Temporal interaction of hybrid signals in various phases of Eu³⁺: BiPO₄ through photon–phonon dressing

In figure 2, we discuss the spectral and temporal outputs from half HP and half LTMP (6:1) of Eu³⁺: BiPO₄. Here, we discuss the one laser broadband (or narrowband) spectral dressing dip (figures 2(a) and (c)) and two lasers spectral profile intensity interactions (figures 2(b) and (d)). Figure 2(a) and (e) shows a single dressing dip at 10 μs of time-gate positions through broadband excitation, which supports thermal phonon ${\left\vert {G}_{pT}\right\vert }^{2}$ dressing from ${\left\vert {G}_{1}\right\vert }^{2}/({{\Gamma}}_{21}+i{{\Delta}}_{1}+{\left\vert {G}_{pT}\right\vert }^{2}/({{\Gamma}}_{22}+i{{\Delta}}_{p1}))$ (equations (1) and (2)). Here E₁ atomic coherence couples the thermal phonon atomic coherence. The narrowband excitation in figures 2(c) and (g) includes only photon2 dressing ${\left\vert {G}_{2}\right\vert }^{2}$ from equations (1) and (2) and results in no dressing dip at 10 μs time-gate positions. The spectral profile (figures 2(b) and (d)) shows two photons and two phonon coupling dressing $d={\left\vert {G}_{1}\right\vert }^{2}/({{\Gamma}}_{10}+i{{\Delta}}_{i}+{\left\vert {G}_{2}\right\vert }^{2}/({{\Gamma}}_{10}+i{{\Delta}}_{i}))+{\left\vert {G}_{pT}\right\vert }^{2}/\left(\right.\,{{\Gamma}}_{20}+i{{\Delta}}_{i}-i{{\Delta}}_{p1}$ $+{\left\vert {G}_{p2}\right\vert }^{2}/({{\Gamma}}_{23}+i{{\Delta}}_{i}+i{{\Delta}}_{p1}-i{{\Delta}}_{pS})\left.\right)$ in the hybrid signal from equation (5).

The dressing dips intensity interaction is the difference in intensities between dressing dips in figures 2(b) and (d), and the sum of dressing dips intensities originates from E_{1 ,} E₂ in figures 2(a) and (c), respectively. The dressing dips interaction is destructive if the intensity of the sum of the dressing dips from the single laser figures 2(a) and (c) is greater than the intensity of dips originated from two lasers destructive dressing figure 2(b). Due to two photons nested destructive dressing ${\left\vert {G}_{1}\right\vert }^{2}/({{\Gamma}}_{10}+i{{\Delta}}_{i}+{\left\vert {G}_{2}\right\vert }^{2}/({{\Gamma}}_{10}+i{{\Delta}}_{i}))$ in the hybrid signal (equation (5)), the dressing dips interaction reduces about 2-times (figure 2(b)) as compared to the above sum exhibited by broadband (figure 4(a)) and narrowband (figure 4(c)) laser excitations. Similarly, destructive spectral dips intensity interaction exhibits in figure 2(d) while the increase in the background owing to contribution from E₁. Such destructive interaction comparison of two lasers is controlled by thermal phonon Rabi frequency (G_pT ) in figures 5(b) and (d) and phase transition phonon detuning (Δ_pS ) in figures 4(b) and (d).

Since the difference between resonance and non-resonance terms (${\mathfrak{R}}_{1}^{(2)}-{\aleph }_{1}^{(2)}={R}_{1}$ or ${\mathfrak{R}}_{2}^{(2)}-{\aleph }_{2}^{(2)}={R}_{2}$) represents the strength of temporal profile interaction, so the destructive temporal profile interaction resonance term ${\mathfrak{R}}_{1}^{2}$ from figure 2(j1) decreases about 3-times in contrast to the one laser resonance term in figure 2(i1) i.e., ${R}_{1}+{N}_{1}=3{\mathfrak{R}}_{1}^{(2)}$, this is due to ${\mathfrak{R}}_{1}^{2}$ suppressed by two lasers destructively dressing. In addition, the difference in ${\mathfrak{R}}_{1}^{2}$ (figure 2(j1)) and ${\aleph }_{1}^{(2)}$ (figure 2(j2)) is almost equal to the difference between ${\mathfrak{R}}_{2}^{(2)}$ (figure 2(l1)) and ${\aleph }_{2}^{(2)}$ (figure 2(l2)), suggesting the strength of the two laser temporal profile interaction is similar in figures 2(j) and (l). Comparing the resonance terms of broadband and narrowband excitations, R₁ + N₁ (figure 2(i1)) is 2.5-times greater than R₂ + N₂ in figure 2(k1). Similarly, about 6-times higher non-resonance exhibits for broadband excitation (figure 2(i2)) than that of narrowband excitation (figure 2(k2)) i.e., N₁ ≫ N₂.

Further, we demonstrate TAT splitting for one laser broadband (or narrowband) and its interaction for two laser excitations. The TAT splitting between the sharp peak and broad shoulder (figure 2(i)) for broadband excitation, can be written as τ_inter < τ₀₁ + τ₀₂ − 2t_cf where t_phot = t_pT + t_pS and t_pT , t_pS represent thermal phonon and phase transition phonon splitting terms, respectively. For narrowband excitation (figure 2(k)) τ₀₂ = t_cf + t_phot. In figure 2(i) τ₀₁ = 10 μs is greater than τ₀₂ = 6 μs (figure 2(k)) due to broadband excitation including phonon splitting. The destructive profile interaction (figures 2(j) and (l)) is sensitive to photon dressing (t_phot) and phonon dressing (t_phon) while insensitive to crystal field splitting (t_cf). Because of τ_inter, the two lasers, TAT splitting (τ⁽²⁾) with destructive nested dressing ${\left\vert {G}_{1}\right\vert }^{2}/{\left\vert {G}_{2}\right\vert }^{2}$ is smaller than the sum of TAT splitting without crystal field splitting of each laser i.e., τ_inter < τ₀₁ + τ₀₂ − 2t_cf (equations (3) and (4)). The TAT splitting interaction is much smaller than the observed difference between TAT splitting in figures 2(j) and (l) and the sum of two single laser TAT splitting in figures 2(i) and (k) i.e., τ_inter ≪ τ₀₁ + τ₀₂.

In figure 3, the spectral-temporal profile interaction is studied for the crystal structure (more HP + less LTMP). Different sample crystals structure determines different lattice vibration frequencies of phase transition phonon. According to G_pj = −μ_mn E_pj /ℏ with constant Ω_mn , ω_pj is increased due to change in the crystal phase (figure 3) as compared to figure 2. The medium ω_pj (figure 2) and larger ω_pj (figure 3) results in Δ_pj (off-resonance) and Δ_pj (near-resonance), respectively.

Next, we show that the temporal intensity profile (figures 3(i)–(l)) is reduced by more phonon dressing as compared to figures 2(i)–(l). Such interaction of resonance term ${\mathfrak{R}}_{1}^{(2)}$ (figure 3(j1)) is caused by the phase transition phonon detuning effect. The temporal profile interaction (figure 3(l1)) resonance term becomes equal to that of figure 3(j1) i.e., ${\mathfrak{R}}_{2}^{(2)}={\mathfrak{R}}_{1}^{(2)}$. The difference in ${\mathfrak{R}}_{1}^{(2)}-{\aleph }_{1}^{(2)}\ne {\mathfrak{R}}_{2}^{(2)}-{\aleph }_{2}^{(2)}$. By comparing figures 3(j1), (j2), (i1) and (i2), we realize that ${\mathfrak{R}}_{1}^{(2)}$ is decreased about 2-times i.e., ${R}_{1}+{N}_{1}=2{\mathfrak{R}}_{1}^{(2)}$ due to two lasers destructive dressing while ${\aleph }_{1}^{(2)}$ is enhanced about 2-times i.e., ${\aleph }_{1}^{(2)}=2N$ due to contributions from R₂ > 0. Furthermore, the TAT splitting of one laser and two laser excitations (figures 3(i)–(l)) can be explained the same as figures 2(i)–(l).

In comparison with figures 2 and 3, figure 4 shows destructive spectral-temporal interaction for the pure HP, suggested by more phase transition phonon dressing. Such crystal structure results in the largest vibration of phonon frequency (ω_pj ) i.e., Δ_pj ≅ 0 (resonance phase transition phonon dressing) when compared to figures 2 and 3(a). The one laser broadband excitation supports ${\left\vert {G}_{1}\right\vert }^{2}/({{\Gamma}}_{10}+i{{\Delta}}_{i}+{\left\vert {G}_{pT}\right\vert }^{2}/({{\Gamma}}_{20}+i{{\Delta}}_{i}-i{{\Delta}}_{p1}+{\left\vert {G}_{p2}\right\vert }^{2}/({{\Gamma}}_{23}+i{{\Delta}}_{i}+i{{\Delta}}_{p1}-i{{\Delta}}_{pS})))$ coupling dressing and exhibits three dressing dips in figure 4(a) due to phonon frequency approaching Δ_pj ≅ 0 in contrast to the single dressing dip in figure 2(e). Here broadband excitation atomic coherence is coupled to thermal phonon and phase transition phonon atomic coherence. As narrowband excitation does not support the phonon dressing, therefore, an increase in ω_pj (figure 4(c)) does not exhibit any change, and the results are consistent with figure 2(c). However, the two laser spectral dips intensity interaction in figure 4(b) decreases about 1.33-times due to phonon nested dressing ${\left\vert {G}_{pT}\right\vert }^{2}/({{\Gamma}}_{20}+i{{\Delta}}_{i}-i{{\Delta}}_{p1}+{\left\vert {G}_{p2}\right\vert }^{2}/({{\Gamma}}_{23}+i{{\Delta}}_{p1}-i{{\Delta}}_{pS}))$. In contrast to figure 2(b). The two laser spectral-temporal profile destructive interaction in figure 4(d) can be explained in a similar fashion as figure 2(d).

Next, we emphasize and suggest that both the resonance and non-resonance terms of the temporal intensity profile interaction are strongly affected by more phase transition phonon dressing. Such interaction resonance term ${\mathfrak{R}}_{1}^{(2)}$ in figure 4(j) reduced about 1.5 times compared to figure 2(j). The phase transition phonon detuning (Δ_pS ≅ 0) is at resonance in figure 4(l) as compared to figure 2(l), which suppresses the interaction through ${\left\vert {G}_{pT}\right\vert }^{2}/{\left\vert {G}_{pS}\right\vert }^{2}$ dressing in the hybrid signal (equation (5)). Here ${\mathfrak{R}}_{2}^{(2)}$ in figure 4(l1) is completely overlapped by ${\aleph }_{2}^{(2)}$ (figure 4(l2)) i.e., ${\mathfrak{R}}_{2}^{(2)}={\aleph }_{2}^{(2)}$. The two lasers temporal profile interaction disappears in figures 4(k) and (l) owing to no difference exhibited between ${\mathfrak{R}}_{2}^{(2)}$ and the sum of the resonance terms from E₁, and E₂. Comparing figures 4(l1) and (l2) to figures 4(k1) and (k2), ${\mathfrak{R}}_{2}^{(2)}$ (figure 4(l1)) is completely suppressed by the non-resonance term due to the significant contribution of ${\aleph }_{2}^{(2)}$ in figure 4(l2) from R₁ > 0. The difference between ${\mathfrak{R}}_{1}^{(2)}$ and ${\aleph }_{1}^{(2)}$ in figure 4(j) is enhanced compared to figure 2(j) due to a decrease in ${\aleph }_{1}^{(2)}$ while ${\mathfrak{R}}_{2}^{(2)}-{\aleph }_{2}^{(2)}$ in figure 4(l) remains almost the same as figure 2(l). Furthermore, phase transition phonon detuning will be gradually increased from figures 2–4. The difference between ${\mathfrak{R}}_{1}^{(2)}$ and ${\aleph }_{1}^{(2)}$ in figure 4(b) becomes very larger than that of ${\mathfrak{R}}_{2}^{(2)}$ and ${\aleph }_{2}^{(2)}$ (figure 4(d)) i.e., ${\mathfrak{R}}_{1}^{(2)}-{\aleph }_{1}^{(2)}\gg {\mathfrak{R}}_{2}^{(2)}-{\aleph }_{2}^{(2)}$. Due to a decrease in N₁, the difference between R₁ + N₁ and N₁ in figure 4(i) is enhanced compared to figure 2(i). Owing to R₂ ≅ N₂ ≅ 0 in figure 4(k), the R₂ and N₂ suggest reduction of about 7-times and 4-times, respectively, as compared to figure 2(k). From the above discussion, it is exhibited that temporal intensity profile interaction efficiently reduces with enhancement in the phase transition phonon dressing.

Further, we show that more phase transition phonon dressing reduces TAT splitting for one laser (figures 4(i) and (k) and two laser TAT splitting interactions (figures 4(j) and (l). The largest ω_pj value influences an increase in t_pS which strongly affects the TAT splitting of narrowband excitation in figure 4(k). Since narrowband excitation does not support Δ_pj , therefore, τ₀₂ approaches zero in figure 4(k). τ⁽²⁾ in figures 4(j) and (l) is equal to τ₀₁ in figure 4(i). The difference between of τ⁽²⁾ in figures 4(j) and (l) and the sum of two single laser TAT splitting (figures 4(i) and (k) is also almost equal to zero. Therefore, TAT-splitting interaction (τ_inter) decreases in figures 4(j) and (l) for the pure HP as compared to that of half HP + half LTMP in figures 2(j) and (l)).

The interaction results in figures 2–4 were based on thermal phonon dressing at 300 K. In the following, we decrease the temperature to 77 K in order to study the interaction without thermal phonon dressing in three phase transitions.

In figure 5, the intensity of spectral-temporal profile for one laser excitation and its interaction for two lasers excitations is dramatically reduced by small thermal phonon dressing. The Rabi frequencies of the thermal phonon dressing is determined by the temperature (thermal phonon number). On the other hand, two laser spectral-temporal profile interaction (figure 5) from the hybrid signal decreases when the temperature is reduced to 77 K (less phonon number) leading to a small Rabi frequency (G_pT ≅ 0) in contrast to 300 K (figure 2).

At 77 K, the broadband excitation (figure 5(a)) is surprisingly similar to narrowband excitation (figure 5(c)). Because the photon-thermal phonon coupling disappears and only ${\left\vert {G}_{1}\right\vert }^{2}$ dressing is left in figure 5(a) as compared to figure 2(a). Due to G_pT ≅ 0 at 77 K, the dressing dips (number) from one laser (figures 5(a), (c) (e) and (g)), and its intensity interaction $({I}_{\mathrm{F}\mathrm{L}+S}^{(2)})$ of the two lasers profile (figures 5(b), (d), (f) and (h)) disappear as compared to figures 2(a)–(d) and (e)–(h).

Next, we show that temporal intensity profile interaction (figures 5(i)–(l)) is decreased due to less thermal phonon dressing. The difference between ${\mathfrak{R}}_{1}^{(2)}$ (figure 5(j1)) and ${\aleph }_{1}^{(2)}$ (figure 5(j2)) decreases 3-times compared to (figure 2(j1)) and (figure 2(j2)), respectively. The difference between ${\mathfrak{R}}_{2}^{(2)}$ (figure 5(l1)) and ${\aleph }_{2}^{(2)}$ (figure 5(l2)) decreases about 2-times as compared to figure 2(l). Comparing ${\mathfrak{R}}_{1}^{(2)}$ (figure 5(j1)) and ${\aleph }_{1}^{(2)}$ (figure 5(j2)) to the R₁ + N₁ (figure 5(i1)) and N₁ (figure 5(i2)), ${\mathfrak{R}}_{1}^{(2)}$ decreases by 2-times i.e., while ${\aleph }_{2}^{(2)}$ enhances by 1.3-times i.e., ${\aleph }_{1}^{(2)}=1.3{N}_{1}$. Furthermore, phonon dressing becomes very small as G_pT ≅ 0 (equation (5)) at 77 K and reduces the intensity of temporal profile interaction resonance and non-resonance curves. A decrease in the thermal phonon dressing results in a decrease in ${\aleph }_{1}^{(2)}$ figure 5(j2) which induces an enhancement in the difference between ${\mathfrak{R}}_{1}^{(2)}$ and ${\aleph }_{1}^{(2)}$ (figure 5) as compared to figure 2. The difference between ${\mathfrak{R}}_{1}^{(2)}$ (figure 5(j1)) and ${\aleph }_{1}^{(2)}$ (figure 5(j2)) is about 2-times more than the difference between ${\mathfrak{R}}_{2}^{(2)}$ (figure 5(l1)) and ${\aleph }_{2}^{(2)}$ (figure 5(l2)) i.e., ${\mathfrak{R}}_{1}^{(2)}-{\aleph }_{1}^{(2)} > {\mathfrak{R}}_{2}^{(2)}-{\aleph }_{2}^{(2)}$. Both the resonance term (figure 5(i1)), and the non-resonance term (figure 5(i2)) of broadband excitation reduced 2-times compared to that of figures 2(i1) and (i2). The temporal profile resonance term in figure 5(k1) dramatically reduced as compared to figure 2(k1).

Further, we exhibit that thermal phonon strongly affect the TAT splitting interaction. The τ⁽²⁾ in figures 5(j) and (l) disappears at low-temperature as τ₀₂ ≅ 0. However, due to no contribution from thermal phonon at 77 K, τ_inter reduces about 2-times in figures 5(j) and (l) compared to figures 2(j) and (l).

In figure 6, the phase transition phonon dressing increases due to a change in the crystal structure while keeping the thermal phonon dressing the same as in figure 5. A large value of phonon frequency (ω_pj ) i.e., Δ_pj (near-resonance) in figure 6 exhibits a minor change in the spectral-temporal profile as compared to figure 5. The spectral signals dip (TAT splitting) from one laser in figures 6(a) and (c), and its intensity interaction $({I}_{\mathrm{F}\mathrm{L}+S}^{(2)})$ of two lasers in figures 6(b) and (d) condense more due to a decrease in Δ_pS , as compared to figure 5.

Next, the destructive temporal interaction reduces more (figures 6(j) and (l)) due to an increase in the phase transition phonon dressing, and can be explained the same as figures 3(j) and (l). We realize that ${\mathfrak{R}}_{2}^{(2)}={\mathfrak{R}}_{1}^{(2)}$ while ${\aleph }_{2}^{(2)}$ (figure 6(l2)) increases 3-times compared to ${\aleph }_{1}^{(2)}$ (figure 6(j2)) i.e. ${\aleph }_{2}^{(2)}=3{\aleph }_{1}^{(2)}$ and can be explained the same as figures 5(j2) and (l2). In figure 7, the destructive spectral and temporal interaction is controlled by thermal phonon Rabi frequency and phase transition phonon detuning. In figure 7, the temperature is reduced to 77 K when compared to figure 4 (300 K). Here the sample changed to pure HP. It should be noted that pure HP have high site symmetry than half HP + half LTMP (figure 5). Since, dressing dip number (figures 7(a) and (e)) through single laser should increase due to largest ω_pj i.e., Δ_pj ≅ 0, as compared to figures 2(a), (e) and (i). However, a decrease in thermal phonon G_pT , destroys the dressing dips in figures 7(a) and (e). In this situation, the phonon dressing (figure 7) has a small G_pT , and largest ω_pj (Δ_pS ≅ 0).

Next, we discuss the comparison of spectral and temporal profile interaction at different phonon detuning and temperatures. The two lasers dressing dips intensity interaction is maximum in figures 2(b), (d), (f) and (h) while completely destroying in figures 7(b), (d), (f) and (h). As compared to figures 2(b), (d), (j) and (l) with strong spectral-temporal profile interaction having smaller Δ_pS ≅ 0 and larger G_pT , here the spectral-temporal interaction (figures 7(b), (d), (j) and (l)) is completely destroyed by larger Δ_pS > 0 and smaller G_pT . For the pure hexagonal phase of Eu³⁺: BiPO₄, the thermal and phase transition phonon destructive dressing ${\left\vert {G}_{p1}\right\vert }^{2}/({{\Gamma}}_{20}+i{{\Delta}}_{i}-i{{\Delta}}_{p1}+{\left\vert {G}_{p2}\right\vert }^{2}/({{\Gamma}}_{23}+i{{\Delta}}_{p1}-i{{\Delta}}_{p2}))\left.\right)$ vanishes the spectral-temporal profile interaction at both 300 K and 77 K.

Further, we compare TAT splitting at different phonon detuning and temperatures. Since t_pT dominates over t_pS , thus Δ_pS ≅ 0 does not bring change in the TAT splitting in figures 7(i) and (k). The TAT splitting in figure 7(i) reduces about 2-time while completely destroying in figure 7(k) as compared to figures 2(i) and (k). Compared to figures 2(i)–(l) with strong TAT splitting interaction owing to larger t_pT and smaller t_pS , the TAT splitting and its interaction (τ_inter) destroys in figures 7(i)–(l) for the pure HP due to a decrease in t_pT .

Based on the above discussions we realized that spectral-temporal intensity profile interaction is depending upon temperature phonon and phase transition phonon. Such interaction is strongest at 300 K and less phase transition phonon dressing, while weakest at 77 K and more phase transition phonon dressing.

Finally, two phase transitions (figure 8) are compared to three phase transitions (figures 2–7) in hybrid signals with photon and phonon dressing. Different phases of a crystal have different lattice vibrational frequencies. A small value of the lattice vibrational frequencies (ω_pj ) i.e., Δ_p2S ≫ 0 (far off-resonance) reduces the phonon dressing than that of figures 2–7. However, the thermal phonon dressing in figures 8(a)–(d) can be explained the same as figures 2–4 and 5–7 respectively.

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Figure 8(a) shows the FL supremacy at the near time-gate position in the hybrid signal and results in a broad peak. The multi-peaks with three dressing dips in figure 8(b) are exhibited due to ${\left\vert {G}_{1}\right\vert }^{2}/({{\Gamma}}_{10}+i{{\Delta}}_{i}+{\left\vert {G}_{p1}\right\vert }^{2}/({{\Gamma}}_{20}+i{{\Delta}}_{i}-i{{\Delta}}_{p1}+{\left\vert {G}_{p2}\right\vert }^{2}/({{\Gamma}}_{23}+i{{\Delta}}_{i}+i{{\Delta}}_{p1}-i{{\Delta}}_{p2})))$ form equation (1), and can be explained the same as figure 4(a). In figure 8(b), the left dressing dip is produced by ${\left\vert {G}_{1}\right\vert }^{2}$, which is further added with two phonon dressing terms on its right produced through ${\left\vert {G}_{pT}\right\vert }^{2}/({{\Gamma}}_{20}+i{{\Delta}}_{i}-i{{\Delta}}_{p1}+{\left\vert {G}_{p2}\right\vert }^{2}/({{\Gamma}}_{23}+i{{\Delta}}_{p1}-i{{\Delta}}_{pS}))\left.\!\right)$. At 77 K, the pure LTMP exhibits a sharp peak with a slight change in the background in figure 8(c). Compared to figure 8(a), the linewidth of the spectrum becomes sharper shown in figure 8(c) due to a decrease in the FL at 77 K. A single dressing dip in figure 8(d) is exhibited by photon dressing ${\left\vert {G}_{1}\right\vert }^{2}/({{\Gamma}}_{21}+i{{\Delta}}_{1})$. Compared to figure 8(b), the dressing dips reduce in figure 8(d) due to ${\left\vert {G}_{p1T}\right\vert }^{2}/{\left\vert {G}_{p2S}\right\vert }^{2}$ nested dressing with G_p1T ≅ 0. Moreover, the strong dressing effect in figures 8(b) and (d) is exhibited due to the high power (7 mW) of the laser as compared to figure 4 and 7 (4 mW).