**Unformatted text preview:**

ARTICLEnATuRE CommunICATIons | 3:882 | DoI: 10.1038/ncomms1872 | www.nature.com/naturecommunications© 2012 Macmillan Publishers Limited. All rights reserved.Received 4 Jan 2012 | Accepted 25 Apr 2012 | Published 6 Jun 2012DOI: 10.1038/ncomms1872Topological phases exhibit some of the most striking phenomena in modern physics. much of the rich behaviour of quantum Hall systems, topological insulators, and topological superconductors can be traced to the existence of robust bound states at interfaces between different topological phases. This robustness has applications in metrology and holds promise for future uses in quantum computing. Engineered quantum systems—notably in photonics, where wavefunctions can be observed directly—provide versatile platforms for creating and probing a variety of topological phases. Here we use photonic quantum walks to observe bound states between systems with different bulk topological properties and demonstrate their robustness to perturbations—a signature of topological protection. Although such bound states are usually discussed for static (time-independent) systems, here we demonstrate their existence in an explicitly time-dependent situation. moreover, we discover a new phenomenon: a topologically protected pair of bound states unique to periodically driven systems. 1 Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA. 2 ARC Centre for Engineered Quantum Systems and ARC Centre for Quantum Computation and Communication Technology, School of Mathematics and Physics, University of Queensland, Brisbane 4072, Australia. 3 Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, USA. *These authors contributed equally to this work. Correspondence and requests for materials should be addressed to T.K. (email: [email protected]) or to M.A.B. (email: [email protected]). observation of topologically protected bound states in photonic quantum walksTakuya Kitagawa1,*, matthew A. Broome2,*, Alessandro Fedrizzi2, mark s. Rudner1, Erez Berg1, Ivan Kassal2,3, Alán Aspuru-Guzik3, Eugene Demler1 & Andrew G. White2ARTICLEnATuRE CommunICATIons | DoI: 10.1038/ncomms1872nATuRE CommunICATIons | 3:882 | DoI: 10.1038/ncomms1872 | www.nature.com/naturecommunications© 2012 Macmillan Publishers Limited. All rights reserved.Phases of matter have long been characterized by their sym-metry properties, with each phase classified according to the symmetries that it possesses1. The discovery of the inte-ger and fractional quantum Hall effects in the 1980s has led to a new paradigm, where quantum phases of matter are characterized by the topology of their ground-state wavefunctions. Since then, topological phases have been identified in physical systems ranging from condensed-matter2–9 and high-energy physics10 to quantum optics11 and atomic physics12–15.Topological phases of matter are parametrized by integer topo-logical invariants. As integers cannot change continuously, a conse-quence is exotic phenomena at the interface between systems with different values of topological invariants. For example, a topological insulator supports conducting states at the surface, precisely because its bulk topology is different to that of its surroundings8,9. Creat-ing and studying new topological phases remains a difficult task in a solid-state setting because the properties of electronic systems are often hard to control. Using controllable simulators may be advantageous in this respect.Here we simulate one-dimensional topological phases using a discrete time quantum walk16, a protocol for controlling the motion of quantum particles on a lattice. We create regions with distinct values of topological invariants and directly image the wavefunction of bound states at the boundary between them. The controllability of our system allows us to make small changes to the Hamiltonian and demonstrate the robustness of these bound states. Finally, using the quantum walk, we can access the dynamics of strongly driven systems far from the static or adiabatic regimes17–19, to which most previous work on topological phases has been restricted. In this regime, we discover a topologically protected pair of non-degenerate bound states, a phenomenon that is unique to periodically driven systems.ResultsSplit-step quantum walks. Discrete time quantum walks have been realized in several physical architectures20–24. Here we use the photonic set-up demonstrated in ref. 24 to implement a variation of these walks, the split-step quantum walk25 of a single photon, with two internal states encoded in its horizontal, |H〉, and vertical, |V〉, polarization states. The quantum walk takes place on a one dimensional lattice (Fig. 1). One step of the split-step quantum walk consists of four steps. First, a polarization rotation R(θ1) of the single photon is achieved with a suitable wave plate (see Methods), then a polarization-dependent translation T1 of |H〉 to the right by one lattice site using a calcite beam displacer. This is followed by a second rotation R(θ2), and finally another translation T2 of |V〉 to the left. The quantum walk is implemented by repeated applications of the one-step operator U(θ1, θ2) = T2R(θ2)T1R(θ1).The propagation of the photon in the static experimental set-up can be described by an effective time-dependent Schrödinger equa-tion with periodic driving. The dynamics of the quantum walk can be understood through the effective Hamiltonian Heff(θ1, θ2), defined through U eiH( , ) =1 2( , ) /1 2q qq q t−eff, where τ is the time required for one step of the quantum walk. Throughout this paper, we chose units such that t / = 1. Therefore, the quantum walk described by the evolution U(θ1, θ2) corresponds to a strobo-scopic simulation of the effective Hamiltonian Heff(θ1, θ2) viewed at unit time intervals. That is, after n steps of the quantum walk, the photon evolves according to U eninH( , ) =1 2( , )1 2q qq q−eff, mean-ing that the evolution under the quantum walk coincides with the evolution under Heff(θ1, θ2) for integer multiples of τ.The topological structure underlying split-step quantum walks is revealed by studying the structure and symmetry of