Someone who is asleep is sleeping.
My four-year-old daughter was asleep on the sofa.
我4岁的女儿在沙发上睡着了。
fall asleep
When you fall asleep, you start sleeping.
Sam snuggled down in his pillow and fell asleep.
萨姆舒适地枕着枕头睡着了。
Someone who is fast asleep or sound asleep is sleeping deeply.
They were both fast asleep in their beds.
他们俩在各自的床上熟睡了。