The quantum-mechanical collapse (alias fall onto the center of particles attracted by potential $\ensuremath{-}{r}^{\ensuremath{-}2}$) is a well-known issue in quantum theory. It is closely related to the quantum anomaly, i.e., breaking of the scaling invariance of the respective Hamiltonian by quantization. We demonstrate that the mean-field repulsive nonlinearity prevents the collapse and thus puts forward a solution to the quantum-anomaly problem that differs from that previously developed in the framework of the linear quantum-field theory. This solution may be realized in the 3D or 2D gas of dipolar bosons attracted by a central charge and in the 2D gas of magnetic dipoles attracted by a current filament. In the 3D setting, the dipole-dipole interactions are also taken into regard, in the mean-field approximation, resulting in a redefinition of the scattering length which accounts for the contact repulsion between the bosons. In lieu of the collapse, the cubic nonlinearity creates a 3D ground state (GS), which does not exist in the respective linear Schr\"odinger equation. The addition of the harmonic trap gives rise to a tristability, in the case when the Schr\"odinger equation still does not lead to the collapse. In the 2D setting, the cubic nonlinearity is not strong enough to prevent the collapse; however, the quintic term does it, creating the GS, as well as its counterparts carrying the angular momentum (vorticity). Counterintuitively, such self-trapped 2D modes exist even in the case of a weakly repulsive potential ${r}^{\ensuremath{-}2}$. The 2D vortical modes avoid the phase singularity at the pivot (
r=0) by having the amplitude diverging at $r\ensuremath{\rightarrow}0$ instead of the usual situation with the amplitude of the vortical mode vanishing at $r\ensuremath{\rightarrow}0$ (the norm of the mode converges despite of the singularity of the amplitude at $r\ensuremath{\rightarrow}0$). In the presence of the harmonic trap, the 2D quintic model with a weakly repulsive central potential ${r}^{\ensuremath{-}2}$ gives rise to three confined modes, the middle one being unstable, spontaneously developing into a breather. In both the 3D and 2D cases, the GS wave functions are found in a numerical form and in the form of an analytical approximation, which is asymptotically exact in the limit of the large norm.
This article provides a focused review of recent findings which demonstrate, in some cases quite counter-intuitively, the existence of bound states with a singularity of the density pattern at the center; the states are physically meaningful because their total norm converges. One model of this type is based on the 2D Gross–Pitaevskii equation (GPE), which combines the attractive potential ∼ r − 2 and the quartic self-repulsive nonlinearity, induced by the Lee–Huang–Yang effect (quantum fluctuations around the mean-field state). The GPE demonstrates suppression of the 2D quantum collapse, driven by the attractive potential, and emergence of a stable ground state (GS), whose density features an integrable singularity ∼ r − 4 / 3 at r → 0 . Modes with embedded angular momentum exist too, but they are unstable. A counter-intuitive peculiarity of the model is that the GS exists even if the sign of the potential is reversed from attraction to repulsion, provided that its strength is small enough. This peculiarity finds a relevant explanation. The other model outlined in the review includes 1D, 2D, and 3D GPEs, with the septimal (seventh-order), quintic, and cubic self-repulsive terms, respectively. These equations give rise to stable singular solitons, which represent the GS for each dimension D, with the density singularity ∼ r − 2 / ( 4 − D ) . Such states may be considered the results of screening a “bare” delta-functional attractive potential by the respective nonlinearities.
It is known that attractive potential ~− 1 / r 2 gives rise to the critical quantum collapse in the framework of the three-dimensional (3D) linear Schrödinger equation. This article summarizes theoretical analysis, chiefly published in several original papers, which demonstrates suppression of the collapse caused by this potential, and the creation of the otherwise missing ground state in a 3D gas of bosonic dipoles pulled by the same potential to the central charge, with repulsive contact interactions between them, represented by the cubic term in the respective Gross–Pitaevskii equation (GPE). In two dimensions (2D), quintic self-repulsion is necessary for the suppression of the collapse; alternatively, this may be provided by the effective quartic repulsion produced by the Lee–Huang–Yang correction to the GPE. 3D states carrying angular momentum are constructed in the model with the symmetry reduced from spherical to cylindrical by an external polarizing field. Interplay of the collapse suppression and miscibility–immiscibility transition is considered in a binary condensate. The consideration of the 3D setting in the form of the many-body quantum system, with the help of the Monte Carlo method, demonstrates that, although the quantum collapse cannot be fully suppressed, the self-trapped states predicted by the GPE exist in the many-body setting as metastable modes protected against the collapse by a tall potential barrier.
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.
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.
In recent work [Sakaguchi and Malomed, Phys. Rev. A 83, 013607 (2011)], a solution to the problem of the quantum collapse (fall onto the center) in the three-dimensional space with the attractive potential $\ensuremath{-}({U}_{0}/2){r}^{\ensuremath{-}2}$ was proposed, based on the replacement of the linear Schr\"odinger equation by the Gross-Pitaevskii (GP) equation with the repulsive cubic term. The model applies to a quantum gas of molecules carrying permanent electric dipole moments, with the attraction center representing a fixed electric charge. It was demonstrated that the repulsive nonlinearity suppresses the quantum collapse and creates the corresponding spherically symmetric ground state (GS), which was missing in the case of the linear Schr\"odinger equation. Here, we aim to extend the analysis to the cylindrical geometry and to eigenstates carrying angular momentum. The cylindrical anisotropy is imposed by a uniform dc field, which fixes the orientation of the dipole moments, thus altering the potential of the attraction to the center. First, we analyze the modification of the condition for the onset of the quantum collapse in the framework of the linear Schr\"odinger equation with the cylindrically symmetric potential for the states with azimuthal quantum numbers
m=0 (the GS) and
m=1, 2. The corresponding critical values of the strength of the attractive potential
(U0)cr(m) are found. Next, a numerical solution of the nonlinear GP equation is developed, which demonstrates the replacement of the quantum collapse by the originally missing eigenstates with
m=0,1,2. Their dynamical stability is verified by means of numerical simulations of the perturbed evolution. For
m=0, the Thomas-Fermi approximation is presented too, in an analytical form. Crucially important for the solution is the proper choice of the boundary conditions at $r\ensuremath{\rightarrow}0$.