  We present an experimental measurement of the cooperative Lamb shift and the
Lorentz shift using an atomic nanolayer with tunable thickness and atomic
density. The cooperative Lamb shift arises due to the exchange of virtual
photons between identical atoms. The interference between the forward and
backward propagating virtual fields is confirmed by the thickness dependence of
the shift which has a spatial frequency equal to $2k$, i.e. twice that of the
optical field. The demonstration of cooperative interactions in an easily
scalable system opens the door to a new domain for non-linear optics.
