From studies of exotic quantum many-body phenomena to applications in spintronics and quantum information processing, topological materials are poised to revolutionize the condensed-matter frontier and the landscape of modern materials science. Accordingly, there is a broad effort to realize topologically nontrivial electronic and photonic materials for fundamental science as well as practical applications. In this work, we demonstrate the first simultaneous site-and time-resolved measurements of a time-reversalinvariant topological band structure, which we realize in a radio-frequency photonic circuit. We control band-structure topology via local permutation of a traveling-wave capacitor-inductor network, increasing robustness by going beyond the tight-binding limit. We observe a gapped density of states consistent with a modified Hofstadter spectrum at a flux per plaquette of ϕ ¼ π=2. In situ probes of the band gaps reveal spatially localized bulk states and delocalized edge states. Time-resolved measurements reveal dynamical separation of localized edge excitations into spin-polarized currents. The radio-frequency circuit paradigm is naturally compatible with nonlocal coupling schemes, allowing us to implement a Möbius strip topology inaccessible in conventional systems. This room-temperature experiment illuminates the origins of topology in band structure, and when combined with circuit quantum electrodynamics techniques, it provides a direct path to topologically ordered quantum matter. [7], and 2DEGs [8,9]. In a condensed-matter context, such "topologically protected" properties include single-particle features of the band structure and many-particle ground-state degeneracies, with the latter typically emerging from the former in conjunction with strong interactions. To explore the nature of topologically derived material properties, it is desirable to develop materials that not only support conserved topological quantities but that may be precisely produced, manipulated, and probed. The aim, then, is to realize material test beds that marry favorable coherence properties, strong interactions, and topologically nontrivial single-particle dynamics.Metamaterials, where interaction strengths and length scales can be engineered, are a promising avenue for studying topological physics. Efforts are ongoing to produce the requisite topological single-particle dynamics in ultracold atomic gases [10][11][12][13][14][15][16], gyrotropic metamaterials [17,18], and photonic systems [17,[19][20][21][22][23][24][25][26].In cold atomic gases, gauge fields are generated either through spatially dependent Raman coupling of internal atomic states [10,14], or time-and space-periodic modulation of lattice tunneling rates [15,27,28]. In the optical domain, synthetic magnetic fields were realized via strain of a honeycomb lattice [29]. A Floquet topological insulator [30,31] was realized under a space-to-time mapping of an array of tunnel-coupled waveguides modulated along their propagation direction [21]. A photonic topologic...

Superconducting circuits have emerged as a competitive platform for quantum computation, satisfying the challenges of controllability, long coherence and strong interactions between individual systems. Here we apply this toolbox to the exploration of strongly correlated quantum matter, building a Bose-Hubbard lattice for photons in the strongly interacting regime. We develop a versatile recipe for dissipative preparation of incompressible many-body phases through reservoir engineering and apply it in our system to realize the first Mott insulator of photons. Site-and time-resolved readout of the lattice allows us to investigate the microscopic details of the thermalization process through the dynamics of defect propagation and removal in the Mott phase. These experiments demonstrate the power of superconducting circuits for studying strongly correlated matter in both coherent and engineered dissipative settings. In conjunction with recently demonstrated superconducting microwave Chern insulators, the approach demonstrated in this work will enable exploration of elusive topologically ordered phases of matter. arXiv:1807.11342v1 [cond-mat.quant-gas] 30 Jul 2018

We present a scalable architecture for the exploration of interacting topological phases of photons in arrays of microwave cavities, using established techniques from cavity and circuit quantum electrodynamics. A time-reversal symmetry-breaking (nonreciprocal) flux is induced by coupling the microwave cavities to ferrites, allowing for the production of a variety of topological band structures including the α ¼ 1=4 Hofstadter model. To induce photon-photon interactions, the cavities are coupled to superconducting qubits; we find these interactions are sufficient to stabilize a ν ¼ 1=2 bosonic Laughlin puddle. Exact diagonalization studies demonstrate that this architecture is robust to experimentally achievable levels of disorder. These advances provide an exciting opportunity to employ the quantum circuit toolkit for the exploration of strongly interacting topological materials.

The Weyl particle is the massless fermionic cousin of the photon [1]. While no fundamental Weyl particles have been identified, they arise in condensed matter [2][3][4] and meta-material [5, 6] systems, where their spinor nature imposes topological constraints on low-energy dispersion and surface properties. Here we demonstrate a topological circuit with Weyl dispersion at low-momentum, realizing a 3D lattice that behaves as a half-flux Hofstadter model in all principal planes [7]. The circuit platform [8] provides access to the complete complex-valued spin-texture of all bulk-and surfacestates, thereby revealing not only the presence of Weyl points and the Fermi arcs that connect their surface-projections, but also, for the first time, the Berry curvature distribution through the Brillouin zone and the associated quantized Chiral charge of the Weyl points. This work opens a path to exploration of interacting Weyl physics [9] in superconducting circuits [10], as well as studies of how manifold topology impacts band topology in three dimensions [11].

Topological and strongly correlated materials are exciting frontiers in condensed matter physics, married prominently in studies of the fractional quantum hall effect [1]. There is an active effort to develop synthetic materials where the microscopic dynamics and ordering arising from the interplay of topology and interaction may be directly explored. In this work we demonstrate a novel architecture for exploration of topological matter constructed from tunnel-coupled, time-reversalbroken microwave cavities that are both low loss and compatible with Josephson junction-mediated interactions [2]. Following our proposed protocol [3] we implement a square lattice Hofstadter model at a quarter flux per plaquette (α = 1/4), with time-reversal symmetry broken through the chiral Wannier-orbital of resonators coupled to Yttrium-Iron-Garnet spheres. We demonstrate site-resolved spectroscopy of the lattice, time-resolved dynamics of its edge channels, and a direct measurement of the dispersion of the edge channels. Finally, we demonstrate the flexibility of the approach by erecting a tunnel barrier and investigating dynamics across it. With the introduction of Josephson junctions to mediate interactions between photons, this platform is poised to explore strongly correlated topological quantum science for the first time in a synthetic system.