In strongly correlated spin systems, the presence of nuclear spins and hyperfine interactions can have highly non-trivial effects. In the case of LiHoF4 as an archetypical 3D dipolar Ising magnet, the strong hyperfine interaction directly modifies its quantum critical behaviours [1]. To elucidate the underlying physics, we perform radio-frequency spectroscopy on the electro-nuclear spin excitation spectra in LiHoF4, enabled by strongly hybridizing a single crystal sample with a high-finesse re-entrant cavity at milli-kelvin temperatures and high magnetic fields. Additionally, by integrating the linear response theory [2] with the input-output formalism [3], we derive an explicit expression that directly connects experimental observations to the spin states. On the one hand, we show quantitative agreement between theoretical and experimental results across a wide range of parameter space [4], thereby providing strong support for existing hypotheses [5]. On the other hand, we also demonstrate further complex behavior near the quantum critical point that is still beyond current understanding.

[1] H. M. Ronnow, et al. Science, 5720, 308 (2005)
[2] J. Jensen. Rare Earth Magnetism (1991)
[3] P. Meystre, et al. Elements of Quantum Optics (2007)
[4] Y. Yang, et al. arXiv:2309.05051 [cond-mat.str-el] (2023)
[5] R. D. McKenzie, et al. Phys. Rev. B, 97, 214430 (2018)