Help adding a 3 way switch
It looks like your current switch is on a switch loop.  The wires you've labeled A1 and A2 are not neutral and hot, they are always-hot and switched-hot.

That means that to convert this to a 3-way configuration, you need to wire it like this:



As @Harper pointed out in the comments, since A1 and B2 aren't acting as neutrals, you should mark them with black electrical tape to designate them as hot.  It's also a good idea to mark the ends of B1 and B3 with yellow electrical tape to flag them as travelers to make it clear what purpose they're serving as well.

I've not drawn the grounds in my diagram, but they should be connected to all of the other grounds.