Lifetime Autler-Townes Splitting of Dressed Multi-Order Fluorescence in Pr3+:YSO

For first time, we study primary and secondary Autler-Townes (AT) splitting of multi-order fluorescence (FL) in time domain. The AT-splitting of multi-order FL signals are controlled by changing power, detuning, and polarization of single and double dressing in a heteronuclear-like molecule system of Pr+3:YSO. The primary and secondary AT-splitting is caused by double cascaded dressing in time domain. The AT-splitting of multi-order FL in time domain is more sensitive than that of in spectral domain. Such results have potential applications in quantum communication and optical information processing on photonic chip.


INTRODUCTION
Recently, quantum coherence excitation and coherence transfer have been thoroughly studied in rare-earth ion doped crystals (like Pr 3+ :Y 2 SiO 5 ) which exhibit various advantages of coherent excitation. The recent research progresses related to atomic coherence in solid-state materials include electromagnetically induced transparency in solid materials [1], optical velocity reduction [2], all-optical routing based on optical storage [3], coherent storage of light pulses [4], and controllable erasing of optically stored information [5]. The EIT processes can be controlled by selecting different transitions among Zeeman sublevels via the polarization states of the laser beams [6]. The polarization dependences of four-wave mixing (FWM) and the enhanced FWM have been studied [7,8]. The atomic Autler-Townes (AT) splitting was first observed on an rf transition [9] and then in calcium atoms [10]. Here we study AT-splitting of fluorescence (FL) in time domain, which exhibit several applications in all-optical communication and optical information processing on photonic chip. The aim of this paper is to investigate primary and secondary AT-splitting of FL by changing power, detuning, and polarization of laser beams in Pr 3+ :Y 2 SiO 5 crystal. We also compare ATsplitting of FL in time domain and frequency domain, which is controlled by multi-parameters. The polarization states of generating fields and dressing fields are achieved by half-waveplate (HWP) and quarter-wave plate (QWP).

EXPERIMENTAL SETUP AND THEORY
To implement this experiment, the sample is held in a cryostat (CFM-102) with a temperature 77 K. The Y 2 SiO 5 crystal is doped with rare-earth Pr 3+ ion with doping concentration of 0.05% to form Pr 3+ : Y 2 SiO 5 . Under the action of crystal field of YSO, the triplet energy-level 3 H 4 and singlet energy-level 1 D 2 split into nine and five stark components, respectively. The Pr 3+ impurity ions occupy two non-equivalent cation sites (denoted as sites I and II) in YSO crystal lattice where the energy-level of site I is labeled without asterisk and the one for site II is labeled with an asterisk. Two tunable dye lasers (with a 0.04 cm −1 linewidth) pumped by an injection locked single-mode Nd:YAG laser (Continuum Powerlite DLS 9010, 10 Hz repetition rate, 5 ns pulse width) are used to generate the pumping fields E a (ω 1 , monitored by two photomultiplier tubes (PMTs) with a fast gated integrator. These laser beams are in xy-plane as shown in Figure 1A.
In this experiment, we use HWP and QWP to change polarization state of E a and E b . The propagation direction of vertically polarized beams E a and E b are opposite as shown in Figure 1A. A PMT is used to detect FL signals. We can obtain spectral signals by scanning laser frequency and time domain signals by fixing laser frequency.
For understanding the influences of the incident beams on AT-splitting of multi-order FL, we report the corresponding experimental observations, which can be effectively controlled by the polarization states of the pumping laser beams by HWP and QWP. Due to the periodic change of the polarization states of the pumping beam, the intensity of the beam also changes periodically. Figure 1D shows transition paths and Clebsch-Gordan (CG) coefficients at different laser polarization states. The HWP is used to investigate dressing effect in different level systems i.e., two-level system (single dressing) and V-type system (double dressing). In quantum theory, the FL signal intensity can be described by the diagonal density matrix elements. In Vtype system, when fields E a and E b are open simultaneously, the fourth-order (or multi-order) FL can be generated, which can be defined by perturbation chain ρ (0) Where ρ aa is probability of diagonal density matrix element at |0>. When E a and E b are open simultaneously, E b excitation from |0> to |2> in ket site, represented by off-diagonal density matrix (ρ (1) ca ) in perturbation chain, is also called coherence. In perturbation chain (E a ) * and (E b ) * are complex conjugate of E a and E b , respectively. The E a beam excitation from |0> to |1> in ket site, represented by ρ (3) ba in perturbation chain [11]. Considering the dressing effect of E a and E b the diagonal density matrix element ρ (4) bb is given by Equation (1) shows density matrix element without polarization. Equation (2) shows density matrix element, when polarization of incident beam E a (or E b ) is changed. Theta θ 2 (or θ 4 ) is used when E a (or E b ) polarization is changed by HWP. Equations (3) and (4) show density matrix element for linear polarization and for circular polarization by QWP, respectively. We use θ 1 (or θ 3 ) when E a (or E b ) polarization is changed by QWP. Here, A 2 = c 2 x cos 2 2θ 2 and A 2 = c 2 y sin 2 2θ 2 for horizontal and vertical polarizations of E b , respectively, and , and G i = −µ ij E i /h is the Rabi frequency of E i with the electric dipole moment µ ij between levels |i and |j , and Ŵ ij is the transverse decay rate. Next, we will discuss the AT-splitting results in twolevel and three-level system.

RESULTS AND DISCUSSION
In Figure 2 we investigate the AT-splitting of FL in time domain. We obtain time domain FL signals by fixing E a frequency at resonant point with different powers (P 1 ) of E a and with different QWP polarization (θ 1 ) of E a as illustrated in Figures 2A,B, respectively, in two-level system. With increasing power of E a from bottom to top we can see in Figure 2A the right peak shifts toward right (shown by dashed line in Figure 2A) and left peak intensity increases first (due to effect of E a power) then it decreases due to dressing effect. The total intensity can be written as ρ = ρ SP + ρ (4) bb and there is a competition between the SP-FWM and FL [8]. In the total process, ρ (4) bb increases first with the increasing of the |G a | 2 and gradually vanishes due to dressing effect of |G a | 2 /Ŵ bb (refer to Equation 1). At first, the power of the pumping field E a is too little (1 mW) to split |1 into |G a± , so we cannot see right peak appear in Figure 2a1. With the increasing of the power (≈8 mW) of E a , the energy level |1 splits into |G a± and right peak appears can be seen in Figure 2a5. We can explain increasing of the splitting distance by observing the Hamiltonian , when we set |1 at the frequency reference point. From the equation H|G a± = λ ± |G a± , we can obtain The obvious splitting distance of the right peak in Figure 2A is caused by the residual particles in |G a+ transferring to |G a− through phonon-assisted non-radiative transition which is mainly determined by acoustic phonons at low temperature. The AT-splitting distance is proportional to the distance between |G a+ and |G a− . With increasing of the G a , a keeps increasing which leads to the increasing of the splitting distance of the right peak. Furthermore, we can say that the left and right peaks of the FL signal correspond to the dressed states |G a+ and |G a− , respectively ( Figure 1C). Here, second peak (right) is moving toward right while first peak (left peak) is not moving horizontally (see Figure 2A). The curve (in Figure 2E) shows that AT-splitting distance corresponding to Figure 2A which is measured from zero delay time to right peak. AT-splitting distance increases as power increases. Figures 2B,F can be defined with same reason as in Figures 2A,E, respectively but here we use QWP. We change QWP polarization of E a (θ 1 ) from right-circular polarization (45 • ) to linear polarization state (90 • ) corresponding Figure 2B from bottom to top. Although there are several transition paths for the FL signals, considering the population of each level, the optical transition from the lowest crystal-field level |±5/2 is the dominant one shown in Figure 1D. The Rabi frequency for right-circularly polarized states G + a−5/2 is smaller than linearly polarized states G 0 a±5/2 , so a is smaller which makes the delay time of the right peak smaller at 45 • than at 90 • . Due to decrease in Rabi frequency the AT-splitting increases from |G + a−5/2 | 2 /(Ŵ b−3/2a−5/2 + i 1 ) to |G 0 a−5/2 | 2 /(Ŵ b−5/2a−5/2 + i 1 ) refer to Equation (4) and Equation (3), respectively. As polarization states approach horizontal then ρ (4) bb decreases, which determines the reducing of the right peak intensity and increasing AT-splitting distance as shown in Figure 2B. The reason for AT-splitting distance

(H)
Shows distance between three peaks in (D) with respect to (w.r.t) polarization. The curve with circles (h1) shows distance between first peak and third peak of (D) w.r.t polarization. The curve with asterisk (h2) shows distance between first peak and second peak of (D) w.r.t polarization, and finally curve with triangles (h3) shows distance between second peak and third peak of (D) w.r.t polarization.
can be given with help of Hamiltonian similarly as Figure 2A.
In Figures 2C,D, we investigate the AT-splitting of FL obtained by changing HWP and QWP polarization, respectively in time domain two-level system (using two beams), where we change polarization state of input polarized beam E a by HWP (θ 2 ) and QWP (θ 1 ) as illustrated in Figures 2C,D the evolution of the FL is similar as Figure 2B. The reason is also similar with twolevel system using QWP in Figure 2B but here the |G a+ is further split by E b . Therefore, Hamiltonian can be written as . When 2 ∼ 0, we can know that the increasing of the |G b | can lead to the increasing of the |λ − | in the cascaded situation here. The splitting distance between |G b− and |G a+ can be expressed as b = 2(|G a | 2 +|G b | 2 ) 1/2 when 1 = 2 = 0. So when we change the polarization angle of E b from vertical polarization to horizontal polarization, the Rabi frequency of the E b changes from c 2 y | G b | 2 to c 2 x | G b | 2 refer to Equation (2) which leads to the increasing of the b and the delay time of the right peak. As the dressing increases from c 2 Equation 2). In Figure 2D describe two-level system using two beams, it also has double cascaded dressing, primary dressing can be described similarly as in two-level system single beam (in Figure 2B) and secondary dressing can be given by terms |G + b−5/2 | 2 /(Ŵ a−3/2b−5/2 + i 2 ) to |G 0 b−5/2 | 2 /(Ŵ a−5/2b−5/2 + i 2 ) for circular and linear polarization states, refer to Equation (4) and Equation (3), respectively which determines the increasing of AT-splitting distance from bottom to top due to increasing of dressing effect as shown in Figure 2D. In Figure 2D when E a primary dressing is changed, the second and third peaks are moving together toward right, however, the distance between second and third peak is not changing because such distance in time domain is only determined by secondary dressing E b . Furthermore, we can say that coupling field E a dresses the state |− > and splits it in to two secondary dressed states |−+ > and |−− >. The three peaks in the FL time domain signal (Figure 2d5), from left to right, can be corresponded to the primary dressed state |+ >, the secondary dressed states |−+ > and |−− >, respectively, shown in Figure 1C. On the other hand, second peak is smaller because of near double resonance dressing at 1 ∼ 2 ∼ 0 [12]. Next, we will study time domain and spectral domain FL AT-splitting in V-type system by changing power and detuning as shown in Figure 3. Figures 3A,B show the variation of composite FL signals intensity at 2 = 0 in the time domain by opening E a and E b in V-type system. The power (P 2 ) of E b is changed in Figure 3A while E b detuning ( 2 ) is changed in Figure 3B. Figure 3A shows the variation of composite FL signals intensity at 2 = 0 in the time domain by increasing E b power from 1 mW (bottom curve) to 8 mw (top curve), and scanning E a . The V-type level system contains two dressing fields (E b primary dressing and E a secondary dressing). The dressings can be explained similar as in two-level two beams system as in Figure 2C, however, due to 2 = 0 therefore, E b became primary dressing. In Figure 3A level |1 is already split into |G b± by E b with increasing power of E a will split into |G a−± with dressing effect of (|G a | 2 /Ŵ bb refer to (1)). Therefore, the splitting distance between |G b−− and |G a+ can be expressed by b = 2(|G a | 2 +|G b | 2 ) 1/2 at 2 = 0. Furthermore, three peaks theory in time domain can be explained similar as two-level system Figure 2D. Figure 3B shows the variation of composite FL signals intensity at 2 = 0 in time domain V-type system, where E b detuning is changed. We can see that by changing detuning from −200 GHz to resonant point (0 GHz) of E b , we can see that the right peak shifts toward right from bottom to top (shown by dashed line in Figure 3B). Here, level |1 is already split into |G a± by E a and when E b detuning approaches to resonant point then |G a− will further split into |G b−± with dressing effect of (|G b | 2 /Ŵ bb refer to (1). Figure 3C shows theoretical predications of time domain FL signals corresponding to Figure 3B. It can be clearly seen that experimental results match theoretical predication. Figures 3D,E show AT-splitting distance of V-type level system by changing power, and detuning corresponding to Figures 3A,B, respectively. In Figure 3F we discussed ATsplitting of FL in spectral. Here, initially 2 (−200 GHz) is far away from the resonant region, the FL signal has no AT-splitting and is nearly not affected by E a as shown in Figures 3f1,2). As 2 get closer to the resonant point, the baseline (dashed-curve) raises gradually due to the competed excitations of the particles by E a and E b . At the two peaks of AT-splitting of the FL signal profile, E a dominates the competition and dresses the FL signal. At the near-resonant region, the competition of E b on particles increases gradually until 1 ∼ 0 due to the self-dressing effect of the FL signal. However, the dressing effect of E a on the FL signal increases gradually and the AT-splitting depth (doted-line in Figure 3F) reaches the maximum at 1 ∼ 2 ∼ 0. Such mutual interaction between self-dressing FL signals and extradressing FL signals can be well-interpreted refer to Equation   Figure 3A) at near resonance stage. The two peaks in the FL spectral signal (Figure 3f5) can be seen clearly while central peak is very small, which cannot seen because of near double resonance dressing at 1 ∼ 2 ∼ 0 [12]. Thus, the time domain (Figure 3A) AT-splitting is more sensitive than spectral ( Figure 3F) [10]. Figure 4A shows the variation of composite FL signal intensity at 2 = 0 in the time domain by changing QWP polarization of E b (θ 3 ) from circular to linear polarization states (bottom to top) and scanning E a . We can observe that when E a polarization approaches to right circular then right peak appears, which can be seen clearly in Figure 4a4, and Figure 4a5 at θ 3 = 75 • , and 90 • , respectively. The QWP polarization can be described similarly as two-level ( Figure 2D). Figure 4B can be explained similarly as two-level system ( Figure 2C). Figure 4B shows the variation of composite FL signal intensity at 2 = 0 in the time domain V-type system, where E b polarization (θ 4 ) is changed by HWP from vertical to horizontal shown from bottom to top. Therefore, suppression increases, the right peak moves to right and AT-splitting also increases as shown in Figure 4B. The increasing of |G a | due to change in polarization state can also lead to the increasing of |λ − |. When 1 = 2 = 0, the splitting distance between |G b−− and |G a+ is b = 2(|G a | 2 +|G b | 2 )1/2. Figure 4C shows the FL signals in V-type three-level (double dressing) system, where polarization angle (θ 4 ) of the dressing field E b is changed by HWP. That can be explained with same reason as in Figure 3F. Figures 4D,E show AT-splitting distance corresponding to Figures 4A,B, respectively. In Figure 4B we use high power E a and E b beams. Therefore, when we change polarization of beam E b by HWP from vertical to horizontal then particles are transferring from ground state to excited state and then a situation will reach, when there will be no particles to transfer from ground state to excited state, that situation is called saturation state can be seen in Figure 4B. The AT-splitting distance saturation can be seen in Figure 4E. The saturation is resulted from the balance between −|G b | 2 /d 1 d 12 − |G b | 2 /d 2 Ŵ cc and |G b | 4 /d 1 d 12 d 2 Ŵ cc refer to (5). Figure 4F shows theoretical predications of frequency domain FL signals corresponding to Figure 4C. It can be clearly seen that experimental results match theoretical predication.

CONCLUSION
It is concluded that AT-splitting of multi-order FL signals can be controlled by changing power, detuning, and polarization of single and double dressing in Pr 3+ :YSO. The primary and secondary AT-splitting are caused by double cascaded dressing in time domain. The AT-splitting of multi-order FL in time domain is more sensitive than that of in spectral domain. Such results have potential applications in quantum communication.