This dissertation focuses on dynamics and transport effects of semiclassical topological defects in systems with important quantum degrees of freedom, which we term "hybrid quantum systems". The topological defects under consideration are skyrmions and magnetic vortices in layered heterostructures of three-dimensional (3D) topological insulators (TI) and magnetic materials, and dark solitons in Bose-Einstein condensates (BEC).

We examine the proximity effect between a 3D TI and two types of insulating magnets: a chiral magnet with a single skyrmion in a ferromagnetic background, and an XY magnet with strong easy-plane anisotropy which undergoes a vortex unbinding transition. The skyrmion magnetic texture leads to confinement of Dirac states at the skyrmion radius, resulting in a charged skyrmion that can be manipulated by an external electric field. We show that the bound states are robust in the presence of an external magnetic field. Magnetic vortices in the XY magnet affect electronic transport at the TI surface. Scattering at classical magnetic fluctuations influences surface resistivity of the TI, and near the transition temperature we find that the resistivity has a clear maximum and scales linearly with temperature on either side of the transition. We discuss the limits of mapping the TI-XY magnet model to the classic theoretical problem of free Dirac fermions in a random magnetic field.

Secondly, we study dark solitons in a BEC coupled to thermal non-interacting impurity atoms acting as a dissipative bath. We calculate the friction coefficient due to scattering and find that it can be tuned with accessible experimental probes. We develop a general theory of stochastic dynamics of the negative-mass dark soliton and solve the corresponding Fokker-Planck equation exactly. From the time-dependent phase-space probability distribution function we find the soliton can undergo Brownian motion only in the presence of friction and a confining potential.

Finally, we numerically study the ground-state properties of a spin-1 BEC gas in the "synthetic dimensions" experimental set-up. Ground state phases depend on the sign of the spin-dependent interaction parameter and the strength of the spin-orbit field. We find "charge"- and spin-density-wave phases related to helical spin order.