Single-photon observables and preparation uncertainty relations (Vol. 46 No. 5-6)

Increase of uncertainty relations for circularly polarized Gaussian states as a function of the momentum spread. In the paraxial limit (ΔP→0) ħ/2 is retrieved

The escalating requests for highly accurate manipulation of single photons call for an appropriate description of their observables. The authors provide a unified procedure for treating all single-photon observables in terms of Positive Operator-Valued Measures (POVMs), allowing for the evaluation of corresponding probability distributions.

The suppression of longitudinal (or equivalently 0-helicity) photon states is identified as a projection from an extended Hilbert space onto the physical one, carrying an irreducible spin-1 mass-0 representation of the Poincaré group.

POVMs are naturally obtained by applying such projections to Projection-Valued Measures (PVMs) associated to operators well-defined on the extended Hilbert space. Such operators are inherited from the familiar relativistic description of spin-1 massive particles and simply adapted to photons. Results show that PVMs of momentum and helicity remain unaltered, while those of position and spin are turned into POVMs by the projection, reflecting their intrinsic unsharpness. Finally, evaluation of uncertainty relations for position and momentum and probability distribution of spin over a broad class of physically relevant states is done, leading to new quantitative and experimentally measurable results.

G. Guarnieri, M. Motta and L. Lanz, Single-photon observables and preparation uncertainty relations, J. Phys. A: Math.Theor., 48, 265302 (2015)