Three-type Fano interference controlled by the phase transition of Eu3+/Pr3+:YPO4

We study the Fano interference from different phase transitions of Eu3+:YPO4 and Pr3+:YPO4 crystals through simultaneous detection of bright and dark states. For this study, we employed tetragonal (T), hexahedral (H), and (H + T)-phases of YPO4 crystals. The Fano interfering bright and dark states are classified based on dressed spontaneous parametric four-wave mixing, dressed multi-order fluorescence, and hybrid signal regimes. Further, the Fano interference between conti-nuous and discrete states can be controlled from partly-(constructive–destructive)-interference through a phase transition, nonlinear phase, and dressing. The (H + T)-phase Pr3+:YPO4 suggests more obvious Fano interference in contrast to T- or H-phase Pr3+:YPO4. We report that the Pr3+:YPO4 and Eu3+:YPO4 demonstrates the dressed multi-level- and dressed single-level Fano interference, respectively, with further three types of Fano interference that can be distinguished at different time delays. The experimental measurement agrees with theoretical simulations. Such results may be used for the design of new types of quantum electronic or spintronic devices such as Fano-transistors.


Introduction
Progress in photonics and nanotechnology brings several examples of resonant optical phenomena associated with the physics of Fano resonances, with applications in optical switching and sensing [1]. Above the ionization threshold of atomic systems, the presence of discrete states leads to autoionization, which is an interference between two quantum paths; direct ionization and excitation of the discrete state coupled to the continuum [2]. The probability for autoionization exhibits a universal Fano intensity profile as a function of excitation energy [3]. The spectral phase variation can be used as a fingerprint of the interactions between the discrete state and the ionization continuum, indicating a new route towards monitoring electron correlations in time [4]. Mikhail et al, discussed resonant electromagnetic effects by using effective coupled oscillators, the Fano resonance, electromagnetically induced transparency, Kerker and Borrmann effects, and symmetry breaking [5][6][7][8]. The dynamics of quantum systems are encoded in the amplitude and phase of wave packets. A universal temporal-phase formalism and mapping of the Fano asymmetry to a phase of the time-dependent dipole response function has been reported [2,9,10]. Fano interference may potentially be used for the design of new types of quantum electronic or spintronic devices such as spin transistors and Fano-filters for polarized electrons [11].
Rare-earth ion-doped crystal has been widely appreciated for its advantages for coherent excitation and because it is more appropriate for the development of integrated quantum circuits. Yttrium phosphate (YPO 4 ) is used as host in luminescence materials optional substitution of Y 3+ sites, a high index of refraction (∼1.72) and insignificant luminescence quenching of Ln 3+ [12]. The crystal structure of YPO 4 is known to in two polymorphic forms mainly, hexagonal (H-) and tetragonal (T-) phases [13,14]. The hexagonal-phase and tetragonal-phase occupy a D 2 and D 2d point-group symmetry site, respectively [15,16]. Eu 3+ and Pr 3+ ions are extremely sensitive to the site symmetry and its surrounding crystal-field of the host material [17,18], which makes them an attractive material for important applications like an optical transistor and router [19,20]. Phase transition of phosphate ion-driven Eu 3+ :BiPO 4 has been reported [21]. The Pr 3+ ion has attracted considerable attention in recent years for its interesting fluorescence features, such as up conversion [22]. Energy transfer and concentration quenching of luminescence have been studied recently in Pr 3+ doped crystals powder [23]. Wang et al, modulated fluorescence (FL) lifetime through dressing effect between Stark levels of Pr 3+ :YPO 4 nanocrystals in different crystal phases [24]. Li et al reported how dressing effect could be controlled through resonant excitation to enhance nonlinear gain in Pr 3+ :Y 2 SiO 5 in view of quantum correlation and squeezing [25]. Autler-Townes (AT) splitting of multi-order fluorescence (MFL) has been investigated in various atomic-like media [18,26].
In this paper, we investigated the multiple types of Fano interference by controlling the transition from bright and dark states in different phases of Eu 3+ :YPO 4 and Pr 3+ :YPO 4 crystals. The bright and dark states result from constructive and destructive interference between continuous and discrete states. We classify these states based on the order and lineshape of the dressed spontaneous parametric four-wave mixing (SP-FWM), the dressed MFL, and the hybrid signals. The first observed type is the narrow Fano interference, which comes from the third-order perfect continuous Stokes (E S ) or anti-Stokes (E AS ) and fifth-order discrete E S/AS (narrow). The second observed type is the broad Fano interference, which comes from perfect continuous second-order FL and broad fourth-order FL. The third observed type comes from interference between E S (narrow third-order) and FL (broad fourth-order) in the hybrid signal (FL + E S ), where the ratio of FL and E S in a hybrid signal determines the intermediate Fano spectral linewidth. We also investigated the transition between the bright and dark states by changing non-linear phase and off-resonant detuning while scanning the dressing fields. The corresponding anti-cross plots have been displayed with the detuning of input fields and non-linear phase. By comparing different doped ion, we observed the dressed single-level Fano interference from Eu 3+ :YPO 4 , whereas high resolution dressed multi-level Fano interference is observed from Pr 3+ :YPO 4 .  3 solution, pH value of the mixture was kept to 1. After additional stirring for 60 min, the resulting mixture was transferred into the Teflon lined stainless steel autoclaves and then heated at temperature of 180 • C for almost 9 h. Then, autoclave was allowed to naturally cooled down to 30 • C. The white precipitates were cleaned from distilled water and absolute ethanol by centrifugation. Lastly, cleaned white precipitates were left to dry at 80 • C for about 12 h and final H-phase YPO 4 samples with D 2 point group symmetry was obtained. We synthesized different phases of YPO 4 crystals such as pure T-phase, and T-/H-mixture phase samples by carefully adjusting the molar ratio of PO 3− 4 :RE 3+ (Y 3+ , Eu 3+ /Pr 3+ ):trisodium citrate (Cit 3− ). The composition of YPO 4 phases used in our experiment are: pure T-phase; pure H-phase; (much H-phase (74%) and less T-phase (26%)) and (much T-phase (54%) and less H-phase (46%). Figure 1 shows the schematic diagram of the experimental setup where the sample (Eu 3+ /Pr 3+ :YPO 4 ) is held in a cryostat (CFM-102) whose temperature was controlled by flowing liquid nitrogen. Dressing-phonon competition can be modulated desirably by controlling temperature. The optical output generation experimental setup is demonstrated in figure 1. In figure 1, the injection-locked single-mode Nd 3+ :YAG laser (Continuum Powerlite DLS 9010, 10 Hz repetition rate, 5 ns pulse width) is employed to pump two dye lasers (DL1 and DL2 narrow scan with a 0.04 cm −1 linewidth). DL1 and DL2 produce the dressing fields E 1 (ω 1 , Δ 1 ) and E 2 (ω 2 , Δ 2 ), respectively, where Δ i = Ω mn − ω i is the frequency detuning, Ω mn is the frequency of atomic transition between states |m and |n and ω j (j = 1, 2) is the frequency of the laser. The pulse generated from Nd 3+ :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 YPO 4 crystal, which is perpendicular to the optical axis. The dressing field E 1 excites the sample and is reflected back from the surface of YPO 4 crystal in its original path, which is named as E 1 with a small angle θ between them. By exciting Eu 3+ /Pr 3+ :YPO 4 crystal with E 1 /E 2 and E 1 (reflection of E 1 ) beams (figure 1), the optical outputs such as the FL emission (figures 2(a) and (b)) and E S/AS (figure 2(d)) signals are generated under  phase-matched condition (k 1 + k 1 = k S + k AS ) while interacting with energy levels of Eu 3+ :YPO 4 and Pr 3+ :YPO 4 . The spectral optical outputs (FL, E S/AS , FL + E S ) are obtained by scanning laser frequency, while the time-resolved OS obtains temporal optical outputs by fixing DL frequency. The grating motor of DL1 and DL2 is scanned by computer to form x-axis (wavelength), and the intensity of the excitation spectrum is the average of ten shots from the gated integrator (figure (1)) appearing on the y-axis. The optical signal generated are detected at photomultiplier tubes (PMT) via confocal lenses (CL). PMT1 is precisely placed to detect hybrid (FL + E S ). FL has random emission and vanishes before reaching the far detector. Similarly, PMT2 is also precisely placed to detect pure anti-Stokes signals due to phase-matched conditions. Figure 2(a) shows fine structure energy levels of hexagonal-phase Eu 3+ :YPO 4 with D 2 point-group symmetry, the ground state 7 F 1 is split into m j=0 and m j=±1 under the crystal field effect of YPO 4 crystal. Figure 2(b) shows fine structure energy levels of T-phase Pr 3+ :YPO 4 . The crystal field of Pr 3+ :YPO 4 and site symmetry of tetragonal phase (D 2d ) partly lifting the (2J + 1) degeneracy, and 3 H 4 (ground state) and 1 D 2 (excited state) levels are split into two levels, as shown in figure 2(b). At PMT1 (figure 1), FL dominates in the hybrid signal. At the same time, signals can be selected from specific energy levels through boxcar gated integrators by controlling gate position (time delay) and gate width (integration duration) as shown in figure 1. One can obtain output signals from different energy levels with different lifetimes by selecting the gate position. Thus, the gate position may control the ratio of FL and E S . Since the FL and SP-FWM have different decay rates, therefore can also be easily distinguished at PMT1 using a boxcar gate position.

Basic theory
Further, gate width can be varied to control emission from the number of energy levels. If the gate width is narrow enough, the observed signal could result from a single energy level with a single lifetime having an obvious dressing effect. All the experiment results in this manuscript are obtained by fixing gate width at 100 ns.
By controlling the gate position, the laser power (P), detector position and temperature, we observe Fano resonance between the discrete and continuous state, which can be classified as dressed MFL, hybrid and dressed SP-FWM Fano interference. These observed Fano interference can be controlled through time delay (ratio of MFL and Stokes). The optimal Fano interference could be achieved by adjusting experimental parameters. Furthermore, results can also be controlled by phase transition (D 2 and D 2d ) of Eu 3+ /Pr 3+ :YPO 4 .

Dressed SP-FWM Fano interference
By opening field E 1 and E 2 , the dressed third-order density matrix element for , respectively, can be expanded using Taylor expansion where (5) S/AS , G i = μ ij E i / is the Rabi frequency with the electric dipole matrix elements μ ij of levels |i and |j . The linewidth of the E S/AS signal is Γ S/AS = Γ 01 + Γ 11 + Γ 21 . The decoherence rate Γ ij = (Γ i + Γ j )/2, Γ i/j = Γ pop + Γ ion−spin + Γ ion−ion + Γ phonon − Γ dres sing , where Γ phonon is related to the sample temperature and Γ dres sing is associated with to dressing. The dressed third-order density matrix ρ (3) S/AS can be approximated as the sum of the third-order E S/AS and fifth-order E S/AS governed by dressed SP-FWM Fano phase Δφ S/AS = φ (3) S/AS − φ (5) S/AS (dressing phase) and detuning (Δ 2 ). The total phase (φ (j) S/AS ) of generated E S/AS is a sum of the initial phase (φ I (j) S/AS ), cross-Kerr non-linear phase (φ x (j) S/AS ), and self-Kerr non-linear phase (φ s (j) S/AS ) written as φ depends upon the dressing of the atomic system [25]. In our experiment, the initial phase depends upon time delay (figure 1). Using equation (1), we can write interference as |ρ (3) S/AS + ρ (5) (5) S/AS |, where I a and I b represent the intensity of φ (3) S/AS and φ (5) S/AS , respectively. Therefore, interference could be constructive (|ρ (3) S/AS | + |ρ (5) S/AS |) or destructive (|ρ (3) S/AS | − |ρ (5) S/AS |) depending upon the total phase to be either zero orπ. Fano contrast for SP-FWM Fano interference can be defined as where C S 1 suggest maximum interference while C S = 1 suggest no interference. From coupling Hamiltonian defined in [21], the nonlinear gain isg = |(−iω S/AS is the nonlinear susceptibility. The nonlinear susceptibility induces nonlinear phase through cross-phase modulation (XPM) or self-phase modulation (SPM), where n S/AS 1 is the linear refractive index of Stokes or anti-Stokes, n s 2 and n x 2 is self-and cross-Kerr nonlinear refractive index, respectively. The intensity of output E S/AS signals using photon number N S/AS = a + S/AS a S/AS can be modeled as I S/AS (t S/AS ) = |N S/AS |exp(−Γ S/AS t S/AS ). Thus, the respective intensities of the output E S and E AS signals are proportional to the photon numbers and can be given as The substitutions of ρ (3) S 1 = Ae iφ 1 s and ρ (3) AS 1 = Be iφ 2 d are taken from equations (1) and (2), where A (B) is the modulus and φ 1 (φ 2 ) are the phase angles of ρ (3) S 1 (ρ (3) AS 1 ) and t is time.

Dressed MFL Fano interference
Similarly, the dressed second-order FL ρ (2) FL in a Λ-type system via perturbation chain can be written and expanded using Taylor expansion as where (4) FL . Similar to equations (1) and (2), Fano phase for dressed MFL Fano interference can be defined as Δφ FL = φ (2) FL − φ (4) FL and the FL phase that can be defined as φ represents cross-Kerr-and self-Kerr-non-linear phase, respectively. The dressed MFL Fano interference results from interference between perfect continuous state ρ (2) FL (φ (2) FL ) and broad state i.e. ρ (4) FL (φ (4) FL ) when detuned with Δ 2 . Similar to section 3.1, we can obtain where I c and I d represent the intensity of second-order FL and fourth-order FL, respectively. Fano contrast for FL Fano interference is
Next, we discuss the dependence of the excitation spectrum on the Fano phase. Figure 4(a) shows the evolution of Stokes when the Fano phase (Δφ S ) is changed from 0 to 2π. At Δφ S = 0 the excitation spectrum is observed with shows a sharp enhancement peak (figure 4(a1)), which results from constructive interference between ρ (3) S and ρ (5) S as explained in figure 3(a1). As Δφ S increases to π/2, similar to figure 3(a2), the suppression dip and enhancement peak appears from destructive and constructive interference, respectively. As Δφ S approaches π, suppression dip (similar to figure 3(a3)) from destructive interference appears (figure 4(a3)). Figure 4(b) shows the same behavior of Fano interference between ρ (2) FL and ρ (4) FL (From equation (5)) as discussed for Stokes in figure 4(a). It is worth mentioning that the FL In figure 5, the spectral signals are obtained by changing E 2 power from 1 mW to 9 mW while fixing E 1 at 1 mW and the gate position at 1 μs (figure 6(d)). Figures 5(a) and (b) shows the spectral intensities of the hybrid signal (PMT1) and anti-Stokes (PMT2), respectively when the temperature is fixed at 77 K. To observe the spectral FL output in figure 5(a), the gate position is fixed at a certain point (1 μs) on the time domain curve (figure 6(d)). As discussed in figure 4, Fano interference depends upon the Fano phase (Δφ FL/S/AS ), which can be controlled through laser power. In figure 5(a), the spectral signal shows the transition from pure enhancement peak (figure 5(a1)) to half-dip-half-peak (figure 5(a2)), to pure-suppression dip (figure 5(a3)), to half-peak-half-dip (figure 5(a4)) and to pure enhancement peak (figure 5(a5)) as E 2 power is changed from low to high. When E 2 power is at 1 mW, the perfect continuous state (ρ (2) FL ) and discrete state (ρ (4) FL ) interfere constructively (|ρ (2) FL | + |ρ (4) FL |) and the dressing enhancement peak (figure 5(a1)) is observed. According to equation (5), MFL Fano phase approaches zero i.e. Δφ FL = 0 due to low contribution of cross-Kerr-(φ x FL = 2k FL n FL 2 |E 2 | 2 e −r 2 z/n FL 1 ) at low power. As power increases (P 2 = 3 mW), the proportion of ρ (4) FL in |ρ (2) FL | + |ρ (4) FL | increases and MFL Fano phase Δφ FL increases to Δφ FL = π/2 due to an increase in the cross-Kerr phase difference (φ x (2) FL − φ x(4) FL ). Hence, Fano interference with half-dip and half-peak (figure 5(a2)) is observed due to destructive (|ρ (2) FL | − |ρ (4) FL |) and constructive interference between ρ (2) FL and ρ (4) FL , respectively. Such dressed MFL Fano interference (corresponds to spectral curve obtained theoretically at Δφ FL = π/2 in figure 4(c2)). When P 2 of E 2 is further increased (5 mW), the FL evolves from AT-like splitting, causing the MFL Fano phase to increase up to Δφ FL = π. Due to the increase in the MFL Fano phase, ρ (2) FL and ρ (4) FL destructively interfere and pure dressing suppression dip (similar to figure 4(c3)) is observed in figure 5(a3). At P 2 = 7 mW, the dressed MFL Fano interference with half-peak-half-dip ( figure 5(a4)) is observed as the MFL Fano FL phase changes from Δφ FL = π to Δφ FL = 3π/2. This can be explained by the strong contribution of φ x FL at high power of E 2 . The MFL Fano interference ( figure 5(a4)) corresponds to the spectral curve calculated at Δφ FL = 3π/2 in figure 4(c4). When P 2 is set at 9 mW, enhancement peak ( figure 5(a5)) is observed again as the FL Fano phase increases to Δφ FL = 2π. Our experiment results (figure 5(a)) agree with theoretical results (figures 3(b) and 4(b)). The maximum MFL Fano contrast C FL is measured at about 4.3 in figure 5(a4) at P 2 = 7 mW. Figure 5(b) shows a spectral intensity signal of the anti-Stokes signal. In case of narrow E AS Fano interference ( figure 5(b)), enhancement peak and dressing suppression dip come from constructive  interference (|ρ (3) AS + ρ (5) . The linewidth of dressed E AS Fano interference is narrow than dressed MFL Fano interference (figures 3(a) and 4(a)) due to coherent nature. Our experiment results ( figure 5(b)) agree with theoretical results (figures 3(a) and 4(a)). The maximum Fano contrast C AS for SP-FWM interference is measured by about 3.5 in figure 5(b4) at P 2 = 7 mW, which is slightly less than C FL ( figure 5(a4)). Figure 5(c) shows the spectral intensity of the FL signal measured at PMT1 under the same conditions as figure 5(a) except temperature, which is 300 K (figure 1(d)). The FL spectral signal measured as figure 5(a). At 300 K, the dressing effect reduces as the phonon effect increases with an increase in temperature from 77 K to 300 K [25]. At 300 K ( figure 5(c)), the linewidth of spectral signal increases in contrast to figure 5(a). This can be explained from the phonon broadening effect at high temperatures. Compared with figure 5(a5), the pure enhancement peak is stronger in figure 5(c5) due to high temperature.
By exploiting spectral intensity results presented in figure 5, we realized Fano transistor as illustrated in figure 2(c), where baseline of MFL signal is input (a in = MFL/E AS ), E 2 is a control signal (analogous to the base current of bipolar junction transistor, while a out is an output of the Fano transistor, gain 'g' is related to Fano interference, N is the internal noise of the transistor. Therefore, we can mathematically model the output of the Fano transistor as a out = g * a in + N. Fano contrast for MFL Fano interference C FL = |I c + I d |/|I c − I d |is measured maximum at 4.3 ( figure 5(a4)). The experimental results of figure 5(c) and 4(b) are also consistent with each other. The Fano contrast C FL is about 3.6 in figure 5(c4). At the same power, Fano contrast for FL at 77 K is best, and C AS is the worst. The bandwidth of the Fano transistor can be controlled by adjusting laser power. Figures 6(a) and (b) show the spectral intensities of the hybrid signal and anti-Stokes signal obtained from (H + T)-phase Eu 3+ :YPO 4 , respectively. When the gate position is changed from 200 ns to 5 μs and power of E 1 and E 2 are fixed at 2 mW, the hybrid signal shows the transition from broad enhancement peak (figure 6(a1)) to broad half peak-half dip (figure 6(a3)) to intermediate half peak-half dip (figure 6(a4)) to sharp half peak-half dip (figure 6(a5)) to pure sharp enhancement peak ( figure 6(a6)). Figures 6(d) and (e) show the time-domain intensity signal obtained from (H + T)-phase Eu 3+ :YPO 4 and H-phase Pr 3+ :YPO 4 , respectively. The time AT-splitting in Pr 3+ :YPO 4 (figure 6(e)) is stronger than Eu 3+ :YPO 4 (figure 6(d)) [17]. Change in gate position (figures 6(d) and (e)) corresponds to change in FL and E S ratio (figures 6(a) and (c)). As discussed earlier in section 3.1, change in gate position effects initial phase difference S ) in the Fano phase (Δφ FL/S ). As both lasers are kept at low power, the contribution of cross-Kerr-and self-Kerr phase can be ignored and the effect of the only initial phase is considered. The total intensity of the hybrid signal can be written as ρ = ρ (3) S + ρ (4) FL . At t 1 = 200 ns (figure 6(d)), the FL emission dominates due to a short lifetime, and hybrid signal behaves as pure FL signal (ρ = ρ (4) FL ). Broad enhancement peak (figure 6(a1)) can be attributed to FL emission, as the FL signal has a low lifetime and incoherent nature. Due to the dominance of FL emission at gate position t 1 and t 2 , the initial phase difference (φ I (2) FL − φ I(4) FL = 0) along with Fano phase approaches zero; hence, the enhancement peak is observed due to constructive interference between continuous and discrete states in figures 6(a1) and (a2), respectively. At t 3 (1 μs), FL dominates in hybrid signal so figure 6(a3) suggests the broad MFL Fano interference, which comes from constructive and destructive interference between ρ (2) FL and, ρ (4) FL as discussed in figure 5(a4). As gate position is changed from t 1 to t 3 , the Fano phase increases from Δφ FL = 0 to Δφ FL = 3π/2 suggested by an increase in the initial phase. So, good MFL Fano interference with a high C FL of about 3.2 is observed at t 3 . As gate position is changed to t 4 (1.5 μs), the proportion of E S increases and becomes equal to FL emission in the hybrid signal, and initial phase difference (φ I(3) S − φ I(4) FL = 3π/2) is modified whereas the Fano phase (Δφ H = 3π/2) remains the same. Hybrid Fano interference at t 4 results from both constructive (|ρ (3) S | + |ρ (4) FL |) and destructive (|ρ (3) S | − |ρ (4) FL |) interference between narrow discrete state (ρ (3) S ) and broad continuous state (ρ (4) FL ), resulting in half peak and half dip, respectively. The hybrid Fano interference (figure 6(a4)) with narrower spectral linewidth than MFL Fano interference (figure 6(a3)) is modeled using equation (9). The Fano contrast C H of about 2.9 (figure 6(a4)) is measured at t 4 . When the gate position changes to t 5 (2 μs), initial phase difference increases to φ I(3) S − φ I(5) S = π due to low proportion of E S in hybrid suggesting narrowest linewidth of E S Fano interference (figure 6(a5)) results from constructive and destructive interference determined by fixed Fano phase Δφ S = 3π/2 as explained in figure 5(b4). The Fano contrast C AS is about 2.8 in figure 6(a5), so E S Fano interference decreases as gate position increases. When gate position is further increased, the sharp peak is observed at t 5 and t 6 , the outputs can be attributed to evolved E S suggested by long lifetime. The evolved intensity of hybrid signal at t 5 and t 6 can be redefined as (ρ = ρ (3) S ). So the sharp enhancement peak (figure 6(a6)) from constructive interference of ρ (3) S and ρ (5) S corresponds to figure 4(a5). Spectral intensity signal detected at PMT2 (figure 6(b)), follows similar behavior to change in gate position as discussed for Eu 3+ :YPO 4 in figure 6(a). However, the linewidth of E AS signal (figure 6(b3)) is significantly narrower in contrast with FL emission observed at the same gate position, as shown in figure 6(a3). SP-FWM Fano interference in figure 6(b3) with phase Δφ AS = 3π/2 has Fano contrast C AS of about 2.95. From figure 6(a), we can say that Fano contrast for MFL (figure 6(a3)) is highest, whereas Fano contrast for E AS (figure 6(a5)) is lowest. However, MFL Fano contrast in hybrid signal (figure 6(a3)) is lower than pure FL emission (figures 5(a4) and (c4)), which can be explained by the presence of E AS in the hybrid signal. Due to pure E AS detection at PMT2, E AS Fano contrast in figure 5(b4) is higher than that of figure 6(a5). Figure 6(c) shows spectral intensity signal from T-phase of Pr 3+ :YPO 4 detected at PMT1 (figure 1) at different gate positions. Unlike Eu 3+ :YPO 4 , FL signal from Pr 3+ :YPO 4 shows dressed MFL Fano interference (figure 6(c2) when the gate position is fixed at t 2 . At t 2 = 500 ns, the constructive interference between ρ (2) FL and ρ (4) FL is significantly stronger than destructive interference which leads to strong enhancement peak and weak suppression dip in figure 6(c2). The result in figure 6(c2) corresponds to the simulated result shown in figure 3(b4). Unlike Eu 3+ :YPO 4 (figures 6(a3) and (b3)), when gate position is placed at point of time AT-splitting (t 3 ), broad suppression dip appears (figure 6(c3)) which is already explained in detail in figure 3(c3). By increasing gate position to t 5 , the proportion of E S increases and becomes equal to FL in a hybrid signal. In contrast to single-level dressed FL Fano interference (figure 6(a4)), multi-level hybrid Fano interference with strong suppression dips and multiple sharp peaks (figure 6(c4)) is observed by changing Fano phase Δφ H (near 3π/2) as calculated in figure 7(d4). Such multi-level Fano interference results from higher dipole moment and stronger dressing of Pr 3+ :YPO 4 than in Eu 3+ :YPO 4 , which corresponds to more energy levels. These multiple sharp peaks (figure 6(c4)) can be attributed to SP-FWM emission due to their narrow linewidth. The Fano contrast C H is about 3.3 in figure 6(c4). The Fano contrast C AS is about 3 in figure 6(c5). At gate position t 6 , E S completely evolves from FL and no Fano interference is observed ( figure 6(c6)). Although, Fano interference in Pr 3+ :YPO 4 follows similar behavior as explained for Eu 3+ :YPO 4 . The spectral linewidth of signal in Pr 3+ :YPO 4 (figure 6(c)) is relatively broader than Eu 3+ :YPO 4 ( figure 6(a)). This can be explained from stronger dressing effect due to higher dipole moment and more atomic-like behavior of Pr 3+ :YPO 4 . From figure 6, it can be concluded that hybrid Fano interference (figure 6(c4)) is more obvious SP-FWM Fano interference (figure 6(a5)).
In figure 7, spectral is obtained by fixing the gate position at 1.5 μs. Figures 7(a1)-(a3) show excitation spectra measured for H-phase Pr 3+ :YPO 4 , T-phase Pr 3+ :YPO 4 and (less-H and much-T)-phase Pr 3+ :YPO 4 , respectively. The difference in point group symmetry affects the cross-Kerr phase (φ x (3) S , φ x(4) FL ). In H-phase Pr 3+ :YPO 4 , hybrid Fano interference is not obvious in figure 7(a1), which can be explained from weak E S emission in |ρ (4) FL + ρ (3) S |. H-phase Pr 3+ :YPO 4 should have strong dressing due to low D 2 point-group symmetry site, yet suppression dip is weak ( figure 7(a1)). This can be explained from low transition probability and weak dipole moment, causing weak dressing effect and low contribution of  figure 7(a2)). Figure 7(b) shows the excitation spectra of Pr 3+ :YPO 4 by varying temperatures from low (77 K) to high (300 K) under the same conditions as defined for figure 7(a). When the temperature is fixed at 77 K, the phonon effect is weak in comparison to the dressing effect. Due to the strong dressing effect, strong φ x(3) S − φ x (4) FL causes Δφ H to increase up to 3π/2. Hence, multi-level hybrid Fano interference is observed in figure 7(b1). As temperature increases, the phonon effect increases and Δφ H decreases due to (4) FL reduced. At 300 K, weak dressing effect results in weak multi-level hybrid Fano interference as shown in figure 7(b3). The Fano phase (Δφ H ) in figures 7(b1)-(b3) are arranged from left to right between 3π/2 and 2π in figures 7(d4) and (d5)), respectively. The maximum multi-level hybrid Fano contrast C H of about 3.28 (figure 7(b1)), is measured at 77 K, which is significantly better than C H measured for figure 6(c4). The lowest value of Fano contrast is observed at 300 K (figure 7(b3)). Figure 7(c) shows the spectral signal obtained from (less H-and much T-)phase Pr 3+ :YPO 4 by changing the power of E 2 from low (1 mW) to high (5 mW). At low power, multi-level hybrid Fano interference is low, which can be explained from low φ x(3) x(4) (as defined in figure 5(a)) decreasing Δφ H . This result corresponds to the theoretical results in figure 7(d4). As power increases, Δφ H increases due to the increase in dressing effect, suggesting strong multi-level hybrid Fano interference in figure 7(c3). The hybrid in Δφ H ≈ 2π interference phases (Δφ H ) in figures 7(c1) and (c2) are arranged from right to left between 3π/2 and 2π in figures 7(d4) and (d5). Figure 7(d) represents the theoretical results of the evolution of hybrid Fano interference plotted against the hybrid Fano phase using equation (9). The Fano contrast C H increases from 2 (figure 7(c1)) to 2.8 ( figure 7(c3)) as the power of E 2 increases from 1 mW to 5 mW. Although multi-level hybrid Fano interference is very prominent at high power, the corresponding Fano contrast measured is worst ( figure 7(c3)).

Conclusion
In summary, we theoretically and experimentally studied the Fano interference from the dressed SP-FWM, the dressed MFL and the hybrid signals, observed from different phases of Eu 3+ /Pr 3+ :YPO 4 crystals. In physics, Fano interference results from constructive and destructive interference of observed continuous and discrete states. Here, we obtained such states by controlling the gate position, temperature and power. We also investigated the symmetry dependent multi-level hybrid Fano interference and observed that (H + T)-phase demonstrated the strongest peak and dip, whereas pure H-phase did not demonstrate any Fano interference at all. Also, the multi-level hybrid Fano interference was observed to be stronger with Pr 3+ :YPO 4 than Eu 3+ :YPO 4 . We also observed that by decreasing temperature from high to low, multi-level hybrid Fano interference increases. Further, the Fano interference can be used for optical sensing and detecting, which also offers a possibility to explore new fundamental physics in analogous atomic systems.