Investigating Bose-Einstein Condensation in a Two-Level System Using the Multi-Orbital Hubbard Model

Bose-Einstein Condensation (BEC) is a fascinating phenomenon in quantum physics, where a dilute gas of bosonic particles transitions into a state characterized by the macroscopic occupation of a single quantum state. In this study, we investigated the intricacies of BEC by examining a simplified two-level system using a multi-orbital Hubbard model. A novel numerical method was employed to compute Hamiltonian matrix elements for 2D multi-orbital Hubbard clusters. By leveraging Python programming and binary/Boolean logic, this approach provided an innovative solution for analyzing the quantum mechanical behavior of interacting electrons in lattice structures. The Hubbard model, a well-established framework in condensed matter physics, is widely used to describe interacting electrons in a lattice. In our investigation, we simplified this model into a two-orbital Hubbard configuration to examine the behavior of particles, including electrons and holes, within these two orbitals across a range of temperatures.