Enhanced quantum coherence of plasmonic resonances with a chiral exceptional points

Abstract

While strategies to enhance the quantum coherence of plasmonic resonances have attracted a lot of attention in the past, the advent of non-Hermitian optics carries promising possibilities in this direction, mostly of which are still unexplored. In this work, we show that the quantum coherence of plasmonic resonances can be enhanced by integrating a plasmonic antenna in a photonic cavity operated at a chiral exceptional point (CEP), where the phase of light offers an additional degree of freedom for flexibly manipulating the quantum dynamics. The few-mode quantization theory is employed to demonstrate the advantages and related quantum-optics applications of the proposed hybrid cavity in both off- and on-resonance plasmon-photon coupling. For the former case, the local density of states evolves into sub-Lorentzian lineshape, resulting in reduced dissipation of polaritonic states. On resonance, we identify two mechanisms improving the quantum yield by two orders of magnitude at room temperature: the reduction of plasmonic absorption through Fano interference and the enhancement of cavity radiation through superscattering. Our results establish CEP-engineered plasmonic resonances as a promising platform for controlling quantum states and building high-performance quantum devices.

Introduction

Plasmonic nanocavities have achieved great success in realizing the room-temperature strong coupling between the localized surface plasmon resonance and a few or even single quantum emitter (QE) at the nanoscale^1,2,3,4. This leads to the formation of hybrid plasmon-QE states termed plexcitons, which enables a rich variety of advanced quantum technologies, such as ultrafast single-photon source^5,6,7, single-qubit control^8,9,10, and ultrasensitive quantum sensing^11,12. Nevertheless, to move toward practical applications, plexcitonic systems still call for further engineer to prolong the coherence time, which is severely limited by the large Ohmic loss of metals. To overcome this challenge, the oscillating photonic building blocks like photonic crystal and whispering-gallery-mode (WGM) cavities, have been employed to reduce the plasmonic dissipation while simultaneously produce larger cooperativity of light-matter interaction^{13,14,15,16,17,18,20,21,22}. This attributes to the constructive interference of scattering paths in hybrid cavity with red-detuned plasmon-photon coupling, which demonstrates the unique benefit of higher Purcell factor than separate entities^14,15,18,23. Such plasmonic-photonic hybrids also emerge as one of successful approaches to manipulate quantum states. In this respect, on resonance the plasmonic absorption can be suppressed by exploiting the principle of bound states in the continuum²⁴ and as a result, dressed bound states with anomalously small decay have been found²⁵. Therefore, previous studies have shown the ability of photonic elements to tailor the electromagnetic environment of localized surface plasmon resonance (LSPR) and modify the plasmon-QE interaction.

Leakage of optical resonators, often considered detrimental to quantum devices, is inevitable but at the same time offers alternative avenues for light manipulation by virtue of the non-Hermitian degeneracies, known as exceptional points (EPs)^26,27,28. With the flourish of non-Hermitian quantum mechanics, the dimensionality reduction of state space at EPs arises as a new strategy to steer the quantum states and spurs the exploration of quantum-optics applications, including tunable photon statistics^29,30, enhanced sensitivity of quantum sensing^31,32,33,34, and emission control of quantum light^35,36,37, to name a few. Nevertheless, the isolated EPs are very sensitive to the external perturbations, this feature hinders the access to EPs in realistic quantum systems. For this reason, extending isolated EPs to higher dimension has emerged as an alternative for the observation of related phenomena^38,39,40,41. Particularly, a chiral EP (CEP), which is a two-dimensional collection of second-order EPs in parameter space, can be implemented in WGM cavity by introducing the chiral coupling between the degenerate counterclockwise (CCW) and clockwise (CW) modes^40,42,43,44. This configuration exhibits a fascinating feature that a CEP hosting in WGM cavity is immune to the inevitable experimental uncertainties and fabrication imperfections, and hence can offer a robust platform for exploring the non-Hermitian physics.

Inspired by these pioneering works, we propose to enhance the coherent light-matter interaction by engineering the local density of states (LDOS) of a plasmonic nanocavity through the integration with a WGM photonic cavity featuring CEP, which we call hybrid CEP cavity hereafter. A full quantum model is first built to formulate the LDOS of hybrid CEP cavity, which allows to obtain the analytical expression of LDOS through the frequencies, decay rates and coupling rate of uncoupled plasmonic and photonic modes. It reveals that the magnitude and linewidth of LDOS is related to the phase of light field at CEP, and hence the LDOS of hybrid mode can be flexibly tuned to enhance while at the same time the linewidth is greatly reduced. It also reveals the mechanisms of order-of-magnitude improvement of quantum yield at CEP, by either reducing the plasmonic absorption or enhancing the photonic radiation. Combined with the electromagnetic simulations of a simple design of hybrid CEP cavity, we analyze the enhanced light-matter interaction and quantum yield by varying the parameters of structure geometry, from which we predict that a high quantum yield $\sim 0.9$ at room temperature is attainable in realistic structures with proper design. We also summarize the design rules for hybrid CEP cavity.

Methods

Model and theory

The hybrid CEP cavity under investigation is schematically sketched in Fig. 1a, which consists of a usual hybrid plasmonic-photonic cavity and a semi-infinite waveguide evanescently coupled to a pair of degenerate WGM modes with opposite propagating direction, i.e., the CW and CCW modes. The mirror at the right end of waveguide results in the presence of a CEP due to the chiral coupling from CCW mode to CW mode^34,40,42, which can be found from the coupled-mode equation of field amplitudes without the QE and plasmonic mode $\frac{d}{{dt}}\left(\begin{array}{c}{a}_{{cw}}\\ {a}_{{ccw}}\end{array}\right)=-i{M}_{{cep}}\left(\begin{array}{c}{a}_{{cw}}\\ {a}_{{ccw}}\end{array}\right)$, with ${M}_{{cep}}{{{{{\boldsymbol{=}}}}}}\left[\begin{array}{cc}{\omega }_{c}-i{\kappa }_{c}/2 & 0\\ -i{\kappa }_{c}{e}^{i\phi } & {\omega }_{c}-i{\kappa }_{c}/2\end{array}\right]$, where the mirror is assumed to have unity reflectivity. ${a}_{{cw}}$ (${a}_{{ccw}}$) stands for the field amplitude of CW (CCW) mode with resonant frequency ${\omega }_{c}$ and ${\kappa }_{c}$ is the decay rate of cavity modes to waveguide. The roundtrip phase factor $\phi =2\beta L$ is induced by the light field of CCW mode propagating from the waveguide-cavity junction to the mirror, with $\beta$ and $L$ being the propagation constant of the waveguide and the distance between the cavity-waveguide junction and the mirror, respectively. The eigenvalues and the associated eigenvectors of ${M}_{{cep}}$ are ${\omega }_{1,2}={\omega }_{c}-i{\kappa }_{c}/2$ and ${{{{{{\boldsymbol{v}}}}}}}_{1,2}={\left({{{{\mathrm{0,1}}}}}\right)}^{T}$, respectively, where the two identical eigenmodes with a null CW component indicate the formation of a CEP⁴⁰. By integrating WGM resonator with a plasmonic antenna, CEP can engineer the plasmonic resonance by tuning LDOS, which is a measure of how efficiently a QE transfers energy to the surrounding electromagnetic environment and hence plays a central role in determining the quantum light-matter interaction⁴⁵.

Fig. 1: Model of the proposed hybrid CEP cavity and the few-mode quantization for local density of states.

a Schematic diagram of the proposed hybrid CEP cavity based on the whispering-gallery mode (WGM) cavity, which supports the degenerate clockwise (CW) and counterclockwise (CCW) modes with unidirectional coupling provided by the mirror. Therefore, the hybrid CEP cavity is different from a usual hybrid cavity without CEP where two WGM modes are uncoupled. ${\kappa }_{c}$ is the decay of WGM modes induced by waveguide and $L$ is the distance between the cavity-waveguide junction and the mirror. b Conceptual sketch of the map** of the local density of states (LDOS) in a hybrid cavity into a quantized few-mode model, where the cavity QED system can be decomposed into a non-Markovian core and the Markovian bath. The gray dashed lines indicate the structured environment of hybrid cavity which is constituted of continuous electromagnetic modes. c Normalized LDOS of a hybrid CEP cavity for various $\phi$ (solid lines). The inset shows the logarithmic plot of the normalized LDOS versus frequency and $\phi$. Normalized LDOS of the usual hybrid cavity is shown for comparison (dashed black line). The parameters are ${\kappa }_{i}={\gamma }_{{{{{{\rm{p}}}}}}}=0$, ${g}_{1}=-20{{{{{\rm{meV}}}}}}$, ${g}_{a}=10{{{{{\rm{meV}}}}}}$, and the plasmon-photon detuning ${\Delta }_{{ac}}={\omega }_{a}-{\omega }_{c}=1{{{{{\rm{eV}}}}}}$. The quality factors of dipolar plasmonic antenna and WGM modes are ${Q}_{a}=18$ and ${Q}_{c}=2\times {10}^{3}$, respectively.

When interacting with a QE, the LDOS of a hybrid CEP cavity in general manifests non-Lorentzian lineshape and can be described as a structured electromagnetic environment that constitutes continuous bosonic modes in quantized representation⁴⁶. On the other hand, the non-Lorentzian LDOS stems from the coupling of plasmonic and photonic resonances, and hence can be generated using the parameters of individual resonances and the coupling rate between them. With this prospect, we develop a quantized LDOS theory for hybrid CEP cavity in this section based on the ideal of few-mode quantization^47,48,49, as Fig. 1b depicts.

Specifically, we consider a dipolar plasmonic antenna where the nonradiative higher-order modes are well separated from the dipolar mode, so that the plasmonic antenna can be treated as a single-mode resonator with resonance frequency ${\omega }_{a}$. This approximation is valid for many plasmonic antennas, such as the elongated rods⁴ and dimer structures^23,50. The dipolar plasmonic mode is coupled to both CCW and CW modes with equal coupling rate ${g}_{1}$. Accordingly, we can obtain a quantized few-mode model of hybrid CEP cavity where the continuous bosonic environment is decomposed into a non-Markovian core and a Markovian bath^{25,47,48,49,51}, as illustrated in Fig. 1(b). The non-Markovian core encodes the interactions between the QE, the photonic and plasmonic modes, while the Markovian bath accounts for the dissipation from photonic and plasmonic modes to the environment. As proven in previous studies^47,48,49,48,, the equivalence of two models lies in the fact that the few-mode model can yield the same reduced quantum dynamics as the original system [the general model in Fig. 1b] if their LDOS are identical. Based on the quantized few-mode model, an extended cascaded quantum master equation of a two-level QE interacting with the proposed hybrid CEP cavity is derived in Supplementary Note 1, and employed to describe the quantum dynamics of the composed cavity QED system

$$\dot{\rho }=-i\left[H,\rho \right]{{+}}{{{{{\mathcal{D}}}}}}\left[\rho \right]$$

(1)

where the total Hamiltonian reads

$$H={H}_{0}+{H}_{I}$$

(2)

with the free Hamiltonian

$${H}_{0}={\omega }_{a}{a}^{{{\dagger}} }a+{\omega }_{c}{c}_{{ccw}}^{{{\dagger}} }{c}_{{ccw}}+{\omega }_{c}{c}_{{cw}}^{{{\dagger}} }{c}_{{cw}}+{\omega }_{0}{\sigma }_{+}{\sigma }_{-}$$

(3)

and the interaction Hamiltonian

$${H}_{I}= \left({g}_{1,\theta }{c}_{{ccw}}^{{{\dagger}} }a+{g}_{1,\theta }^{* }{a}^{{{\dagger}} }{c}_{{ccw}}\right)+\left({g}_{1,\theta }^{* }{c}_{{cw}}^{{{\dagger}} }a+{g}_{1,\theta }{a}^{{{\dagger}} }{c}_{{cw}}\right)+{g}_{a}\left({a}^{{{\dagger}} }{\sigma }_{-}+{\sigma }_{+}a\right) \\ +\left({g}_{c,\theta }{c}_{{ccw}}^{{{\dagger}} }{\sigma }_{-}+{g}_{c,\theta }^{* }{\sigma }_{+}{c}_{{ccw}}\right)+\left({g}_{c,\theta }^{* }{c}_{{cw}}^{{{\dagger}} }{\sigma }_{-}+{g}_{c,\theta }{\sigma }_{+}{c}_{{cw}}\right)$$

(4)

where $a$ and ${c}_{{ccw}}$ (${c}_{{cw}}$) are the bosonic annihilation operators for dipolar plasmonic mode and CCW (CW) mode, respectively, while ${\sigma }_{-}$ is the lowering operator of QE with transition frequency ${\omega }_{0}$. ${g}_{1,\theta }={g}_{1}{e}^{i\theta },{g}_{a}$ and ${g}_{c,\theta }={g}_{c}{e}^{i\theta }$ denote the complex coupling rates for plasmon-photon, plasmon-QE and photon-QE interactions, respectively, where $\theta$ is the angular position of plasmonic antenna and QE. The plasmonic antenna and QE are placed at the point perpendicular to the waveguide direction and bisects the WGM cavity so that $\theta =0$ and the coupling rates ${g}_{1,\theta }$ and ${g}_{c,\theta }$ can be taken as real numbers^52,53, i.e., ${g}_{1,\theta }={g}_{1}$ and ${g}_{c,\theta }={g}_{c}$. For a small distance between the direct QE and plasmonic antenna, ${g}_{a}$ is also a real number since the coupling of QE to the plasmonic mode is dominant over the coupling mediated through the free-space continuum^14,15,54. It should be emphasized that the bare (uncoupled) modes are considered in the quantized few-mode model, while the ‘plasmon-like’ and ‘photon-like’ hybrid modes are involved in the quasi-normal mode approach^54,55,56, where the diagonalization gives rise to complex coupling rates. In the later part of the work, we show that the few-mode model is accurate at predicting the LDOS of a realistic structure due to the weak Fano effect in the proposed hybrid CEP cavity.

The Lindblad operator ${{{{{\mathcal{D}}}}}}\left[\rho \right]$ in Eq. (1) is given by

$${{{{{\mathcal{D}}}}}}\left[\rho \right]= \;{\gamma }_{p}{{{{{\mathcal{L}}}}}}\left[{\sigma }_{z}\right]\rho +{\gamma }_{0}{{{{{\mathcal{L}}}}}}\left[{\sigma }_{-}\right]\rho +{\kappa }_{a}{{{{{\mathcal{L}}}}}}\left[a\right]\rho +\kappa \left({{{{{\mathcal{L}}}}}}\left[{c}_{{cc}w}\right]\rho {{+}}{{{{{\mathcal{L}}}}}}\left[{c}_{{cw}}\right]\rho \right) \\ +{\kappa }_{c}\left({e}^{i\phi }\left[{c}_{{ccw}}\rho ,{c}_{{cw}}^{{{\dagger}} }\right]+{e}^{-i\phi }\left[{c}_{{cw}},\rho {c}_{{ccw}}^{{{\dagger}} }\right]\right)$$

(5)

where ${{{{{\mathcal{L}}}}}}\left[O\right]\rho =O\rho {O}^{{{\dagger}} }-\left\{{O}^{{{\dagger}} }O,\rho \right\}/2$ is the Liouvillian superoperator for the dissipation of operator $O$. The first line of Eq. (5) introduces the dissipation for individuals, where ${\sigma }_{z}={\sigma }_{+}{\sigma }_{-}-{\sigma }_{-}{\sigma }_{+}$. ${\gamma }_{p}$ accounts for the dephasing rate of QE, which is typically $10-20{{{{{\rm{meV}}}}}}$ for QEs at room temperature^57,58. ${\gamma }_{0}$ is the free-space emission rate of QE. ${\kappa }_{a}={\kappa }_{o}+{\kappa }_{r}$ stands for the total decay rate of dipolar plasmonic mode, with ${\kappa }_{o}$ and ${\kappa }_{r}$ being the nonradiative and radiative decay rates, respectively. $\kappa ={\kappa }_{c}+{\kappa }_{i}$ denotes the total decay rate of WGM modes, where ${\kappa }_{c}$ stems from the evanescent coupling of cavity to the guided mode of waveguide, which can be tuned by adjusting the cavity-waveguide separation. ${\kappa }_{i}$ is the intrinsic decay of CCW and CW modes resulting from the material absorption and the coupling of cavity to the modes other than the guided modes of waveguide. In the following discussion, we first omit the intrinsic decay ${\kappa }_{i}$ in consideration of the high-Q feature of WGM modes, but retrieve it when analyzing the quantum yield of realistic structures. The second line of Eq. (5) describes the cascaded interaction between the CW and CCW modes, i.e., the CW mode is driven by the output field from the CCW mode through the mirror reflection.

For the cases of spontaneous emission (SE) and weak drive studied in this work, we can apply the factorization $\left\langle {\sigma }_{z}O\right\rangle \, \approx \, \left\langle {\sigma }_{z}\right\rangle \left\langle O\right\rangle$ and the equations of motion can be linearized by the approximation $\left\langle {\sigma }_{z}\right\rangle \, \approx \, -1$^59,60$.$ From Eqs. (1)–(5), we can obtain the equation of motion $\frac{d}{{dt}}{{{{{{\boldsymbol{p}}}}}}}_{{{{{{\boldsymbol{0}}}}}}}=-i{M}_{0}{{{{{{\boldsymbol{p}}}}}}}_{{{{{{\boldsymbol{0}}}}}}}$, with ${{{{{{\boldsymbol{p}}}}}}}_{{{{{{\boldsymbol{0}}}}}}}={\left[\left\langle {\sigma }_{-}\right\rangle ,\left\langle a\right\rangle ,\left\langle {c}_{{ccw}}\right\rangle ,\left\langle {c}_{{cw}}\right\rangle \right]}^{T}$ and the matrix ${M}_{0}$ being

$${M}_{0}{{{{{\boldsymbol{=}}}}}}\left[\begin{array}{cccc}{\omega }_{0}-i \frac{\gamma }{2} & {g}_{a} & {g}_{c} & {g}_{c}\\ {g}_{a} & {\omega }_{a}-i\frac{{\kappa }_{a}}{2} & {g}_{1} & {g}_{1}\\ {g}_{c} & {g}_{1} & {\omega }_{c}-i\frac{\kappa }{2} & 0\\ {g}_{c} & {g}_{1} & -i{\kappa }_{c}{e}^{i\phi } & {\omega }_{c}-i\frac{\kappa }{2}\end{array}\right]$$

(6)

where $\gamma ={\gamma }_{0}+2{\gamma }_{p}$. We then change the basis of WGM modes into the representation of standing wave modes⁵³ by substituting ${c}_{1}=\left({c}_{{cw}}+{c}_{{ccw}}\right)/\sqrt{2}$ and ${c}_{2}=\left({c}_{{cw}}-{c}_{{ccw}}\right)/\sqrt{2}$ into Eq. (6), and obtain $d{{{{{\boldsymbol{p}}}}}}/{dt}=-i{M}_{p}{{{{{\boldsymbol{p}}}}}}$ with ${{{{{\boldsymbol{p}}}}}}={\left[\left\langle {\sigma }_{-}\right\rangle ,\left\langle a\right\rangle ,\left\langle {c}_{1}\right\rangle ,\left\langle {c}_{2}\right\rangle \right]}^{T}$. The matrix ${M}_{p}$ is given by

$${M}_{p}{{{{{\boldsymbol{=}}}}}}\left[\begin{array}{cccc}{\omega }_{0}-i\frac{{\gamma }_{0}}{2} & {g}_{a} & \sqrt{2}{g}_{c} & 0\\ {g}_{a} & {\omega }_{a}-i\frac{{\kappa }_{a}}{2} & \sqrt{2}{g}_{1} & 0\\ \sqrt{2}{g}_{c} & \sqrt{2}{g}_{1} & {\omega }_{c}-i\frac{{\kappa }_{+}}{2} & i\frac{{\kappa }_{c}}{2}{e}^{i\phi }\\ 0 & 0 & -i\frac{{\kappa }_{c}}{2}{e}^{i\phi } & {\omega }_{c}-i\frac{{\kappa }_{-}}{2}\end{array}\right]$$

(7)

where ${\kappa }_{\pm }={\kappa }_{i}+{\kappa }_{c}\left(1\pm {e}^{i\phi }\right).$ We can see that without the mirror (i.e., ${\kappa }_{c}=0$), the standing wave mode ${c}_{2}$ becomes a dark (uncoupled) mode and Eq. (7) returns to the single-mode treatment of WGM cavity that is widely used in the literatures^61,62,63. In the presence of mirror, ${c}_{2}$ is still decoupled from the QE and plasmon, which can simplify the subsequent derivation of LDOS. What is more important, in this case the decay and the coupling of two standing wave modes can be tuned by $\phi$, indicating the key role of CEP in controlling the light-matter interaction.

LDOS of hybrid CEP cavity

We consider the normalized LDOS, which is equal to Purcell factor⁶⁴ and linked to the spectral density $J\left(\omega \right)$ through the relation $P\left(\omega \right)=J\left(\omega \right)/{J}_{0}\left(\omega \right)$, with ${J}_{0}\left(\omega \right)={\omega }^{3}{\mu }^{2}/6{\pi }^{2}\hslash {\varepsilon }_{0}{c}^{3}$⁶⁵ being the spectral density of a QE with dipole moment μ in the free space, where $c$ is the speed of light and ${\varepsilon }_{0}$ is the permittivity of vacuum. The spectral density of hybrid CEP cavity is given by $J\left(\omega \right)={{{{\mathrm{Re}}}}}[{i\chi }_{{sys}}\left(\omega \right)]={{{{\mathrm{Re}}}}}{\int }_{\!\!-\infty }^{+\infty }d \, \tau {e}^{i\omega \tau }\left\langle {\Lambda }^{{{\dagger}} }\left(0\right)\Lambda \left(\tau \right)\right\rangle$⁴⁹, with $\Lambda \left(t\right)={g}_{{{{{{\rm{a}}}}}}}a\left(t\right)+\sqrt{2}{g}_{{{{{{\rm{c}}}}}}}{c}_{1}\left(t\right)$, where ${\chi }_{{sys}}\left(\omega \right)$ defines the polarizability of hybrid CEP cavity. We can see that the spectral density can be separated into three parts, $J\left(\omega \right)={J}_{a}\left(\omega \right)+{J}_{c}\left(\omega \right)+{J}_{ac}\left(\omega \right)$, with ${J}_{a}\left(\omega \right)={g}_{a}^{2}{{{{\mathrm{Re}}}}}\left\{{{{{{\mathcal{F}}}}}}\left[\left\langle {a}^{{{\dagger}} }\left(0\right)a\left(\tau \right)\right\rangle \right]\right\}$ and ${J}_{c}\left(\omega \right)=2{g}_{c}^{2}{{{{\mathrm{Re}}}}}\{{{{{{\mathcal{F}}}}}}[\langle {c}_{1}^{{{\dagger}} }\left(0\right){c}_{1}\left(\tau \right)\rangle ]\}$ being the modified plasmon and cavity responses, respectively, while ${J}_{ac}\left(\omega \right)=2\sqrt{2}{g}_{{{{{{\rm{a}}}}}}}{g}_{{{{{{\rm{c}}}}}}}{{{{\mathrm{Re}}}}}\left\{{{{{{\mathcal{F}}}}}}\left[\left\langle {a}^{{{\dagger}} }\left(0\right){c}_{1}\left(\tau \right)\right\rangle \right]\right\}$ containing the plasmon-photon interference, where ${{{{{\mathcal{F}}}}}}\left[\cdot \right]$ represents the Fourier transform. The two-time correlation functions $\left\langle {a}^{{{\dagger}} }\left(0\right)a\left(\tau \right)\right\rangle$, $\langle {c}_{1}^{{{\dagger}} }\left(0\right){c}_{1}\left(\tau \right)\rangle$ and $\left\langle {a}^{{{\dagger}} }\left(0\right){c}_{1}\left(\tau \right)\right\rangle$ can be calculated using the quantum regression theorem⁶⁵. Taking $\left\langle {a}^{{{\dagger}} }\left(0\right)a\left(\tau \right)\right\rangle$ as an example, its dynamics follows the equation $\frac{d}{d\tau }\left\langle {a}^{{{\dagger}} }\left(0\right){{{{{\boldsymbol{c}}}}}}\left(\tau \right)\right\rangle =-i{M}_{s}\left\langle {a}^{{{\dagger}} }\left(0\right){{{{{\boldsymbol{c}}}}}}\left(\tau \right)\right\rangle$, where ${{{{{\boldsymbol{c}}}}}}\left(\tau \right)={\left[a\left(\tau \right),{c}_{1}\left(\tau \right),{c}_{2}\left(\tau \right)\right]}^{T}$. The matrix ${M}_{s}$ takes the form

$${M}_{s}{{{{{\boldsymbol{=}}}}}}\left[\begin{array}{ccc}{\omega }_{a}-i\frac{{\kappa }_{a}}{2} & \sqrt{2}{g}_{1} & 0\\ \sqrt{2}{g}_{1} & {\omega }_{c}-i\frac{{\kappa }_{+}}{2} & i\frac{{\kappa }_{c}}{2}{e}^{i\phi }\\ 0 & -i\frac{{\kappa }_{c}}{2}{e}^{i\phi } & {\omega }_{c}-i\frac{{\kappa }_{-}}{2}\end{array}\right]$$

(8)

With initial condition $\left\langle {a}^{{{\dagger}} }\left(0\right)a\left(0\right)\right\rangle =1$, $\left\langle {a}^{{{\dagger}} }\left(0\right){c}_{1}\left(0\right)\right\rangle =0$ and $\left\langle {a}^{{{\dagger}} }\left(0\right){c}_{2}\left(0\right)\right\rangle =0$, the equation can be solved through the Fourier transform. Other correlation functions can be obtained with the similar fashion. The analytical expressions of each component are as follows (see Supplementary Note 2 for detailed derivation):

$${J}_{\!\!a}\left(\omega ,\phi \right)=-{g}_{a}^{2}{{{{{\rm{I}}}}}}{{{{{\rm{m}}}}}}\left[\frac{i{\chi }_{a}\left(\omega \right)}{1-{g}_{1}^{2}{\chi }_{a}\left(\omega \right){\chi }_{{EP}}\left(\omega ,\phi \right)}\right]$$

(9)

$${J}_{\!\!c}\left(\omega ,\phi \right)=-{g}_{c}^{2}{{{{{\rm{Im}}}}}}\left[{\chi }_{c}\left(\omega \right)\frac{1-i\frac{{\kappa }_{c}}{2}{e}^{i\phi }{\chi }_{c}\left(\omega \right)}{1-{g}_{1}^{2}{\chi }_{a}\left(\omega \right){\chi }_{{EP}}\left(\omega ,\phi \right)}\right]$$

(10)

$${J}_{\!\!ac}\left(\omega ,\phi \right)=-\sqrt{2}{g}_{\!a}{g}_{\!c}{{{{{\rm{Im}}}}}}\left[\sqrt{2}{g}_{1}{\chi }_{a}\left(\omega \right){\chi }_{c}\left(\omega \right)\frac{1-i\frac{{\kappa }_{c}}{2}{e}^{i\phi }{\chi }_{c}\left(\omega \right)}{1-{g}_{1}^{2}{\chi }_{a}\left(\omega \right){\chi }_{{EP}}\left(\omega ,\phi \right)}\right]$$

(11)

where the polarizabilities of uncoupled plasmonic antenna and CEP cavity are characterized by ${\chi }_{a}\left(\omega \right)={\left[\omega -{\omega }_{a}+i{\kappa }_{a}/2\right]}^{-1}$ and ${\chi }_{{EP}}\left(\omega ,\phi \right)=2{\chi }_{c}\left(\omega \right)-i{\kappa }_{c}{{{{{{\rm{e}}}}}}}^{{{{{{\rm{i}}}}}}\phi }{\chi }_{c}^{2}\left(\omega \right)$, respectively, with ${\chi }_{c}\left(\omega \right)={\left[\omega -{\omega }_{c}+i{\kappa }_{c}/2\right]}^{-1}$ being the polarizability of WGM modes. Both $-{{{{{\rm{I}}}}}}{{{{{\rm{m}}}}}}\left[{\chi }_{a}\left(\omega \right)\right]$ and $-{{{{{\rm{I}}}}}}{{{{{\rm{m}}}}}}\left[{\chi }_{c}\left(\omega \right)\right]$ present the conventional Lorentzian lineshape, while the squared Lorentzian term [${{{{{{\rm{e}}}}}}}^{{{{{{\rm{i}}}}}}\phi }{\chi }_{c}^{2}\left(\omega \right)$] in ${\chi }_{{EP}}\left(\omega ,\phi \right)$ is a hallmark of second-order EP, where the phase $\phi$ originating form CEP provides an extra degree of freedom to tune LDOS. ${J}_{a}\left(\omega ,\phi \right)$ and ${J}_{c}\left(\omega ,\phi \right)$ stand for the modified plasmon and cavity responses, while ${J}_{ac}\left(\omega ,\phi \right)$ is the cross term corresponding to plasmon-cavity interference, see Supplementary Fig. 2 for illustrating the physical meanings of three components of spectral density. We notice that there are other methods of obtaining the spectral density of hybrid CEP cavity, e.g., the general Fano model⁶⁶ and the recently proposed formalism of few-mode quantization⁴⁸, which can yield the same results as Eqs. (9)–(11), see Supplementary Note 2 for detailed discussion. Therefore, the validity of analytical spectral density $J\left(\omega \right)$ has been confirmed by different models. To clarify the role of CEP in engineering the plasmonic resonance, in the following discussion we take ${g}_{{{{{{\rm{c}}}}}}}=0$ unless special noted.

We first consider the case of red-detuned plasmon-photon interaction, with coupling rates ${g}_{1}=-20{{{{{\rm{meV}}}}}}$ and ${g}_{a}=10{{{{{\rm{meV}}}}}}$. This value of ${g}_{1}$ is comparable to the theoretical values previously reported in the literature^25,48, while ${g}_{a}=10{{{{{\rm{meV}}}}}}$ is far below the achievable plasmon-QE coupling rate in experiments, which can exceed $40{{{{{\rm{meV}}}}}}$ for molecule QEs^1,4,16 and is over $100{{{{{\rm{meV}}}}}}$ for semiconductor quantum dots^3,67. Therefore, the aforementioned parameters are attainable in realistic structures. The normalized LDOS of hybrid CEP cavity versus frequency is in the inset of Fig. 1c, where it demonstrates distinct lineshapes as $\phi$ varies. Figure 1c shows this dependence of normalized LDOS on $\phi$ more explicitly. We can see that the frequency, magnitude, and linewidth of normalized LDOS varies greatly with $\phi$. For example, two peaks appear in the normalized LDOS for $\phi \sim 0$, with a dip around cavity resonance resulting from the deconstructive interference between the CCW and CW modes at CEP. On the contrary, for $\phi ={{{{{\rm{\pi }}}}}}$ a single peak is presented close to the resonance frequency of hybrid cavity without CEP, but with a linewidth narrower than the corresponding Lorentzian function with the same maximum, demonstrating a sub-Lorentzian lineshape. In particular, the normalized LDOS reaches the maximum with $\phi \sim 3{{{{{\rm{\pi }}}}}}/4$, accompanied by an eightfold enhancement and order-of-magnitude linewidth narrowing compared to that of hybrid cavity without CEP. This is explained by the complex constructive interference of cavity modes at CEP. It should be emphasized that in hybrid CEP cavity, the lineshape of LDOS is no longer Lorentzian, in stark contrast to the hybrid cavity where the normalized LDOS can still be well approximated by Lorentzian function in the case of far red-detuned plasmon-photon coupling, which is a special case of Fano resonance with a large Fano parameter q^14,18,68$.$

Distinguished from the lower quality factor (Q) of plasmonic antenna, which is ${Q}_{a}\equiv {\omega }_{a}/{\kappa }_{a}=10 \sim 20$ for common structures^1,2,4,16,50, the WGM cavity covers a wide range of Q varying from several hundred to million^62,69,70, according to the different materials and structure geometries. Therefore, in Fig. 2a–d we investigate the LDOS enhancement ${F}_{L}/{F}_{L}^{0}$ and the corresponding linewidth narrowing ${\Gamma }_{{EP}}/{\Gamma }_{{EP}}^{0}$ of hybrid CEP cavity versus the quality factor ${Q}_{c}={\omega }_{c}/\kappa$ of WGM modes and ${g}_{1}$, respectively, where ${F}_{L}=\max \left[J\left(\omega \right)/{J}_{0}\left(\omega \right)\right]$ is the maximal normalized LDOS and ${\Gamma }_{{EP}}$ is the linewidth of hybrid CEP cavity, while the superscript ‘0’ denotes the corresponding quantities of hybrid cavity without CEP. The results exhibit two important features. One is that the LDOS enhancement can be always achieved in the chosen parameter ranges, as Fig. 2a, c show, and a maximum of eightfold (tenfold) enhancement can be obtained for arbitrary ${Q}_{c}$ with plasmon-photon detuning ${\Delta }_{{ac}}={\omega }_{a}-{\omega }_{c}=1{{{{{\rm{eV}}}}}}$ ($1.4{{{{{\rm{eV}}}}}}$) and an optimal ${g}_{1}$, which is denoted as ${g}_{1}^{{{{{{\rm{opt}}}}}}}$ and indicated by the white dashed line in Fig. 2a–d. However, a smaller ${Q}_{c}$ requires stronger plasmon-photon interaction to achieve the maximal LDOS enhancement. On the other hand, Fig. 2a–d show that ${g}_{1}^{{{{{{\rm{opt}}}}}}}$ is exactly corresponding to the greatest linewidth narrowing, where we find ${\Gamma }_{{EP}}/{\Gamma }_{{EP}}^{0} \sim 0.1$ and is almost unchanged as ${Q}_{c}$ varies. From the analytical expression of spectral density [Eq. (9)], we find ${g}_{1}^{{{{{{\rm{opt}}}}}}}\approx -\sqrt{3{\Delta }_{{ac}}{\kappa }_{c}}/2$ and the maximal LDOS enhancement (also the greatest linewidth narrowing) is achieved at $\omega \, \approx \, {\omega }_{c}-3{\kappa }_{c}/2$. In Fig. 2e, we compare the analytically predicted ${g}_{1}^{{{{{{\rm{opt}}}}}}}$ (solid line) with the numerical results (hollow circles), where good agreement can be seen. It shows that WGM cavity with a moderate quality factor of ${Q}_{c}={10}^{3} \sim {10}^{4}$ is suitable to demonstrate the enhanced LDOS at CEP since it yields $\left|{g}_{1}^{{{{{{\rm{opt}}}}}}}\right|=10 \sim 30{{{{{\rm{meV}}}}}}$, which is attainable in realistic plasmonic-photonic cavities.

Fig. 2: LDOS enhancement and linewidth narrowing of hybrid CEP cavity.

a LDOS enhancement ${F}_{p}/{F}_{p}^{0}$ and (b) linewidth narrowing ${\Gamma }_{{EP}}/{\Gamma }_{{EP}}^{0}$ of hybrid CEP cavity versus ${Q}_{c}$ and ${g}_{1}$ for plasmon-photon detuning ${\Delta }_{{ac}}={\omega }_{a}-{\omega }_{c}=1{{{{{\rm{eV}}}}}}$. The white dashed line in (a) and (b) traces the maximal LDOS enhancement. c, d are the same as (a) and (b), respectively, but for ${\Delta }_{{ac}}=1.4{{{{{\rm{eV}}}}}}$. e Comparison of the optimal plasmonic-photonic coupling ${g}_{1}^{{{{{{\rm{opt}}}}}}}$ for maximal LDOS enhancement obtained from the numerical results (circles) and predicted by the analytical expression (solid lines). f Comparison of the normalized LDOS of hybrid CEP cavity (solid lines) and usual hybrid cavity (dashed lines) for ${\Delta }_{{ac}}=0.6{{{{{\rm{eV}}}}}}$ (pink lines) and $1.4{{{{{\rm{eV}}}}}}$ (blue lines). Parameters not mentioned are the same as Fig. 1c.

It is worth noting that a larger ${\Delta }_{{ac}}$ gives rise to the greater LDOS enhancement, thus high LDOS of hybrid CEP cavity can be achieved over a wide range of frequency. This feature benefits the optical applications in near-infrared region since the maximal LDOS of hybrid CEP cavity is comparable in different frequencies, while the LDOS without CEP is only a half of that in the visual band, as Fig. 2f shows.

Results and discussion

Red-detuned plasmon-photon coupling: CEP-enhanced coherent light-matter interaction

The LDOS enhancement and linewidth narrowing is expected to enhance the coherent light-matter interaction at CEP. This can be revealed from the SE spectrum, which reflects the local quantum dynamics of a QE and is defined as $S\left(\omega \right)={{{{{\mathrm{lim}}}}}}_{t\to \infty }2{{{{\mathrm{Re}}}}}[{\int }_{\!\!0}^{\infty }d\tau \left\langle {\sigma }_{+}\left(t\right){\sigma }_{-}\left(t+\tau \right)\right\rangle {e}^{i\omega \tau }]$⁶⁵$,$ where the two-time correlation function $\langle {\sigma }_{+}(t){\sigma }_{-}(t+\tau )\rangle$ can be calculated using the quantum regression theorem in a fashion similar to Eq. (8). The SE spectrum is expressed as (see Supplementary Note 3 for detailed derivation)

$$S\left(\omega \right)=\frac{1}{\pi }\frac{\gamma +\Gamma \left(\omega \right)}{{\left[\omega -{\omega }_{0}-\Delta \left(\omega \right)\right]}^{2}+{\left[\frac{\gamma +\Gamma \left(\omega \right)}{2}\right]}^{2}}$$

(12)

where the local coupling strength $\Gamma \left(\omega \right)$ and the photon-induced Lamb shift $\Delta \left(\omega \right)$ are related to LDOS and given by $\Gamma \left(\omega \right)=-2{g}_{a}^{2}{{{{{\rm{Im}}}}}}[{\chi }_{{sys}}\left(\omega \right)]$ and $\Delta \left(\omega \right)={g}_{a}^{2}{{{{\mathrm{Re}}}}}[{\chi }_{{sys}}\left(\omega \right)]$, respectively. The temporal dynamics of QE can be retrieved from the Fourier transform of SE spectrum.

Figure 3a plots the SE spectrum corresponding to the maximal LDOS in hybrid CEP cavity (i.e., $\phi =3{{{{{\rm{\pi }}}}}}/4$), where the strong-coupling anticrossing can be clearly seen by varying the transition frequency of QE. While without CEP, no Rabi splitting can be observed in the SE spectrum, as Fig. 3b shows. By comparing with the eigenenergies (dashed lines), we can see that the peak location of SE spectrum is exactly corresponding to the transition frequency of QE, with broaden linewidth around cavity resonance due to the Purcell effect. This reflects the fact that the QE-cavity interaction is under the weak-coupling regime. In contrast, the strong light-matter interaction is achieved in hybrid CEP cavity, which is evident by the well-separated Rabi splitting in the SE spectrum [Fig. 3c] and the prominent Rabi oscillation in temporal domain [Fig. 3d].

Fig. 3: CEP-enhanced quantum coherence.

a, b SE spectrum versus QE-cavity detuning of hybrid CEP cavity for $\phi =3\pi /4$ and the usual hybrid cavity without CEP, respectively. The white dashed lines trace the eigenenergies of ${M}_{0}$[Eq. (6)]. c, d SE spectrum and the corresponding temporal dynamics, respectively. The circles in (c) plot the numerical results of SE spectrum calculated by the quantum master equation [Eqs. (1)–(5)] using QuTip⁸³, while the lines represent the analytical results of emission spectrum based on the quantum regression theorem using Eq. (6). The parameters are ${\gamma }_{0}=3\mu {{{{{\rm{eV}}}}}}$, ${g}_{1}=-24{{{{{\rm{meV}}}}}}$ and ${Q}_{c}={10}^{3}$. e SE spectrum in the strong-coupling regime with (solid line) and without CEP (dashed line). The pink solid line indicates the SE spectrum with $\phi ={\phi }^{{{{{{\rm{opt}}}}}}}$ corresponding to the maximal LDOS, while the blue dashed line shows the SE spectrum of usual hybrid cavity without CEP. The inset shows the normalized LDOS of hybrid CEP cavity as the function of frequency and $\phi$, where the horizontal dashed line indicates ${\phi }^{{{{{{\rm{opt}}}}}}}$. The parameters are ${g}_{1}=-20{{{{{\rm{meV}}}}}}$, ${g}_{a}=40{{{{{\rm{meV}}}}}}$ and ${Q}_{c}={10}^{4}$. f Shows the corresponding temporal dynamics of QE. In (d) and (f), the circles represent the results obtained by numerically calculating the quantum master equation using QuTip⁸³ while the lines show the results of Fourier transforming the SE spectrum based on the analytical spectral density [Eqs. (9)–(11)]. Parameters not mentioned are the same as Fig. 1c.

Figure 3d shows that the Rabi oscillation decays with a rate $\Gamma$ smaller than the emission rate ${\gamma }_{{{{{{\rm{eff}}}}}}}$ of QE in the weak-coupling regime. It is a counterintuitive behavior because in the standard cavity QED system (i.e., the Jaynes–Cummings model)⁶⁵, we have $\Gamma ={\kappa }_{c}/2$ and ${\gamma }_{{{{{{\rm{eff}}}}}}}=4{g}^{2}/{\kappa }_{c}$ when ${\gamma }_{0} \, \ll \, {\kappa }_{c}$, where $g$ is the QE-cavity coupling rate. On the other hand, the critical coupling rate of the onset of strong coupling is ${g}_{0}={\kappa }_{c}/4$⁷¹$,$ which yields $\Gamma \, > \, {\gamma }_{{{{{{\rm{eff}}}}}}}$, i.e., the Rabi oscillation should decay faster than a QE weakly coupled to the cavity. Therefore, the results of Fig. 3d implies that the composed system enters into the strong-coupling regime mainly attributed to the effect of linewidth narrowing at CEP. This feature is more evident for a composed system without CEP but the QE-cavity interaction is already in the strong-coupling regime, see Fig. 3e for an example. In such a case, we find the locations of Rabi peaks in the SE spectrum remain approximately unchanged in the presence of CEP, so does the period of Rabi oscillation, as we see in Fig. 3f. This indicates that the effective coupling rate between the QE and cavity is similar to the case without CEP. However, the linewidth of Rabi peaks is greatly narrow and as a consequence, the Rabi oscillation manifests a slower decay. The results demonstrate the quantum coherence can be enhanced by CEP in both the weak- and strong-coupling regimes.

Resonant plasmon-photon coupling: enhanced quantum yield at CEP

In the above subsection, we discuss the enhanced coherent light-matter interaction in hybrid CEP cavity with off-resonance plasmon-photon coupling. In the following, we demonstrate the possibility of enhancing the quantum yield⁷² by CEP in the case of resonant plasmon-photon coupling, where low quantum yield is expected due to the severe nonradiative loss of plasmonic resonance. We adopt the following definition of quantum yield¹⁵: $\eta ={\Phi }_{r}/\left({\Phi }_{r}+{\Phi }_{d}\right)$, with radiation power ${\Phi }_{r}=\left\langle \left(\sqrt{{\kappa }_{r}}{a}^{{{\dagger}} }+\sqrt{{\gamma }_{0}}{\sigma }_{+}\right)\left(\sqrt{{\kappa }_{r}}a+\sqrt{{\gamma }_{0}}{\sigma }_{-}\right)\right\rangle +{\kappa }_{c}\left(\left\langle {c}_{{ccw}}^{{{\dagger}} }{c}_{{ccw}}\right\rangle +\left\langle {c}_{{cw}}^{{{\dagger}} }{c}_{{cw}}\right\rangle \right)$ and absorption power of plasmonic modes ${\Phi }_{d}={\kappa }_{o}\left\langle {a}^{{{\dagger}} }a\right\rangle +{\gamma }_{m}\left\langle {\sigma }_{+}{\sigma }_{-}\right\rangle$, where ${\gamma }_{m}$ accounts for the dissipation of QE to higher-order plasmonic modes, i.e., the decay rate $\gamma$ of QE is now $\gamma ={\gamma }_{0}+2{\gamma }_{{{{{{\rm{p}}}}}}}+{\gamma }_{m}$ in Eq. (5). It is worth noting that the first term in ${\Phi }_{r}$ stands for the radiation from both plasmonic antenna and QE, considering that the differences of emission from these two constituents to detector can be neglected due to their subwavelength dimensions^15,73. Furthermore, although the evaluation of quantum yield requires calculating the expectation values of two operators, the factorization $\langle {O}_{A}^{{{\dagger}} }{O}_{B}\rangle \, \approx \, \langle {O}_{A}^{{{\dagger}} }\rangle \langle {O}_{B}\rangle$, where ${O}_{A}$ and ${O}_{B}$ stand for two operators, can be applied to simplify the calculations and obtain the analytical results with the approximation $\left\langle {\sigma }_{z}\right\rangle \approx -\!\!1$ under the weak coherent pump. As a result, the quantum yield can be evaluated from the calculations of single-time averages of operators. The validity of this treatment is verified by the results of numerical simulations (circles) shown in Fig. 4. Accordingly, the quantum yield can be evaluated from the equation $\frac{d}{{dt}}{{{{{{\boldsymbol{p}}}}}}}_{{{{{{\boldsymbol{0}}}}}}}=-i{M}_{0}{{{{{{\boldsymbol{p}}}}}}}_{{{{{{\boldsymbol{0}}}}}}}-i{{{{{\boldsymbol{\Omega }}}}}}{{{{{\boldsymbol{,}}}}}}$ where the frequency in the diagonal elements of ${M}_{0}$ is replaced by the frequency detuning ${\Delta }_{L}={\omega }_{X}-{\omega }_{L}$ between the system constituents and the driving field, with $X=0,a,{ccw},{cw}$ and ${\omega }_{L}$ being the frequency of driving field. The composed system is initially in the ground state and ${{{{{\boldsymbol{\Omega }}}}}}={\left[{p}_{{in}},{{{{\mathrm{0,0,0}}}}}\right]}^{T}$ accounts for a weak coherent drive of QE with amplitude ${p}_{{in}}$, which is introduced by implementing a driving Hamiltonian ${H}_{d}={p}_{{in}}\left({e}^{-i{\omega }_{L}t}{\sigma }_{+}+{\sigma }_{-}{e}^{i{\omega }_{L}t}\right)$ in Eq. (2).

Fig. 4: Quantum yield η of hybrid CEP cavity.

a Quantum yield $\eta$ of hybrid CEP cavity versus ${Q}_{c}$ and $\phi$. The red dotted line surrounds the region of enhanced quantum yield at CEP, i.e., $\eta \, > \, {\eta }_{0}$. The black dashed line shows the parameter area of cavity radiation enhancement ${\eta }_{r}=\max \left[{\Phi }_{r}\right]/\max \left[{\Phi }_{r}^{0}\right] \, > 10$, while the white dashed dotted line shows the parameter area of plasmon absorption reduction ${\eta }_{d}=\min \left[{\Phi }_{d}^{0}\right]/\min \left[{\Phi }_{d}\right] > 10$ for dipolar plasmonic mode. The green triangle and yellow star indicate the parameters of (c, d) and (e, f), respectively. b Quantum yield ${\eta }_{0}$ of hybrid cavity without CEP. c, d Quantum yield and plasmon absorption of dipolar mode of hybrid CEP cavity for ${Q}_{c} \, \approx \, {1.5\times 10}^{4}$ [see yellow star in (a)], respectively. e, f Quantum yield and cavity radiation of hybrid CEP cavity for ${Q}_{c}={10}^{5}$ [see green triangle in (a)], respectively. The inset shows a close-up of quantum yield around cavity resonance. In (c, d) and (e, f), the results of usual hybrid cavity without CEP are also shown for comparison (black lines). The circles represent the results obtained by numerically calculating the expectation values of two operators, e.g., $\left\langle {a}^{{{\dagger}} }a\right\rangle$, using QuTip⁸³. Other parameters are ${g}_{c}=0.144{{{{{\rm{meV}}}}}}$, ${g}_{a}=7.2{{{{{\rm{meV}}}}}}$, ${g}_{1}=-2.9{{{{{\rm{meV}}}}}}$, ${\kappa }_{r}=2.45{{{{{\rm{meV}}}}}}$, ${\kappa }_{o}=200{{{{{\rm{meV}}}}}}$, ${Q}_{c}={10}^{5}$, ${\gamma }_{0}=3\mu {{{{{\rm{eV}}}}}}$, ${\gamma }_{m}=83\mu {{{{{\rm{eV}}}}}}$ and ${\gamma }_{{{{{{\rm{p}}}}}}}=0$¹⁵.

To better illustrate the enhancement of quantum yield at CEP, we first consider the parameters of a hybrid cavity studied by Peng Pai et al.¹⁵ at low temperature. The results are presented in Fig. 4a, where it shows the quantum yield $\eta$ with CEP versus ${Q}_{c}$ and $\phi ,$ while the result without CEP is provided in Fig. 4b for comparison. We can see that the maximal quantum yield without CEP is ${\eta }_{\max }^{0} \, \approx \, 0.636$, achieved at ${Q}_{c} \, \approx \, 2\times {10}^{4}$; while in a hybrid CEP cavity, the enhanced quantum yield (i.e., $\eta \, > \, {\eta }_{\max }^{0}$) can be found with ${Q}_{c}$ varying from $2\times {10}^{3}$ to ${10}^{6}$, indicated by the red dashed dotted line. In addition, high quantum yield with $\eta \, > \, 0.8$ can be achieved in a wide range of parameters.

To shed insight into the physical mechanism of enhanced quantum yield at CEP, we plot $\eta$ as the function of frequency detuning ${\Delta }_{L}={\omega }_{c}-{\omega }_{L}$ in Fig. 4c, e, where two situations are analyzed. The parameters of the first are similar to hybrid cavity achieving ${\eta }_{\max }^{0}$, with ${Q}_{c} \, \approx \, {1.5\times 10}^{4}$ and $\phi =3{{{{{\rm{\pi }}}}}}/2$, as indicated by the yellow star in Fig. 4a. The maximal quantum yield ${\eta }_{\max }$ of hybrid CEP cavity is enhanced by $50 \%$ and can reach $0.92$, as we see in Fig. 4c. This enhancement stems from the CEP-reduced absorption of dipolar plasmonic mode, see the absorption power ${\Phi }_{d}$ shown in Fig. 4d. It shows that the minimum of absorption power is reduced by one order of magnitude in the presence of CEP, and the valley of absorption power is exactly corresponding to ${\eta }_{\max }$. Furthermore, LDOS also presents a strong reduction at this region and approaches to zero for hybrid CEP cavity. These features indicate that the enhanced quantum yield originates from the Fano interference, as found in previous study¹⁵, while in our setup this mechanism is strengthened by CEP. The white dotted line in Fig. 4a surrounds the region of ${\eta }_{d}=\min \left[{\Phi }_{d}^{0}\right]/\min \left[{\Phi }_{d}\right]\, > \, 10$, indicating the parameter area of Fano-reduced plasmonic absorption. We can see that for hybrid CEP cavity with high ${Q}_{c}$, high quantum yield is still hard to achieve by means of reducing the plasmonic absorption. In such a case, we find that the improvement of quantum yield is hindered by the low radiation power of cavity.

We then focus on the region of high quantum yield extended to high ${Q}_{c}$ at the upper right corner of Fig. 4a, with $\phi$ around zero. The quantum yield $\eta$ versus ${\Delta }_{L}$ for $\phi =0$ and ${Q}_{c}={10}^{5}$ [green triangle in Fig. 4a] is analyzed in Fig. 4e, f. It shows that high quantum yield can be achieved in a narrow range of frequency around ${\Delta }_{L}=0$, where $\eta$ can reach $0.98$ while the maximal quantum yield without CEP is below $0.5$. We notice that in hybrid CEP cavity, the frequency corresponding to the maximal quantum yield ${\eta }_{\max }$ deviates from that of hybrid cavity; actually, the quantum yield with $\phi =0$ is slightly reduced at the frequency of ${\eta }_{\max }^{0}$. The result clearly indicates that the underlying mechanism of ${\eta }_{\max }$ achieved at CEP is essentially different from the Fano-reduced plasmonic absorption of hybrid cavity. By inspecting the output power of cavity radiation ${\Phi }_{r}$ [see Fig. 4f], we find a sharp peak appearing exactly at the frequency of ${\eta }_{\max }$, with an enhancement of about three orders of magnitude compared to that without CEP. It reveals that the great improvement of quantum yield with $\phi =0$ comes from the enhanced cavity radiation at CEP, which can be interpretated as a phenomenon analogous to superscattering where the partial radiation of eigenmodes is added up to produce an extremely sharp peak in ${\Phi }_{r}$ without interference⁷⁴. Another evidence of superscattering is that LDOS also exhibits a prominent enhancement around ${\Delta }_{L}=0$, as we see in Fig. 4f. To the best of our knowledge, this mechanism of enhancing quantum yield, i.e., the superscattering at CEP, has not been reported in plasmonic-photonic cavity. The black dashed line in Fig. 4a indicates the region of ${\eta }_{r}=\max \left[{\Phi }_{r}\right]/\max \left[{\Phi }_{r}^{0}\right]\, > \, 10$, where we see that the prominent superscattering is responsible for high quantum yield around $\phi =0$.

To better understand how the cavity radiation is enhanced by the superscattering at CEP, we derive a formalism of eigenmode decomposition for the scattering of hybrid CEP cavity⁷⁴. In the absence of QE, the dynamic of cavity modes is described by $d\left\langle {{{{{\boldsymbol{c}}}}}}\left(t\right)\right\rangle /{dt}=-i{M}_{c}\left\langle {{{{{\boldsymbol{c}}}}}}\left(t\right)\right\rangle -i{{{{{{\boldsymbol{\Omega }}}}}}}_{{{{{{\boldsymbol{p}}}}}}}$, where the frequency in the diagonal elements of ${M}_{c}$ is again replaced by the frequency detuning ${\Delta }_{L}$, and ${{{{{{\boldsymbol{\Omega }}}}}}}_{{{{{{\boldsymbol{p}}}}}}}={\left[{q}_{{in}},{{{{\mathrm{0,0}}}}}\right]}^{T}$ accounts for the driving field for dipolar plasmonic mode. This equation can be rewritten as $i\frac{d}{{dt}}\left\langle {{{{{\boldsymbol{c}}}}}}\left(t\right)\right\rangle ={V}^{-1}{BV}\left\langle {{{{{\boldsymbol{c}}}}}}\left(t\right)\right\rangle +{{{{{{\boldsymbol{\Omega }}}}}}}_{{{{{{\boldsymbol{p}}}}}}}$, where $B$ and $V$ are the diagonal matrix formed from the eigenvalues of ${{{{{{\rm{M}}}}}}}_{{{{{{\rm{c}}}}}}}$ and the corresponding left eigenvectors in matrix form, respectively, ${M}_{c}\,{{{{{\boldsymbol{=}}}}}}\,{V}^{-1}{BV}$. The above equation can be solved through the Fourier transform. The formal solution is given by ${{{{{\boldsymbol{c}}}}}}\left({\Delta }_{L}\right)={V}^{-1}{\left({\Delta }_{L}I-B\right)}^{-1}V{{{{{{\boldsymbol{\Omega }}}}}}}_{{{{{{\boldsymbol{p}}}}}}}$. The scattering of hybrid CEP cavity is expressed as $\sigma \left({\Delta }_{L}\right)={s}^{{{{{{\boldsymbol{{{\dagger}}}} }}}}}\left({\Delta }_{L}\right)s\left({\Delta }_{L}\right)$, with $s\left({\Delta }_{L}\right)=K{{{{{\boldsymbol{c}}}}}}\left({\Delta }_{L}\right)$, where $K$ defines the coupling $\Gamma$ between different scattering channels, which are independent in our setup

$$\Gamma ={K}^{{{{{{\boldsymbol{{{\dagger}}}} }}}}}K=\left[\begin{array}{ccc}{\kappa }_{a} & 0 & 0\\ 0 & {\kappa }_{c} & 0\\ 0 & 0 & {\kappa }_{c}\end{array}\right]$$

(13)

After lengthy calculation (see Supplementary Note 4 for details), we can obtain the expression of $\sigma \left({\Delta }_{L}\right)$ at ${\Delta }_{L}=0$

$${\sigma }_{0}\equiv \sigma \left(0\right)= \frac{p}{{\gamma }_{a}{\gamma }_{b}{\gamma }_{c}}\left\{{h}_{{aa}}{\left|{C}_{a}\right|}^{2}+{h}_{{bb}}{\left|{C}_{b}\right|}^{2}+{h}_{{cc}}{\left|{C}_{c}\right|}^{2}\right. \\ \left.+\,2{{{{\mathrm{Re}}}}}\left[{h}_{{ab}}{C}_{a}^{* }{C}_{b}+{h}_{{ac}}{C}_{a}^{* }{C}_{c}+{h}_{{bc}}{C}_{b}^{* }{C}_{c}\right]\right\}$$

(14)

where $p={{{{{{\rm{Det}}}}}}\left[V\right]}^{-2}$. ${C}_{i}$ is the radiation pattern of eigenmodes, with $i=a,b,c$, while ${h}_{{ij}}$ and ${h}_{{ii}}$ encode the interaction and the weighted coefficients of eigenmodes, respectively. For the specific expressions of ${C}_{i}$, ${h}_{{ii}}$ and ${h}_{{ij}}$, we refer to Supplementary Note 4. ${\gamma }_{i}$ is the imaginary part of eigenvalues of ${M}_{c}$ in decreasing order, i.e., ${\gamma }_{a} \, > \, {\gamma }_{b} \, > \, {\gamma }_{c}$. Therefore, eigenmode $a$ is superradiant. We denote the first term as ${\sigma }_{\sup }$ while the remaining terms as ${\sigma }_{{{{{{\rm{so}}}}}}}$, thus ${\sigma }_{0}={\sigma }_{\sup }+{\sigma }_{{{{{{\rm{so}}}}}}}$. If the condition ${\gamma }_{a} \, \gg \, {\gamma }_{b},{\gamma }_{c}$ is satisfied, which holds true for hybrid CEP cavity, the superscattering will occur when ${\sigma }_{\sup } \, > \, {\sigma }_{{{{{{\rm{so}}}}}}} \, > \, 0$, giving rise to a peak in the scattering spectrum.

Figure 5a compares $\sigma \left({\Delta }_{L}\right)$ with $\phi =0$ and $3\pi /2$, while other parameters are the same as Fig. 4c. We can see that a sharp peak shows around ${\Delta }_{L}=0$ for both cases, but the intensity of $\phi =0$ is much higher than that of $\phi =3{{{{{\rm{\pi }}}}}}/2$. On the contrary, no peak can be found around ${\Delta }_{L}=0$ without CEP. In Fig. 5b, c, we plot the corresponding eigenmode decomposition of $\sigma \left({\Delta }_{L}\right)$ around ${\Delta }_{L}=0$. We find ${\sigma }_{{{{{{\rm{so}}}}}}} \, \gg \, {\sigma }_{\sup }$ for $\phi =0$; furthermore, the scattering peak is mainly contributed by ${\sigma }_{{{{{{\rm{so}}}}}}}$ and demonstrates a subnatural linewidth. These features signify the occurrence of superscattering at CEP. While for $\phi =3{{{{{\rm{\pi }}}}}}/2$, Fig. 5c shows a scattering peak appearing at the local maximum of ${\sigma }_{{{{{{\rm{so}}}}}}}$, however, in this case ${\sigma }_{{{{{{\rm{so}}}}}}}$ is negative around ${\Delta }_{L}=0$, i.e., ${\sigma }_{\sup } \, > \, 0 \, > \, {\sigma }_{{{{{{\rm{so}}}}}}}$. Therefore, it belongs to the intermediate mechanism of electromagnetically induced transparency and superscattering.

Fig. 5: Superscattering and near-unity quantum yield at room temperature.

a Scattering spectrum $\sigma \left({\Delta }_{L}\right)$ of hybrid CEP cavity for plasmon driving. The inset shows a close-up of $\sigma \left({\Delta }_{L}\right)$ around ${\Delta }_{L}=0$. b, c Decomposition of cavity scattering $\sigma \left({\Delta }_{L}\right)$ into the superscattering (${\sigma }_{\sup }$) and other (${\sigma }_{{{{{{\rm{so}}}}}}}$) terms for the cases of $\phi =0$ and $3\pi /2$, respectively. Parameters of these two cases are indicated by the green triangle and yellow star in Fig. 4a, respectively. The results of hybrid cavity without CEP are also shown for comparison (dashed lines). d, e Room-temperature quantum yield $\eta$ of hybrid CEP cavity versus ${g}_{a}$ and ${g}_{1}$ for ${Q}_{c}={10}^{4}$ and ${10}^{3}$, respectively. ${\eta }_{0}$ denotes the quantum yield of hybrid cavity without CEP. $\phi =0$ and ${\gamma }_{{{{{{\rm{p}}}}}}}=30{{{{{\rm{meV}}}}}}$, while other parameters are the same as Fig. 4a. f Comparison of room-temperature quantum yield versus laser frequency for various ${Q}_{c}$. The parameters are $\phi =0$, ${g}_{1}=-15{{{{{\rm{meV}}}}}}$, ${g}_{a}=20{{{{{\rm{meV}}}}}}$ and ${\gamma }_{{{{{{\rm{p}}}}}}}=30{{{{{\rm{meV}}}}}}$ (solid lines). The quantum yield of hybrid cavity without CEP (black dashed line) and hybrid CEP cavity with ${\gamma }_{{{{{{\rm{p}}}}}}}=60{{{{{\rm{meV}}}}}}$ (dashed lines with the same color) are shown for comparison.

In view of the advance of hybrid CEP cavity in enhancing quantum yield at low temperature, we investigate the performance at room temperature. In this case, the free-space quantum yield ${\gamma }_{0}/({\gamma }_{0}+{\gamma }_{m}+2{\gamma }_{{{{{{\rm{p}}}}}}})$ of QE approaches zero, while the quantum yield without CEP is dramatically reduced by the QE dephasing ${\gamma }_{{{{{{\rm{p}}}}}}}$. We evaluate that without CEP, the quantum yield decreases from $0.58$ to less than $0.008$ for ${Q}_{c}={10}^{4}$ and ${\gamma }_{{{{{{\rm{p}}}}}}}=30{{{{{\rm{meV}}}}}}$^75,76$.$ With the same ${Q}_{c}$ and ${\gamma }_{{{{{{\rm{p}}}}}}}$, Fig. 5d displays the quantum yield of hybrid CEP cavity as the function of ${g}_{a}$ and ${g}_{1}$, where it shows that $\eta \, > \, 0.9$ can be achieved with ${g}_{a},\left|{g}_{1}\right| > 20{{{{{\rm{meV}}}}}}$. It also shows that strong plasmon-photon and plasmon-QE interactions are beneficial to improve quantum yield. $\eta$ rapidly grows from $\sim 0.4$ to $\sim 0.9$ as the coupling rates ${g}_{a}$ and $\left|{g}_{1}\right|$ increase from $10{{{{{\rm{meV}}}}}}$ to $20{{{{{\rm{meV}}}}}}$. With ${g}_{a},\left|{g}_{1}\right| \sim 30{{{{{\rm{meV}}}}}}$, which are attainable in realistic structures, near-unity quantum yield can be achieved at room temperature, demonstrating over hundredfold enhancement of quantum yield. As ${Q}_{c}$ reduces to ${10}^{3}$, Fig. 5e shows that the quantum yield greatly decreases in a wide range of parameters, but hybrid CEP cavity with stronger plasmon-photon and plasmon-QE interactions manifests higher robustness. For example, the quantum yield drops from $0.92$ to $0.56$ for ${g}_{a},\left|{g}_{1}\right|=20{{{{{\rm{meV}}}}}}$, while high quantum yield ${\eta }_{\max } \sim 0.84$ can still be maintained with ${g}_{a},\left|{g}_{1}\right|=30{{{{{\rm{meV}}}}}}$.

The results presented in Fig. 5d, e indicate that a high ${Q}_{c}$ facilitates to improve the quantum yield of hybrid CEP cavity. Figure 5f investigates the quantum yield for various ${Q}_{c}$ versus ${\Delta }_{L}$, where ${\eta }_{\max } \, > \, 0.9$ can be achieved with ${Q}_{c}={10}^{4}$ while ${\eta }_{\max }$ can reach $0.97$ at room temperature with ${Q}_{c}={10}^{5}$. It indicates that this near-unity quantum yield is achieved by the superscattering at CEP, while the mechanism of Fano interference fails to produce high quantum yield at room temperature. It also indicates that a high ${Q}_{c}$ is beneficial to obtain high quantum yield for QEs with large dephasing rate. As we see in Fig. 5f, when ${\gamma }_{{{{{{\rm{p}}}}}}}$ increases from $30{{{{{\rm{meV}}}}}}$ to $60{{{{{\rm{meV}}}}}}$, ${\eta }_{\max }$ drops from $\sim 0.9$ to $\sim 0.8$ for ${Q}_{c}={10}^{4}$ while it is hardly reduced for ${Q}_{c}={10}^{5}$. Therefore, the results of Fig. 5f demonstrate the unique advance of hybrid CEP cavity in realizing high quantum yield and great potential in practical applications, such as building single-photon sources^77,78. In this respect, we consider two scenarios for single-photon generation, the single-photon blockade through weak coherent drive and the indistinguishable single photons under the pulse drive of QE. For the former case, we implement a driving Hamiltonian ${H}_{d}={p}_{{in}}\left({e}^{-i{\omega }_{L}t}{\sigma }_{+}+{\sigma }_{-}{e}^{i{\omega }_{L}t}\right)$ in Eq. (5) and numerically calculate the zero-time-delay correlation function ${g}^{\left(2\right)}\left(0\right)=\left\langle {c}^{{{\dagger}} }\left(0\right){c}^{{{\dagger}} }\left(0\right)c\left(0\right)c\left(0\right)\right\rangle /{n}_{c}^{2}$, where $t=0$ means the steady state and ${n}_{c}=\left\langle {c}^{{{\dagger}} }c\right\rangle$ is the photon number of mode $c$. ${g}^{\left(2\right)}\left(0\right)=0$ corresponds to perfect single photons⁷⁹. As Fig. 6a shows, the introduction of CEP can reduce ${g}^{\left(2\right)}\left(0\right)$ of cavity mode ${c}_{2}$ by over four orders of magnitude with $\phi =0$ and optimized parameters. This results from the chiral coupling of two-photon state of cavity mode ${c}_{2}$ created by CEP; combined with enhanced quantum yield, the CEP greatly reduces the ${g}^{\left(2\right)}\left(0\right)$ of ${c}_{2}$. Besides the giant enhancement of photon correlations, the inset of Fig. 6 shows that the great tunability of photon statistics can be achieved at CEP by varying $\phi$, where we can see that the strongly antibunched photon [${g}^{\left(2\right)}\left(0\right) \, \approx \, {10}^{-6}$] becomes bunched [${g}^{\left(2\right)}\left(0\right) \, > \, 10$] at $\phi =3{{{{{\rm{\pi }}}}}}/2$. For the case of QE prepared at the excited state by a pulse drive, the degree of indistinguishability $I$ is the key figure of merit of the emitted photon ($I=1$ stands for indistinguishable single photons)⁸⁰, which is defined as:

$$I=\frac{{\int }_{\!\!0}^{\infty }{dt}{{\int }_{\!\!0}^{\infty }d\tau \left|\left\langle {c}^{{{\dagger}} }\left(t+\tau \right)c\left(t\right)\right\rangle \right|}^{2}}{{\int }_{\!\!0}^{\infty }{dt}{\int }_{\!\!0}^{\infty }d\tau \left\langle {c}^{{{\dagger}} }\left(t+\tau \right)c\left(t+\tau \right)\right\rangle \left\langle {c}^{{{\dagger}} }\left(t\right)c\left(t\right)\right\rangle }$$

(15)

Fig. 6: Single-photon generation in hybrid CEP cavity at ϕ = 0.

a Comparison of ${g}^{\left(2\right)}\left(0\right)$ with and without CEP under weak coherent drive of QE. The inset plots ${g}^{\left(2\right)}\left(0\right)$ as the function of frequency detuning and $\phi$. Other parameters are ${\omega }_{a}=2.3{{{{{\rm{eV}}}}}}$, ${\omega }_{c}=1.3{{{{{\rm{eV}}}}}}$, ${\omega }_{0}=1.284{{{{{\rm{eV}}}}}}$, ${g}_{c}=0.07m{{{{{\rm{eV}}}}}}$, ${g}_{1}=-12.6m{{{{{\rm{eV}}}}}}$, ${\kappa }_{a}=100m{{{{{\rm{eV}}}}}}$, ${\kappa }_{c}=0.1m{{{{{\rm{eV}}}}}}$, ${\gamma }_{p}=0$ and ${\gamma }_{0}=10\mu {{{{{\rm{eV}}}}}}$. b, c Room-temperature indistinguishability and efficiency of emitted photons from an initial excited QE prepared by a pulse drive. The cavity and QE are set to be resonant. Other parameters are ${g}_{c}=0$, ${g}_{a}=31m{{{{{\rm{eV}}}}}}$, ${g}_{1}=-11m{{{{{\rm{eV}}}}}}$, ${\kappa }_{a}=100m{{{{{\rm{eV}}}}}}$, ${\kappa }_{c}=2m{{{{{\rm{eV}}}}}}$, ${\gamma }_{p}=20m{{{{{\rm{eV}}}}}}$ and ${\gamma }_{0}=10\mu {{{{{\rm{eV}}}}}}$. The results are obtained from numerical simulation using QuTip⁸³ at truncated states space with three excitations.

Accordingly, the efficiency is given by $\xi ={\kappa }_{c}{\int }_{\!\!0}^{\infty }d\tau \left\langle {c}^{{{\dagger}} }\left(t\right)c\left(t\right)\right\rangle$. Figure 6b, c compares $I$ and $\xi$ of hybrid CEP cavity with that of hybrid cavity without CEP, respectively. The results reveal that at $\phi =0$, the indistinguishability of the output field of waveguide is nearly unchanged after introducing CEP, while the efficiency is increased by $\sim 30 \%$ and reaches $\xi \, \approx \, 6.35 \%$ at room temperature. Therefore, the hybrid CEP cavity can generate single photons more efficiently than the hybrid cavity without CEP. Furthermore, the efficiency of hybrid CEP cavity shows an eightfold increase compared to the scheme of cascaded photonic cavities⁸¹, which requires a cavity with $Q \sim 50000$ to achieve $I=0.95$, while in our model a photonic (WGM) cavity with $Q \sim 800$ can generate single photons with $I=0.93$, which greatly reduce the requirement of $Q$ factor.

A physical realization

In the above discussion, we omit the intrinsic decay ${\kappa }_{i}$ of WGM modes. We will address this issue in this subsection by evaluating the quantum yield of QE in a realistic structure of the proposed hybrid CEP cavity. To do this, we need to extract the system parameters from the simulation data by applying the analytical LDOS. Therefore, we should first validate our LDOS theory.

Figure 7a depicts the geometry of hybrid CEP cavity under study, which is based on a SiN microdisk with a gold dimer placed on the top surface. The refractive index of microdisk is $n=2$, with an imaginary component of $4\times {10}^{-6}$ to include the material absorption¹⁴. The permittivity $\epsilon \left(\omega \right)$ of gold is characterized by the Drude model $\epsilon \left(\omega \right)=1-{\omega }_{g}^{2}/\omega (\omega +i{\gamma }_{g})$, where ${\omega }_{g}=1.26\times {10}^{16}$ rad/s and ${\gamma }_{g}=1.41\times {10}^{14}$ rad/s are the plasma frequency and the collision rate, respectively²³. The gold dimer on SiN slab supports a single localized plasmon resonance with the peak appearing at ${\omega }_{a}=2.254{{{{{\rm{eV}}}}}}$ and a linewidth (total decay rate) of ${\kappa }_{a}=255{{{{{\rm{meV}}}}}}$, where the radiative decay rate is evaluated as ${\kappa }_{r}=13{{{{{\rm{meV}}}}}}$. A point-dipole QE is located at the gap center of the gold dimer.

Fig. 7: Enhanced coherent light-matter interaction and room-temperature quantum yield of a realistic hybrid CEP cavity.

a Schematic diagram of a hybrid CEP cavity based on WGM microdisk. A gold dimer consisting of two nanorods is placed on the top of WGM cavity, with a QE in the gap center and aligned matching the polarization of dipolar plasmonic mode. The radius of nanorods is 40 nm. Other geometry parameters are indicated in the figure. LDOS can by tuned by adjusting the edge-to-edge separation $s$ between the cavity and the waveguide and the distance $L$ between the cavity-waveguide junction and the mirror. The width of waveguide is 400 nm. The orange color illustrates the field profiles for $\phi ={{{{{\rm{\pi }}}}}}$ and ${\omega }_{c}=1.462{{{{{\rm{eV}}}}}}$ obtained from electromagnetic simulation. b LDOS of the physical realization shown in (a) for various $s$ obtained from electromagnetic simulations. The dashed lines with the same color plot the corresponding analytical results given by the quantized few-mode model [Eqs. (9)–(11)]. c Temporal dynamics of initially excited QE with a dipole moment $\mu =48$ Debye for various LDOS shown (b). The inset shows the logarithmic plot of QE population. d Room-temperature quantum yield $\eta$ of hybrid CEP cavity as the function of ${Q}_{c}$ and ${\kappa }_{c}/{\kappa }_{i}$. ${\eta }_{0}$ denotes the quantum yield of hybrid cavity without CEP. The red star indicates ${Q}_{c}$ and ${\kappa }_{c}/{\kappa }_{i}$ of WGM modes with frequency ${\omega }_{c}=2.246{{{{{\rm{eV}}}}}}$. Other parameters are ${g}_{1}=-11{{{{{\rm{meV}}}}}}$, ${g}_{a}=23.6{{{{{\rm{meV}}}}}}$, ${g}_{c}=0.171{{{{{\rm{meV}}}}}}$ and ${\gamma }_{{{{{{\rm{p}}}}}}}=30{{{{{\rm{meV}}}}}}$. e Room-temperature quantum yield $\eta$ versus ${g}_{1}$ for various ${Q}_{c}$. The frequencies and decay rates of uncoupled gold dimer and WGM cavity are provided in the text. The dashed lines represent the quantum yield obtained by numerically calculating the expectation values of two operators using QuTip⁸³. f Room-temperature quantum yield $\eta$ for various ${g}_{1}$. The parameters are $\kappa ={\omega }_{c}/{10}^{4}$ and ${\kappa }_{c}={10}^{2}{\kappa }_{i}$. Other parameters are the same as (d).

According to the LDOS theory developed in previous section, the Purcell effect felt by QE can be modified by adjusting the cavity-waveguide separation $s$ and the distance $L$ between the cavity-waveguide junction and the mirror, as the former determines ${\kappa }_{c}$ while the latter controls $\phi$.

To verify our LDOS theory, we consider a pair of degenerate WGM modes with frequency ${\omega }_{c}=1.462{{{{{\rm{eV}}}}}}$ and intrinsic linewidth ${\kappa }_{i}=0.387{{{{{\rm{meV}}}}}}$ in the absence of waveguide, see the blue line in Fig. 7b. The frequency of cavity resonance is unaffected by the coupling to waveguide with $s=100{{{{{\rm{nm}}}}}}$, but the linewidth is broadened to $\kappa$, from which we obtain ${\kappa }_{c}=0.867{{{{{\rm{meV}}}}}}$ by subtracting the intrinsic linewidth ${\kappa }_{i}$ from $\kappa$. With the parameters of uncoupled constituents at hand, we can apply the analytical expressions Eqs. (9)–(11) to determine the plasmon-photon coupling rate ${g}_{1}$ by fitting the normalized LDOS obtained from electromagnetic simulations. We evaluate ${g}_{1}=-11{{{{{\rm{meV}}}}}}$ and find good agreement between the analytical and simulation results, see the blue dashed and solid lines in Fig. 7b. Then we introduce the mirror at the right end of waveguide to create a CEP with $\phi ={{{{{\rm{\pi }}}}}}$, which corresponds to the maximal LDOS enhancement of this setup. We find the analytically predicted LDOS accords well with the simulation results, see the pink solid and dashed lines in Fig. 7b. In addition, for large cavity-waveguide separation $s=160{{{{{\rm{nm}}}}}}$, the analytical model can also give a correct prediction of LDOS. Therefore, the results of Fig. 7b validate the applicability of our LDOS theory for hybrid CEP cavity.

Figure 7b shows that the normalized LDOS is increased by 65% after introducing a CEP and is double as the cavity-waveguide separation enlarges to $160{{{{{\rm{nm}}}}}}$. In Fig. 7c, we plot the corresponding temporal dynamics of an initially excited QE, where it shows the greater and faster Rabi oscillation with slower decay compared to that without CEP. This indicates the stronger coherent energy exchange between the cavity and QE as a result of the enhanced LDOS with slightly narrow linewidth at CEP.

The intrinsic decay ${\kappa }_{i}$ of cavity modes is inevitable in realistic structures, which hinders the improvement of quantum yield. Figure 7d displays the quantum yield of hybrid CEP cavity versus ${Q}_{c}$ and ${\kappa }_{c}/{\kappa }_{i}$ with $\phi =0$, where the QE is resonantly coupled to WGM modes with frequency ${\omega }_{c}=2.246{{{{{\rm{eV}}}}}}$. The corresponding intrinsic and waveguide-induced decay rates for $s=40{{{{{\rm{nm}}}}}}$ are indicated by the red star in Fig. 7d, which yields $\eta \, \sim \, 0.32$. Figure 7d shows that the enhancement of quantum yield requires an increase of ${Q}_{c}$ while simultaneously reducing the intrinsic decay of WGM modes. This can be achieved by using a WGM cavity with large radius⁴⁰ and made of high refractive index⁴⁸. With the fixed ${\kappa }_{c}/{\kappa }_{i}=100$, Fig. 7e shows that the quantum yield for $\left|{g}_{1}\right| > 5{{{{{\rm{meV}}}}}}$ can be greatly enhanced as ${Q}_{c}$ increases from ${10}^{3}$ to ${10}^{4}$, and reaches $0.9$ as $\left|{g}_{1}\right|$ increases to $30{{{{{\rm{meV}}}}}}$. However, a high ${Q}_{c}$ may also have negative impact on the quantum yield. When $\left|{g}_{1}\right| > 30{{{{{\rm{meV}}}}}}$, the increase of ${Q}_{c}$ from ${10}^{4}$ to ${10}^{5}$ leads to the reduction of quantum yield. It is because the radiation through cavity is less efficient for a high ${Q}_{c}$, then a substantial portion of energy is absorbed by plasmon with a large ${g}_{1}$ since the radiation of QE is extremely weak for ${\gamma }_{{{{{{\rm{p}}}}}}} \, \gg \, {\gamma }_{0}$. Therefore, the results show that ${g}_{1}$ also plays an important role in determining the quantum yield, but is not always in a positive manner. In addition, Fig. 7f shows that a large ${g}_{1}$ narrows the linewidth of radiation spectrum, thus may lead to the reduced quantum yield outside the cavity resonance, which is undesirable to realize the broadband enhancement of quantum yield.

Finally, we briefly summarize and discuss the findings from this simple design of hybrid CEP cavity. To achieve high quantum yield at room temperature, a hybrid CEP cavity with moderate $\left|{g}_{1}\right| \sim 10{{{{{\rm{meV}}}}}}$, relatively high ${Q}_{c} \sim {10}^{5}$ and low ${\kappa }_{i}$ are desirable. We note that it is not difficult to find a realistic structure satisfying one or two of them, but is challenging to meet all of these requirements in WGM cavity. It is because a high ${Q}_{c}$ in general means weak energy leakage, resulting in a small overlap between the plasmonic and photonic modes when the plasmonic antenna is placed outside the cavity. This results in the tradeoff between ${Q}_{c}$ and ${g}_{1}$ in hybrid CEP cavity. To overcome this obstacle, the plasmonic antenna can be embedded in the cavity, with an analogs design of the structure studied by Medina et al.⁴⁸; however, in such a case the QE location cannot be feasibly controlled in experiments. A possible solution is to introduce a thin air slot at the antinode of the selected WGM modes⁸², where the plasmonic antenna and QE can be placed with precise control. The presence of air slot in WGM microdisk will not degrade ${Q}_{c}$ since it has a physical volume as small as the plasmonic antenna, thus can be seen as a perturbation of WGM modes. Such hybrid CEP cavity design can fulfill the requirements of achieving high quantum yield at room temperature and can also be integrated with other on-chip optoelectronic devices.

Conclusions

In this work, we propose to engineer the plasmonic resonance by virtue of a CEP hosting in a WGM cavity and demonstrate the great tunability of LDOS provided by CEP. A LDOS theory based on few-mode quantization description is established for a prototypic model and applied to a realistic structure. It reveals that the phase of light field can offer a new degree of freedom to flexibly tune the magnitude and linewidth of LDOS and as a result, the quantum coherence of plasmonic resonance can be greatly enhanced. As the examples of quantum-optics applications, we demonstrate the advantages of hybrid CEP cavity in achieving strong light-matter interaction, enhancing quantum yield and photon correlations. To be specific, we find the enhanced coherent light-matter interaction while simultaneously reduced dissipation of polaritonic states, and a new mechanism of enhancing the quantum yield through the superscattering at CEP is identified, which holds great promise for realizing high quantum yield and generating indistinguishable single photons at room temperature. A physical implement is utilized to analyze the possible factors that hinder the experimental demonstration of the predicted high quantum yield. One direction of future study is the optimization of structure design with moderate plasmon-photon interaction, high quality factor and low intrinsic decay. We believe that our work can provide insights on harnessing the non-Hermitian physics of open quantum systems in quantum states control, which may benefit diverse quantum-optics applications.