The low energy lattice dynamics of the quasi-periodic spin-ladder cuprate Sr$_{14-x}$Ca$_x$Cu$_{24}$O$_{41}$ are investigated using terahertz frequency synchrotron radiation. A high density of low-lying optical excitations are present in the 1-3 THz energy range, while at least two highly absorbing excitations stemming from rigid acoustic oscillations of the incommensurate chain and ladder sublattices, are observed at sub-terahertz frequencies. The effects of Ca substitution on the sub-terahertz quasi-acoustic sliding mode gaps is investigated using coherent synchrotron radiation. Analysis of the results suggest increasing substitution of Sr for Ca is accompanied by a transfer of spectral weight between sliding modes associated with different chain-ladder dynamics. The observation is consistent with a transfer of hole charges from the chains to the ladders and modification of the sublattice dimensions following Ca substitution. The results are discussed in context to the significance of low-lying vibrational dynamics and electron-phonon coupling in the superconducting state of certain quasi-periodic systems.
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