be asleep

spát
"He waited until his daughter was asleep, then tiptoed quietly out of the room." (Čekal, dokud jeho dcera neusne/nebude spát a potom odešel tiše po špičkách z pokoje.)

fall asleep

usnout
"I fell asleep as soon as my head hit the pillow." (Já jsem usnul v momentě, kdy má hlava dopadla na polštář.)