Ground State and Collective Modes of Bose-Einstein Condensates in Newtonian and MOND-inspired gravitational potentials

TL;DR

The paper studies a Bose-Einstein condensate in two gravitational traps—Newtonian and a deep-MOND-inspired logarithmic potential—using Gaussian variational methods for both ground-state and monopole-dynamics analyses. It shows bound states exist only in the deep-MOND regime, with the MOND condensate being larger and more weakly bound than in Newtonian gravity. In the strong-interaction limit, the MOND case exhibits a clean scaling $\tilde\sigma_0^M \propto \beta^{1/3}$ (captured by the TF approximation) and a monopole frequency scaling $\tilde\Omega_M \propto \beta^{-1/3}$, providing a distinct experimental signature from the Newtonian case, where no simple TF scaling holds. These results offer actionable benchmarks for quantum-simulation experiments aiming to emulate modified gravity, enabling tests of MOND-like dynamics with ultracold atoms.

Abstract

We analytically and numerically study the ground state and collective dynamics of Bose-Einstein condensates in two traps: a Newtonian potential and a logarithmic potential inspired by Modified Newtonian Dynamics (MOND). In the ground state, the MOND potential supports bound states only in the deep-MOND regime, where the condensate becomes significantly larger than its Newtonian counterpart. The size increases with repulsive coupling parameter $β$ in both potentials. A clear scaling law of the size with $β^{1/3}$ emerges in the MOND case and is confirmed numerically over a wide parameter range, while for the Newtonian potential no simple scaling law exists as the Thomas-Fermi approximation ceases to be valid. For the dynamics, we derive and solve equations for the monopole collective mode. The larger MOND-bound condensate oscillates at a lower frequency, which scales as $β^{-1/3}$ in the strong-interaction limit. These scaling laws provide insights for quantum-simulation experiments aiming to probe modified-gravity scenarios with cold atoms.

Figures (3)

• Figure 1: Dimensionless energy per atom $\tilde{e}$ as a function of the normalized width $\tilde{\sigma}$ for the Newtonian (solid lines) and MOND (dashed lines) potentials. Different colors correspond to different interaction strengths: $\beta=0.1$ (green), $1$ (blue), $10$ (red), and $100$ (black). Energy minima are marked by circles of the corresponding color.
• Figure 2: Scaling of the condensate width $\tilde{\sigma}_0^M$ with the interaction parameter $\beta$ in the MOND potential. The dashed lines show the theoretical scaling $\tilde{\sigma}_0^M \propto \beta^{1/3}$ for $\eta=1$ (red) and $\eta=0.1$ (green). The symbols represent exact numerical solutions, demonstrating excellent agreement with the theoretical prediction across the entire range $\beta=10^1 \sim 10^4$.
• Figure 3: Width oscillations for $\beta=1$. Newtonian potential (blue solid). MOND potential for $\eta=0.1$ (red dashed), $\eta=0.5$ (black dashed), and $\eta=1$ (green dashed). The dashed horizontal lines indicate the respective equilibrium widths from Table \ref{['tab:ground_state_properties']}.