  Josephson Traveling Wave Parametric Amplifiers (J-TWPAs) are promising
platforms for realizing broadband quantum-limited amplification of microwave
signals. However, substantial gain in such systems is attainable only when
strict constraints on phase matching of the signal, idler and pump waves are
satisfied -- this is rendered particularly challenging in the presence of
nonlinear effects, such as self- and cross-phase modulation, which scale with
the intensity of propagating signals. In this work, we present a simple J-TWPA
design based on `left-handed' (negative-index) nonlinear Josephson
metamaterial, which has phase matching native to its design precluding the need
for any complicated circuit or dispersion engineering. The resultant efficiency
of four-wave mixing process can implement gains in excess of 20 dB over few GHz
bandwidths with much shorter lines than previous implementations. Furthermore,
the autonomous nature of phase matching considerably simplifies the J-TWPA
design than previous implementations based on `right-handed' (positive index)
Josephson metamaterials, making the proposed architecture particularly
appealing from a fabrication perspective. The left-handed JTL introduced here
constitutes a new modality in distributed Josephson circuits, and forms a
crucial piece of the unified framework that can be used to inform the optimal
design and operation of broadband microwave amplifiers.
