So they have a big stream of sub atomic particles, how do they actually bring it altogether perfectly? I mean their is zillions of particles, what do they use to put all the particles into atoms, then molecules, and so on into the perfect form of the subject?
Are you looking for a in-universe answer or a real-universe answer? If you seek for the last one:
We all know that many parts of Star Trek are imaginable in todays physics. This may be Impulse drive, phasers, warp drive or leaps of time. But Beaming, unfortunatelly, is fantasy. Gene Roddenberry once stated for TOS that Beaming has been "invented" for two reasons:
- You don't want to see shuttles starting, flying and landing instead of one Beaming action every episode.
- Creating a shuttle scene every time instead of beaming simply has been to expensive back in those days.
Sorry, if youre looking for a in-universe answer, because I cant answer that. Maybe it is stated in the Technical Manual, which I didn't read yet.
In-universe, the transporter's functions are quite well described.
The subject is scanned, dematerialised, converted into a stream of matter and then "transported" to the remote destination along an 'annular confinement beam'. At the destination end, this beam then creates a 'matrix' which the particles fill, a bit like pouring water into a jug.
This excerpt from the "TNG Technical Manual" gives a little more info;