Issue 
EPJ Appl. Metamat.
Volume 5, 2018
Metamaterials'2017 – Metamaterials and Novel Wave Phenomena: Theory, Design and Application



Article Number  6  
Number of page(s)  6  
DOI  https://doi.org/10.1051/epjam/2018004  
Published online  04 June 2018 
https://doi.org/10.1051/epjam/2018004
Research Article
Fluorescence enhancement and strongcoupling in faceted plasmonic nanocavities
^{1}
Blackett Laboratory, Prince Consort Road, Imperial College London,
London
SW7 2AZ, UK
^{2}
School of Physics and Astronomy, University of Birmingham,
Edgbaston,
Birmingham B15 2TT, UK
^{3}
Cavendish Laboratory, University of Cambridge,
Cambridge
CB3 0HE, UK
^{*} email: n.kongsuwan15@imperial.ac.uk
Received:
7
September
2017
Accepted:
15
February
2018
Published online: 4 June 2018
Emission properties of a quantum emitter can be significantly modified inside nanometresized gaps between two plasmonic nanostructures. This forms a nanoscopic optical cavity which allows singlemolecule detection and singlemolecule strongcoupling at room temperature. However, plasmonic resonances of a plasmonic nanocavity are highly sensitive to the exact gap morphology. In this article, we shed light on the effect of gap morphology on the plasmonic resonances of a faceted nanoparticleonmirror (NPoM) nanocavity and their interaction with quantum emitters. We find that with increasing facet width the NPoM nanocavity provides weaker field enhancement and thus less coupling strength to a single quantum emitter since the effective mode volume increases with the facet width. However, if multiple emitters are present, a faceted NPoM nanocavity is capable of accommodating a larger number of emitters, and hence the overall coupling strength is larger due to the collective and coherent energy exchange from all the emitters. Our findings pave the way to more efficient designs of nanocavities for roomtemperature lightmatter strongcoupling, thus providing a big step forward to a noncryogenic platform for quantum technologies.
Key words: Nanoplasmonics / Nanophotonics / Lightmatter Strongcoupling / Fluorescence Enhancement / Quenching
© N. Kongsuwan et al., published by EDP Sciences, 2018
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The decay lifetime of an excited quantum emitter is determined by both its intrinsic properties and its optical environment, as theoretically proposed by Purcell [1]. Since then, the effect has been verified experimentally on various opticalfield enhancing environments [2–5]. One example of such an environment is a plasmonic nanostructure which has the ability to provide subwavelength light confinement and dramatically enhance the excitation and emission of an emitter [6]. However, placing an emitter too close to a single metal nanostructure quenches its farfield emission as the emitter dominantly decays into the nonradiative channels of the nanostructure [7–8]. To suppress this quenching, the emitter can be placed inside a nanocavity formed by two closely spaced metallic nanostructures, i.e. a gap of few nanometres [9].
In our previous study, we experimentally and numerically demonstrated a suppression of fluorescence quenching for an emitter placed in a nanoparticleonmirror (NPoM) nanocavity [10]. In that study, we considered perfectly spherical nanoparticles (NPs). However, this does not always reflect experimental setups where NPs are in most cases faceted. Indeed, the spectral behavior of NPoM nanocavities also shows a highly sensitive relationship with its gap morphology. The facet widths of the NPoM nanocavities are hence also expected to greatly influence the field enhancement experienced by quantum emitters.
In this article, we use finitedifference timedomain (FDTD) method to investigate the effects of facet widths on the fluorescence enhancement, fluorescence quenching and strongcoupling of emitters in the NPoM nanocavity. We find that, by increasing the facet width, the NPoM nanocavity provides a weaker field enhancement and less coupling strength to a single emitter since the effective mode volume increases with the facet width. However, if multiple emitters are present, the faceted NPoM nanocavities are capable of accommodating more emitters, and hence the coupling strength is larger due to the collective and coherent energy exchange from all the emitters [11].
2 Fluorescence enhancement and quenching
Fluorescence emission of an emitter is a twostep process involving excitation and relaxation decay. When the excitation rate γ_{exc} of the emitter is much slower than its total decay rate γ_{tot}, the excited emitter decays to its ground state before the next excitation event. In this weak excitation regime, fluorescence emissions are limited by the excitation events, and therefore the fluorescence enhancement is given by [8]: $${\gamma}_{\text{em}}={\gamma}_{\text{exc}}\eta \text{\hspace{0.17em}}\text{,}$$(1) where η is the emitter's quantum yield. For an emitter located at position r_{0} with transition dipole moment μ and transition frequency ω_{0}, its excitation rate γ_{exc} is determined by the local electric field E(r_{0}) in the direction of μ, ${\gamma}_{\text{exc}}\propto \mu \cdot \mathbf{\text{E}}({\mathbf{\text{r}}}_{0}){}^{2}$. In vacuum, the emitter can only experience the incident field E_{0} (r_{0}) whereas, in nonvacuum, the emitter experiences the local field E(r_{0}) which is the combination of E_{0}(r_{0}) and its secondary field reflected from the environment. Let us represent the excitation enhancement through the normalized rate: $${{\displaystyle \tilde{\gamma}}}_{\mathrm{e}\mathrm{x}\mathrm{c}}=\frac{{\gamma}_{\text{exc}}}{{\gamma}_{\text{exc}}^{0}}={\left\frac{{\displaystyle \stackrel{\u02c6}{\mu}}\cdot \mathbf{\text{E}}({\mathbf{\text{r}}}_{0})}{{\displaystyle \stackrel{\u02c6}{\mathit{\mu}}}\cdot {\mathbf{\text{E}}}_{0}({\mathbf{\text{r}}}_{0})}\right}^{2}\text{,}$$(2)where ${\gamma}_{\text{exc}}^{0}$ is the excitation rate of the emitter in vacuum and $\stackrel{\u02c6}{\mathit{\mu}}$ is a unit vector in direction of μ.
Each decay event either channels energy into a farfield radiation or is dissipated in the environment. The quantum yield η = γ_{rad}/γ_{tot} provides a measure of this behavior and is defined as the ratio between farfield radiative decay rate γ_{rad} and the total decay rate γ_{tot} of the emitter. On the basis of Fermi's golden rule, γ_{tot} can be expressed in terms of the electromagnetic density of states $\rho \mathrm{}{(r}_{0}\mathrm{}{\omega}_{0})\mathrm{}$ as [12]: $${\gamma}_{\mathrm{t}\mathrm{o}\mathrm{t}}=\frac{2{\omega}_{0}}{3\mathrm{\hslash}{\epsilon}_{0}}\left{\mathit{\mu}}^{2}\right\rho ({\mathbf{r}}_{0},{\omega}_{0})\text{.}$$(3) Assuming that the emitter does not have an intrinsic loss, its radiative decay rate γ_{rad} = γ_{tot} − γ_{nr} can be indirectly found by calculating the nonradiative decay rate γ_{nr}. In the case of a plasmonic environment, γ_{nr} is predominantly determined by Ohmic losses [12]: $${\gamma}_{\text{nr}}\text{\hspace{0.17em}}\propto \text{\hspace{0.17em}}{\int}_{V}\mathrm{Re}\left\{{\mathbf{j}}_{\text{ind}}(\mathbf{\text{r}})\cdot {\mathbf{E}}_{\mathrm{e}\mathrm{m}}^{*}(\mathbf{\text{r}})\right\}d{\mathbf{\text{r}}}^{3}\text{,}$$(4)where j_{ind} is the induced current density within the volume V and E_{em} is the field emitted by the emitter. We define the radiative and nonradiative decay enhancements as the emitter's decay rates normalized to its decay rate in vacuum ${\gamma}_{\text{rad}}^{0}$ as ${{\displaystyle \tilde{\gamma}}}_{\mathrm{r}\mathrm{a}\mathrm{d}}={\gamma}_{\text{rad}}/{\gamma}_{\text{rad}}^{0}$ and ${{\displaystyle \tilde{\gamma}}}_{\mathrm{n}\mathrm{r}}={\gamma}_{\text{nr}}/{\gamma}_{\text{rad}}^{0}$ respectively. The fluorescence enhancement is similarly defined as ${{\displaystyle \tilde{\gamma}}}_{\text{em}}\mathrm{}={{\displaystyle \tilde{\gamma}}}_{\text{exc}}\eta $.
A plasmonic nanostructure provides an ideal environment for fluorescence enhancement as it substantially amplifies both γ_{exc} and γ_{tot} of an emitter by confining light to a subwavelength volume and therefore substantially amplifying the electromagnetic density of states. However, the emitter in a plasmonic environment also experiences a large γ_{nr} due to Ohmic losses, which reduces η. Because of these competing factors, a specific plasmonic nanostructure can either potentially enhance or quench fluorescence.
3 Suppression of fluorescence quenching in faceted plasmonic nanocavities
Fluorescence quenching is observed when an emitter is placed too close to an isolated plasmonic nanostructure [7]. In the case of an isolated NP, for example, its optical response can be described as a multipole expansion of its surface plasmon (SP) modes [13]. The lowestorder dipole SP is the only mode that has a nonzero dipole moment and couples to the farfield radiation. On the other hand, the higher order modes confine light more strongly within a smaller volume surrounding the NP than the first order mode. As the emitter gets closer than 10 nm to the NP, it becomes dominantly coupled to the nonradiative higherorder modes of the NPs, and as a result, its fluorescence is quenched through energy dissipation in metal [8]. This is, however, not a general feature for all plasmonic nanostructures and in Reference. [10] we demonstrate the suppression of fluorescence quenching in a plasmonic nanocavity.
A plasmonic nanocavity emerges when two plasmonic nanostructures are brought together forming a gap of just a few nanometres. At such small gaps, the nonradiative SP modes of one nanostructure can couple with the radiative modes of the other nanostructure, and vice versa [13]. As a consequence, the hybridized higherorder modes of the combined structure gain radiative nature, and fluorescence quenching is hence suppressed in a plasmonic nanocavity. The hybridized SPs, called bonding SPs, also provide a massive field confinement inside the gap, allowing ultrasensitive detection down to singleemitter level [14,15]. Indeed, plasmonic structures have been developed for fluorescence enhancement and singleemitter detection, such as coreshell nanodumbbells [14], bowtie nanoantennas [16,17] and filmcoupled nanocubes [18]. In this article, we focus on the NPoM geometry, which is experimentally realized by assembling molecular layers inbetween a spherical NP and a metal film (mirror) to form a nanocavity. The NPoM nanocavity has the advantage of providing subnanometre precision control over the spacing between a metallic NP and its mirror by using molecular layers, such as graphene [19], Cucurbituril [20] and DNAorigami [21].
3.1 Faceted NPoM nanocavities
So far we assumed that NPs are perfectly spherical. However, SEM images and experimental results [22,23] show that NPs are always faceted. Here, we investigate the fluorescence emission of an emitter inside a faceted NPoM nanocavity formed by a gold NP with diameter 2R = 80 nm and facet widths f = 0, 5 and 10 nm atop a dielectric spacer with thickness d = 5 nm and a gold mirror, see Figure 1(a). As we previously demonstrated [10], such a nanocavity can be fabricated using DNA origami (refractive index n = 2.1) to precisely control the positions of emitters in the nanocavity.
The bonding SPs of the NPoM nanocavity provide a large field enhancement $\mathrm{}\left\mathbf{E}\right/\left{\mathbf{E}}_{0}\right$ in the gap region. The electric field distribution of the lowestorder bonding plasmon, the bonding dipole plasmon (BDP), is shown in Figure 1(b) for an NPoM with f = 10 nm. The spectral positions of the BDPs for NPoM nanocavities with f = 0, 5 and 10 nm are summarized in Figure 1(c) in the form of scattering crosssections σ_{scat}. The dominant peaks σ_{scat} correspond to the BDPs, which redshift from 690 to 730 nm as facet widths increase. The peaks around 570 nm correspond to secondorder bonding quadrupole plasmons (BQP), and they are spectrally less affected by the facet widths. For comparison, σ_{scat} for NPs placed on plain dielectric substrates are also shown in Figure 1(c) and also exhibit firstorder localized SPs at ∼570 nm.
Fig. 1 . (a) Schematic diagram of a nanoparticleonmirror (NPoM) nanocavity with diameter 2R = 80 nm, facet diameter 2f and gap size d = 5 nm. (b) The field enhancement distribution $\left\mathbf{\text{E}}\right/\left{\mathbf{\text{E}}}_{0}\right$ of the bonding dipole plasmon (BDP) of the NPoM with f = 10 nm (c) Scattering crosssection σ_{scat} for the NPoM with f = 0, 5 and 10 nm and for nanoparticles (NPs) on plain dielectric substrates with f = 0 and 10 nm. 
3.2 Suppression of fluorescence quenching
When a quantum emitter is placed in a NPoM nanocavity, the large field enhancement of the BDP massively enhances the excitation rate ${{\displaystyle \tilde{\gamma}}}_{\text{exc}}$. However, the fluorescence enhancement of the emitter also depends on the quantum yield η, and a large ${{\displaystyle \tilde{\gamma}}}_{\text{exc}}$ alone is not a sufficient condition for suppressing fluorescence quenching.
Here, we investigate the fluorescence emission in faceted NPoM nanocavities. Figure 2 shows the rate enhancements ${{\displaystyle \tilde{\gamma}}}_{\text{rad}}$, ${{\displaystyle \tilde{\gamma}}}_{\text{nr}}$, ${{\displaystyle \tilde{\gamma}}}_{\text{exc}}$ and ${{\displaystyle \tilde{\gamma}}}_{\text{em}}$ computed by placing a classical dipole emitter at 2.5 nm below the NPs for faceted NPoM nanocavities and NPs on dielectric substrates. The results for each nanostructure are evaluated at the resonance frequency of the nanostructure's dominant plasmon modes, as shown in Figure 1(c).
As the facet width increases, the NPs experience the lightning rod effect at their sharp facet edges, and consequently the rate enhancements of the emitters reach maxima near the facet edges, as seen in Figure 2(b–e), dashed and dashdotted lines. By contrast, the rate enhancements for the NPoM nanocavities are maximum at the center of the nanocavity for all facet widths. This is due to the nature of the BDPs which spatially confines the light field at the nanocavities' center. The NPoM nanocavities with larger facet widths also confine light less efficiently and possess larger effective mode volumes. Hence, the rate enhancements of the emitters spatially broaden with increasing facet width.
In the case of the NPs, its fluorescence ${{\displaystyle \tilde{\gamma}}}_{\text{em}}$ is quenched as ${{\displaystyle \tilde{\gamma}}}_{\text{nr}}$ dominates ${{\displaystyle \tilde{\gamma}}}_{\text{rad}}$ and gives a diminishing quantum yield η < 0.03 for both f = 0 and 10 nm. On the other hand, plasmon hybridization in the NPoM nanocavities provides sufficiently large ${{\displaystyle \tilde{\gamma}}}_{\text{rad}}$ and significant reductions in ${{\displaystyle \tilde{\gamma}}}_{\text{nr}}$, giving η ≈ 0.46 at the center of the nanocavity for all facet widths. As shown in Figure 2(e), the faceted NPoM nanocavities suppress fluorescence quenching of the emitters and provide more than two orders of magnitude increase in ${{\displaystyle \tilde{\gamma}}}_{\text{em}}$, compared to the faceted NPs without the mirror. Hence, a faceted NPoM nanocavity with f ≤ 10 nmretains the ability to suppress quenching. Unlike single NPs, the large η of NPoM nanocavities, with facet width f < 10 nm, allow information of the emitterplasmon interaction to be observed by farfield detectors. Hence, the NPoM nanocavities, faceted or not, provides an ideal environment to observe strongcoupling.
Fig. 2 . (a) The radiative decay ${{\displaystyle \tilde{\gamma}}}_{\text{rad}}$, (b) nonradiative decay ${{\displaystyle \tilde{\gamma}}}_{\text{nr}}$, (c) excitation ${{\displaystyle \tilde{\gamma}}}_{\text{exc}}$ and (d) fluorescence enhancements ${{\displaystyle \tilde{\gamma}}}_{\text{em}}$ for the NPoM nanocavities with facet diameter f = 0, 5 and 10 nm and the NPs on dielectric substrates with f = 0 and 10 nm. For clarity, ${{\displaystyle \tilde{\gamma}}}_{\text{rad}}$, ${{\displaystyle \tilde{\gamma}}}_{\text{exc}}$ and ${{\displaystyle \tilde{\gamma}}}_{\text{em}}$ for the NPs are multiplied by the indicated factors. 
4 Strongcoupling in faceted NPoM nanocavities
The emission properties of an emitter are influenced by its interaction with its optical environment. If the interaction is weak, only the emission rate is modified whereas the emission frequency remains unaltered. This is known as the weakcoupling regime. However, when the interaction is sufficiently strong, energy is reversibly exchanged between the two systems, and the electronic state of the emitter becomes inextricably mixed with the optical environment. In this strongcoupling regime, the resulting hybrid states split in energy and greatly differ from the original state of either constituent above.
To appreciate this process in the NPoM systems, let us consider an electronic (optical) state with resonance frequency ω_{0} (ω_{p}) and spectral linewidth Γ_{0} (Γ_{p}). The hybrid state frequencies Ω_{+} and Ω_{−} can be expressed as complex numbers of which the real parts ω_{+} and ω_{−} correspond to their resonance frequencies and imaginary parts Γ_{+} and Γ_{−} to their spectral linewidths [24]. At resonance, ω_{0} = ω_{p}, these are given by: $${\mathrm{\Omega}}_{\pm}={\omega}_{0}i\frac{1}{2}({\mathrm{\Gamma}}_{\text{p}}+{\mathrm{\Gamma}}_{0})\pm \frac{1}{2}\sqrt{{(2g)}^{2}{({\mathrm{\Gamma}}_{\text{p}}{\mathrm{\Gamma}}_{0})}^{2}}\text{,}$$(5) where $g\text{\hspace{0.17em}}\propto \text{\hspace{0.17em}}\mu \sqrt{N/V}$ is the coupling strength which depends on the number N of emitters and the optical mode volume V [25]. The criteria for strongcoupling is when the energy splitting is larger than the new spectral width $\sqrt{({2g\mathrm{}}^{2}){\mathrm{}({\mathrm{\Gamma}}_{\text{p}}{\mathrm{\Gamma}}_{0}\mathrm{})}^{2}}\mathrm{}>{\mathrm{\Gamma}}_{\text{p}}+{\mathrm{\Gamma}}_{0}$ which corresponds to $g>\sqrt{({\mathrm{\Gamma}}_{\text{p}}^{2}+{\mathrm{\Gamma}}_{0}^{2})/2}$.
In plasmonics, although metal nanostructures provide a large field confinement 1/V and coupling strength g, they suffer from rapid Ohmic dissipation which results in a large Γ_{p}. Consequently, strongcoupling in plasmonics is often achieved through a coherent coupling with a collection of emitters. However, recent experiments by Chikkaraddy et al. [20] shows that certain NPoM nanocavities can provide such a massive field enhancement (∼10^{3}) to sufficiently compensate for its Ohmic dissipation and achieve singlemolecule strongcoupling even at room temperature. Here, we focus on the farfield and nearfield spectral responses of the emitterplasmon strongcoupling in faceted NPoM nanocavities.
4.1 Farfield spectral response
Due to the nanoscale size of plasmonic structures, strongcoupling in this field is often observed via farfield measurements [20,26]. Therefore, we consider a collection of N emitters in the NPoM nanocavity. We place the emitters in a hexagonal lattice with lattice separation 2.5 nm. For N = 1, a single emitter is placed at the nanocavity's center. As more emitters are placed in the nanocavities, the emitters occupy the immediate neighboring sites outwards. The transitional frequency for the emitters is set to the corresponding BDP mode for each facet width of NPoM.
The energy splitting due to the coupling between the BDP and N emitters can be observed as a peak splitting in the scattering crosssection σ_{scat}, shown in Figure 3(a) for an NPoM nanocavity with f = 10 nm. The corresponding energy splitting Δω for facet widths f = 0, 5 and 10 nm are also summarized in Figure 3(b). For small emitter numbers $\sqrt{N}<5$, the NPoM nanocavities with smaller facets provide stronger field enhancements and therefore exhibit larger Δω. As N increases, the energy splitting follows a linear relationship with $\sqrt{N}$, implying that all emitters coherently exchange energy with plasmons. On the other hand, for $\sqrt{N}>10$, Δω becomes saturated for all faceted NPoM nanocavities. The saturation points indicate the extent of the BDP mode volume beyond which further addition of emitters no longer changes the coupling strength. The NPoM nanocavities with larger facets provide larger mode volumes which can accommodate more emitters.
Fig. 3 . (a) Scattering crosssections σ_{scat} of the NPoM with f = 10 nm accommodating N quantum emitters as indicated. Each line is offset by 5πR^{2}. (b) Spectral splitting Δω = ω_{+} − ω_{−} of the hybrid states between N emitters and BDPs of the NPoM with f = 0, 5 and 10 nm. The N emitters are organized in a hexagonal lattice with lattice spacing 2.5 nm. 
4.2 Nearfield spectral response
The scattering crosssections σ_{scat} provides partial information on the nanostructures' plasmonic response. These quantities only reflect the farfield collective behavior of all emitters inside and outside the NPoM nanocavities. In order to gain an insight into the individual coupling of each emitter, we investigate the nearfield response.
The nearfield optical responses of the NPoM nanocavity with N emitter are shown, in Figure 4, as spatiospectral field profile $\mathbf{\text{E}}(x,\lambda )$ evaluated at 1 nm above the mirror and along the xdirection, Figure 2(a), of the nanocavity. For empty (N = 0) nanocavities, the f =10 nm faceted nanocavity exhibits a red shift and lower spatial field confinement compared to f=0 nm, in accordance with the earlier analyses in Figure 1(c) and 3(b). When small numbers of emitters, N = 19, are placed at the center of the nanocavity, Figure 4(b) and (e), the spectral distributions split and show the spatiospectral field profile of the hybrid states. The field distributions also reveal the exact locations of the emitters as sharp, discrete peaks due to the emitters' abilities to absorb and store electromagnetic energy. As N increases from 19 to 183, the nanocavity with f = 0 nm does not show a significant change in its field distribution. By contrast, the f =10 nm nanocavity shows a greater participation of emitters in the coupling. This confirms the analysis in the previous section in which the NPoM nanocavities with larger facets are capable of accommodating more emitters.
Fig. 4 . Spatiospectral field distributions $\mathbf{E}(\mathit{x},\mathit{\lambda})$ of the emitterplasmon hybrid states for (a,d) N = 0, (b,e) N = 19, (c,f) N = 183 quantum emitters and (a,b,c) f = 0 nm and (d,e,f) f = 10 nm. The field distributions are evaluated at 1 nm above the mirror. 
5 Conclusion
In conclusion, we numerically demonstrate how the gap morphology of a NPoM nanocavity affects its interaction with quantum emitters. By increasing the NP's facet width at its contact with the mirror, the field enhancement inside the nanocavity is weakened in exchange for a larger effective volume. The analysis shows that, in the weak coupling regime, a NPoM nanocavity retains its ability to suppress fluorescence quenching of an emitter for facet radius <10 nm. In the strongcoupling regime, we show that a nanocavity with a smaller facet provides a larger field enhancement and couples more strongly with a small number of emitters placed at or close to the center of the nanocavity. On the other hand, a nanocavity with a larger facet can accommodate more emitters and couple more strongly with a large collection of emitters. We envisage numerous applications, including fastemitting singlephoton sources, nonlinear optics, quantum chemistry and quantum technologies.
Acknowledgments
We acknowledge support from EPSRC grants EP/G060649/1 and EP/L027151/1 and European Research Council grant LINASS 320503.
Appendix MaxwellBloch description
In Sections 2 and 3, a classical dipole emitter is used to study the Purcell enhancement effect in the weakcoupling regime. This classical model is not sufficient to study the lightmatter strongcoupling dynamics as we require a model which simultaneously describes the excitation of plasmons and their coupling to emitters. In Section 4, the emitters are considered semiclassically using twolevel MaxwellBloch description with Hamiltonian [26]: $$\stackrel{\u02c6}{H}}={{\displaystyle \stackrel{\u02c6}{H}}}_{0}+{\displaystyle \stackrel{\u02c6}{V}}(t)=\mathrm{\hslash}{\omega}_{0}{{\displaystyle \stackrel{\u02c6}{\sigma}}}^{\mathrm{\u2020}}{\displaystyle \stackrel{\u02c6}{\sigma}}{\displaystyle \stackrel{\u02c6}{\mu}}\cdot E\text{,$$ where $\stackrel{\u02c6}{\sigma}$ denotes the exciton annihilation operator, $\stackrel{\u02c6}{\mathit{\mu}}}\mathrm{}\mathit{\mu}\mathrm{}{{\displaystyle \stackrel{\u02c6}{\sigma}}}^{\mathrm{\u2020}}+{\displaystyle \stackrel{\u02c6}{\sigma}}\mathrm{$ is the transitional dipole moment operator and ω_{0} the transition frequency. The emitter is modeled after the Cy5 molecule with μ = 10.1 D. We can treat the system of an emitter inside a NPoM nanocavity as an open quantum system and express the problem using the density matrix $\stackrel{\u02c6}{\rho}$ approach and Lindblad master equation: $$\frac{\partial {\displaystyle \stackrel{\u02c6}{\rho}}}{\partial t}=\frac{i}{\mathrm{\hslash}}\left[{\displaystyle \stackrel{\u02c6}{H}},{\displaystyle \stackrel{\u02c6}{\rho}}\right]+\frac{{\gamma}_{r}}{2}(2{\displaystyle \stackrel{\u02c6}{\sigma}}{\displaystyle \stackrel{\u02c6}{\rho}}{{\displaystyle \stackrel{\u02c6}{\sigma}}}^{\mathrm{\u2020}}{{\displaystyle \stackrel{\u02c6}{\sigma}}}^{\mathrm{\u2020}}{\displaystyle \stackrel{\u02c6}{\sigma}}{\displaystyle \stackrel{\u02c6}{\rho}}{\displaystyle \stackrel{\u02c6}{\rho}}{{\displaystyle \stackrel{\u02c6}{\sigma}}}^{\mathrm{\u2020}}{\displaystyle \stackrel{\u02c6}{\sigma}})+\frac{{\gamma}_{p}}{2}(2{{\displaystyle \stackrel{\u02c6}{\sigma}}}^{\mathrm{\u2020}}{\displaystyle \stackrel{\u02c6}{\rho}}{\displaystyle \stackrel{\u02c6}{\sigma}}{\displaystyle \stackrel{\u02c6}{\sigma}}{{\displaystyle \stackrel{\u02c6}{\sigma}}}^{\mathrm{\u2020}}{\displaystyle \stackrel{\u02c6}{\rho}}{\displaystyle \stackrel{\u02c6}{\rho}}{\displaystyle \stackrel{\u02c6}{\sigma}}{{\displaystyle \stackrel{\u02c6}{\sigma}}}^{\mathrm{\u2020}})+\frac{{\gamma}_{d}}{2}({{\displaystyle \stackrel{\u02c6}{\sigma}}}_{z}{\displaystyle \stackrel{\u02c6}{\rho}}{{\displaystyle \stackrel{\u02c6}{\sigma}}}_{z}{\displaystyle \stackrel{\u02c6}{\rho}})\text{,}$$where ${{\displaystyle \stackrel{\u02c6}{\sigma}}}_{z}={{\displaystyle \stackrel{\u02c6}{\sigma}}}^{\u2020}{\displaystyle \stackrel{\u02c6}{\sigma}}{\displaystyle \stackrel{\u02c6}{\sigma}}{{\displaystyle \stackrel{\u02c6}{\sigma}}}^{\u2020}$ and γ_{r}, γ_{p} and γ_{d} denote incoherent relaxation rate, incoherent pumping rate and pure dephasing rate respectively. Solving the above equation, we obtain: $$\frac{\partial {\rho}_{22}}{\partial t}=\frac{\partial {\rho}_{11}}{\partial t}=\gamma [{\rho}_{22}{\rho}_{22}^{SS}]\frac{2}{\mathrm{\hslash}}\mathit{\mu}\cdot \mathbf{E}\mathrm{}\text{\hspace{0.17em}}\mathrm{Im}({\rho}_{12})\text{,}$$ $$\frac{\partial {\rho}_{12}}{\partial t}\mathrm{}\frac{\partial {\rho}_{21}^{\mathrm{}}}{\partial t}\mathrm{}\mathrm{}{\mathrm{\Gamma}}_{0}i{\omega}_{0}\mathrm{}{\rho}_{12}+\frac{i}{\mathrm{\hslash}}\mathit{\mu}\cdot E(2{\rho}_{22}1),$$where we define the total relaxation rate γ = γ_{r} + γ_{p} = 0.66 μeV and the total dephasing rate Γ_{0} = γ_{d} + γ/2 =3.98meV. The steadystate excited state population ${\rho}_{22}^{SS}={\gamma}_{p}/({\gamma}_{r}+{\gamma}_{p})$ is negligible in our system where γ_{p} ≪ γ_{r} at an optical frequency and room temperature.
In our FDTD calculations, the simulation space is divided into a grid, and the plasmons (Efield) in each grid cell is obtained by solving the Maxwell equations. The emitters are driven by the plasmons and inject photons back to the simulation through a macroscopic polarization $\mathbf{P}={N}_{\mathrm{d}}\mathrm{T}\mathrm{r}({\displaystyle \stackrel{\u02c6}{\rho}}{\displaystyle \stackrel{\u02c6}{\mathit{\mu}}})=2{N}_{\mathrm{d}}{\displaystyle \stackrel{\u02c6}{\mathit{\mu}}}\mathrm{Re}({\rho}_{12})$, where N_{d} is the total carrier density. The dynamic equations for ρ_{ij} can then be expressed in terms of the polarization P, ground state population N_{1} = N_{d} ρ_{11} and excited state population N_{2} = N_{d} ρ_{22} [27]: $$\frac{{\partial}^{2}P}{\partial {t}^{2}}+2{\mathrm{\Gamma}}_{0}\frac{\partial P}{\partial t}+({\mathrm{\Gamma}}_{0}^{2}+{\omega}_{0}^{2})\mathbf{P}=\frac{2{\omega}_{0}}{\mathrm{\hslash}}{\mu}^{2}({N}_{2}{N}_{1})\mathbf{E}(t)$$ $$\frac{\partial {N}_{2}}{\partial t}=\frac{\partial {N}_{1}}{\partial t}=\gamma {N}_{2}+\frac{1}{\mathrm{\hslash}{\omega}_{0}}\left(\frac{\partial P}{\partial t}+{\mathrm{\Gamma}}_{0}\mathbf{P}\right)\cdot \mathbf{E}(t)\text{.}$$
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Cite this article as: Nuttawut Kongsuwan, Angela Demetriadou, Rohit Chikkaraddy, Jeremy J. Baumberg, Ortwin Hess, Fluorescence enhancement and strongcoupling in faceted plasmonic nanocavities, EPJ Appl. Metamat. 5, 6 (2018)
All Figures
Fig. 1 . (a) Schematic diagram of a nanoparticleonmirror (NPoM) nanocavity with diameter 2R = 80 nm, facet diameter 2f and gap size d = 5 nm. (b) The field enhancement distribution $\left\mathbf{\text{E}}\right/\left{\mathbf{\text{E}}}_{0}\right$ of the bonding dipole plasmon (BDP) of the NPoM with f = 10 nm (c) Scattering crosssection σ_{scat} for the NPoM with f = 0, 5 and 10 nm and for nanoparticles (NPs) on plain dielectric substrates with f = 0 and 10 nm. 

In the text 
Fig. 2 . (a) The radiative decay ${{\displaystyle \tilde{\gamma}}}_{\text{rad}}$, (b) nonradiative decay ${{\displaystyle \tilde{\gamma}}}_{\text{nr}}$, (c) excitation ${{\displaystyle \tilde{\gamma}}}_{\text{exc}}$ and (d) fluorescence enhancements ${{\displaystyle \tilde{\gamma}}}_{\text{em}}$ for the NPoM nanocavities with facet diameter f = 0, 5 and 10 nm and the NPs on dielectric substrates with f = 0 and 10 nm. For clarity, ${{\displaystyle \tilde{\gamma}}}_{\text{rad}}$, ${{\displaystyle \tilde{\gamma}}}_{\text{exc}}$ and ${{\displaystyle \tilde{\gamma}}}_{\text{em}}$ for the NPs are multiplied by the indicated factors. 

In the text 
Fig. 3 . (a) Scattering crosssections σ_{scat} of the NPoM with f = 10 nm accommodating N quantum emitters as indicated. Each line is offset by 5πR^{2}. (b) Spectral splitting Δω = ω_{+} − ω_{−} of the hybrid states between N emitters and BDPs of the NPoM with f = 0, 5 and 10 nm. The N emitters are organized in a hexagonal lattice with lattice spacing 2.5 nm. 

In the text 
Fig. 4 . Spatiospectral field distributions $\mathbf{E}(\mathit{x},\mathit{\lambda})$ of the emitterplasmon hybrid states for (a,d) N = 0, (b,e) N = 19, (c,f) N = 183 quantum emitters and (a,b,c) f = 0 nm and (d,e,f) f = 10 nm. The field distributions are evaluated at 1 nm above the mirror. 

In the text 
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