Study On Phase Of Bose-Einstein Condensate With Time

The quantum invariant theory proposed by Lewis and Riesenfeld is a powerful tool for treating systems with time-dependent Hamiltonians. We first introduced the theory of Bose-Einstein Condensate and its history, and then introduced the method of unitary transformation based on the quantum invariant theory, geometric phase theory, the general solution of question with time, the quasi-algebra contract by subspace of engine state of invariant. By using of the Lewis-Riesenfeld invariant theory, Geometric phase for a time-dependent system of double-well Bose-Einstein condensate the dynamical and geometric phases of a weakly interacting Bose system with a time spontaneous U(1) symmetry breaking were studied, and their dynamical phases ,geometric phases and geometric Aharonov-Anandan phases were gotten.