What does it mean to "sleep soundly"? What does it mean to "sleep deep"?
We can unpack these phrases with the scientific discoveries about sleep physiology of the past several decades.
"Sound" means whole and possessing integrity. Sound sleep is a description given to a night’s sleep that is satisfying to the sleeper. This is subjective, of course, and probably in relation to the sleeper’s experience of recent sleep. A sound sleep for a 50-year old might be described as troubling by a 20-year old. And that’s a good word for the opposite: troubling. The opposite of sound sleep is troubled sleep.
If we were to hook up a polysomnography machine to a person getting sound sleep, we would see few nighttime awakenings, no significant breathing problems, and a decent amount of time spent in both slow-wave deep sleep and REM sleep. "Decent amount" is subjective, but we can ballpark it as 90 to 120 min per night of deep sleep and 90 to 130 min per night of REM sleep for adults. (Too much REM is associated with depression. There doesn't seem to be any such thing as too much deep sleep.) Sound also somewhat tangentially refers to the depth of sleep and how susceptible the sleeper is to awakening by noise. This concept is probably close to what people mean by sleep deeply.
At Sleepdex, we like to divide the night's sleep into light sleep, deep sleep, and REM sleep (we didn't invent these classifications.) Deep sleep is slow-wave sleep (called so because of the patterns of the EEG printouts) or Stage 3 sleep (according to one classification scheme) or Stages 3 and 4 sleep (according to other classification schemes).
What is the difference between quiet wakefulness and sleep? One difference is the ability or tendency to react mindfully to external stimuli. What is the difference between sleep and coma? Sleep is reversible; coma is permanent. But coma is also a state in which the subject cannot be awakened by shaking or loud noises, the way a sleeper can.
The depth of sleep is generally taken to mean how likely the sleeper is to be awakened from an external stimulus. Children are difficult to awaken not because they are lazy but because they are sleeping deeply. Even within a single stage of sleep (as measured by EEG) the depth varies as measured by sensitivity to acoustic stimuli. It has also been shown that people who have more "sleep spindles" on the EEG readings during a normal night of sleep have more tolerance for noise.
Also a hypnogram — the output of a polysomnogram — is typically printed so Stages 3 and 4 (the deep sleep) are shown lower on the page than Stages 1 and 2. This is just convention, but it further promotes the idea that Stages 3 and 4 are deeper sleep than the earlier stages.
By this measure, some people do sleep deeper than others. Compare two people in stage 3 slow-wave sleep and one may be easier to awaken than the other. Does this mean the deeper sleeper is getting more of the slow-wave sleep benefits than the lighter sleeper? No. The characteristics and benefits of different stages of sleep are relative to each other, not compared among people. Although some people are lighter or deeper sleepers than others (as measured by susceptibility to awakening), that doesn't mean they are getting more or less out of sleep.
On an EEG measure, alpha waves (8-13 Hz) are a signature characteristic of the waking state. They significantly decline when the person falls asleep, but they do not disappear. Scientists have found that the level of alpha waves correlates with the depth of sleep. When alpha level is stronger, the sleeper is more susceptible to awaking from external stimuli.
Now what produces alpha waves is not understood. The thalamus section of the brain influences activity in the cortex that give rise to alpha activity, it appears, leading to a hypothesis that thalamus relays external stimuli to "cortical processing centers where it is capable of interrupting sleep".