The production of a sterile species via active-sterile mixing in a thermal medium is studied in an exactly solvable model. The exact time evolution of the sterile distribution function is determined by the dispersion relations and damping rates ${\Gamma }_{1,2}$ for the quasiparticle modes. These depend on $\tilde{\gamma }={\Gamma }_{aa}/2\Delta E$, with ${\Gamma }_{aa}$ the interaction rate of the active species in absence of mixing and $\Delta E$ the oscillation frequency in the medium without damping. $\tilde{\gamma }\ll 1$, $\tilde{\gamma }\gg 1$ describe the weak and strong damping limits, respectively. For $\tilde{\gamma }\ll 1$, ${\Gamma }_1={\Gamma }_{aa}{cos}^2{\theta }_m$; ${\Gamma }_2={\Gamma }_{aa}{sin}^2{\theta }_m$ where ${\theta }_m$ is the mixing angle in the medium and the sterile distribution function does not obey a simple rate equation. For $\tilde{\gamma }\gg 1$, ${\Gamma }_1={\Gamma }_{aa}$ and ${\Gamma }_2={\Gamma }_{aa}{sin}^{2}2{\theta }_m/4{\tilde{\gamma }}^2$, is the sterile production rate. In this regime sterile production is suppressed and the oscillation frequency vanishes at an Mikheyev-Smirnov-Wolfenstein (MSW) resonance, with a breakdown of adiabaticity. These are consequences of quantum Zeno suppression. For active neutrinos with standard model interactions the strong damping limit is only available near an MSW resonance if $\sin 2\theta \ll {\alpha }_w$ with $\theta$ the vacuum mixing angle. The full set of quantum kinetic equations for sterile production for arbitrary $\tilde{\gamma }$ are obtained from the quantum master equation. Cosmological resonant sterile neutrino production is quantum Zeno suppressed relieving potential uncertainties associated with the QCD phase transition.

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