By your paper, it seems that THH (A) is simply given by the realization of the usual bar complex A \wedge A ... \wedge A (the wedge is over S here) by using a similar argument as the one given in the nlab (by the definition HH(X) := O(LX), they conclude that HH (X) for X = Spec (A) can be computed from the usual bar complex). However in
math.mit.edu/~nrozen/juvitop/dundas.pdf pag. 125, it's said that this naive definition is incorrect (the corrected one is defined in pag.126)