Do they physically enter paradise after death or is it just spiritual experience?
Imam Reza(a.s.) quoted from his fathers (ancestors) that Prophet Muhammad (SAWW) said:
Shahid(martyre) is the initial person who enters the Janna (paradise)
(Majmoo'eh Varam Vol.2, Pg.121 )
Of course I assume we ought to pay attention that seemingly there are two kinds of Heaven(paradise). A heaven which is related to Barzakh, and the other heaven which apparently is the main heaven and eventually good people will go there (after the Day of Judgment) (so, seemingly, is has not mentioned that which paradise…)
I suggest you to see the site(source) below. So it consists of further info. Regarding Shahid. (note: unfortunately this is a Persian site (not English), hence I hope it could help you anyhow)