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Circuit Quantum Electrodynamics Simulator of Flat Band Physics in Lieb lattice

The concept of flat band plays an important role in strongly-correlated many-body physics. However, the demonstration of the flat band physics is highly nontrivial due to intrinsic limitations in conventional condensed matter materials. Here we propose a circuit quantum electrodynamics simulator of the 2D Lieb lattice exhibiting a flat middle band. By exploiting the simple parametric conversion method, we design a photonic Lieb lattice with \textit{in situ} tunable hopping strengths in a 2D array of coupled superconducting transmissionline resonators. Moreover, the flexibility of our proposal enables the immediate incorporation of both the artificial gauge field and the strong photon-photon interaction in a time- and site-resolved manner. To unambiguously demonstrate the synthesized flat band, we further investigate the observation of the flat band localization of microwave photons through the pumping and the steady-state measurements of only few sites on the lattice. Requiring only current level of technique and being robust against imperfections in realistic circuits, our scheme can be readily tested in experiments and may pave a new way towards the future realization of exotic photonic quantum Hall fluids including anomalous quantum Hall effect and bosonic fractional quantum Hall states without magnetic fields.