The Bose-Hubbard model as an example of diagonalization techniques in many body systems

U ovom radu razmatra se određivanje vlastitih stanja Hamiltonijana i kvantna dinamika u višečestičnim sistemima na primjeru Bose-Hubbardovog modela. Proučavaju se metode generiranja baze prostora stanja sistema te izračuna matrice Hamiltonijana u odabranim reprezentacijama. Nakon potpune specifikacije sistema određuje se energijski spektar i svojstvena stanja. Korištenjem pogodnih tehnika obavlja se povezivanje indeksa vektora stanja i kvantnih brojeva radi učinkovitoga pristupanja i manipuliranja. Ove tehnike su posebno pogodne pri odrezivanju fizičkih veličina te simuliranju dinamike u kvantnim međudjelujućim sistemima. Nadalje, promatra se ponašanje karakterističnih veličina poput energijskih procjepa, pobuđenja ovakvog sistema u ovisnosti o parametrima Hamiltonijana, očekivane raspodjele čestica i drugih veličina. U radu je dan i pregled formalizma druge kvantizacije te izvod Bose- Hubbardovog modela u tom formalizmu. Isto tako, promatra se i granica beskonačno jakih odbojnih interakcija izjedu bozona kada sustav pokazuje odredbene sličnosti sa sustavom neinteragirajućih fermiona te se radi mapiranje jednog sustava na drugi. Na kraju se promatra vremenska evolucija sustava bozona u optičkoj rešetci u prisustvu vanjskog potencijala za slučaj stalnog i vremenski promjenljivog potencijala.This thesis deals with the determination of the eigenstates of the Hamiltonian and quantum dynamics in many-body interacting systems. As an example, the Bose- Hubbard model is used. Methods of generating the basis of system state space and calculation of Hamiltonian matrix in selected representations are studied. After the complete specification of the system, the energy spectrum and the eigenstates are determined. By using the appropriate techniques, the mapping of the vector index and quantum numbers is established for efficient access and manipulation for physical quantities evaluations and simulating dynamics in quantum interacting systems. Furthermore, the expected distribution of particles and momenta are examined, studying the behaviour of characteristic quantities such as energy gaps, the excitation spectra of such a system as a function of the Hamiltonian parameters is determined for a couple of cases. An overview of the formalism of second quantization is given in the paper as well as the derivation of the Bose-Hubbard model in this formalism. Also, the limit of infinitely strong repulsive interactions between the bosons is investigated when the system shows certain similarities with the system of non-interacting fermions onto which it is mapped. In the end, the time evolution of the system of bosons in an optical lattice is investigated in the presence of an external potential in the case of a constant and a time-varying potential