  We show, in the context of single photon detection, that an atomic
three-level model for a transmon in a transmission line does not support the
predictions of the nonlinear polarisability model known as the cross-Kerr
effect. We show that the induced displacement of a probe in the presence or
absence of a single photon in the signal field, cannot be resolved above the
quantum noise in the probe. This strongly suggests that cross-Kerr media are
not suitable for photon counting or related single photon applications. Our
results are presented in the context of a transmon in a one dimensional
microwave waveguide, but the conclusions also apply to optical systems.
