Reincarnation is nonsense. There is not a single evidence nor any scientific reason to believe reincarnation is anything but a stupid dogmatic and romantic idea that one can live forever.
While waiting at a bus stop about 25 years ago a girl with a clipboard asked if I would help with a quick survey. One of the questions was “If possible who would I return if I got a second life?” I answered by saying “Myself”. She seemed upset by this and ended the survey abruptly. Turned out from overhearing the next person’s answers that she was a Scientologist looking for fresh recruits. They still accost people at the same spot so I must do another survey. My answer will be Tom Cruise.
I will be reincarnated in accordance with the laws of conservation of mass and conservation of energy.
Before you can have a discussion about reincarnation, you need to define a soul.
That being said, reincarnation on atomic level is a true phenomenon (e.g. iron atoms in your body were born in supernova(s)).
To believe in reincarnation, you pretty much have to believe in spirits (souls, ghosts). I would expect that anyone who doesn't believe in the big spirit in the sky doesn't believe in ghosts, spirits, or souls. They're kind of necessary for reincarnation, as I understand it.