I have a 13‑day‑old baby myself, and I can say this with confidence. If your wife feels well enough, sta y up through the night and sleep during the day. That way she gets the rest she needs, you get the rest you need, and you can watch over the baby.
The first few days are terrifying, but once the baby’s weight steadies around day 7, it only climbs from there.
Also, be sure the baby burps, sanitize the bottles carefully, and keep track of feedings and diaper changes your pediatrician will want that information to decide whether to refer you to a lactation specialist or confirm that the baby is completely healthy.