Quantized quasinormal-mode description of nonlinear cavity-QED effects from coupled resonators with a Fano-like resonance

We employ a recently developed quantization scheme for quasinormal modes (QNMs) to study a nonperturbative open cavity–QED system consisting of a hybrid metal-dielectric resonator coupled to a quantum emitter. This hybrid cavity system allows one to explore the complex coupling between a low-$Q$ (quality factor) resonance and a high-$Q$ resonance, manifesting in a striking Fano resonance, an effect that is not captured by traditional quantization schemes using normal modes or a Jaynes-Cummings (JC)–type model. The QNM quantization approach rigorously includes dissipative coupling between the QNMs and is supplemented with generalized input-output relations for the output electric field operator for multiple modes in the system and correlation functions outside the system. The role of the dissipation-induced mode coupling is explored in the strong coupling regime between the photons and emitter beyond the first rung of the JC dressed-state ladder. Important differences in the quantum master equation and input-output relations between the QNM quantum model and phenomenological dissipative JC models are found. For the hybridized high-$Q$ cavity mode, we show how the dissipation-induced coupling causes a significant reduction in the cavity-emitter coupling rate, and the cavity decay rate, compared to a simpler JC model. In a second step, numerical results for the Fock distributions and system as well as output correlation functions obtained from the quantized QNM model for the hybrid structure are compared with results from a phenomenological approach. We demonstrate explicitly how the quantized QNM model manifests in multiphoton quantum correlations beyond what is predicted by the usual JC models.

Figure 1

(a) Top view on the metal-dielectric hybrid structure, consisting of a metallic dimer on top of a photonic crystal beam. (b) Side view on the metallic dimer, supporting a quantum dipole ($z$ polarized) in the middle of the gap. (c) Schematic of the quantum subsystems: The photonic crystal-like mode (violet) is coupled to the plasmonic-like cavity mode (yellow) and the TLS (red) is coupled to both symmetrized QNMs. (d) Excitation and input-output scheme. The TLS is driven by a continous wave (cw)-pump field with strength ${\mathrm{\Omega }}_{\mathrm{L}}$ and decays with a rate ${\gamma }_{\mathrm{SE}}$, while the QNM subsystem is characterized by the decay rates ${\mathrm{\Gamma }}_{\mathrm{pl}}$ and ${\mathrm{\Gamma }}_{\mathrm{pc}}$, naturally entering the model through the input fields $A_{\mathrm{pl}}^{\mathrm{in}}$ and $A_{\mathrm{pc}}^{\mathrm{in}}$, respectively (see text).

Figure 2

Classical Purcell factors (first and second rows) and classical radiative $\beta$ factors (third and fourth row) from an embedded dipole emitter in the metal-dielectric hybrid structure from Fig. 1 with gap sizes of 2 and 5 mn, obtained from full Maxwell simulations over a frequency regime covering both hybrid resonances (left) with an enlargement close to the high-$Q$ resonance (right). Note that for the 5-nm (2-nm) gap case, the numerical calculations were done for 81 (93) nonequidistant frequency points in the interval $\hslash \omega \in [1.55,1.95]\mathrm{eV}$, and that the corresponding $F_p^{\mathrm{num}}$ and ${\beta }_{\mathrm{class}}^{\mathrm{rad}}$ values are linearly interpolated. Also note that the maximum Purcell factor of the 2-nm gap hybrid is one order of magnitude higher compared to the 5-nm gap hybrid. The value of ${\beta }_{\mathrm{class}}^{\mathrm{rad}}$ for the 2-nm gap has additional peaks for frequencies towards the low-$Q$ mode, which likely come from nonmodal quasistatic coupling and a constant background term with higher order modes, whose influence reduces with decreasing gap size. For all simulations, the classical dipole is at the gap center of the dimer and is $z$ polarized.

Figure 3

Purcell factor of a $z$-polarized dipole emitter in the metal-dielectric hybrid structure from Fig. 1, shown over a broad frequency range from full dipole simulations (dots), the two-mode QNM-JC model (solid magenta line), and the phenomenological dissipative JC model (dashed green line). Two peaks appear around the hybridized QNM frequencies $\hslash {\tilde{\omega }}_{\mathrm{pl}}=1.6999-0.0479i(\mathrm{eV})$ and $\hslash {\tilde{\omega }}_{\mathrm{pc}}=1.6052-0.0007i(\mathrm{eV})$, originating from the metallic ellipsoidal dimer and the photonic-crystal beam, respectively. The highly non-Lorentz interference effect (dip in $F_p$) is located near one of the eigenfrequencies, ${\epsilon }_{\mathrm{em}}^{\left(\mathrm{pc}\right)}$, of the full electromagnetic Hamiltonian $H_{\mathrm{em}}$ of the QNM-JC model and is fully recaptured by the intermode coupling terms.

Figure 4

Quantum modal $\beta$ factor of the metal-dielectric hybrid structure from Fig. 1, shown over a broad frequency range for the two-mode quantum QNM model [solid magenta line, Eq. (73)]. For the phenomenological dissipative JC model, a phenomenological constant is shown (green dashed line with $\beta \approx 0.14$), which mimics the behavior of the radiative output at TLS frequencies around ${\omega }_{\mathrm{pc}}$ with respect to the classical ${\beta }_{\mathrm{class}}^{\mathrm{rad}}$ solution from Fig. 2. The inset shows an enlargement around the PC-like mode frequency $\hslash {\omega }_{\mathrm{pc}}=1.6052(\mathrm{eV})$. In the QNM-JC model, a peak close to the PC-like mode with frequency ${\omega }_{\mathrm{pc}}$ appears, which is located at the position of the highly non-Lorentz interference effect (dip in $F_p$) from Fig. 3.

Figure 5

Absolute square of the basis coefficients $c_{k_{\mathrm{pc}},k_{\mathrm{pl}},l}^{i_N}$ for the one- and two-excitation manifold ($N=1,2$) with the TLS frequency ${\omega }_{\mathrm{a}}={\epsilon }_{\mathrm{em}}^{\left(2\right)}$ aligned with the eigenvalue of the electromagnetic Hamiltonian $H_{\mathrm{em}}$ close to the PC frequency. The solid (QNM-JC model) and dashed (phenomenological dissipative JC model) bars reflect the contribution of the (bare) eigenstates $|k_{\mathrm{pl}},k_{\mathrm{pc}},l\rangle$ of the uncoupled photon-exciton system to the eigenstates $|{\phi }_{N,i_N}\rangle$ of the actual coupled photon-exciton system. The eigenstates of the coupled system are similar to the respective bare states, since the coupling constants indicate a regime below ultrastrong coupling (cf. Table 1).

Figure 6

(a) First (threefold) and second (fivefold) rung of the QNM-JC energy ladder with ${\omega }_{\mathrm{a}}$ aligned to the eigenfrequency of the electromagnetic Hamiltonian $H_{\mathrm{em}}$ close to the PC mode frequency: The solid lines show the real part of the complex eigenenergies $\left({\epsilon }_{N,i_N}\right)$, while the gray area covers the imaginary part and is bounded by $\mathrm{Re}\left({\epsilon }_{N,i_N}\right)\pm 0.2\mathrm{Im}\left({\epsilon }_{N,i_N}\right)$. (b) Eigenenergies ${\epsilon }_{1,2}$ and ${\epsilon }_{2,2}$: The solid (light blue dashed) line shows the real part of the eigenenergies in the QNM-JC model (phenomenological dissipative JC model), while the gray (light blue dashed) area reflects the imaginary part and is bounded by $\mathrm{Re}\left({\epsilon }_{N,i_N}\right)\pm \mathrm{Im}\left({\epsilon }_{N,i_N}\right)$. Direct two-photon transitions via an external laser with frequency ${\omega }_{\mathrm{L}}\left({\omega }_{\mathrm{L}}^{\prime }\right)$ are sketched. Note that in both subfigures, there is a sudden jump on the energy axis from the first to second rung and that the scaling is different in the upper and lower halves.

Figure 7

Occupation numbers $n_{\mathrm{a}},n_{\mathrm{pc}},n_{\mathrm{pl}}$ and probabilities $P_{\mathrm{a}},P_{\mathrm{pc}},P_{\mathrm{pl}}$ of the TLS (corresponding to upper level $|e\rangle$), plasmon, and PC-like mode in the steady state, obtained with the QNM-JC model (solid) and phenomenological dissipative JC model (dashed) as functions of laser frequency ${\omega }_{\mathrm{L}}$ around ${\omega }_{\mathrm{pc}}$. The TLS frequency $\hslash {\omega }_{\mathrm{a}}={\epsilon }_{\mathrm{em}}^{\left(2\right)}$ is aligned to the eigenenergy of the full electromagnetic Hamiltonian $H_{\mathrm{em}}$ (${\omega }_{\mathrm{pc}}$ in the phenomenological dissipative JC model) and the TLS is pumped with an external laser with Rabi frequency $\mathrm{\Omega }\sim |{\tilde{G}}_{\mathrm{pc}}|$, where $|{\tilde{G}}_{\mathrm{pc}}|\approx 8{\mathrm{\Gamma }}_{\mathrm{pc}}$ is the TLS-PC coupling constant (in the strong photon-exciton coupling regime) and ${\mathrm{\Gamma }}_{\mathrm{pc}}$ is the PC mode decay rate (cf. Table 1). Additionally, we show $P_{\mathrm{pc}}$ for the QNM-JC model in the limit ${\tilde{g}}_{\mathrm{em}}=0$ (dotted line); cf. Eq. (55).

Figure 8

Probabilities $P_{a-\mathrm{pc}},P_{\mathrm{pc}-\mathrm{pc}},P_{a-\mathrm{pl}}$ corresponding to the two-excitation manifold in the steady state obtained with the QNM-JC model (solid) and phenomenological dissipative JC model (dashed) as functions of laser frequency ${\omega }_{\mathrm{L}}$ around the PC mode resonance ${\omega }_{\mathrm{pc}}$. The TLS frequency $\hslash {\omega }_{\mathrm{a}}={\epsilon }_{\mathrm{em}}^{\left(\mathrm{pc}\right)}$ is aligned to the eigenenergy of the full electromagentic Hamiltonian $H_{\mathrm{em}}$ (${\omega }_{\mathrm{pc}}$ in the phenomenological dissipative JC model) and the TLS is pumped with an external laser with Rabi frequency ${\mathrm{\Omega }}_{\mathrm{L}}\sim \left|{\tilde{G}}_{\mathrm{pc}}\right|$, where $|{\tilde{G}}_{\mathrm{pc}}|\approx 8{\mathrm{\Gamma }}_{\mathrm{pc}}$ is the TLS-PC coupling constant (in the strong photon-exciton coupling regime) and ${\mathrm{\Gamma }}_{\mathrm{pc}}$ is the PC mode decay rate (cf. Table 1). We also show $P_{\mathrm{pc}-\mathrm{pc}}$ for the QNM-JC model in the limit ${\tilde{g}}_{\mathrm{em}}=0$ (dotted line), cf. Eq. (55), and the occupation number $n_{\mathrm{pc}}$ normalized to the same height as $0.1P_{\mathrm{pc}-\mathrm{pc}}$. Note, that all other contributions corresponding to the two-excitation manifold are negligible and not shown here.

Figure 9

Output intensity function $I^{\mathrm{out}}$ measured on a sphere in the far field as function of laser frequency ${\omega }_{\mathrm{L}}$ for the QNM-JC model (solid) and the phenomenological dissipative JC model (dashed) in the steady state with the TLS frequency $\hslash {\omega }_{\mathrm{a}}={\epsilon }_{\mathrm{em}}^{\left(\mathrm{pc}\right)}$ aligned to the eigenenergy of the full electromagentic Hamiltonian $H_{\mathrm{em}}$. The Rabi frequency of the external pump is ${\mathrm{\Omega }}_{\mathrm{L}}\sim \left|{\tilde{G}}_{\mathrm{pc}}\right|$, where the TLS-PC coupling is in the strong coupling regime, i.e., $|{\tilde{G}}_{\mathrm{pc}}|\approx 8{\mathrm{\Gamma }}_{\mathrm{pc}}$ (cf. Table 1). The lower plot shows an enlargement close to the PC-mode frequency ${\omega }_{\mathrm{pc}}$. The quantities are normalized to $\max \left(I^{\mathrm{out}}\right)$ from the QNM-JC model.

Figure 10

Normalized second-order quantum correlation functions in the steady-state regime for the plasmonic-like mode (blue), photonic-crystal mode (green), and the output electric field (magenta) for the QNM-JC model (solid) and the phenomenologically dissipative JC model (dashed) over laser frequency ${\omega }_{\mathrm{L}}$. Lower plot shows an enlargement close to the PC-mode frequency ${\omega }_{\mathrm{pc}}$.

Figure 11

Excited state occupation $n_{\mathrm{a}}=\langle {\sigma }^{+}{\sigma }^{-}\rangle$ over time for different dipole strengths in units of debye (D), using the quantized QNM model for the hybrid structure with initial occupation $n_{\mathrm{a}}(t=0)=1$ and without external pump. The time axis is normalized to the cavity-enhanced spontaneous emission rate $\mathrm{\Gamma }$ from Eq. (68) scaled with the respective dipole strength. The value of $n_{\mathrm{a},\mathrm{Bad}}$ (black dashed line) reflects the exponential decay in the bad cavity limit, i.e., $n_{\mathrm{a},\mathrm{Bad}}\left(t\right)=exp[-(\mathrm{\Gamma }+{\gamma }_{\mathrm{SE}})t]$.

Figure 12

The one- and two-excitation probability of the PC-like eigenstate at the laser frequency of maximum probability peak for the QNM-JC (solid) and phenomenological dissipative JC model (dashed) over Rabi frequency ${\mathrm{\Omega }}_{\mathrm{L}}$. Lower plot shows the same result as upper plot with scaled ${\mathrm{\Omega }}_{\mathrm{L}}$ axis with $g_{\mathrm{pc}}^{\mathrm{eff}}={\tilde{g}}_{\mathrm{pc}}$ for the phenomenological dissipative JC model and $g_{\mathrm{pc}}^{\mathrm{eff}}={\tilde{G}}_{\mathrm{pc}}$ for the QNM-JC model. ${\mathrm{\Omega }}_{\mathrm{L}}=g_{\mathrm{pc}}^{\mathrm{eff}}$ signifies the intermediate excitation regime. Both models have different intermediate excitation regimes, since $|{\tilde{g}}_{\mathrm{pc}}|/|{\tilde{G}}_{\mathrm{pc}}|\approx 5$.