  On the background of the Born-Oppenheimer (adiabatic) approximation, we
investigate the geometrical and topological structure in the theory of quantum
gravity by using the path integral method and halfclassical approximation. As
we know, Berry curvature can be extracted from the linear response of a driven
two-level system to nonadiabatic manipulations of its Hamiltonian. In parameter
space of the Hamiltonian, magnetic monopoles can be artificially simulated.
Ripples occur in Hilbert space when the monopole travels from inside to outside
the surface of energy manifold spanned by system parameters. From this point of
view, we set up the connection between the ripples characterized by the
fidelity of quantum states in Hilbert space and gravitational-like waves in
minisuperspace. This might open a window for the study of geometrical and
topological properties of quantum gravity with the help of quantum physical
systems.
