# Optical scattering imaging with sub-nanometer precision based on position-
ultra-sensitive giant Lamb shift

Zeyang Liao  1,# Yuwei Lu  1,# and Xue-Hua Wang1  1State Key Laboratory of
Optoelectronic Materials and Technologies, School of Physics, Sun Yat-sen
University, Guangzhou, 510275, Peoples Republic of China
<EMAIL_ADDRESS>
###### Abstract

The Lamb shift of a quantum emitter very close to a plasmonic nanostructure,
mainly induced by the higher-order plasmonic dark modes, can be three or more
orders of magnitude larger than that in the free space and it is ultra-
sensitive to the emitter position and polarization. We show that this giant
Lamb shift can be sensitively observed from the scattering spectrum dip shift
of coupled system when the plasmonic nanoparticle or tip scans through the
emitter. Based on these observations, we propose an optical localization and
polarization microscopy scheme with sub-nanometer precision for a quantum
emitter via detecting the scattering spectrum instead of fluorescence. Our
method is free of fluorescence quenching problem and it is relatively easier
to be implemented in the plasmon-emitter coupling system. Moreover, the sample
in our method does not need to be placed inside a plasmonic picocavity to
enhance the radiative fluorescence rate and it also works even if the quantum
emitter is slightly below a dielectric surface which can bring about broader
applications in various fields, such as physics, chemistry, medicine, life
science and materials science.

††journal: opticajournal††articletype: Research Article

#These authors contributing equally

##  1 Introduction

The optical microscope is widely used to localize and image objects which
cannot be directly observed by the human eyes. However, the resolution of the
conventional optical microscope is subjected to the well-known Abbe’s
diffraction limit, i.e., the resolvable smallest distance is about half
wavelength of the detection light [1]. For visible light, the spatial
resolution is about 250-300 nm. In the past few decades, several
superresolution methods have been proposed to surpass the diffraction limit,
such as the stimulated emission depletion microscopy (STED) [2, 3], structured
illumination microscopy (SIM) [4, 5, 6, 7], the single-molecule localization
microscopy (SMLM) [8, 9], and stochastic optical fluctuation imaging (SOFI)
[10]. These methods have been widely used for biological imaging with typical
resolution being about 20-50 nm. The near-field scanning optical microscope
(NSOM) can collect the evanescent field and is in principle not subjected to
the diffraction limit, but the achievable minimum resolution to date is about
10 nm in practice [11, 12, 13]. With the help of quantum effects, such as
using quantum entanglement [14, 15, 16], quantum Rabi oscillations [17, 18,
19], quantum statistical imaging [20, 21, 22], and spatial mode demultiplexing
techniques [23, 24, 25], the optical diffraction limit can also be overcome,
but the resolutions demonstrated by these methods are still far from a few
nanometers. To achieve atomic-level resolution, the scanning tunneling
microscopy (STM) and atomic force microscopy (AFM) are commonly used [26, 27,
28]. The resolution gains of STM and AFM are, however, at the expense of
losing the spectroscopic properties of the sample and the requirement of
extreme environment conditions. Using the highly localized tunneling electrons
as a source of excitation, a technique called STM-induced luminescence (STML)
can achieve spectral characterization with sub-nanometer spatial resolution
[29, 30, 31, 32]. However, since the electronic tunneling excitation has some
limitations, it is a long-standing pursuing goal to achieve sub-nanometer
resolution using all-optical methods.

It is well-known that the plasmonic system can confine the light field in an
extremely tiny volume [33, 34, 35], which is widely used to enhance the
interaction between light and quantum emitters (atom,molecule,quantum dot and
so on) [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47]. The typical STM junction
is actually a plasmonic picocavity which can confine the light field down to
sub-nanometer scale [48, 49] and it can also enhance the field strength orders
of magnitude which has been used for demonstrating the tip-enhanced Raman
spectroscopy (TERS) with atomic resolution [50, 51, 52, 53]. In addition, the
plasmonic picocavity can act as an antenna which can enhance the fluorescence
emission rate of a quantum emitter inside the cavity [54, 55, 56, 57] and it
was also applied to demonstrate the tip-enhanced photoluminescence (TEPL) with
sub-nanometer resolution [58]. In both the TERS and TEPL, the target emitters
need to be placed inside the plasmonic picocavity which may limit its
applications. Considering that the scattering spectrum is much easier to be
observed in the plasmonic-emitter coupling system [59, 60], it is thus very
interesting to devise a plasmonic scanning localization microscopy with ultra-
high resolution via its scattering spectrum.

In this article, we propose an optical scattering imaging (OSI) method with
sub-nanometer resolution based on the scattering spectrum shift induced by the
plasmon-enhanced Lamb shift when a plasmonic nanoparticle or tip scans through
a quantum emitter. As is known, quantum theory predicts that vacuum is not
truly empty, but full of fluctuations where virtual particles are constantly
created and annihilated [61] which can lead to many interesting quantum
effects. Among the observable effects of the electromagnetic vacuum, the Lamb
shift (LS) is one of the most important effects which directly stimulated the
emerging of modern quantum electrodynamics theory, and its precious
measurement becomes an important testbed for quantum field theory [61, 63].
Through vacuum engineering, the LS of a quantum emitter can be significantly
modified [64, 65, 66, 67, 68, 69, 70], e.g., a QE near band edge of photonic
crystal can have energy shift one to two orders of magnitudes larger than that
in the normal vacuum [71, 72, 73]. Due to the much stronger field confinement,
the Lamb shift of an emitter very close to a plasmonic nanostructure can be
strongly amplified [74, 75, 76] and has also been experimentally observed [32,
58]. Here, we show that the giant Lamb shift in the plasmon-emitter coupling
system, mainly induced by the higher-order plasmonic dark modes when the
emitter is very close to the plasmonic nanostructure, is ultra-sensitive to
the emitter position and it can be observed from the scattering spectrum dip
shift of the coupled system with variation of several meV/nm when the metal
nanoparticle or tip scans through the quantum emitter. Due to these
observations, we propose that this quantum effect can be exploited to localize
the position of an emitter precisely and construct the Lamb shift imaging by
scanning the plasmonic nanoparticle or tip through the emitter. The obtained
imaging spot is of angstrom size and its shape evidently depends on the dipole
orientation of the emitter. Since our method is based on scattering spectrum
instead of fluorescence, it is free of fluorescence quenching problem and
compared with the usual STML and TEPL methods, our scheme does not require
that the sample is placed inside a plasmonic picocavity. The findings here can
in principle be developed as a Lamb-shift-based superresolution scanning
optical microscope with atomic-level resolution and it also works even when
the emitter is embedded slightly below a dielectric substrate, which can bring
about broader applications in various fields.

The article is organized as follows. In Sec. 2, we first describe the model
under study and illustrate the theory used to calculate the emission and
scattering spectra of this system including the effect of Lamb shift.
Particularly, we study the scattering spectrum shift as a function of emitter-
nanoparticle distance and propose an experimentally feasible method to observe
the giant Lamb shift in this system. In Sec. 3, we propose a possible
experimental scheme based on tip-scattering method to detect the giant Lamb
shift which can be used to localize the position of an emitter and its
polarization even if it is embedded inside a substrate with ultra-high
sensitivity. Finally, we summarize our results.

##  2 Model and theory

The schematic system considered here is shown in Fig. 1(a) where a QE
interacts with a MNP with radius $R$ and $\mathbf{r}_{a}$ is the position of
the QE with distance $h$ away from the surface of the MNP. In this paper, we
mainly consider that $h$ is larger than $1nm$ where the electron tunnelling
effect and nonlocal optical response can be neglected [77, 78]. The MNP can
support many localized surface plasmon modes with the dipole mode being the
bright mode and the higher-order modes (HOMs) being the dark modes [46, 47].
Considering that the dipole mode can be effectively excited by the incident
light and be scattered into the far field while the HOMs do not [79], it is
convenient to treat the dipole mode as a quantized pseudo-mode separately [80,
81, 82] and leave the HOMs as background reservoir fields which can be well
described by the macroscopic quantization method based on dyadic Green’s
function [83, 84]. The effective Hamiltonian is then given by

$H=\hbar\left(\omega_{e}-i\frac{\gamma_{e}}{2}\right)\hat{\sigma}^{+}\hat{\sigma}^{-}+\hbar\left(\omega_{d}-i\frac{\gamma_{d}}{2}\right)\hat{d}^{+}\hat{d}+\hbar
g_{de}\left(\hat{\sigma}^{+}\hat{d}+\hat{d}^{+}\hat{\sigma}^{-}\right)+H_{hi},$
(1)

where the first term is the effective emitter energy, the second term is the
effective plasmon dipole energy, the third term is the interaction between the
emitter and the dipole mode, and the four term desribes the effect of the HOMs
with

$H_{hi}=\hbar\int d\boldsymbol{r}\int
d\omega_{\lambda}\omega_{\lambda}\hat{f}^{\prime+}\left(\boldsymbol{r},\omega_{\lambda}\right)\hat{f}^{\prime}\left(\boldsymbol{r},\omega_{\lambda}\right)+\left[\hat{\sigma}^{+}+\hat{\sigma}^{-}\right]\boldsymbol{\mu}_{e}\cdot\widehat{\boldsymbol{E}}^{\prime}\left(\boldsymbol{r}_{e}\right).$
(2)

The first term in Eq. (2) is the Hamiltonian of the HOMs and the second term
is the interaction between the HOMs and QE where the non-rotating interaction
terms are retained for the correct calculation of Lamb shift.
$\hat{\sigma}^{+}\left(\hat{\sigma}^{-}\right)$ is the Pauli lowering
(raising) operator of the QE with transition frequency $\omega_{e}$, and
$\gamma_{e}=\gamma_{e}^{0}+\gamma_{e}^{nr}$ is the emitter decay rate
including the radiative part $\gamma_{e}^{0}$ and nonradiative part
$\gamma_{e}^{nr}$; $\hat{d}^{+}(\hat{d})$ is the creation (annihilation)
operator of the plasmon dipole mode with frequency $\omega_{d}$ and linewidth
$\gamma_{d}$ which is the combination of radiative decay $\gamma_{d}^{0}$ and
nonradiative decay $\gamma_{d}^{nr}$;
$g_{de}=\omega_{\mathrm{d}}^{2}\mu_{0}\boldsymbol{\mu}_{e}\cdot\boldsymbol{G}_{0}\left(\boldsymbol{r}_{e},\boldsymbol{r}_{d};\omega_{d}\right)\cdot\boldsymbol{\mu}_{d}/\hbar$
is the coupling strength between the emitter and the plasmon dipole mode, here
$\mathbf{\mu_{e}}$ and $\mathbf{\mu_{d}}$ are the dipole moments of the QE and
the nanoparticle, respectively. Here, we assume that $|\mathbf{\mu_{e}}|=24D$
[85] and the magnitude of the effective plasmon dipole moment
$\left|\boldsymbol{\mu}_{d}\right|=\epsilon_{b}\sqrt{12\pi\epsilon_{0}\hbar\eta_{1}R^{3}}$
where
$\eta_{1}=\left\\{\left.\frac{d}{d\omega}Re\left[\epsilon_{m}(\omega)\right]\right|_{\omega=\omega_{d}}\right\\}^{-1}$
and
$\epsilon_{m}(\omega)=\epsilon_{\infty}-\omega_{p}^{2}/\left(\omega^{2}+i\omega\gamma_{p}\right)$
is the effective permittivity of the nanoparticle in the Drude model with
$\epsilon_{b}$ being the relative permittivity of the surrounding environment
and $\epsilon_{0}$ is the vacuum permittivity [86]. In all the numerical
calculations throughout this paper, we choose typical parameters for silver
with $\epsilon_{\infty}=6.0$, $\omega_{p}=7.9$eV, $\gamma_{p}=51$meV and
$\gamma_{e}=15$meV [75]. Since the free vacuum and the dipolar field have been
accounted separately,
$\hat{f}^{\prime+}\left(r,\omega_{\lambda}\right)\left(\hat{f}^{\prime}\left(r,\omega_{\lambda}\right)\right)$
denotes the continuum bosonic-field creation (annihilation) operator of the
HOMs and
$\widehat{\boldsymbol{E}}^{\prime}(\boldsymbol{r})=\widehat{\boldsymbol{E}}^{\prime+}(\boldsymbol{r})+\widehat{\boldsymbol{E}}^{\prime-}(\boldsymbol{r})$
is the electric field operator of the HOMs with

$\hat{E}^{\prime+}\left(\boldsymbol{r}_{e}\right)=i\sqrt{\frac{\hbar}{\pi\epsilon_{0}}}\int\frac{\omega_{\lambda}^{2}}{c^{2}}d\omega_{\lambda}\int
d\boldsymbol{r}^{\prime}\sqrt{\epsilon_{I}\left(\boldsymbol{r}^{\prime},\omega_{\lambda}\right)}\sum_{n\geq
2}^{\infty}\mathbf{G}_{n}^{S}\left(\boldsymbol{r},\boldsymbol{r}^{\prime};\omega_{\lambda}\right)\cdot\hat{f}^{\prime}\left(\boldsymbol{r}^{\prime},\omega_{\lambda}\right)$
(3)

being the positive part of the field where
$\epsilon_{I}\left(\boldsymbol{r}^{\prime},\omega_{\lambda}\right)$ is the
imaginary part of the material permittivity and $\mathbf{G}_{n\geq
2}^{\mathbf{S}}\left(\boldsymbol{r},\boldsymbol{r}^{\prime};\omega_{\lambda}\right)$
is the higher-order scattering Green function with $\mathrm{n}\geq 2$. The
dyadic Green function
$\boldsymbol{G}\left(\boldsymbol{r},\boldsymbol{r}^{\prime};\omega_{\lambda}\right)$
satisfies the equation

$\boldsymbol{\nabla}\times\boldsymbol{\nabla}\times\mathbf{G}\left(\boldsymbol{r},\boldsymbol{r}^{\prime};\omega_{\lambda}\right)-\frac{\omega_{\lambda}^{2}}{c^{2}}\epsilon\left(\boldsymbol{r},\omega_{\lambda}\right)\mathbf{G}\left(\boldsymbol{r},\boldsymbol{r}^{\prime};\omega_{\lambda}\right)=\mathbf{I}\delta\left(\boldsymbol{r}-\boldsymbol{r}^{\prime}\right).$
(4)

Without plasmonic structure,
$\boldsymbol{G}\left(\boldsymbol{r},\boldsymbol{r}^{\prime};\omega\right)=\boldsymbol{G}_{0}\left(\boldsymbol{r},\boldsymbol{r}^{\prime};\omega\right)$.
In the presence of plasmonic structure,
$\boldsymbol{G}\left(\boldsymbol{r},\boldsymbol{r}^{\prime};\omega\right)=\boldsymbol{G}_{0}\left(\boldsymbol{r},\boldsymbol{r}^{\prime};\omega\right)+\boldsymbol{G}^{S}\left(\boldsymbol{r},\boldsymbol{r}^{\prime};\omega\right)$
where
$\boldsymbol{G}_{0}\left(\boldsymbol{r},\boldsymbol{r^{\prime}};\omega\right)$
and
$\boldsymbol{G}^{S}\left(\boldsymbol{r},\boldsymbol{r}^{\prime};\omega\right)$
are the free and scattering parts of the dyadic Green function, respectively
and their analytical expressions are given in the Supplementary Information
(SI) Sec. I. The normalized spectral density of the nth-order plasmonic mode
at position $\mathbf{r}$ is given by
$\mathrm{J}_{\mathrm{n}}(\boldsymbol{r},\omega)=\operatorname{Im}\left[\hat{\boldsymbol{e}}_{\boldsymbol{i}}\cdot\boldsymbol{G}_{n}^{S}(\boldsymbol{r},\boldsymbol{r};\omega)\cdot\hat{\boldsymbol{e}}_{\boldsymbol{i}}\right]/G_{0}$
where $\boldsymbol{G}_{n}^{S}(\boldsymbol{r},\boldsymbol{r};\omega)$ is the
nth-order scattering Green function of the nanoparticle, $G_{0}=k/6\pi$ and
$\hat{\boldsymbol{e}}_{\boldsymbol{i}}$ is a unit vector [75]. When the
distance between the emitter and the nanoparticle is of the order of or larger
than the radius of the nanoparticle, the dipole mode dominates (see upper
panel of Fig. 1(b)). However, when the emitter is very close to the
nanoparticle, the HOMs can have larger spectral density than that of the
dipole mode (see lower panel of Fig. 1(b)) and therefore they should be taken
into accounts.

Figure 1: (a) QE couples to a nanoparticle. (b) The spectral densities of
different plasmonic modes with two different separations (upper panel: $h=10$
nm and lower panel: $h=2$ nm). (c) The spontaneous decay rate and LS of the QE
induced by the dipole plasmonic mode. The solid lines are the results
calculated by the scattering Green function with $n=1$ and the symbols are
those calculated by the effective dipole model. (d) The decay rate (blue
thinner line) and Lamb shift (red thicker line) of the QE due to higher-order
modes. The total decay rate (green squares) and the total LS (black circles)
are also shown. The dashed dotted lines in both (c) and (d) mark the plasmon
dipole frequency when $R=10$ nm and $h=2$ nm. The decay rate (e) and Lamb
shift (f) of quantum emitter as a function of $h/R$ with two different
nanoparticle radius (i.e., $R=10$ nm and $R=15$ nm) where the curve with
symbols are the results considering all orders of plasmonic modes while the
dashed curves are those in the dipolar approximation. The insets of (e) and
(f) show the plasmonic dipolar frequency as a function of $R$ and $h$,
respectively.

For spontaneous decay process where only the emitter is initially excited, we
can obtain the emission spectrum of the emitter given by (see SI Sec. II)

$S_{emi}(\omega)\propto\left|\omega-\omega_{e}^{\prime}-\frac{g_{de}^{2}}{\omega-\omega_{d}^{\prime}}\right|^{-2}.$
(5)

Here, we define the effective transition frequency of the quantum emitter
$\omega_{e}^{\prime}=\omega_{e}+\Delta_{e}^{\prime}(\omega)-i\left[\gamma_{e}+\gamma_{\mathrm{e}}^{\prime}(\omega)\right]/2$
in which

$\Delta_{e}^{\prime}(\omega)=-\frac{\omega^{2}}{\hbar\epsilon_{0}c^{2}}\boldsymbol{\mu}_{\mathbf{e}}\cdot\operatorname{Re}\sum_{n\geq
2}\mathbf{G}_{n}^{S}\left(\boldsymbol{r}_{a},\boldsymbol{r}_{a};\omega\right)\cdot\boldsymbol{\mu}_{e}$
(6)

and

$\gamma_{\mathrm{e}}^{\prime}(\omega)=\frac{2\omega^{2}}{\hbar\epsilon_{\mathrm{o}}c^{2}}\boldsymbol{\mu}_{e}\cdot\operatorname{Im}\sum_{n\geq
2}\mathbf{G}_{n}^{S}\left(\boldsymbol{r}_{a},\boldsymbol{r}_{a};\omega\right)\cdot\boldsymbol{\mu}_{e}$
(7)

are the Lamb shift and decay rate induced by the HOMs vacuum fields,
respectively. The term $g_{de}^{2}/\left(\omega-\omega_{d}^{\prime}\right)$
with $\omega_{d}^{\prime}=\omega_{d}-i\gamma_{d}/2$ is the contribution from
the plasmon dipole mode. The total LS of the emitter is given by
$\Delta_{e}^{\prime}(\omega)+\operatorname{Re}\left[g_{de}^{2}/\left(\omega-\omega_{d}^{\prime}\right)\right]$
and the total decay rate is given by
$\gamma_{e}+\gamma_{\mathrm{e}}^{\prime}(\omega)-2\operatorname{Im}\left[g_{de}^{2}/\left(\omega-\omega_{d}^{\prime}\right)\right]$.
The emission spectrum shown in Equation (5) is actually equivalent to that
obtained from the original Hamiltonian of the system in the rotating wave
approximation, i.e.
$S_{emi}(\omega)=\left\\{\left[\omega-\omega_{e}-\Delta_{e}(\omega)\right]^{2}+\Gamma_{e}^{2}(\omega)/4\right\\}^{-1}$
[87] where
$\Delta_{e}(\omega)=-\frac{\omega^{2}}{\hbar\epsilon_{0}c^{2}}\boldsymbol{\mu}_{e}\cdot\operatorname{Re}\left[\boldsymbol{G}^{S}\left(\boldsymbol{r}_{e},\boldsymbol{r}_{e};\omega\right)\right]\cdot\boldsymbol{\mu}_{e}$
and
$\Gamma_{e}(\omega)=\frac{2\omega^{2}}{\hbar\epsilon_{0}c^{2}}\boldsymbol{\mu}_{e}\cdot\operatorname{Im}\left[\boldsymbol{G}^{S}\left(\boldsymbol{r}_{e},\boldsymbol{r}_{e};\omega\right)\right]\cdot\boldsymbol{\mu}_{e}$
are the LS and the effective decay rate calculated by the full scattering
Green function. The equivalence can be seen from Fig. 1(c) where the well
agreement of both the real and imaginary parts of
$g_{de}^{2}/\left(\omega-\omega_{d}^{\prime}\right)$ with
$-\left(\omega^{2}/\hbar\epsilon_{0}c^{2}\right)\boldsymbol{\mu}_{e}\cdot\boldsymbol{G}_{n=1}^{S}\left(\boldsymbol{r}_{e},\boldsymbol{r}_{e};\omega\right)\cdot\boldsymbol{\mu}_{e}$
are shown (also see SI Sec.IV). Around the dipolar mode frequency
$\omega_{d}$, the LS induced by the dipole mode vanishes but the decay rate is
maximum. On the contrary, the HOMs have little effect on the decay rate but
have significant effect on the LS even though the bare emitter transition
frequency is far-off resonant from the HOMs’ frequencies, as shown in Fig.
1(d). The LS induced by the HOMs can be high up to 34.1 meV in current example
which is more than three orders of magnitude larger than that in the free
vacuum. The decay rate and Lamb shift of the quantum emitter as a function of
$h/R$ for two different nanoparticle radii (i.e., $R=10$ nm and $R=15$ nm) are
shown in Figs. 1(e) and (f), respectively. For comparisons, the results in the
dipolar approximations are also shown as the dashed curves. We can see that
both the total decay rate and the Lamb shift increase rapidly when $h/R$
decreases and their values especially the Lamb shift when $h/R$ is small can
be very different from those in the dipole approximation. We also find that
both the decay rate and the Lamb shift increase when $R$ decreases for the
same $h/R$. By extracting the peak of the density of state of the dipolar
component, we find that the plasmon dipolar frequency does not depend on $h$
but is red-shifted when $R$ increases (see the inset of Fig. 1(e) and Fig.
1(f)). In the small radius limit, the dipolar frequency approaches the value
of $\omega_{p}/\sqrt{\epsilon_{\infty}+2\epsilon_{b}}$ in the quasistatic-
limit which is $2.793$ eV in our example. Since
$\boldsymbol{G}^{S}\left(\boldsymbol{r}_{e},\boldsymbol{r}_{e};\omega\right)$
strongly depends on the position of the emitter, the LS of the emitter varies
with emitter’s position. From Fig. 2(a), we can see that the emission spectrum
peak is red-shifted and the linewidth is also broaden when the nanoparticle
moves towards the emitter which provides a method to observe the LS in this
system and similar phenomena have also been experimentally observed in the
plasmonic nanocavity system [58]. Although the coupling strength increases
when the emitter is closer to the nanoparticle, we do not observe fluorescence
splitting because the damping rate also increases rapidly and thus no spectrum
splitting is observed in Fig. 2(a). We should also mention that in Fig. 2(a)
we show the emitter fluorescence spectrum shape but its real intensity in the
far field also depends on the quantum yield. In this coupling system, since
the emitter radiative decay $\gamma_{e}^{0}$ is usually much less than its
non-radiative decay rate $\gamma_{e}^{nr}$ especially when the quenching
effect is considered in the case that the emitter is very close to the metal
nanoparticle [88], the quantum yield of the fluorescence is actually very
small and most of the emitted energy are absorbed by the metal nanoparticle.
In addition, the spectrum shown in Equation (5) is the fluorescence spectrum
when only the emitter is initially excited which is not easy to be realized in
this system because the cross section of the nanoparticle is usually much
larger than that of the quantum emitter. To enhance the quantum yield of the
fluorescence, a plasmonic cavity together with a dielectric spacer is usually
required [58] which may limit its application scenarios. Next, we show that
the LS can also be observed from the scattering spectrum of the system under
weak driving field, which is much easier to be experimentally implemented.

Figure 2: The emission (a) and scattering (b) spectrum of the coupled system
for different emitter-nanoparticle distances (the curves with filling area).
The black dotted curves are the results without Lamb shift. In both (a) and
(b), the emitter is assumed to be along the direction of dipole polarization.
(c) The Lamb shift, scattering spectrum dip shift, and emission peak shift as
a function of distance. (d) The scattering spectrum changes with different
polarization orientations of the emitter with $h=2$ nm. Inset: the scattering
spectrum dip shift as a function of $\theta$. $R=10$ nm and
$\omega_{e}=\omega_{d}=2.785$ eV.

According to the input-output theory [89], the output field operator
$\hat{a}_{out}(t)=\sqrt{\gamma_{d}^{0}}\hat{d}(t)+\sqrt{\gamma_{e}^{0}}\sigma^{-}(t)$
and the scattering light spectrum
$S(\omega)\propto\langle\hat{a}^{+}_{out}(\omega)\hat{a}_{out}(\omega)\rangle$
where $\hat{a}_{out}(\omega)$ is the Fourier transformation of
$\hat{a}_{out}(t)$. In the typical nanoparticle-emitter coupling system,
$\gamma_{d}^{0}\gg\gamma_{e}^{0}$ and therefore the output field is mainly due
to the emission of the palsmon dipole, i.e.,
$\hat{a}_{out}(t)\approx\sqrt{\gamma_{d}^{0}}\hat{d}(t)$ [88]. Thus, the
scattering spectrum of the system in the stationary limit
$\mathrm{S}_{\mathrm{sca}}(\omega)\propto\lim_{t\rightarrow\infty}\operatorname{Re}\left[\int_{0}^{\infty}\left\langle
d^{+}(t)\mathrm{d}(t+\tau)\right\rangle e^{i\omega\tau}dt\right]$ and in the
weak excitation limit is given by (see Sec. III in SM)

$S_{sca}(\omega)\propto-\operatorname{Im}\left[\omega-\omega_{d}^{\prime}-\frac{g_{de}^{2}}{\left(\omega-\omega_{e}^{\prime}\right)}\right]^{-1}.$
(8)

If the coupling strength vanishes (i.e., $g_{de}=0$), the scattering spectrum
$S_{sca}(\omega)\propto-\operatorname{Im}\left(\omega-\omega_{d}^{\prime}\right)^{-1}$
which has Lorentzian lineshape with frequency $\omega_{d}$ and linewidth
$\gamma_{d}$. Without coupling, the scattering spectrum of the nanoparticle
does not contain any information of the emitter. However, if $g_{de}\neq 0$,
the effective transition frequency of the plasmon dipole is modified in the
presence of the QE and hence $S_{sca}(\omega)$ includes the emitter’s spectral
information. The scattering spectrum shown in Equation (8) can be rewritten as

$S_{sca}(\omega)\propto-\operatorname{Im}\left[\frac{f_{+}}{\omega-\omega_{+}}+\frac{f_{-}}{\omega-\omega_{-}}\right],$
(9)

where
$\omega_{\pm}=\frac{1}{2}\left(\omega_{e}^{\prime}+\omega_{d}^{\prime}\pm\Delta_{ls}\right)$
are two eigenfrequencies of the coupled system, with
$\Delta_{ls}=\sqrt{\left(\omega_{d}^{\prime}-\omega_{e}^{\prime}\right)^{2}+4g_{de}^{2}}$
and
$\mathrm{f}_{\pm}=\frac{1}{2}\pm\frac{\omega_{d}^{\prime}-\omega_{e}^{\prime}}{\Delta_{ls}}$
are two constant coefficients [60]. It is clearly seen that the scattering
spectrum is the superposition of two eigen-channels and quantum interference
between these two channels can induce a spectrum dip even if the system is in
the pseudo-strong coupling regime. If $f_{+}=f_{-}$ and they are real, the
spectrum dip occurs at exactly the center of the two eigen-frequencies, i.e.,
$\frac{1}{2}\left(\omega_{e}+\Delta_{e}^{\prime}+\omega_{d}\right)$ which is a
linear function of the LS. However, in the usual case, $f_{+}\neq f_{-}$ and
both of them may be complex number, the spectrum dip usually deviates from
$\frac{1}{2}\left(\omega_{e}+\Delta_{e}^{\prime}+\omega_{d}\right)$ but it is
still a monotonic function of the emitter LS (see SI Sec. V).

The scattering spectrum as a function of the emitter distance is shown in Fig.
2(b). Different from the spontaneous emission spectrum, there is usually a
spectrum dip in the scattering spectrum due to the Fano-like interfernce even
if the strong coupling condition is not met [32, 60]. From Fig. 2(b) we can
see that the LS can be observed from the shift of the spectrum dip when the
nanoparticle moves towards the emitter (the curves with filling area). In
contrast, without considering the LS, the position of spectrum dip does not
change with the emitter position (dotted curves) although the peak separation
increases with decreasing distance. The LS, emission peak shift and scattering
spectrum dip shift as a function of emitter position are shown in Fig. 2(c).
It is clearly seen that both the emission peak and the scattering spectrum dip
are rapidly red-shifted together with the LS when $h$ decreases and the
gradients are about 50$\sim$60 meV/nm when $h\approx 2$ nm. Therefore, either
the emission peak shift or the scattering dip shift can reveal the LS of the
emitter. The method discussed above also works when the emitter transition
frequency is slightly different from the plasmon dipole frequency (see SI Sec.
VI).

Since the LS highly depends on the relative position between the emitter and
the nanoparticle, it is possible to determine an emitter’s position from the
scattering spectrum. From Fig. 2(c) we can see that when the distance
decreases from 20 nm to 1.5 nm, the magnitude of LS increases from almost 0 to
77.4 meV which can be observed from the scattering spectrum dip shift of about
53 meV. In particular, when the emitter-nanoparticle distance changes from 2
nm to 1.5 nm, the magnitude of LS increases by about 48 meV and the spectrum
dip is red-shifted by about 27 meV. In a typical high-resolution spectrometer,
0.02 nm wavelength difference (about 0.1 meV energy shift) can be resolved.
Thus, from the scattering spectrum dip shift we can in principle measure
emitter position change with angstrom or even sub-angstrom precision which is
only limited by the precision position control of the nanoparticle. The
precision of the scheme may be further improved if the emitter-nanoparticle
distance is less than 1 nm, while in this regime the quantum effects such as
quantum tunneling and nonlocal effects may play an important role which needs
to be considered [90, 77, 78, 33, 91, 92].

In addition to the distance, the polarization direction of the quantum emitter
can also affect the LS and therefore the scattering spectrum is changed as the
emitter polarization angle. In the near-field regime,
$\mathbf{G}_{0}\left(\boldsymbol{r}_{e},\boldsymbol{r}_{d};\omega\right)\approx
e^{ikr}\left(-\boldsymbol{I}+3\mathbf{e}_{\mathrm{r}}\mathbf{e}_{\mathrm{r}}\right)/4\pi
k^{2}r^{3}$ where
$\mathrm{r}=\left|\boldsymbol{r}_{e}-\boldsymbol{r}_{d}\right|$. The coupling
strength
$g_{\mathrm{de}}\approx\mu_{d}\mu_{e}\sqrt{1+3\cos^{2}\theta}/4\pi\epsilon_{0}\hbar(R+h)^{3}$
which decreases when $\theta$ increases from 0 to $\pi$/2. In addition to the
coupling strength, the magnitude of LS also decreases when $\theta$ increases
from 0 to $\pi$/2 (see SI Sec. VII). The scattering spectra for different
emitter polarizations are shown in Fig. 2(d) where it is shown that the
spectrum dip is shifted when $\theta$ varies. The dip shift is maximum when
the polarization of the emitter is along the z-direction while it is minimum
when it is along the x-direction (Fig. 2(d)). The spectrum dip shift with
respect to $\theta$ is also shown in the inset of Fig. 2(d). The average
gradient of the shift is about 0.17 meV per degree. Hence it is possible to
detect the polarization of the emitter with about $1^{\text{o}}$ resolution
from the scattering spectrum.

Figure 3: (a) A schematic setup for ultrasensitive emitter localization based
on tip plasmonics induced LS (left) and the field intensity distribution
around the tip when the tip is 1 nm above the emitter (right). The tip has a
semi elliptical shape and the major axis is 20 nm and minor axis is 13 nm. (b)
The scattering spectrum of the coupled plasmonic system for different lateral
offsets ($0\rightarrow 1.4$ nm) of the tip positions when the z-offset is
fixed to be 1 nm. (c,d) The LS (black stars) and the spectrum dip shift (red
down triangles) for different lateral tip positions with 1 nm z-offset (c) and
for different $z$ offset when the tip is right on top of the emitter (d).
(e,f) The extracted 2D image of the emitter based on the dip shift of the
scanning scattering spectrum when the emitter is x-polarized (e) or
z-polarized (f) when the z-offset is 1 nm, where the color represents the
amount of the spectrum dip shift. In all these figures, the emitter is assumed
to be 0.5 nm below the SiN substrate surface.

##  3 Experiment proposal

Finally, we propose a possible realization based on the theory described above
to detect the position of a quantum emitter sitting on or inside a substrate
(such as SiN) as shown in Fig. 3(a). Here, as an example the emitter is
assumed to be x-polarized and 0.5 nm below the surface. The field intensity
distribution when the tip is 0.5 nm above the surface is also shown in right
panel of Fig. 3(a), where we can see that the field is strongly localized
around the tip. A light beam is applied to the tip and the scattering spectra
for different tip lateral positions are shown in Fig. 3(b) when the z distance
between the emitter and tip (z-offset) is fixed to be 1 nm. When the tip is
closer to the emitter, the spectrum dip is shifted to the lower frequency due
to the LS discussed above. The dip shift and the LS as a function of lateral
offset are shown in Fig. 3(c) when the z-offset is fixed to be 1 nm. When the
tip is right on top of the emitter, the spectrum dip has a maximum shift. The
gradient around the maximum shift is about 1 meV/nm near the center. When the
tip is right on top of the emitter, the spectrum dip shift as a function of z
distance is shown in Fig. 3(d). When the tip moves towards the emitter, the
spectrum dip is red-shifted with gradient about 15.4 meV/nm, which indicates a
much higher sensitivity for detecting the longitudinal distance. For a
spectrometer with 0.02 nm resolution, the lateral and longitudinal resolutions
in this setup can in principle be about 1 $\mathring{A}$ and 0.1
$\mathring{A}$, respectively. If the substrate is replaced by a material with
lower reflective index or the emitter is placed above the substrate surface,
the LS can be even larger and the resolution can be further enhanced (see SI
Sec. VIII). We can also extract a two-dimensional image of the emitter from
the dip shift of the scattering spectra at different positions when the
emitter is x-polarized (Fig. 3(e)) or z-polarized (Fig. 3(f)). When the
emitter is z-polarized, the image is symmetric. However, when the emitter is
x-polarized, the image has an elliptical shape with major axis along the
polarization direction. Thus, in our scheme, we can not only detect the
position of the emitter but also its polarization direction. When the z-offset
increases, the major axis of the imaging spot also increases which indicates
the reduction of resolution (see Fig. S6 in SM). Different from the typical
STML and TEPL methods [29, 30, 31, 58], here our scheme does not require the
formation of plasmonic picocavity to enhance the fluorescence rate and it also
works even when the emitter is embedded slightly below a dielectric substrate
which may find more application scenarios.

##  4 Summary and discussion

To conclude, we show that a quantum emitter very close to a metal nanoparticle
or tip can have huge Lamb shift mainly induced by the higher-order plasmonic
dark modes and this energy shift is ultra-sensitive to the emitter position.
We also show that this giant Lamb shift can be sensitively observed from the
scattering spectrum dip shift when the metal nanoparticle or tip scans through
the emitter. Moreover, we propose that this quantum effect can be exploited
for all-optical detection of an emitter position with angstrom precision which
is comparable to the typical STM, STML and TEPL methods and we can also
determine the polarization direction of the quantum emitter. In contrast to
the typical STML and TEPL methods which are usually surface bound and only
detect the sample in a plasmonic picocavity, we can detect the position of the
emitter precisely even if it is slightly below a dielectric substrate. Due to
the fact that the experimental measurement of the scattering spectrum is much
easier than the fluorescence in the plasmon-emitter coupling system, our work
here can find important applications in many areas and can stimulate extensive
theoretical and experimental researches in the future. Finally, we should
mention that the emitter is treated as a point dipole in this work to
demonstrate the basic principle, the full scattering spectrum considering the
non-dipolar effect of the quantum emitter and other quantum effects should be
further studied in the future [93].

Funding The authors thank R. Liu and X. Zeng for helpful discussions. This
work was supported by the National Key R&D Program of China (Grant No.
2021YFA1400800), the Key-Area Research and Development Program of Guangdong
Province (Grant No.2018B030329001), the Guangdong Special Support Program
(Grant No.2019JC05X397), the Natural Science Foundations of Guangdong (Grant
Nos. 2021A1515010039 and 2018A030313722).

Disclosures The authors declare no conflicts of interest.

Data availability Data underlying the results presented in this paper are not
publicly available at this time but may be obtained from the authors upon
reasonable request.

Supplemental document See Supplement 1 for supporting content.

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