Should my old employer know the name of my new one?

Please bear with me as I'm going through all this for the first time.

I resigned, and I'd have to have a discussion with my manager where she might ask me who my new employer is. I'm trying to negotiate my notice period to a lesser number of days as my joining date is earlier and there's no scope for changing that. The new company is ready for a NP buyout too.

Would I need to tell her the new company I'm joining or would saying "I'm taking a break and travelling" work too

? I want to leave my current company in good terms and ofc don't want to have problems with experience letter.