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Analysis and Applications

Dynamics of dipolar quantum droplets in an extended Gross–Pitaevskii equation in the presence of time-dependent harmonic trapping potential and a damping term

Brahim Alouini, Hichem Hajaiej

Published Jun 23, 2022• 0 citations
Publisher

Abstract

The purpose of this paper is to study the dynamics of solutions to an extended Gross–Pitaevskii equation that models the formation of droplets in a dipolar Bose–Einstein condensate (BEC). The formation of these droplets has been recently discovered by driving the BEC into the strongly dipolar regime. Surprisingly, instead of collapsing, the system formed stable droplets. So far, no rigorous mathematical explanation has been proved. To the best of our knowledge, only experimental results have been obtained. The goal of this paper is to validate this breakthrough discovery. Many predictions/ conjectures properties of these droplets have been stated by some research groups in physics and engineering. In particular, it has been claimed that the stability of these droplets is a consequence of the presence of the damping term in the extended Gross–Pitaevskii equation under study. This term describes the three-body loss process. To accurately model the dynamics of formation of these droplets, it is necessary to consider a time-dependent harmonic trapping potential as well as other terms with different types of nonlinearity among them that describe the Lee–Huang–Yang (LHY). This presents some challenges that will be solved in this paper.

DOI
10.1142/s0219530522500117

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The purpose of this paper is to study the dynamics of solutions to an extended Gross–Pitaevskii equation that models the formation of droplets in a dipolar Bose–Einstein condensate (BEC). The formation of these droplets has been recently discovered by driving the BEC into the strongly dipolar regime. Surprisingly, instead of collapsing, the system formed stable droplets. So far, no rigorous mathematical explanation has been proved. To the best of our knowledge, only experimental results have been obtained. The goal of this paper is to validate this breakthrough discovery. Many predictions/ conjectures properties of these droplets have been stated by some research groups in physics and engineering. In particular, it has been claimed that the stability of these droplets is a consequence of the presence of the damping term in the extended Gross–Pitaevskii equation under study. This term describes the three-body loss process. To accurately model the dynamics of formation of these droplets, it is necessary to consider a time-dependent harmonic trapping potential as well as other terms with different types of nonlinearity among them that describe the Lee–Huang–Yang (LHY). This presents some challenges that will be solved in this paper.
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.
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.
We consider quantum droplets in dipolar Bose-Einstein condensates embedded in optical lattices within the framework of Gross-Pitaevskii equations. In dipolar BECs, the long-range and isotropic dipole-dipole interaction provides an additional mechanism for self-binding. We analyze the linear stability as well dynamics of quantum droplets. We find an effective potential for the width and show that the optimum width for formation of quantum droplet increases as the dipole-dipole interaction increases. We study dynamics of the stable droplets and see that its width oscillates with time, and the amplitude of oscillation increases with the increase of dipole-dipole interaction. In presence of optical lattices, width of a stable droplet changes quasi-periodically while the location of density profile changes periodically in space. The frequency of oscillation depends sensitively on the lattice parameters. We check dynamical stability of a quantum droplet through numerical simulation.
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.
Abstract We consider quantum droplets in dipolar Bose–Einstein condensates embedded in optical lattices (OLs) within the framework of Gross–Pitaevskii equations. In dipolar BECs, the long-range and isotropic dipole–dipole interaction (DDI) provides an additional mechanism for self-binding. We analyze the linear stability as well dynamics of quantum droplets. We find an effective potential for the width and show that the optimum width for formation of quantum droplet increases as the DDI increases. We study dynamics of the stable droplets and see that its width oscillates with time, and the amplitude of oscillation increases with the increase of DDI. In presence of OLs, width of a stable droplet changes quasi-periodically while the location of density profile changes periodically in space. The frequency of oscillation depends sensitively on the lattice parameters. We check dynamical stability of a quantum droplet through numerical simulation.
We address the existence and stability of multipole quantum droplets in symmetric binary Bose-Einstein condensates described by the amended Gross-Pitaevskii equation with Lee-Huang-Yang quantum corrections. Quantum droplets trapped in a weakly anharmonic potential can be composed of different numbers of globally linked poles with an azimuthally periodic distribution. Due to the competing Lee-Huang-Yang-augmented nonlinearity, the norm of two branches of droplets with slopes of opposite sign merges together at a lower cutoff of chemical potential. The lower and upper branches of droplets at the same chemical potential, but with a different norm, can evolve stably in certain parameter regions simultaneously. The stability domain of droplets shrinks with the growth of the number of poles. Even unstable necklacelike droplets can survive for a very long time.