A non-Markovian approach for two dissipative quantum walks

We study the non-Markovian evolution of two free spinless distinguishable particles in a 1D lattice using a completely positive map. The Renewal theory is used to introduce, in a phenomenological way, the concept of disorder in the random time-interventions of the environment. If the waiting-time of the Renewal approach is exponential, we recover a semigroup description for the density matrix. Introducing a non-Poissonian random-time bath-interventions a non-Markovian evolution for the density matrix has been worked out. Under this scenario, we have studied the time evolution of the Quantum Coherence and Negativity measures for local and non-local initial conditions. We show the relevance of the (weak) non-Markovian evolution in calculating short-time correlations, while in the long-time regime a renormalized Markovian evolution appears.