Frequency ratio of the $^{229\mathrm{m}}$Th nuclear isomeric transition and the $^{87}$Sr atomic clock

Optical atomic clocks$^{1,2}$ use electronic energy levels to precisely keep track of time. A clock based on nuclear energy levels promises a next-generation platform for precision metrology and fundamental physics studies. Thorium-229 nuclei exhibit a uniquely low energy nuclear transition within reach of state-of-the-art vacuum ultraviolet (VUV) laser light sources and have therefore been proposed for construction of the first nuclear clock$^{3,4}$. However, quantum state-resolved spectroscopy of the $^{229m}$Th isomer to determine the underlying nuclear structure and establish a direct frequency connection with existing atomic clocks has yet to be performed. Here, we use a VUV frequency comb to directly excite the narrow $^{229}$Th nuclear clock transition in a solid-state CaF$_2$ host material and determine the absolute transition frequency. We stabilize the fundamental frequency comb to the JILA $^{87}$Sr clock$^2$ and coherently upconvert the fundamental to its 7th harmonic in the VUV range using a femtosecond enhancement cavity. This VUV comb establishes a frequency link between nuclear and electronic energy levels and allows us to directly measure the frequency ratio of the $^{229}$Th nuclear clock transition and the $^{87}$Sr atomic clock. We also precisely measure the nuclear quadrupole splittings and extract intrinsic properties of the isomer. These results mark the start of nuclear-based solid-state optical clock and demonstrate the first comparison of nuclear and atomic clocks for fundamental physics studies. This work represents a confluence of precision metrology, ultrafast strong field physics, nuclear physics, and fundamental physics.