repulsive cases in one and three dimensions for pure harmonic as well as harmonic plus optical-lattice traps. We also demonstrate the stability of this scheme under different perturbing forces.Comment: 7 pages, 15 ps figures, Final version accepted in J Low Temp Phy

Two phase transitions in (s+id)-wave Bardeen-Cooper-Schrieffer superconductivity

We establish universal behavior in temperature dependencies of some observables in $(s+id)$-wave BCS superconductivity in the presence of a weak $s$ wave. There also could appear a second second-order phase transition. As temperature is lowered past the usual critical temperature $T_c$, a less ordered superconducting phase is created in $d$ wave, which changes to a more ordered phase in $(s+id)$ wave at $T_{c1}$ ($< T_c$). The presence of two phase transitions manifest in two jumps in specific heat at $T_c$ and $T_{c1}$. The temperature dependencies of susceptibility, penetration depth, and thermal conductivity also confirm the new phase transition.Comment: 6 pages, 5 post-script figures

Localization of a Bose-Einstein condensate vortex in a bichromatic optical lattice

By numerical simulation of the time-dependent Gross-Pitaevskii equation we show that a weakly interacting or noninteracting Bose-Einstein condensate (BEC) vortex can be localized in a three-dimensional bichromatic quasi-periodic optical-lattice (OL) potential generated by the superposition of two standing-wave polarized laser beams with incommensurate wavelengths. This is a generalization of the localization of a BEC in a one-dimensional bichromatic OL as studied in a recent experiment [Roati et al., Nature 453, 895 (2008)]. We demonstrate the stability of the localized state by considering its time evolution in the form of a stable breathing oscillation in a slightly altered potential for a large period of time. {Finally, we consider the localization of a BEC in a random 1D potential in the form of several identical repulsive spikes arbitrarily distributed in space