The phrase "profound sense" refers to a deep or intense feeling or understanding about something. It usually signifies that the person experiencing it has a strong and meaningful connection to the subject or has gained important insight from it.
Out of these shared feelings — and despite the challenges of raising a child — couples often experience a new, profound sense of intimacy. (powerofpositivity.com)
That was contemplation in its simplest but also most profound sense. (religion-online.org)
In a very profound sense, these paintings break us open to the religious questions of our time. (religion-online.org)