being pedantic about the question :
Emergency descent is in my experience a memory drill to be used in a cabin hi-alt / loss of cabin pressure. The contents of this drill are typically :
crew masks don, flow rate 100%, crew communications establish, announce emergency descent, thrust levers flight idle, speed MMO/vMO, airbrakes out
check MSA , inform atc etc ...
Rapid descent ... as above but with no emergency, therefore no drill and no masks and with clearance from or at the request of atc.