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Quantising one-dimensional electromagnetic fields in position space

Daniel Hodgson, Jake Southall, Robert Purdy, Almut Beige

Published Apr 9, 2021• 2 citations
OpenAlex

Abstract

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.

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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.
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.
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.
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.
The notion that the electromagnetic field is quantised is usually inferred from observations such as the photoelectric effect and the black-body spectrum. However accounts of the quantisation of this field are usually mathematically motivated and begin by introducing a vector potential, followed by the imposition of a gauge that allows the manipulation of the solutions of Maxwell's equations into a form that is amenable for the machinery of canonical quantisation. By contrast, here we quantise the electromagnetic field in a less mathematically and more physically-motivated way. Starting from a direct description of what one sees in experiments, we show that the usual expressions of the electric and magnetic field observables follow from Heisenberg's equation of motion. In our treatment, there is no need to invoke the vector potential in a specific gauge and we avoid the commonly-used notion of a fictitious cavity that applies boundary conditions to the field.