Quantum electrodynamics description of localized surface plasmons at a metal nanosphere

Letter

Quantum electrodynamics description of localized surface plasmons at a metal nanosphere

Kuniyuki Miwa and George C. Schatz

Phys. Rev. A 103, L041501 – Published 20 April 2021

Abstract

A canonical quantization scheme for localized surface plasmons (LSPs) in a metal nanosphere is presented based on a microscopic model composed of electromagnetic fields, oscillators that describe plasmons, and a reservoir that describes excitations other than plasmons. The eigenmodes of this fully quantum electrodynamic theory show a spectrum that includes radiative depolarization and broadening, including redshifting from the quasistatic LSP modes with increasing particle size. These spectral profiles correctly match those obtained with exact classical electrodynamics (Mie theory). The present scheme provides the electric fields per plasmon in both near- and far-field regions whereby its utility in the fields of quantum plasmonics and nano-optics is demonstrated.

Received 9 July 2020
Revised 20 January 2021
Accepted 5 April 2021

DOI:https://doi.org/10.1103/PhysRevA.103.L041501

Physics Subject Headings (PhySH)

Dielectric properties Electric polarization Permittivity Plasmonics Surface plasmons

Nanoparticles

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Kuniyuki Miwa ^* and George C. Schatz ^†

Department of Chemistry, Northwestern University, Evanston, Illinois 60208-3113, USA

^*Present address: Institute for Molecular Science, Okazaki, Aichi 444-8585, Japan; kuniyukimiwa@ims.ac.jp
^†g-schatz@northwestern.edu

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Issue

Vol. 103, Iss. 4 — April 2021

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Images

Figure 1
Spectral function $ρ_{qs} (ω)$ for the dipolar mode $(l = 1)$ for (a) Al, (b) Ag, and (c) Au nanospheres. The real part of the permittivity of (d) Al taken from the experimental data reported in Ref. [54] and (e) Ag and (f) Au in Ref. [55] are displayed. The spectral function $ρ_{qs} (ω)$ for multipolar modes in (g) Al, (h) Ag, and (i) Au nanospheres are plotted. The red solid, blue dashed, green dotted, and purple dashed-dot lines indicate $l = 1 - 4$ modes, respectively.
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Figure 2
Comparison of the spectral function $ρ_{full} (ω)$ of the total system (red solid line), the spectral function $ρ_{qs} (ω)$ in the quasistatic approximation (green dotted line), and the extinction spectra $σ_{ext}$ calculated using Mie theory (blue dashed line) for Al nanosphere with radii $R$ of (a) and (d) 30 nm, (b) and (e) 60 nm, and (c) and (f) 90 nm, respectively. The dipolar (a)–(c) and quadrupolar modes (d)–(f) are displayed. The permittivity $ε (ω)$ of Al is taken from the experimental data [54].
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Figure 3
Comparison of $ρ_{full} (ω)$ (red solid line), $ρ_{qs} (ω)$ (green dotted line), and $σ_{ext}$ (blue dashed line) for Ag and Au nanospheres with radii $R$ of (a) and (d) 30 nm, (b) and (e) 60 nm, and (c) and (f) 90 nm, respectively. The dipolar mode is displayed. The permittivity $ε (ω)$ of both Ag and Au is taken from the experimental data [55].
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Figure 4
Comparison of $ρ_{full} (ω)$ (red solid line), $ρ_{qs} (ω)$ (green dotted line), and $σ_{ext}$ (blue dashed line) for a metal nanosphere with radii $R$ of (a) and (d) 30 nm, (b) and (e) 60 nm, and (c) and (f) 90 nm, respectively. The (a)–(c) dipolar and (d)–(f) quadrupolar modes are displayed. The metal permittivity $ε (ω)$ is parametrized utilizing the Drude model. The parameters $(ε_{\infty}, ℏ ω_{p}, ℏ γ)$ are assumed to be (1.00, 9.04 eV, and 21.25 meV) as has been used to model the permittivity of silver [58].
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Figure 5
Spatial distribution of electric-field $E_{ω}^{(s l m)} (r)$ per plasmon for Al nanosphere with the radii $R$ for (a) and (d) 30 nm, (b) and (e) 60 nm, and (c) and (f) 90 nm. Absolute values of the electric fields are plotted. The angular frequency $ω$ is set at the resonance excitation energy of (a) and (d) 5.67 eV, (b) and (e) 3.08 eV, and (c) and (f) 2.04 eV. The permittivity of Al is taken from Ref. [54].
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Figure 6
(a) and (b) Frequency dependence of the coupling constant for the quantum emitter and LSP modes ( $l = 1 - 3$ ). The radius of the metal nanosphere is 50 nm and the quantum emitter is located 5 nm from the metal nanosphere. The permittivity $ε (ω)$ of the metal is given by the Drude model with $(ε_{\infty}, ℏ ω_{p}, ℏ γ) = (6.00, 7.90 eV, and 51.00 meV)$ [60].
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