Scattering process from a disklike obstacle of massive vortex pairs in binary BECs

In this work the scattering process of a massive vortex-antivortex dipole from a disklike obstacle has been investigated. The term “massive” refers to the fact that vortices of one species host the atoms of the other species, which thus play the role of massive cores. The motion of massless vortices is described by first-order motion equations, while the inclusion of core mass introduces a second-order time derivative that doubles the number of independent dynamical variables needed and is responsible for a much richer phenomenology. Infact, while complete integrability is reached in the massless case due to the presence of a sufficient number of conserved quantities to ensure such character, the introduction of core masses makes the system not-integrable as the number of degrees of freedom overtakes the number of conserved quantities. The main purpose of this work is then analyzing the dynamical features of a massive pair scattered against a circular obstacle, shedding a light on eventual chaotic behaviour. After an introductory chapter devoted to the phenomenology of ultracold bosons, including Bose-Einstein condensation and the emergence of the Superfluid regime, the second chapter will contain a brief review of the second quantization picture and bosonic field theories, the GPE in the mean-field limit, together with the hydrodynamic picture of bosonic fluids, advantageous to investigate topological excitations in the fluid medium. In the third chapter the Point-like model will be derived through the approximation of the usual mean-field Hamiltonian accounting for two-body interactions adopting a flat density profile for the condensate that neglects the presence of vortex singularities. The resulting Hamiltonian of N free vortices will be extended to account for the presence of a circular confining boundary by means of the Virtual Images Method.The passage to the massive model is performed by introducing a Lagrangian that contains a quadratic kinetic term depending on vortices’ masses, appropriately justified thanks to the method of the time-dependent variational lagrangian by means of trial quantum-mechanical wavefunctions. In the fourth chapter a different description of the system will be proposed: by making use of the dynamical-algebra approach, it is possible to express the model Hamiltonian as a linear combination of generators of a Lie Algebra that not only allows to diagonalize the Hamiltonian (whenever this is possible) but also to directly find conserved quantities from algebra’s invariants. Locating the model in a specific algebraic framework, aside from giving deeper insights in the mathematical characterization of the system by exhibiting the presence of eventual symmetries, permits a more rigorous justification of numerical results by means of conserved quantities. In chapter five infact, results from numerical simulations will be presented and discussed. The resulting motion of the vortex dipole will be analyzed and justified firstly through the consideration of dynamical equations in two asymptotic regimes and then in the light of the algebraic description presented in the previous chapter. The presence of a few conserved quantities infact can conduct to the derivation of a relation between initial velocities and their values long after the influence of the circular obstacle, justifying the existence of a deflection angle after the dipole’s interaction with the obstacle.