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arXiv:2002.08877 · Physical Review A

Detecting a logarithmic nonlinearity in the Schrödinger equation using Bose-Einstein condensates

Sascha Vowe, Cláus Lämmerzahl, Markus Krutzik

Published Apr 22, 2020• 14 citations
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Abstract

We study the effect of a logarithmic nonlinearity in the Schr\"odinger equation (SE) on the dynamics of a freely expanding Bose-Einstein condensate (BEC). The logarithmic nonlinearity was one of the first proposed nonlinear extensions to the SE which emphasized the conservation of important physical properties of the linear theory, e.g., the separability of noninteracting states. Using this separability, we incorporate it into the description of a BEC obeying a logarithmic Gross-Pitaevskii equation. We investigate the dynamics of such BECs by using variational and numerical methods and find that, by using experimental techniques like $\ensuremath{\delta}$-kick collimation, experiments with extended free-fall times as available on microgravity platforms could be able to lower the bound on the strength of the logarithmic nonlinearity by at least one order of magnitude.

DOI
10.1103/physreva.101.043617

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•Sep 16, 2009•1 citation
We study the effect of a logarithmic nonlinearity in the Schr"odinger equation (SE) on the dynamics of a freely expanding Bose-Einstein condensate (BEC). The logarithmic nonlinearity was one of the first proposed nonlinear extensions to the SE which emphasized the conservation of important physical properties of the linear theory, e.g., the separability of noninteracting states. Using this separability, we incorporate it into the description of a BEC obeying a logarithmic Gross-Pitaevskii equation. We investigate the dynamics of such BECs by using variational and numerical methods and find that, by using experimental techniques like \ensuremathδ\ensuremath{\delta}\ensuremathδ-kick collimation, experiments with extended free-fall times as available on microgravity platforms could be able to lower the bound on the strength of the logarithmic nonlinearity by at least one order of magnitude.
A variational technique is applied to solve the time-dependent nonlinear Schr\"odinger equation (Gross-Pitaevskii equation) 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 Schr\"odinger 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.
We study a relativistic scalar field model for self-bound Bose-Einstein condensates (BECs) by analyzing a nonlinear Klein-Gordon equation with cubic and logarithmic interactions. This framework captures essential features of quantum droplets, such as self-trapping and finite energy configurations, which emerge from the interplay between attractive and repulsive terms. By performing the non-relativistic limit, we derive a generalized Gross-Pitaevskii equation with a logarithmic correction, consistent with recent models used to describe ultra-cold atomic gasses beyond mean-field theory. We construct the corresponding Lagrangian density, identify conserved quantities via Noether's theorem, and compute the energy-momentum tensor. Numerical solutions of the BEC parameters are shown, establishing the foundations for a field theoretical description of relativistic condensates with a logarithmic interaction. This model provides a unified approach to investigate relativistic effects in quantum droplets and enriches the theoretical landscape of Bose-Einstein condensates with non-standard interactions. The resulting dynamics exhibit stable oscillatory regimes consistent with self-bound condensate configurations.
The Gross-Pitaevskii equation (GPE) plays an important role in the description of Bose-Einstein condensates (BECs) at the mean-field level. The GPE belongs to the class of nonlinear Schr\"odinger equations which are known to feature dynamical instability and collapse for attractive nonlinear interactions. We show that the GPE with repulsive nonlinear interactions typical for BECs features chaotic wave dynamics. We find positive Lyapunov exponents for BECs expanding in periodic and aperiodic smooth external potentials, as well as disorder potentials. Our analysis demonstrates that wave chaos characterized by the exponential divergence of nearby initial wave functions is to be distinguished from the notion of nonintegrability of nonlinear wave equations. We discuss the implications of these observations for the limits of applicability of the GPE, the problem of Anderson localization, and the properties of the underlying many-body dynamics.
Abstract. The motto of this paper is: Let's face Bose-Einstein condensation through nonUnear dynamics. We do this by choosing variational forms of the condensate wave functions (of given symmetry classes), which convert the Bose-Einstein condensates via the time-dependent Gross-Pitaevskii equation into Hamiltonian systems that can be studied using the methods of nonUnear dynamics. We consider in particular cold quantum gases where long-range interactions between the neutral atoms are present, in addition to the conventional short-range contact interaction, viz. gravity-Uke interactions, and dipole-dipole interactions. The residts obtained serve as a useful giude in the search for nonlinear dynamics effects in numerically exact quantum calcinations for Bose-Einstein condensates. A main residt is the prediction of the existence of stable islands as well as chaotic regions for excited states of dipolar condensates, which coidd be checked experimentally. Keywords: Bose-Einstein condensation, Gross-Pitaevskii equation, nonlinear dynamics effects PACS: 03.75.Kk, 34.20.Cf, 02.30.-f, 47.20.Ky 1.