[Paper Review] Self-bound vortex lattice in a rapidly rotating quantum droplet

Qi Gu, Xiaoling Cui|arXiv (Cornell University)|Jun 26, 2023

Cold Atom Physics and Bose-Einstein CondensatesPhysics and Astronomy49 references3 citations

TL;DR

This paper demonstrates a self-bound, visible triangular vortex lattice in a rapidly rotating two-dimensional Bose-Bose quantum droplet under the quantum Hall limit ($\Omega \approx \omega$). Using numerical and variational methods, it shows that quantum fluctuations (LHY corrections) stabilize the vortex lattice against melting, enabling a smooth crossover from a needled surface with small cores to a lowest-Landau-level state with extended Gaussian cores, with surface density universally dependent on $\Omega/U$.

ABSTRACT

A rapidly rotating Bose gas in the quantum Hall limit is usually associated with a melted vortex lattice. In this work, we report a self-bound and visible triangular vortex lattice without melting for a two-dimensional Bose-Bose droplet rotating in the quantum Hall limit, i.e., with rotation frequency $Ω$ approaching the trapping frequency $ω$. Increasing $Ω$ with respect to interaction strength $U$, we find a smooth crossover of the vortex lattice droplet from a needling regime, as featured by small vortex cores and an equilibrium flat-top surface, to the lowest-Landau-level regime with Gaussian-extended cores spreading over the whole surface. The surface density of such a rotating droplet is higher than that of a static one, and their ratio is found to be a universal function of $Ω/U$. We have demonstrated these results by both numerical and variational methods. The results pave the way for future experimental exploration of rapidly rotating ultracold droplets into the quantum Hall limit.

Motivation & Objective

• To investigate the stability and structure of vortex lattices in rapidly rotating 2D quantum droplets near the quantum Hall limit ($\Omega \approx \omega$).
• To determine whether self-bound droplets can sustain a visible, non-melted vortex lattice despite extreme rotation.
• To explore the role of Lee-Huang-Yang (LHY) corrections in stabilizing the vortex lattice against melting.
• To establish a universal scaling relationship between surface density and the ratio $\Omega/U$.
• To provide a theoretical and numerical framework for future experimental observation of rotating quantum droplets in the quantum Hall regime.

Proposed method

• Numerical solution of the extended Gross-Pitaevskii equation using imaginary time evolution with a split-step Fourier method on a $512 \times 512$ grid.
• Implementation of a self-bound droplet initial state via phase imprinting on a static vacuum droplet, using a triangular lattice of vortices with $C_6$ symmetry.
• Use of a complex coordinate parametrization $z = x + iy$ to define vortex positions based on the Feynman-Onsager relation and theoretical spacing $l_0 = \sqrt{\pi/(2\sqrt{3})} l_\Omega$.
• Variational approach to complement numerical results and confirm universal scaling behavior.
• Fixed-atom-number evolution with $\Delta\tau = 10^{-5}$ to ensure convergence to the steady-state vortex lattice.
• Calculation of key observables: chemical potential $\mu$, root-mean-square radius $\sqrt{\langle r^2 \rangle}$, filling factor $\nu = N/N_v$, and density profiles.

Experimental results

Research questions

• RQ1 Can a self-bound quantum droplet sustain a stable, visible vortex lattice in the quantum Hall limit ($\Omega \approx \omega$) without melting?
• RQ2 How does the vortex core structure evolve as $\Omega/U$ increases from the weak-rotation (needled) to the strong-rotation (LLL) regime?
• RQ3 What is the role of Lee-Huang-Yang (LHY) corrections in stabilizing the vortex lattice against quantum melting?
• RQ4 Is the surface density of the rotating droplet universally related to $\Omega/U$, and how does it compare to the static droplet?
• RQ5 What is the filling factor $\nu = N/N_v$ of the vortex lattice across the crossover, and how does it vary with $\Omega/U$?

Key findings

• A self-bound, visible triangular vortex lattice is stabilized in a rapidly rotating 2D quantum droplet at $\Omega \approx \omega$, with no melting observed.
• The vortex lattice undergoes a smooth crossover from a 'needled' regime ($\Omega/U = 0.008$) with small cores and flat-top surface to a lowest-Landau-level regime ($\Omega/U = 3.085$) with extended Gaussian cores.
• The surface density of the rotating droplet is universally higher than that of a static droplet, with the ratio depending only on $\Omega/U$.
• The filling factor $\nu$ ranges from $\sim 63,000$ to $\sim 150$ across the $\Omega/U$ range studied, with $\nu \approx 5780$ at $\Omega/U = 0.03$ and $\nu \approx 155$ at $\Omega/U = 1.00$.
• The root-mean-square radius $\sqrt{\langle r^2 \rangle}$ remains nearly constant ($\sim 8.4$ to $8.7$ in units of $l_\Omega$) across the crossover, indicating structural stability.
• The chemical potential $\mu$ increases from $-30.7345$ to $0.04546$ (in units of $\Omega$) as $\Omega/U$ increases, reflecting the transition from weak to strong rotation.