Quantum walk, as a natural and straightforward generalization of the classical random walk to the quantum world, has nowadays been a topic of great interest to both theorists and experimentalists. Actually, quantum walks has already been realized in a variety of quantum systems. For example, an implementation of continuous-time quantum walk of two identical bosons on a one-dimensional optical lattice has been reported in [Science 347, 1229 (2015)]. Also, quantum walks of phonons has been realized in an array of coupled single-mode waveguides [Phys. Rev. Lett. 109, 106402(2012)]. These exciting experimental breakthroughs have stimulated a lot of theoretical works. Fundamental effects of quantum statistics, interactions, disorders, defects, and hopping modulations on quantum walks have been intensively investigated by theorists. In collaboration with Prof. Shu Chen from the Institute of Physics, Chinese Academy of Sciences, in our latest work, we investigated the intriguing effect of periodically modulated hopping on the continuous-time quantum walks of one and two identical particles in one-dimensional lattices.

The quantum system investigated here is described by commensurate off-diagonal Aubry-Andre-Harper model under periodic or open boundary conditions (Left and Right in the above Figure). Due to the topological nature of this model, exotic edge states have been found existing in its spectrum. Close and careful observation of single-particle quantum walks in this model reveals that these exotic topologically protected edge states have an interesting trapping effect on the quantum walker initiated from the boundary sites of the lattice making the quantum walker localized and also, an intriguing repulsion effect on the quantum walker set out from lattice sites inside the bulk prohibiting the quantum walker from reaching the boundary sites.

For quantum walks of two identical fermions, it is found that the nearest-neighbor interaction could dramatically enhance the repulsion effect of these exotic edge states（See the Left two figures below）. Furthermore, an interesting pinning effect is also revealed in the quantum walks of two identical fermions initially positioned on the two leftmost sites of lattice（See the Right two figures below).

These interesting effects should be observable with existing experimental platforms, for example, an array of coupled photonic waveguides written in bulk glass using femtosecond laser micro-fabrication technology as used in [Phys. Rev. Lett. 109, 106402(2012)]. And they may have potential applications in the designing of micro-architectures for quantum information and quantum computing. Imagine that two optical signals, one is injected into the photonic waveguide at the boundary, the other is injected into a photonic waveguide in bulk. By modulating the hoppings, these two signals could be made to meet each other or transmit separately. Predictions that call for future experimental verification are also given therein.

This work has been published, see http://journals.aps.org/pra/abstract/10.1103/PhysRevA.95.013619.