Photon localization: a comparative study

We compare different measures for photon localization in terms of two-dimensional Gaussian wave packets. We find that all measures start to coalesce if the wave packet has evolved for times which are larger than a few times the inverse momentum-space width of the package. However, the Landau-Peierls wave function yields the largest positive offset <r>-ct>0 while the scalar Fourier transform of the wave packet yields the smallest offset and converges towards <r>-ct=0 fastest. We also discuss local detection of photons through a model detector consisting of ions in ion traps. The position-dependent detection probabilities are inferred from the scattering matrix. We find that the local detection probability for photons can be expressed in terms of three of the proposed localization measures, viz. the Landau-Peierls wave function, the energy wave function, and the Hawton density. Those three porposals also remain close throughout the time evolution of the single-photon wave packet.