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In-plane optical conductivity of La$_{2-x}$Sr$_x$CuO$_4$: Reduced superconducting condensate and residual Drude-like response

Temperature dependences of the optical spectra of La$_{2-x}$Sr$_x$CuO$_4$ with $x$=0.12 and 0.15 were carefully examined for a polarization parallel to the CuO$_2$-plane over a wide frequency range down to 8 cm$^{-1}$. Selection of well-characterized crystals enabled us to measure purely in-plane polarized spectra without any additional peak. The weight of superconducting (SC) condensate estimated from the missing area in $σ_1(ω$) well agrees with the estimate from the slope of $σ_2(ω$) vs 1/$ω$ plot, showing no evidence that the Ferrell-Glover-Tinkham sum-rule is violated in the optical spectrum. We demonstrate that the optically estimated SC condensate is much smaller than the value obtained from the $μ$SR measurement of magnetic penetration depth. We also find an anomalous increase of conductivity in sub-millimeter region towards $ω$=0 below $T_c$, which suggests the microscopic inhomogeneity in the superconducting state. Both observations are discussed in relation with the inhomogeneous electronic state that might be inherent to high-$T_c$ cuprates.