It's "addicted to." I don't know why and can't give a rule. We say someone is "on" a drug--medical or recreational, legal or illegal, meaning simply the person is taking and is under the influence of the drug. "He's on chemo, and having a tough time." "She's groggy after the surgery, they have her on morphine." We say someone is "under" sedation or anesthesia.

Always addicted to