No such mediums does not exist physically but when we are doing Physics through mathematical analysis we may obtain negative values.

My suggestion is to apply Diarac theory  for negative kerr medium and find its constitue ts.

I guess you are looking for refractive index  less than one, which I believe is possible with a rectified LC resonator in vacuum and in a few liquids and solids. It is not in agreement with FLRW, which I have suggested is in need of revision about the curvature of EM.

My recent work is predicting local vacuum experiences a negative Kerr effect in vicinity of a fast moving vehicle of large mass.

Rectified in this case means the inductors of an LC oscillator are in the DC branch of a full wave rectifier, while the capacitors reside in the AC branch with the power supply. An enormous DC magnetic field can be created with relatively little total energy contained when resonance is achieved by design and operation, if the magnetic flux is conserved in a closed loop..

I have considered two cases. In one case the magnetic flux core is in the shape of a ring, but gradually tapered to be thicker on one side than the other. Flux is forced to a limiting extreme in the thin part. In the other case the thin section of the flux ring is surrounded by a super conducting collar that further compresses the flux density.

In any case the Kerr effect is a type of local curvature modification by EM.