你说的情况，应该是形容词asleep，不是动词sleep。

1. sleep作为动词，当然可以用deeply修饰，表示“睡得深沉”，例如你的句子。

2. asleep是表语形容词，这个词的基本含义是“睡着的”，指已经进入了睡眠的状态，与awake相对。

3. 除了be以外，asleep经常跟fall连用：When you fall asleep, you start sleeping. 例如：

Her breathing became steady and she fell asleep.她的呼吸变得均匀，便睡着了。

Sam snuggled down in his pillow and fell asleep.萨姆舒服地枕着枕头进入了梦乡。

4. fast和sound两个程度副词，经常放在asleep之前修饰该词。Someone who is fast asleep or sound asleep is sleeping deeply. 例如：

They were both fast asleep in their cots.他们俩在婴儿床上酣睡着。

Turning over, she was soon sound asleep again.她翻了个身，很快又沉沉睡去。

5.如果表示“很快入睡”，较少说sleep fast，应说：

Because I am tired, I fall asleep quickly.因为我很疲劳，因此很快就睡着了。

He lay down on the sofa and soon fall asleep.她躺在沙发上很快就睡著了。