Josephson dynamics of two-dimensional Bose-Einstein condensates in a dual-core trap: Homogeneous, droplet-droplet, and vortex-vortex regimes

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>π</a:mi> </a:math> -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 <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>π</b:mi> </b:math> -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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>S</c:mi> </c:math> and sufficiently small particle number are typically unstable, breaking up into <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mi>S</d:mi> <d:mo>+</d:mo> <d:mn>1</d:mn> </d:mrow> </d:math> (and occasionally <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mi>S</e:mi> <e:mo>+</e:mo> <e:mn>2</e:mn> </e:mrow> </e:math> ) 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 <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mi>S</f:mi> <f:mo>=</f:mo> <f:mn>1</f:mn> <f:mo>,</f:mo> <f:mn>2</f:mn> </f:mrow> </f:math> , and 3 are investigated.