We deal with the three-dimensional Gross-Pitaevskii equation, which is used to describe a cloud of dilute bosonic atoms that interact under competing two- and three-body scattering potentials. We study the case where the cloud of atoms is strongly confined in two spatial dimensions, allowing us to build an unidimensional nonlinear equation, controlled by the nonlinearities and the confining potentials that trap the system along the longitudinal coordinate. We focus attention on specific limits, dictated by the cubic and quintic coefficients, and we implement numerical simulations to help us to quantify the validity of the procedure.
Overlap of exact and Gross-Pitaevskii wavefunctions in Bose-Einstein condensates of dilute gases
对于在稀释极限中具有两体排斥相互作用势的玻色子，已经在理论上证明了Gross-Pitaevskii方程提供了每个粒子的精确能量和密度，基本的多粒子薛定谔方程也是如此[Lieb和Seiringer，Phys. Rev. Lett. 88，170409（2002）]。在这里，我们调查的重叠的Gross-Pitaevskii和确切的基态波函数。据发现，这种重叠总是小于单位，甚至可能消失，尽管事实上，这两个波函数提供相同的能量和密度每个粒子。讨论了后果。
Quantum phenomena appear in a macroscopic scale in Bose-Einstein condensates. The Gross-Pitaevskii (GP) equation describes the dynamics of the weakly interacting Bose-Einstein condensates. The GP equation has a form of the Schroedinger equation with self-interaction. The coupled Gross-Pitaevskii equations are used to describe some mixtures of Bose-Einstein condensates. In this paper, we will show some numerical results of coupled Gross-Pitaevskii equations without self-interaction, which has a form of nonlinearly-coupled Schroedinger equations. We demonstrate that the macroscopic oscillation a…
A prescription is given to obtain some exact results for certain external potentials $V\left({\vec r}\right)$ of the time-independent Gross-Pitaevskii and Schrödinger equations. The study motivation is the ability to program $V\left({\vec r}\right)$ experimentally in Bose-Einstein condensates. Rather than derive wavefunctions that are solutions for a given $V\left({\vec r}\right)$, we ask which $V\left({\vec r}\right)$ will have a given pdf (probability density function) $P\left({\vec r}\right)$. Several examples in 1D and 2D are presented for well-known pdfs and for the hydrogen atom in momen…
