  Many applications in photonics require all-optical manipulation of plasma
waves, which can concentrate electromagnetic energy on sub-wavelength length
scales. This is difficult in metallic plasmas because of their small optical
nonlinearities. Some layered superconductors support weakly damped plasma
waves, involving oscillatory tunneling of the superfluid between capacitively
coupled planes. Such Josephson plasma waves (JPWs) are also highly nonlinear,
and exhibit striking phenomena like cooperative emission of coherent terahertz
radiation, superconductor-metal oscillations and soliton formation. We show
here that terahertz JPWs in cuprate superconductors can be parametrically
amplified through the cubic tunneling nonlinearity. Parametric amplification is
sensitive to the relative phase between pump and seed waves and may be
optimized to achieve squeezing of the order parameter phase fluctuations or
single terahertz-photon devices.
