用作副词的 sound 和 soundly 都可以表示睡觉睡得很沉或很酣畅，它们俩的分工通常是：修饰动词sleep 时通常用副词 soundly（美国英语也用 sound），而修饰形容词 asleep 时通常用副词 sound（很少用 soundly）。请看相关例句（均摘自词典）：

1. sleep soundly

The baby slept soundly all night. 宝宝整晚都睡得很香甜。

She was too soundly asleep to hear Stefano’s return. 她睡得太熟，没有听见斯蒂芬诺回来。

She was sleeping so soundly that we had difficulty arousing her. 她睡得太香了，我们很难叫醒她。

One moment she was sleeping soundly—the next she was wide awake. 她前一刻还睡得深沉，下一刻就十分清醒了。

2. sleep sound

He sleeps sound every night. 他每晚都睡得很熟。

一般认为 sleep sound 主要用于美国英语，且不如 sleep soundly 常见，如下面一句摘自《韦氏高阶》：

She slept sound. [=(more commonly) soundly] 她睡得很香。

3. sound asleep

The children were all sound asleep in bed. 孩子们都熟睡在床。

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

He was sound asleep within moments of getting into bed. 他上床没多久就进入了梦乡。

He had fallen sound asleep in the chair by the fire. 他坐在火炉边的椅子上酣然入睡。

至于用副词 soundly 修饰动词 asleep，确实比较少见，但也不是完全不可以。如：

She was too soundly asleep to hear Stefano’s return. 她睡得太熟，没有听见斯蒂芬诺回来。（柯林斯词典）

另外，《牛津英语搭配词典》（Oxford Collocations Dictionary for Students of English, 2nd Edition）也提供了类似的搭配：