  We analyze nonlinear optics schemes for generating pairs of quantum entangled
plasmons in the terahertz-infrared range in graphene. We predict that high
plasmonic field concentration and strong optical nonlinearity of monolayer
graphene enables pair-generation rates much higher than those of conventional
photonic sources. The first scheme we study is spontaneous parametric down
conversion in a graphene nanoribbon. In this second-order nonlinear process a
plasmon excited by an external pump splits into a pair of plasmons, of half the
original frequency each, emitted in opposite directions. The conversion is
activated by applying a dc electric field that induces a density gradient or a
current across the ribbon. Another scheme is degenerate four-wave mixing where
the counter-propagating plasmons are emitted at the pump frequency. This
third-order nonlinear process does not require a symmetry-breaking dc field. We
suggest nano-optical experiments for measuring position-momentum entanglement
of the emitted plasmon pairs. We estimate the critical pump fields at which the
plasmon generation rates exceed their dissipation, leading to parametric
instabilities.
