  Dynamics of the double-well Bose-Einstein condensate subject to energy
dissipation is studied by solving a reduced one-dimensional time-dependent
Gross-Pitaevskii equation numerically. We first reproduce the phase space
diagram of the system without dissipation systematically, and then calculate
evolutionary trajectories of dissipated systems. It is clearly shown that the
dissipation can drive the system to evolve gradually from the $\pi$-mode
quantum macroscopic self-trapping state, a state with relatively higher energy,
to the lowest energy stationary state in which particles distribute equally in
the two wells. The average phase and phase distribution in each well are
discussed as well. We show that the phase distribution varies slowly in each
well but may exhibit abrupt changes near the barrier. This sudden change occurs
at the minimum position in particle density profile. We also note that the
average phase in each well varies much faster with time than the phase
difference between two wells.
