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...

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Much of the richness in nature emerges because the same simple constituents can form an endless variety of ordered states [1]. While many such states are fully characterized by their symmetries [2], interacting quantum systems can also exhibit topological order, which is instead characterized by intricate patterns of entanglement [3,4]. A paradigmatic example of such topological order is the Laughlin state [5], which minimizes the interaction energy of charged particles in a magnetic field and underlies the fractional quantum Hall effect [6]. Broad efforts have arisen to enhance our understanding of these orders by forming Laughlin states in synthetic quantum systems, such as those composed of ultracold atoms [7,8] or photons [9][10][11]. In spite of these efforts, electron gases remain essentially the only physical system in which topological order has appeared [6,[12][13][14]. Here, we present the first observation of optical photon pairs in the Laughlin state. These pairs emerge from a photonic analog of a fractional quantum Hall system, which combines strong, Rydbergmediated interactions between photons [15-18] and synthetic magnetic fields for light, induced by twisting an optical resonator [11,19,20]. Photons entering this system undergo collisions to form pairs in an angular momentum superposition consistent with the Laughlin state. Characterizing the same pairs in real space reveals that the photons avoid each other, a hallmark of the Laughlin state. This work heralds a new era of quantum many-body optics, where strongly interacting and topological photons enable exploration of quantum matter with wholly new properties and unique probes.

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].

We demonstrate hybridization of optical cavity photons with atomic Rydberg excitations using electromagnetically induced transparency (EIT). The resulting dark state Rydberg polaritons exhibit a compressed frequency spectrum and enhanced lifetime indicating strong light-matter mixing. We study the coherence properties of cavity Rydberg polaritons and identify the generalized EIT linewidth for optical cavities. Strong collective coupling suppresses polariton losses due to inhomogeneous broadening, which we demonstrate by using different Rydberg levels with a range of polarizabilities. Our results point the way towards using cavity Rydberg polaritons as a platform for creating photonic quantum materials.PACS numbers: 42.50. Gy, 42.50.Pq, 32.80.Ee, 71.36.+c Coupling photons to electronic excitations of a medium leads to hybrid quasiparticles, or polaritons, that carry properties of both light and matter. The photonic component allows polaritons to propagate like light, while the material component enables interactions between polaritons. An important example is exciton polaritons in semiconductor microcavities, which exhibit an effective mass and two-dimensional motion arising from the photonic component, while the exciton component leads to a mean-field interaction, allowing Bose-Einstein condensation [1][2][3]. Rydberg polaritons in atomic gases enable strong interactions even at the few-quantum level [4][5][6][7][8][9][10][11][12], a key ingredient for producing highly correlated states, including fractional quantum Hall states [13][14][15][16][17] and emergent quantum crystals [17][18][19][20][21]. While previous work on Rydberg polaritons has focused on onedimensional free-space light fields, photons in optical cavities provide access to two-dimensional motion, harmonic trapping [22], and effective magnetic fields [17,23]. In addition, optical cavities can enhance the optical nonlinearity arising from Rydberg interactions [24,25].Rydberg polaritons are formed by coherently coupling light to a highly excited atomic Rydberg level using electromagnetically induced transparency (EIT) [26]. At EIT resonance, destructive interference prevents population of a lossy intermediate atomic level, resulting in a dark state polariton [27] that consists of a superposition of a photon and a collective atomic Rydberg excitation. A large admixture of the (long-lived) atomic excitation in a dark state polariton slows all photonic dynamics [27]. In an optical cavity, this results in a polariton whose lifetime can exceed the empty-cavity lifetime by orders of magnitude, and an energy that is pulled toward the EIT resonance [28][29][30][31][32][33]. In a multimode cavity, hybridization rescales the trap frequency and effective mass of the polariton [17].We experimentally observe Rydberg polaritons in an optical cavity and explore the spectral and coherence properties of these collective states in cavity transmission spectroscopy. While Doppler decoherence and inhomogeneous Stark shifts [35][36][37] are more significant for Rydbe...