Quantum spin mixing dynnamics in a ferromagnetic spin-1 Bose-Einstein condensate beyond single mode approximation

Precision measurements, based on spinor Bose-Einstein condensates and beyond the standard quantum limit, often employ a single mode approximation, i. e., all spin components share the same spatial mode. However, the single spatial mode approximation in a ferromagnetically interacting atomic spinor condensate becomes inaccurate due to magnetic field gradients or strong spin interactions. By assuming three modes of a spin-1 condensate, we derive spin mixing Hamiltonian of the system beyond the conventional single mode approximation. At zero bias magnetic field, the fraction of spin zero component and the condensate spin length are found decreasing with the separation factor increasing. For nonzero bias field, the regime of the broken-axis phase shrinks as the separation factor increases. In addition, final conversion efficiency and effective spin length are noticeably alternated during the generation of twin-Fock and balanced Dicke state. These results provide a useful clue to further improve the quantum metrology based on spinor condensates.