Ground-state solution of quantum droplets in Bose-Bose mixtures

Abstract

In this paper, we present a systematic study on the ground state computation of quantum droplets in homonuclear Bose-Bose mixtures, governed by the extended Gross-Pitaevskii equations (eGPEs) with Lee-Huang-Yang (LHY) corrections. This model captures the formation of self-bound droplets stabilized by the delicate balance between the attractive mean-field interaction and the repulsive quantum fluctuations. We formulate dimensionless energy functionals for both the general two-component system and the reduced single-component density-locked model. To compute the ground states efficiently, we adapt and benchmark various gradient flow discretization schemes, identifying a backward-forward sine-pseudospectral scheme based on the gradient flow with Lagrange multiplier method (GFLM-BFSP) as the robust solver for our simulations. Utilizing this method, we report three main numerical observations: (i) the density-locked model is quantitatively validated as a reliable approximation for ground state properties; (ii) the dimension-dependent convergence rates of the Thomas-Fermi approximation are established in the strong-coupling regime; and (iii) the critical particle number for self-binding in free space is numerically determined, providing a precise correction to the analytical prediction by Petrov [Phys. Rev. Lett. 115, 155302 (2015)].

Figures (4)

• Figure 1: Accuracy validation of the single-component density-locked model compared to the full two-component GPE. The left panels show the relative energy error $\mathcal{E}_E$, and the right panels show the relative $L^2$-norm wavefunction error $\mathcal{E}_{\phi}$. The results demonstrate that the reduced model maintains high precision ($\sim 10^{-3}$ relative error) across a wide range of experimentally relevant parameters.
• Figure 2: Ground state profiles of 1D density-locked droplets. (a) Wave function $\phi_g(r)$ in free space with scaling parameters $\omega=50$ and $\omega_{\perp}=100$. (b) Wave function $\phi_g(r)$ under a harmonic trap with $\omega_r=20$ and $\omega_{\perp}=2000$. In both panels, the six curves correspond to particle numbers $N$ starting from $2.5 \times 10^4$ and doubling successively up to $8 \times 10^5$. The profiles with wider spatial support correspond to larger particle numbers.
• Figure 3: Ground state profiles of 2D radially symmetric droplets. (a) Radial wave function $\phi_g(r)$ in free space with scaling parameters $\omega=50$ and $\omega_z=100$. (b) Radial wave function $\phi_g(r)$ under a harmonic trap with $\omega_r=20$ and $\omega_z=2000$. The curves correspond to particle numbers $N$ starting from $10^5$ and doubling successively up to $2.56\times10^7$ in (a), and from $10^4$ up to $1.28\times10^6$ in (b). The profiles with wider spatial support correspond to larger particle numbers.
• Figure 4: Ground state profiles of 3D spherically symmetric droplets. (a) Radial wave function $\phi_g(r)$ in free space with scaling parameter $\omega=50$. (b) Radial wave function $\phi_g(r)$ under a harmonic trap with $\omega_r=100$. In both panels, the curves correspond to particle numbers $N$ starting from $4\times10^5$ and doubling successively up to $1.28\times 10^7$. The profiles with wider spatial support correspond to larger particle numbers.

Theorems & Definitions (13)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 3.1
• Remark 3.2
• Theorem 3.3
• Proof 1
• Theorem 3.4: bao2004computing
• Remark 3.5
• Remark 3.6