Quantum Phase Transitions Of Bose Systems In Artificial Lattices

For the past decade,the techniques for quantum simulation have achieved great pro-gresses.By means of trapping ultracold atoms in optical lattices,the Hubbard model,geo-metric frustration and artificial gauge field systems et al.have been realized in experiments.Meanwhile,the quantum electrodynamics(QED)cavity arrays are recognized as another promising platform for realizing quantum simulation.In this thesis,four kinds of systems are studied analytically:scalar Bose gases trapped in optical lattices,an extended Bose-Hubbard model in hexagonal optical lattices,spin-1 Bosons trapped in hexagonal optical lattice system and the Jaynes-Cummings-Hubbard model in the QED-cavity arrays as well.By treating hopping terms as a perturbation,with the help of cumulant expansion technique,we calculate the single-particle Green's functions of an extended Bose-Hubbard model and a spin-1 Bose-Hubbard model on honeycomb lattices.Under single-chain ap-proximation,we obtain the quantum phase boundary equations and the corresponding phase diagrams of these two models by using resummed Green's function method.For the purpose of comparing our results with possible experimental data in the near future,the Time-of-Flight(TOF)pictures of both systems are also calculated.Furthermore,we develop a Green's function d' Alembert ratio test method which can be used to investigate different kinds of Bose-Hubbard systems on multipartite lattices.As an example,we calculate the phase diagrams of the Bose-Hubbard model on a Kagom6 lattice,which turn out to be exactly the same as that obtained from the so called effective potential approach.On the other hand,we also apply this newly-developed Green's func-tion approach to the scalar Bose systems on triangular optical lattices.By calculating the quantum loop correction of Green's function,the corresponding TOF pictures and visibil-ity are determined analytically,and the quantum correction effect on the visibility is also discussed.Meanwhile,by means of our Green's function d'Alembert ratio test method,we in-vestigate systematically the quantum phase diagrams of Jaynes-Cummings Hubbard model in triangular,square,honeycomb and Kagome lattice.By treating the hopping term in Jaynes-Cummings Hubbard model as perturbation,we calculate the phase boundaries ana-lytically,which are beyond mean field results for different polaritons number.