Laser frequency stabilization for narrow linewidth cooling of $^6$Li atoms*

Laser cooling to micro-Kelvin temperatures requires a laser with active frequency stabilization. The linewidth $\Gamma$ of an atomic transition sets a lower bound on the Doppler cooling temperature $k_B T_D = \hbar \Gamma/2$. The $2s-2p$ transition in $^6$Li has a lower bound temperature of $T_D \approx 140 \, \mu$K. In contrast, the $2s-3p$ transition has a narrower linewidth and thus provides a lower temperature limit of $T_D \approx 20 \, \mu$K. We present a method for stabilizing a laser to an atomic line in a vapor cell using modulation transfer spectroscopy and a home-built lock-in amplifier. Our results demonstrate successful locking of a 323 nm laser to the $2s-3p$ transition. The stabilized laser provides a second stage of magneto-optical trapping that results in an order of magnitude increase in the phase space density before evaporating to degeneracy in an optical dipole trap.