The ranks/rates for this role would obviously vary from navy to navy and from submarine to submarine based on the role and size of the boat.
In the US Navy, the chief sonar operator on an SSN, SSBN, or SSGN would probably be rated as a Chief Sonar Technician (Submarine). This is at the E7 (Chief Petty Officer) pay grade, so he'd probably be referred to informally as "Chief." His rate would be as an STS (Sonar Technician (Submarine).
The chief would likely play more of a supervisory role during his watches and he might or might not not actively play a role as an "operator" actively working the sonar gear. I don't know what the exact Navy policy on this is.