Effective theory of surface oscillations in self-bound superfluid droplets

Abstract

We investigate the low-energy dynamics of small-amplitude surface oscillations of spherical superfluid droplets in vacuum. Starting from the effective field theory of superfluid phonons, we derive an effective action governing the surface oscillations under a fixed particle-number constraint. The normal-mode eigenfrequencies $ω_{\ell}$ for each angular momentum quantum number $\ell$ are determined and shown to depend on a dimensionless parameter measuring the ratio of surface tension to bulk compressibility energy. We identify a critical value of this parameters at which the breathing mode ($\ell = 0$) becomes mechanically unstable, and show that all multipole surface modes with $\ell \geq 2$ enter the low-energy regime when the surface tension is sufficiently small. Within this regime, we further quantize the surface oscillations, whose quanta correspond to ripplons, allowing the construction of general multi-ripplon states obeying angular-momentum selection rules. We also apply our formalism to a concrete example: a weakly interacting two-component Bose mixture realizing a self-bound superfluid droplet. The resulting description is universal in the sense that it applies to surface dynamics of generic nonrelativistic superfluids with a free interface, independent of microscopic details.

Figures (4)

• Figure 1: Schematic cross-sectional view of the droplet, illustrating the time-dependent radius $R(t,\theta,\phi)$, the equilibrium radius $R_0$, and the radial displacement field $u(t,\theta,\phi)$.
• Figure 2: Schematic illustration of the surface deformation modes. (a) The original surface ($|\alpha_{\ell,m}|=0$). (b) Deformed surface with $|\alpha_{\ell,m}|=0.3$, illustrating (i) the breathing mode ($\ell=0, m=0$), (ii) the quadrupole mode ($\ell=2, m=0$), and (iii) the octupole mode ($\ell=3, m=0$).
• Figure 3: Numerical solutions to Eq. \ref{['eq:frequency-condition']} of the dimensionless frequencies $z_\ell$ as functions of $\ell$ for $\xi=0.1$ (top-left), $\xi=0.5$ (top-right) and $\xi=1.0$ (bottom-left). Different markers correspond to distinct normal-mode branches, and the dashed line indicates the low-frequency approximation given by Eq. \ref{['eq:low-energy-freq']}.
• Figure 4: Numerical solutions of the dimensionless frequencies $z_\ell$ as functions of $\xi$ for $\ell=0$ (top-left) , $\ell=1$ (top-right) , $\ell=2$ (bottom-left) and $\ell=3$ (bottom-right). Different curves correspond to distinct normal-mode branches, and the dashed curve indicates the low-frequency approximation given by Eq. \ref{['eq:low-energy-freq']}]. For $\ell=0$, the lowest branch vanishes at $\xi=3$. In the upper-right figure for $\ell=1$, both the first branched solution and the approximate solution are always zero and lie exactly on the same line.