  In this work, we consider a gravitational wave interacting with a quantum
harmonic oscillator in the transverse-traceless gauge. We take the
gravitational wave to be carrying the signatures of both plus and cross
polarization at first. We then try to obtain a suitable form of the Lewis
invariant using the most general form possible while considering only quadratic
order contributions from both position and momentum variables. In order to
progress further, we then drop the cross terms obtaining a separable
Hamiltonian in terms of the first and the second spatial coordinates. We then
obtain two Lewis invariants corresponding to each separable parts of the entire
Hamiltonian of the system. Using both Lewis invariants, one can obtain two
Ermakov-Pinney equations, from which we finally obtain the corresponding Lewis
phase and eventually the Berry phase for the entire system. Finally, we obtain
some explicit expressions of the Berry phase for a plane polarized
gravitational wave with different choices of the harmonic oscillator frequency.
