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Superconducting circuits are a competitive platform for quantum computation because they offer controllability, long coherence times and strong interactions - properties that are essential for the study of quantum materials comprising microwave photons. However, intrinsic photon losses in these circuits hinder the realization of quantum many-body phases. Here we use superconducting circuits to explore strongly correlated quantum matter by building a Bose-Hubbard lattice for photons in the strongly interacting regime. We develop a versatile method for dissipative preparation of incompressible many-body phases through reservoir engineering and apply it to our system to stabilize a Mott insulator of photons against losses. 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. Our 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, we expect that our approach will enable the exploration of topologically ordered phases of matter., Comment: Maintext 10 pages, 5 figures; Supplementary Information with 12 figures and 3 tables. Very similar to the published version

The Weyl particle is the massless fermionic cousin of the photon. While no fundamental Weyl particles have been identified, they arise in condensed matter and meta-material 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. The circuit platform provides access to the complete complex-valued spin-texture of all bulk- and surface- states, 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 in superconducting circuits, as well as studies of how manifold topology impacts band topology in three dimensions., Comment: 5+14 pages, 3+16 Figures, for Text+Supplement

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 ({\alpha} = 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 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., Comment: 11 pages, 9 Figures

We suggest a simple approach to populate photonic quantum materials at non-zero chemical potential and near-zero temperature. Taking inspiration from forced evaporation in cold-atom experiments, the essential ingredients for our low-entropy thermal reservoir are (a) inter-particle interactions, and (b) energy-dependent loss. The resulting thermal reservoir may then be coupled to a broad class of Hamiltonian systems to produce low-entropy quantum phases. We present an idealized picture of such a reservoir, deriving the scaling of reservoir entropy with system parameters, and then propose several practical implementations using only standard circuit quantum electrodynamics tools, and extract the fundamental performance limits. Finally, we explore, both analytically and numerically, the coupling of such a thermalizer to the paradigmatic Bose-Hubbard chain, where we employ it to stabilize an $n=1$ Mott phase. In this case, the performance is limited by the interplay of dynamically arrested thermalization of the Mott insulator and finite heat capacity of the thermalizer, characterized by its repumping rate. This work explores a new approach to preparation of quantum phases of strongly interacting photons, and provides a potential route to topologically protected phases that are difficult to reach through adiabatic evolution., Comment: 11 Figures, 8 Figures

In this letter we introduce a novel approach to Hamiltonian tomography of non-interacting tight-binding photonic lattices. To begin with, we prove that the matrix element of the low-energy effective Hamiltonian between sites $i$ and $j$ may be obtained directly from $S_{ij}(\omega)$, the (suitably normalized) two-port measurement between sites $i$ and $j$ at frequency $\omega$. This general result enables complete characterization of both on-site energies and tunneling matrix elements in arbitrary lattice networks by spectroscopy, and suggests that coupling between lattice sites is actually a topological property of the two-port spectrum. We further provide extensions of this technique for measurement of band-projectors in finite, disordered systems with good flatness ratios, and apply the tool to direct real-space measurement of the Chern number. Our approach demonstrates the extraordinary potential of microwave quantum circuits for exploration of exotic synthetic materials, providing a clear path to characterization and control of single-particle properties of Jaynes-Cummings-Hubbard lattices. More broadly, we provide a robust, unified method of spectroscopic characterization of linear networks from photonic crystals to microwave lattices and everything in-between., Comment: 7 pages, 5 figures

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 (non-reciprocal) flux is induced by coupling the microwave cavities to ferrites, allowing for the production of a variety of topological band structures including the $\alpha=1/4$ Hofstadter model. Effective photon-photon interactions are included by coupling the cavities to superconducting qubits, and are sufficient to produce a $\nu=1/2$ bosonic Laughlin puddle. We demonstrate by exact diagonalization 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.

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 non-trivial 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 reversal invariant topological band-structure, which we realize in a radio frequency (RF) 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 $\phi=\pi/2$. In-situ probes of the band-gaps reveal spatially-localized bulk-states and de-localized edge-states. Time-resolved measurements reveal dynamical separation of localized edge-excitations into spin-polarized currents. The RF circuit paradigm is naturally compatible with non-local coupling schemes, allowing us to implement a M\"{o}bius 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 (QED) techniques, provides a direct path to topologically-ordered quantum matter., Comment: 15 pages, 4 figures & 6 supplementary figures