  We present a new model of two electrically coupled optomechanical cavities.
This model is based on the recently presented [Physical Review A \textbf{103}
(2021) 043509]. We found that coupling two optomechanical cavities via Coulomb
force leads to cross-Kerr interactions between those cavities. We show that
such systems may be ideal for a protocol of quantum non-demolition measurement
because it is easy to eliminate the self-phase modulation effect. Moreover,
nonlinearities in our model are based on easily adjustable parameters, and
therefore, given recent experimental studies, we believe that experimental
realization of a cross-Kerr interaction via Coulomb force coupling is feasible.
