Gigahertz-Bandwidth Optical Memory in Pr$^{3+}$:Y$_2$SiO$_5$

We experimentally study a broadband implementation of the atomic frequency comb (AFC) rephasing protocol with a cryogenically cooled Pr$^{3+}$:Y$_2$SiO$_5$ crystal. To allow for storage of broadband pulses, we explore a novel regime where the input photonic bandwidth closely matches the inhomogeneous broadening of the material $(sim5,textrm{GHz})$, thereby significantly exceeding the hyperfine ground and excited state splitting $(sim10,textrm{MHz})$. Through an investigation of different AFC preparation parameters, we measure a maximum efficiency of $10%$ after a rephasing time of $12.5,$ns. With a suboptimal AFC, we witness up to 12 rephased temporal modes.