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We have investigated the time-evolution of a free particle in interactionwith a phonon thermal bath, using the tight-binding approach. A dissipativequantum walk can be defined and many important non-equilibrium decoherenceproperties can be investigated analytically. The non-equilibrium statistics ofa pure initial state have been studied. Our theoretical results indicate thatthe evolving wave-packet shows the suppression of Anderson's boundaries(ballistic peaks) by the presence of dissipation. Many important relaxationproperties can be studied quantitatively, such as von Neumann's entropy andquantum purity. In addition, we have studied Wigner's function. Thetime-dependent behavior of the quantum entanglement between a free particle -inthe lattice- and the phonon bath has been characterized analytically. Thisresult strongly suggests the non-trivial time-dependence of the off-diagonalelements of the reduced density matrix of the system. We have established aconnection between the quantum decoherence and the dissipative parameterarising from interaction with the phonon bath. The time-dependent behavior ofquantum correlations has also been pointed out, showing continuous transitionfrom quantum random walk to classical random walk, when dissipation increases.

Non-equilibrium transition from dissipative quantum walk to classical random walk