Quantum Random Walks in Honeycomb Lattice

The term quantum random walks (QRW) appeared in the early 1990 as the quantum analogous of classical random walks (CRW) (Physical Review A 48, 1687, 1993). Thus, while the CRW is defined by probability of a stochastic event occurs, the QRW is defined by probability amplitudes associated with unitary transformations. One feature that differs a QRW of a CRW is the mean square displacement $[\Delta r]^2$. For QRW $[\Delta r]^2 \sim t^2$ and to CRW $[\Delta r]^2 \sim t$, where $t$ is time. For an introduction to QRW we suggest the following reference (Contemporary Physics, v. 44, n. 4, p. 307-327, 2003). The QRW can be divided into two groups, one with an discrete time evolution, and other with a continuous time evolution. However, the system is always implemented on discrete space (lattices or graphs). The discrete time QRW version can be formulated in terms of two entirely equivalent models, the coin model and the scattering model, we will adopt the scattering model, which we consider more physically intuitive. This model consists of a particle moving through the edges lattice and the sites act as scattering centers. Thus, in a time step, the particle to focus on a site is scattering towards the edges connected to site. These processes of scattering are weighted by probability amplitudes given by scattering matrix (Physical Review A 68, 3, 032314 (2003). Since they were proposed the QRW are well studied in the linear lattice. However, there are few studies that dedicated to analyze them in the honeycomb lattice. Furthermore, since the QRW have been employed in the study of electronic transport properties of some quantum systems (Physical Review Letters 94, 100602, 2005), we will study the in the honeycomb lattice in order to a possible application in the theoretical study of the properties electronic and the transport system involving graphene and other materials that exhibit the honeycomb pattern, for example carbon nanotubes and fullerenes. In this work we present a formulation for the QRW in honeycomb lattice, compare the classical and quantum random walks and found the expected behavior for mean square displacement in those two systems.