We construct strongly anisotropic quantum droplets with embedded vorticity in the 3D space, with mutually perpendicular vortex axis and polarization of atomic magnetic moments. Stability of these anisotropic vortex quantum droplets (AVQDs) is verified by means of systematic simulations. Their stability area is identified in the parametric plane of the total atom number and scattering length of the contact interactions. We also construct vortex-antivortex-vortex bound states and find their stability region in the parameter space. The application of a torque perpendicular to the vorticity axis gives rise to robust intrinsic oscillations or rotation of the AVQDs. The effect of three-body losses on the AVQD stability is considered too. The results show that the AVQDs can retain the topological structure (vorticity) for a sufficiently long time if the scattering length exceeds a critical value.
We construct strongly anisotropic quantum droplets with embedded vorticity in the 3D space, with mutually perpendicular vortex axis and polarization of atomic magnetic moments. Stability of these anisotropic vortex quantum droplets (AVQDs) is verified by means of systematic simulations. Their stability area is identified in the parametric plane of the total atom number and scattering length of the contact interactions. We also construct vortex-antivortex-vortex bound states and find their stability region in the parameter space. The application of a torque perpendicular to the vorticity axis gives rise to robust intrinsic oscillations or rotation of the AVQDs. The effect of three-body losses on the AVQD stability is considered too. The results show that the AVQDs can retain the topological structure (vorticity) for a sufficiently long time if the scattering length exceeds a critical value.
Creation of stable intrinsically anisotropic self-bound states with embedded vorticity is a challenging issue. Previously, no such states in Bose–Einstein condensates (BECs) or other physical settings were known. Dipolar BEC suggests a unique possibility to predict stable two dimensional anisotropic vortex quantum droplets (2D-AVQDs). We demonstrate that they can be created with the vortex axis oriented perpendicular to the polarization of dipoles. The stability area and characteristics of the 2D-AVQDs in the parameter space are revealed by means of analytical and numerical methods. Further, the rotation of the polarizing magnetic field is considered, and the largest angular velocities, up to which spinning 2D-AVQDs can follow the rotation in clockwise and anti-clockwise directions, are found. Collisions between moving 2D-AVQDs are studied too, demonstrating formation of bound states with a vortex-antivortex-vortex structure. A stability domain for such stationary bound states is identified. Unstable dipolar states, that can be readily implemented by means of phase imprinting, quickly transform into robust 2D-AVQDs, which suggests a straightforward possibility for the creation of these states in the experiment.
Anisotropic interactions profoundly affect topological excitations in quantum fluids. Motivated by recent advances in the studies of microwave-shielded polar molecules, we introduce self-trapped modes in the form of pancake-shaped quantum droplets (QDs) with embedded vorticity, which are maintained by strongly anisotropic dipole-dipole interactions. The stability region of singly charged (S=1) vortices nearly coincides with that of the ground-state QDs (S=0), demonstrating the robustness of the vortex states. The strong anisotropy of the system splits the core (pivot) of vortex QDs with S=2 into separated unitary ones. The angular momentum and stability of the state with S=2 are affected by the separation between the unitary cores. Higher-charge vortex QDs with S>2 are stable too, for sufficiently large particle numbers and inter-core separations. On the other hand, bound vortex-antivortex pairs with S=±1 are unstable. Head-on collisions between the vortex QDs exhibit distinct regimes, including rebound, merger, and fragmentation. Tuning the cylindrically symmetric component of the dipolar interaction reveals a pronounced sign-dependent response: positive tuning preserves the self-bound vortex, whereas negative tuning drives expansion and fragmentation. The results demonstrate that the microwave-dressed molecular QDs offer a robust platform for the realization of self-trapped vortex states, demonstrating how the strong anisotropy reshapes their structure, stability, and dynamics.
This concise review summarizes recent advancements in theoretical studies of vortex quantum droplets (VQDs) in matter-wave fields. These are robust self-trapped vortical states in two- and three-dimensional (2D and 3D) Bose–Einstein condensates (BECs) with intrinsic nonlinearity. Stability of VQDs is provided by additional nonlinearities resulting from quantum fluctuations around mean-field states, often referred to as the Lee–Huang–Yang (LHY) corrections. The basic models are presented, with emphasis on the interplay between the mean-field nonlinearity, LHY correction, and spatial dimension, which determines the structure and stability of VQDs. We embark by delineating fundamental properties of VQDs in the 3D free space, followed by consideration of their counterparts in the 2D setting. Additionally, we address stabilization of matter-wave VQDs by optical potentials. Finally, we summarize results for the study of VQDs in the single-component BEC of atoms carrying magnetic moments. In that case, the anisotropy of the long-range dipole-dipole interactions endows the VQDs with unique characteristics. The results produced by the theoretical studies in this area directly propose experiments for the observation of novel physical effects in the realm of quantum matter, and suggest potential applications to the design of new schemes for processing classical and quantum information.
Stability of elongated (``slender") quantum droplets (QDs) with embedded unitary and multiple vorticity is a problem that was not solved previously. In this work, we propose a solution which relies upon the use of the spatial modulation of the inter-species scattering length in the binary Bose-Einstein condensates, in the form of a two-dimensional axisymmetric Gaussian, shaped by means of the optical Feshbach resonance. The corresponding effective nonlinear trapping potential supports completely stable elongated QDs with vorticity S=0 and partly stable families of elongated QDs with S=1,2,3,4 (other nonlinear systems do not maintain stability of vortex droplets with ≥2). We systematically analyze effects of the amplitude and width of the Gaussian modulation, as well as the total number of atoms, on the shape and stability of the QDs, some effects being explained analytically. Collisions between identical QDs with moving in opposite directions along the central axis leads to their merger into still more elongated breathing QDs with the same vorticity, while collisions between QDs with S=±1 are quasi-elastic. Moving modulation profiles are able to adiabatically rotate the trapped elongated QDs. Application of a torque to the vector QD sets in the gyroscopic regime of robust precession, which realizes a macroscopic spin-orbit-coupling effect.
Toroidal Confinement and Beyond: Vorticity-Defined Morphologies of Dipolar 164Dy Quantum Droplets
We investigate the formation, stability, and dynamics of 3D ring-shaped and multipole vortical quantum droplets (QDs) in non-rotating 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 (DDI) 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 necklace-like modes, with the number of \textquotedblleft beads" (multipole's 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 annular density 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 non-rotating configurations.
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.