No, our souls don’t sleep after death. Upon death, our soul leaves our body, is judged and goes to either Heaven, Purgatory (a place of preparation for eventual entry into Heaven) or Hell. From there, if a soul is in Purgatory, it can advance to Heaven when God determines it has been suitably prepared. If a soul is in Heaven or Hell, it just stays there till the end of the world, at which point our bodies will be resurrected and rejoined with our souls, then sent on to the new world God will create or condemned once again by God to eternal punishment.
No sleeping involved.
Sleep as we know it on earth is a bodily function designed as part of the physical operation of the body. Souls after death are not in a physical body and have no need to sleep.