  Based on a variational principle with a stochastic forcing, we indicate that
the stochastic Schr\"odinger equation in Stratonovich sense is an
infinite-dimensional stochastic Hamiltonian system, whose phase flow preserves
symplecticity. We propose a general class of stochastic symplectic Runge-Kutta
methods in temporal direction to the stochastic Schr\"odinger equation in
Stratonovich sense and show that the methods preserve the charge conservation
law. We present a convergence theorem on the relationship between the
mean-square convergence order of a semi-discrete method and its local accuracy
order. Taking stochastic midpoint scheme as an example of stochastic symplectic
Runge-Kutta methods in temporal direction, based on the theorem we show that
the mean-square convergence order of the semi-discrete scheme is 1 under
appropriate assumptions.
