The nonlinear dynamics of quantum droplets in a quasi-one-dimensional BEC loaded in a two-core trap potential are investigated. The cases of standing and moving quantum droplets are considered. The frequencies of the Josephson oscillations and the condition for self-trapping are found. An analytical description is based on the variational approach to a system of coupled extended Gross-Pitaevskii (EGP) equations, including Lee-Huang-Yang correction terms. In the case of moving droplets, the system of equations describing the coupling of the relative motion of droplets in two cores with the dynamics of the relative phase and population imbalance (Josephson currents) is derived. Different regimes of droplet dynamics are found, including attraction and repulsion in dependence on the relative phase evolution. These phenomena are the realization of the Andreev-Bashkin effect for superfluids in a two-core trap potential. Analytical predictions of the model are confirmed by full numerical simulations of the coupled extended Gross-Pitaevskii equations.
Abstract Bose–Bose droplets are self-bound objects emerging from a mixture of two interacting Bose–Einstein condensates when their interactions are appropriately tuned. During droplet formation three continuous symmetries of the system’s Hamiltonian are broken: translational symmetry and two U(1) symmetries, allowing for arbitrary choice of phases of the mean-field wavefunctions describing the two components. Breaking of these symmetries must be accompanied by appearance of zero-energy excitations in the energy spectrum of the system recovering the broken symmetries. Normal modes corresponding to these excitations are the zero-energy modes. Here we find analytic expressions for these modes and introduce Hamiltonians generating their time evolution—dynamics of the droplet’s centers of mass as well as dynamics of the phases of the two droplet’s wavefunctions. When internal types of excitations (quasiparticles) are neglected then the very complex system of a quantum droplet is described using only a few ‘global’ degrees of freedom—the position of the center of mass of the droplet and two phases of two wave-functions, all these being quantum operators. We believe that our work might be useful in describing in a relatively easy way the low energy collisions of quantum droplets in situations where coherent flow of atoms between the droplets takes place.
An ultradilute quantum droplet is a self-bound liquidlike state recently observed in ultracold Bose-Einstein condensates. In most previous theoretical studies, it is described by a phenomenological low-energy effective theory, termed as the extended Gross-Pitaevskii equation. Here, we microscopically derive the Gross-Pitaevskii equation for the condensate and also for a pairing field in an inhomogeneous quantum droplet realized by Bose-Bose mixtures with attractive interspecies interaction. We show that the inclusion of the pairing field is essential, in order to have a consistent description of the droplet state. We clarify that the extended Gross-Pitaevskii equation used earlier should be understood as the equation of motion for the pairing field, rather than the condensate. The fluctuations of the pairing field give rise to low-energy collective excitations of the droplet. We also present the Bogoliubov equations for gapless phonon modes and gapped modes due to pairing in real space, which characterizes single-particle-like excitations of the droplet. The equations of motion derived in this work for the condensate and the pairing field serve as an ideal starting point to understand the structure and collective excitations of nonuniform ultradilute quantum droplets in ongoing cold-atom experiments.
The dynamics of a two-dimensional Bose-Einstein condensate mixture, loaded into a dual-core trap, when the beyond-mean-field effects are taken into account, are considered. The effects of quantum fluctuations are described by the Lee-Huang-Yang correction terms in the extended coupled Gross-Pitaevskii equations (GPE). The spatially uniform and inhomogeneous condensate cases are studied. In the first case, the parameter regimes associated with macroscopic quantum tunneling, macroscopic self-trapping, and revival-like localization dynamics are found. The Josephson oscillation frequencies for both the zero-phase and the π -phase modes are derived. As the total atom number is varied, the dynamics exhibit a nontrivial bifurcation structure: along the zero-phase branch, two successive pitchfork bifurcations generate bistability and hysteresis, while the π -phase branch shows a single pitchfork bifurcation. In the second case, the Josephson dynamics for quantum droplets and vortices are investigated. The analytical predictions for the oscillation frequencies of the atomic population between quantum droplets are found, and results are validated by direct numerical simulations of the system of coupled extended GPE. The existence of the Andreev-Bashkin nondissipative drag through simulations of droplet-droplet interactions is shown. The Josephson dynamics of vortex states are studied. It was found that vortices with topological charge S and sufficiently small particle number are typically unstable, breaking up into S + 1 (and occasionally S + 2 ) fundamental, nonvortical fragments, with the time to breakup increasing as the particle number grows. It is also found that unstable asymmetric vortices exhibit the splitting and/or crescentlike instability. For vortices with sufficiently large norms, long-time simulations confirm robust stability against small perturbations. In this stable regime, the properties of Josephson oscillations and Andreev-Bashkin-type entrainment for vortex states with charges S = 1 , 2 , and 3 are investigated.
We investigate the dynamics of Bose-Einstein condensate droplets composed of
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=3μm, we consider the relative dipolar-to-contact interaction strengths
εdd=1.4,
1.45, and
1.5. Symmetry-broken lowest-energy configurations are found for
εdd=1.5 and, over a reduced range of atom numbers, for
εdd=1.45, whereas no symmetry-broken states are found for
εdd=1.4 up to
N=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 and
2d=3μm, merging is found for
N=8000 and
8800, whereas the droplets remain separated for the investigated cases with
N≥9000. Increasing the initial separation to
2d=5μm shifts the approximate upper atom number for merging to
N≃1.65×104, demonstrating that the crossover is controlled by the competition between the post-quench excess energy and the repulsive inter-droplet barrier.
The collisional dynamics of two symmetric droplets with equal intraspecies scattering lengths and particle number density for each component is studied by solving the corresponding extended Gross—Pitaevskii equation in two dimensions by including a logarithmic correction term in the usual contact interaction. We find the merging droplet after collision experiences a quadrupole oscillation in its shape and the oscillation period is found to be independent of the incidental momentum for small droplets. With increasing collision momentum the colliding droplets may separate into two, or even more, and finally into small pieces of droplets. For these dynamical phases we manage to present boundaries determined by the remnant particle number in the central area and the damped oscillation of the quadrupole mode. A stability peak for the existence of droplets emerges at the critical particle number N c ≃ 48 for the quasi-Gaussian and flat-top shapes of the droplets.