Locally acting mirror Hamiltonians

Abstract

Photons, i.e. the basic energy quanta of monochromatic waves, are highly non-localized and occupy all available space in one dimension. This non-local property can complicate the modelling of the quantized electromagnetic field in the presence of optical elements that are local objects. Therefore, in this paper, we take an alternative approach and quantize the electromagnetic field in position space. Taking into account the negative-and the positive-frequency solutions of Maxwell's equations, we construct annihilation operators for highly localized field excitations with bosonic commutator relations. These provide natural building blocks of wave packets of light and enable us to construct locally acting interaction Hamiltonians for two-sided semi-transparent mirrors.

Locally-acting mirror Hamiltonians

“…( 13)-are sufficiently large. Otherwise only some of the incoming light is reflected [33]. Moreover, being an exchange Hamiltonian, H I (t) transfers coherent into coherent states.…”

The quantum optics of asymmetric mirrors with coherent light absorption

“…Here H atom = ω 0 σ + σ − describes the energy of the atom and H mirr denotes the energy of the EM field in the presence of an optical interface which can be found in Eq. (15). Moreover, H int describes the atom-field interaction and equals H int = e d • E mirr (r) in the usual dipole approximation [54].…”

“…Unfortunately, this approach can result in the prediction of unphysical interference effects when modelling light approaching a mirror from both sides [11]. If one wants to avoid such interference problems, adjustments have to be made [12,13,14,15], for example by doubling the usual Hilbert space of the quantised EM field in the presence of a semi-transparent mirror [14]. However, this immediately raises the question where the doubling of the Hilbert space comes from.…”

“…However, this immediately raises the question where the doubling of the Hilbert space comes from. For a detailed discussion of this question see a recent paper by Southall et al [15] which models two-sided semi-transparent mirrors with the help of locally-acting mirror Hamiltonians and a recent paper by Hodgson et al [16] which quantises the electromagnetic field in position space.…”

The Quantum Optics of Asymmetric Mirrors With Coherent Light Absorption

“…However, this immediately raises the question where the doubling of the Hilbert space comes from. For a detailed discussion of this question see a recent paper by Southall et al (2021) which models two-sided semi-transparent mirrors with the help of locally-acting mirror Hamiltonians and a recent paper by which quantises the electromagnetic field in position space.…”