  High-temperature cuprates superconductors are characterised by the complex
interplay between superconductivity (SC) and charge density wave (CDW) in the
context of intertwined competing orders. In contrast to abundant studies for
hole-doped cuprates, the exact nature of CDW and its relationship to SC was
much less explored in electron-doped counterparts. Here, we performed resonant
inelastic x-ray scattering (RIXS) experiments to investigate the relationship
between CDW and SC in electron-doped La$_{2-x}$Ce$_x$CuO$_4$. The short-range
CDW order with a correlation length $\sim35$~\AA~was found in a wide range of
temperature and doping concentration. Near the optimal doping, the CDW order is
weakened inside the SC phase, implying an intimate relationship between the two
orders. This interplay has been commonly reported in hole-doped La-based
cuprates near the optimal doping. We reconciled the diverging behaviour of CDW
across the superconducting phase in various cuprate materials by introducing
the CDW correlation length as a key parameter. Our study paves the way for
establishing a unified picture to describe the phenomenology of CDW and its
relationship with SC in the cuprate family.
