  Two-dimensional terahertz spectroscopy (2DTS), a terahertz analogue of
nuclear magnetic resonance, is a new technique poised to address many open
questions in complex condensed matter systems. The conventional theoretical
framework used ubiquitously for interpreting multidimensional spectra of
discrete quantum level systems is, however, insufficient for the continua of
collective excitations in strongly correlated materials. Here, we develop a
theory for 2DTS of a model collective excitation, the Josephson plasma
resonance in layered superconductors. Starting from a mean-field approach at
temperatures well below the superconducting phase transition, we obtain
expressions for the multidimensional nonlinear responses that are amenable to
intuition derived from the conventional single-mode scenario. We then consider
temperatures near the superconducting critical temperature $T_c$, where
dynamics beyond mean-field become important and conventional intuition fails.
As fluctuations proliferate near $T_c$, the dominant contribution to nonlinear
response comes from an optical parametric drive of counter-propagating
Josephson plasmons, which gives rise to 2D spectra that are qualitatively
different from the mean-field predictions. As such, and in contrast to
one-dimensional spectroscopy techniques, such as third harmonic generation,
2DTS can be used to directly probe thermally excited finite-momentum plasmons
and their interactions. Our theory provides a clear interpretation of recent
2DTS measurements on cuprates, and we discuss implications beyond the present
context of Josephson plasmons.
