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Motivated by the recent proposals for unconventional emergent physics in twisted bilayers of nodal superconductors, we study the peculiarities of the Josephson effect at the twisted interface between d-wave superconductors. We demonstrate that for clean interfaces with a twist angle θ0 in the range 0◦ < θ0 < 45◦, the critical current can exhibit nonmonotonic temperature dependence with a maximum at a nonzero temperature as well as a complex dependence on the twist angle at low temperatures. These effects are shown to reflect the destructive interference between the d-wave order parameters near the nodes at nonzero twist angle. Close to 45◦ we find that the critical current does not vanish due to Cooper pair cotunneling, which can lead to the transition to a time-reversal breaking superconducting d + id phase, which can be suppressed by the interface roughness. We provide a comprehensive theoretical analysis of experiments that can reveal this cotunneling for twisted superconductors close to θ0 = 45◦. In particular, we demonstrate that both the emergence of the Fraunhofer interference pattern near θ0 = 45◦ and fractional Shapiro steps yield unambiguous evidence of Cooper pair cotunneling, necessary for topological superconductivity. 

Twisted interfaces between stacked van der Waals (vdW) cuprate crystals present a platform for engineering superconducting order parameters by adjusting stacking angles. Using a cryogenic assembly technique, we construct twisted vdW Josephson junctions (JJs) at atomically sharp interfaces between Bi₂Sr₂CaCu₂O_8+xcrystals, with quality approaching the limit set by intrinsic JJs. Near 45° twist angle, we observe fractional Shapiro steps and Fraunhofer patterns, consistent with the existence of two degenerate Josephson ground states related by time-reversal symmetry (TRS). By programming the JJ current bias sequence, we controllably break TRS to place the JJ into either of the two ground states, realizing reversible Josephson diodes without external magnetic fields. Our results open a path to engineering topological devices at higher temperatures.