None of the above. If you want to describe 'sleeping well,' then the best expression is simply to say 'he slept well,' or 'he had a really good sleep.' You could also say 'he had a really refreshing sleep.' 'Heavy sleep,' and 'dead sleep,' are not used in our language. You might, very occasionally hear somebody say 'he was sleeping like the dead,' to indicate that a person was in a deep sleep. 'Deep sleep,' has two meanings. Firstly it can mean that somebody will sleep through almost any noise. Secondly, it is a recognised stage of the nightly sleep cycle. All normal individuals enter a phase of deep sleep every night. See, for example: https://en.m.wikipedia.org/wiki/Slow-wave_sleep

I would use "deep sleep", because it means "to sleep well". I`m not so sure about the two others.

I think sleep well are better than those u mentioned.

I would use deep sleep/Heavy sleep sounds awkward and dead sleep sounds like you're not having proper sleep.

As an American, my first choice would probably be deep sleep, but all three could be used.