Abstract

Discrete-time Quantum walks (DTQW) are a quantum generalisation of classical random walks. They receive many attention these last few years as they provide a powerful platform for quantum computation and quantum algorithm. They also can be understood as stroboscopic simulator for hamiltonian systems and, in that sense, it is possible to mimic well-known results of tight-binding hamiltonians. By studying DTQW under a transverse magnetic field, and plotting the corresponding quasi-energies with respect to the magnetic flux, we obtain the Floquet-Hofstadter butterfly, a phase diagram for topological insulator. Another result of hamiltonian systems one can reproduce with DTQW are the Aharonov-Bohm cages which correspond to an extreme confinement of electrons on some special regular graphs such that the diamond chain or the T_3 tiling in a transverse magnetic field . When the dimensionless magnetic flux per plaquette f equals a critical value f_c=1/2, the destructive interference forbids the particle to diffuse away from a small cluster. The corresponding energy levels pinch into a set of highly degenerate discrete levels as f->f_c. Cages also occur for discrete-time quantum walks but require specific coin operators. The Floquet-Hofstadter butterfly displaying pinching at a critical flux f_c and that may be tuned away from 1/2. The spatial extension of the associated cages can also be engineered. A study of different disorder to break cages will also be discussed.