  In this manuscript, we propose efficient stochastic semi-explicit symplectic
schemes tailored for nonseparable stochastic Hamiltonian systems (SHSs). These
semi-explicit symplectic schemes are constructed by introducing augmented
Hamiltonians and using symmetric projection. In the case of the artificial
restraint in augmented Hamiltonians being zero, the proposed schemes also
preserve quadratic invariants, making them suitable for developing
semi-explicit charge-preserved multi-symplectic schemes for stochastic cubic
Schr\"odinger equations with multiplicative noise. Through numerical
experiments that validate theoretical results, we demonstrate that the proposed
stochastic semi-explicit symplectic scheme, which features a straightforward
Newton iteration solver, outperforms the traditional stochastic midpoint scheme
in terms of effectiveness and accuracy.
