Static and dynamical properties of quadrupolar quantum droplets in quasi-2D condensates

TL;DR

This work addresses the stabilization, structure, and dynamics of quadrupolar quantum droplets in a quasi-2D two-component BEC by combining nonlocal quadrupole–quadrupole interactions (QQIs) with beyond-mean-field Lee–Huang–Yang (LHY) corrections. A symmetric reduction yields a single-component extended GPE with a nonlocal QQI kernel $R(\mathbf{r}-\mathbf{r}')$ and a density-dependent LHY term, analyzed via Thomas–Fermi baselines and full eGPE simulations to characterize stationary states and collisions. The authors find flat-top density profiles, an incompressible self-bound liquid with $A_{\mathrm{eff}} \propto N$, and density/chemical potential saturation at large $N$, with vortex droplets ($S=1$) exhibiting anisotropic elliptical shapes and a finite norm threshold $N_{cr} \approx 140$. Collision dynamics reveal rich behaviors: ground-state droplets transition from inelastic merging to quasi-elastic scattering and then to quantum penetration with increasing impact velocity, while vortex droplets display phase-induced repulsion, fragmentation, and topologically protected tunneling, highlighting the versatility of QQIs in shaping anisotropic, topological quantum fluids and guiding future experiments with polar molecules.

Abstract

Quantum droplets, stabilized by beyond-mean-field effects, represent a novel state of matter in quantum many-body systems. While previous studies have focused primarily on dipolar and contact-interacting systems, quadrupolar condensates remain relatively unexplored. In this work, we explore the formation, structural properties, and dynamical behaviors of quantum droplets in a two-component quadrupolar Bose-Einstein condensate confined to a quasi-two-dimensional geometry. Analytical results obtained via the Thomas-Fermi approximation predict flat-topped density profiles and linear scaling between effective area and particle number. These predictions are corroborated by numerical simulations, which also reveal the saturation of peak density and chemical potential at large norm. Furthermore, vortex quantum droplets exhibit anisotropic elliptical morphologies due to the directional nature of QQIs, with their aspect ratios significantly tunable by varying the particle number and quadrupolar interaction strength. Collision dynamics demonstrate rich behavior modulated by velocity and topology: ground-state droplets transition from inelastic merging to quasi-elastic scattering and quantum penetration, while vortex droplets exhibit phase-induced repulsion, fragmentation, and topologically protected tunneling. These findings offer a comprehensive understanding of how higher-order interactions and quantum fluctuations govern the formation and stability of quadrupolar droplets. This work lays a theoretical foundation for experimental realization and opens new directions for exploring anisotropic quantum fluids, topological excitations, and applications in quantum sensing and simulation.

Figures (7)

• Figure 1: (Color online) Schematic illustration of the proposed experimental setup for realizing quadrupolar quantum droplets in a quasi-2D two-component Bose–Einstein condensate. Molecules with intrinsic electric quadrupole moments are uniformly aligned along the vertical $z$-axis by a spatially varying external electric field generated by a tapered capacitor. The system is tightly confined in the axial direction, resulting in effective two-dimensional dynamics in the $x-y$ plane. Each quadrupole is modeled as a dipole–antidipole pair (green up and down arrows), and the tunability of the QQI is achieved via the spatial modulation of the field gradient. This configuration allows for the exploration of self-bound quantum droplet states stabilized by the interplay between long-range attractive QQIs and repulsive quantum fluctuations. Similar mechanisms involving field-induced long-range interactions and spin-orbit coupling have been shown to support stable and even excited solitonic states in other two-dimensional condensate systems Huang2018Jiang2016.
• Figure 2: (Color online) Stationary properties of ground-state quantum droplets (QDs) under attractive QQIs and their dependence on the quadrupolar interaction strength. (a–c) Peak density $I_{\mathrm{max}}$, chemical potential $\mu$, and effective area $A_{\mathrm{eff}}$ as functions of the total norm $N$ at fixed quadrupolar strength $\kappa = 0.05$. With increasing $N$, $I_{\mathrm{max}}$ and $\mu$ gradually saturate, while $A_{\mathrm{eff}}$ grows approximately linearly in the large-$N$ regime, consistent with the incompressibility characteristic of quantum droplets. (d-f) Variation of $I_{\mathrm{max}}$, $\mu$, and $A_{\mathrm{eff}}$ as functions of $\kappa$ at fixed norm $N = 500$. As $\kappa$ increases, $I_{\mathrm{max}}$ increases and $A_{\mathrm{eff}}$ decreases, indicating stronger spatial localization of the droplets. Meanwhile, the chemical potential $\mu$ exhibits a monotonically decreasing trend. The red dashed curves indicate the analytical predictions based on the Thomas--Fermi approximation derived in Eqs. (\ref{['Eq:ne']}) and (\ref{['Eq:mue']}), and valid primarily in the large-$N$ regime.
• Figure 3: (Color online) Stationary properties of vortex quantum droplets (vQDs) under attractive quadrupole–quadrupole interactions (QQIs) and the influence of the interaction strength. (a1–a3) Peak density $I_{\mathrm{max}}$, chemical potential $\mu$, and effective area $A_{\mathrm{eff}}$ as functions of total norm $N$ at fixed quadrupolar strength $\kappa = 0.05$. The peak density and chemical potential exhibit gradual saturation as $N$ increases. Compared to the ground-state droplets, although it’s not obvious, vQDs feature a slightly larger effective area $A_{\mathrm{eff}}$ at the same $N$, resulting from the central density depletion that displaces particles outward. (b1–b3) Dependence of $I_{\mathrm{max}}$, $\mu$, and $A_{\mathrm{eff}}$ on the quadrupolar interaction strength $\kappa$ at a fixed norm $N = 500$. With increasing $\kappa$, $I_{\mathrm{max}}$ increases while $\mu$ decreases monotonically, indicating enhanced self-binding. Meanwhile, $A_{\mathrm{eff}}$ shrinks due to the stronger attractive interactions. Red dashed lines represent theoretical predictions from the TF approximation, using Eqs. (\ref{['Eq:ne']}) and (\ref{['Eq:mue']}).
• Figure 4: (Color online) Semi-major ($a$) and semi-minor ($b$) axes of the inner vortex core and outer boundary of vortex quantum droplets. Panels (a) and (b) show the dependence on total norm $N$ at fixed quadrupolar strength $\kappa = 0.05$, while panels (c) and (d) illustrate the dependence on $\kappa$ at fixed $N = 1500$. As $N$ increases, the outer boundary expands monotonically, whereas the core size remains nearly unchanged. In contrast, increasing $\kappa$ leads to overall droplet contraction, with the outer boundary shrinking more noticeably than the vortex core. The anisotropic nature of QQIs is evident in (c,d), where the semi-minor axis ($b$) exhibits a more pronounced reduction than the semi-major axis ($a$). In all panels, blue solid lines denote the inner boundary dimensions, and red dashed lines denote the outer boundary dimensions.
• Figure 5: (Color online) Typical examples of dynamically stable ground-state and vortex-state quantum droplets (QDs) under attractive QQIs. (a1) Density distribution of the ground-state QD corresponding to point A in Fig. \ref{['Properties_GS']}, with parameters $N=1500$ and $\kappa=0.05$. (b1) Density distribution of the vortex-state QD under the same parameters, corresponding to point B in Fig. \ref{['Properties_VS']}. A clear central density depletion is observed due to the embedded vortex core. The inset in (b1) shows the corresponding phase pattern after real-time evolution, featuring a $2\pi$ phase winding that confirms the presence of a singly charged vortex. (a2) and (b2) show the results of real-time simulations with 1% random perturbations added to the initial states shown in (a1) and (b1), respectively. The persistent density profiles over time confirm that both the ground-state and vortex-state QDs exhibit robust dynamical stability.