One-dimensional discrete-time quantum walks show a rich spectrum of topological phases that have so far been exclusively analysed in momentum space. In this work we introduce an alternative approach to topology which is based on the scattering matrix of a quantum walk, adapting concepts from time-independent systems. For gapped quantum walks, topological invariants at quasienergies 0 and π probe directly the existence of protected boundary states, while quantum walks with a non-trivial quasienergy winding have a discrete number of perfectly transmistting unidirectional modes. Our classification provides a unified framework that includes all known types of topology in one dimensional discrete-time quantum walks and is very well suited for the analysis of finite size and disorder effects. We provide a simple scheme to directly measure the topological invariants in an optical quantum walk experiment.
We simulate various topological phenomena in condense matter, such as formation of different topological phases, boundary and edge states, through two types of quantum walk with step-dependent coins. Particularly, we show that one-dimensional quantum walk with step-dependent coin simulates all types of topological phases in BDI family, as well as all types of boundary and edge states. In addition, we show that step-dependent coins provide the number of steps as a controlling factor over the simulations. In fact, with tuning number of steps, we can determine the occurrences of boundary, edge st…
We study the simulation of the topological phases in three subsequent dimensions with quantum walks. We are mainly focused on the completion of a table for the protocols of the quantum walk that could simulate different family of the topological phases in one, two dimensions and take the first initiatives to build necessary protocols for three-dimensional cases. We also highlight the possible boundary states that can be observed for each protocol in different dimensions and extract the conditions for their emergences or absences. To further enrich the simulation of the topological phenomenas, …
Topological insulators host topology-linked boundary states, whose spin and charge degrees of freedom could be exploited to design topological devices with enhanced functionality. We experimentally observe that dissipationless chiral edge states in a spin-orbit coupled anomalous Floquet topological phase exhibit topological spin texture on boundaries, realized via a two-dimensional quantum walk. Our experiment shows that, for a walker traveling around a closed loop along the boundary in real space, its spin evolves and winds through a great circle on the Bloch sphere, which implies that edge-s…
We calculate the hybrid entanglement entropy between coin and walker degrees of freedom in a non-unitary quantum walk. The model possesses a joint parity and time-reversal symmetry or PT-symmetry and supports topological phases when this symmetry is unbroken by its eigenstates. An asymptotic analysis at long times reveals that the quantum walk can indefinitely sustain hybrid entanglement in the unbroken symmetry phase even when gain and loss mechanisms are present. However, when the gain-loss strength is too large, the PT-symmetry of the model is spontaneously broken and entanglement vanishes.…
Bulk-boundary correspondence is a fundamental principle for topological phases where bulk topology determines gapless boundary states. On the other hand, it has been known that corner or hinge modes in higher order topological insulators may appear due to "extrinsic" topology of the boundaries even when the bulk topological numbers are trivial. In this paper, we find that Floquet anomalous boundary states in quantum walks have similar extrinsic topological natures. In contrast to higher order topological insulators, the extrinsic topology in quantum walks is manifest even for first-order topol…
