Generalized Gross-Pitaevskii Equation for 2D Bosons with Attractive Interactions

TL;DR

This work introduces a generalized Gross-Pitaevskii equation (GPE) for 2D attractive Bose systems by incorporating a density-dependent coupling $g$, capturing universal droplet physics and enabling analysis beyond the ground state. The authors derive and validate a practical form $g \approx -\frac{4\pi}{\ln\left(\alpha\,|\psi|^2/B_2\right)}$ with $\alpha\approx 2.607$, leading to a tractable GPE with an effective coupling $G = g + g^2/(8\pi)$. They develop and benchmark numerical and analytical tools, including an IM-SRG flow equation framework for cross-validation against the GPE, a robust numerical scheme with a dynamic grid for the GPE, and a variational Townes-soliton approach. The study predicts universal excited states (e.g., vortices), analyzes breathing and quench dynamics in traps, and establishes a unified framework for exploring static and dynamical 2D universality in attractive Bose gases, with implications for future experiments. Overall, the work provides a cohesive theoretical platform linking droplets, vortex states, and non-equilibrium phenomena in 2D attractive bosonic systems.

Abstract

We introduce and investigate a generalized Gross-Pitaevskii equation for two-dimensional (2D) attractive Bose systems. First, we demonstrate that it accurately captures key properties of universal bound states in free space commonly referred to as quantum droplets. We then apply the framework to predict the existence of universal excited states, including vortex configurations, which may be more accessible to experimental investigation than the ground state. Additionally, we investigate breathing modes and quench dynamics in trapped geometries. Our results establish a robust theoretical foundation for exploring both static and dynamical phenomena in finite 2D Bose systems, offering guidance for the design of future experimental protocols.

Figures (10)

• Figure 1: (a) Energy, $\ln \abs{E_N/B_2}$, as a function of the number of bosons $N$ for the lowest energy states with $s=0$ and $s=1$ in the limit $W\to0$. The solid green line shows numerical solution of Eq. \ref{['eq:GPE']} for $s=0$; the black dashed line shows the expected asymptote $\ln \abs{E_N/B_2}=\frac{4\pi N}{\mathcal{G}}-1.91$Petrov2024. The red dots demonstrate the numerically exact solution of Eq. \ref{['eq:eq1']} available for 'small' values of $N$BazakIOP2018. The blue line shows the lowest energies for $s=1$. The blue dashed line is a linear fit to this result, $\ln \abs{E_N/B_2}=0.5174N+1.218$. The insets sketch the radial densities for the states. (b) The lowest energy solution of Eq. \ref{['eq:GPE']} for weak/intermediate interactions (solid lines) together with the numerical solution to Eq. \ref{['eq:eq1']} obtained using flow equations for bosons (dots with error bars).
• Figure 2: (a) The density profile in the transition between the weakly- and strongly-interacting limits. The dimensionless density, $R^2n(r)$, for the ground state with $s=0$ and $N=50$ is plotted as a function of $r/R$ for different interaction strengths $\ln B_2/N$. The dotted black line shows the non-interacting harmonic oscillator solution, and the dashed line represents the Townes-soliton profile. (b) Energy, $E_N$, as a function of the interaction parameter $\ln B_2/N$. The solid curves show the exact numerical result, the dashed black curves present the result of minimization with respect to the Townes-soliton profile. Different curves correspond to different numbers of particles. The vertical dotted line shows the transition point in the limit $N\to\infty$Tononi2024Brauneis2024.
• Figure 3: Breathing mode frequencies $\Omega$ as a function of interaction strength $\ln B_2/N$ for different numbers of bosons $N$ in (a) strong and (b) weak interaction regimes. Dots demonstrate frequencies from the numerical solutions. The dashed curves show analytical results valid for $N\gg1$: (a) $3.8|E_N|/\sqrt{N}$; (b) $2+N/(\ln B_2)^2$.
• Figure 4: Quench dynamics for systems with $N=20$ (a-b) and $50$ (c-d) from a non-interacting system to the interaction strength $\ln B_2/N=-3.0$ and $-2.14$ . Panels (a,c) demonstrate the size of the system, $R^2$, as a function of time $t$. Panels (b,d) demonstrate the Fourier transforms of the $R^2$ signal (dots). To facilitate interpretation, we fit each peak with a Gaussian function (solid lines with shaded areas).
• Figure S1: (a) Energy $\ln |E_N/B_2|$, (b) energy difference to expected asymptote Petrov2024$\Delta \ln |E_N/B_2|=\ln |E_N/B_2|-(4N/C+c_1)$, (c) energy ratio $E_{N+1}/E_N$ for different $\alpha(N)$. Solid lines represent solutions of the generalized GPE. The black dashed line represents the solution of Ref. Petrov2024, and the blue dots with a dotted line show the results of Ref. BazakIOP2018. In panel (b), the dashed colored lines show $\Delta \ln |E_N/B_2|$ where $E_N/B_2$ is calculated from Eq. \ref{['eq:BN_alpha']} with $\kappa=1.448$.