Internal Josephson Effects in Spinor Dipolar Bose--Einstein Condensates

We theoretically study the internal Josephson effect, which is driven by spin exchange interactions and magnetic dipole-dipole interactions, in a three-level system for spin-1 Bose--Einstein condensates, obtaining novel spin dynamics. We introduce single spatial mode approximations into the Gross--Pitaevskii equations and derive the Josephson type equations, which are analogous to tunneling currents through three junctions between three superconductors. From an analogy with two interacting nonrigid pendulums, we identify unique varied oscillational modes, called the 0--$\pi$, 0--$running$, $running$--$running$, 2n\pi & running--$2\pi$, $single nonrigid pendulum$, and $two rigid pendulums$ phase modes. These Josephson modes in the three states are expected to be found in real atomic Bose gas systems.