Nonlinear Schrödinger equations for Bose-Einstein condensates (original) (raw)
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Fluid Dynamics Research, 2003
The Gross-Pitaevskii equation, also called the nonlinear Schrödinger equation (NLSE), describes the dynamics of low-temperature superflows and Bose-Einstein Condensates (BEC). We review some of our recent NLSE-based numerical studies of superfluid turbulence and BEC stability. The relations with experiments are discussed.
Dynamics of Bose-Einstein condensates: Variational solutions of the Gross-Pitaevskii equations
Physical Review A, 1997
A variational technique is applied to solve the time-dependent nonlinear Schrödinger equation ͑Gross-Pitaevskii equation͒ with the goal to model the dynamics of dilute ultracold atom clouds in the Bose-Einstein condensed phase. We derive analytical predictions for the collapse, equilibrium widths, and evolution laws of the condensate parameters and find them to be in very good agreement with our numerical simulations of the nonlinear Schrödinger equation. It is found that not only the number of particles, but also both the initial width of the condensate and the effect of different perturbations to the condensate may play a crucial role in the collapse dynamics. The results are applicable when the shape of the condensate is sufficiently simple.
Bose–Einstein condensates: Analytical methods for the Gross–Pitaevskii equation
Physics Letters A, 2006
We present simple analytical methods for solving the Gross-Pitaevskii equation (GPE) for the Bose-Einstein condensation (BEC) in the dilute atomic alkali gases. Using a soliton variational Ansatz we consider the effects of repulsive and attractive effective nonlinear interactions on the BEC ground state. We perform a comparative analysis of the solutions obtained by the variational Ansatz, the perturbation theory, the Thomas-Fermi approximation, and the Green function method with the numerical solution of the GPE finding universal ranges where these solutions can be used to predict properties of the condensates. Also, a generalization of the soliton variational approach for two-species of alkali atoms of the GPE is performed as a function of the effective interaction λ i (i = 1, 2) and the inter-species λ 12 and λ 21 constants.
Numerical Methods for the study of Bose-Einstein Condensates using the Gross-Pitaevskii Equation
Universitat Politècnica de Catalunya, 2019
The Gross-Pitaevskii equation (GPE) is a mean field approximation used to study Bose-Einstein condensates (BEC). Here, we present a brief derivation of the GPE without the use of second quantization, as well as some of its consequences. Next, we present the implementation in Python of some numerical methods to compute the ground state of BEC. We compare the results with well known theoretical limits and the numerical results obtained using GPELab, a MATLAB toolbox to compute stationary and dynamic solutions of the GPE.
Bose-Einstein condensation dynamics from the numerical solution of the Gross-Pitaevskii equation
Journal of Physics B: Atomic, Molecular and Optical Physics, 2003
We study certain stationary and time-evolution problems of trapped Bose-Einstein condensates using the numerical solution of the Gross-Pitaevskii equation with both spherical and axial symmetries. We consider time-evolution problems initiated by changing the interatomic scattering length or harmonic trapping potential suddenly in a stationary condensate. These changes introduce oscillations in the condensate which are studied in detail. We use a time iterative split-step method for the solution of the time-dependent Gross-Pitaevskii equation, where all nonlinear and linear nonderivative terms are treated separately from the time propagation with the kinetic energy terms. Even for an arbitrarily strong nonlinear term this leads to extremely accurate and stable results after millions of time iterations of the original equation.
Nonlinear waves in Bose–Einstein condensates: physical relevance and mathematical techniques
Nonlinearity, 2008
The aim of the present review is to introduce the reader to some of the physical notions and of the mathematical methods that are relevant to the study of nonlinear waves in Bose-Einstein Condensates (BECs). Upon introducing the general framework, we discuss the prototypical models that are relevant to this setting for different dimensions and different potentials confining the atoms. We analyze some of the model properties and explore their typical wave solutions (plane wave solutions, bright, dark, gap solitons, as well as vortices). We then offer a collection of mathematical methods that can be used to understand the existence, stability and dynamics of nonlinear waves in such BECs, either directly or starting from different types of limits (e.g., the linear or the nonlinear limit, or the discrete limit of the corresponding equation). Finally, we consider some special topics involving more recent developments, and experimental setups in which there is still considerable need for developing mathematical as well as computational tools. Recall that the underlying model, namely the completely integrable NLS equation, has infinitely many symmetries, including translational and Galilean invariances.
Gravity, Bose-Einstein Condensates and Gross-Pitaevskii Equation
We explore the effect of mutual gravitational interaction between ultra-cold gas atoms on the dynamics of Bose-Einstein condensates (BEC). Small amplitude oscillation of BEC is studied by applying variational technique to reduce the Gross-Pitaevskii equation, with gravity included, to the equation of motion of a particle moving in a potential. According to our analysis, if the s-wave scattering length can be tuned to zero using Feshbach resonance for future BEC with occupation numbers as high as approx1020\approx 10^{20}approx1020, there exists a critical ground state occupation number above which the BEC is unstable, provided that its constituents interact with a 1/r31/r^3 1/r3 gravity at short scales.
Soliton dynamics in an inhomogeneous Bose–Einstein condensate driven by linear potential
Communications in Nonlinear Science and Numerical Simulation, 2012
In this paper we study the propagation of solitons in a Bose-Einstein condensate governed by the time dependent one dimensional Gross-Pitaevskii equation managed by Feshbach resonance in a linear external potential. We give the Lax pair of the Gross-Pitaevskii equation in Bose-Einstein condensates and obtain exact N-soliton solution by employing the simple, straightforward Darboux transformation. As an example, we present exact one and two-soliton solution and discuss their transmission, interaction and dynamic properties. We further calculate the particle number, momentum and energy of the solitons and discuss their conservation laws. Knowledge of soliton dynamics helps us in understanding the physical nature of the condensate and in the calculation of the thermodynamic properties.
International Journal of Modern Physics B, 2012
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 threebody 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.
Dynamics of Bose-Einstein Condensates with Long-Range Attractive Interactions
We solve the time-dependent Gross-Pitaevskii equation for the Bose-Einstein condensate with non-local dipole-dipole interaction potential as well as with the attractive gravity-like potential numerically. We observe formation of supersolid structure above the critical intensity in harmonic traps. Simple Linear Combination of Gaussian Orbitals (LCGO) theory is provided. We also observe self-bound structures for the condensate with gravity-like potentials.