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Distinguishing photons and anti-photons for highly-localised mirror Hamiltonians

Jake Southall, Robert Purdy, Almut Beige

Published Aug 20, 2019• 0 citations
OpenAlex

Abstract

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

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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
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.
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.
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.
The local observables of the quantised electromagnetic field near a mirror-coated interface depend strongly on the properties of the media on both sides. In macroscopic quantum electrodynamics, this fact is taken into account with the help of optical Green’s functions which correlate the position of an observer with all other spatial positions and photon frequencies. Here we present an alternative, more intuitive approach and obtain the local field observables with the help of a quantum mirror image detector method. In order to correctly normalise electric field operators, we demand that spontaneous atomic decay rates simplify to their respective free space values far away from the reflecting surface. Our approach is interesting, since mirror-coated interfaces constitute a common basic building block for quantum photonic devices.
Non-classical light sources are now being developed in more and more laboratories. Their noise reduction properties are potentially important for technology. Examples are squeezed and antibunched light sources. It is tempting to picture in some of these processes single photons flying through space. In non-relativistic quantum mechanics it is easy to discuss the properties of particle localisation in a general context without recourse to a particular experimental configuration. A position operator exists and the localised states are just the eigenstates of this operator. The situation in relativistic quantum mechanics (or field theory) is not as straightforward. The photon is relativistic and it has the addi-tional property of being massless. It has no position operator associated with it. Yet, nevertheless, it is possible to pose the question of finding a generic construction for a wave packet for the photon which is localised as far as possible in some (imaginary) box in space. The difficulty of the construction follows from the fact that in an electromagnetic wave the polarisation is perpendicular to the direction of propagation. Thus we need to construct a vector field which is localised and has this transverse property related to the masslessness of the photon.