  We present an experimental and theoretical analysis of the self- and
cross-Kerr effect of extended plasma resonances in Josephson junction chains.
We calculate the Kerr coefficients by deriving and diagonalizing the
Hamiltonian of a linear circuit model for the chain and then adding the
Josephson non-linearity as a perturbation. The calculated Kerr-coefficients are
compared with the measurement data of a chain of 200 junctions. The Kerr effect
manifests itself as a frequency shift that depends linearly on the number of
photons in a resonant mode. By changing the input power on a low signal level,
we are able to measure this shift. The photon number is calibrated using the
self-Kerr shift calculated from the sample parameters. We then compare the
measured cross-Kerr shift with the theoretical prediction, using the calibrated
photon number.
