Positronium density measurements using polaritonic effects

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Positronium density measurements using polaritonic effects

Erika Cortese, David B. Cassidy, and Simone De Liberato

Phys. Rev. A 107, 023306 – Published 10 February 2023

Abstract

Recent experimental advances in positronium (Ps) physics have made it possible to produce dense Ps ensembles in which Ps-Ps interactions may occur, leading to the production of ${Ps}_{2}$ molecules and paving the way to the realization of a Ps Bose-Einstein condensate (BEC). In order to achieve this latter goal it would be advantageous to develop new methods to measure Ps densities in real time. Here we describe a possible approach to do this using polaritonic methods: Using realistic experimental parameters, we demonstrate that a dense Ps gas can be strongly coupled to the photonic field of a distributed Bragg reflector microcavity. In this strongly coupled regime, the optical spectrum of the system is composed of two hybrid positronium-polariton resonances separated by the vacuum Rabi splitting, which is proportional to the square root of the Ps density. Given that polaritons can be created on a subcycle timescale, a spectroscopic measurement of the vacuum Rabi splitting could be used as an ultrafast Ps density measurement in regimes relevant to Ps BEC formation. Moreover, we show how positronium polaritons could potentially enter the ultrastrong light-matter coupling regime, introducing a platform to explore its nonperturbative phenomenology.

Received 7 November 2022
Accepted 23 January 2023

DOI:https://doi.org/10.1103/PhysRevA.107.023306

Physics Subject Headings (PhySH)

Bound states Cavity quantum electrodynamics Collective effects in atomic physics Polaritons Quantum electrodynamics Ultrafast optics

Atomic gases Bose-Einstein condensates Positronium Positrons

Rabi model Transfer matrix calculations

Atomic, Molecular & Optical

Authors & Affiliations

Erika Cortese ¹, David B. Cassidy², and Simone De Liberato ^1,*

¹School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, United Kingdom
²Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom

^*s.de-liberato@soton.ac.uk

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Vol. 107, Iss. 2 — February 2023

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Images

Figure 1
(a) Sketch of the Ps CQED system, representing Ps atoms interacting with the electromagnetic field trapped in a cavity formed by two DBRs. (b) Energy levels of Ps with the Lyman- $α$ and Rydberg-Rydberg transitions explicitly considered in this paper highlighted (not to scale). (c) Dipole moment $d_{n \to n + 1}$ in debyes as a function of the principal quantum number $n$ of the initial energy state.
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Figure 2
(a) Sketch of a proposed experimental setup, composed of AlGaN/AlN DBRs composed of $r = 30$ repetitions of layers of thicknesses $l_{1} = 22.7$ nm (AlGaN) and $l_{2} = 25.2$ nm (AlN), embedding a Ps formation material of thickness $L_{c} = 83.8$ nm. (b)–(d) Reflectance spectra of the cavity-Ps system as a function of the bare cavity frequency (violet dashed line), calculated via the transfer matrix at the Ps atom densities (b) $ρ_{L} = 10^{16} {cm}^{- 3}$ , (c) $ρ_{M} = 10^{18} {cm}^{- 3}$ , and (d) $ρ_{H} = 10^{19} {cm}^{- 3}$ . The bare transition energy is marked in all the panels by a green dashed line and the calculated polariton modes are blue (lower polariton) and red (upper polariton) solid curves. (e) Plots of polariton linewidths $ℏ γ_{M, \pm}$ at density $ρ_{M}$ as a function of the cavity energy (blue stars for the lower polariton and red dots for the upper polariton) compared to the bare cavity (violet dashed line) and bare matter loss (green dashed line). (f) Reflectance spectra at the resonance condition $ω_{c} = ω_{A}$ (black dashed line). The polaritonic resonances have been fitted by Lorentzian functions (magenta lines) from which the linewidths of the different resonances have been extracted.
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Figure 3
Same as in Fig. 2 but for the $12 \to 13$ Rydberg-Rydberg transition, considering a DBR composed of $r = 2$ Si layers in air, of thickness $l_{1} = 13 μ m$ and spacing $l_{2} = 44.4 μ m$ . The Ps formation material is here considered to be $L_{c} = 61.2 μ m$ thick. The chosen densities are one order of magnitude lower than those considered in Fig. 2: (b) ${\tilde{ρ}}_{L} = 10^{15} {cm}^{- 3}$ , (c) ${\tilde{ρ}}_{M} = 10^{17} {cm}^{- 3}$ , and (d) ${\tilde{ρ}}_{H} = 10^{18} {cm}^{- 3}$ , in order to take into account the lower efficiency of Rydberg state creation.
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Figure 4
(a) Plot of the relation in Eq. (2) between the VRF and Ps density. The area achieving SC is shaded in yellow. Cyan dots represent the values of the VRF derived from the numerical simulations in Sec. 2b, with blue horizontal error bars describing the uncertainties on the splitting measure and vertical magenta error bars representing the propagated error on the Ps density. (b)–(e) Sketches of the different target geometries as described in the text, all featuring a vertical thickness of the porous silica substrate compatible with the DBRs spacing for the Lyman transition $L_{c} = 83.8$ nm. However, the structure overall thickness may slightly vary according to the specific design. In (b) we show the Swiss cheese pore structure, with typical pore diameter $2 - 5$ nm. In (c) we show a collection of small disconnected cavities made by seeding a porous silica target film with monodisperse approximately ( $15 - 20$ )-nm-diam polyethylene porogen particles. In (d) we represent a straw-hat-shaped cavity, of proposed dimensions 30 nm for the maximum central height and 7 nm for the brim thickness. Finally, in (e) we show the proposed design for slowed Ps atoms, featuring a few-tens-of-nanometers-thick diamond layer facing a cavity placed within the silica substrate. The four colored symbols mark the Ps density achieved or expected for each one.
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