From Feshbach-Resonance Managed Bose-Einstein Condensates to Anisotropic Universes: Some Applications of the Ermakov-Pinney equation with Time-Dependent Nonlinearity

TL;DR

This paper uses the Ermakov-Pinney equation to analytically study the dynamics of two-dimensional Bose-Einstein condensates with time-dependent parameters, revealing conditions for dispersion, breathing, or collapse, and explores related cosmological models.

Contribution

It introduces an analytical approach using the Ermakov-Pinney equation to control BEC dynamics with time-dependent interactions and confinement, also linking to cosmological models.

Findings

Analytical solutions match numerical simulations of BEC dynamics.

Engineered conditions lead to dispersing, breathing, or collapsing condensates.

Potential applications in anisotropic scalar field cosmologies.

Abstract

In this work we revisit the topic of two-dimensional Bose-Einstein condensates under the influence of time-dependent magnetic confinement and time-dependent scattering length. A moment approach reduces the examination of moments of the wavefunction (in particular, of its width) to an Ermakov-Pinney (EP) ordinary differential equation (ODE). We use the well-known structure of the solutions of this nonlinear ODE to ``engineer'' trapping and interatomic interaction conditions that lead to condensates dispersing, breathing or even collapsing. The advantage of the approach is that it is fully tractable analytically, in excellent agreement with our numerical observations. As an aside, we also discuss how similar time-dependent EP equations may arise in the description of anisotropic scalar field cosmologies.