  Non-local quasiparticles in correlated quantum materials can exhibit the
proximity effect. For instance, in metal superconductor hybrid systems, the
leaking of cooper pairs to the metallic region induces superconducting
correlations in a standard metal. This paper explores the proximity effects of
charge density wave (CDW) on metal using the attractive Hubbard model, which
harbors CDW state at half-filling. Our fully self-consistent calculations
demonstrate that periodic charge modulations develop in a metal due to the
tunneling of finite momentum particle-hole pairs from the CDW region. Upon
doping the normal region, the commensurate CDW changes to an incommensurate one
by incorporating regular phase shifts. Furthermore, the induced CDW produces a
soft gap in the density of states and thus can be detected in tunneling
experiments. We discuss our results in light of recent reports of such
proximity-induced charge order in different two-dimensional heterostructures.
