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arXiv:2104.04499 · Frontiers in Photonics

Local photons

Daniel Hodgson, Jake Southall, Robert Purdy, Almut Beige

Published Sep 15, 2022• 15 citations
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Abstract

The classical free-space solutions of Maxwell’s equations for light propagation in one dimension include wave packets of any shape that travel at the speed of light. This includes highly-localised wave packets that remain localised at all times. Motivated by this observation, this paper builds on recent work by Southall et al. [J. Mod. Opt. 68, 647 (2021)] and shows that a local description of the quantised electromagnetic field, which supports such solutions and which must overcome several no-go theorems, is indeed possible. Starting from the assumption that the basic building blocks of photonic wave packets are so-called bosons localised in position (blips), we identify the relevant Schrödinger equation and construct Lorentz-covariant electric and magnetic field observables. In addition we show that our approach simplifies to the standard description of quantum electrodynamics when restricted to a subspace of states.

DOI
10.3389/fphot.2022.978855

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The classical free-space solutions of Maxwell’s equations for light propagation in one dimension include wave packets of any shape that travel at the speed of light. This includes highly-localised wave packets that remain localised at all times. Motivated by this observation, this paper builds on recent work by Southall et al. [J. Mod. Opt. 68, 647 (2021)] and shows that a local description of the quantised electromagnetic field, which supports such solutions and which must overcome several no-go theorems, is indeed possible. Starting from the assumption that the basic building blocks of photonic wave packets are so-called bosons localised in position (blips), we identify the relevant Schrödinger equation and construct Lorentz-covariant electric and magnetic field observables. In addition we show that our approach simplifies to the standard description of quantum electrodynamics when restricted to a subspace of states.
An intuitive and straightforward modelling of many locally-interacting quantum system requires a local description of the quantised electromagnetic field. In a recent paper [Southall et al., J. Mod. Opt. 68, 647 (2021)], we showed that such a description, which must overcome several no-go theorems, is indeed possible. To better justify our approach, this paper presents a systematic quantisation of one-dimensional electromagnetic fields in position space. Starting from the assumption that its basic building blocks are local bosonic particles with a clear direction of propagation--so-called bosons localised in position (BLiPs)--we identify the relevant Schroedinger equation and construct Lorentz-covariant electric and magnetic field observables. In addition we show that our approach simplifies to the standard description of quantum electrodynamics when restricted to a subspace of monochromatic photon Fock states.
In a recent paper [Southall et al., Locally-acting mirror Hamiltonians, arXiv:1908.07597 (2021)], we showed that it is possible to design locally-acting mirror Hamiltonians which only affect incoming but do not change the dynamics of outgoing wave packets. To better justify our approach, this paper presents a systematic quantisation of the electromagnetic field in position space. Starting from the assumption that the basic building blocks of the electromagnetic field in one dimension are localised bosonic wave packets with a clear direction of propagation -- so-called bosons localised in position (BLiPs) -- we identify the relevant Schroedinger equation and construct Lorentz covariant electric and magnetic field observables. Our description contains the standard description of the quantised electromagnetic field which is shown to apply to a subspace of states.
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.
Photons, i.e. the basic energy quanta of monochromatic waves, are highly non-localised and occupy all available space in one dimension. This can complicate the modelling of the quantised electromagnetic field in the presence of optical elements which correspond to localised objects. In this paper, we therefore quantise the electromagnetic field in position space and introduce annihilation operators for highly-localised field excitations. It is shown that these are well-defined, if we quantise the negative as well as the positive frequency solutions of Maxwell's equations. Highly-localised field excitations are natural basic building blocks of wave packets of light and enable us to construct locally-acting interaction Hamiltonians for two-sided semi-transparent mirrors
In recent years, non-Hermitian quantum physics has gained a great deal of popularity in the quantum optics and condensed matter communities in order to model quantum systems with varying symmetries. In this paper, we identify a non-standard inner product that implies bosonic commutator relations for local electric and magnetic field observables and leads to a natural local biorthogonal description of the quantised electromagnetic field. When comparing this description with an alternative local Hermitian description, in which the states of local photonic particles, i.e., of so-called bosons localised in position (blips), are orthogonal under the conventional Hermitian inner product, we find that there is an equivalence between the two approaches. Careful consideration needs to be given to the physical interpretation of the different descriptions. Whether a Hermitian or a non-Hermitian approach is more suitable depends on the circumstances that we want to model.
In this chapter we examine the quantised electromagnetic (EM) field in the context of a Schrödinger equation for single photons. For clarity we consider only a one-dimensional system. As a universal tool for calculating the time-evolution of quantum states, a Schrödinger equation must exist that describes the propagation of single photons. Being inherently relativistic, however, critical aspects of both special relativity and quantum mechanics must be combined when quantising the EM field. By taking the approach of a Schrödinger equation for localised photons, we will show how novel and previously overlooked features of the quantised EM field become a necessary part of a complete description of photon dynamics. In this chapter, I shall provide a thorough examination of new features and discuss their significance in topics such as quantum relativity and photon localisation.