Vendredi 14 février 2014 à 14h15
Salle 234, Ecole de Physique

Scattering theory of topological phases in discrete-time quantum walks

Janos Asboth, Wigner Research Centre for Physics, Institute of Solid State Physics and Optics, Budapest

Quantum Walks are quantum mechanical generalizations of the random walk, where the "walker" is an object described by a wavefunction. Examples are single cold atoms trapped by standing wave laser beams, or photons in waveguide arrays. Quantum walks spread faster than the classical walk, and so could be useful for some quantum information processing tasks. Quantum Walks can be used to simulate topological insulators via their effective Hamiltonians. This is interesting, because this new class of materials has many curious properties, but few physical realizations. However, Quantum Walks also display unique topological phases, which can only be described by going beyond the standard theory of topological insulators [1]. We [2] give a complete classification of the unique topological phases of Quantum Walks using the scattering matrix approach, generalizing a technique that was developed for band insulators [3]. I show how to define a scattering matrix for the quantum walk, and how to express the topological invariants as matrix invariants of this matrix, computed at 0 and pi quasienergies (in natural units). For every combination of symmetries where the corresponding insulator is topological, this gives a pair of topological invariants controlling the number of topologically protected end states at 0 and pi quasienergy. The invariants can be efficiently calculated, and also directly measured in optical realizations of the quantum walk. [1]: Asboth and Hideaki, Phys. Rev. B 88, 121406(R) (2013) [2]: Tarasinski, Asboth, Dahlhaus, [3]: Fulga, Hassler, Akhmerov, Beenakker, Phys. Rev. B 83, 155429 (2011)