How do doctors sleep babies for surgerys?

Same as adults: The same way that they do for adults, but the doses are smaller and the equipment is tinier.

General anesthesia : In the vast majority of cases when an infant requires surgery the anesthesia administered is general anesthesia by inhalation. After monitors are attached to the baby a face mask is gently held close to the babies airway so he/she can breathe in nitrous oxide or other anesthesia gas such as sevoflurane (a derivative of ether). These gases have a sweet odor and they gradually fall asleep.