Quantum walks and orbital states of a Weyl particle

Abstract

The time-evolution equation of a one-dimensional quantum walker is exactly mapped to the three-dimensional Weyl equation for a zero-mass particle with spin $1∕2$, in which each wave number $k$ of the walker's wave function is mapped to a point $\mathbf{q}(k)$ in the three-dimensional momentum space and $\mathbf{q}(k)$ makes a planar orbit as $k$ changes its value in $[\ensuremath{-}\ensuremath{\pi},\ensuremath{\pi})$. The integration over $k$ providing the real-space wave function for a quantum walker corresponds to considering an orbital state of a Weyl particle, which is defined as a superposition (curvilinear integration) of the energy-momentum eigenstates of a free Weyl equation along the orbit. Konno's novel distribution function of a quantum walker's pseudovelocities in the long-time limit is fully controlled by the shape of the orbit and how the orbit is embedded in the three-dimensional momentum space. The family of orbital states can be regarded as a geometrical representation of the unitary group U(2) and the present study will propose a new group-theoretical point of view for quantum-walk problems.