The quantum-mechanical counterpart of a classical random walk offers a rich dynamics that has recently been shown to include topologically protected bound states (zero modes) at boundaries or domain walls. Here we show that a topological zero mode may acquire a dynamical role in the presence of nonlinearities. We consider a one-dimensional discrete-time quantum walk that combines zero modes with a particle-conserving nonlinear relaxation mechanism. The presence of both particle-hole and chiral symmetry converts two zero modes of opposite chirality into an attractor-repeller pair of the nonlinear dynamics. This makes it possible to steer the walker towards a domain wall and trap it there.
The quantum-mechanical counterpart of a classical random walk offers a rich dynamics that has recently been shown to include topologically protected bound states (zero modes) at boundaries or domain walls. Here we show that a topological zero mode may acquire a dynamical role in the presence of nonlinearities. We consider a one-dimensional discrete-time quantum walk that combines zero modes with a particle-conserving nonlinear relaxation mechanism. The presence of both particle-hole and chiral symmetry converts two zero modes of opposite chirality into an attractor-repeller pair of the nonlinear dynamics. This makes it possible to steer the walker towards a domain wall and trap it there.
Discrete-time quantum walks have been shown to simulate all known topological phases in one and two dimensions. Being periodically driven quantum systems, their topological description, however, is more complex than that of closed Hamiltonian systems. We map out the topological phases of the particle-hole symmetric one-dimensional discrete-time quantum walk. We find that there is no chiral symmetry in this system: its topology arises from the particle-hole symmetry alone. We calculate the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ topological invariant in a simple way that is consistent with a general definition for one-dimensional periodically driven quantum systems. These results allow for a transparent interpretation of the edge states on a finite lattice via the the bulk-boundary correspondence. We find that the bulk Floquet operator does not contain all the information needed for the topological invariant. As an illustration to this statement, we show that in the split-step quantum walk, the edges between two bulks with the same Floquet operator can host topologically protected edge states.
Discrete-time quantum walks (DTQW) have topological phases that are richer than those of time-independent lattice Hamiltonians. Even the basic symmetries, on which the standard classification of topological insulators hinges, have not yet been properly defined for quantum walks. We introduce the key tool of time frames, i.e., we describe a DTQW by the ensemble of time-shifted unitary time-step operators belonging to the walk. This gives us a way to consistently define chiral symmetry (CS) for DTQW's. We show that CS can be ensured by using an ``inversion symmetric'' pulse sequence. For one-dimensional DTQW's with CS, we identify the bulk $\mathbb{Z}\ifmmode\times\else\texttimes\fi{}\mathbb{Z}$ topological invariant that controls the number of topologically protected 0 and $\ensuremath{\pi}$ energy edge states at the interfaces between different domains, and give simple formulas for these invariants. We illustrate this bulk-boundary correspondence for DTQW's on the example of the ``4-step quantum walk,'' where tuning CS and particle-hole symmetry realizes edge states in various symmetry classes.
We investigate one-dimensional (1D) discrete-time quantum walks (QWs) with spatially or temporally random defects as a consequence of interactions with random environments. We focus on the QWs with chiral symmetry in a topological phase, and reveal that chiral symmetry together with the bipartite nature of the QWs brings about intriguing behaviors such as coexistence of topologically protected edge states at zero energy and Anderson transitions in the 1D chiral class at nonzero energy in their dynamics. In contrast to results of previous studies, therefore, the spatially disordered QWs can avoid complete localization due to the Anderson transition. It is further confirmed that the edge states are robust to spatial disorder but not to temporal disorder.
Discrete-time quantum walks (QWs) represent robust and versatile platforms for the controlled engineering of single particle quantum dynamics, and have attracted special attention due to their algorithmic applications in quantum information science. Even in their simplest 1D architectures, they display complex topological phenomena, which can be employed in the systematic study of topological quantum phase transitions [1]. Due to the exponential scaling in the number of resources required, most experimental realizations of QWs to date have been limited to single particles, with only a few implementations involving correlated quantum pairs. In this paper we study applications of QWs in the controlled dynamical engineering of entanglement in bipartite bosonic systems. We show that QWs can be employed in the transition from mode entanglement, where indistinguishability of the quantum particles plays a key role, to the standard type of entanglement associated with distinguishable particles. We also show that by carefully tailoring the steps in the QWs, as well as the initial state for the quantum walker, it is possible to preserve the entanglement content by topological protection. The underlying mechanism that allows for the possibility of both entanglement engineering and entanglement protection is the strong ‘spin–orbit’ coupling induced by the QW. We anticipate that the results reported here can be employed for the controlled emulation of quantum correlations in topological phases.
Chirality in active and passive fluids gives rise to odd transport properties, most notably the emergence of robust edge currents that defy standard dissipative dynamics. While these phenomena are well-described by continuum hydrodynamics, a microscopic framework connecting them to their topological origins has remained elusive. Here, we present a lattice model for an isotropic chiral random walk that bridges the gap between classical stochastic diffusion and unitary quantum evolution. By equipping the walker with an internal degree of freedom and a tunable chirality parameter, p, we interpolate between a standard diffusive random walk and a deterministic, topologically non-trivial quantum walk. We show that the topological protection characteristic of the unitary limit (p=1) remarkably persists into the dissipative regime (p<1). This correspondence allows us to theoretically ground the robustness of edge flows in classical chiral systems using the bulk-boundary correspondence of Floquet topological insulators. Our results provide a discrete microscopic description for odd diffusion, offering a powerful toolkit to predict transport in confined geometries and disordered chiral media.
The quantum walk was originally proposed as a quantum-mechanical analog of the classical random walk, and has since become a powerful tool in quantum information science. In this paper, we show that discrete-time quantum walks provide a versatile platform for studying topological phases, which are currently the subject of intense theoretical and experimental investigations. In particular, we demonstrate that recent experimental realizations of quantum walks with cold atoms, photons, and ions simulate a nontrivial one-dimensional topological phase. With simple modifications, the quantum walk can be engineered to realize all of the topological phases, which have been classified in one and two dimensions. We further discuss the existence of robust edge modes at phase boundaries, which provide experimental signatures for the nontrivial topological character of the system.