Abstract We address the existence and stability of one-dimensional (1D) holes and kinks and two-dimensional (2D) vortex-holes nested in extended binary Bose mixtures, which emerge in the presence of Lee–Huang–Yang (LHY) quantum corrections to the mean-field energy, along with self-bound quantum droplets. We consider both the symmetric system with equal intra-species scattering lengths and atomic masses, modeled by a single (scalar) LHY-corrected Gross–Pitaevskii equation (GPE), and the general asymmetric case with different intra-species scattering lengths, described by two coupled (spinor) GPEs. We found that in the symmetric setting, 1D and 2D holes can exist in a stable form within a range of chemical potentials that overlaps with that of self-bound quantum droplets, but that extends far beyond it. In this case, holes are found to be always stable in 1D and they transform into pairs of stable out-of-phase kinks at the critical chemical potential at which localized droplets turn into flat-top states, thereby revealing the connection between localized and extended nonlinear states. In contrast, we found that the spinor nature of the asymmetric systems may lead to instability of 1D holes, which tend to break into two gray states moving in the opposite directions. Remarkably, such instability arises due to spinor nature of the system and it affects only holes nested in extended modulationally-stable backgrounds, while localized quantum droplet families remain completely stable, even in the asymmetric case, while 1D holes remain stable only close to the point where they transform into pairs of kinks. We also found that symmetric systems allow fully stable 2D vortex-carrying single-charge states at moderate amplitudes, while unconventional instabilities appear also at high amplitudes. Symmetry also strongly inhibits instabilities for double-charge vortex-holes, which thus exhibit unexpectedly robust evolutions at low amplitudes.
We study nonlinear holes, bubbles, kink fronts, and quantum-droplet-like states in an elongated Bose mixture governed by an extended Gross-Pitaevskii equation with attractive cubic mean-field interactions and repulsive quartic Lee-Huang-Yang contribution. We classify the constant-amplitude backgrounds: the Bogoliubov-de Gennes spectrum shows that the upper branch is modulationally stable, whereas the lower branch is unstable to long-wavelength perturbations. Grand-potential and pressure determine the vacuum-finite-density coexistence point and the unique background supporting stationary kink and antikink fronts. Symmetric holes on the stable background have a density deficit and integral width that diverge near coexistence as the hole separates into two nonlinear fronts. A multiplicative kink-antikink construction connects bell-shaped states to broad flat-top droplets and approaches the stationary quantum-droplet family as the front separation increases. An antikink-kink sum with equal signs generates bubble-like density depletions without a phase jump, whereas their difference produces dark-hole-like profiles with a
π phase difference between the asymptotic backgrounds. Bubble depth increases with front separation, distinguishing gray-like and dark-like profiles. Real-time simulations show that deep bubbles persist under weak random perturbations, demonstrating finite-time robustness without establishing spectral stability. In the asymmetric two-component model, localized relative-density modes can induce spinor-driven breakup on a modulationally stable background, while robust holes persist near the kink limit. Collisions of kink-antikink droplet-like states display phase-dependent merging, asymmetric particle transfer, and effective reflection. These results connect nonlinear fronts with bubbles, phase-jump holes, and self-bound droplets in beyond-mean-field Bose mixtures.
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.
The dynamics of a two-dimensional Bose-Einstein condensate mixture, loaded into a dual-core trap, when the beyond-mean-field effects are taken into account, are considered. The effects of quantum fluctuations are described by the Lee-Huang-Yang correction terms in the extended coupled Gross-Pitaevskii equations (GPE). The spatially uniform and inhomogeneous condensate cases are studied. In the first case, the parameter regimes associated with macroscopic quantum tunneling, macroscopic self-trapping, and revival-like localization dynamics are found. The Josephson oscillation frequencies for both the zero-phase and the π -phase modes are derived. As the total atom number is varied, the dynamics exhibit a nontrivial bifurcation structure: along the zero-phase branch, two successive pitchfork bifurcations generate bistability and hysteresis, while the π -phase branch shows a single pitchfork bifurcation. In the second case, the Josephson dynamics for quantum droplets and vortices are investigated. The analytical predictions for the oscillation frequencies of the atomic population between quantum droplets are found, and results are validated by direct numerical simulations of the system of coupled extended GPE. The existence of the Andreev-Bashkin nondissipative drag through simulations of droplet-droplet interactions is shown. The Josephson dynamics of vortex states are studied. It was found that vortices with topological charge S and sufficiently small particle number are typically unstable, breaking up into S + 1 (and occasionally S + 2 ) fundamental, nonvortical fragments, with the time to breakup increasing as the particle number grows. It is also found that unstable asymmetric vortices exhibit the splitting and/or crescentlike instability. For vortices with sufficiently large norms, long-time simulations confirm robust stability against small perturbations. In this stable regime, the properties of Josephson oscillations and Andreev-Bashkin-type entrainment for vortex states with charges S = 1 , 2 , and 3 are investigated.
Abstract The static and dynamic properties of self-bound quantum droplets in a one-dimensional Bose-Bose mixture are discussed in the spirit of the Hartree–Fock-Bogoliubov theory. This latter enables us to provide beyond the Lee-Huang-Yang (LHY) quantum corrections to the equation of state at both zero and finite temperatures. In the uniform case our results for the ground-state energy and the critical temperature are confirmed through comparison with Quantum Monte-Carlo simulation and with available theoretical results. The density profiles are supported by numerical simulations of the generalized Gross-Pitaevskii equation which selfconsistently includes higher-order terms originating from the normal and anomalous fluctuations under the local density approximation. We show that the density exhibits a dip near its center in the flat-top plateau region for large interspecies interactions. We exemplify the impact of the beyond LHY corrections on the spatiotemporal evolution of the self-bound droplet in the presence of excitation induced by periodic density modulation. It is found that higher-order corrections may lead to the formation of a train of small droplets. We then extend our study for the case of inhomogeneous droplets in quasi one-dimensional Bose mixtures.
We study two-dimensional (2D) matter-wave solitons in spinor Bose–Einstein condensates under the action of the spin–orbit coupling and opposite signs of the self- and cross-interactions. Stable 2D two-component solitons of the mixed-mode type are found if the cross-interaction between the components is attractive, while the self-interaction is repulsive in each component. Stable solitons of the semi-vortex type are formed in the opposite case, under the action of competing self-attraction and cross-repulsion. The solitons exist with the total norm taking values below a collapse threshold. Further, in the case of the repulsive self-interaction and inter-component attraction, stable 2D self-trapped modes, which may be considered as quantum droplets (QDs), are created if the beyond-mean-field Lee–Huang–Yang terms are added to the self-repulsion in the underlying system of coupled Gross–Pitaevskii equations. Stable QDs of the mixed-mode type, of a large size with an anisotropic density profile, exist with arbitrarily large values of the norm, as the Lee–Huang–Yang terms eliminate the collapse. The effect of the spin–orbit coupling term on characteristics of the QDs is systematically studied. We also address the existence and stability of QDs in the case of SOC with mixed Rashba and Dresselhaus terms, which makes the density profile of the QD more isotropic. Thus, QDs in the spin-orbit-coupled binary Bose–Einstein condensate are for the first time studied in the present work.
Bose-Einstein condensation(BEC) has been a research hotspot in cold atom physics for the past few decades. Recently, experiments on BECs have reported the observation of liquid-like quantum droplets, attracting significant interest. The physics for forming quantum droplets can be explained by zero-point quantum fluctuation which effectively prohibit the collapse caused by the mean-field interactions between the atoms. The stabilization mechanism beyond the mean-field theory, induced by quantum fluctuations, can be described by the Lee–Huang–Yang(LHY) correction. Normally, forming stable 2D or 3D solitons in free space is a challenging problem. Additionally, an intriguing question is how to generate 2D or 3D solitons with embedded vorticity. The LHY stability mechanism offers a new approach to producing stable solitons and vortex solitons in BECs. In this thesis, two-dimensional vortex quantum droplets in Bose-Einstein condensates, including those with and without external potential, are examined. The study of solitons and vortex solitons in BECs under mean-field interactions and external potential has yielded significant results. This thesis expands this field by introducing the LHY correction to study stable quantum droplets and vortex droplets in BECs. The work chapters begin with studying binary quantum droplets in free space, considering the case where interspecies interactions are weakly attractive and intraspecies interactions are weakly repulsive. The zero-vorticity, semi-vortex, and binary vortex quantum droplets are studied in Chapter 3. Stable semi-vortex quantum droplets with embedded vorticity up to 8 are found in this system. This result gives a way to form stable vortex droplets with high topological charges in free space by controlling one component with zero vorticity and the other component with embedded vorticity. The stability and properties of these droplets are systematically studied. Chapter 4 explores the vortex quantum droplets and hidden vortex droplets confined within a radial lattice trap. It is found that the existence curves of the droplets confined within this optical trap could violate the Vakhitov-Kolokolov(VK) criterion, which is a necessary condition to form a stable soliton. A remarkable finding is that the vortex and hidden vortex quantum droplets in this radial lattice remain stable when they carry the embedded vorticity up to 11. The double-ring vortex and hidden vortex quantum droplets confined within this radial lattice are also discussed. The work expands from non-magnetic atoms to magnetic atoms in Chapter 5, which needs to consider the dipole-dipole interactions(DDIs). The anisotropic dipolar vortex quantum droplets trapped in an annular potential are studied. Under the effect of the ring shape potential, the double peak ring-shaped vortex droplets remain stable when they carry the embedded vorticity up to 11. It is the first time stable droplets with high topological charges have been formed in a dipolar system. Additionally, the shape transformation of dipolar vortex droplets induced by the strength of the mean-field interaction is examined. Finally, the behavior of dipolar vortex droplets confined within a weak-depth annular potential is also discussed. The results of this thesis may prove highly useful in creating stable vortex quantum droplets in Bose-Einstein condensates and advancing the broader field of ultracold atoms.