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Two-dimensional terahertz spectroscopy (2DTS), a terahertz analogue ofnuclear magnetic resonance, is a new technique poised to address many openquestions in complex condensed matter systems. The conventional theoreticalframework used ubiquitously for interpreting multidimensional spectra ofdiscrete quantum level systems is, however, insufficient for the continua ofcollective excitations in strongly correlated materials. Here, we develop atheory for 2DTS of a model collective excitation, the Josephson plasmaresonance in layered superconductors. Starting from a mean-field approach attemperatures well below the superconducting phase transition, we obtainexpressions for the multidimensional nonlinear responses that are amenable tointuition derived from the conventional single-mode scenario. We then considertemperatures near the superconducting critical temperature $T_c$, wheredynamics beyond mean-field become important and conventional intuition fails.As fluctuations proliferate near $T_c$, the dominant contribution to nonlinearresponse comes from an optical parametric drive of counter-propagatingJosephson plasmons, which gives rise to 2D spectra that are qualitativelydifferent from the mean-field predictions. As such, and in contrast toone-dimensional spectroscopy techniques, such as third harmonic generation,2DTS can be used to directly probe thermally excited finite-momentum plasmonsand their interactions. Our theory provides a clear interpretation of recent2DTS measurements on cuprates, and we discuss implications beyond the presentcontext of Josephson plasmons.

Principles of 2D terahertz spectroscopy of collective excitations: the case of Josephson plasmons in layered superconductors