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EPJ Web of Conferences

Non-local property of single-photon states: Illustration with spontaneously emitted photon from a Hydrogen atom

Maxime Federico, Hans Rudolf Jauslin

Published Jan 1, 2023• 0 citations
Publisher

Abstract

We discuss a new proof of the non-locality of single-photons using a concrete physical observable such as the local energy density. As an illustration of this non-locality, we compute the mean value of the local energy observable for the spontaneous emission from a Hydrogen atom. We find that, in the subspace of single-photon states, the mean value of the local energy density observable is non-zero everywhere and decreases as a power of the distance.

DOI
10.1051/epjconf/202328704034

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We discuss a new proof of the non-locality of single-photons using a concrete physical observable such as the local energy density. As an illustration of this non-locality, we compute the mean value of the local energy observable for the spontaneous emission from a Hydrogen atom. We find that, in the subspace of single-photon states, the mean value of the local energy density observable is non-zero everywhere and decreases as a power of the distance.
Abstract We analyze through the expectation value of the energy density the spatial nonlocality of single photons emitted by the spontaneous decay of a Hydrogen atom. By using a minimal coupling between the quantized electromagnetic field and the atom, we compute the state of the photon under the assumption that only a single-photon is produced. The calculations are thus performed in the subspace of single-photon states which is essentially equivalent to the rotating wave approximation. We obtain a characterization of the spatial decay of the energy density. We compute the asymptotic limit of large distances from the atom at each given time, and find an algebraic behavior of 1 / r 6 . This result confirms that the energy density of single-photon states is nonlocal and the algebraic decay is far from the maximal quasiexponential localization predicted by the theory.
The nonlocality of single-photon states has been analyzed from several different but interrelated perspectives. In this article we propose a demonstration based on the electromagnetic energy density observable and on the antilocal property of the frequency operator $\mathrm{\ensuremath{\Omega}}=c{(\ensuremath{-}\mathrm{\ensuremath{\Delta}})}^{1/2}$. The present proof is based on the standard quantization of the electromagnetic field, which can be formulated equivalently in the momentum representations or in the position representations of Landau and Peierls [Z. Phys. 62, 188 (1930)] and of Bia\l{}ynicki-Birula [edited by E. Wolf (Elsevier, Amsterdam, 1996)]. Our proof extends to all single-photon states the results of Bia\l{}ynicki-Birula, which were formulated for two particular classes of states, those involving a uniform localization [Phys. Rev. Lett. 80, 5247 (1998)] or alternatively states that are electrically or magnetically localized [Phys. Rev. A 79, 032112 (2009)]. Our approach is formulated in terms of Knight's definition of strict localization [J. Math. Phys. 2, 459 (1961)], based on the comparison of expectation values of single-photon states of local observables with those of the vacuum.
We calculate spatial localization and phase properties of spontaneously emitted photons. Our model is simple yet fully quantized: the emitting atom is a two-level atom located in a one-dimensional multimode optical cavity. Although the photon state vector does not have a position-space representation, the expectation value of the square of the electric field operator (intensity) is spatially localized and this pattern shifts at the speed of light. The emitted photon exhibits classical-like phase properties in the intensity expectation value when it “interferes with itself” after reflection. The phase properties of the emitted radiation are also evident at times sufficiently long for the radiation to have returned to the emitting atom before the decay of excited atomic state is complete.
Recent works have shown that collective single photon spontaneous emission from an ensemble of N resonant two-level atoms is a rich field of study. Superradiance describes emission from a completely symmetric state of N atoms, with a single excited atom prepared with a given phase, for instance imprinted by an external laser. Instead, subradiance is associated with the emission from the remaining N−1 asymmetric states, with a collective decay rate less than the single-atom value. Here, we discuss the properties of the orthonormal basis of symmetric and asymmetric states and the entanglement properties of superradiant and subradiant states.
In this paper we investigate the non-Markovian dynamics of a giant atom coupled to a one-dimensional photonic lattice with synthetic gauge fields. By engineering a complex-valued hopping amplitude, we break reciprocity and explore how chiral propagation and phase-induced interference affect spontaneous emission, bound-state formation, and atom–field entanglement. The giant atom interacts with the lattice at multiple, spatially separated sites, leading to rich interference effects and decoherence-free subspaces. We derive an exact expression for the self-energy and perform real-time Schrödinger simulations in the single-excitation subspace, for the atomic population, von Neumann entropy, field localization, and asymmetry in emission. Our results show that the hopping phase ϕ governs not only the directionality of emitted photons but also the degree of atom–bath entanglement and photon localization. Remarkably, we observe robust bound states inside the photonic band and directional asymmetry, due to interference from spatially separated coupling points. These findings provide a basis for engineering non-reciprocal, robust, and entangled light–matter interactions in structured photonic systems.