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 $\sigma_1(\omega$) well agrees with the estimate from the slope of $\sigma_2(\omega$) vs 1/$\omega$ 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 $\mu$SR measurement of magnetic penetration depth. We also find an anomalous increase of conductivity in sub-millimeter region towards $\omega$=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. 
S. Tajima, Y. Fudamoto, T. Kakeshita, B. Gorshunov, V. \v{Z}elezn\'y,   K. M. Kojima, M. Dressel, and S. Uchida