  We study numerically the two-dimensional coherent spectrum (2DCS) of the
Tomonaga-Luttinger liquid hosted by the $S=1/2$ XY spin chain. The 2DCS
characterizes the system's third order nonlinear magnetic response triggered by
three pulses, separated successively by the delay time and the waiting time. It
exhibits a photon echo signal resulting from a lensing process of the
fractional excitations: A pair of photon-excited fractional excitations,
initially moving apart, reverse their direction of motion, and annihilate each
other. In the XY chain, the nonlinearity in the dispersion relation of the
Jordan-Wigner fermions leads to the dispersion of the fractional excitation
wave packets and thereby suppresses lensing. The magnitude of the echo signal
decreases exponentially with increasing delay time. The decay rate scales with
the temperature $T$ as $T^n$ at low temperature, where $n$ is the leading order
of the Jordan-Wigner fermion dispersion, and as $T$ at high temperature. By
contrast, as the waiting time increases, the magnitude of the echo signal
saturates, reflecting the integrability of the system. Our results illustrate
the effectiveness of the 2DCS in detecting subtle dynamical properties of
optical excitations in spin chains.
