To the little tray....
In Japan, you should not hand in the cash directly to the casher's hands. Always leave the cash into the little tray at the casher or if they do not have the tray, you should leave it to the casher-desk. This is to avoid any direct conflicts with the person in charge of the casher in case of lost cash. It will be the same when you are receiving changes too.
Although there is not direct contact, don't forget the "Arigato-Gozaimasu" when leaving :) And enjoy shopping!!! Yay!