Momentum-Resolved Two-Dimensional Spectroscopy as a Probe of Nonlinear Quantum Field Dynamics

Emergent collective excitations constitute a hallmark of interacting quantum many-body systems, yet in solid-state platforms their study has been largely limited by the constraints of linear-response probes and by finite momentum resolution. We propose to overcome these limitations by combining the spatial resolution of ultracold atomic systems with the nonlinear probing capabilities of two-dimensional spectroscopy (2DS). As a concrete illustration, we analyze momentum-resolved 2DS of the quantum sine-Gordon model describing the low-energy dynamics of two weakly coupled one-dimensional Bose-Einstein condensates. This approach reveals distinctive many-body signatures, most notably asymmetric cross-peaks reflecting the interplay between isolated (B_{2} breather) and continuum (B_{1} pair) modes. The protocol further enables direct characterization of anharmonicity and disorder, establishing momentum-resolved 2DS as both a powerful diagnostic for quantum simulators and a versatile probe of correlated quantum matter.