Bose mixture quantum droplets display a fascinating stability that relies on quantum fluctuations to prevent collapse driven by mean-field effects. Most droplet research focuses on untrapped or weakly trapped scenarios, where the droplets exhibit a liquidlike flat density profile. When weakly trapped droplets rotate, they usually respond through center-of-mass motion or splitting instability. Here, we study rapidly rotating droplets in the strong external confinement limit where the external potential prevents splitting and the center-of-mass excitation. We find that quantum droplets form a triangular vortex lattice as in single-component repulsive Bose-Einstein condensates (BECs), but the overall density follows the analytical Thomas-Fermi profile obtained from a cubic equation. We observe three significant differences between rapidly rotating droplets and repulsive BECs. First, the vortex core size changes markedly at finite density, visible in numerically obtained density profiles. We analytically estimate the vortex core sizes from the droplets' coherence length and find good agreement with the numerical results. Second, the change in the density profile gives a slight but observable distortion to the lattice, which agrees with the distortion expected due to nonuniform superfluid density. Lastly, unlike a repulsive BEC, which expands substantially as the rotation frequency approaches the trapping frequency, rapidly rotating droplets show only a fractional change in their size. We argue that this last point can be used to create clouds with lower filling factors, which may facilitate reaching the elusive strongly correlated regime.
Bose mixture quantum droplets display a fascinating stability that relies on quantum fluctuations to prevent collapse driven by mean-field effects. Most droplet research focuses on untrapped or weakly trapped scenarios, where the droplets exhibit a liquidlike flat density profile. When weakly trapped droplets rotate, they usually respond through center-of-mass motion or splitting instability. Here, we study rapidly rotating droplets in the strong external confinement limit where the external potential prevents splitting and the center-of-mass excitation. We find that quantum droplets form a triangular vortex lattice as in single-component repulsive Bose-Einstein condensates (BECs), but the overall density follows the analytical Thomas-Fermi profile obtained from a cubic equation. We observe three significant differences between rapidly rotating droplets and repulsive BECs. First, the vortex core size changes markedly at finite density, visible in numerically obtained density profiles. We analytically estimate the vortex core sizes from the droplets' coherence length and find good agreement with the numerical results. Second, the change in the density profile gives a slight but observable distortion to the lattice, which agrees with the distortion expected due to nonuniform superfluid density. Lastly, unlike a repulsive BEC, which expands substantially as the rotation frequency approaches the trapping frequency, rapidly rotating droplets show only a fractional change in their size. We argue that this last point can be used to create clouds with lower filling factors, which may facilitate reaching the elusive strongly correlated regime.
We consider a “symmetric” quantum droplet in two spatial dimensions, which rotates in a harmonic potential, focusing mostly on the limit of “rapid” rotation. We examine this problem using a purely numerical approach, as well as a semianalytic Wigner-Seitz approximation (first developed by Baym, Pethick, and their co-workers) for the description of the state with a vortex lattice. Within this approximation we assume that each vortex occupies a cylindrical cell, with the vortex-core size treated as a variational parameter. Working with a fixed angular momentum, as the angular momentum increases and depending on the atom number, the droplet accommodates none, few, or many vortices, before it turns to center-of-mass excitation. For the case of a “large” droplet, working with a fixed rotational frequency of the trap Ω , as Ω approaches the trap frequency ω , a vortex lattice forms, the number of vortices increases, the mean spacing between them decreases, while the “size” of each vortex increases as compared to the size of each cell. In contrast to the well-known problem of contact interactions, where we have melting of the vortex lattice and highly correlated many-body states, here no melting of the vortex lattice is present, even when Ω = ω . This difference is due to the fact that the droplet is self-bound. For Ω = ω , the “smoothed” density distribution becomes a flat top, very much like the static unconfined droplet. When Ω exceeds ω , the droplet maintains its shape and escapes to infinity, via center-of-mass motion. Published by the American Physical Society 2024
A rapidly rotating Bose gas in the quantum Hall limit is usually associated with a melted vortex lattice. In this work, we report a self-bound and visible triangular vortex lattice without melting for a two-dimensional Bose-Bose droplet rotating in the quantum Hall limit, i.e., with rotation frequency $\mathrm{\ensuremath{\Omega}}$ approaching the trapping frequency $\ensuremath{\omega}$. Increasing $\mathrm{\ensuremath{\Omega}}$ with respect to interaction strength U, we find a smooth crossover of the vortex lattice droplet from a needling regime, as featured by small vortex cores and an equilibrium flat-top surface, to the lowest-Landau-level regime with Gaussian-extended cores spreading over the whole surface. The surface density of such a rotating droplet is higher than that of a static one, and their ratio is found to be a universal function of $\mathrm{\ensuremath{\Omega}}/U$. We have demonstrated these results by both numerical and variational methods. The results pave the way for future experimental exploration of rapidly rotating ultracold droplets into the quantum Hall limit.
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.
The interacting Bose-Einstein condensates are mechanically stabilized by an external confinement such as magnetic, or optical traps. When the confinement is removed, they exhibit an expansion or collapse behavior depending on the interaction being repulsive, or attractive, respectively. The Gross-Pitaevskii mean-field theories are generally sufficient to provide a qualitative understanding of the condensate equilibrium properties and dynamics, therefore, beyond mean-field fluctuations are considered as minor quantitative corrections. The quantum droplet phases are self-stabilized phases of the condensate, in which the attractive mean-field interaction is stabilized by the repulsive Lee-Huang-Yang fluctuations without any external trap. The droplets are studied theoretically for binary Bose mixture and dipolar condensates, and both are recently realized in the ultracold atomic gas experiments. Here we propose the spin-1 condensate as another candidate to exhibit a quantum droplet phase. We argue that the spin-1 condensates in the polar and anti-ferromagnetic order parameters can exhibit a similar self-stabilizing mechanism. We also realize an analogy between the polarized spin-1 and Rabi-coupled binary mixture condensates. The Rabi frequency ωR and quadratic Zeeman energy q can tune the Lee-Huang-Yang energy for the Rabi-coupled mixture and spin-1 gas, respectively. Furthermore, the detuning δ and linear Zeeman energy p are external parameters to tune the polarization in Rabi-coupled mixture, and spin-1 gas, respectively, which can significantly change the mean-field energy. We create the phase diagram for the mechanical stability of the Rabi-coupled mixture and spin-1 gas on the parameter planes of ωR−δ and q−p, respectively. We finally address how the Bose mixture quantum droplets respond to the rotation. We find that the strongly confined quantum droplets exhibit triangular vortex lattices under rapid rotation. The properties of the vortex lattices differ from the repulsive condensates in three points. First, the change in the droplet size is only fractional at the rapid rotation limit, unlike the diverging radius of the repulsive condensate. This property can lead the way to realizing the lowest Landau level states and strongly correlated phases in the rotating ultracold gases. Secondly, the change in the core size throughout the vortex lattice is more tractable due to the finite density at the edge of the condensate. Finally, the vortex lattices are slightly distorted, which agrees with the expected vortex density profile due to the nonuniform superfluid density. In the rapid rotation limit, the droplet approaches the flat-top density profile, in which the vortex lattice is closest to a perfect periodicity.
Abstract This study presents a detailed analysis of the stationary properties and dynamics of an anharmonically trapped, rapidly rotating Bose–Einstein liquid droplet. We investigate the effects of the particle number, confining potential, and rotation speed on the formation of the energetically favored bead, multiple quantized vortex, off-center vortex, and center-of-mass states. The multi-periodic trajectories and breathing provide evidence of the collective excitations of the surface mode in the vortex states. Observation of the self-trapping phenomenon and tendency toward the lowest Landau level in rapid rotation regimes coincides well with the quantum-Hall limit. Modifying the topological charges and destroying the potential flow of the vortex state can occur if an external disturbance is imposed upon the quantum droplet.