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RESEARCH PRODUCT

Unveiling two-dimensional discrete quantum walks dynamics via dispersion relations

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The discrete, or coined, quantum walk (QW) [1] is a process originally introduced as the quantum counterpart of the classical random walk (RW). In both cases there is a walker and a coin: at every time step the coin is tossed and the walker moves depending on the toss output. Unlike the RW, in the QW the walker and coin are quantum in nature what allows the coherent superpositions right/left and head/tail happen. This feature endows the QW with outstanding properties, such as making the standard deviation of the position of an initially localized walker grow linearly with time t, unlike the RW in which this growth goes as t1/2. This has strong consequences in algorithmics and is one of the reasons why QWs are receiving so much attention from the past decade. However the relevance of QW's is being recognized to go beyond this specific arena and, for example, some simple generalizations of the standard QW have shown unsuspected connections with phenomena such as Anderson localization and quantum chaos. Moreover, theoretical and experimental studies evidence that the QW finds applications in outstanding systems, such as Bose-Einstein condensates, atoms in optical lattices, trapped ions, or optical devices, just to mention a few (see [2] and references therein). Continuous versions of the QW exist as well, whose relationship with the discrete QW has been discussed in [3].