Collision dynamics and entanglement generation of two initially independent and indistinguishable boson pairs in one-dimensional harmonic confinement

We investigate finite number effects in collisions between two states of an initially well known number of identical bosons with contact interactions, oscillating in the presence of harmonic confinement in one dimension. We investigate two N/2 (interacting) ground states, which are initially displaced from the trap center, and the effects of varying interaction strength. The numerics focus on the simplest case of N=4. In the non-interacting case, such a system would display periodic oscillation with a half harmonic oscillator period (due to the left-right symmetry). With the addition contact interactions between the bosons, collisions generate entanglement between each of the states and distribute energy into other modes of the oscillator. We study the system numerically via an exact diagonalization of the Hamiltonian with a finite basis, investigating left/right number uncertainty as our primary measure of entanglement. Additionally we study the time-evolution and equilibration of the single-body von Neumann entropy for both the attractive and repulsive cases. We identify parameter regimes for which attractive interactions create qualitatively different behavior to repulsive interactions, due to the presence of bound states (quantum solitons) and explain the processes behind this.