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Coupling of Klein-Andreev Resonant States in Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$-graphene-Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ Devices

Quantum devices require coherent coupling over macroscopic distances. Recently, resonances due to Klein tunneling and Andreev reflection states (KARS) have been observed in a naturally occurring p-n junction at the interface between Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ (BSCCO), a high-Tc superconductor (HTS), and graphene. The resonances appear as conductance oscillations with gating. Here, we show coupling between the KARS in BSCCO-graphene-BSCCO devices of varying separation (L). The coupling is evidenced by a power-law decay of resonance period as L increases from tens of nanometers to single microns. These results demonstrate the long-distance coupling of KARS cavities in graphene-HTS junctions. The length dependence seen in experiments is supported by single-particle spectral functions which show KARS are coupled by transport modes in graphene. The strong coupling between KARS in BSCCO-graphene-BSCCO junctions showcases the novelty of HTS-graphene junctions for quantum circuits and unconventional Josephson junctions.