Scaling Laws Governing the Collapse of a Bose-Einstein Condensate

TL;DR

We study collapse in a 3D cubic-quintic nonlinear Schrödinger equation with dissipation, $ i \frac{\partial \psi}{\partial t} = -\nabla^2 \psi + \alpha |\psi|^2 \psi - i \eta |\psi|^4 \psi$, focusing on single-collapse events and comparing to box-trapped $^{39}$K measurements. The main finding is that the total single-collapse loss $\\Delta N$ does not follow the Zakharov prediction $\\Delta N \\propto \\eta^{1/2}/|\\alpha|^2$, whereas the loss up to collapse time $\\Delta N_{t_c}$ does, and post-collapse dynamics exhibit hotspot formation with a universal shoulder and a new scaling controlled by $\\eta^{1/2}/|\\alpha|^3$. The authors quantify Zakharov scaling for $n_{\\mathrm{d}}$, $r_{\\mathrm{d}}$, $t_{\\mathrm{d}}$, and $\\dot N_{\\max}$, obtaining explicit fits in the weak-dissipation regime, and demonstrate that elastic three-body interactions with $\\kappa>0$ slow dissipation and attenuate hotspot formation, offering a route to probe beyond-mean-field effects. Collectively, the results reconcile simulations with experiments, extend Zakharov's weak-collapse framework to atomic gases, and position collapse dynamics as a sensitive probe of three-body interactions in quantum fluids.

Abstract

We study the collapse of an attractive Bose-Einstein condensate, where an unstable system evolves towards a singularity, by numerically solving the underlying cubic-quintic nonlinear Schrödinger equation. We find good agreement between our simulations and the atom-loss measurements with a $^{39}$K condensate. Our simulations reveal an interplay of weak collapse and the propensity of the system to form a hotspot, and we uncover new scaling laws that govern this behavior. We also identify promising signatures of the theoretically predicted, but so far experimentally elusive, elastic three-body interactions.

Figures (8)

• Figure 1: Weak collapse. (a) Sketch of the effective energy landscape for a wavefunction of extent $\ell$. The interaction energy is $\propto \alpha/\ell^3 < 0$ and collapse occurs (arrow) for $\alpha<\alpha_{\textrm{c}}$. (b) Typical evolution of the density profile $n(r,t)$ after a quench of $\alpha$ from above to below $\alpha_{\textrm{c}}$. As the system approaches the singularity (light to dark colors), the central density diverges at a time $t_{\textrm{s}}$ (see inset), and a diminishing fraction of the wave approaches the singularity, with stationary tails left behind. (c) Simulation results for a spherical box with $\alpha_{\textrm{c}}=-3.89(1)$, $\alpha=-4.2$, and nonzero dissipation strength $\eta=3\times 10^{-4}$ [see Eq. (\ref{['eq:ndegpe']})]. We show the dynamics of the radial population density $4\pi r^2 n$ (top), the central density $n_0$ (middle), and the fractional loss $1-N$ (bottom), where $N=\int 4\pi r^2 n\, {\rm d}r \leq 1$; $t_{\textrm{c}}$ is the time when $n_0$ first peaks (note that $t_{\textrm{c}}\to t_{\textrm{s}}$ for $\eta\to0$tcvsts). The dashed line indicates $\Delta N$, the total single-collapse loss.
• Figure 2: Single-collapse atom-loss scaling laws. Plot of $\Delta N$ versus $\eta^{1/2}/\alpha^2$, comparing the experimental data (open triangles) to our simulations with different $\alpha$ (colored symbols) and $\eta$ (color bar). The inset shows the simulated $\Delta N_{t_{\textrm{c}}}$, the loss up to $t_{\textrm{c}}$, which follows the prediction in Eq. (\ref{['eq:weakcollapseloss']}) (solid line), and suggests that the different scaling of $\Delta N$ arises due to dynamics in the aftermath of the singularity; the dotted line shows $\Delta N = 2\Delta N_{t_{\textrm{c}}}$. Note that the systematic uncertainty in experimentally determining $\eta^{1/2}/\alpha^2$ is $\pm20\%$.
• Figure 3: Time-resolved loss dynamics revealing the scaling laws that govern the propensity to form a hotspot. (a) Normalized loss rate $\dot{N}/\dot{N}_{\rm max}$ versus $(t-t_{\textrm{c}})/t_{\textrm{d}}$ for $\alpha=-4.0$ and varying $\eta$; here $t_{\textrm{d}}$ is the full-width-half-maximum of $|\dot{N}(t)|$. For smaller $\eta$, the universal weak-collapse peak is accompanied by a growing 'shoulder' (for $t > t_{\textrm{c}}$). The insets show $n(r,t)$ for $\eta= 5\times 10^{-5}$ (top) and $\eta= 10^{-3}$ (bottom), where for the smaller $\eta$ the density remains high and localized for a prolonged time, a key signature of hotspot behavior; here $n_{\textrm{d}}=n(0, t_{\textrm{c}})$ and $r_{\textrm{d}}$ is defined so that $n(r_{\textrm{d}}, t_{\textrm{c}}) = n_{\textrm{d}}/2$. (b) Plotting $\dot{N}/\dot{N}_{\rm max}$ at three characteristic $(t-t_{\textrm{c}})/t_{\textrm{d}}>0$ (panels) versus $\eta^{1/2}/|\alpha|^3$ for different $\alpha$ (legend) and $\eta$ [colors, as in (a)] reveals remarkably universal behavior. (c) The loss asymmetry $A=2\Delta N_{t_{\textrm{c}}}/\Delta N-1$ also shows universal behavior in $\eta^{1/2}/|\alpha|^3$. For $\eta^{1/2}/|\alpha|^3\to 0$ we find that $A$ approaches $\approx-0.4$; the dashed line shows a linear fit for $\eta^{1/2}/|\alpha|^3<1\times 10^{-4}$.
• Figure 4: Effects of elastic three-body interactions. (a) Plot of $\Delta N$ versus $\eta^{1/2}/\alpha^2$ for $\kappa/\eta=2$ and $\kappa=0$ (colors) on log-log scale, also varying both $\alpha$ (symbols) and $\eta\in[0.3, 10] \times 10^{-4}$ as in Fig. \ref{['fig2']}. For reference, the dashed line shows $\propto \eta^{1/2}/\alpha^2$, and the $x$ error bar in the bottom right corner indicates the $\pm 20\%$ systematic uncertainty in experimentally determining $\eta^{1/2}/\alpha^2$alphackappashift. (b) Time-resolved $\dot{N}$ for different $\kappa/\eta$ (legend) and fixed $\alpha=-4.4$ and $\eta=10^{-4}$, highlighting that the dissipation time $t_{\textrm{d}}$ and hotspot behavior are highly sensitive to $\kappa/\eta$, even though the differences in $\Delta N$ are small. The main panel shows $\dot{N}/\dot{N}^{\star}_{\rm max}$ versus $(t-t_{\textrm{c}})/t_{\textrm{d}}^{\star}$, where the stars ($^{\star}$) indicate $\kappa=0$ values, while the inset shows individually normalized curves.
• Figure S1: Self-similar wavefunction evolution throughout the collapse. (a) Log-log plot of $n(r)$ at different times $t<t_{\textrm{c}}$ (light to dark colors), for $\alpha=-4$ and $\eta\approx0$. The gray dashed line shows $n(r,0)$. (b) Plot of $n_0$ versus $1/(\alpha r_0)^2$ for all our $\alpha$ (legend) and $\eta$ (color bar), evenly sampled from $-5<(t-t_{\textrm{c}})/t_{\textrm{d}}<0$ (top) and $0<(t-t_{\textrm{c}})/t_{\textrm{d}}<5$ (bottom), revealing a universal scaling trajectory. For reference, the solid lines show $n_0 \propto 1/(\alpha r_0^2)$. (c) Plot of all the normalized density profiles $n(r/r_0,t)/n_0(t)$ for the same $\alpha$, $\eta$, and $t$ as shown in (b). Each individual curve is shown with $3\%$ opacity and the dashed lines show squared Lorentzians.