Rotational-hyperfine cooling of $^{205}$TlF in a cryogenic beam

The aim of CeNTREX (Cold Molecule Nuclear Time-Reversal Experiment) is to search for time-reversal symmetry violation in the thallium nucleus, by measuring the Schiff moment of $^{205}$Tl in the polar molecule thallium fluoride (TlF). CeNTREX uses a cryogenic beam of TlF with a rotational temperature of 6.3(2) K. This results in population spread over dozens of rotational and hyperfine sublevels of TlF, while only a single level is useful for the Schiff moment measurement. Here we present a protocol for cooling the rotational and hyperfine degrees of freedom in the CeNTREX beam, transferring the majority of the Boltzmann distribution into a single rotational and hyperfine sublevel by using a single ultraviolet laser and a pair of microwave beams. We achieve a factor of $20.1(4)$ gain in the population of the $J=0$, $F=0$ hyperfine sublevel of the TlF ground state.