Research Of Bose-Hubbard Mode In Dilute Ultracold Gases

Research of dilute ultracold atomic gases system is an important direction of microscopic quantum physics. It is a great path which we can take to understand the microscopic world of quantum physics, and covers atomic and molecular physics, quantum optics, condensed matter physics and other fields. Quantum phase transition in optical lattice is an important part of dilute ultracold gases research, such as phase transition of bosons from superfluid phase to Mott insulator phase, and it can be theoretically studied by using the Bose-Hubbard model. Due to the quantum uncertainty principle, the momentum and coordinate of micro-particle cannot be determined at the same time, so even at absolute zero temperature, the particles also have "zero-point energy", which can lead to quantum fluctuation. Therefore, at absolute zero temperature, the kinetic energy of particles is not zero, the competition between their kinetic energy and potential energy will lead to the existence of different phases and phase transition between them, which we call the quantum phase transition. As we can't achieve absolute zero temperature under the experimental conditions, so in experiment it is really the situation of finite temperature.In this paper, we use the Bose-Hubbard model to research the phase transition of bosons from superfluid phase to Mott insulator phase in optical lattice, and use Landau theory to deal with the quantum phase transition problem. In Landau theory, due to symmetry breaking, the value of order parameter varies from zero in disordered phase to non-zero in ordered phase. In theoretical calculation, starting from the Hamiltonian of system, we can find the free energy, and expand free energy about its order parameter, when the coefficient of the square part of order parameter equals zero, it corresponds to the phase transition boundary.In this paper, we study the zero temperature and finite temperature Bose-Hubbard model using different methods, including the mean-field method, variational method and field theory method, and obtain the quantum phase diagrams of Bose-Hubbard model in both the zero and finite temperature. Mean-field approach is direct, effective and relatively simple, but overlooks the impact of quantum fluctuations; and the variational method is a quantum correction to mean-field method; while the field theory is more general, it gives the corresponding classical solution and quantum corrections of each order. We compare the diagrams obtained by the three different methods with the numerical simulation results, and compare the pros and cons of three methods. The comparisons show that the results of field theory method is closest to the numerical simulation results, it is the most accurate analytical solution of superfluid to Mott insulator phase transition using Bose-Hubbard model so far.