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Electromagnetically induced transparency (EIT) has been often proposed forgenerating nonlinear optical effects at the single photon level; in particular,as a means to effect a quantum non-demolition measurement of a single photonfield. Previous treatments have usually considered homogeneously broadenedsamples, but realisations in any medium will have to contend with inhomogeneousbroadening. Here we reappraise an earlier scheme [Munro \textit{et al.} Phys.Rev. A \textbf{71}, 033819 (2005)] with respect to inhomogeneities and show analternative mode of operation that is preferred in an inhomogeneousenvironment. We further show the implications of these results on a potentialimplementation in diamond containing nitrogen-vacancy colour centres. Ourmodelling shows that single mode waveguide structures of length $200\mu\mathrm{m}$ in single-crystal diamond containing a dilute ensemble of NV$^-$of only 200 centres are sufficient for quantum non-demolition measurementsusing EIT-based weak nonlinear interactions.

Single photon quantum non-demolition in the presence of inhomogeneous broadening