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arXiv:2510.12390 · Physical Review A

Empty and filled vortices in squeezed K 39 Bose-Bose liquid drops

Ivan Poparić, Leandra Vranješ Markić, Jordi Boronat

Published Nov 13, 2025• 1 citation
arXivHTMLPDF

Abstract

Using density functional theory, we have theoretically studied the formation and the stability of vortices in quantum liquid droplets composed of a mixture of hyperfine states of potassium. Following the experimental setup that produced quantum droplets for the first time, we work with squeezed drops that are compressed in one direction. By squeezing the drops even more, toward a quasi-two-dimensional geometry, we study the minimum atom number able to show a stable vortex and obtain that this number is significantly smaller than previous predictions for spherical droplets. The reduction of the critical atom number for forming a stable vortex could make their experimental observation in these droplets, which is still lacking, more feasible. Contrary to results obtained in heteronuclear mixtures, where the energetically preferred vortices are partially filled with the species not participating in the rotation, our results show a relevant stability island of fully empty vortices. Increasing the number of particles in the drop and the speed of rotation, we estimate the transition line between empty and filled vortices.

DOI
10.1103/bvgw-9q78

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Using density functional theory, we have theoretically studied the formation and the stability of vortices in quantum liquid droplets composed of a mixture of hyperfine states of potassium. Following the experimental setup that produced quantum droplets for the first time, we work with squeezed drops that are compressed in one direction. By squeezing the drops even more, toward a quasi-two-dimensional geometry, we study the minimum atom number able to show a stable vortex and obtain that this number is significantly smaller than previous predictions for spherical droplets. The reduction of the critical atom number for forming a stable vortex could make their experimental observation in these droplets, which is still lacking, more feasible. Contrary to results obtained in heteronuclear mixtures, where the energetically preferred vortices are partially filled with the species not participating in the rotation, our results show a relevant stability island of fully empty vortices. Increasing the number of particles in the drop and the speed of rotation, we estimate the transition line between empty and filled vortices.
We have theoretically investigated the structure of spinning self-bound droplets made of a $^{41}\mathrm{K}\text{\ensuremath{-}}^{87}\mathrm{Rb}$ Bose mixture by solving the Gross-Pitaevskii equation including beyond-mean-field correction in the Lee-Huang-Yang form. The structure and energetics of vortex formation in the self-bound mixture have been elucidated, showing that the formation of linear vortices in the heavier species is energetically favored over other configurations. A fake (partially filled) core develops as a consequence in the other species, resulting in a hole which might be imaged in experiments. We computed the minimal size of $^{41}\mathrm{K}\ensuremath{-}^{87}\mathrm{Rb}$ droplets which can host stable vortex lines in their interior, which is important information for experiments aimed at the observation of vortices in such systems. The different role of quantized vortices and capillary waves, which are the two ways angular momentum can be stored in a swirling superfluid, is addressed in detail by computing the relation between angular momentum and rotational frequency. The results show intriguing similarities with the case of a prototypical superfluid, i.e., 4He^{4}\mathrm{He}4He droplets when set into rotation. A two-branches curve in the stability diagram, qualitatively similar to the one expected for classical (incompressible and viscous) rotating liquid droplets, is obtained when vortices are present in the droplets, while prolate (i.e., nonaxisymmetric) shapes are only permitted in vortex-free droplets.
We explore ultradilute Bose-Bose liquid droplets squeezed by an external harmonic potential in one spatial direction. Our theoretical study is based on a functional that is built using quantum Monte Carlo results of the bulk phase and incorporates finite-range effects. A characteristic feature of these drops is the existence of a critical atom number, which is the minimum number of particles to have a many-body bound state. We report results on the critical atom numbers for different magnetic fields and applying confinement strengths towards a quasi-two-dimensional setup. In the regime where the local-density approximation is expected to be valid, we find that the critical atom number decreases linearly with the harmonic oscillator length of the confining potential. With the largest squeezing explored in our work, we predict stable drops containing from approximately 1000 to 2000 atoms for magnetic fields from 56.230 to 56.574 G, respectively. Our functional reduces the critical numbers for any confinement and applied magnetic field with respect to the estimations based on the Lee-Huang-Yang model. We observe saturated drops when the number of atoms in the drop is much larger than the critical value, their central density being higher for the quantum Monte Carlo functional than for the Lee-Huang-Yang one.
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.
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.
We investigate the lowest excitation modes of binary 39^{39}39K ultradilute droplets confined harmonically in one direction, which reduces the critical number of particles needed to sustain a self-bound state. Using the density functional built from first-principles quantum Monte Carlo calculations (QMC), which incorporates the effects of finite range interactions, we study the crossover of the oscillation modes from three to two dimensions. For weak squeezing, we observe the coupling between breathing and quadrupole modes, resulting from the breaking of spherical symmetry. Increasing the squeezing, the higher-frequency mode corresponding to three-dimensional breathing mode becomes unstable. For squeezings stronger and comparable to experiments, we analyze the dependence of the excitation modes on the droplet size for several interactions and show the reduction of the number of particles needed to sustain stable oscillations. The differences between the results obtained with the QMC functional and the extended Gross-Pitaevskii theory are increased with respect to the case of previously studied spherical droplets.
The rotation of a quantum liquid induces vortices to carry angular momentum. When the system is composed of multiple components that are distinguishable from each other, vortex cores in one component may be filled by particles of the other component, and coreless vortices form. Based on evidence from computational methods, here we show that the formation of coreless vortices occurs very similarly for repulsively interacting bosons and fermions, largely independent of the form of the particle interactions. We further address the connection to the Halperin wave functions of non-polarized quantum Hall states.