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We present the properties of 16 faint Lyman-$\alpha$ emitting galaxies (LAEs)at $z>5.8$ from the JWST Advanced Deep Extragalactic Survey (JADES)spectroscopic data in the Hubble Ultra Deep Field/GOODS-S. These LAEs span aredshift range $z\approx5.8-8.0$ and UV magnitude range $M_{\textrm{UV}}\approx -17$ to $-20.6$, with Ly$\alpha$ equivalent width (EW) in the range$\approx 25-350$ \AA. The detection of other rest-optical emission lines in thespectra of these LAEs enables the determination of accurate systemic redshiftsand Ly$\alpha$ velocity offsets, as well as the physical and chemicalcomposition of their stars and interstellar media. These faint LAEs areconsistent with metal-poor systems with high ionization parameters, similar tothe general galaxy population at $z>6$. We measure an average ionizing photonproduction efficiency, log($\xi_{\textrm{ion}}$/erg$^{-1}$ Hz) $\approx 25.56$across our LAEs, which does not evolve strongly with redshift. We report ananti-correlation between Ly$\alpha$ escape fraction and velocity offset fromsystemic, consistent with model expectations. We further find that the strengthand velocity offset of Ly$\alpha$ are not correlated with galaxy spectroscopicproperties nor with $\xi_{\textrm{ion}}$. We find a decrease in Ly$\alpha$escape fractions with redshift, indicative of decreasing sizes of ionizedbubbles around LAEs at high redshifts. We use a range of galaxy properties topredict Lyman continuum escape fractions for our LAEs, finding that theionizing photon output into the intergalactic medium from our LAEs remainsroughly constant across the observed UV magnitude and Ly$\alpha$ equivalentwidth, showing a mild increase with redshift. We derive correlations betweenthe ionizing photon output from LAEs and UV magnitude Ly$\alpha$ strengths andredshift, which can be used to build realistic reionization models.

JADES: The production and escape of ionizing photons from faint Lyman-alpha emitters in the epoch of reionization