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Physical Review Research

Topological router induced via long-range hopping in a Su-Schrieffer-Heeger chain

Lu Qi, Yu Yan, Yan Xing, Xuedong Zhao, Shutian Liu, Wen‐Xue Cui, Xue Yan Han, Shou Zhang, Hong‐Fu Wang

Published Apr 12, 2021• 37 citations
Publisher

Abstract

We show how to implement the various kinds of topological routers via introducing the specific long-range hopping in the one-dimensional modulated Su-Schrieffer-Heeger chain.We find that the long-range hopping with hopping amplitudes identical to those of the intercell hopping between the first site and other odd sites holds a distinctive topological channel (gap state), by which the particle initially prepared at the last site can appear at the first site and all of the even sites with approximately equal probabilities.This extraordinary performance indicates that, from the perspective of treating the last site as the input port and treating the first site and other even sites as output ports, the present system can be naturally equivalent to a distribution device with multiple output ports, i.e., the topological router.We show that the number of the output ports for the present topological router can be flexibly tuned via reducing the long-range hopping terms.Especially, the system experiences a phase transition with the increasing of the long-range hopping amplitudes, in which the original gap state becomes a topological channel to implement the topological router with the output ports only at all even sites.We stress that both the two kinds of topological routers are protected by the energy gap and hence are robust to the mild disorder added into the system.Moreover, together with the construction of the topological interface, we find that the topological routers can own the output ports in a symmetrical way about the interface site.The topological routers may play the pivotal role in the large-scale entanglement distributions, which significantly expands and supplies the potential applications of the topological materials in quantum information processing.

DOI
10.1103/physrevresearch.3.023037

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We show how to implement the various kinds of topological routers via introducing the specific long-range hopping in the one-dimensional modulated Su-Schrieffer-Heeger chain.We find that the long-range hopping with hopping amplitudes identical to those of the intercell hopping between the first site and other odd sites holds a distinctive topological channel (gap state), by which the particle initially prepared at the last site can appear at the first site and all of the even sites with approximately equal probabilities.This extraordinary performance indicates that, from the perspective of treating the last site as the input port and treating the first site and other even sites as output ports, the present system can be naturally equivalent to a distribution device with multiple output ports, i.e., the topological router.We show that the number of the output ports for the present topological router can be flexibly tuned via reducing the long-range hopping terms.Especially, the system experiences a phase transition with the increasing of the long-range hopping amplitudes, in which the original gap state becomes a topological channel to implement the topological router with the output ports only at all even sites.We stress that both the two kinds of topological routers are protected by the energy gap and hence are robust to the mild disorder added into the system.Moreover, together with the construction of the topological interface, we find that the topological routers can own the output ports in a symmetrical way about the interface site.The topological routers may play the pivotal role in the large-scale entanglement distributions, which significantly expands and supplies the potential applications of the topological materials in quantum information processing.
We propose a scheme to implement the phase-robust topological router based on a one-dimensional dimerized superconducting circuit lattice with long-range hopping. We show that the proposed dimerized superconducting circuit lattice can be mapped into an extended chiral-symmetric Su-Schrieffer-Heeger (SSH) model with long-range hopping, in which the existence of long-range hopping induces a special zero-energy mode. The peculiar distribution of the zero-energy mode enables us to engineer a phase-robust topological router, which can achieve quantum state transfer (QST) from one site (input port) to multiple sites (output ports). Benefiting from the topological protection of chiral symmetry, we demonstrate that the presence of the mild disorder in nearest-neighbor and long-range hopping has no appreciable effects on QST in the lattice. Particularly, after introducing another long-range hopping into the extended SSH lattice, we propose an optimized protocol of the phase-robust topological router, in which the number of the output ports can be efficiently increased. Resorting to the Bose statistical properties of the superconducting circuit lattice, the input port and output ports assisted by the zero-energy mode can be detected via the mean distribution of the photons. Our work breaks the traditional QST form with only one outport by the zero-energy mode and opens a pathway to construct large-scale quantum information processing in the SSH chains with long-range hopping.
We propose a scheme to implement the phase-robust topological router based on a one-dimensional dimerized superconducting circuit lattice with long-range hopping. We show that the proposed dimerized superconducting circuit lattice can be mapped into an extended chiral-symmetric Su-Schrieffer-Heeger (SSH) model with long-range hopping, in which the existence of long-range hopping induces a special zero-energy mode. The peculiar distribution of the zero-energy mode enables us to engineer a phase-robust topological router, which can achieve quantum state transfer (QST) from one site (input port) to multiple sites (output ports). Benefiting from the topological protection of chiral symmetry, we demonstrate that the presence of the mild disorder in nearest-neighbor and long-range hopping has no appreciable effects on QST in the lattice. Especially, after introducing another new long-range hopping into the extended SSH lattice, we propose an optimized protocol of the phase-robust topological router, in which the number of the output ports can be efficiently increased. Resorting to the Bose statistical properties of the superconducting circuit lattice, the input port and output ports assisted by the zero-energy mode can be detected via the mean distribution of the photons. Our work breaks the traditional QST form with only one outport by the zero-energy mode and opens a pathway to construct large-scale quantum information processing in the SSH chains with long-range hopping.
Abstract We present a feasible scheme to implement a planar and tunable quantum state transfer (QST) via topologically protected zero-energy mode in a splicing Y-junction Su–Schrieffer–Heeger (SSH) chain. The introduction of the elaborate nearest-neighbor (NN) hopping enables one to generate a topological interface at the central site of the Y-junction. By modulating the NN hopping adiabatically in the chain, the quantum state initially prepared at the central site can be simultaneously transferred to the three endpoints of the Y-junction with the equal/unequal probabilities. The planar distribution of QST is expected to realize a quantum router, whose function is to make the quantum information on the central site (input port) appear equally/unequally at the three endpoints (output ports) with different directions. Moreover, the numerical simulations demonstrate that the scheme possesses the robustness on the fluctuations of the NN hopping and the on-site potential in the system. Furthermore, we show that the number of the output ports with different directions can be flexibly increased in an extended X-junction SSH chain, and the experimental feasibility for implementing special QST in a superconducting qubit-resonator system is briefly discussed. Our work extends the space distribution of QST from linear distribution to planar distribution and promotes the construction of large-scale quantum networks.
In the usual Su–Schrieffer–Heeger (SSH) model with an even number of lattice sites, the topological pumping between left and right edge states cannot be easily realized since the edge states occupy two-end sites simultaneously. Here we propose a scheme to investigate the topological edge pumping in an even-sized periodically modulated SSH model mapped by a one dimensional superconducting transmission line resonators array. We find that the photon initially prepared in the first resonator can be finally observed at the two-end resonators with a certain proportion. The final photon splitting at the two-end resonators indicates that the present superconducting circuit is expected to realize the topological beam splitter. Further, we demonstrate that the splitting proportion between the two-end resonators can be arbitrarily tuned from 1 to 0, implying the potential feasibility of implementing the tunable topological beam splitter. Meanwhile, we also show that the tunable topological beam splitter is immune to the mild disorder added into the system due to the topology protection of the zero energy modes, and find that the tunable topological beam splitter is much more robust to the global on-site disorder compared with the nearest neighbor disorder. Our work greatly extends the practical application of topological matter in quantum information processing and opens up a new way towards the engineering of topological quantum optical device.
We address the effect of nearest-neighbor (NN) interactions on the topological properties of the Su-Schrieffer-Heeger (SSH) chain, with alternating hopping amplitudes t1{t}_{1}t1​ and t2{t}_{2}t2​. Both numerically and analytically, we show that the presence of interactions induces phase transitions between topologically different regimes. In the particular case of one-hole excitations in a half-filled SSH chain, the V/t2V/{t}_{2}V/t2​ versus t1/t2{t}_{1}/{t}_{2}t1​/t2​ phase diagram has topological phases at diagonal regions of the phase plane. The interaction acts in this case as a passivation potential. For general filling of the SSH chain, different eigensubspaces of the SSH Hamiltonian may be classified as topologically trivial and nontrivial. The two-hole case is studied in detail in the large interaction limit, and we show that a mapping can be constructed of the two-hole SSH eigensubspaces into one-particle states of a noninteracting one-dimensional (1D) tight-binding model, with interfaces between regions with different hopping constants and local potentials. The presence of edge states of topological origin in the equivalent chain can be readily identified, as well as their correspondence to the original two-hole states. Of these states only some, identified by us, are protected and, therefore, truly topological. Furthermore, we found that the presence of the NN interaction generates a state where two holes occupy two consecutive edge states. Such many-body states should also occur for arbitrary filling leading to the possibility of a macroscopic hole gathering at the surface (at consecutive edge states).
We use the Lindblad equation approach to investigate topological phases hosting more than one localized state at each side of a disordered Su-Schrieffer-Heeger chain with properly tuned long-range hoppings. Inducing a nonequilibrium steady state across the chain, we probe the robustness of each phase and the fate of the edge modes looking at the distribution of electrons along the chain and the corresponding standard deviation in the presence of different kinds of disorder either preserving the symmetries of the Hamiltonian or not.