  The generation of quantum entanglement between phonons in photoirradiated
remote electron-phonon systems is numerically studied. Upon excitation by a
visible/ultraviolet laser pulse, the entanglement of electrons is immediately
generated and that of phonons follows via electron-phonon interactions, i.e.,
the entanglement generation of phonons is a two-step process. Therefore, it is
important to design the temporal properties of incident optical pulses in order
to control the entanglement of electrons and/or phonons. These features are
revealed by the quantum mutual information and the composite modes derived from
the Heisenberg equation of motion. The calculated results also show that the
dynamics of the phonon entanglement can be observed by time-resolved
spectroscopy on the scattered light.
