Quantum Entanglement in Spinor Bose-Einstein Condensates and the Preparation of Dicke State

The spin-1 Bose Einstein Condensate system can exhibit many interesting phenomena under the effect of linear and quadratic Zeeman effects, resulting from the rich spin texture and the long range correlation in the system. The manybody ground state is shown and phase transition and entanglement properties are discribed. We also propose a method to prepare the Dicke state, with time evolution from a state relatively easy to realise from current experimental cooling methods. The entanglement depth, which based only on the measuring of collective spin operators, is used to characterise the entanglement in the final prepared state. Various noises in the preparation process are also discussed.