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Physical Review A

Beyond mean-field effects in dynamics of Bose-Einstein condensate in two-core trap

F. Kh. Abdullaev, R. M. Galimzyanov, Akbar M. Shermakhmatov

Published Jun 30, 2025• 6 citations
Publisher

Abstract

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.

DOI
10.1103/hr45-tcyb

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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.
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.
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.
We study coherent dynamics of two interacting Bose-Bose droplets by means of the extended Gross-Pitaevskii equation. The relative motion of the droplets couples to the phases of their components. The dynamics can be understood in terms of the evolution of zero-energy modes recovering symmetries spontaneously broken by the mean-field solution. These are translational symmetry and two U(1) symmetries, associated with the phases of the droplets' two components. A phase-dependent interaction potential and double Josephson-junction equations are introduced to explain the observed variety of different scenarios of collision. We show that the evolution of the droplets is a macroscopic manifestation of the hidden dynamics of their phases. The occurrence of nondissipative drag between the two supercurrents (Andreev-Bashkin effect) is mentioned.
Dynamics of the repulsive Bose–Einstein condensate (BEC) in a double-well trap is explored within the 3D time-dependent Gross–Pitaevskii equation. The model avoids numerous common approximations (two-mode treatment, time-space factorization, fixed values of the chemical potential and barrier penetrability, etc) and thus may provide a realistic description of BEC dynamics, including both weak-coupling (sub-barrier) and strong-coupling (above-barrier) regimes and their crossover. The strong coupling regime is achieved by increasing the number N of BEC atoms and thus the chemical potential. The evolution with N of Josephson oscillations (JO) and macroscopic quantum self-trapping (MQST) is examined and the crucial impact of the BEC interaction is demonstrated. At weak coupling, the calculations well reproduce the JO/MQST experimental data. At strong coupling with a significant overlap of the left and right BECs, we observe a remarkable persistence of the Josephson-like dynamics: the JO and MQST converge to a high-frequency JO-like mode where both population imbalance and phase difference oscillate around the zero averages.
This thesis presents a numerical study of the beyond mean field extended Gross-Pitaevskii equation for a two-component self-bound Bose-Einstein condensate, supplemented by analytical work. We extend the quasi-two-dimensional system's mean field particle-particle contact interaction by quantum fluctuations (LHY-correction). We investigate the formation of self-bound droplets without external confinement and confirm their ground-state stability. Furthermore, we examine and discuss the real-time stability and dynamics of vortices in self-bound droplets and find a size dependence for stability and their dynamics of their release from a confined trap.
Abstract The static and dynamic properties of self-bound quantum droplets in a one-dimensional Bose-Bose mixture are discussed in the spirit of the Hartree–Fock-Bogoliubov theory. This latter enables us to provide beyond the Lee-Huang-Yang (LHY) quantum corrections to the equation of state at both zero and finite temperatures. In the uniform case our results for the ground-state energy and the critical temperature are confirmed through comparison with Quantum Monte-Carlo simulation and with available theoretical results. The density profiles are supported by numerical simulations of the generalized Gross-Pitaevskii equation which selfconsistently includes higher-order terms originating from the normal and anomalous fluctuations under the local density approximation. We show that the density exhibits a dip near its center in the flat-top plateau region for large interspecies interactions. We exemplify the impact of the beyond LHY corrections on the spatiotemporal evolution of the self-bound droplet in the presence of excitation induced by periodic density modulation. It is found that higher-order corrections may lead to the formation of a train of small droplets. We then extend our study for the case of inhomogeneous droplets in quasi one-dimensional Bose mixtures.