1. Introduction

Strong light-matter interaction between quantum emitters and plasmonic nanocavity at room temperature has attracted great interests due to its promising applications in single-atom source, modulation of chemical reaction rate, and quantum information processing [1–6]. When strong coupling regime is reached, the energy exchanging between emitter and plasmon becomes stronger than the decay of system, and an energy-level splitting shows up in frequency-domain spectrums such as extinction spectrum [7,8], scattering spectrum [9,10], and photoluminescence (PL) spectrum [11,12]. This can lead to the formation of plexciton which is the hybridized states of emitter and nanocavity and can be used in quantum-state manipulation.

Recent years, many efforts have been made to realize single-emitter strong coupling with plasmon at room temperature [13–19], and various criterion of strong coupling are proposed [20,21]. Although strong coupling between single molecule and open plasmonic nanocavity has been realized in scattering spectrum [22–24], it is still a challenge to realize an obvious splitting in PL spectrum which is necessary for studying the quantum dynamics of plexciton. A few previous works refer that only Rabi splitting in PL spectrum can represent the realization of strong coupling rather than scattering spectrum [25,26], because the splitting in scattering spectrum may be disturbed by other factors such as Fano effect, electromagnetic induced transparency and so on. As for extinction spectrum, which is the sum of absorption and scattering, it may also be misled by scattering. Recently, researchers found that the splitting in scattering spectrum is not a pure quantum Rabi splitting [27], and the splitting in absorption spectrum is highly correlated to the dissipative decays of the coupling subsystems [28]. Therefore, it’s necessary to explore the essential principle of plexciton in PL spectrum.

Generally, the fluorescence of quantum emitters is weak and hard to be detected. Surface-enhanced fluorescence, utilizing the strong electric-field to enhance fluorescence, has been widely studied in recent years [29–31]. Researchers find that the fluorescence’s lifetime of emitter near a plasmonic nanocavity can shorten to tens of femtoseconds and the spontaneous emission rate of emitter is strongly enhanced, which is well known as Purcell effect. Actually, the local density of states (LDOS), describing the confinement of electric field near emitter, not only change the spontaneous emission but also determine the excitation rate which is proportional to fluorescence emission rate [32,33]. In weak coupling regime, Purcell effect is in good agreement with experiment. However, things are quite different in strong coupling regime. The intensity of fluorescence may sharply decrease due to the stronger Ohmic dissipation [32] in strong coupling regime and the lifetime is sharply shortened. On the other hand, the expected Rabi splitting is hardly found in PL spectrum. It usually shows up a single peak at uncoupled resonance with broadening linewidth in PL spectrum, while scattering spectrum appears an obvious splitting [34]. There is still a lack of recognized explanation and quantitative calculation for this phenomenon.

In this paper, we investigate the light emission of quantum emitter placed near a plasmonic nanocavity. A full-quantum method is used to describe the coupling between emitter and LSP mode. We find that the fluorescence enhancement produced by plasmonic nanocavity is not proportional to the coupling strength in strong coupling regime, instead, there is an optimum near intermediate coupling regime. Furthermore, we discuss the nonradiative decay of plexcitonic system in strong coupling regime, which will cause the broadening of linewidth in spectra. By using finite-difference time-domain (FDTD) method, we numerically calculate the specific coupling strength in a nanostructure of Ag bowties with an emitter, and obtain the relation between coupling strength and nonradiative decay of plexciton. Unlike previous works that the Rabi splitting is proportional to the coupling strength [26,34], we surprisedly find that the broadening linewidth may cover Rabi splitting in PL spectrum with the increasing of coupling strength, though there is a clear splitting in scattering spectrum. Our method gives a quantitative calculation for nonradiative decay of plexciton in PL spectrum, providing a promising way for both revealing single-emitter quantum optics and its applications in quantum devices, and quantum information processing.

2. Theory

We utilize a full-quantum method to describe the interaction between a quantum emitter and Ag bowties nanostructure composed by two metallic triangle nanoparticles shown in Fig. 1(a). Two regular triangle nanoparticles are set at the same plane with a gap. The quantum emitter is placed at the gap of bowties. The electromagnetic environment of localized surface plasmon (LSP) is set to vacuum. The non-Hermitian Hamiltonian of interaction between emitter and LSP contains three parts: $H=H_{e}+H_{p}+H_{I}$. Three parts are written as follows:

With initial state $\left |c_{e}(0)\right |=1$ and $\left |c_{g}(0)\right |=0$, we can solve the probability amplitudes for slowly varying written as

 

Fig. 1. (a) The diagram of bowties nanostructure composed by two Ag regular triangles with a quantum emitter placed at the center of gap. (b) The probability of $|E, 0\rangle$ and $|G, 1\rangle$. The parameters are set to $g=0.3$eV, $\gamma _{0}=0.03$eV, $\gamma _{p}=0.2$eV, $\Delta =0$.

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It should be noticed that the energy eigenvalue of this hybrid system is a complex because the Hamiltonian is non-Hermitian. This result can also be obtained from coupled oscillator model [18] written as $\omega _{\pm }=\frac {1}{2}\left \{\omega _{0}+\omega _{p}-i\left (\gamma _{0}+\gamma _{p}\right ) / 2 \pm \Omega _{R}\right \}$ where $\hbar =1$. $\Omega _{R}$ represent the Rabi frequency (or Rabi splitting in frequency domain) and display as Rabi oscillation which indicates the reaching of strong coupling regime. In weak coupling regime $g<\left |\gamma _{p}-\gamma _{0}\right | / 2$, the energy splitting can be written as $\omega _{\pm }=\frac {1}{2}\left \{\omega _{0}+\omega _{p}-i\left (\gamma _{0}+\gamma _{p}\right ) / 2 \pm i\Omega _{R}\right \}$. In this situation, the coupling between emitter and plasmon won’t cause the splitting in energy level but just influence the spectral linewidths of system. In intermediate coupling regime $\left |\gamma _{p}-\gamma _{0}\right | / 2<g<\left |\gamma _{p}+\gamma _{0}\right | / 2$, the energy splitting appears but may be covered by linewidth of plasmon. In strong coupling regime $g>\left |\gamma _{p}+\gamma _{0}\right | / 2$, the energy splitting is larger than linewidth and we can observe Rabi splitting in experimental spectrum. These criteria also works for PL spectrum, and We will explain more explicitly later.

Typically, spontaneous emission is calculated by integrating a two-time correlation function written as

Equations (5)–(8) describe the hybrid system with master equation and $\mathscr {L}$ is the Lindblad term and $[\cdot, \cdot ]\left ([\cdot, \cdot ]_{+}\right )$ denotes the commutator (anticommutator). We can express $\rho (t) \sigma ^{\dagger }$ into $\rho (t) \sigma ^{\dagger }=c_{e}^{*}(t)|\psi (t)\rangle \langle g, 1|$, and

Then the two-time correlation function can be written as

Therefore, we obtain the PL spectrum of emitter

At resonance $\Delta =0$, the total PL spectrum can be calculated by

It should be noticed that our method provides a simple way to calculate the PL spectrum which only needs a single integration instead of double integration of correlation function where quantum regression theorem is needed [26].

3. Simulation and discussion

The emitter side emission $P L_{\mathrm {side}}$, nanocavity axis emission $P L_{\mathrm {axis}}$ and total emission $P L_{\mathrm {total}}$ spectra are shown in Fig. 2(a).

 

Fig. 2. PL spectra calculated from eq. (12)–(14). (a) Emitter side emission $P L_{\mathrm {side}}$, nanocavity axis emission $P L_{\mathrm {axis}}$ and total emission $P L_{\mathrm {total}}$ spectra with $g=0.15$eV, $\omega _{0}=2.388$eV, $\gamma _{0}=0.03$eV, $\gamma _{p}=0.15$eV, $\Delta =0$. (b) The total PL spectrum $P L_{\mathrm {total}}$ with $\omega _{0}=2.388$eV, $\gamma _{0}=0.03$eV, $\gamma _{p}=0.15$eV, $\Delta =0$. Coupling coefficients $g$ is set to 0.01eV, 0.07eV, 0.1eV, 0.15eV. (c) $P L_{\mathrm {side}}$ and $P L_{\mathrm {axis}}$ with changing $\gamma _{0}$ while $\gamma _{p}$ is fixed at 0.15eV. (d) $P L_{\mathrm {side}}$ and $P L_{\mathrm {axis}}$ with changing $\gamma _{p}$ while $\gamma _{0}$ is fixed at 0.03eV.

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With $g=0.15$eV, $\omega _{0}=2.388$eV, $\gamma _{0}=0.03$eV, $\gamma _{p}=0.15$eV, $\Delta =0$, $PL_{\mathrm {axis}}$ shows stronger intensity than $PL_{\mathrm {side}}$. This indicates that the emission of single emitter is weaker than nanocavity’s, though emitter’s decay $\gamma _{0}$ is smaller than nanocavity’s decay $\gamma _{p}$. In Fig. 2(c) and Fig. 2(d), $P L_{\mathrm {side}}$ and $P L_{\mathrm {axis}}$ are given with different $\gamma _{0}$ and $\gamma _{p}$, respectively. We find that the Rabi splitting of $P L_{\mathrm {side}}$ and $P L_{\mathrm {axis}}$ decreases while $\gamma _{0}$ and $\gamma _{p}$ increase, because larger decays would weaken the energy exchanging. This can also be proved by $\Omega _{R}=\sqrt {4 g^{2}-\left [i \Delta +\left (\gamma _{p}-\gamma _{0}\right ) / 2\right ]^{2}}$. When it comes to intensity, $P L_{\mathrm {axis}}$ decreases obviously with the increasing of $\gamma _{0}$, while the intensity of $P L_{\mathrm {side}}$ barely changes. It means that the emission intensity of emitter barely changes in the same coupling strength and the dissipation in nonradiative channel of emitter increases with larger $\gamma _{0}$. The emission intensity of $P L_{\mathrm {axis}}$ decreases obviously because energy decays in nonradiative channel of emitter before energy exchanging. On the other hand, the increasing of $\gamma _{p}$ not only decreases the intensity of $P L_{\mathrm {side}}$ but also changes the intensity of $P L_{\mathrm {axis}}$. To explore the total PL intensity with different coupling strength, we obtain the total PL spectra as shown in Fig. 2(b). When coupling coefficient is set to 0.01eV with $\Delta =0$, $\gamma _{0}=$0.03eV and $\gamma _{p}=$0.15eV, $g<\left |\gamma _{p}-\gamma _{0}\right | / 2$ and the system is in weak coupling regime showing a single peak in PL spectrum (black line). When the system achieves intermediate coupling regime, where $\left |\gamma _{p}-\gamma _{0}\right | / 2<g<\left |\gamma _{p}+\gamma _{0}\right | / 2$, both intensity and linewidth of single peak increases shown as bule line in Fig. 2(b). In this regime, there should be a Rabi splitting in spectrum because $\Omega _{R}>0$. However, the large linewidth of nanocavity covers the small splitting. This phenomenon is often seen in experiment. When it comes to strong coupling regime ($g>\left |\gamma _{p}+\gamma _{0}\right | / 2$), the splitting becomes larger than linewidth and a clear splitting can be observed in PL spectrum. We also notice that intensity of fluorescence doesn’t increase with coupling strength, instead it has an optimum for surface-enhanced fluorescence in intermediate coupling regime. This result can be explained as follows. There are usually two decay channels in emitter-nanocavity system: radiative decay and nonradiative decay which are both proportional to electric-field enhancement. When the coupling coefficient increases from weak coupling regime, the energy exchanging increases and radiative decay of this system dominates, though both radiative decay and nonradiative decay are enhanced. When the coupling coefficient increases from intermediate coupling regime, the nonradiative decay of emitter $\gamma _{nr}$ increases exponentially and becomes dominating [32]. Besides, more energy takes part in energy exchanging between emitter and nanocavity, which finally decays in nonradiative channel, instead of emission. In this situation, fluorescence intensity decreases and a clear Rabi splitting may also not be observed in PL spectrum because of the increasing dissipation in plexciton. In Fig. 2(d), we can find obvious Rabi splitting in PL spectrum because the nonradiative decay dependent on coupling strength has not been considered yet. We will discuss it later.

There are many reasons responsible for nonradiative decay such as Ohmic loss, electron tunnelling and nonlocal effect [39]. Here, we focus on nonradiative decay caused by Ohmic dissipation. The fluorescence rate of emitter placed near a nanoparticle can be written as $\gamma _{\mathrm {em}}=\gamma _{\mathrm {exc}}\left [\gamma _{r} /( \gamma _{r}+\gamma _{nr})\right ]$, where $\gamma _{\mathrm {exc}}$ is the excitation rate depending on local electric field. $\gamma _{r}(\gamma _{nr})$ represents radiative (nonradiative) decay rate of emitter. By obtaining stronger enhancement of electric field, fluorescence rate can be enhanced, and $\gamma _{\mathrm {exc}}, \gamma _{r}, \gamma _{nr}$ increases at the same time. However, $\gamma _{nr}$ raises much faster than $\gamma _{r}$ and offsets the profit from $\gamma _{exc}$ when the electric-field enhancement exceeds a critical value. In this region, Ohmic dissipation dominates, where $\gamma _{nr}$ increases exponential, leading to broadening linewidth in PL spectrum totally changing the observable spectrum. In strong coupling regime, where $g>\left |\gamma _{p}+\gamma _{0}\right | / 2$, an obvious splitting should be found in spectrum, but the larger linewidth of emitter may cover the Rabi splitting in PL spectrum. This explains why Rabi splitting is hardly found in PL spectrum at room temperature which greatly limits the application of strong coupling in quantum devices. To further explore the characteristics of fluorescence, we calculate the coupling coefficient which is written as $g=\hbar \omega _{0} \mu E /\left (\sqrt {2 \hbar \varepsilon _{0} \varepsilon _{r} V} E_{\max }\right )$ where $V$ is the mode volume of electric-filed mode. $\mu$ is the dipole moment of emitter and $\varepsilon _{0} ( \varepsilon _{r})$ is vacuum (relative) permittivity. $E/E_{\max }$ denotes the normalization of electric field. We use finite-difference time-domain (FDTD) method to obtain the exact value of electric field enhancement and mode volume in emitter-bowties coupling system, and the electric-filed enhancement distribution of Ag bowties nanostructure are shown in Fig. 3(a) and Fig. 3(b).

 

Fig. 3. Simulation obtained from FDTD method. (a) Electric-field enhancement of single Ag bowties structure when the polarization of incident light is parallel to x axis. (b) Electric-field enhancement when the polarization of incident light is parallel to y axis. (c) Electric-field enhancement along x axis when y=0. (d) Simulated scattering spectrum of single Ag bowties structure with different sizes of gaps.

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The Ag bowties is composed by two triangle nanoparticles with 60nm in length and 20nm in thickness. The corner radius is set to 2nm. We adopted the dielectric constant for the Ag bowties structure from Johnson and Christy [40]. There is a 10nm gap between two nanoparticles, which can produce strong electric-field enhancement to realize strong light-matter interaction. Different electric-field modes of Ag bowties can be excited by incident light with specific polarization. For example, when the polarization of incident light is parallel to x axis, a gap mode, shown in Fig. 3(a), can be obtained. In this mode, electric-field is confined in the gap of two triangles structures and the maximal enhancement can be achieved. The electric-field enhancement reduces to minimum while the polarization is parallel to y axis because the gap mode is not excited and energy no longer gathers in gaps shown in Fig. 3(b). Figure 3(c) shows the electric-field enhancement distribution along x axis when y axis is fixed at 0. We find that the maximum point of enhancement is near the surface of nanoparticles’ tip instead of gathering at the middle of gap. We simulate the scattering response of Ag bowties with different gaps shown in Fig. 3(d). As the size of gap increases, the resonance of gap mode blue shifts closing to single Ag triangle nanocavity’s resonance.

Here we choose a CdSe/ZnS quantum dot as the quantum emitter, which has a diameter of 8nm. This quantum dot can be modeled by Lorentzian model written as $\varepsilon _{q e}=\varepsilon _{\infty }+f \omega _{0}^{2} /\left (\omega _{0}^{2}-\omega ^{2}-i \gamma _{0} \omega \right )$ where $\varepsilon _{\infty }$ is set to 6.1 and $\omega _{0}$ is set to 527 nm while $\gamma _{0}, f$ is set to 80meV and 0.6, respectively. The orange line in Fig. 4(a) represents the scattering result from simulation.

 

Fig. 4. (a) Scattering spectrum from simulation (orange line) and its fitting (blue line) with $g$=66meV, $\gamma _{0}$=80meV, $\gamma _{p}$=150meV. Yellow line represents the PL spectrum which uses the same parameters. (b) $P L_{\textrm {side }}$ (blue dashed line), $P L_{\textrm {axis }}$ (orange dashed line) and $P L_{\textrm {total }}$ (black dashed line) with $g$=150meV, $\gamma _{0}$=80meV, $\gamma _{p}$=150meV. Blue solid line, orange solid line and black solid line represent the three PL spectra ($P L_{\textrm {side }}$, $P L_{\textrm {axis }}$, and $P L_{\textrm {total }}$) after broadening, respectively. (c) PL spectrum after broadening (solid line) and PL spectrum (dashed line) with $g$=10meV (blue), 50meV (red), 65meV (pink) and 100meV (green). $\gamma _{0}$=80meV, $\gamma _{p}$=150meV. (d) The Rabi splitting in scattering spectrum (blue line), PL spectrum (red line), and PL spectrum after broadening (pink line) changes with coupling coefficient$g$ .

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There is a red shift of bowties structure’s resonance, because the appearance of quantum dot has changed the permittivity of the gap environment where electric field is confined. We fit it with scattering spectrum calculation proposed in [41] (blue line), and obtain the coupling coefficient $g=66$meV. Here we should emphasize that the coupling coefficient is a statistical result of collective oscillation of many small oscillators. As shown in Fig. 3(c), the electric-field enhancement changes rapidly which means that each part of 8nm quantum dot may be enhanced by different electric field intensity. Therefore, the total coupling coefficient can be considered as a collective oscillation of many small oscillators. Another evidence that collective oscillation works here is that the scattering spectrum and PL spectrum are determined by the coupling coefficient of collective oscillation instead of the largest coupling coefficient of single small oscillator. We obtain the PL spectrum (yellow line) in Fig. 4(a) from Eq. (14) instead of simulation because quantum process cannot be modeled by FDTD method. Obviously, when $g$ $\gamma _{0}$, $\gamma _{p}$ are 66meV, 80mV and 150meV, $\left |\gamma _{p}-\gamma _{0}\right | / 2 <g<\left |\gamma _{p}+\gamma _{0}\right | / 2$ indicating that the system is in intermediate coupling regime. However, the scattering spectrum has shown obvious Rabi splitting because the first criterion $g>\left |\gamma _{p}-\gamma _{0}\right | / 2$ has been satisfied. For PL spectrum, the second criterion $g>\left |\gamma _{p}+\gamma _{0}\right | / 2$ is not satisfied, so the Rabi splitting is covered by the linewidth. We have discussed above that the emission rate of fluorescence is strongly effected by electric-filed enhancement, i.e. coupling strength. Too strong enhancement may produce strong Ohmic dissipation and lead to strong nonradiative decay in plexcitonic system. Therefore, we develop a broadening linewidth to describe and evaluate the nonradiative decay channel of system. Equation (14) can be revised as

4. Conclusion

In this paper, we theoretically investigate the fluorescence of single quantum emitter in an open plasmonic nanocavity. By utilizing full-quantum method, we obtain the PL spectrum of complex system in different coupling regime and find that fluorescence usually gets maximally enhanced in intermediate coupling. In strong coupling regime, strong electric-field enhancement will increase the nonradiative decay and lead to the broadening of linewidth in plexciton. To obtain quantitative calculation for the hybrid system, FDTD method is used to obtain the electric-field enhancement and scattering spectrum. By comparing PL spectrum with scattering spectrum, we find that the Rabi splitting in PL spectrum is usually smaller than that in scattering spectrum because of nonradiative decay in plexciton, and large coupling strength can greatly increase the nonradiative decay. This leads to the broadening of linewidth in plexciton covering the Rabi splitting in PL spectrum. As previous works have proved that Rabi splitting is proportional to the coupling strength [22,26], our results may seem confusing. It is because large coupling strength can lead to strong nonradiative decay broadening the linewidth. We further give the optimum of observing Rabi splitting in PL spectrum, which provides a theoretical support for the experiments and has applications in single photon source, quantum devices, and quantum information processing.