Low Density Limit and the Quantum Langevin Equation for the Heat Bath

We consider a repeated quantum interaction model describing a small system H(S) in interaction with each of the identical copies of the chain circle times(N*) C(n+1), modeling a heat bath, one after another during the same short time intervals [0, h]. We suppose that the repeated quantum interaction Hamiltonian is split into two parts: a free part and an interaction part with time scale of order h. After giving the GNS representation, we establish the connection between the time scale h and the classic allow density limit. We introduce a chemical potential mu related to the time has follows: h(2) = e(beta mu). We further prove that the solution of the associated discrete evolution equation converges strongly, when h tends to 0, to the unitary solution of a quantum Langevin equation directed by the Poisson processes.