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Superconductivity-induced transverse plasma mode and phonon anomaly in the $c$-axis response of the bilayer compound RbCa$_2$Fe$_4$As$_4$F$_2$

With infrared spectroscopy we studied the out-of-plane ($c$-axis) response of the iron arsenide superconductor ACa$_2$Fe$_4$As$_4$F$_2$ (A = Rb,Cs) which has a bilayer structure similar to the high $T_c$ cuprates YBa$_2$Cu$_3$O$_7$ (YBCO) and Bi$_2$Sr$_2$CaCu$_2$O$_8$ (Bi2212). In analogy to the cuprates, we observe a superconductivity-induced transverse plasma mode (tPM) and a phonon anomaly that are both signatures of local electric field effects that arise from a large difference between the local conductivities in the intra- and inter-bilayer regions. Using a multilayer model developed for the cuprates, we obtain a good description of the $c$-axis response and derive the local conductivities at $T \simeq T_c$ of $σ_1^{\mathrm{bl}}(ω\rightarrow 0) \simeq$ 1\,000 $Ω^{-1}\mathrm{cm}^{-1}$ and $σ_1^{\mathrm{int}}(ω\rightarrow 0) \simeq$ 15 $Ω^{-1}\mathrm{cm}^{-1}$, respectively, that are similar to the ones previously found in underdoped YBCO. Different from the cuprates, we find no evidence of a normal state pseudogap in terms of a partial suppression of the low-energy electronic states that sets in already well above $T_c$. There is also no clear sign of an onset of precursor superconducting pairing correlations well above $T_c \simeq$ 30~K. This highlights that the pseudogap and the precursor superconducting pairing well above $T_c$ are unique features of the cuprates with their strong electronic correlations and, for example, not just the result of a strongly anisotropic electronic response due to the layered crystal structure.