# Nonlocal effects in atom-plasmon interactions

Mikkel Have Eriksen  0000-0002-0159-0896 POLIMA—Center for Polariton-driven
Light–Matter Interactions, University of Southern Denmark, Campusvej 55,
DK-5230 Odense M, Denmark  Christos Tserkezis  0000-0002-2075-9036
POLIMA—Center for Polariton-driven Light–Matter Interactions, University of
Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark  N. Asger Mortensen
0000-0001-7936-6264 POLIMA—Center for Polariton-driven Light–Matter
Interactions, University of Southern Denmark, Campusvej 55, DK-5230 Odense M,
Denmark  Danish Institute for Advanced Study, University of Southern Denmark,
Campusvej 55, DK-5230 Odense M, Denmark  Joel D. Cox  0000-0002-5954-6038
<EMAIL_ADDRESS>POLIMA—Center for Polariton-driven Light–Matter Interactions,
University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
Danish Institute for Advanced Study, University of Southern Denmark, Campusvej
55, DK-5230 Odense M, Denmark

###### Abstract

Nonlocal and quantum mechanical phenomena in noble metal nanostructures become
increasingly crucial when the relevant length scales in hybrid nanostructures
reach the few-nanometer regime. In practice, such mesoscopic effects at metal-
dielectric interfaces can be described using exemplary surface-response
functions (SRFs) embodied by the Feibelman $d$-parameters. Here we show that
SRFs dramatically influence quantum electrodynamic phenomena—such as the
Purcell enhancement and Lamb shift—for quantum emitters close to a diverse
range of noble metal nanostructures interfacing different homogeneous media.
Dielectric environments with higher permittivities are shown to increase the
magnitude of SRFs calculated within the specular-reflection model. In
parallel, the role of SRFs is enhanced in nanostructures characterized by
large surface-to-volume ratios, such as thin planar metallic films or shells
of core-shell nanoparticles. By investigating emitter quantum dynamics close
to such plasmonic architectures, we show that decreasing the width of the
metal region, or increasing the permittivity of the interfacing dielectric,
leads to a significant change in the Purcell enhancement, Lamb shift, and
visible far-field spontaneous emission spectrum, as an immediate consequence
of SRFs. We anticipate that fitting the theoretically modelled spectra to
experiments could allow for experimental determination of the $d$-parameters.

##  I Introduction

Nanoscale light-matter interactions, and the strong coupling of light with
atoms and molecules, are abundant sources of fundamental physical insights,
while offering applications in fields such as optical sensing, photocatalysis,
and quantum optics [1, 2, 3]. In these and other areas, metallic
nanostructures that support plasmons—the collective oscillations of conduction
electrons—are sought as light-focusing elements that enhance the interaction
of atomic systems with external optical fields [4, 5, 6, 7]. Meanwhile, in a
quantum-electrodynamical context, plasmonic nanostructures are actively
explored as subwavelength optical cavities that control the generation of
single photons [8, 9, 10, 11], a key resource for future quantum optical
information and communication technologies [12]. Steady progress in the
aforementioned frontier research topics has relied crucially on the framework
of classical electrodynamics within the local-response approximation (LRA)
[13]. However, as the fabrication of plasmonic nanostructures such as metallic
nanoparticles (NPs) becomes increasingly advanced, such that the feature size
and distance between structures approach the few-nanometer regime, the LRA can
no longer accurately estimate the response of the system, as it neglects
nonlocal and quantum mechanical corrections in the bulk and at the surface of
the metal [14, 15, 16, 17, 13].

Nonlocal and quantum mechanical phenomena at metal surfaces have been modelled
through a series of methods such as descriptions of the bulk response through
semi-classical hydrodynamic models [18, 19, 20, 21, 22, 23, 24, 25, 26] or
_ab-initio_ methods such as time-dependent density-functional theory (TDDFT)
[27, 28, 29, 30]. The standard hydrodynamic Drude model (HDM) that relies on
the Thomas–Fermi theory with hard-wall boundary conditions describes the
motion of the compressible electron gas as a convective fluid and needs
amendments to account for surface phenomena such as electron spill-out or
spill-in [20, 31, 21], while TDDFT, constituting a more sophisticated method
for modelling quantum mechanical phenomena in plasmonic nanostructures,
demands huge computational costs and is practically restricted to few-atom
structures. The mesoscopic regime, bridging the micro- and macroscopic realms,
necessitates a description that goes beyond the classical LRA and yet is less
computationally demanding than _ab-initio_ approaches [32].

Including surface-response functions (SRFs) [33, 34, 35, 36] in the otherwise
classical or semiclassical constitutive relations when solving Maxwell’s
equations allows us to take into account quantum mechanical phenomena when
electrons are confined to small structures in the mesoscopic regime, while
still maintaining the simple classical or semiclassical bulk response
functions [37, 38, 39, 40, 41]. Feibelman $d$-parameters are such SRFs that
permit the modelling of surface-enabled Landau damping, nonlocality, and
electron spill-out or spill-in effects to leading order [42, 43, 44]. The
Feibelman $d$-parameters, ${d_{\perp}}$ and ${d_{\parallel}}$, are associated
with an interface between two materials, and depend on the properties of these
two materials constituting the interface. The $d$-parameters are often
computed using atomistic or _ab-initio_ methods for metal-vacuum interfaces,
an approach that is prohibitively time-intensive to tabulate for arbitrary
metal-dielectric interfaces. We here focus on noble metal interfaces, where
d-band screening and spill-in effects cannot be captured by jellium models
[29]; we thus resort to finding analytical expressions for the $d$-parameters
using the specular-reflection model (SRM) in combination with HDM for the
longitudinal component of the dielectric tensor [45, 46].

When positioned near a plasmonic nanostructure, a quantum emitter (QE), such
as a quantum dot or an atom, will exhibit altered emission properties
associated with the modified local photonic density of states (LDOS) [47, 48,
49, 50]. In the specific case of a sodium (Na) NP or interface, the quantum
mechanical and nonlocal corrections captured by SRFs result in clear changes
to the light emission spectrum of the QE [38], as well as quantum
electrodynamical phenomena in the form of the Lamb shift and Purcell
enhancement of the QE [42, 40]. Here, we investigate the influence of the
$d$-parameters in plasmonic nanostructures comprised of noble metals such as
gold (Au) or silver (Ag). We show that the influence of the SRFs on the
emission properties of a nearby QE increases when the permittivity of the
interfacing dielectric increases, or the surface area becomes larger compared
to the volume of the nanostructure by—for example—increasing the number of
interfaces when going from a single extended interface to a thin film, or
analogously by substituting a solid spherical NP with a core-shell NP. Our
results showcase configurations and situations where the inclusion of quantum
mechanical corrections in the mesoscopic regime is of particular importance.
These large non-classical corrections might allow for experimental
determination of the $d$-parameters by, e.g., fitting the theory presented
here to experimental results [51].

##  II Results and discussion

Figure 1: The effect of Feibelman $d$-parameters from SRM. (a) Schematic of a
QE (e.g., an atom) located a distance $h$ from a metal with permittivity
$\epsilon_{\rm m}$ interfacing a dielectric material with permittivity
$\epsilon_{\rm d}$. The electron-density profile $n(z)$ (black solid curve) is
calculated using a quantum infinite-barrier model [45], while the induced-
charge density (dashed curves, for the two different $\epsilon_{\rm d}$
indicated in the legend) and the associated perpendicular
${d_{\perp}}$-parameters qualitatively show the spill-in depending on
$\epsilon_{\rm d}$. (b) Real (solid curves) and imaginary (dashed curves)
parts of the Feibelman ${d_{\perp}}$-parameter computed from Eq. (4) for
dielectric media with permittivity $\epsilon_{\rm d}$ (indicated by the color-
coded legend) interfacing the metal Au with permittivity $\epsilon_{\rm m}$
characterized by the model of Ref. [52]. Panels (c-f) show the Purcell factor
$\Gamma/\Gamma_{0}$ and Lamb shift $\delta\varepsilon$ of the emitter as a
function of its transition energy $\hbar\varepsilon$ when omitting (dashed
curves) and including (solid curves) the Feibelman $d$-parameters in the metal
film response; the effects of separation $h$ (for an emitter in vacuum) and
dielectric environment $\epsilon_{\rm d}$ (for a fixed separation $h=2$ nm)
are explored in panels (c,d) and (e,f), respectively, for an emitter with
transition dipole moment $d=1$ $e\cdot$nm.

Surface response functions for arbitrary metal-dielectric interfaces. For a
particular dielectric–metal interface spanning the ${\bf R}=(x,y)$ plane, the
associated Feibelman $d$-parameters can be obtained regarding the quantum
mechanical induced charge and current density in the metal, $\rho({\bf
r})=\rho(z){\rm e}^{{\rm i}{\bf Q}\cdot{\bf R}}$ and ${\bf j}({\bf r})={\bf
j}(z){\rm e}^{{\rm i}{\bf Q}\cdot{\bf R}}$, respectively, with $Q$ being the
in-plane wavevector, from which the parameters are found as [42, 43, 33]:

$\displaystyle{d_{\perp}}(\omega)$
$\displaystyle=\frac{\int_{-\infty}^{\infty}dzz\rho^{\rm
ind}(z,\omega)}{\int_{-\infty}^{\infty}dz\rho^{\rm ind}(z,\omega)},$ (1a)
$\displaystyle{d_{\parallel}}(\omega)$
$\displaystyle=\frac{\int_{-\infty}^{\infty}dzz\partial_{z}j_{\parallel}^{\rm
ind}(z,\omega)}{\int_{-\infty}^{\infty}dz\partial_{z}j_{\parallel}^{\rm
ind}(z,\omega)}.$ (1b)

In the above expressions, $z$ is taken as the axis normal to the interface,
while the system is assumed isotropic in the $xy$ plane. An intuitive physical
understanding of the perpendicular parameter, ${d_{\perp}}$, is here apparent,
corresponding to the centroid of the induced charge density as displayed in
the schematic of a metal interfacing a dielectric material in Fig. 1(a), where
the equilibrium density of electrons in the metal, $n$, is calculated using a
quantum infinite-barrier model and is normalized to the equilibrium density
for the infinite electron gas, $n_{0}$ [45]. The sign of the real part then
differentiates between electron spill-out ($\text{Re}\\{{d_{\perp}}\\}>0$) or
the contrary situation of spill-in ($\text{Re}\\{{d_{\perp}}\\}<0$) associated
with a high work function. Correspondingly, ${d_{\parallel}}$ is the centroid
of the normal derivative of the in-plane current.

Using these general definitions, one can compute the Feibelman $d$-parameters
through _ab-initio_ methods, but this requires a new computation for any
interface between two different materials. Alternatively, the parameters can
be computed using SRM, also known as the semi-classical infinite-barrier
(SCIB) model, which assumes specular reflection of the conduction electrons at
the interface [53]. This results in the parameters [45]:

$\displaystyle d_{\perp}^{\text{SRM}}=-\frac{2}{\pi}\frac{\epsilon_{\rm
m}\epsilon_{\rm d}}{\epsilon_{\rm m}-\epsilon_{\rm
d}}\int_{0}^{\infty}\frac{dk_{\rm L}}{k_{\rm
L}^{2}}\left[\frac{1}{\epsilon_{\rm L}(k_{\rm
L},\omega)}-\frac{1}{\epsilon_{\rm m}}\right],$ (2a) $\displaystyle
d_{\parallel}^{\text{SRM}}=0,$ (2b)

where ${d_{\parallel}}=0$ owing to the fact that the interface is
intrinsically charge-neutral in the model, $\epsilon_{\rm m}$ is the classical
or semi-classical bulk-metal permittivity, $\epsilon_{\rm d}$ the
corresponding permittivity of the dielectric, and $\epsilon_{\rm L}$ is the
longitudinal permittivity of the spatially dispersive metal—here computed
using HDM. Finally, $\omega$ is the angular frequency of the incident light.

In what follows, we use for $\epsilon_{\rm m}$ a Drude model, while the
longitudinal permittivity is obtained within the HDM,

$\displaystyle\epsilon_{\rm m}(\omega)$ $\displaystyle=\epsilon_{\rm
b}(\omega)-\frac{\omega_{\rm p}^{2}}{\omega^{2}+{\rm i}\omega\gamma},$ (3a)
$\displaystyle\epsilon_{\rm L}(k_{\rm L},\omega)$ $\displaystyle=\epsilon_{\rm
b}(\omega)-\frac{\omega_{\rm p}^{2}}{\omega^{2}+{\rm
i}\omega\gamma-\beta^{2}k_{\rm L}^{2}},$ (3b)

where $\beta\propto v_{\rm F}$ contains the dependence on the Fermi velocity
$v_{\rm F}$, being the characteristic velocity of conduction electrons in the
metal. In the high (optical) frequency limit $\beta^{2}=3v_{\rm F}^{2}/5$ [54,
26]. Finally, $\epsilon_{\rm b}(\omega)$ is the background permittivity that
includes all more complicated contributions such as interband transitions, and
is obtained by fitting to experimental data [55, 52].

With these bulk response functions at hand, an analytical expression for the
$d$-parameters can be found in the SRM by using Eq. (2a) [56]:

$d_{\perp}^{\text{SRM}}={\rm i}\frac{\epsilon_{\rm m}\epsilon_{\rm
d}}{\epsilon_{\rm m}-\epsilon_{\rm d}}\frac{\beta}{\omega_{\rm
p}\sqrt{\epsilon_{\rm b}}}\left(\frac{\epsilon_{\rm b}}{\epsilon_{\rm
m}}-1\right)^{3/2}.$ (4)

The magnitude and behaviour of the SRFs are investigated by computing
$d_{\perp}^{\text{SRM}}$ analytically for the noble metal Au, while changing
the permittivity of the interfacing dielectric material, in Fig. 1(b). For all
frequencies under consideration here, $\text{Re}\\{{d_{\perp}}\\}<0$,
indicating electron spill-in for the conduction electrons in gold, whose work
function is relatively high. Such spill-in has previously been associated with
core electron screening in noble metals [57], and is at odds with what is
typically seen in jellium metals with lower work functions (e.g., Na), which
exhibit spill-out and therefore a positive perpendicular Feibelman parameter
[42]. Fig. 1(b) also reveals that the magnitude of $d_{\perp}^{\text{SRM}}$
becomes larger for dielectric media with higher permittivity, resulting in a
pronounced increase of the spill-in at lower frequencies. For instance
$d_{\perp}^{\text{SRM}}$ approaches $d_{\perp}^{\text{SRM}}\approx-0.35$ $\AA$
at lower frequencies when the dielectric permittivity is $\epsilon_{\rm d}=1$,
but goes towards $d_{\perp}^{\text{SRM}}\approx-2.8$ $\AA$ for $\epsilon_{\rm
d}=8$.

Figure 2: Enhanced surface effects in the quantum electrodynamic response of
thin metal films. (a) Schematic of a QE at distance $h$ from a thin metal film
of width $L$. In panels (b-g) the calculated Purcell factors and Lamb shifts
experienced by a QE with transition dipole moment $d=1$ $e\cdot$nm placed a
distance $h=2$ nm above a film are shown for cases with (solid curves) and
without (dashed curves) incorporating SRFs in the optical response. Panels
(b,c) show results for a QE in vacuum ($\epsilon_{1}=1)$ close to an Au film
of varying thickness, on a Si substrate with permittivity $\epsilon_{3}$
interpolated from experimental data [58]. For a film with $L=3$ nm, panels
(d,e) show the effect of substrate permittivity $\epsilon_{3}$ when the QE is
in vacuum, while panels (f,g) show the quantum electrodynamic response when
the Au film is embedded in a medium with varying permittivity
$\epsilon_{1}=\epsilon_{3}$.

Semi-infinite metal films. The influence of the SRFs can manifest as a change
in the emission properties of a QE positioned near a metal film at ${\bf
r}=(x,y,h)$, i.e., a distance $h$ from a metallic film extended in the $xy$
plane, as illustrated schematically in Fig. 1a. Invoking the macroscopic
quantum electrodynamics formalism detailed in the SI [59, 60, 61, 62, 63] for
a QE characterized as a two-level system with transition energy
$\hbar\varepsilon$ and dipole moment ${\bf p}$, the total QE spontaneous
emission rate is

$\Gamma=\Gamma_{0}+\frac{2\mu_{0}}{\hbar}\varepsilon^{2}{\rm
Im}\mathopen{}\left\\{{\bf p}^{*}\cdot\mathcal{G}_{\varepsilon}^{\rm ref}({\bf
r},{\bf r})\cdot{\bf p}\right\\}\mathclose{},$ (5)

where $\Gamma_{0}=\varepsilon^{3}\mathopen{}\left|{\bf
p}\right|\mathclose{}^{2}/3\pi\epsilon_{0}\hbar c^{3}$ is the vacuum decay
rate, while the shift in the bare atomic transition frequency due to the
photonic environment—the Lamb shift—is

$\delta\varepsilon=\frac{\mu_{0}}{\pi\hbar}\mathcal{P}\int_{0}^{\infty}d\omega\omega^{2}\frac{{\rm
Im}\mathopen{}\left\\{{\bf p}^{*}\cdot\mathcal{G}_{\omega}^{\rm ref}({\bf
r},{\bf r})\cdot{\bf p}\right\\}\mathclose{}}{\varepsilon-\omega},$ (6)

with $\mathcal{P}$ denoting the principal value of the integral. The above
expressions depend on the reflected part of the Green’s tensor
$\mathcal{G}_{\omega}^{\rm ref}$ at the QE location, which quantifies the QE
self-interaction mediated by the metal-dielectric interface [64]. For a dipole
oriented perpendicular to the metal surface, i.e., ${\bf p}=p\hat{\bf z}$, the
nonvanishing Green’s tensor component entering Eqs. (5) and (6) reads as

$\mathcal{G}_{\omega,\perp}^{\rm ref}=\frac{{\rm i}}{4\pi k_{\rm
d}^{2}}\int_{0}^{\infty}d{k_{\parallel}}k_{\parallel}^{3}\frac{{\rm e}^{{\rm
i}2k_{{\rm d},z}z}}{k_{{\rm d},z}}r_{\rm p}({k_{\parallel}},\omega),$ (7)

where the Fresnel reflection coefficient for p-polarized light,

$r_{\rm p}=\frac{\epsilon_{\rm m}k_{{\rm d},z}-\epsilon_{\rm d}k_{{\rm
m},z}+(\epsilon_{\rm m}-\epsilon_{\rm d})({\rm
i}k_{\parallel}^{2}{d_{\perp}}-{\rm i}k_{{\rm d},z}k_{{\rm
m},z}{d_{\parallel}})}{\epsilon_{\rm m}k_{{\rm d},z}+\epsilon_{\rm d}k_{{\rm
m},z}-(\epsilon_{\rm m}-\epsilon_{\rm d})({\rm
i}k_{\parallel}^{2}{d_{\perp}}+{\rm i}k_{{\rm d},z}k_{{\rm
m},z}{d_{\parallel}})},$ (8)

depends on the optical wavevector ${\bf k}_{j}$ in a medium of permittivity
$\epsilon_{j}$ for $j\in\\{{\rm d},{\rm m}\\}$, with normal component
$k_{j,z}=\sqrt{\epsilon_{j}\omega^{2}/c^{2}-{k_{\parallel}}^{2}+{\rm i}0^{+}}$
and conserved parallel component ${k_{\parallel}}$, as well as on the SRFs
that modify the electromagnetic boundary conditions at the metal-dielectric
interface.

Using the above expressions, we compute the enhancement in spontaneous
emission—quantified by the Purcell factor $\Gamma/\Gamma_{0}$—along with the
Lamb shift $\delta\varepsilon$ for a QE in a medium with permittivity
$\epsilon_{1}$ that is placed a distance $h$ from a gold interface. In Figs.
1c and 1d, the Purcell factor and Lamb shift are respectively presented in
calculations adopting a fixed vacuum permittivity $\epsilon_{1}=1$ at
separations $h$ indicated in the legend of Fig. 1c when omitting (dashed
curves) and including (solid curves) SRFs. In the cases considered, the SRFs
are found to introduce damping and spectral shifts in prominent plasmonic
features of the Purcell factor and Lamb shift that become increasingly
important as the QE is brought extremely close to the metal–dielectric
interface. The effect of the surface response in quantum electrodynamical
phenomena is further amplified when the permittivity of the dielectric medium
is increased to enhance spill-in of the interfacing metal electron gas, as
revealed in Figs. 1d and 1e by the change in Purcell factor and Lamb shift,
respectively, for a QE with fixed separation $h=2$ nm and varying environment-
permittivity $\epsilon_{1}$. The above results underscore the importance of
nonclassical surface effects in quantum electrodynamics at a metal-dielectric
interface, particularly for dielectric media with high permittivity.

Surface response of thin films. Thin metal films present larger surface-to-
volume ratios that should enhance SRF contributions in nanoscale light–matter
interactions, while also exhibiting higher sensitivity to screening from
interfacing dielectric media. In Fig. 2 we explore the interaction of a QE in
a dielectric medium $\epsilon_{1}$ placed a distance $h$ above a thin gold
film with permittivity $\epsilon_{2}$ and thickness $L$ on a substrate with
permittivity $\epsilon_{3}$, as illustrated schematically in Fig. 2a. The film
reflection coefficients are

$R_{\alpha}=r^{(12)}_{\alpha}+\frac{t^{(21)}_{\alpha}t^{(12)}_{\alpha}r^{(23)}_{\alpha}{\rm
e}^{{\rm i}2k_{2,z}L}}{1-r^{(21)}_{\alpha}r^{(23)}_{\alpha}{\rm e}^{{\rm
i}2k_{2,z}L}}$ (9)

for $\alpha\in\\{{\rm s},{\rm p}\\}$ polarization, where
$r^{(jj^{\prime})}_{\alpha}$ and $t^{(jj^{\prime})}_{\alpha}$ are reflection
and transmission coefficients, respectively, associated with light impinging
from a medium with permittivity $\epsilon_{j}$ on an interface with
permittivity $\epsilon_{j^{\prime}}$, which contain the dependence on SRFs as
detailed in the SI.

Figure 3: Surface effects in quantum light emission near a spherical plasmonic
nanoparticle. (a) Schematic of a QE located a distance $h$ from the surface of
a spherical NP with radius $a$. Panels (b,c) show the Purcell factor and Lamb
shift of a QE in vacuum ($\epsilon_{1}=1$) close to Au NPs with radii
indicated in (c), while panels (d,f) show results obtained for a small NP with
fixed radius $a=3$ nm and varying background dielectric constant. Results are
obtained for a QE transition dipole moment $d=1$ $e\cdot$nm placed at $h=2$ nm
from the outer surface of the NP, with solid and dashed curves indicating
results obtained with and without including SRFs, respectively.

We explore the influence of SRFs on the Purcell factor and Lamb shift in Figs.
2b and 2c, respectively, for a QE in vacuum ($\epsilon_{1}=1$) above gold
films ($\epsilon_{2}=\epsilon_{\rm Au}$) with varying thickness $L$ deposited
on silicon ($\epsilon_{3}=\epsilon_{\rm Si}$, obtained from interpolated
experimental data in Ref. [58]) by replacing the single-interface reflection
coefficient in Eq. (7) with that of Eq. (9) obtained with (solid curves) and
without (dashed curves) SRFs. In thin metal films, the largest deviation from
a classical description of the photonic environment is predicted for the
thinnest film, where a second plasmonic peak is present at low energies coming
from the plasmonic mode at the lower Si-Au interface. Interestingly, low-
energy features in the Purcell factor and Lamb shift are enhanced by SRFs in
the thinnest film, presumably due to the large negative real values of
${d_{\perp}}$ that describe spill-in of the metal film charge density that
effectively reduces the thickness of the free electron gas and boosts
plasmonic field confinement. In contrast, high-energy features in the emission
spectra experience greater surface damping quantified by the imaginary part of
${d_{\perp}}$. Similar behavior is seen by considering substrates with
different dielectric permittivities, as shown in Figs. 2d and 2e, where an
additional peak emerges when substrates with high permittivities are
considered. In Figs. 2(f,g) we consider the impact of SRFs on the QE self-
interaction when $\epsilon_{1}=\epsilon_{2}$ and the metal film is embedded in
a high-permittivity dielectric, where plasmon hybridization between the two
surfaces may yield a low-energy bonding mode and a higher energy antibonding
mode [65, 66]. For $\epsilon_{1}=\epsilon_{3}=8$ in Fig. 2f one can see the
splitting associated with the hybridization between the two surfaces when
including $d$-parameters but not when classically computing the Purcell
factor. This can be interpreted as the SRFs effectively decreasing the width
of the thin film and hence increasing the hybridization effect through
electron spill-in (a negative ${d_{\perp}}$). This is most pronounced when
considering high dielectric environments to the gold film.

Figure 4: Dependence of the Lamb shift and Purcell factor on surface effects
in a plasmonic CSNP. (a) Schematic of a QE at distance $h$ from the surface of
a CSNP with core radius $a$ and shell radius $b$. (b,c) show, respectively,
the Purcell enhancement and Lamb shift of a QE close to a CSNP with a Si core
($\epsilon_{3}$ from Ref. [58]) and Au shell ($b=25$ nm) at varying core-radii
as indicated in (c). (d,e) consider a core with fixed radius of $a=23$ nm and
varying permittivity. Lastly, in panels (f,g) the core is fixed with
$\epsilon_{3}=2.13$ to mimic SiO2 and the environment permittivity
$\epsilon_{1}$ is varied. Results are obtained for a QE with transition dipole
moment $d=1$ $e\cdot$nm placed a distance $h=2$ nm from the outer NP surface
and the background dielectric material is $\epsilon_{1}=1$ except for in
(f,g).

Spherical metallic nanoparticles. Metallic NPs supporting localized plasmon
resonances are quintessential light-focusing elements in nanophotonics that
are conveniently described using Mie theory. As depicted in Fig. 3a, we
consider the light emission properties of an emitter characterized by a
radially-oriented dipole moment in a medium with permittivity $\epsilon_{1}$
at a distance $r$ from the center of a spherical metallic NP with radius $a$
and permittivity $\epsilon_{2}$. The quantum electrodynamic response is
quantified by the reflected part of the Green’s function

$\mathcal{G}_{\varepsilon}^{\rm ref}(r)=-\frac{{\rm
i}k_{1}}{4\pi}\sum_{l}l(l+1)(2l+1)a_{l}\left[\frac{h_{l}^{(1)}(k_{1}r)}{k_{1}r}\right]^{2},$
(10)

where

$\displaystyle
a_{l}=\frac{\epsilon_{2}\Psi_{l}^{\prime}(x_{1})j_{l}(x_{2})-\epsilon_{1}j_{l}(x_{1})\Psi_{l}^{\prime}(x_{2})+(\epsilon_{2}-\epsilon_{1})\left[\bar{d}_{\perp}j_{l}(x_{1})j_{l}(x_{2})+\bar{d}_{\parallel}\Psi_{l}^{\prime}(x_{1})\Psi_{l}^{\prime}(x_{2})\right]}{\epsilon_{2}j_{l}(x_{2})\xi_{l}^{\prime}(x_{1})-\epsilon_{1}h_{l}^{(1)}(x_{1})\Psi_{l}^{\prime}(x_{2})+(\epsilon_{2}-\epsilon_{1})\left[\bar{d}_{\perp}h_{l}^{(1)}(x_{1})j_{l}(x_{2})+\bar{d}_{\parallel}\Psi_{l}^{\prime}(x_{2})\xi_{l}^{\prime}(x_{1})\right]}$
(11)

are Mie coefficients that are linearized in the Feibelman $d$-parameters, as
reported in Ref. [42], normalized according to $\bar{d}_{\perp}\equiv
l(l+1){d_{\perp}}/a$ and $\bar{d}_{\parallel}\equiv{d_{\parallel}}/a$, while
$j_{l}$ and $h_{l}^{(1)}$ are spherical Bessel and Hankel-of-the-first-kind
functions, respectively, of the normalized parameter $x_{j}\equiv k_{j}a$ that
also enters the derivatives of the Riccati–Bessel functions
$\Psi_{l}^{\prime}(x)=\partial_{x}\mathopen{}\left[xj_{l}(x)\right]\mathclose{}$
and $\xi_{l}^{\prime}(x)=\partial_{x}[xh_{l}^{(1)}(x)]$.

Figure 5: Detection of surface response effects in spontaneous emission
produced near a plasmonic nanostructure. (a) Schematic illustration of a QE in
a medium with permittivity $\epsilon_{1}$ at distance $h$ from the outer
surface of a CSNP, with shell radius (permittivity) $b$ ($\epsilon_{2}$) and
core radius (permittivity) $a$ ($\epsilon_{3})$, emitting light that is
detected in the far field at a distance $r_{\rm D}$. Panels (b-g) show the
spontaneous emission spectra $S(\omega)$ detected at $r_{\rm D}=1$ $\mu$m for
a QE with transition frequency $\varepsilon$, dipole moment $d=1e\cdot$nm, and
intrinsic broadening $\hbar\gamma_{\rm d}=15$ meV (typical for a quantum-dot
exciton at room temperature) positioned at $h=2$ nm from a CSNP with Si core
($\epsilon_{3}$ from Ref. [58]) of radii $a$ indicated in each column and Au
shell with radius $b=10$ nm; the upper row of panels (b,d,f) shows contours of
$S(\omega)$ that sweep the detected light energy $\hbar\omega$ on horizontal
axes and the QE transition energy on the vertical axes, while the lower row of
panels (c,e,g) shows the emission spectrum for specific QE transition energies
$\hbar\varepsilon$—indicated by the color-coded dashed horizontal lines in the
panels immediately above—when including (solid curves) or omitting (dashed
curves) SRFs (each set of curves is appropriately re-scaled for clarity).

With the above expressions at hand, the Purcell enhancement and Lamb shift of
a QE near a spherical Au NP are found for a series of radii in Figs. 3(b,c).
The $d$-parameters in Eq. (11) go as $l(l+1){d_{\perp}}/a$ and
${d_{\parallel}}/a$, meaning that their relative contribution increases for
smaller NP radius and higher multipolar modes. As the higher multipolar modes
influence the Green’s function for smaller distances between the QE and the
NP, as seen in Eq. (10), one would expect the influence of SRFs to increase
for smaller radii, in agreement with the results in Figs. 3(b,c). In addition,
when varying the background permittivity rather than the radius of the NP, as
in Fig. 3(d,f), one sees that the larger permittivity ensures a larger
deviation from the classical results when including the SRF, an effect which
is particularly noticeable in the Lamb shift. Incidentally, as we show in the
SI, the above results obtained using Mie theory are qualitatively reproduced
by incorporating SRFs in the multipolar polarizability obtained in the
quasistatic approximation, which is well-justified when the size of the NP and
the QE-NP separation are small.

Surface response effects in spherical core-shell nanoparticles. Compared to
spherical NPs, the additional metal-dielectric interface in spherical
dielectric core–metal shell NPs (CSNPs) leads to a more involved dependence on
surface effects in the optical response, as indicated by the lengthy
analytical expressions reported in the SI for both the associated Mie
coefficients and the quasistatic multipolar polarizability. For a dipole
oriented in the radial direction and placed a distance $h$ from the center of
a CSNP with core radius $a$ and shell radius $b$, as illustrated in Fig. 4a,
the Purcell factor and Lamb shift corresponding to the solutions from Mie
theory are plotted in Figs. 4(b-g). The sharper spectral features in the
classical estimation of the Purcell factor and Lamb shift are slightly damped
and blueshifted by SRFs, which can also produce qualitative changes around
these multipolar modes. Here, the $d$-parameters appear in the polarizability
as $l(l+1){d_{\perp}}/R$ for $R\in\\{a,b\\}$, with the higher-order modes
tending to exhibit larger SRF-contributions resulting in larger blueshift and
damping. Similarly to the spherical NP, the influence of SRFs is most
prominent in higher-order multipolar modes, with the $d$-parameters
corresponding to the different interfaces contributing more when the radius to
the corresponding interface is small. This can be seen in Figs. 4(b,c), where
the introduction of $d$-parameters to the CSNP with the thinnest shell
generally damps and influences the Purcell enhancement and Lamb shift more as
compared to CSNPs with thicker shells. We remark that qualitatively similar
results are obtained by introducing SRFs in a quasistatic description of the
CSNP response, which conveniently leads to analytical expressions for the
multipolar polarizability (see SI).

In Figs. 4(d,e) and Figs. 4(f,g) we explore the impact of SRFs in the quantum
electrodynamic response of the CSNP when the core permittivity and the
surrounding permittivity are varied. In the former case, the oscillations in
the Purcell enhancement and Lamb shift of the QE associated with the core-
shell interface are significantly blue-shifted and damped in the higher-
frequency regime when introducing the SRF for high-permittivity cores. In the
latter situation, choosing silicon dioxide (SiO2)–also known as silica–as the
core while the surrounding permittivity is varied, the high-frequency peak
associated with the shell-surrounding interface more pronounced but similar
behaviour is observed by the inclusion of the SRF.

The spontaneous emission spectrum of a QE positioned in a photonic environment
can be written in terms of the Green’s function characterizing the
environment:

$\displaystyle S({\bf r}_{\rm D},\omega)=\left|\frac{\mu_{0}\omega^{2}{\bf
p}^{*}\cdot\mathcal{G}_{\omega}({\bf r}_{\rm D},{\bf
r})(\omega+\varepsilon)}{\varepsilon^{2}-\omega^{2}-{\rm i}\omega\gamma_{\rm
d}-2\mu_{0}\varepsilon\omega^{2}{\bf p}^{*}\cdot\mathcal{G}_{\omega}({\bf
r},{\bf r})\cdot{\bf p}/\hbar}\right|^{2},$ (12)

where ${\bf r}_{\rm D}$ denotes the detector position in the far field. The
above expression is found in the weak excitation approximation by using the
Wiener–Khinchin theorem and accounting for non-Markovian effects [67, 64, 68].
These spectra allow for the investigation of possible strong coupling regimes
accessible in these systems, and of the impact of SRFs on experimental
observables in these regimes.

The spectrum of the QE computed from Eq. (12) is displayed in Fig. 5 for a
CSNP with an Au shell with fixed radius $b=10$ nm and a Si core in Figs.
5(b-g) with varying radii of the core as described in the figure caption.
Silicon is used, since it is a dielectric material with a large permittivity
in the frequency window of interest, and any effects from SRFs should be more
pronounced than for a lower permittivity material such as SiO2. The
$d$-parameters introduce a large visible blue-shift in all cases as seen in
Fig. 5, which is as expected for the gold shell as the $d$-parameters
introduces spill-in.

The spectra displayed in Fig. 5 exhibit two peaks at some QE transition
frequencies, which can be seen for $\hbar\varepsilon=1.5$ eV in Fig. 5(g).
These peaks are significantly influenced by the SRFs; the amplitude of the
spectrum is here enhanced by the SRFs and the two classical peaks coalesces
into one peak. It is clear from Fig. 5 that the inclusion of $d$-parameters
influences the spectrum dramatically with a generally more pronounced effect
for the thinnest shell in Figs. 5(g). This supports the claim that SRFs are
particularly important for thin films or shells, where the surface-to-volume
ratio is larger, allowing the SRFs to play a vital role in the optical
response.

##  III Conclusions

Quantum mechanical and nonlocal phenomena at metal-dielectric interfaces
become important in the description of light-matter interactions on $\lesssim
10$ nm length scales. These phenomena impact the quantum electrodynamic
response of a quantum emitter in close proximity to noble metal
nanostructures, as revealed here by the large differences in the Lamb shift
and Purcell factor that emerge when comparing the classical response of
metallic nanostructures to the nonlocal and quantum mechanical response found
from the Feibelman $d$-parameter formalism. In particular, the $d$-parameters
obtained from the SRM predict a sizeable increase in ${d_{\perp}}$ when high-
permittivity media interface the noble metal, which is especially important
when regarding complicated nanophotonic systems with multiple interfaces and
increased surface-to-volume ratios. This finding is showcased for metal films
and CSNPs that exhibit pronounced differences in the Lamb shift and Purcell
enhancement of a nearby QE when thinner films or shells are considered. The
influence of $d$-parameters on the spontaneous emission spectrum of a QE close
to a noble metal nanostructure has similarly been investigated, where a large
difference in the spectrum can be seen in the case of Si–Au CSNPs. The QE
spectrum may be observed in experiments, and constitutes a route for
experimentally determining the $d$-parameters by comparing the experimental
results with theoretical modelling, similar to the strategy proposed in Ref.
[51] for determining $d$-parameters using electron-beam spectroscopy. The
analytical results presented here facilitate straightforward evaluation of the
spectrum, Purcell enhancement, and Lamb shift of QEs close to commonly
explored metallic nanostructure morphologies in both quasistatic and retarded
regimes.

##  IV Acknowledgements

We thank P. A. D. Gonçalves and C. Wolff for stimulating discussions. J. D. C.
is a Sapere Aude research leader supported by Independent Research Fund
Denmark (grant no. 0165-00051B). N. A. M. is a VILLUM Investigator supported
by VILLUM FONDEN (Grant No. 16498). The Center for Polariton-driven
Light–Matter Interactions (POLIMA) is funded by the Danish National Research
Foundation (Project No. DNRF165).

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