Finite-momentum nematic fluctuations soften phonons in the superconducting state of Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$
Nematic order is ubiquitous in liquid crystals and is characterized by a rotational symmetry breaking in an otherwise uniform liquid. Recently a similar phenomenon has been observed in some electronic phases of quantum materials related to high temperature superconductivity, particularly in the Fe-based superconductors. While several experiments have probed nematic fluctuations, they have been primarily restricted to the uniform nematic susceptibility, i.e. q = 0 fluctuations. Here, we investigate the behavior of finite-momentum nematic fluctuations by measuring transverse acoustic phonon modes with wavelengths of up to 25 unit cells in the prototypical Fe-based compound Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$. While the slope of the phonon dispersion gives information about the uniform nematic susceptibility, deviations from this linear behavior at finite but small wave-vectors are attributed to finite-momentum nematic fluctuations. Surprisingly, these non-zero q fluctuations lead to a softening of the phonon mode below the superconducting transition temperature, in contrast to the behavior of the phonon velocity at q = 0, which increases below $T_c$. Our work not only establishes a sound method to probe long wavelength nematic fluctuations, but also sheds light on the unique interplay between nematicity and superconductivity in Fe-based compounds.