Few Interacting Particles in Low-Dimensional Quantum Systems: From Harmonic Oscillators to Fractal Lattices
[eng] 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, co...
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| Tipo de recurso: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2025 |
| País: | España |
| Institución: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/221131 |
| Acceso en línea: | https://hdl.handle.net/2445/221131 http://hdl.handle.net/10803/694470 |
| Access Level: | acceso abierto |
| Palabra clave: | Teoria quàntica Entrellaçament quàntic Estadística quàntica Fermions Quantum theory Quantum entanglement Quantum statistics |
| Sumario: | [eng] 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, considering either an external harmonic confinement or a lattice potential. These systems can be created nowadays in ultracold atom laboratories worldwide, where they can control the dimensionality, the external potential, as well as the number of particles and the interactions. Our goal is to understand the static and dynamic properties of different systems involving few particles and a wide range of interactions. In particular, we focus on the one-dimensional harmonic trap, fractal lattices, and one-dimensional three-site lattices. In each system, we consider a contact interaction potential defined by a delta function in the continuum space or an on-site interaction for the lattices. In addition, we compute different properties, such as energy spectra, densities, pair correlations, mean square displacement, and population of each site, in both static and time-dependent cases. We combine analytical and numerical techniques to study the different systems. In particular, the numerical part is mostly done using exact diagonalization techniques. With this approach, we can only examine systems with few particles due to the intrinsic limitations of the method, although it provides the capacity to obtain any property needed. The case of the harmonic oscillator trap includes an additional limitation due to the exact diagonalization method, as the basis must be truncated, resulting in upper-bound energies. We show how to correct this error and obtain a better estimate of the energy and the density by using the analytical solution of the two-particle system. We demonstrate that this correction works for a larger number of particles by computing results for up to eight particles. For the systems with harmonic oscillator confinement, we compute the energy spectrum as a function of the interaction strength, the density for different interaction values, and pair correlations. We present the results for the symmetric SU (N ) case and then study systems with broken interaction symmetry. We present different interaction configurations, and we explore the ground state structure, where the correlations play an important role. We also consider the impurity case, where one particle interacts with all the bath particles but the bath particles do not interact with each other. We show that the impurity system can be mapped to an effective one particle in a double-well model, showing two phases: the miscible and the immiscible. Afterwards, we study fractal lattice systems, where the sites and the tunneling connections are configured in a non-standard scheme creating an effective finite representation of a fractal structure. 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 motion of the particle and that this effect is robust against a random potential. Using the slower dynamics, we show how the fractal system can preserve information about the initial phase of the wavefunction for much longer times than a regular lattice. We also explore the entanglement of a two-particle system and how the interactions affect 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. |
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