Ellerman bombs (EBs) with significant H$\alpha$ wing emissions and ultraviolet bursts (UV bursts) with strong Si IV emissions are two kinds of small transient brightening events that occur in the low solar atmosphere.We numerically investigated the magnetic reconnection process between the emerging arch magnetic field and the lower atmospheric background magnetic field. We aim to find out if the hot UV emissions and much colder H$\alpha$ wing emissions can both appear in the same reconnection process and how they are located in the reconnection region. The open-source code NIRVANA was applied to perform the 2.5D magnetohydrodynamic (MHD) simulation. We developed the related sub-codes to include the more realistic radiative cooling process for the photosphere and chromosphere and the time-dependent ionization degree of hydrogen. The initial background magnetic field is 600 G, and the emerged magnetic field in the solar atmosphere is of the same magnitude, meaning that it results in a low- $\beta$ magnetic reconnection environment. We also used the radiative transfer code RH1.5D to synthesize the Si IV and H$\alpha$ spectral line profiles based on the MHD simulation results. Magnetic reconnection between emerged and background magnetic fields creates a thin, curved current sheet, which then leads to the formation of plasmoid instability and the nonuniform density distributions. The mix of hot tenuous and much cooler dense plasmas in the turbulent reconnection region can appear at about the same height, or even in the same plasmoid. The turbulent current sheet is always in a dense plasma environment with an optical depth larger than 6.5$\times$10$^{-5}$ due to the emerged magnetic field pushing high-density plasmas upward.
