scholarly journals PATTERN FORMING DYNAMICAL INSTABILITIES OF BOSE–EINSTEIN CONDENSATES

2004 ◽  
Vol 18 (05n06) ◽  
pp. 173-202 ◽  
Author(s):  
P. G. KEVREKIDIS ◽  
D. J. FRANTZESKAKIS
Keyword(s):  
Modulational Instability ◽  
Mean Field ◽  
Three Dimensional ◽  
Two Dimensions ◽  
One Dimension ◽  
Pitaevskii Equation ◽  
Long Wavelength ◽  
Pattern Forming ◽  
Bose Einstein ◽  
Dynamical Instabilities

In this short topical review, we revisit a number of works on the pattern-forming dynamical instabilities of Bose–Einstein condensates in one- and two-dimensional settings. In particular, we illustrate the trapping conditions that allow the reduction of the three-dimensional, mean field description of the condensates (through the Gross–Pitaevskii equation) to such lower dimensional settings, as well as to lattice settings. We then go on to study the modulational instability in one dimension and the snaking/transverse instability in two dimensions as typical examples of long-wavelength perturbations that can destabilize the condensates and lead to the formation of patterns of coherent structures in them. Trains of solitons in one dimension and vortex arrays in two dimensions are prototypical examples of the resulting nonlinear waveforms, upon which we briefly touch at the end of this review.

THREE-BODY INTERACTIONS BEYOND THE GROSS–PITAEVSKII EQUATION AND MODULATIONAL INSTABILITY OF BOSE–EINSTEIN CONDENSATES

2012 ◽  
Vol 26 (32) ◽  
pp. 1250202 ◽  
Author(s):  
DIDIER BELOBO BELOBO ◽  
GERMAIN HUBERT BEN-BOLIE ◽  
TIMOLEON CREPIN KOFANE
Keyword(s):  
Modulational Instability ◽  
Mean Field Theory ◽  
Mean Field ◽  
External Potential ◽  
Pitaevskii Equation ◽  
Theoretical Predictions ◽  
The Mean ◽  
Three Body ◽  
Bose Einstein ◽  
Condensate System

Beyond the mean-field theory, a new model of the Gross–Pitaevskii equation (GPE) that describes the dynamics of Bose–Einstein condensates (BECs) is derived using an appropriate phase-imprint on the old wavefunction. This modified version of the GPE in addition to the two-body interactions term, also takes into account effects of the three-body interactions. The three-body interactions consist of a quintic term and the delayed nonlinear response of the condensate system term. Then, the modulational instability (MI) of the new GPE confined in an attractive harmonic potential is investigated. The analytical study shows that the three-body interactions destabilize more the condensate system while the external potential alleviates the instability. Numerical results confirm the theoretical predictions. Further numerical investigations of the behavior of solitons reveal that the three-body interactions enhance the appearance of solitons, increase the number of solitons generated and deeply change the lifetime of solitons. Moreover, the external potential delays the appearance of solitons. Besides, a new initial condition is introduced which enables to increase the number of solitons created and deeply affects the trail of chains of solitons generated. Moreover, the MI of a condensate without the external potential, and in a repulsive potential is also investigated.

scholarly journals GROSS–PITAEVSKII MODEL FOR NONZERO TEMPERATURE BOSE–EINSTEIN CONDENSATES

2006 ◽  
Vol 16 (09) ◽  
pp. 2713-2719 ◽  
Author(s):  
KESTUTIS STALIUNAS
Keyword(s):  
Power Spectra ◽  
Pitaevskii Equation ◽  
Nonzero Temperature ◽  
Ginzburg Landau ◽  
Long Wavelength ◽  
Occupation Numbers ◽  
Ginzburg Landau Equations ◽  
Bose Einstein ◽  
Canonical Ensembles ◽  
Grand Canonical

Momentum distributions and temporal power spectra of nonzero temperature Bose–Einstein condensates are calculated using a Gross–Pitaevskii model. The distributions are obtained for micro-canonical ensembles (conservative Gross–Pitaevskii equation) and for grand-canonical ensembles (Gross–Pitaevskii equation with fluctuations and dissipation terms). Use is made of equivalence between statistics of the solutions of conservative Gross–Pitaevskii and dissipative complex Ginzburg–Landau equations. In all cases the occupation numbers of modes follow a 〈Nk〉 ∝ k-2 dependence, which corresponds in the long wavelength limit (k → 0) to Bose–Einstein distributions. The temporal power spectra are of 1/fα form, where: α = 2 - D/2 with D the dimension of space.

scholarly journals Accurate and Efficient Numerical Methods for Computing Ground States and Dynamics of Dipolar Bose-Einstein Condensates via the Nonuniform FFT

2016 ◽  
Vol 19 (5) ◽  
pp. 1141-1166 ◽  
Author(s):  
Weizhu Bao ◽  
Qinglin Tang ◽  
Yong Zhang
Keyword(s):  
Numerical Methods ◽  
Ground State ◽  
High Efficiency ◽  
Three Dimensional ◽  
Ground States ◽  
Dipole Interaction ◽  
Polar Coordinates ◽  
Pitaevskii Equation ◽  
Nonuniform Fft ◽  
Bose Einstein

AbstractWe propose efficient and accurate numerical methods for computing the ground state and dynamics of the dipolar Bose-Einstein condensates utilising a newly developed dipole-dipole interaction (DDI) solver that is implemented with the non-uniform fast Fourier transform (NUFFT) algorithm. We begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with a DDI term and present the corresponding two-dimensional (2D) model under a strongly anisotropic confining potential. Different from existing methods, the NUFFT based DDI solver removes the singularity by adopting the spherical/polar coordinates in the Fourier space in 3D/2D, respectively, thus it can achieve spectral accuracy in space and simultaneously maintain high efficiency by making full use of FFT and NUFFT whenever it is necessary and/or needed. Then, we incorporate this solver into existing successful methods for computing the ground state and dynamics of GPE with a DDI for dipolar BEC. Extensive numerical comparisons with existing methods are carried out for computing the DDI, ground states and dynamics of the dipolar BEC. Numerical results show that our new methods outperform existing methods in terms of both accuracy and efficiency.

LOD-MS for Gross-Pitaevskii Equation in Bose-Einstein Condensates

2013 ◽  
Vol 14 (1) ◽  
pp. 219-241 ◽  
Author(s):  
Linghua Kong ◽  
Jialin Hong ◽  
Jingjing Zhang
Keyword(s):  
Energy Conservation ◽  
Computational Cost ◽  
Three Dimensional ◽  
Schrodinger Equations ◽  
Pitaevskii Equation ◽  
Schrödinger Equations ◽  
One Dimensional ◽  
Exactly Solvable ◽  
Bose Einstein ◽  
Energy Conservation Law

AbstractThe local one-dimensional multisymplectic scheme (LOD-MS) is developed for the three-dimensional (3D) Gross-Pitaevskii (GP) equation in Bose-Einstein condensates. The idea is originated from the advantages of multisymplectic integrators and from the cheap computational cost of the local one-dimensional (LOD) method. The 3D GP equation is split into three linear LOD Schrödinger equations and an exactly solvable nonlinear Hamiltonian ODE. The three linear LOD Schrödinger equations are multisymplectic which can be approximated by multisymplectic integrator (MI). The conservative properties of the proposed scheme are investigated. It is mass-preserving. Surprisingly, the scheme preserves the discrete local energy conservation laws and global energy conservation law if the wave function is variable separable. This is impossible for conventional MIs in nonlinear Hamiltonian context. The numerical results show that the LOD-MS can simulate the original problems very well. They are consistent with the numerical analysis.

scholarly journals STABILITY OF BOSE-EINSTEIN CONDENSATES IN A PT-SYMMETRIC DOUBLE-δ POTENTIAL CLOSE TO BRANCH POINTS

2014 ◽  
Vol 54 (2) ◽  
pp. 133-138 ◽  
Author(s):  
Andreas Löhle ◽  
Holger Cartarius ◽  
Daniel Haag ◽  
Dennis Dast ◽  
Jörg Main ◽  
...  
Keyword(s):  
Ground State ◽  
Mean Field ◽  
Einstein Condensate ◽  
Pitaevskii Equation ◽  
Branch Points ◽  
Mean Field Limit ◽  
Bose Einstein Condensate ◽  
Stability Change ◽  
The Difference ◽  
Bose Einstein

A Bose-Einstein condensate trapped in a double-well potential, where atoms are incoupled to one side and extracted from the other, can in the mean-field limit be described by the nonlinear Gross-Pitaevskii equation (GPE) with a <em>PT</em> symmetric external potential. If the strength of the in- and outcoupling is increased two <em>PT</em> broken states bifurcate from the <em>PT</em> symmetric ground state. At this bifurcation point a stability change of the ground state is expected. However, it is observed that this stability change does not occur exactly at the bifurcation but at a slightly different strength of the in-/outcoupling effect. We investigate a Bose-Einstein condensate in a <em>PT</em> symmetric double-δ potential and calculate the stationary states. The ground state’s stability is analysed by means of the Bogoliubov-de Gennes equations and it is shown that the difference in the strength of the in-/outcoupling between the bifurcation and the stability change can be completely explained by the norm-dependency of the nonlinear term in the Gross-Pitaevskii equation.

scholarly journals Soliton Solutions and Collisions for the Multicomponent Gross–Pitaevskii Equation in Spinor Bose–Einstein Condensates

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Ming Wang ◽  
Guo-Liang He
Keyword(s):  
Soliton Solutions ◽  
Auxiliary Function ◽  
Einstein Condensate ◽  
Polarization Parameter ◽  
One Dimension ◽  
Pitaevskii Equation ◽  
Bose Einstein Condensate ◽  
The One ◽  
Bose Einstein ◽  
Two Peaks

In this paper, we investigate a five-component Gross–Pitaevskii equation, which is demonstrated to describe the dynamics of an F=2 spinor Bose–Einstein condensate in one dimension. By employing the Hirota method with an auxiliary function, we obtain the explicit bright one- and two-soliton solutions for the equation via symbolic computation. With the choice of polarization parameter and spin density, the one-soliton solutions are divided into four types: one-peak solitons in the ferromagnetic and cyclic states and one- and two-peak solitons in the polar states. For the former two, solitons share the similar shape of one peak in all components. Solitons in the polar states have the one- or two-peak profiles, and the separated distance between two peaks is inversely proportional to the value of polarization parameter. Based on the asymptotic analysis, we analyze the collisions between two solitons in the same and different states.

MODULATIONAL INSTABILITY AND EXACT SOLITON AND PERIODIC SOLUTIONS FOR TWO WEAKLY COUPLED EFFECTIVELY 1D CONDENSATES TRAPPED IN A DOUBLE-WELL POTENTIAL

2010 ◽  
Vol 24 (14) ◽  
pp. 2211-2227 ◽  
Author(s):  
E. KENGNE ◽  
R. VAILLANCOURT ◽  
B. A. MALOMED
Keyword(s):  
Periodic Solutions ◽  
Modulational Instability ◽  
Wave Number ◽  
Pitaevskii Equation ◽  
Nonlinear Dispersion ◽  
Nonlinear Dispersion Relation ◽  
Weakly Coupled ◽  
Modulational Stability ◽  
Bose Einstein ◽  
Double Well Potential

The modulational instability of the coupled Gross–Pitaevskii equation (alias nonlinear Schrödinger equation), which describes two Bose–Einstein condensates trapped in an asymmetric double-well potential, is investigated. The nonlinear dispersion relation that relates the frequency and wave number of the modulating perturbations is found and its analysis shows several possibilities for the modulational stability region. Exact soliton and periodic solutions are constructed via elliptic ordinary differential equations.

scholarly journals Mean-field regime of trapped dipolar Bose-Einstein condensates in one and two dimensions

2010 ◽  
Vol 82 (4) ◽  
Author(s):  
Yongyong Cai ◽  
Matthias Rosenkranz ◽  
Zhen Lei ◽  
Weizhu Bao
Keyword(s):  
Mean Field ◽  
Two Dimensions ◽  
Bose Einstein ◽  
Mean Field Regime

scholarly journals THEORY OF NONLINEAR MATTER WAVES IN OPTICAL LATTICES

2004 ◽  
Vol 18 (14) ◽  
pp. 627-651 ◽  
Author(s):  
V. A. BRAZHNYI ◽  
V. V. KONOTOP
Keyword(s):  
Optical Lattices ◽  
Evolution Equations ◽  
Modulational Instability ◽  
Spatial Scales ◽  
Tight Binding ◽  
Matter Wave ◽  
Pitaevskii Equation ◽  
Tight Binding Approximation ◽  
Matter Waves ◽  
Bose Einstein

We consider several effects of the matter wave dynamics which can be observed in Bose–Einstein condensates embedded into optical lattices. For low-density condensates, we derive approximate evolution equations, the form of which depends on relation among the main spatial scales of the system. Reduction of the Gross–Pitaevskii equation to a lattice model (the tight-binding approximation) is also presented. Within the framework of the obtained models, we consider modulational instability of the condensate, solitary and periodic matter waves, paying special attention to different limits of the solutions, i.e. to smooth movable gap solitons and to strongly localized discrete modes. We also discuss how the Feshbach resonance, a linear force and lattice defects affect the nonlinear matter waves.

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