  In the framework of linearized quantum gravity, we study the quantum
gravitational interaction between two nonpointlike objects induced by
fluctuating gravitomagnetic fields in vacuum. We find that, in addition to the
quantum gravitational interaction induced by fluctuating gravitoelectric fields
previously studied, there exists a quantum gravitomagnetic interaction. This
interaction originates from the interaction between the instantaneous localized
mass currents in nonpointlike objects induced by the fluctuating
gravitomagnetic fields. Using fourth-order perturbation theory, we derive the
explicit form of the quantum gravitomagnetic interaction energy, which shows an
$r^{-10}$ dependence in the near regime and an $r^{-11}$ dependence in the far
regime, where $r$ is the distance between the two objects. This interaction
energy is expected to be significant when the gravitomagnetic polarizability of
the objects is large.
