The spectral properties and dynamical behaviors of three-dimensional quantum droplets loaded in an optical lattice within a Lee–Huang–Yang fluid are investigated. Using a beyond-mean-field model combined with variational analysis and numerical simulations, we explore the system's ground states and dynamics. Our results demonstrate that lattice anisotropy and quantum fluctuations effectively tune the excitation spectrum and lift the degeneracy of the breathing modes. Through Floquet dynamics, we identify distinct nonlinear features, including higher-order harmonics and frequency-mixing modes. Quench dynamics studies show center-of-mass oscillations characterized by a linear frequency and an additional nonlinear frequency. Additionally, we find that the collision dynamics of droplet pairs strongly depend on their relative phase, resulting in diverse dynamical behaviors. These findings provide valuable insights for understanding and manipulating quantum droplets in optical lattices, offering some reference for exploring quantum fluctuations in periodically confined geometries.
© 2020 Elsevier B.V. We study two-dimensional (2D) vortex quantum droplets (QDs) trapped by a thicker transverse confinement with a⊥>1μm. Under this circumstance, the Lee-Huang-Yang (LHY) term should be described by its original form in the three-dimensional (3D) configuration. Previous studies have demonstrated that stable 2D vortex QDs can be supported by a thin transverse confinement with a⊥≪1μm. In this case, the LHY term is described by a logarithm. Hence, two kinds of confinement features result in different mechanisms of the vortex QDs. The stabilities and characteristics of the vortex QDs must be re-identified. In the current system, we find that stable 2D vortex QDs can be supported with topological charge number up to at least 4. We reformulated their density profile, chemical potential and threshold norm for supporting the stable vortex QDs according to the new condition. Unlike the QDs under thin confinement, the QDs in the current system strongly repel each other because the LHY term features a higher-order repulsion than that of the thin confinement system. Moreover, elastic and inelastic collisions between two moving vortex QDs are studied throughout the paper. Two kinds of collisions can be characterized by exerting different values of related speed. The dynamics of the stable nested vortex QD, which is constructed by embedding one vortex QD with a smaller topological number into another vortex QD with a larger number of topological charge, can be supported by the system.
It was recently found that the Lee-Huang-Yang (LHY) correction to the mean-field Hamiltonian of binary atomic boson condensates suppresses the collapse and creates stable localized modes (two-component quantum droplets, QDs) in two and three dimensions (2D and 3D). In particular, the LHY effect modifies the effective Gross-Pitaevskii equation (GPE) in 2D by adding a logarithmic factor to the usual cubic term. In the framework of the accordingly modified two-component GPE system, we construct 2D self-trapped modes in the form of QDs with vorticity
S embedded into each component. Because of the effect of the logarithmic factor, the QDs feature a flat-top shape, which expands with the increase of
S and norm
N. An essential finding, produced by a systematic numerical investigation and analytical estimates, is that the vortical QDs are stable (which is a critical issue for vortex solitons in nonlinear models) up to
S=5, for
N exceeding a certain threshold value, which is predicted to scale as ${N}_{\mathrm{th}}\ensuremath{\sim}{S}^{4}$ for large
S (for three-dimensional QDs, the scaling is ${N}_{\mathrm{th}}\ensuremath{\sim}{S}^{6}$). The prediction is corroborated by numerical findings. Pivots of QDs with $S\ensuremath{\ge}2$ are subject to structural instability, as specially selected perturbations can split the single pivot in a set of
S or
S+2 pivots corresponding to unitary vortices; however, the structural instability remains virtually invisible, as it occurs in a broad central ``hole'' of the vortex soliton, where values of fields are very small, and it does not cause any dynamical instability. In the condensate of
39K atoms, in which QDs with
S=0 and a quasi-2D shape were created recently, the vortical droplets may have radial size $\ensuremath{\lesssim}30\phantom{\rule{0.16em}{0ex}}\ensuremath{\mu}\mathrm{m}$, with the number of atoms in the range of
104−−105. The role of three-body losses is considered too, demonstrating that they do not prevent the creation of the vortex droplets but may produce a noteworthy effect, leading to sudden splitting of ``light'' droplets. In addition, hidden-vorticity (HV) states in QDs, with topological charges ${S}_{+}=\ensuremath{-}{S}_{\ensuremath{-}}=1$ in their components, which are prone to strong instability in other settings, have their stability region too. Unstable HV states tend to spontaneously merge into zero-vorticity solitons. Collisions of QDs, which may lead to their merger, and dynamics of elliptically deformed QDs (which form rotating elongated patterns or ones with oscillations of the eccentricity) are briefly considered too.
We predict a scheme for the creation of isotropic three-dimensional quantum droplets in Rydbeg-dressed Bose gases, which contain both repulsive contact interactions and attractive soft-core interactions. Via manipulating laser detuning, the droplet size and particle number density could be engineered. Quantum fluctuation induced Lee-Huang-Yang (LHY) correction should be considered to stabilize droplets when LHY energy is comparable to the mean-field energy. Self-trapped droplets possess flat-top density distribution for a large particle number, and the collective excitations are calculated to determine the self-evaporation threshold. We further distinguish quantum droplets from solitons according to condensate compressibility. Finally, the dynamic stability of droplets is examined by pairwise-droplet collision. Our system could realize the most dilute quantum droplets, and establish a promising platform for further simulating topological nontrivial droplets. Published by the American Physical Society 2024
This brief review summarizes recent theoretical and experimental results which predict and establish the existence of quantum droplets (QDs), i.e., robust two- and three-dimensional (2D and 3D) self-trapped states in Bose-Einstein condensates (BECs), which are stabilized by effective self-repulsion induced by quantum fluctuations around the mean-field (MF) states [alias the Lee-Huang-Yang (LHY) effect]. The basic models are presented, taking special care of the dimension crossover, 2D → 3D. Recently reported experimental results, which exhibit stable 3D and quasi-2D QDs in binary BECs, with the inter-component attraction slightly exceeding the MF self-repulsion in each component, and in single-component condensates of atoms carrying permanent magnetic moments, are presented in some detail. The summary of theoretical results is focused, chiefly, on 3D and quasi-2D QDs with embedded vorticity, as the possibility to stabilize such states is a remarkable prediction. Stable vortex states are presented both for QDs in free space, and for singular but physically relevant 2D modes pulled to the center by the inverse-square potential, with the quantum collapse suppressed by the LHY effect.
We investigate the formation, stability, and dynamics of three-dimensional ring-shaped and multipole vortical quantum droplets (QDs) in nonrotating dipolar Bose-Einstein condensates held in a toroidal trapping potential. The QD dynamics are investigated in the framework of the extended Gross-Pitaevskii equation, which includes long-range dipole-dipole interactions (DDIs) and the beyond-mean-field Lee-Huang-Yang (LHY) term, revealing the emergence of self-bound states. Stable stationary solutions for multipole QDs with different values of the topological charge (vorticity S) are shaped as necklacelike modes, with the number of "beads" (multipole order) n=2S, up to S=6. The stability area of the multipoles shrinks with the increase of S. For higher values of S the centrifugal effect associated with the phase winding destabilizes the ring-shaped QDs and drives the formation of fragmented multipole droplet states. The dependence of the chemical potential, total energy, and peak density on the norm (number of particles) and S is produced. These findings uncover the stabilizing effect of the LHY correction and DDI anisotropy in maintaining complex QD states in the nonrotating configurations.