Effective equation for quasi-one dimensional tube-shaped Bose–Einstein condensates

In this letter we derive an effective 1D equation that describes the axial dynamics of a tube-shaped Bose–Einstein condensate. The dimensional reduction is achieved by using a variational approach starting from the 3D Gross–Pitaevskii equation (GPE) in presence of a nonharmonic external potential in the transverse direction and generic in the axial one. The resulting equation is a time-dependent 1D nonpolynomial Schrödinger equation (NPSE). In view to check the accuracy of our 1D-NPSE, we numerically investigated the ground state properties of such a system that are in perfect agreement with the results produced by the 3D-GPE. We also compare the results with those from an 1D cubic nonlinear Schrödinger equation and the Thomas–Fermi approximation. Finally, the dynamics of ground states obtained from our 1D-NPSE is verified numerically by considering a small change in the strength of the axial confining potential.     

Mathematical and physical aspects of controlling the exact solutions of the 3D Gross-Pitaevskii equation

The possibility of the decomposition of the three-dimensional (3D) Gross-Pitaevskii equation (GPE) into a pair of coupled Schrödinger-type equations, is investigated. It is shown that, under suitable mathematical conditions, it is possible to construct the exact controlled solutions of the 3D GPE from the solutions of a linear 2D Schrödinger equation coupled with a 1D nonlinear Schrödinger equation (the transverse and longitudinal components of the GPE, respectively). The coupling between these two equations is the functional of the transverse and the longitudinal profiles. The applied method of nonlinear decomposition, called the controlling potential method (CPM), yields the full 3D solution in the form of the product of the solutions of the transverse and longitudinal components of the GPE. It is shown that the CPM constitutes a variational principle and sets up a condition on the controlling potential well. Its physical interpretation is given in terms of the minimization of the (energy) effects introduced by the control. The method is applied to the case of a parabolic external potential to construct analytically an exact BEC state in the form of a bright soliton, for which the quantitative comparison between the external and controlling potentials is presented.     

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) in the presence of a spatio-temporally varying external potential. The controlled 3D GPE is decomposed into a two-dimensional (2D) linear Schr?dinger equation (called the `transverse equation’) and a one-dimensional (1D) nonlinear Schr?dinger equation (called the `longitudinal equation’), constrained by a variational condition for the controlling potential. The latter corresponds to the requirement for the minimization of the control operation in the transverse plane. Then, the above class of localized solutions are constructed as the product of the solutions of the transverse and longitudinal equations. A consistency condition between the transverse and longitudinal solutions yields a relationship between the transverse and longitudinal restoring forces produced by the external trapping potential well through a `controlling parameter’ (i.e. the average, with respect to the transverse profile, of the nonlinear inter-atomic interaction term of the GPE). It is found that the longitudinal profile supports localized solutions in the form of bright, dark or grey solitons with time-dependent amplitudes, widths and centroids. The related longitudinal phase is varying in space and time with time-dependent curvature radius and wavenumber. In turn, all the above parameters (i.e. amplitudes, widths, centroids, curvature radius and wavenumbers) can be easily expressed in terms of the controlling parameter. It is also found that the transverse profile has the form of Hermite-Gauss functions (depending on the transverse coordinates), and the explicit spatio-temporal dependence of the controlling potential is self-consistently determined. 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.     

Effective one-dimensional dynamics of elongated Bose-Einstein condensates

By using a variational approach in combination with the adiabatic approximation we derive a new effective 1D equation of motion for the axial dynamics of elongated condensates. For condensates with vorticity ∣q∣ = 0 or 1, this equation coincides with our previous proposal [A. Muñoz Mateo, V. Delgado, Phys. Rev. A 77 (2008) 013617]. We also rederive the nonpolynomial Schrödinger equation (NPSE) in terms of the adiabatic approximation. This provides a unified treatment for obtaining the different effective equations and allows appreciating clearly the differences and similarities between the various proposals. We also obtain an expression for the axial healing length of cigar-shaped condensates and show that, in the local density approximation and in units of the axial oscillator length, it coincides with the inverse of the condensate axial half-length. From this result it immediately follows the necessary condition for the validity of the local density approximation. Finally, we obtain analytical formulas that give the frequency of the axial breathing mode with accuracy better than 1%. These formulas can be relevant from an experimental point of view since they can be expressed in terms only of the axial half-length and remain valid in the crossover between the Thomas-Fermi and the quasi-1D mean-field regimes. We have corroborated the validity of our results by numerically solving the full 3D Gross-Pitaevskii equation.     

1D stability analysis of filtering and controlling the solitons in Bose-Einstein condensates

We present one-dimensional (1D) stability analysis of a recently proposed method to filter and control localized states of the Bose–Einstein condensate (BEC), based on novel trapping techniques that allow one to conceive methods to select a particular BEC shape by controlling and manipulating the external potential well in the three-dimensional (3D) Gross–Pitaevskii equation (GPE). Within the framework of this method, under suitable conditions, the GPE can be exactly decomposed into a pair of coupled equations: a transverse two-dimensional (2D) linear Schr?dinger equation and a one-dimensional (1D) longitudinal nonlinear Schr?dinger equation (NLSE) with, in a general case, a time-dependent nonlinear coupling coefficient. We review the general idea how to filter and control localized solutions of the GPE. Then, the 1D longitudinal NLSE is numerically solved with suitable non-ideal controlling potentials that differ from the ideal one so as to introduce relatively small errors in the designed spatial profile. It is shown that a BEC with an asymmetric initial position in the confining potential exhibits breather-like oscillations in the longitudinal direction but, nevertheless, the BEC state remains confined within the potential well for a long time. In particular, while the condensate remains essentially stable, preserving its longitudinal soliton-like shape, only a small part is lost into “radiation”.     

Bright,dark optical and other solitons to the generalized higher-order NLSE in optical fibers

In this paper, we study the generalized higher-order nonlinear Schrödinger equation analytically. We use two integral schemes for conducting this study. Dark, bright, combined dark–bright optical, singular soliton, soliton-like and trigonometric function solutions are successfully constructed. We give the constraint conditions for the existence of valid solutions. The 2D, 3D and the contour graphs for the dark and bright solitons are plotted.     

Solitons in Bose–Einstein condensates

The Gross–Pitaevskii equation (GPE) describing the evolution of the Bose–Einstein condensate (BEC) order parameter for weakly interacting bosons supports dark solitons for repulsive interactions and bright solitons for attractive interactions. After a brief introduction to BEC and a general review of GPE solitons, we present our results on solitons that arise in the BEC of hard-core bosons, which is a system with strongly repulsive interactions. For a given background density, this system is found to support both a dark soliton and an antidark soliton (i.e., a bright soliton on a pedestal) for the density profile. When the background has more (less) holes than particles, the dark (antidark) soliton solution dies down as its velocity approaches the sound velocity of the system, while the antidark (dark) soliton persists all the way up to the sound velocity. This persistence is in contrast to the behaviour of the GPE dark soliton, which dies down at the Bogoliubov sound velocity. The energy–momentum dispersion relation for the solitons is shown to be similar to the exact quantum low-lying excitation spectrum found by Lieb for bosons with a delta-function interaction.     

二维线性与非线性光晶格中物质波孤立子的稳定性

利用变分法和数值计算方法研究了二维线性和非线性光晶格中二维玻色-爱因斯坦凝聚体系中物质波孤立子的存在及其稳定性. 利用定态变分原理及Vakhitov-Kolokolov判据总结了线性和非线性结合光晶格中几种参数组合下定态定域解的稳定性. 结果表明, 当存在二维非线性光晶格时, 在吸引和排斥相互作用的原子体系中均可以存在稳定的物质波孤立子. 另外, 利用含时变分法研究了线性和非线性光晶格中物质波孤立子随时间的传播特性, 使波包参数对时间的一阶导数等于零, 可以给出稳定状态对应的参数, 结论和定态变分法给出的结果一致. 最后用数值计算方法研究变分结果的正确性, 把变分结果作为初始条件代入Gross-Pitaevskii方程研究其随时间传播特征, 得到了稳定的传播过程, 所得到的结果和变分分析结果一致. 关键词： 线性非线性光晶格 玻色-爱因斯坦凝聚 孤立子 稳定性     

High-order time-splitting Hermite and Fourier spectral methods

In this paper, we are concerned with the numerical solution of the time-dependent Gross–Pitaevskii Equation (GPE) involving a quasi-harmonic potential. Primarily, we consider discretisations that are based on spectral methods in space and higher-order exponential operator splitting methods in time. The resulting methods are favourable in view of accuracy and efficiency; moreover, geometric properties of the equation such as particle number and energy conservation are well captured.Regarding the spatial discretisation of the GPE, we consider two approaches. In the unbounded domain, we employ a spectral decomposition of the solution into Hermite basis functions; on the other hand, restricting the equation to a sufficiently large bounded domain, Fourier techniques are applicable. For the time integration of the GPE, we study various exponential operator splitting methods of convergence orders two, four, and six.Our main objective is to provide accuracy and efficiency comparisons of exponential operator splitting Fourier and Hermite pseudospectral methods for the time evolution of the GPE. Furthermore, we illustrate the effectiveness of higher-order time-splitting methods compared to standard integrators in a long-term integration.     

Spectral collocation methods using sine functions for a rotating Bose–Einstein condensation in optical lattices

We study spectral-Galerkin methods (SGM) and spectral collocation methods (SCM) for parameter-dependent problems, where the Fourier sine functions are used as the basis functions. When the SGM and the SCM are incorporated in the context of a Taylor predictor–inexact Newton corrector continuation algorithm for tracing solution curves of the Gross–Pitaevskii equation (GPE), they can efficiently provide accurate numerical solutions for the GPE. We show how the inexact Newton method outperforms the classical Newton method in the continuation algorithm. In our numerical experiments, the centered difference method (CDM), the SGM and SCM are exploited to compute energy levels and wave functions of a rotating Bose–Einstein condensation (BEC) and a rotating BEC in optical lattices in 2D. Sample numerical results are reported.     

Bright and dark solitons in a quasi-1D Bose-Einstein condensates modelled by 1D Gross-Pitaevskii equation with time-dependent parameters

We investigate the exact bright and dark solitary wave solutions of an effective 1D Bose-Einstein condensate (BEC) by assuming that the interaction energy is much less than the kinetic energy in the transverse direction. In particular, following the earlier works in the literature Pérez-García et al. (2004) [50], Serkin et al. (2007) [51], Gurses (2007) [52] and Kundu (2009) [53], we point out that the effective 1D equation resulting from the Gross-Pitaevskii (GP) equation can be transformed into the standard soliton (bright/dark) possessing, completely integrable 1D nonlinear Schrödinger (NLS) equation by effecting a change of variables of the coordinates and the wave function. We consider both confining and expulsive harmonic trap potentials separately and treat the atomic scattering length, gain/loss term and trap frequency as the experimental control parameters by modulating them as a function of time. In the case when the trap frequency is kept constant, we show the existence of different kinds of soliton solutions, such as the periodic oscillating solitons, collapse and revival of condensate, snake-like solitons, stable solitons, soliton growth and decay and formation of two-soliton bound state, as the atomic scattering length and gain/loss term are varied. However, when the trap frequency is also modulated, we show the phenomena of collapse and revival of two-soliton like bound state formation of the condensate for double modulated periodic potential and bright and dark solitons for step-wise modulated potentials.     

Bessel型光晶格中双组分玻色-爱因斯坦凝聚体的基态解

研究了平面Bessel型光晶格(BL)中双组分玻色-爱因斯坦凝聚(BECs)体系的基态解.从描述三维(3D)BECs体系的动力学方程Gross-Pitaevskii方程(GPE)出发,当垂直方向囚禁频率远大于平面上囚禁频率时,得到了描述2D-BECs体系的动力学方程.利用双组分BECs体系中原子之间相互作用与BL强度相互平衡的条件,得到了平面BL光晶格中2D-GPE的一组基态精确解,给出了基态的原子数分布,总原子数和能量与原子之间相互作用强度及BL势的关系.相对于单组分BEC体系,由于不同组分原子相互作用的存在,使得BL光晶格中双组分BECs基态具有更丰富的结构.当不存在不同组分原子之间的相互作用时,模型简化到单组分体系,并给出了相应的基态解,原子数分布和能量. 关键词： Bessel型光晶格 基态解 双组分玻色-爱因斯坦凝聚     

Electrical studies on ionic liquid-based gel polymer electrolyte for its application in EDLCs

Ionic liquid-based gel polymer electrolyte (GPE) has been synthesized using standard solution cast technique. Different weight percent of ionic liquid, 1-Butyl-3-methylimidazolium chloride (BMIMCl) and liquid electrolyte, ethylene carbonate (EC)–propylene carbonate (PC)–tetra ethyl ammonium tetra fluoro borate (TEABF₄) was incorporated in polymer, poly(vinylidene fluoride-co-hexafluoro propylene (PVdF-HFP) to obtain mechanically stable gel polymer electrolyte film (GPE) having maximum conductivity of ~10⁻³ S cm⁻¹ at room temperature, which is acceptable from device fabrication point of view. Potential window and ionic transference number has been obtained to examine the potential limit and ionic characteristics of optimized GPE system. Temperature dependence behavior of electrical conductivity curve follows Arrhenius nature in the temperature range of 303–373 K. Pattern of dielectric constant and its loss as a function of frequency and temperature have been studied and is being explained on the basis of electrode interfacial polarization effect. Frequency-dependent conductivity spectra obey the Jonscher’s power law. Further, optimized composition of GPE has been tested successfully for its application in supercapacitor fabrication with activated charcoal as an electrode material. Maximum specific capacitance of 118.6 mF cm⁻² equivalent to single electrode specific capacitance of 61.7 F g⁻¹ have been observed for the optimized GPE film.

     

One-dimensional bosons in three-dimensional traps

Recent experimental and theoretical work has indicated conditions in which a trapped, low density Bose gas ought to behave like the 1D delta-function Bose gas solved by Lieb and Liniger. Up until now, the theoretical arguments have been based on variational/perturbative ideas or numerical investigations. There are four parameters: density, transverse and longitudinal dimensions, and scattering length. In this paper we explicate five parameter regions in which various types of 1D or 3D behavior occur in the ground state. Our treatment is based on a rigorous analysis of the many-body Schr?dinger equation.     

Dynamically compressed bright and dark solitons in highly anisotropic Bose-Einstein condensates

Bright and dark matter wave solitons are constructed analytically in a three-dimensional (3D) highly anisotropic Bose-Einstein condensate (BEC) with a time-dependent parabolic potential, and numerical simulations are performed to confirm the existence and dynamics of such analytical solutions. Different classes of bright and dark solitons are discovered among the solutions of the generalized anisotropic (3+1)D Gross-Pitaevskii equation. Our results demonstrate that the bright and dark solitary waves can be manipulated and controlled by changing the scattering length, which can be used to compress the second-order bright and dark solitons of BECs into desired peak density.     

Output from an atom laser: theory vs. experiment

Atom lasers based on rf-outcoupling can be described by a set of coupled generalized Gross–Pitaevskii equations (GPE). We compare the theoretical predictions obtained by numerically integrating the time-dependent GPE of an effective one-dimensional model with recently measured experimental data for the F=2 and F=1 states of Rb-87. We conclude that the output of a rf atom-laser can be described by this model in a satisfactory way. Received: 15 June 1999 / Revised version: 9 September 1999 / Published online: 10 November 1999     

基于径向剪切干涉仪的三维位移测量技术

本文提出一种基于双圆光栅径向剪切干涉仪的三维位移测量方法,其测量原理是径向剪切干涉仪所形成的莫尔条纹不仅由二维平面内位移决定,轴向位移会在+1和–1级莫尔条纹之间产生一个特定的相移.首先,基于标量衍射理论对双圆光栅径向剪切干涉仪的+1和–1级莫尔条纹强度分布进行推导,建立了三维位移量与莫尔条纹强度分布的精确解析关系;其次,在频谱分析的基础上,利用半圆环滤波器进行空间滤波,实现+1和–1级莫尔条纹的同时成像;然后,提出了从莫尔条纹图中定量提取三维位移的算法,并通过数值模拟进行验证;最后,实验结果验证了该方法测量平面内位移的最大绝对误差为4.8×10^–3 mm,平均误差为2.0×10^–4 mm,轴向位移的最大绝对误差为0.25 mm,平均误差为8.6×10^–3 mm.该方法具有装置简单、测量精度高、非接触、瞬时测量等特点,可实现三维位移的同时测量.     

Analytic Study of a Bose--Einstein Condensate in Waveguide with an Obstacle Potential

We investigate the exact solutions of one-dimensional (1D) time-independent Gross-Pitaevskii equation (GPE), which governs a Bose-Einstein condensate (BEC) in the magnetic waveguide with a square-Sech potential. Both the bound state and transmission state are found and the corresponding spatial configurations and transport properties of BEC are analyzed. It is shown that the well-known absolute transmission of the linear system can occur in the considered nonlinear system.     

Exploring ground states and excited states of spin-1 Bose–Einstein condensates by continuation methods

A pseudo-arclength continuation method (PACM) is employed to compute the ground state and excited state solutions of spin-1 Bose–Einstein condensates (BEC). The BEC is governed by the time-independent coupled Gross–Pitaevskii equations (GPE) under the conservations of the mass and magnetization. The coupling constants that characterize the spin-independent and spin-exchange interactions are chosen as the continuation parameters. The continuation curve starts from a ground state or an excited state with very small coupling parameters. The proposed numerical schemes allow us to investigate the effect of the coupling constants and study the bifurcation diagrams of the time-independent coupled GPE. Numerical results on the wave functions and their corresponding energies of spin-1 BEC with repulsive/attractive and ferromagnetic/antiferromagnetic interactions are presented. Furthermore, we reveal that the component separation and population transfer between the different hyperfine states can only occur in excited states due to the spin-exchange interactions.     

Formal analytical solutions for the Gross-Pitaevskii equation

Considering the Gross-Pitaevskii integral equation we are able to formally obtain an analytical solution for the order parameter Φ(x) and for the chemical potential μ as a function of a unique dimensionless non-linear parameter Λ. We report solutions for different ranges of values for the repulsive and the attractive non-linear interactions in the condensate. Also, we study a bright soliton-like variational solution for the order parameter for positive and negative values of Λ. Introducing an accumulated error function we have performed a quantitative analysis with respect to other well-established methods as: the perturbation theory, the Thomas-Fermi approximation, and the numerical solution. This study gives a very useful result establishing the universal range of the Λ-values where each solution can be easily implemented. In particular, we showed that for Λ<−9, the bright soliton function reproduces the exact solution of GPE wave function.