Non-locality of the energy density for all single-photon states

Abstract

The non-locality of single-photon states has been analyzed from several different but interrelared perspectives. In this article, we propose a demonstration based on the electromagnetic energy density observable and on the anti-local property of the frequency operator $\Omega=c(-\Delta)^{1/2}$. The present proof is based on the standard quantization of the electromagnetic field, which can be formulated equivalently in the momentum representations or in the position representations of Landau-Peierls and of Bia{\l}ynicki-Birula. Our proof extends to all single-photon states the results of Bia{\l}ynicki-Birula, that were formulated for two particular classes of states, involving either a uniform localization [I. Bia{\l}ynicki-Birula, Phys. Rev. Lett. {\bf80} 5247 (1998)], or alternatively, states that are electrically or magnetically localized, as defined in [I. Bia{\l}ynicki-Birula, Z. Bia{\l}ynicka-Birula, Phys.Rev. A {\bf79} 032112 (2009)]. Our approach is formulated in terms of Knight's definition of strict localization, based on the comparison of single-photon states expectation values of local observables with that of the vacuum