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arXiv:2411.01554 · Physical review. E

Tightly self-trapped modes and vortices in three-dimensional bosonic condensates with electromagnetically induced gravity

Zibin Zhao, Guilong Li, Huan-Bo Luo, Bin Liu, Guihua Chen, Boris A. Malomed, Yongyao Li

Published Jun 9, 2025• 4 citations
arXivHTMLPDF

Abstract

The 1/r long-range interaction, induced by laser illumination, offers a mechanism for the implementation of stable self-trapping in Bose-Einstein condensates (BECs) in the three-dimensional free space. Using the variational approximation and numerical solutions, we find that self-trapped states in this setting, with attractive nonlocal and repulsive local interactions, resemble tightly bound compactons. However, these are not true compactons but rather tightly self-trapped modes (TSTMs), with small-amplitude nonvanishing tails. The structure of the self-trapped states is explained by an analytical solution for their tails. Further, we demonstrate that stable TSTMs with embedded vorticity exist in the same setting, with winding numbers up to S=6 (at least). Addressing two-TSTM interactions, we find that pairs of ground states (GSs, with S=0), as well as vortex-vortex and vortex-antivortex pairs (with S_{1}=S_{2} and S_{1}=-S_{2}, respectively), form stably rotating bound states. Head-on collisions between vortex TSTMs, set in slow motion by kicks, are inelastic, resulting in their merger into a GS soliton, that may either remain at the collision position or move aside, shedding the angular momentum with emitted radiation, or, alternatively, lead to the formation of a vortex that also moves aside.

DOI
10.1103/physreve.111.064207

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The 1/r long-range interaction, induced by laser illumination, offers a mechanism for the implementation of stable self-trapping in Bose-Einstein condensates (BECs) in the three-dimensional free space. Using the variational approximation and numerical solutions, we find that self-trapped states in this setting, with attractive nonlocal and repulsive local interactions, resemble tightly bound compactons. However, these are not true compactons but rather tightly self-trapped modes (TSTMs), with small-amplitude nonvanishing tails. The structure of the self-trapped states is explained by an analytical solution for their tails. Further, we demonstrate that stable TSTMs with embedded vorticity exist in the same setting, with winding numbers up to S=6 (at least). Addressing two-TSTM interactions, we find that pairs of ground states (GSs, with S=0), as well as vortex-vortex and vortex-antivortex pairs (with S_{1}=S_{2} and S_{1}=-S_{2}, respectively), form stably rotating bound states. Head-on collisions between vortex TSTMs, set in slow motion by kicks, are inelastic, resulting in their merger into a GS soliton, that may either remain at the collision position or move aside, shedding the angular momentum with emitted radiation, or, alternatively, lead to the formation of a vortex that also moves aside.
By means of variational methods and systematic numerical analysis, we demonstrate the existence of metastable solitons in three dimensional (3D) free space, in the context of binary atomic condensates combining contact self-attraction and spin-orbit coupling, which can be engineered by available experimental techniques. Depending on the relative strength of the intra- and intercomponent attraction, the stable solitons feature a semivortex or mixed-mode structure. In spite of the fact that the local cubic self-attraction gives rise to the supercritical collapse in 3D, and hence the setting produces no true ground state, the solitons are stable against small perturbations, motion, and collisions.
It is commonly known that two-dimensional mean-field models of optical and matter waves with cubic self-attraction cannot produce stable solitons in free space because of the occurrence of collapse in the same setting. By means of numerical analysis and variational approximation, we demonstrate that the two-component model of the Bose-Einstein condensate with the spin-orbit Rashba coupling and cubic attractive interactions gives rise to solitary-vortex complexes of two types: semivortices (SVs, with a vortex in one component and a fundamental soliton in the other), and mixed modes (MMs, with topological charges 0 and ±1 mixed in both components). These two-dimensional composite modes can be created using the trapping harmonic-oscillator (HO) potential, but remain stable in free space, if the trap is gradually removed. The SVs and MMs realize the ground state of the system, provided that the self-attraction in the two components is, respectively, stronger or weaker than the cross attraction between them. The SVs and MMs which are not the ground states are subject to a drift instability. In free space (in the absence of the HO trap), modes of both types degenerate into unstable Townes solitons when their norms attain the respective critical values, while there is no lower existence threshold for the stable modes. Moving free-space stable solitons are also found in the present non-Galilean-invariant system, up to a critical velocity. Collisions between two moving solitons lead to their merger into a single one.
We study the matter-wave solitons in Bose-Einstein condensate (BEC) trapped on a M\"obius st rip (MS), based on the respective Gross-Pitaevskii (GP) equation with the mean-field theory. In the linear regime, vortex states are characterized by quantum numbers, nnn and mmm, corresponding to the transverse and circumferential directions, with the phase structure determined by the winding number (WN) mmm. Odd and even values of nnn must associate, respectively, with integer and half-integer values of mmm, the latter ones requiring two cycles of motion around MS for returning to the initial phase. Using variational and numerical methods, we solve the GP equation with the attractive nonlinearity, producing a family of ground-state (GS) solitons for values of the norm below the critical one, above which the collapse sets in. Vortex solitons with n=1,m=1n=1,m=1n=1,m=1 and n=2,m=1/2n=2,m=1/2n=2,m=1/2 are obtained in a numerical form. The vortex solitons with n=1,m=1n=1,m=1n=1,m=1 are almost uniformly distributed in the azimuthal direction, while ones with n=2,m=1/2n=2,m=1/2n=2,m=1/2 form localized states. The Vakhitov-Kolokolov criterion and linear-stability analysis for the GS soliton solutions and vortices with n=1,m=1n=1,m=1n=1,m=1 demonstrates that they are completely stable, while the localized states with n=2,m=1/2n=2,m=1/2n=2,m=1/2 are completely unstable. Finally, the motion of solitons on the MS and the collision of two solitons are discussed.
We predict the existence of stable fundamental and vortical bright solitons in dipolar Bose-Einstein condensates (BECs) with repulsive dipole-dipole interactions (DDI). The condensate is trapped in the 2D plane with the help of the repulsive contact interactions whose local strength grows ∼r4\sim r^{4}∼r4 from the center to periphery, while dipoles are oriented perpendicular to the self-trapping plane. The confinement in the perpendicular direction is provided by the usual harmonic-oscillator potential. The objective is to extend the recently induced concept of the self-trapping of bright solitons and solitary vortices in the pseudopotential, which is induced by the repulsive local nonlinearity with the strength growing from the center to periphery, to the case when the trapping mechanism competes with the long-range repulsive DDI. Another objective is to extend the analysis for elliptic vortices and solitons in an anisotropic nonlinear pseudopotential. Using the variational approximation (VA) and numerical simulations, we construct families of self-trapped modes with vorticities ℓ=0\ell =0ℓ=0 (fundamental solitons), ℓ=1\ell =1ℓ=1, and ℓ=2\ell =2ℓ=2. The fundamental solitons and vortices with ℓ=1\ell =1ℓ=1 exist up to respective critical values of the eccentricity of the anisotropic pseudopotential, being stable in the entire existence regions. The vortices with ℓ=2\ell =2ℓ=2 are stable solely in the isotropic model.
We study the dynamics of nonlinear localized excitations (‘solitons’) in two-dimensional (2D) Bose–Einstein condensates (BECs) with repulsive interactions, loaded into an optical lattice (OL), which is combined with an external parabolic potential. First, we demonstrate analytically that a broad (‘loosely bound’, LB) soliton state, based on a 2D Bloch function near the edge of the Brillouin zone (BZ), has a negative effective mass (while the mass of a localized state is positive near the BZ centre). The negative-mass soliton cannot be held by the usual trap, but it is safely confined by an inverted parabolic potential ( anti-trap ). Direct simulations demonstrate that the LB solitons (including those with intrinsic vorticity) are stable and can freely move on top of the OL. The frequency of the elliptic motion of the LB-soliton's centre in the anti-trapping potential is very close to the analytical prediction which treats the solition as a quasi-particle. In addition, the LB soliton of the vortex type features real rotation around its centre. We also find an abrupt transition, which occurs with the increase of the number of atoms, from the negative-mass LB states to tightly bound (TB) solitons. An estimate demonstrates that for the zero-vorticity states, the transition occurs when the number of atoms attains a critical number N cr ∼ 10 3 , while for the vortex the transition takes place at N cr ∼ 5 × 10 3 atoms. The positive-mass LB states constructed near the BZ centre (including vortices) can also move freely. The effects predicted for BECs also apply to optical spatial solitons in bulk photonic crystals.
We study formation of rotating three-dimensional high-order solitons (azimuthons) in Bose Einstein condensate with attractive nonlocal nonlinear interaction. In particular, we demonstrate formation of toroidal rotating solitons and investigate their stability. We show that variational methods allow a very good approximation of such solutions and predict accurately the soliton rotation frequency. We also find that these rotating localized structures are very robust and persist even if the initial condensate conditions are rather far from the exact soliton solutions. Furthermore, the presence of repulsive contact interaction does not prevent the existence of those solutions, but allows one to control their rotation. We conjecture that self-trapped azimuthons are generic for condensates with attractive nonlocal interaction.