In our experiment, the molar ratio (PO₄³⁻ : [Bi³⁺ + Eu³⁺]) of doped ions is adjusted with different concentrations of Hexagonal (H) and monoclinic (M) phase to obtain different phase transitions of BiPO₄ nanocrystals (7 : 1, 6 : 1, 1 : 1, 0.5 : 1) as shown in Fig. S1(a-f) (See Supplementary materials for detailed physical images of the samples). The (7 : 1, 6 : 1, 1 : 1) BiPO₄ crystals with C₁ point group site symmetry has a proportion of M-phase, whereas the concentration of H-phase is high in (0.5 : 1) the BiPO₄ sample (C₂ symmetry), which results in more lattice-vibrating phonon^36,37. The Eu³⁺ : NaYF₄ (YF₄³⁻ : [Na³⁺ + Eu³⁺] = 1 : 1/4) exhibits pure H-phase with Cs symmetry. The transition between H-phase to M-phase has been achieved by increasing the phosphate ion (PO₄³⁻) concentration in the Eu³⁺ : BiPO₄ system, while different crystal phases are synthesized by adjusting the pH values^38-42.

In the experiment, samples were held in a cryostat (CFM-102) with temperatures varying from 300 K (strong phonon dressing) to 77 K (weak phonon dressing) by flowing liquid nitrogen. The left and middle of Fig. 1a show crystal field (CF) splitting levels of Eu³⁺ : BPO₄ and Eu³⁺ : NaYF₄, respectively. The right of Fig. 1a shows photon-assisted CF splitting in Eu³⁺ : BPO₄/NaYF₄. The ⁷F₁ fine structure (J = 1) splits into three levels: ⁷F_{1, MJ = −1} (587.3 nm), ⁷F_{1, MJ = 0} (592.3 nm), and ⁷F_{1, MJ = +1} (597.3 nm) under the dressing and CF effect of BiPO₄. Fig. 1b illustrates the subenergy levels corresponding to the right of Fig.1a. Two dye lasers (narrow scan with a 0.04 cm⁻¹ linewidth and broad scan with a 0.08 cm⁻¹ linewidth, R610 dye) pumped by an injection-locked single-mode Nd : YAG laser (Continuum Powerlite DLS 9010, 10 Hz repetition rate, 5 ns pulse width) were employed to generate the two pumping fields $E_{1}\left(\omega_{1}\right)$ and $E_{2}\left(\omega_{2}\right)$. By exciting the Eu³⁺:BiPO₄ crystal with E₁ and $\boldsymbol{E}_{1}^{\prime}$ (reflection of E₁) (Fig. 1d), the output Stokes (E_S)/anti-Stokes (E_AS) signals are generated under a phase-matched condition ( $K_{1}^{\prime}$ + $K_{1}$ = $K_{S}$ + $K_{AS}$) due to interaction with Eu³⁺ : BiPO₄ energy levels. The out-of-phase FL1 and FL2 are generated through E₁ (broadband laser) and E₂ (narrowband laser), respectively. The pulse generated from the Nd³⁺ : YAG laser is used to simultaneously trigger a boxcar-gated integrator (G) and oscilloscope (OS). The input laser beams are along the [010] axis of the crystal, which is perpendicular to the optical axis. The spectral optical outputs are obtained by scanning laser frequency, while the time-resolved OS achieves the temporal optical outputs by fixing dye laser (DL) frequency. The grating motor of DL1 and DL2 is scanned by computer to form the x-axis (wavelength), and the intensity of the excitation spectrum is the average of ten shots from the gated integrator appearing on the y-axis. The optical signal generated is detected at photomultiplier tubes (PMTs) via confocal lenses (CL), as shown in Fig. S1(g) (See Supplementary materials for detailed experimental setup). While PMT2 is positioned close to the sample to detect hybrid signal (FL + E_S/AS) with dominant FL emission, PMT1 (near-detector position) and PMT3 (far-detector position) are precisely positioned to detect the generated in-phase spontaneous four-wave mixing (SFWM) (Fig. 1b). By adjusting the GP, one can obtain output signals from different energy levels with different lifetimes. The ratio of FL to E_S/AS in a hybrid signal is controlled by GP. Due to the distinct decay rates exhibited by the FL and SFWM, they can be easily differentiated at the PMT by adopting a boxcar GP. Furthermore, Fano resonance strongly depends upon point-group site symmetry of different phases of Eu³⁺ : BiPO₄. Fig. 1c shows the proposed model of a non-Hermitian multichannel optical router. Fig. 1d shows the dressing-like Zeeman splitting of ⁷F₁ energy level. For the ⁷F₁ energy level by dressing field quantized rotation, the following splitting energy levels can be obtained: ⁷F_{1, MJ =−1,} ⁷F_{1, MJ = 0,} and ⁷F_{1, MJ = +1.}

The proposed model shows that more phonons can couple more energy levels, as shown in Fig. 1a3. The excitations from single laser and two lasers show a single Fano dip (self-dressing) and multi-Fano dips (external dressing), respectively. It is worth mentioning that multidressing can only be generated from photon-phonon multi-Fano interference, not through excitations of one or two lasers. For the Eu³⁺ : BiPO₄ sample, unique lattice vibrations can produce phonons with different frequencies which are coupled to different CF splitting levels ⁵D₁ →⁷F₁, ⁵D₀ →⁷F₁, and ⁵D₀ →⁷F₃ in the ion Eu³⁺. Therefore, more phonon results in effective dressing³⁶. The findings in the current work reveal the five sharp dips, which can only be explained by the combined effect of photon dressing and phonon.

By opening field E₁, the dressed third-order density matrix element for E_S ( $\rho_{S}^{\prime(3)}$) and E_AS ( $\rho_{AS}^{\prime(3)}$) via perturbation chains $\rho_{00}^{(0)} \xrightarrow{E_{1}} \rho_{20}^{(1)} \xrightarrow{E_{\text {AS }}} \rho_{00}^{(2)} \xrightarrow{E_{1}^{\prime}} \rho_{20(\mathrm{~S})}^{(3)}$ and $\rho_{00}^{(0)} \xrightarrow{E_{1}^{\prime}} \rho_{20}^{(1)} \xrightarrow{E_{S}} \rho_{00}^{(2)} \xrightarrow{E_{1}} \rho_{20(\mathrm{AS})}^{(3)}$, respectively, can be written as follows:

where, $d_{1}=\left|G_{1}\right|^{2} /\left(\Gamma_{20}+i \Delta_{1}-i \Delta_{\mathrm{AS}}+d_{3}\right)$, $d_{3}=\left|G_{\mathrm{p} 1}\right|^{2} /\left(\Gamma_{10}+i \Delta_{1}-i \Delta_{\mathrm{AS}}-i \Delta_{\mathrm{p} 1}\right)$, $d_{2}=\left|G_{1}\right|^{2} /\left(\Gamma_{20}+i \Delta_{1}^{\prime}-i \Delta_{\mathrm{S}}+d_{4}\right), d_{4}=\left|G_{\mathrm{p} 1}\right|^{2} /\left(\Gamma_{10}+i \Delta_{1}^{\prime}-i \Delta_{\mathrm{S}}-i \Delta_{\mathrm{p} 1}\right),$ $\left|\rho_{\mathrm{S} / \mathrm{AS}}^{\prime(3)}\right|=\left(A^{2}+B^{2}+C^{2}+2 A B \cos \left(\Delta \varphi_{1}\right)+2 A C \cos \left(\Delta \varphi_{2}\right)+2 B C \cos \left(\Delta \varphi_{3}\right)\right)^{1 / 2}$.The linewidth of the E_S/AS signal is $\Gamma_{\mathrm{S} / \mathrm{AS}}=\Gamma_{20}+\Gamma_{00}+\Gamma_{20}$. The transverse dephase rate $\Gamma_{\mathrm{ij}}=\left(\Gamma_{\mathrm{i}}+\Gamma_{\mathrm{j}}\right) / 2$, where, $\Gamma_{\text {phonon }}$ is related to the sample temperature and $\Gamma_{\text {dressin }} g$ is associated with dressing. Here, two Fano interferences are controlled by three phases ( $\Delta \varphi_{1}=\pi, \Delta \varphi_{2}=\pi, \Delta \varphi_{3}=0$). The aforementioned equations show nested double dressing suggesting two-Fano interference is originated from one continuous state $\rho_{\mathrm{S} / \mathrm{AS}}^{\prime(3)}$ and two bound states (photon1 $\rho_{\text {S/AS }}^{\prime(5)}$ photon1–phonon1 $\rho_{\text {S/AS }}^{\prime \prime(7)}$). $G_{\mathrm{i}}=-\mu_{\mathrm{H}} H / \hbar$ is the photon Rabi frequency, where, $H$ is the magnetic field for Rabi frequency, $\mu_{H}$ is the magnetic dipole matrix elements levels $|m\rangle$ and $|n\rangle$ and $|m\rangle$ and $|n\rangle$ are the CF-splitting energy levels of ⁵D₀ and ⁷F₁, respectively. $\Gamma_{\mathrm{mn}}$ is the transverse decay rate between levels $|m\rangle$ and $|n\rangle$. The frequency detuning $\Delta_{\mathrm{i}}=\Omega_{\mathrm{mn}}-\omega_{\mathrm{i}}$, where, $\Omega_{\mathrm{kld}}$ is the corresponding atomic transition frequency between levels $|m\rangle$ and $|n\rangle$. $\omega_{i}$ is the frequency of the laser field $\boldsymbol{E}_{i}$.

Photon excitation is resulted from the interaction of laser light with the host material (Eu³⁺ : BiPO₄). On the other hand, phonon excitation arises from the interaction of the doped ion with a source of phonons (crystal lattice vibration), which are vibrations in the crystal lattice. The phonon Rabi frequency is described as the phonon Rabi frequency $G_{\mathrm{pi}}=-\mu_{\mathrm{kl}} E_{\mathrm{pi}} / \hbar, \mu_{\mathrm{kl}},$ which is the dipole moment between $|k\rangle$ and $|l\rangle$ of CF energy levels in ⁷F₁. $\omega_{\mathrm{pi}}$ is the phonon frequency of phonon field $E_{\mathrm{pi}}$, which is determined by the vibration frequency of crystal lattice state mode. The $\Gamma_{\mathrm{kl}}$ is the transverse decay rate between $|K\rangle$ and $|l\rangle$. $G_{\mathrm{i}}=-\mu_{\mathrm{kl}} E_{\mathrm{i}} / \hbar$ is the frequency detuning, where, $\Omega_{\mathrm{kl}}$ is the resonant frequency between $|K\rangle$ and $|l\rangle$. $\omega_{\mathrm{i}}$ is the frequency of photon field $E_{i}$, which is determined by the vibration frequency of crystal lattice state mode. For the three dark states of nested three dressing, the expression (Fig. 5e) can be written as follows:

Where, $\begin{array}{l} \text d_{00}=\left|G_{1}\right|^{2} /\left(\Gamma_{20}+i \Delta_{1}+\left|G_{p 1}\right|^{2} /\left(\Gamma_{10}+i \Delta_{1}-i \Delta_{\mathrm{p} 1}+\right.\right. \\ \left.\left|G_{\mathrm{p} 2}\right|^{2} /\left(\Gamma_{13}+i \Delta_{1}-i \Delta_{\mathrm{p} 1}-i \Delta_{\mathrm{p} 2}\right)\right), d_{00}^{\prime}\left|G_{1}\right|^{2} /\left(\Gamma_{20}+i \Delta_{1}^{\prime}+\left|G_{\mathrm{p} 1}\right|^{2} /\right. \\ \left.\left(\Gamma_{10}+i \Delta_{1}^{\prime}-i \Delta_{\mathrm{p} 1}+\left|G_{\mathrm{p} 2}\right|^{2} /\left(\Gamma_{13}+i \Delta_{1}^{\prime}-i \Delta_{\mathrm{p} 1}-i \Delta_{\mathrm{p} 2}\right)\right)\right). \end{array}$ $\begin{array}{l} \left|\rho_{\mathrm{S} / \mathrm{AS}}^{\prime(3)}+\rho_{\mathrm{S} / \mathrm{AS}}^{\prime(5)}+\rho_{\mathrm{S} / \mathrm{AS}}^{\prime(7)}+\rho_{\mathrm{S} / \mathrm{AS}}^{\prime(9)}\right|=\left(A^{2}+B^{2}+C^{2}+D^{2}+2 A B \cos \left(\Delta \varphi_{1}\right)+\right. \\ \quad 2 A C \cos \left(\Delta \varphi_{2}\right)+2 A D \cos \left(\Delta \varphi_{3}\right)+2 B C \cos \left(\Delta \varphi_{4}\right)+2 B D \cos \left(\Delta \varphi_{5}\right) \\ \left.\quad+2 C D \cos \left(\Delta \varphi_{6}\right)\right)^{1 / 2} \end{array}$ Eq. (3)–(4) represent nested three dressing resulting in three-Fano interference, which is originated from an imperfect continuous state ( $\rho_{\mathrm{S} / \mathrm{AS}}^{\prime(3)}$) and three bound (photon1 $\rho_{\text {S/AS }}^{\prime(5)}$ photon1–phonon1 $\rho_{\text {S/AS }}^{\prime(7)}$ photon1–phonon1–phonon2 $\rho_{\text {S/AS }}^{\prime(9)}$) states. The three Fano interferences are governed by six phases ( $\Delta \varphi_{1}=\pi, \Delta \varphi_{2}=\pi, \Delta \varphi_{3}=\pi, \Delta \varphi_{4}=0, \Delta \varphi_{5}=0, \Delta \varphi_{6}=0$). For the four dark states of nested four dressing, the expression can be written as follows

Where, $\begin{array}{l} \text d_{3}=\left|G_{1}\right|^{2} / \Gamma_{20}+i \Delta_{1}+\left|G_{\mathrm{p} 1}\right|^{2} /\left(\Gamma_{10}+i \Delta_{1}-i \Delta_{\mathrm{p} 1}+\left|G_{\mathrm{p} 2}\right|^{2} /\right. \\ \left.\left.\left(\Gamma_{30}+i \Delta_{1}-i \Delta_{\mathrm{p} 1}+i \Delta_{\mathrm{p} 2}+\left|G_{\mathrm{p} 3}\right|^{2} /\left(\Gamma_{31}+i \Delta_{1}-i \Delta_{\mathrm{p} 1}+i \Delta_{\mathrm{p} 2}-i \Delta_{\mathrm{p} 3}\right)\right)\right)\right) \\ d_{4}=\left|G_{1}\right|^{2} / \Gamma_{20}+i \Delta_{1}^{\prime}+i \Delta_{1}+\left|G_{\mathrm{p} 1}\right|^{2} /\left(\Gamma_{10}+i \Delta_{1}^{\prime}+i \Delta_{1}-i \Delta_{\mathrm{p} 1}+\left|G_{\mathrm{p} 2}\right|^{2} /\right. \\ \left(\Gamma_{13}+i \Delta_{1}^{\prime}+i \Delta_{1}-i \Delta_{\mathrm{p} 1}-i \Delta_{\mathrm{p} 2}+\left|G_{\mathrm{p} 3}\right|^{2} /\left(\Gamma_{03}+i \Delta_{1}^{\prime}+i \Delta_{1}-i \Delta_{\mathrm{p} 1}-\right.\right. \\ \left.\left.\left.\left.i \Delta_{\mathrm{p} 2}-i \Delta_{\mathrm{p} 3}\right)\right)\right)\right). \end{array}$ By using Eq. (1) and (2), the interference terms can be written as follows: $\begin{array}{l} \left|\rho_{\mathrm{S} / \mathrm{AS}}^{\prime(3)}+\rho_{\mathrm{S} / \mathrm{AS}}^{\prime}+\rho_{\mathrm{S} / \mathrm{AS}}^{\prime(7)}+\rho_{\mathrm{S} / \mathrm{AS}}^{\prime(9)}+\rho_{\mathrm{S} / \mathrm{AS}}^{\prime(11)}\right|=\left(A^{2}+B^{2}+C^{2}+D^{2}+E^{2}+2 A B \cos (\Delta\right. \\ \left.\varphi_{1}\right)+2 A C \cos \left(\Delta \varphi_{2}\right)+2 A D \cos \left(\Delta \varphi_{3}\right)+2 A E \cos \left(\Delta \varphi_{4}\right)+2 B C \cos \left(\Delta \varphi_{5}\right)+2 B D \cos \\ \left.\left.\left(\Delta \varphi_{6}\right)+2 B E \cos \left(\Delta \varphi_{7}\right)+2 C D \cos \left(\Delta \varphi_{8}\right)+2 C E \cos \left(\Delta \varphi_{9}\right)+2 D E \cos \left(\Delta \varphi_{10}\right)\right)\right)^{1 / 2} \\ \end{array}$ The dressed third-order density matrix can be approximated as the sum of the third-, fifth-, seventh-, ninth-, and eleventh-order E_S/AS, governed by dressed SFWM multi-Fano phase $\Delta \varphi_{\mathrm{S} / \mathrm{AS}}=\varphi_{\mathrm{S} / \mathrm{AS}}^{(\mathrm{L})}-\varphi_{\mathrm{S} / \mathrm{AS}}^{(\mathrm{K})}$ (L < K). The total phase $\varphi_{\mathrm{S} / \mathrm{AS}}^{(\mathrm{j})}$ of generated E_S/AS is a sum of the initial phase $\varphi^{I}{ }_{S / A S}^{(j)}$, cross-Kerr nonlinear phase $\varphi^{x}{ }_{S / A S}^{(j)}$, and self-Kerr nonlinear phase $\varphi^{s}{ }_{S / A S}^{(j)}$ ( $\varphi_{\mathrm{S} / \mathrm{AS}}^{(\mathrm{j})}=\varphi_{\mathrm{S} / \mathrm{AS}}^{\mathrm{I}}+\varphi_{\mathrm{S} / \mathrm{AS}}^{\mathrm{s}(\mathrm{j})}+\varphi_{\mathrm{S} / \mathrm{AS}}^{\mathrm{x}}$). Therefore, multi-Fano interference could be constructive or destructive depending upon the total phase when the condition is set as $\Delta \varphi_{S / A S}=0$ (i = 1 to 10), which results in 10 bright states. However, at 0 or $\pi$ (i = 1 to 10), the result shows 10 dark states. For the four dark states, the expression can be written as $\left|\rho_{\mathrm{S} / \mathrm{AS}}^{\prime(3)}-\rho_{\mathrm{S} / \mathrm{AS}}^{\prime(5)}-\rho_{\mathrm{S} / \mathrm{AS}}^{\prime(7)}-\rho_{\mathrm{S} / \mathrm{AS}}^{\prime(9)}-\rho_{\mathrm{S} / \mathrm{AS}}^{\prime(11)}\right|$. The four-Fano is controlled by ten phases ( $\Delta \varphi_{1}=\pi, \Delta \varphi_{2}=\pi, \Delta \varphi_{3}=\pi, \Delta \varphi_{4}=\pi, \Delta \varphi_{5}=0, \Delta \varphi_{6}=0, \Delta \varphi_{7}=0, \Delta \varphi_{8}=0, \Delta \varphi_{9}=0, \Delta \varphi_{10}=0$)). Eq. (1) and (2) show four-Fano interference, which is originated from an approximate continuous state $\rho_{\text {S/AS }}^{\prime(3)}$, photon1 $\rho_{\text {S/AS }}^{\prime(5)}$, photon1–phonon1 $\rho_{\text {S/AS }}^{\prime(7)}$, photon1–phonon1–phonon2 $\rho_{\text {S/AS }}^{\prime(9)}$, photon1–phonon1–phonon2–phonon3 $\rho_{\text {S/AS }}^{\prime(11)}$ states. Moreover, the SFWM multi-Fano interference between continuous and multibound states is very obvious when the Δ₁ is scanned.

In a Λ-type three-level system, third-order $\rho_{\text {S/AS }}^{\prime(3)}$ via $\rho_{00}^{(0)} \xrightarrow{E_{1}} \rho_{20}^{(1)} \xrightarrow{E_{\text {AS }}} \rho_{00}^{(2)} \xrightarrow{E_{1}^{\prime}} \rho_{20(\mathrm{~S})}^{(3)}$ and $\rho_{00}^{(0)} \xrightarrow{E_{1}^{\prime}} \rho_{20}^{(1)} \xrightarrow{E_{s}} \rho_{00}^{(2)} \xrightarrow{E_{1}} \rho_{20(\mathrm{AS})}^{(3)}$, respectively, can be written as follows:

where, $\begin{array}{l} \text d_{5}=\left|\mathrm{G}_{2}\right|^{2} /\left(\Gamma_{20}+i \Delta_{1}+i \Delta_{2}+\left|\mathrm{G}_{\mathrm{p} 1}\right|^{2} /\left(\Gamma_{10}+i \Delta_{1}+i \Delta_{2}-i \Delta_{\mathrm{p} 1}\right.\right. \\ \left.\left.+\left|\mathrm{G}_{\mathrm{p} 2}\right|^{2} /\left(\Gamma_{13}+i \Delta_{1}+i \Delta_{2}-i \Delta_{\mathrm{p} 1}-i \Delta_{\mathrm{p} 2}\right)\right)\right) d_{6}=\left|\mathrm{G}_{2}\right|^{2} /\left(\Gamma_{20}+i \Delta_{1}^{\prime}+\right. \\ i \Delta_{2}+\left|\mathrm{G}_{\mathrm{p} 1}\right|^{2} /\left(\Gamma_{10}+i \Delta_{1}^{\prime}+i \Delta_{2}-i \Delta_{\mathrm{p} 1}+\left|\mathrm{G}_{\mathrm{p} 2}\right|^{2} /\left(\Gamma_{13}+i \Delta_{1}^{\prime}+i \Delta_{2}-i \Delta_{\mathrm{p} 1}\right.\right. \\ \left.\left.\left.-i \Delta_{\mathrm{p} 2}\right)\right)\right). \end{array}$ The equations represent three-Fano interference, which is originated from the photon1–photon2 continuum state $\rho_{S / A S}^{\prime \prime x(3)}$ and three bound states (photon1 $\rho_{S / A S}^{\prime \prime (5)}$, photon1–phononon1 $\rho_{S / A S}^{\prime \prime (7)}$, and photon1–phonon1–phonon2 $\rho_{S / A S}^{\prime \prime (9)}$). Three-Fano has six phases ( $\Delta \varphi_{1}=\pi, \Delta \varphi_{2}=\pi, \Delta \varphi_{3}=\pi \Delta \varphi_{4}=0, \Delta \varphi_{5}=0, \Delta \varphi_{6}=0$). The interference between continuous and multibound states of SFWM multi-Fano appears when the Δ₂ is scanned.

Through the Eu³⁺ : BiPO₄ crystal lattice vibrations, the phonon dressing can control multi-Fano interference. Fig. 2a shows the spectral evolution of hybrid signal (FL + SFWM) from single Fano to triple Fano obtained from the (0.5 : 1) Eu³⁺ : BiPO₄ sample by changing GP (5 μs, 500 μs, 1 ms) when E₁ is scanned at 300 K. The (0.5 : 1) Eu³⁺ : BiPO₄ crystal exhibits three distinct fine-structure energy levels named as ⁵D₁ →⁷F₁ (magnetic dipole transition), ⁵D₀ →⁷F₁ (magnetic dipole transition), and ⁵D₀ →⁷F₃ (induced electric dipole transition) at the CF of about 10⁶ V/cm. The spectral signal recorded at PMT1 (Fig. 2a11) shows three Fano interference resulting from the coexistence of dressing and fine-structure energy levels splitting (⁵D₀ →⁷F₁ in Fig. 5g1), whereas three Fano dips (three dark states) come from destructive interference between approximate continuous state ( $\rho^{\prime}{ }_{\mathrm{AS}(3)}^{(3)}$) and three bound states ( $\rho^{\prime}{ }_{\mathrm{AS}(3)}^{(5)}$, $\rho^{\prime}{ }_{\mathrm{AS}(3)}^{(7)}$, $\rho^{\prime}{ }_{\mathrm{AS}(3)}^{(9)}$) are controlled by Fano phases ( $\Delta \varphi_{1}=\pi, \Delta \varphi_{2}=\pi$, and $\Delta \varphi_{3}=\pi$) modeled through Eq. (3) and (4). It is noteworthy that the magnitudes of the two Fano peaks are greater than those of the three Fano dips because of more pronounced splitting of energy levels within the ⁷F₁ state than the nested three dressing. $\begin{array}{l} \left|G_{1}\right|^{2} \\ /\left(\Gamma_{20}+i \Delta_{1}+\left|G_{\mathrm{p} 1}\right|^{2} /\left(\Gamma_{10}+i \Delta_{1}-i \Delta_{\mathrm{p} 1}+\left|G_{\mathrm{p} 2}\right|^{2} /\left(\Gamma_{13}+i \Delta_{1}-i \Delta_{\mathrm{p} 1}-i \Delta_{\mathrm{p} 2}\right)\right)\right) \end{array}$ from Eq. (3) and (4). The validation of this is supported by our theoretical findings as illustrated in Fig.2 h3 at Δ₁ = − 50. When the GP approaches 1 ms, the three suppression Fano dips (Fig. 2a31) tends to be more prominent than the enhancement Fano peaks. This observation can be attributed to the dominance of nested three dressing in the completion between the splitting and dressing of the energy levels. In Fig. 2a21, the first Fano dip is stronger than the third Fano dip, which can be attributed to the greater influence of the linear polarization dipole moment in energy level ⁷F₁, MJ = ₀ than that of the circular partial dipole moment in the energy level ( $\mu_{\mathrm{L}} \mu_{\mathrm{C}}$). The theoretical results (Fig. 2g and h) validate the multi-Fano evolution (Fig. 2a). In Fig. 2a12, two Fano interference is resulted from the coexistence of double dressing and energy levels’ splitting (⁵D₀ →⁷F₁), where two Fano dips come from interference in an approximate continuous state $\rho_{\mathrm{S}(2)}^{\prime(3)}$ and two bound states $\rho_{\mathrm{S}(2)}^{\prime(5)}$, $\rho_{\mathrm{S}(2)}^{\prime(7)}$ controlled by Fano phase ( $\Delta \varphi_{1}=\pi, \Delta \varphi_{2}=\pi$). The practical results obtained align precisely with the theoretical results depicted in Fig. 2h2 (simulated at Δ₁ = − 50). When the GP is increased to 500 μs (Fig. 2a22), the two Fano dips evolve into three Fano dips due to sensitive phonon dressing ( $|G_{\mathrm{p1}}|^{2}|G_{\mathrm{p2}}|^{2}$) and easy distinction for in-phase SFWM. When GP is at 1 ms (Fig. 2a23), the right first Fano dip is the strongest due to more dressing interaction of ⁷F_{1, MJ = 0} and ⁷F_{1, MJ = +1}. The single Fano interference (Fig. 2a13) occurs due to the coexistence of single dressing and energy level splitting. In contrast, a single Fano dip emerges from the interference between dark states and bright states, a phenomenon regulated by the Fano phase ( $\Delta\varphi=\pi$) and modeled by $\rho_{ \mathrm{AS}(1)}^{\prime (3)}+\rho_{ \mathrm{AS}(1)}^{\prime (5)}$. When GP is increased to 1 ms, two Fano peaks are changed into single Fano dip due to photon dressing ( $\left.| G_{1}\right|^{2}$) dominance.

The spectral evolution of two sharp peaks to five Fano interference, recorded at PMT2 by adjusting GP, is displayed in Fig. 2b. The two sharp peaks observed in Fig. 2b11-2b13 can be attributed to the splitting of energy levels during the transitions ⁵D₁ → ⁷F₁, ⁵D₀ → ⁷F₁, and ⁵D₁ → ⁷F₃, respectively. Multi-Fano interference seen at PMT2 (Fig. 2b) is stronger than that at PMT1 (Fig. 2a) due to the strong SFWM at the near detector. The spectral evolution observed at PMT3 (Fig. 2c) exhibits behavior that is identical to that observed at PMT1. The Fano interference in Fig. 2d-f, resulting from the interaction of two lasers, is stronger than that of a single laser (Fig. 2a-c), which is mostly ascribed to two lasers dressing enhancement. However, the left first Fano dip is stronger than the left third Fano dip in Fig. 2d, f, which is mainly attributed to more dressing interaction effect between ⁷F_{1, MJ = +1} and ⁷F_{1, MJ = 0}.

Fig. 3a shows the spectral evolution of multi-Fano interference from three Fano dips to a single Fano dip in the (7 : 1) Eu³⁺ : BiPO₄ crystal corresponding to Fig. 3c when E₁ is scanned. In Fig. 3a11, two Fano peaks and three Fano dips come from nested three dressing $\binom{\left|G_{1}\right|^{2} / \Gamma_{20}+i \Delta_{1}+\left|G_{\mathrm{p} 1}\right|^{2} /\left(\Gamma_{10}+i \Delta_{1}-i \Delta_{\mathrm{p} 1}+\right.}{\left|G_{\mathrm{p} 2}\right|^{2} /\left(\Gamma_{30}+i \Delta_{1}-i \Delta_{\mathrm{p} 1}+i \Delta_{\mathrm{p} 2}\right)}$ and can be controlled through the Fano phase ( $\Delta \varphi_{1}=\pi, \Delta \varphi_{2}=\pi, \Delta \varphi_{3}=\pi$), corresponding to the simulation in Fig. 2g3. The Hamiltonian ( $H=i \hbar \kappa_{1} \alpha_{1}^{\dagger} \alpha_{\mathrm{p} 1}^{\dagger} \alpha_{\mathrm{p} 2}^{\dagger}$ where $\kappa_{1}=-i \varpi \chi_{1}^{(9)} E_{\mathrm{S}} E_{\mathrm{p} 1} E_{\mathrm{p} 2} E_{1}^{3} / 2$) shows strong dressing coupling between photon1 ( $\alpha_{1}^{\dagger}$), sample phonon1 ( $\alpha_{\mathrm{p} 1}^{\dagger}$), and sample phonon2 ( $\alpha_{\mathrm{p} 2}^{\dagger}$). The energy level splitting (⁵D₀–⁷F₃) and dressing coexistence lead to two Fano interferences in Fig. 3a12, where, two Fano dips can be explained by nested two dressing ( $\left|G_{1}\right|^{2} / \Gamma_{20}+i \Delta_{1}+\left|G_{\mathrm{p} 1}\right|^{2} /\left(\Gamma_{10}+i \Delta_{1}-i \Delta_{\mathrm{p} 1}\right)$) controlled by Fano phase ( $\Delta \varphi_{1}=\pi, \Delta \varphi_{2}=\pi$) mentioned in Eq. (1) and (2). The amplitude of three Fano dips (Fig. 3a11) is reduced as a consequence of the weak dressing resulting from the interaction of two lasers, when compared to Fano dips illustrated in Fig. 3a11. Similarly, the two Fano interference tends to be weak in Fig. 3a22, which aligns with the simulation results depicted in Fig. 2g2. When compared to the Fano interference in Fig. 3a11, the Fano interference seen in Fig. 3a3 is weak. This difference can be attributed to the influence of a narrower excitation band, which controls the extent of crystal vibration phonons in the system.

In Fig. 3b1, two sharp peaks are resulted from competition between energy level splitting (⁵D₀–⁷F₁) dominance and dressing, which can be further explained by two lasers interaction and low-intensity phonon dressing ( $\left|G_{\mathrm{p} 1}\right|^{2},\left|G_{\mathrm{p} 2}\right|^{2}$) at low temperatures. Moreover, the spectral linewidth of the peak is sharp (Fig. 3b1). The four Fano dips (Fig. 3b2) come from destructive interference ( $\rho_{\mathrm{S} / \mathrm{AS}}^{\prime(3)}+\rho_{\mathrm{S} / \mathrm{AS}}^{\prime(5)}+\rho_{\mathrm{S} / \mathrm{AS}}^{\prime(7)}+\rho_{\mathrm{S} / \mathrm{AS}}^{\prime(9)}+\rho_{\mathrm{S} / \mathrm{AS}}^{\prime(11)}$ from Eq. (5) and (6). The control of Fano dips and peaks can be achieved by manipulating the Fano phase, where the dips are attributed to the presence of four dark states that are observed at $\Delta \varphi_{1}=\Delta \varphi_{2}=\Delta \varphi_{3}=\Delta \varphi_{4}=\pi$. Such four dressing dips come from photon1 dressing $\left|G_{1}\right|^{2}$ and three phonons dressing ( $\left|G_{\mathrm{p} 1}\right|^{2},\left|G_{\mathrm{p} 2}\right|^{2},\left|G_{\mathrm{p} 3}\right|^{2}$) with Hamiltonian $H=i \hbar \kappa_{1} \alpha_{1}^{\dagger} \alpha_{\mathrm{p} 1}^{\dagger} \alpha_{\mathrm{p} 2}^{\dagger} \alpha_{\mathrm{p} 3}^{\dagger}+H. c.$, where, $\kappa_{1}=-i \varpi \chi_{1}^{(11)} E_{\mathrm{AS}} E_{S} E_{\mathrm{p} 1} E_{1}^{3} E_{\mathrm{p} 2} E_{\mathrm{p} 3} / 2$.It shows the strong dressing coupling among photon1 ( $G_{1}^{\dagger}$)–phonon1 ( $G_{\mathrm{p} 1}^{\dagger}$)–phonon2 ( $G_{\mathrm{p} 2}^{\dagger}$)–phonon3 ( $G_{\mathrm{p} 3}^{\dagger}$). In Fig. 3b2, broad profile linewidth is caused by three phonons dressing in (0.5 : 1) BiPO₄. The Fano phase for four dark states and six bright states can be written as $\Delta \varphi_{1}=\Delta \varphi_{2}=\Delta \varphi_{3}=\Delta \varphi_{4}=\pi$ and $\Delta \varphi_{5}=\Delta \varphi_{6}=\Delta \varphi_{7}=\Delta \varphi_{8}=\Delta \varphi_{9}=\Delta \varphi_{10}=0$, respectively. The presence of six bright states surpasses the number of four dark states due to the additional bright state being integrated into the existing five bright states. The left first Fano dip is stronger than the left fourth Fano dip in Fig. 3b2 due to $\mu_{\mathrm{C}}>\mu_{\mathrm{L}}$. These results coincide with our theoretical results obtained for nested four dressing at Δ₁ = 50.

In Fig. 3b3, the signal observed by scanning E₁ while fixing E₂ (588 nm) shows multi-Fano interference with three Fano dips resulting from nested three dressings ( $\begin{array}{l} \left|G_{2}\right|^{2} /\left(\Gamma_{20}+i \Delta_{2}+\left|G_{\mathrm{p} 1}\right|^{2} /\left(\Gamma_{10}+i \Delta_{2}-i \Delta_{\mathrm{p} 1}\right)+\left|G_{\mathrm{p} 2}\right|^{2}\right. \\ \left.\left./\left(\Gamma_{30}+i \Delta_{2}-i \Delta_{\mathrm{p} 1}+i \Delta_{\mathrm{p} 2}\right)\right)\right) \end{array}$ in Eq. (7) and (8) and four Fano peaks from energy level splitting and Autler–Townes (AT) splitting. In addition, it is observed that single-laser excitation (Fig. 3b2) exhibits a greater number of dressing dips than two lasers excitation (Fig. 3b3). This can be attributed to the interaction between the two lasers within the crystal, which reduces the dressing effect, as illustrated in Fig. 2e. The observed outcome is consistent with the findings obtained from the nested three-dressing simulation at Δ₁ = 50. The spectral signal depicted in Fig. 3b4 exhibits strong three Fano interference at 15 μs gate width for the (0.5 : 1) Eu³⁺ : BiPO₄ sample mostly attributed to the dominance of SFWM dominance. The three multi-Fano dips can be explained by photon2 and two phonons nested dressing ( $\left|G_{2}\right|^{2},\left|G_{\mathrm{p} 1}\right|^{2},\left|G_{\mathrm{p} 2}\right|^{2}$). By comparing the Fano dips observed in Fig. 3b3 to those in Fig. 3b4, it is evident that the latter exhibits three consecutive Fano dips. This can be attributed to a more phonon dressing. As gate width increases from 200 μs (Fig. 3b3) to 500 ns (Fig. 3b4), the increasing ratio of SFWM in hybrid signal leads to high spectral resolution. The bright and dark states can be controlled by adjusting the phase between zero and $\pi$. As the gate width expands, the number of bright states rises, as illustrated in Fig. 3b4. The three Fano dips and four Fano peaks are originated from the two-laser excitation with a Fano phase of three dark states ( $\Delta \varphi_{1}=\Delta \varphi_{2}=\Delta \varphi_{3}=\pi$) and three bright states ( $\Delta \varphi_{4}=\Delta \varphi_{5}=\Delta \varphi_{6}=0$), respectively. The additional peak is originated from pure constructive states at Fano phases $\Delta \varphi_{i}=0 $ ( $i$ = 1, 2, 3, 4, 5, 6). When the GP reaches to 1 ms, Fig. 3b5 depicts clearly defined sharp peaks that can be attributed to the dominance of CF splitting, as illustrated in Fig. 1a2. Based on the findings obtained from our research, it can be concluded that the (0.5 : 1) H-phase Eu³⁺ : BiPO₄ exhibits strong dressing Fano interference compared to (7 : 1) M-phase Eu³⁺ : BiPO₄ resulting from more phonon dressing.

Figs. 4a and b discuss the evolution of SFWM from FL in a hybrid signal for (6 : 1) and (1 : 1) Eu³⁺ : BiPO₄, respectively. At near GP, the dressing effect is weak, leading to a spectral peak characterized by low resolution and a broad linewidth, as shown in Fig. 4a1. Upon increasing the GP (20 μs), an emergence of three Fano dips and two Fano peaks is apparent, (Fig. 4a2), resembling the pattern depicted in Fig. 2a22. The observed phenomenon can be attributed to the reduced impact of dressing detuning, resulting in a decreased photon dressing effect and high SFWM resolution. By further increasing GP (200 μs), the three Fano dips tend to be stronger in Fig. 4a3 due to one-photon dressing ( $\left|G_{1}\right|^{2}$) and two-phonon dressings ( $\left|G_{\mathrm{p} 1}\right|^{2},\left|G_{\mathrm{p} 2}\right|^{2}$) as modeled from Eq. (3) and (4). As the GP reaches 500 μs, the five Fano-dips in Fig. 4a4 are originated from energy levels ⁷F₁ symmetrical splitting of one laser excitation.

Fig. 4b shows the evolution from a single broad dip to sharp multi-dips for (1 : 1) Eu³⁺ : BiPO₄ under the same experimental conditions as described in Fig. 4a. In comparison to Fig. 4a1, the broad dip is shown in Fig. 4b1 due to higher frequency ( $w_{\mathrm{pi}}$) phonon for (1 : 1) than (6 : 1) Eu³⁺ : BiPO₄, which results in strong phase transition phonon dressing ( $\left|G_{\mathrm{pi}}\right|^{2} /\left(i \Omega_{\mathrm{mn}}-i w_{\mathrm{pi}}\right)$) with $\Delta_{1}-\Delta_{\mathrm{p} 1} \cdot \Delta_{1}-\Delta_{\mathrm{p} 1}+\Delta_{\mathrm{p} 2} \cdot \Delta_{1}-\Delta_{\mathrm{p} 1}+\Delta_{\mathrm{p} 2}-\Delta_{\mathrm{p} 3}$, which is close to resonance compared to the (6 : 1) sample, as shown in Fig. 4a. As GP reaches 500 μs, the five Fano dips are shown in Fig. 4b4. So, the phase transition phonon dressing is stronger in the (1 : 1) sample than that in the (6 : 1) sample. In summary, in contrast to the FL signal, the SFWM signal exhibits higher resolution. The presence of multiple Fano dips can be readily observed in the signal generated by SFWM.

From our results, we have realized the wavelength division multiplexing of both classical and coherent channels. The non-classical SFWM may be effectively multiplexed with divided Fano dip across a range of 1 μs to 500 μs. This multiplexing technique enables the routing of identical information across many channels, ranging from one channel (Fig. 4b1) to five channels (Fig. 4b4). A non-Hermitian multichannel router can be realized by non-Hermitian control real-part quantization (Fig. 1c), and the routing process can be achieved by changing the boxcar GP. Similarly, when GP is at 500 μs, the coherent channel output can be multiplexed to five divided Fano dips for routing the same information to different channels. Fig. 4b shows two adjacent dips with relative distance between many dips. Such a phenomenon (pure dip to five Fano dip) is analogous to routing. Such results are used for routing. Here we adopt channel equalization ratio ( $P=1-\sqrt{\sum_{1}^{N-1}\left(s_{1}-a\right)^{2} / a}$) to measure demultiplexing, where $a$ is the area of one dip and s_i is the area of each dip or gap between dips. When $s_{i} \approx a$, P is maximum (100%), more balanced and stable spatial channels were obtained. From Fig. 4b, the channel equalization ratio P has the potential to reach values ranging from 60% to 80%.

Fig. 4c and d represent the spectrum evolution from multi-Fano interference to two sharp peaks for (0.5 : 1, 1 : 1) BiPO₄, respectively, versus changing power. In Fig. 4c, the spectral intensity for the (0.5 : 1) Eu³⁺ : BiPO4 sample is shown at the near detector (PMT2). The laser power was changed from 9 mW to 1 mW at 300 K. When the power is high, strong $\left|G_{1}\right|^{2}$ results in five dressing dips (Fig. 4c1, 4c2) from ⁷F₁ symmetrical splitting (similar to Fig. 4a4), which is corresponding to the simulation result at $\Delta_{1}=0$ in Fig. 4e1–e4, respectively. Also, the left first to third Fano dip for energy level ⁷F_{1, MJ = ±1} as shown in Fig. 4c1–c2, come from nested three dressings, where the left fourth to fifth Fano dips come for energy level ⁷F_{1, MJ = ±1} as shown in Fig. 4c1–c2, owing to nested double dressing. Fig. 4d shows the spectral intensity for the (1 : 1) Eu³⁺ : BiPO₄ at PMT2 by changing power (9 mW, 7 mW, 5 mW, 3 mW, 1 mW). When the power of E₁ is decreased to 1 mW, the two sharp peaks are observed, as shown in Figs. 4c5 and d5, due to weak dressing ( $\left|G_{1}\right|^{2} \approx 0$) at low power.

The nested dressing multi-Fano dips observed in our studies can be associated with third-order quantization, as depicted in Fig. 1d. Herein, EP or non-Hermitian control originating from real and imaginary quantization alignment is discussed, as shown in Fig. 1d. The non-Hermitian EP manifests itself via CF splitting independently of any dressing effects. Fig. 4a–d exhibit full- and half-EP control, respectively. The multi-Fano real-part quantization channels can be defined as $\mathrm{N}=N_{\mathrm{m}} / N_{\mathrm{EP}}$, where $N_{\mathrm{m}}$ represents the number of dressing dips in SFWM signal, and $N_{\mathrm{EP}}$ is the number of peaks at EP control (two CF peaks), where, each peak represents one level (⁷F_{1, MJ = ±1}, or ⁷F_{1, MJ = 0}). The SFWM multi-Fano quantization leads to different channels (N = 3 (fourth-order) at ⁷F_1, ⁷F_{1, MJ = ±1} for the left EP peak (Fig. 4a4–a2 and $N=2 $ (third-order) at ⁷F_{1, MJ = 0} for the right EP peak (Fig. 4c5–c1). Similarly, N = 1 (second-order) at ⁷F_{1, MJ = 0,} and third-order N = 2 at ⁷F_{1, MJ = ±1} exhibited from each level in Fig. 4b and N = 2 (third-order) exhibited in each level from Fig. 4d.

Fig. 5 shows the evolution from multiple dips to two peaks when the laser of E₁ is kept high. Fig. 5a and b show five Fano dips resulting from five nested dressings (three dressing in ⁷F_{1, MJ = ±1} and two dressing in ⁷F_{1, MJ = 0}) in a manner similar to that described in Fig. 4c1. When the laser power is lowered, there is a reduction in photon dressing, which leads to the diminishing of three Fano dips resulting from three distinct fine structure energy levels (⁷F_{1, MJ = -1}, ⁷F_{1, MJ = 0}, ⁷F_{1, MJ = 1}). This reduction in photon dressing also contributes to the decrease in phenomenon illustrated in Fig. 4c4. When the power reaches 1 mW, only two sharp peaks appear as shown in Fig. 5a4-c4 due to weak photon dressing. From Fig. 5a1-a4, it can be seen that the center of the peak remains unchanged due to strong phonon dressing. When comparing the signal measurements of PMT2 with PMT1 and PMT3, it is observed that the signal measured at PMT2 exhibits a stronger dressing effect. This can be attributed to its proximity to the sample, allowing it to detect FL with a higher amplitude, resulting in a more significant dressing dip. Theoretical results are displayed in Fig. 5d in accordance with experimental results (Fig. 5b).

Owing to the earlier explanations, Fig. 5 shows similar EP control for SFWM multi-Fano quantization (half-EP control). Fig. 5b1 shows channel N = 3 (fourth-order) at ⁷F_{1, MJ = ±1} for the left EP peak b4 and exhibits N = 2 (third-order) at ⁷F_{1, MJ = 0} in b1 for the right EP peak Fig. 5b4. Similar results were observed for Fig. 5a, c.

Fig. 6a shows the EP evolution vs the ratio (G_p1/Γ₂₀) between photon-phonon dressing Rabi frequency and transverse dephase rate between level |2> and |0>. The theoretical results are corresponding to nested double dressing. At G_p1/Γ₂₀ = 0.5, one energy level splits into a real part and an imaginary part. At internal dressing G₁/Γ₂₀ = 0.5, the real and imaginary parts are split as shown in Fig. 6a3, a4, respectively. The splitting of the real part and the imaginary part of the nested double dressing does not occur simultaneously. When the third-order splitting (G₁/Γ₂₀) occurs in the real part (Fig. 6a3), the corresponding imaginary part (Fig. 6a4) will have the second-order splitting (G_p1/Γ₂₀). Thus, the real-part splitting is larger than the imaginary part splitting. The imaginary part is nonlocal, more degenerate and non-Abelian, and highly symmetric. The real part is local, less degenerate and Abelian, and low symmetric. More importantly, the real part is changed from local, Abelian, and more degenerate to nonlocal, non-Abelian, and less degenerate in third-order systems. The dominance of real-part splitting (G > Г) shows high-order router; however, the dominance of imaginary part splitting (Г > G) shows low-order router. The non-Hermitian EP is a point without dressing. Such an EP can be achieved by varying various external parameters. Such an EP can be achieved by changing different external parameters. The GP is the primary parameter. When the GP is changed from middle (hybrid) to far (SFWM), there exists the EP between hybrid and SFWM. For the nested double dressing, the dressing term is already solved, and it is converted into a univariate cubic equation to be solved, so three eigenvalues are obtained, but in our model, the photon-dressing-like Zeeman splits three energy levels, as shown in Fig. 1a4. The ⁷F_{1, MJ = -1} and ⁷F_{1, MJ = +1} energy levels are dressed by two phonons (phonon1 and phonon2) and the other two phonons (phonon3 and phonon4), respectively. However, the ⁷F_{1, MJ = 0} energy level is dressed by one photon (photon1). So, eight eigenvalues can be obtained by solving the equations of two nested double dressings and one single dressing. Moreover, the experimental results are a subset of the theoretical simulation.

Figs. 6c and d show the spectral intensity signal obtained from (0.5 : 1) Eu³⁺ : BiPO₄ (C₂ symmetry) and H-phase Eu³⁺ : NaYF₄ (Cs symmetry), respectively. In this experiment, the competition between real $a$ and imaginary ib eigenvalues based on nested photon–phonon dressing is discussed. The detailed model of EPs’ control based on different laser dressings is depicted in Tables S1-S4 (Supplementary materials). It is worthy to be mentioned that the GP affects the dephase rate (Γ) at the spectral signal. In other words, the ratio of G and Γ can be tuned by changing the GP. In Fig. 4c, Fig. 5b1, and Fig. 6c1, the visible five Fano dips can be observed due to the dominance of the nested dressing ( $\left|G_{1}\right|^{2},\left|G_{\mathrm{p} 1}\right|^{2},\left|G_{\mathrm{p} 2}\right|^{2},\left|G_{\mathrm{p} 3}\right|^{2}$ in Eq. (5) and (6). The left first and second Fano dips are originated from phonon1 and phonon2 dressing. The central Fano dip is resulted from photon1 dressing. The fourth and fifth Fano dips can be explained by phonon3 and phonon4 dressing. Compared to the other four Fano dips, the fifth Fano dip is the strongest (Fig. 4a4) due to photon dressing (similar to Fig. 4c1, c2 and Fig. 5a1, b1).

With the GP reaching 5 ms, the two sharp peaks can be observed in Fig. 6c3 corresponding to EP, where G and Γ are equal due to the dominance of CF splitting (Fig. 1a2), which is validated by theoretical results in Fig. 6a. The visible three Fano dips observed in Fig. 2a12, Fig. 4a2, b2, and Fig. 6c3 are attributed to the stronger nested three dressing (photon1-phonon1-phonon2 atomic coherence coupling in Fig. 1a4) than CF splitting. The evolution from three Fano dips to two sharp EP peaks to five Fano dips is corresponding to Fig. 1a2, a4, Fig. 6b1, b2 are the three-dimensional simulation results corresponding to the real and imaginary parts of 2D simulations (Fig. 6a,b), respectively.

Lastly, Fig. 6c exhibits fourth-order (N = 3) at ⁷F_{1, MJ = ±1}, which corresponds to four eigenvalues in a univariate quartic with three dressing, and one third-order (N = 2) EP control at ⁷F_{1, MJ = ±1} corresponding to four eigenvalues in a univariate cubic equation (S3 Galois group) with two dressing for left and right peaks, respectively. Fig. 6d1-d4 exhibits one third-order (N = 2) EP control for each peak.

The samples were held in a cryostat with different temperatures. Two dye lasers were used to generate the two pumping fields. The spectral optical outputs are obtained by scanning laser frequency, while the temporal optical outputs are fixed by DL frequency. The optical signal generated is detected at PMT. The ratio of FL to ES/AS is changed by different gate positions. Moreover, The point group site symmetry of different phases of samples can control multi-Fano resonance.