The bulk modulus of three-dimensional quantum droplets

TL;DR

This work addresses the elasticity of three-dimensional quantum droplets stabilized by the Lee-Huang-Yang correction within a dimensionless Gross-Pitaevskii framework. By combining variational approximations with a super-Gaussian ansatz and numerical simulations, the authors derive analytical expressions for the intrinsic vibration frequency $\Omega$ and the bulk modulus $B$, and verify them against BdG calculations. They introduce the ratio $\eta=B/\Omega^{2}$ and connect it to system parameters, using both VA and Thomas-Fermi approximations to show how $\eta$ scales with atom number $\mathcal{N}$ and attraction strength $-g$, enabling practical estimation of BM from vibrational measurements. The results reveal that BM can be extremely small ($<1~\mathrm{μPa}$), indicating ultra-soft quantum elasticity governed by the LHY effect, and suggest experimental routes for probing and exploiting this elasticity in quantum fluids.

Abstract

Quantum droplets (QDs), formed by ultra-dilute quantum fluids under the action of Lee-Huang-Yang (LHY) effect, provide a unique platform for investigating a wide range of macroscopic quantum effects. Recent studies of QDs' breathing modes and collisional dynamics have revealed their compressibility and extensibility, which suggests that their elasticity parameters can be identified. In this work, we derive the elastic bulk modulus (BM) of QDs by means of the theoretical analysis and numerical simulations, and establish a relation between the BM and the eigenfrequency of QD's intrinsic vibrations. The analysis reveals the dependence of the QD's elasticity on the particle number and the strength of interparticle interactions. We conclude that the BM of QDs can be less than $1~\mathrm{μPa}$, implying that QDs are ultra-soft quantum elastic media. These findings suggest new perspectives for realizing elastic media governed by the LHY effect.

Figures (5)

• Figure 1: (a) The radial density distribution of the stationary isotropic QD for $\mathcal{N}=250$ and $g=-6$. The blue solid curve represents the numerical result, while the purple dashed curve shows the VA prediction, obtained with the super-Gaussian ansatz. (b,c) Heatmaps of the VA-predicted values of the eigenfrequency of the internal oscillations $\Omega (\mathcal{N},g)$ and BM $B(\mathcal{N},g)$ [see Eqs. (\ref{['Omega2']}) and (\ref{['VA-Bulkmodulus']})], in the plane of norm $\mathcal{N}$ and reduced MF interaction strength $g$. The color shading from light to dark indicates increasing values of $\Omega$ and $B$, with the black dashed curves representing their contour lines. The two white dashed lines correspond to $\mathcal{N}=250$ and $g=-6$, which represent the cases shown in Figs. \ref{['B-omega-gN']} (a,c) and (b,d), respectively.
• Figure 2: The quench dynamics of the QD with $\mathcal{N}=250$ and $g=-6$, inititated by the weak perturbation, $g\rightarrow g+\delta g$ with $\delta g/g=0.01$. Panels (a,b) show the evolution of chemical potential $\mu$ and volume $V$, respectively, while panel (c) depicts the corresponding trajectory in the $\left( V,\mu \right)$ plane, which is actually a straight line. In (a,b), red dashed lines mark two adjacent peaks of $\mu$ and $V$, corresponding to an oscillation period of $t=0.4$.
• Figure 3: (a,b) and (c,d) The oscillation eigenfrequency $\Omega$ and BM $B$, respectively, vs. $g$ and $\mathcal{N}$. Green solid lines represent the VA results, while chains of blue solid spheres denote the numerical results. The red triangles in panels (a,b) correspond to the BdG calculations. In panels (a,c), the norm is fixed as $\mathcal{N}=250$, whereas in panels (b,d), the MF attraction strength is fixed as $g=-6$.
• Figure 4: (a) The heatmap of values $\eta_{\mathrm{VA}} (\mathcal{N},g)$ of the BM/eigenfrequency ratio (\ref{['eta_formula']}). The color shading from light to dark indicates increasing values of $\eta_{\mathrm{VA}}$, while the black dashed curves correspond to the contour lines of $\eta_{\mathrm{VA}}$. The vertical and horizontal white dashed lines indicate $\mathcal{N}=250$ and $g=-6$, corresponding to the cases shown in panels (b) and (c), respectively. Panels (b,c) display the dependence of $\eta$ on $g$ and $\mathcal{N}$, where the green solid lines represent the VA result (\ref{['eta_relation']}), the purple dashed-dotted line correspond to the TF approximation given by Eq. (\ref{['eta-TF']}), and the chain of red triangles denotes the numerical results.
• Figure 5: The relation between $r_{\mathrm{TF}}$ and $\bar{r}$.