Abstract: Quantum mechanics is a fascinating field to explore, where its effects are usually non-trivial and counterintuitive. In this Thesis, we focus on systems with a small number of particles in low dimensions, where the quantum effects are enhanced. We consider experimentally realistic systems that can be created nowadays in ultracold atom laboratories, where they can control the dimensionality, the external potential, the number of particles, and the interactions.

We combine analytical and numerical techniques to study the different systems. The harmonic oscillator trap includes an additional limitation due to the exact diagonalization method, which obtains upper-bound energies. We show how to correct this error and better estimate the energy and density by using the analytical solution of the two-particle system. 

We compute different properties for systems with harmonic oscillator confinement. We consider the symmetric SU(N) case and systems with broken interaction symmetry. We explore the ground state structure, where the correlations play an important role. We also consider the impurity case, which can be mapped to an effective one particle in a double-well model, showing two phases.

Afterwards, we study fractal lattice systems. Under this situation, we explore the effect on the transport of a single particle obtaining the diffusion exponent and relating it to spectral properties. We demonstrate that the fractal slows down the particle motion, and how the fractal system can preserve information about the initial phase of the wavefunction. We also explore the entanglement of a two-particle system and how the interactions affect the entanglement creation.

Finally, we study a three-site lattice where each site has a different energy and the couplings are time-dependent. We implement the spatial adiabatic passage protocol, that transfers a single particle from the first to the third site, and generalize it for a few particle systems with interactions. Due to the interactions, the adiabaticity of the protocol is lost, but we demonstrate that it is possible to populate the third state for certain interaction strength values. As a result, since the final state populates the most energetic site of the system, we propose this setup as a quantum battery model.