Detecting Collective Excitations in Self-Gravitating Bose-Einstein Condensates via Faraday Waves

TL;DR

This paper addresses how to detect collective excitations in self-gravitating Bose-Einstein condensates by using Faraday waves. A semi-classical approach based on linear stability of the Gross-Pitaevskii-Newton equations yields a damped Mathieu equation with parameters $(\lambda, q, \bar{\gamma})$, enabling a Floquet-stability analysis that separates Jeans instability from parametric resonance via the Jeans boundary $k_J$. Key results include a monotonic Jeans growth rate $\Gamma_{\text{Jeans}}(k)$ that vanishes at $k_J$ and a non-monotonic parametric-resonance growth within the first Mathieu tongue, along with a formula for the Faraday wavevector $k_F$ under driving; numerical simulations corroborate the transition from Faraday waves to Jeans collapse as gravity strengthens. The work provides explicit experimental guidelines—operating in the parametric regime with suitable $k$ and driving parameters—to observe Faraday waves in SGBECs and offers a tractable framework for probing the excitation spectrum of self-gravitating quantum fluids with potential astrophysical analogs.

Abstract

We propose Faraday waves as a probe for collective excitations in self-gravitating Bose-Einstein condensates (SGBECs). Using a semi-classical approach based on linear stability analysis of the Gross-Pitaevskii-Newton equations, we derive a damped Mathieu equation governing parametric instabilities. Our analysis reveals well-separated regions of parametric resonance and Jeans instability in parameter space, with distinct growth rate characteristics: Jeans instability decreases monotonically to zero at the critical wavenumber $k_J$, while parametric resonance exhibits non-monotonic behavior with a clear maximum. These findings provide explicit experimental guidelines for accessing the parametric resonance regime. Numerical simulations demonstrate the transition from Faraday wave formation to Jeans collapse as gravitational strength increases, validating our theoretical framework.

Figures (7)

• Figure 1: (Color online) Collective excitation spectrum of SGBECs. Parameters: $\Omega_0 = g_0 \bar{n}_0/\hbar$, $\xi = \hbar/\sqrt{2 m g_0 \bar{n}_0}$. The vertical dashed line indicates $k_J$ where $\Omega_k^2 = 0$, marking the onset of Jeans instability. For $k < k_J$ (specifically, $k_J \xi = \sqrt{\sqrt{1+\Omega_J^2/\Omega_0^2}-1}$), $\Omega_k^2 < 0$ indicates exponential growth, defining the quantum Jeans scale chavanis2020jeans.
• Figure 2: (Color online) Stability phase diagrams in the $(k\xi, \alpha)$ parameter space for different gravitational strengths: (a) $\Omega_J/\Omega_0 = 0.2$, (b) $\Omega_J/\Omega_0 = 0.5$, and (c) $\Omega_J/\Omega_0 = 1.0$. Blue regions denote Jeans instability ($\lambda < 0$), while purple regions indicate parametric resonance within the first Mathieu instability tongue ($\lambda > 0$). The vertical dashed lines mark the critical Jeans wavenumber $k_J\xi$, which separates the two distinct instability regimes. Parameters: $\omega/\Omega_0 = 0.5$.
• Figure 3: (Color online) Comparison of instability growth rates for different gravitational strengths with fixed modulation depth $\alpha = 0.4$: (a) $\Omega_J/\Omega_0 = 0.2$ and (b) $\Omega_J/\Omega_0 = 1.0$. Blue solid curves: Jeans growth rate $\Gamma_{\text{Jeans}}/\Omega_0$; purple dashed curves: parametric resonance growth rate $\Gamma_{\text{parametric}}/\Omega_0$ obtained from numerical Floquet analysis. The vertical dashed lines indicate the critical Jeans wavenumber $k_J\xi$, where $\Gamma_{\text{Jeans}}$ vanishes as required by theory.
• Figure 4: (Color online) Stability phase diagrams at different driving frequencies with fixed gravitational strength $\Omega_J/\Omega_0 = 1.0$: (a) $\omega/\Omega_0 = 0.5$, (b) $\omega/\Omega_0 = 1.0$, and (c) $\omega/\Omega_0 = 2.0$. The parametric resonance tongue (purple) shifts systematically toward larger wavenumbers with increasing frequency, while the Jeans instability boundary (blue) remains fixed at $k_J$. This frequency control provides an experimental approach to optimize parametric resonance accessibility.
• Figure 5: (Color online) Faraday wavevector versus driving frequency for $\nu=1,2,3$ with $\Omega_J/\Omega_0 = 1.2$.