High-energy neutrinos from late-time jets of gamma-ray bursts seeded with cocoon photons

In gamma-ray bursts (GRBs), $\sim$ 100 - 1000 s after the prompt emission, afterglow observations have consistently shown X-ray excesses detected in the form of flares (XFs; in long GRBs) or extended emission (EEs; in short GRBs). These observations are interpreted as emissions from jets launched by late central engine activity. However, the characteristics of these late-time jets, particularly the dissipation radius ($r_{\rm diss}$), Lorentz factor ($\Gamma$), and cosmic-ray loading factor ($\xi_p$), remain unknown despite their importance. Here, in order to understand the properties of the late-time jets with future multi-messenger observations, we estimate the detectability of neutrinos associated with late-time emissions for a wide range of $r_{\rm diss}$ and $\Gamma$, assuming $\xi_p=10$. We take into account external seed photons from the cocoon around the jets, which can enhance the neutrino production through photohadronic interaction in the jet dissipation region. Our results are still consistent with the upper limit obtained by IceCube. Our calculations indicate a promising prospect for neutrino detection with IceCube-Gen2 through the stacking of $\sim 1000-2000$ events, for a wide range of $r_{\rm diss}$ and $\Gamma$. We found that setting an optimal energy threshold of 10 TeV can significantly reduce noise without negatively affecting neutrino detection. Furthermore, even in the case of non-detection, we show that meaningful constraints on the characteristics of the late-time jets can be obtained.