Locally-acting mirror Hamiltonians

Abstract

Photons with well-defined energy are highly non-localised and occupy all available space in one dimension. On the other hand, optical elements, like mirrors, are highly-localised objects. To model the electromagnetic field in the presence of optical elements, we therefore introduce annihilation operators for highly-localised field excitations. These are different from single-particle annihilation operators and arise naturally if we quantise the negative as well as the positive solutions of Maxwell's equations. Moreover, they enable us to obtain locally-acting interaction Hamiltonians for two-sided semi-transparent mirrors with a wide range of applications in quantum optics.

Local photons

“…As in Ref. [42], we find that Lorentz covariance demands the presence of a position-independent superoperator R. Nevertheless, we hope that our formalism will find a wide range of applications, from the Fermi problem [1] to an improved modelling of local light-light and light-matter interactions [18,60] and the modelling of the quantised EM field on curved spacetimes [68].…”

“…As we have recently shown in Ref. [60], the annihilation and creation operators of these particles may obey different commutator relations. Like Cook [42,51], we assume in the following bosonic commutators and refer to the corresponding particles as BLiPs which stands for bosons localised in position.…”

“…Doing so is equivalent to the introduction of negative-frequency particle states with positive energies which, as pointed out above, is a way of overcoming locality issues. Negative-frequencies already attracted some interest in the literature [46,[55][56][57][58][59][60]; sometimes they are simply added for convenience and to simplify calculations [15,61,62]. For example, Hawton and Debière [46] introduced negative-frequency states by extending the well-known expressions for the gauge field and canonical momenta of a canonical formalism.…”

“…Recently, in Ref. [60], we presented a local description of the quantised EM field in position space. Like Hawton and Debière [46], we introduce negative-frequency photons.…”

“…The primary purpose of this paper is to derive the local description of the quantised EM field in Ref. [60] from basic principles by following ideas of an approach taken earlier in Ref. [63].…”