Properties Of One Dimension Bose System On Periodic Boundary Condition At Absolute Zero Temperature

Cold atomic physics is a discipline to research the characteristics of atoms in ul-tracold environment. Because of the realization of Bose-Einstein condensation(BEC)on experiment, the theories and experiments of cold atomic physics has quickly beendeveloped and cold atomic physics has also become a important research direction. Incold atomic systems, the quantum effect is very strong but the classical thermodynamicseffect is very week, so people try to use cold atoms to simulate many quantum systems.And the theories and experiments of quantum simulation drive the developments ofmany body physics, condensate matters and so on. Specially the experimental realiza-tion of cold atoms trapped in optical lattices. This experiment make people can use coldatomic systems to effectively simulate the controllable strongly interaction system. Italso make the research of quantum phase transition in strongly interaction system withthe method of quantum simulation become the focus of the physical research.Quantum information theories is a very important discipline to research the cor-relation and entanglement of quantum systems. Because of the development of thequantum simulation using cold atomic systems, the research of quantum correlationand entanglement of condensate matters become more and more important. But inalmost all many-body systems, there are interactions between particles. It makes theHamiltonian of the system become very complex and can not be solved by analyticalmethods. In this situation, we always use mean-field theory to research these systemsand it is very useful in may situation. But the mean-field theory averages the quantum?uctuation of systems, it can not describe the quantum correlation and entanglementof systems. So we have to develop some new methods beyond mean-field theory toresearch the quantum many-body systems.The model we will research is Bose-Einstein condensates on a ring with periodicscattering length. There is quantum phase transition in this system. We use two meth-ods beyond mean-field theory to research the characteristics of this system includingenergy, correlation, entanglement and so on. We also analyse the reason of the quan-tum phase transition.1. Because this system has periodic boundary condition, we can expand the Hamil-tonian of this system on angular momentum basis. Then we can use a modified Bogoli-ubov method to search any possible Bose condensate ground state of the system and research the properities of the excitation spectrums. We found that quantum phase tran-sition of this system have two different mechanism from researching the transformationof ground state and Bogoliubov excitation with the change of the outside parameters.The first is Bogoliubov excitation spectrums become pure imaginaries and the secondis that different energy spectrums cross with each other. Compare with the result ofmean-field imaginary time evolution, we can find that for different modified periods,the mechanisms of quantum phase transition are different. When the modified periodis two, the critical points of two different mechanisms of phase transitions is adjacentwith each other so two different mechanisms compete with each other. But when themodified period is larger than two, only the cross of different energy spectrums canbring on the quantum phase transiton.2. From diagonalize of Hamiltonian, we can get the accurate ground state of thesystem when the particle number is small. But when we increase the particle num-ber, the dimensions of Hilbert space of this system increase exponentially. So we cannot use diagonal method to calculate the ground state using classical computer whenparticle number increase. For overcoming this difficulty, we use a new method, TimeEvolving Block Decimation(TEBD) algorithm. In TEBD algorithm, state is describedby the entanglement between particles. If the state involve all entanglements betweenparticles, the answer of TEBD algorithm is the same with the diagonal method. Butin many situation, we can ignore the less entanglement part of the system and the newstate is very similar with the original state. But the description of the new state is sim-pler than the original state. Using this method, we can calculate the system with moreparticles.In this paper, we use TEBD algorithm to calulate the ground state of the system.Because the state is described by entanglement in this method, we can easily calculatethe correlation and entanglement of this system. From researching the quantum groundstate of this system approaching the critical point, we find that the Bose condensatestate in mean-field theory is close to a superposition state of the quantum ground stateand excited states which ignore the quantum correlation. Close to the critical point, thestructure of the ground state will change obviously. And this change will in?uence thechange of quantum correlation and dynamic evolution. Using the TEBD algorithm, wecan not only get the quantum correlation of this system, but also accurately calculatethe dynamic evolution of this system. From calculating these physical characteristicsclose to the critical point, we can get these is a obvious change of the structure of theground state at the critical point. So we can find that there is a quantum phase transition.