Quantum phase transition inside the superconducting dome of Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ probed by optical magneto-sensing using NV-centers in diamond

While unusual normal state properties, such as non-Fermi liquid behavior of the resistivity, are commonly associated with strong quantum fluctuations, evidence for its presence inside the superconducting dome are much scarcer. In this paper, we use sensitive and minimally invasive optical magnetometry based on NV-centers in diamond to probe the doping evolution of the $T=0$ penetration depth in the electron-doped iron-based superconductor Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$. A non-monotonic evolution with a pronounced peak in the vicinity of the putative magnetic QPT is found. This behavior is reminiscent to that previously seen in isovalently-substituted BaFe$_2$(As$_{1-x}$P$_x$)$_2$ compounds, despite the notable differences between these two systems. Whereas the latter is a very clean system that displays nodal superconductivity and a single simultaneous first-order nematic-magnetic transition above, and even somewhat below, $T_c$, the former is a significantly dirtier system with fully gapped superconductivity and split second-order nematic and magnetic transition above $T_c$. Thus our observation that such distinct systems display remarkably similar penetration depth peaks, combined with the theoretical result that a QPT alone does not ensure the existence of a peak, unveils a puzzling and seemingly universal manifestation of quantum fluctuations in the iron pnictides.