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Fabrice Debbasch

Université Paris 6

Discrete Time Quantum Walks In (1 + 1) Dimensions: Wave Propagation And Diffusive Transport

I will focus on discrete time quantum walks defined in one space dimension with a quantum coin acting on a two-dimensional Hilbert space. The quantum coin is then an operator in U(2) and depends on 4 angles, which are allowed to depend on both time and space.

I will first revisit the continuous limits of these walks. A new systematic and mathematically rigorous limiting procedure will be introduced. It will be shown that the continuous limit, in most cases where it exists, is represented by a Dirac equation in (1 + 1) dimensional space-time. The space-time dependence of the angles defining the walks transcribes into a coupling between the Dirac fermion and both an electric and a relativistic gravitational field. This new result opens up the possibility of performing laboratory experiments simulating relativistic charged spin 1/2 particles in arbitrary electromagnetic and gravitational fields.

The second part of the talk will be devoted to quantum walks loosing quantum coherence because at least one of the angles defining the walk is chosen randomly at each time step. I will present numerical evidence that these walks, as they loose their coherence, are best approximated in classical terms by relativistic stochastic or diffusion processes.

The seminar will end by my listing several open problems suggested by these results, highlighting in particular connections between quantum walks and geometry.