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Physical Review A

Internal modes of two-dimensional quantum droplets

Liangwei Dong, Kai Shi, Changming Huang

Published Nov 3, 2022• 17 citations
Publisher

Abstract

We address the effects of internal modes on the evolution dynamics of quantum droplets in symmetric binary Bose-Einstein condensates. Angle-dependent internal modes and the corresponding eigenvalues are solved numerically for both fundamental droplets and vortex droplets with different topological charges. When internal modes corresponding to the largest eigenvalues are excited, perturbed droplets are deformed and appear as regular polygons. While neutrally internal modes make the evolution of a droplet like a breather, the internal modes with higher angular indices result in a persistently periodic rotation of droplets.

DOI
10.1103/physreva.106.053303

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We address the effects of internal modes on the evolution dynamics of quantum droplets in symmetric binary Bose-Einstein condensates. Angle-dependent internal modes and the corresponding eigenvalues are solved numerically for both fundamental droplets and vortex droplets with different topological charges. When internal modes corresponding to the largest eigenvalues are excited, perturbed droplets are deformed and appear as regular polygons. While neutrally internal modes make the evolution of a droplet like a breather, the internal modes with higher angular indices result in a persistently periodic rotation of droplets.
We discuss the stability of topological quantum droplets with the shape of two-dimensional soliton rings endowed with angular momentum that stem from a symmetric binary mixture in a Bose–Einstein condensate, with a strong trapping in one of the three spatial dimensions. We show that, in the lossless symmetric case, modeled by a Schrödinger equation with a Shannon-type nonlinear potential function, stable eigenstates can exist for arbitrarily large values of their topological charge l, provided the number of atoms is above a certain threshold. By comprehensive numerical computations, we analyze in detail cases up to l=50. We have found the perturbation modes and their eigenvalues, determining in each case which one dominates and destabilizes the solutions that lie below the stability threshold. We compare these results to the fate of the eigenstates that evolve in time. We also study the stability of the droplets under dynamical conditions by simulating collisions with potential barriers.
We predict a new type of two- and three-dimensional stable quantum droplets persistently rotating in broad external two-dimensional and weakly anharmonic potential. Their evolution is described by the system of the Gross-Pitaevskii equations with Lee-Huang-Yang quantum corrections. Such droplets resemble whispering-gallery modes localized in the polar direction due to nonlinear interactions and, depending on their chemical potential and rotation frequency, they appear in rich variety of shapes, ranging from nearly flat-top or strongly localized rotating wave packets, to crescentlike objects extending nearly over the entire range of polar angles. Above critical rotation frequency quantum droplets transform into vortex droplets (in two dimensions) or vortex tori (in three dimensions), whose topological charge gradually increase with the increase of the modulus of chemical potential, and therefore they belong to the family of nonlinear modes connecting fundamental and vortex quantum droplets. Rotating quantum droplets are exceptionally robust objects, stable practically in the entire range of their existence.
Quantum droplets represent a novel state of matter achieved by tailoring a binary Bose–Einstein condensate, where the collapse driven by self-attraction is arrested by quantum effects. This opens the possibility of producing interesting novel physical phenomena, which require accurate mathematical models for its correct description. In this work, we apply variational procedures to the mean field equation governing two-dimensional symmetric quantum droplets, to find simple yet very precise formulae for the droplet profile and the relationships between its main parameters, such as internal and external radii of the bright rings, the chemical potential or the number of atoms of the droplets. The comparison with direct numerical calculations demonstrates that our model provides a highly accurate description of the self-trapped eigenstates with flat-top profiles. These results should aid in the exploration of additional effects in liquid-like solitons and vortex states endowed with angular momentum.
Quantum droplets may form out of a gaseous Bose-Einstein condensate, stabilized by quantum fluctuations beyond mean field. Determining the ground state of a rotating binary condensate, we show that multiple singly quantized vortices may form in these droplets at moderate angular momenta in two dimensions. Droplets carrying these precursors of an Abrikosov lattice remain self-bound for certain timescales after switching off an initial harmonic confinement. Intriguingly, we find evidence of a metastable persistent current in these new types of binary condensates. We discuss how this finding can be used to experimentally generate vortex-carrying quantum droplets.
Abstract This study presents a detailed analysis of the stationary properties and dynamics of an anharmonically trapped, rapidly rotating Bose–Einstein liquid droplet. We investigate the effects of the particle number, confining potential, and rotation speed on the formation of the energetically favored bead, multiple quantized vortex, off-center vortex, and center-of-mass states. The multi-periodic trajectories and breathing provide evidence of the collective excitations of the surface mode in the vortex states. Observation of the self-trapping phenomenon and tendency toward the lowest Landau level in rapid rotation regimes coincides well with the quantum-Hall limit. Modifying the topological charges and destroying the potential flow of the vortex state can occur if an external disturbance is imposed upon the quantum droplet.
We analyze quantum droplets formed in a two-dimensional symmetric mixture of Bose-Einstein condensed atoms. For sufficiently large atom numbers, these droplets exhibit a flattop density profile with sharp boundaries governed by surface tension. Within the bulk of the droplet, traveling matter waves-localized density dips-can propagate at constant velocity while maintaining their shape. Using numerical simulations and qualitative analysis, we investigate the rich phenomenology that arises when such excitations reach the boundary of a finite droplet. We show that they can emit a small outgoing droplet, excite internal modes of the host soliton, or, in the case of vortex-antivortex pairs, split into individual vortices propagating backward near the edge. Furthermore, we demonstrate that traveling waves can be dynamically generated near the boundary through the collision of distinct droplets, and we discuss their trajectories and interactions.