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arXiv:2604.02011 · Physical Review A

Merging and oscillations of dipolar Bose-Einstein-condensate droplets

Anonymous, B. Szafran

Published Aug 21, 2026• 0 citations
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Abstract

We investigate the dynamics of Bose-Einstein condensate droplets composed of $^{164}$Dy atoms formed in a double-well potential following removal of the interwell barrier. By solving the dipolar Gross-Pitaevskii equation, we determine phase diagrams of ground-state configurations as functions of the atom number confined in the double-well potential. For an interwell separation $2d=3\,μ\mathrm{m}$, we consider the relative dipolar-to-contact interaction strengths $\varepsilon_{dd}=1.4$, $1.45$, and $1.5$. Symmetry-broken lowest-energy configurations are found for $\varepsilon_{dd}=1.5$ and, over a reduced range of atom numbers, for $\varepsilon_{dd}=1.45$, whereas no symmetry-broken states are found for $\varepsilon_{dd}=1.4$ up to $N=5\times10^4$. We analyze the subsequent time evolution after removal of the central barrier, revealing both droplet oscillations and merger events leading to the formation of larger droplets. The oscillations are driven by the external potential and by the repulsive tails of the in-plane component of the dipolar interaction. For the two-droplet states at $\varepsilon_{dd}=1.5$ and $2d=3\,μ\mathrm{m}$, merging is found for $N=8000$ and $8800$, whereas the droplets remain separated for the investigated cases with $N\geq9000$. Increasing the initial separation to $2d=5\,μ\mathrm{m}$ shifts the approximate upper atom number for merging to $N\simeq1.65\times10^4$, demonstrating that the crossover is controlled by the competition between the post-quench excess energy and the repulsive inter-droplet barrier.

DOI
10.1103/km38-xwbz

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We investigate the dynamics of Bose-Einstein condensate droplets composed of 164^{164}164Dy atoms formed in a double-well potential following removal of the interwell barrier. By solving the dipolar Gross-Pitaevskii equation, we determine phase diagrams of ground-state configurations as functions of the atom number confined in the double-well potential. For an interwell separation 2d=3 μm2d=3\,μ\mathrm{m}2d=3μm, we consider the relative dipolar-to-contact interaction strengths εdd=1.4\varepsilon_{dd}=1.4εdd​=1.4, 1.451.451.45, and 1.51.51.5. Symmetry-broken lowest-energy configurations are found for εdd=1.5\varepsilon_{dd}=1.5εdd​=1.5 and, over a reduced range of atom numbers, for εdd=1.45\varepsilon_{dd}=1.45εdd​=1.45, whereas no symmetry-broken states are found for εdd=1.4\varepsilon_{dd}=1.4εdd​=1.4 up to N=5×104N=5\times10^4N=5×104. We analyze the subsequent time evolution after removal of the central barrier, revealing both droplet oscillations and merger events leading to the formation of larger droplets. The oscillations are driven by the external potential and by the repulsive tails of the in-plane component of the dipolar interaction. For the two-droplet states at εdd=1.5\varepsilon_{dd}=1.5εdd​=1.5 and 2d=3 μm2d=3\,μ\mathrm{m}2d=3μm, merging is found for N=8000N=8000N=8000 and 880088008800, whereas the droplets remain separated for the investigated cases with N≥9000N\geq9000N≥9000. Increasing the initial separation to 2d=5 μm2d=5\,μ\mathrm{m}2d=5μm shifts the approximate upper atom number for merging to N≃1.65×104N\simeq1.65\times10^4N≃1.65×104, demonstrating that the crossover is controlled by the competition between the post-quench excess energy and the repulsive inter-droplet barrier.
We investigate the quench dynamics of quasi-one- and two-dimensional dipolar Bose-Einstein condensates of 164Dy^{164}\mathrm{Dy}164Dy atoms under the influence of a fast rotating magnetic field. The magnetic field thus controls both the magnitude and sign of the dipolar potential. We account for quantum fluctuations, critical to formation of exotic quantum droplet and supersolid phases in the extended Gross-Pitaevskii formalism, which includes the so-called Lee-Huang-Yang correction. An analytical variational ansatz allows us to obtain the phase diagrams of the superfluid and droplet phases. The crossover from the superfluid to the supersolid phase and to single and droplet arrays is probed with particle number and dipolar interaction. The dipolar strength is tuned by rotating the magnetic field with subsequent effects on phase boundaries. Following interaction quenches across the aforementioned phases, we monitor the dynamical formation of supersolid clusters or droplet lattices. We include losses due to three-body recombination over the crossover regime, where the three-body recombination rate coefficient scales with the fourth power of the scattering length (as)({a}_{s})(as​) or the dipole length (add)({a}_{dd})(add​). For fixed values of the dimensionless parameter, ${\ensuremath{\epsilon}}_{dd}={a}_{dd}/{a}_{s}$, tuning the dipolar anisotropy leads to an enhancement of the droplet lifetimes.
We study the dynamics of merging two initially independent Bose–Einstein condensates by adiabatically deforming the trap potential from a double well into a single well. The process of slowly decreasing the height of the central barrier of the double well is simulated by a full 3D calculation of the Gross–Pitaevskii equation. We investigate the dynamics by considering some factors relevant to experiments, e.g., the atom number, the initial state in each well, the initial relative phase between the two condensates. For a small number of 4400 atoms in each well and the same initial phase, we find for both condensates in the ground state or the p-wave soliton state that the final state is the ground state or the p-wave soliton state of the final trap potential. For cases in which both condensates are vortex states, we find much richer dynamics with the appearance of an extra vortex. We also discuss the cases for different initial relative phases. For comparison, we present results for 44 000 atoms corresponding to a larger nonlinear term in the Gross–Pitaevskii equation. In support of future experimental observations, we show both probability density and current maps.
We study the statics and dynamics of a binary dipolar Bose–Einstein condensate soliton for repulsive inter and intraspecies contact interactions with the two components subject to different spatial symmetries—distinct quasi-one-dimensional and quasi-two-dimensional shapes—using numerical solution and variational approximation of a three-dimensional mean-field model. The results are illustrated with realistic values of parameters in the binary 164 Dy- 168 Er mixture. The possibility of forming robust dipolar solitons of a very large number of atoms make them of great experimental interest. The existence of the solitons is illustrated in terms of stability phase diagrams. Exotic shapes of these solitons are illustrated in isodensity plots. The variational results for statics (size and chemical potential) and dynamics (small oscillation) of the binary soliton compare well with the numerical results. A way of preparing and studying these solitons in the laboratory is suggested.
We study the statics and dynamics of a dipolar Bose-Einstein condensate (BEC) droplet bound by interspecies contact interaction in a trapped nondipolar BEC. Our findings are demonstrated in terms of stability plots of a dipolar 164{}^{164}164Dy droplet bound in a trapped nondipolar 87{}^{87}87Rb BEC with a variable number of 164{}^{164}164Dy atoms and interspecies scattering length. A trapped nondipolar BEC of a fixed number of atoms can bind only a dipolar droplet containing fewer atoms than a critical number for the interspecies scattering length between two critical values. The shape and size (statics) as well as the small breathing oscillation (dynamics) of the dipolar BEC droplet are studied using numerical and variational solutions of a mean-field model. We also suggest an experimental procedure for achieving such a 164{}^{164}164Dy droplet by relaxing the trap on the 164{}^{164}164Dy BEC in a trapped binary 87{}^{87}87Rb-164{}^{164}164Dy mixture.
We investigate the formation, stability, and dynamics of three-dimensional ring-shaped and multipole vortical quantum droplets (QDs) in nonrotating 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 (DDIs) 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 necklacelike modes, with the number of "beads" (multipole 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 ring-shaped QDs 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 nonrotating configurations.
Toroidal Confinement and Beyond: Vorticity-Defined Morphologies of Dipolar 164^{164}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 SSS) are shaped as necklace-like modes, with the number of \textquotedblleft beads" (multipole's order) n=2Sn=2Sn=2S, up to S=6S=6S=6. The stability area of the multipoles shrinks with the increase of SSS. For higher values of SSS 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 SSS 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.