Topological phases of matter offer a new approach to quantum technologies based on disorder-robust quantum states. Here we present the topological characterization of the two-photon bands in a dissipative array of qubits described by the extended Bose-Hubbard model which includes density-dependent coupling. While density-dependent coupling forms the nontrivial two-photon topological phase, single-photon states remain topologically trivial. We discuss the connection between the topological invariant of the band and the number of states in this band that allows us to determine the topological phase of the system with reflection symmetry using only the information on the number of eigenstates retrieved from the experiment in the case of small dissipation. We also discuss another method of topological order probing based on the analysis of a time evolution that can be used in a regime with stronger dissipation.
Topological phases of matter offer a new approach to quantum technologies based on disorder-robust quantum states. Here we present the topological characterization of the two-photon bands in a dissipative array of qubits described by the extended Bose-Hubbard model which includes density-dependent coupling. While density-dependent coupling forms the nontrivial two-photon topological phase, single-photon states remain topologically trivial. We discuss the connection between the topological invariant of the band and the number of states in this band that allows us to determine the topological phase of the system with reflection symmetry using only the information on the number of eigenstates retrieved from the experiment in the case of small dissipation. We also discuss another method of topological order probing based on the analysis of a time evolution that can be used in a regime with stronger dissipation.
Topological protection of quantum correlations opens new horizons and opportunities in quantum technologies. A variety of topological effects has recently been observed in qubit networks. However, the experimental identification of the topological phase still remains challenging, especially in the entangled many-body case. Here, we propose an approach to independently probe single- and two-photon topological invariants from the time evolution of the two-photon state in a one-dimensional array of qubits. Extending the bulk-boundary correspondence to the two-photon scenario, we show that an appropriate choice of the initial state enables the retrieval of the topological invariant for the different types of the two-photon states in the interacting Su-Schrieffer-Heeger model. Our analysis of the Zak phase reveals additional facets of topological protection in the case of collapse of bound photon pairs.
We investigate two-photon topological states in the array of nearest-neighbor coupled qubits for two model systems based on the extended version of the Bose-Hubbard Hamiltonian, one of which includes direct two-photon tunneling and an- other one incorporates density-dependent hopping. For both models, we demonstrate that the effective photon-photon interaction gives rise to the topological bandgap and topologically protected edge states of bound photon pairs. We design effective density- dependent hopping by inserting auxiliary frequency-detuned qubits in the appropriate positions in the array, which is feasible to realize, for instance, in existing arrays of transmon qubits. Furthermore, we map our system onto the paradigmatic Su-Schrieffer- Heeger model in the limiting cases illustrating our conclusions by the results of numerical simulations.
Cold-atom experiments in optical lattices offer a versatile platform to realize various topological quantum phases. A key challenge in those experiments is to unambiguously probe the topological order. We propose a method to directly measure the characteristic topological invariants (order) based on the time-of-flight imaging of cold atoms. The method is generally applicable to detection of topological band insulators in one, two, or three dimensions characterized by integer topological invariants. Using detection of the Chern number for the two-dimensional anomalous quantum Hall states and the Chern-Simons term for the three-dimensional chiral topological insulators as examples, we show that the proposed detection method is practical, and robust to typical experimental imperfections such as limited imaging resolution, inhomogeneous trapping potential, and disorder in the system.
Quenching a quantum system involves three basic ingredients: the initial phase, the postquench target phase, and quantum dynamics, which may carry the information of the former two. Here we propose a dynamical theory, based on an interaction quench, to characterize both the equilibrium symmetry-breaking order and topological phases by nonequilibrium correlated quantum dynamics. We illustrate the theory with the Haldane-Hubbard model, which is quenched from an initial correlated magnetic phase to a topologically nontrivial regime. We show that the quench dynamics exhibit profound universal behaviors on the so-called band-inversion surfaces (BISs), from which both the topological phase in the weakly interacting regime and the correlated magnetic phase in the strongly interacting regime can be extracted. In particular, the topology is characterized by dynamical topological patterns emerging on BISs, which are robust against interaction-induced dephasing and heating; the symmetry-breaking order can be read out from a universal dynamical scaling behavior, which is valid beyond the mean-field theory. This work uncovers the first paradigm of nonequilibrium characterization of equilibrium symmetry-breaking and topological phases.
One-dimensional discrete-time quantum walks show a rich spectrum of topological phases that have so far been exclusively analyzed using the Floquet operator 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 quantum walks with gaps in the quasienergy spectrum at 0 and $\ensuremath{\pi}$, we find three different types of topological invariants, which apply dependent on the symmetries of the system. These determine the number of protected boundary states at an interface between two quantum-walk regions. Quantum walks with an unequal number of leftward and rightward shifts per cycle are characterized by the number of perfectly transmitting unidirectional modes they support, which is equal to their nontrivial quasienergy winding. 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.
How to characterize topological quantum phases is a fundamental issue in the broad field of topological matter. Based on dimension reduction approach, here we propose a new theory to characterize equilibrium topological phases by far-from-equilibrium quantum dynamics, and further report the experimental observation. For a d-dimensional (dD) topological phase with integer invariants, we show that its characterization can be reduced to lower-dimensional topological invariants in the high-order band inversion surfaces (BIS-s), of which the nth-order BIS is a (d-n)D interface in momentum space. In quenching the system from trivial phase to topological regime, we show that the quantum dynamics exhibits nontrivial topological pattern in the arbitrary nth-order BISs, which universally corresponds to and so characterizes the equilibrium topological phase of the post-quench Hamiltonian. This prediction unveils a high-order dynamical bulk-surface correspondence in the momentum space, and provides new dynamical schemes with experimental advantages to detect topological states. In particular, based on the highest-order BISs which are of zero dimension, the detection of topological phase is achieved with minimal measurements. We experimentally build up a quantum simulator using nitrogen-vacancy center to measure the high-order dynamical bulk-surface correspondence in 3D chiral topological insulator, and show the advantages of characterizing the topological phase with high-order BISs in experiment.