A Study On The Characteristics Of An Entangled Bose - Einstein Condensate System

The special properties of an entangled Bose-Einstein condensate system have been explored in this paper. This system is composed of two different types of atoms each type of which has two pseudo-spins. Differences between this system and general multi-component system is that there are spin exchanges between two species. Spin exchange interaction gives rise to many special properties of the entangled system, such as existence of entanglement in a wide range of parameter, polarization and etc.Theoretical and experimental achievements related to our work are introduced in the first three chapters.Characterization of this polarized entangled BEC is studied in the fourth chap-ter. The Hamiltonian of the system is simplified by giving some special condition of parameter so as to solve in an analytical way. We have derived all ground state of the simplified Hamiltonian. Many of these ground states have the character of polar-ized condensation, and we can defined them as "polarized entangled condensed state" whose feature is that in addition to condensate of interspecies singlet pairs, there are unpaired atoms with spins polarized in the same direction. We draw the phase dia-grams of ground state distribution of all parameter regime from which we can see that there are phase transitions occurred at some boundaries when to cross. All the ground states are expressed in terms of Bose operator. We calculate the entanglement entropy of two typical ground states. we have found the generalized Gross-Pitaevskii equations governing the condensate wave function. The interspecies entanglement and polariza-tion significantly affects the generalized Gross-Pitaevskii equations as an interesting interplay between spin and orbital degrees of freedom.Distributions of physical quantities of system in a harmonic trap are studied in the fifth chapter. We get the relationship between energy and condensate atom number and draw the density profiles of atom and spin from the generalized Gross-Pitaevskii equation. We can see that entanglement parameter and condensate atom number both affect the atom density distribution. The atom density profiles have some similar prop-erties to general two component system, and but new characters of its own which is that entanglement make the four density profiles become closer and the region where the four component coexist become larger. Spin density distributions also have its own features. Density distributions of total spin and spin z component become to zero in a very long distance from origin. In view of the over all situation, the peak density of total spin is near by the origin and the density of spin z component have an experience from down to up then to zero. These intuitionistic pictures provide good references to the experimental realization of this peculiar system.