  Josephson meta-materials have recently emerged as very promising platform for
superconducting quantum science and technologies. Their distinguishing
potential resides in ability to engineer them at sub-wavelength scales, which
allows complete control over wave dispersion and nonlinear interaction. In this
article we report a versatile Josephson transmission line with strong third
order nonlinearity which can be tuned from positive to negative values, and
suppressed second order non linearity. As an initial implementation of this
multipurpose meta-material, we operate it to demonstrate reversed Kerr
phase-matching mechanism in traveling wave parametric amplification. Compared
to previous state of the art phase matching approaches, this reversed Kerr
phase matching avoids the presence of gaps in transmission, can reduce gain
ripples, and allows in situ tunability of the amplification band over an
unprecedented wide range. Besides such notable advancements in the
amplification performance with direct applications to superconducting quantum
computing and generation of broadband squeezing, the in-situ tunability with
sign reversal of the nonlinearity in traveling wave structures, with no
counterpart in optics to the best of our knowledge, opens exciting experimental
possibilities in the general framework of microwave quantum optics,
single-photon detection and quantum limited amplification.
