Recent observations of time-reversal breaking superconductivity at twisted cuprate interfaces motivate the development of new approaches to better characterize this emergent phenomenon. Here we study the dynamical properties of the order parameters at the twisted unconventional superconductor interfaces. We reveal the emergence of a soft collective mode (Josephson plasmon) at the time-reversal breaking transition, which can be tuned by temperature, twist angle, or magnetic field. Furthermore, nonlinear dynamical responses contain direct signatures of both the transition and the broken symmetry itself. In particular, we show that the generation of a second harmonic voltage under alternating current driving is a necessary and sufficient signature of time-reversal symmetry breaking. Finally, we demonstrate that strong nonlinear driving induces dynamical phase transitions between phases with and without spontaneous symmetry breaking, introducing a tool for their out-of-equilibrium control. We discuss the signatures of our predictions in ac current-driven experiments on twisted ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+x}$interfaces.

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• Josephson effect
• High-temperature superconductors
• Josephson junctions
• Superconductivity
• Twisted heterostructures
• Unconventional superconductors