Spin-wave directional anisotropies in antiferromagnetic Ba3NbFe3Si2O14

${\mathrm{Ba}}_{3}{\mathrm{NbFe}}_{3}{\mathrm{Si}}_{2}{\mathrm{O}}_{14}$ (langasite) is structurally and magnetically single-domain chiral with the magnetic helicity induced through competing symmetric exchange interactions. Using neutron scattering, we show that the spin waves in antiferromagnetic langasite display directional anisotropy. On applying a time-reversal symmetry breaking magnetic field along the $c$ axis, the spin-wave energies differ when the sign is reversed for either the momentum transfer $\ifmmode\pm\else\textpm\fi{}\stackrel{P\vec}{Q}$ or applied magnetic field $\ifmmode\pm\else\textpm\fi{}{\ensuremath{\mu}}_{0}\mathrm{H}$. When the field is applied within the crystallographic $ab$ plane, the spin-wave dispersion is directionally isotropic and symmetric in $\ifmmode\pm\else\textpm\fi{}{\ensuremath{\mu}}_{0}\mathrm{H}$. However, a directional anisotropy is observed in the spin-wave intensity. We discuss this directional anisotropy in the dispersion in langasite in terms of a field-induced precession of the dynamic unit cell staggered magnetization resulting from a broken twofold symmetry. Directional anisotropy, often referred to as nonreciprocal responses, can occur in antiferromagnetic phases in the absence of the Dzyaloshinskii-Moriya interaction or other effects resulting from spin-orbit coupling.