Generalized one-dimensional nonpolynomial Schrödinger equation for Bose-Einstein condensates with generic transverse confinement

TL;DR

We study dimensional reduction of a Bose-Einstein condensate described by the 3D-GPE under generalized transverse confinement parameterized by the positive exponent $n$. Using a variational ansatz with a transverse width $\sigma(z,t)$ and an adiabatic approximation yields the generalized 1D nonpolynomial Schrödinger equation (1D-EFG) together with an analytic expression for $\sigma^2_+$, while the $\sigma^2_-$ branch is discarded as unphysical. Across $n$ and interaction strength $g$, the 1D-EFG reproduces ground-state properties and dynamical scattering results with high fidelity to full 3D-GPE, outperforming the simple 1D cubic reduction, especially at small $n$. This provides a unified framework bridging cigar- and funnel-like geometries and enabling systematic exploration of confinement effects and trap optimization in experiments.

Abstract

This work presents a dimensional reduction of Bose-Einstein condensates confined by generalized transverse potentials, parametrized by an exponent $n$. Starting from the three-dimensional Gross-Pitaevskii equation, we employ a variational ansatz to derive an effective one-dimensional nonpolynomial Schrödinger equation, which self-consistently determines the transverse width dynamics. The model generalizes existing formalisms for cigar- and funnel-shaped geometries. We validate the approach through comprehensive numerical tests, demonstrating excellent agreement with full 3D simulations for ground-state properties across various interaction regimes. Finally, real-time simulations of matter-wave scattering at potential barriers verify the model's dynamical robustness, successfully replicating the spatiotemporal evolution and energy-dependent transmission characteristics observed in full 3D calculations.

Figures (8)

• Figure 1: Transverse potential defined by Eq. (\ref{['potgeralcharuto']}), expressed in units of $m\omega_{\perp}^2$, as a function of the radial coordinate $r$ for $n=1$, $2$, $3$, $4$, $5$, and $6$. The parameters are fixed at $\alpha=1$ and $\beta=1$.
• Figure 2: Solutions for the variational field $\sigma^2_+$ and $\sigma^2_-$ in the upper and lower quadrants, respectively, as a function of $n$ for different values of $g|f|^2$.
• Figure 3: Axial density for the effective 1D-EFG (\ref{['efetivancharuto']}) for different values of the interaction strength, i.e., $g = 1$, $10$, and $100$.
• Figure 4: Maximum value of the axial density as a function of $n$ for the 1D-EFG (\ref{['efetivacharutogeral']}), the 1D-CN (\ref{['reduzida']}) and the 3D-GPE (\ref{['charuto3dn']}), for different values of the interaction strength, i.e., $g = 1$, $10$, and $100$.
• Figure 5: Chemical potential (left) and expectation value of the squared longitudinal length (right) for the 1D-EFG (\ref{['efetivacharutogeral']}), the 1D-CN (\ref{['reduzida']}) and the 3D-GPE (\ref{['charuto3dn']}), as a function of $n$ for different values of the interaction strength, viz., $g = 1$, $10$, and $100$.