Order Parameter at the Boundary of a Trapped Bose Gas

Through a suitable expansion of the Gross-Pitaevskii equation near the classical turning point, we obtain an explicit solution for the order parameter at the boundary of a trapped Bose gas interacting with repulsive forces. The kinetic energy of the system, in terms of the classical radius $R$ and of the harmonic oscillator length $a_{_{HO}}$, follows the law $E_{kin}/N \propto R^{-2} [\log (R/a_{_{HO}}) + \hbox{const.}]$, approaching, for large $R$, the results obtained by solving numerically the Gross-Pitaevskii equation. The occurrence of a Josephson-type current in the presence of a double trap potential is finally discussed.Comment: 11 pages, REVTEX, 4 figures (uuencoded-gzipped-tar file) also available at http://anubis.science.unitn.it/~dalfovo/papers/papers.htm

Stability of Attractive Bose-Einstein Condensates in a Periodic Potential

Using a standing light wave trap, a stable quasi-one-dimensional attractive dilute-gas Bose-Einstein condensate can be realized. In a mean-field approximation, this phenomenon is modeled by the cubic nonlinear Schr\"odinger equation with attractive nonlinearity and an elliptic function potential of which a standing light wave is a special case. New families of stationary solutions are presented. Some of these solutions have neither an analog in the linear Schr\"odinger equation nor in the integrable nonlinear Schr\"odinger equation. Their stability is examined using analytic and numerical methods. Trivial-phase solutions are experimentally stable provided they have nodes and their density is localized in the troughs of the potential. Stable time-periodic solutions are also examined.Comment: 12 pages, 18 figure

Bose condensates in a harmonic trap near the critical temperature

The mean-field properties of finite-temperature Bose-Einstein gases confined in spherically symmetric harmonic traps are surveyed numerically. The solutions of the Gross-Pitaevskii (GP) and Hartree-Fock-Bogoliubov (HFB) equations for the condensate and low-lying quasiparticle excitations are calculated self-consistently using the discrete variable representation, while the most high-lying states are obtained with a local density approximation. Consistency of the theory for temperatures through the Bose condensation point requires that the thermodynamic chemical potential differ from the eigenvalue of the GP equation; the appropriate modifications lead to results that are continuous as a function of the particle interactions. The HFB equations are made gapless either by invoking the Popov approximation or by renormalizing the particle interactions. The latter approach effectively reduces the strength of the effective scattering length, increases the number of condensate atoms at each temperature, and raises the value of the transition temperature relative to the Popov approximation. The renormalization effect increases approximately with the log of the atom number, and is most pronounced at temperatures near the transition. Comparisons with the results of quantum Monte Carlo calculations and various local density approximations are presented, and experimental consequences are discussed.Comment: 15 pages, 11 embedded figures, revte

Reasons for participation in a child development study: Are cases with developmental diagnoses different from controls?

Background: Current knowledge about parental reasons for allowing child participation in research comes mainly from clinical trials. Fewer data exist on parents’ motivations to enrol children in observational studies. Objectives: Describe reasons parents of preschoolers gave for participating in the Study to Explore Early Development (SEED), a US multi-site study of autism spectrum disorder (ASD) and other developmental delays or disorders (DD), and explore reasons given by child diagnostic and behavioural characteristics at enrolment. Methods: We included families of children, age 2–5 years, participating in SEED (n = 5696) during 2007–2016. We assigned children to groups based on characteristics at enrolment: previously diagnosed ASD; suspected ASD; non-ASD DD; and population controls (POP). During a study interview, we asked parents their reasons for participating. Two coders independently coded responses and resolved discrepancies via consensus. We fit binary mixed-effects models to evaluate associations of each reason with group and demographics, using POP as reference. Results: Participants gave 1–5 reasons for participation (mean = 1.7, SD = 0.7). Altruism (48.3%), ASD research interest (47.4%) and perceived personal benefit (26.9%) were most common. Two novel reasons were knowing someone outside the household with the study conditions (peripheral relationship; 14.1%) and desire to contribute to a specified result (1.4%). Odds of reporting interest in ASD research were higher among diagnosed ASD participants (odds ratio [OR] 2.89, 95% confidence interval [CI] 2.49–3.35). Perceived personal benefit had higher odds among diagnosed (OR 1.92, 95% CI 1.61–2.29) or suspected ASD (OR 3.67, 95% CI 2.99–4.50) and non-ASD DD (OR 1.80, 95% CI 1.50–2.16) participants. Peripheral relationship with ASD/DD had lower odds among all case groups. Conclusions: We identified meaningful differences between groups in parent-reported reasons for participation. Differences demonstrate an opportunity for future studies to tailor recruitment materials and increase the perceived benefit for specific prospective participants

Bose-Einstein condensates in a one-dimensional double square well: Analytical solutions of the Nonlinear Schr\"odinger equation and tunneling splittings

We present a representative set of analytic stationary state solutions of the Nonlinear Schr\"odinger equation for a symmetric double square well potential for both attractive and repulsive nonlinearity. In addition to the usual symmetry preserving even and odd states, nonlinearity introduces quite exotic symmetry breaking solutions - among them are trains of solitons with different number and sizes of density lumps in the two wells. We use the symmetry breaking localized solutions to form macroscopic quantum superpositions states and explore a simple model for the exponentially small tunneling splitting.Comment: 11 pages, 11 figures, revised version, typos and references correcte

Condensate fluctuations in finite Bose-Einstein condensates at finite temperature

A Langevin equation for the complex amplitude of a single-mode Bose-Einstein condensate is derived. The equation is first formulated phenomenologically, defining three transport parameters. It is then also derived microscopically. Expressions for the transport parameters in the form of Green-Kubo formulas are thereby derived and evaluated for simple trap geometries, a cubic box with cyclic boundary conditions and an isotropic parabolic trap. The number fluctuations in the condensate, their correlation time, and the temperature-dependent collapse-time of the order parameter as well as its phase-diffusion coefficient are calculated.Comment: 29 pages, Revtex, to appear in Phys.Rev.

Correlated N-boson systems for arbitrary scattering length

We investigate systems of identical bosons with the focus on two-body correlations and attractive finite-range potentials. We use a hyperspherical adiabatic method and apply a Faddeev type of decomposition of the wave function. We discuss the structure of a condensate as function of particle number and scattering length. We establish universal scaling relations for the critical effective radial potentials for distances where the average distance between particle pairs is larger than the interaction range. The correlations in the wave function restore the large distance mean-field behaviour with the correct two-body interaction. We discuss various processes limiting the stability of condensates. With correlations we confirm that macroscopic tunneling dominates when the trap length is about half of the particle number times the scattering length.Comment: 15 pages (RevTeX4), 11 figures (LaTeX), submitted to Phys. Rev. A. Second version includes an explicit comparison to N=3, a restructured manuscript, and updated figure

Stability and collapse of localized solutions of the controlled three-dimensional Gross-Pitaevskii equation

On the basis of recent investigations, a newly developed analytical procedure is used for constructing a wide class of localized solutions of the controlled three-dimensional (3D) Gross-Pitaevskii equation (GPE) that governs the dynamics of Bose-Einstein condensates (BECs). The controlled 3D GPE is decomposed into a two-dimensional (2D) linear Schr\"{o}dinger equation and a one-dimensional (1D) nonlinear Schr\"{o}dinger equation, constrained by a variational condition for the controlling potential. Then, the above class of localized solutions are constructed as the product of the solutions of the transverse and longitudinal equations. On the basis of these exact 3D analytical solutions, a stability analysis is carried out, focusing our attention on the physical conditions for having collapsing or non-collapsing solutions.Comment: 21 pages, 14 figure

Solutions of Gross-Pitaevskii equations beyond the hydrodynamic approximation: Application to the vortex problem

We develop the multiscale technique to describe excitations of a Bose-Einstein condensate (BEC) whose characteristic scales are comparable with the healing length, thus going beyond the conventional hydrodynamical approximation. As an application of the theory we derive approximate explicit vortex and other solutions. The dynamical stability of the vortex is discussed on the basis of the mathematical framework developed here, the result being that its stability is granted at least up to times of the order of seconds, which is the condensate lifetime. Our analytical results are confirmed by the numerical simulations.Comment: To appear in Phys. Rev.

Nonadiabatic Dynamics of Atoms in Nonuniform Magnetic Fields

Dynamics of neutral atoms in nonuniform magnetic fields, typical of quadrupole magnetic traps, is considered by applying an accurate method for solving nonlinear systems of differential equations. This method is more general than the adiabatic approximation and, thus, permits to check the limits of the latter and also to analyze nonadiabatic regimes of motion. An unusual nonadiabatic regime is found when atoms are confined from one side of the z-axis but are not confined from another side. The lifetime of atoms in a trap in this semi-confining regime can be sufficiently long for accomplishing experiments with a cloud of such atoms. At low temperature, the cloud is ellipsoidal being stretched in the axial direction and moving along the z-axis. The possibility of employing the semi-confining regime for studying the relative motion of one component through another, in a binary mixture of gases is discussed.Comment: 1 file, 17 pages, RevTex, 2 table