## Abstract

We report an observation of the self- and external-dressed Autler-Townes (AT) splitting in six-wave mixing (SWM) within an electromagnetically induce transparency window, which demonstrates the interaction between two coexisting SWM processes. The multi-dressed states induced by the nested interactions between many dressing fields and the five-level atomic system lead to the primary, secondary and triple AT splittings in the experiment. Such controlled multi-channel splitting of nonlinear optical signals can be used in a range of applications, e.g. the wavelength-demultiplexer in optical communication and quantum information processing.

©2011 Optical Society of America

## 1. Introduction

Aulter-Townes (AT) splitting was first observed on a radio-frequency transition more than sixty years ago [1]. Such effect has been clearly demonstrated in a four-wave mixing on a simple atomic system dressed with a single laser beam [2]. With the cw triple-resonant spectroscopy and the ultrashort intense laser pulses, respectively, the AT splitting effect in lithium molecules [3] and a semiconductor material [4] was investigated. Theoretical studies also indicate that the AT-split Rydberg population can lead to an antiblockade effect [5] and such phenomenon was experimentally demonstrated with two-photon excitation in a three-level atomic system [6]. Recently, a great deal of attention has been paid to observe and understand the phenomenon of electromagnetically induced transparency (EIT) [7–10] and related effects in multi-level atomic systems interacting with two or more electromagnetic fields [11,12].

The interaction of double-dark states (nested scheme of doubly-dressing) and splitting of dark state (the secondarily-dressed state) in a four-level atomic system with EIT were studied theoretically by Lukin, *et al* [13]. Then doubly-dressed states in cold atoms were observed, in which triple-photon absorption spectrum exhibits a constructive interference between the excitation pathways of two closely-spaced, doubly-dressed states [14]. A similar result was obtained in the inverted-Y [15] and double-Λ [16] atomic systems.

In this letter, we present the first experimental observation of the self-, doubly- and triply-dressed AT splitting states of the SWM process within the EIT window in a five-level atomic system. Theoretical calculations are carried out and used to well explain the observed results, giving a full physical understanding of the interesting multiple AT splittings in the high-order nonlinear optical processes. On the basis of our previous study of the AT splitting in the four-wave mixing (FWM) process [17], we go further to investigate the complex AT splitting phenomena in the SWM process.

## 2. Theoretical model and experimental scheme

The experimental demonstration of the AT splitting of SWM within the EIT window is carried out in the atomic system of ^{85}Rb. The energy levels of $5{\text{S}}_{1/2}(F=3)$, $5{\text{S}}_{1/2}(F=2)$, $5{P}_{3/2}(F=3)$, $5{D}_{3/2}$, and $5{D}_{5/2}$ form the five-level atomic system, as shown in Fig. 1(b)
. The atomic vapor cell temperature is set at ${60}^{\circ}C$. The probe laser beam ${E}_{1}$ (with frequency ${\omega}_{1}$, wave vector **k**_{1}, Rabi frequency ${G}_{1}$, and wavelength of 780.245 nm, connecting the transition $5{\text{S}}_{1/2}-5{P}_{3/2}$) is from an external cavity diode laser (Toptica DL100L), which is horizontally polarized and has a power of about ${P}_{1}\approx 1.3\mathrm{mW}$. The coupling laser beams ${E}_{2}$ (${\omega}_{2}$, ${\mathbf{k}}_{2}$, ${G}_{2}$, and wavelength 775.978 nm, connecting the transition $5{P}_{3/2}-5{D}_{5/2}$) and ${E}_{4}$ (${\omega}_{4}$, ${\mathbf{k}}_{4}$, ${G}_{4}$, and wavelength 776.157 nm, connecting the transition $5{P}_{3/2}-5{D}_{3/2}$) are from two external cavity diode lasers (Hawkeye Optoquantum and UQEL100), respectively. The pump laser beams ${E}_{3}$ (${\omega}_{3}$, ${\mathbf{k}}_{3}$, ${G}_{3}$) and ${E}_{3}^{\prime}$ (${\omega}_{3}$,${\mathbf{k}}_{3}^{\prime}$, ${G}_{3}^{\prime}$), which are split from a tapered-amplifier diode laser (Thorlabs TCLDM9) with equal power (${P}_{3}\approx {P}_{3}^{\prime}$) and vertical polarization, drive the transition $5{\text{S}}_{1/2}-5{P}_{3/2}$. The diameters of the probe, pump and coupling beams are about 0.3, 0.5, 0.5 mm at the cell center, respectively. The pump and coupling laser beams (${E}_{3}$, ${E}_{3}^{\prime}$, ${E}_{2}$, ${E}_{4}$) are spatially aligned in a square-box pattern as shown in Fig. 1(a), which propagate through the atomic medium in the same direction with small angles ($~0.{3}^{\circ}$) between them (the angles are exaggerated in the figure). The probe beam (${E}_{1}$) propagates in the opposite direction with a small angle from the other beams. Under this configuration the diffracted FWM signal (${E}_{\text{F}}$) and two SWM signals (${E}_{\text{S}1}$ and ${E}_{\text{S}2}$) with same horizontal polarization are in the directions determined by the phase-matching conditions ${\mathbf{k}}_{\text{F}}={\mathbf{k}}_{1}+{\mathbf{k}}_{3}-{\mathbf{k}}_{3}^{\prime}$, ${\mathbf{k}}_{\text{S}1}={\mathbf{k}}_{1}+{\mathbf{k}}_{2}-{\mathbf{k}}_{2}+{\mathbf{k}}_{3}-{\mathbf{k}}_{3}^{\prime}$and ${\mathbf{k}}_{\text{S}2}={\mathbf{k}}_{1}+{\mathbf{k}}_{4}-{\mathbf{k}}_{4}+{\mathbf{k}}_{3}-{\mathbf{k}}_{3}^{\prime}$, respectively. These signals are in the same direction as ${E}_{\text{F}}$ (at the lower right corner of Fig. 1(a)), and are detected by an avalanche photodiode detector. The transmitted probe beam is simultaneously detected by a silicon photodiode.

For the five-level atomic system as shown in Fig. 1(b), if two strong coupling laser fields (${E}_{2}$ and ${E}_{4}$) drive two separate upper transitions (|1> to |2> and |1> to |4>), respectively, and a weak laser field (..) probes the lower transition (|0> to |1>), two ladder-type EIT subsystems will form with two-photon Doppler-free configuration [8] and two EIT windows appear. Depending on the frequency detunings of the two coupling laser beams, these two EIT windows can either overlap or be separated in frequency on the probe beam transmission signal. On the other hand, if the probe field ${E}_{1}$ drives the transition (|0> to |1>) and the two pump fields (${E}_{3}$ and ${E}_{3}^{\prime}$) drive another transition (|3> to |1>) in the three-level Λ-type subsystem, as shown in Fig. 1(b), there will be a corresponding FWM signal generated at frequency ${\omega}_{1}$ (satisfying ${\mathbf{k}}_{F}={\mathbf{k}}_{1}+{\mathbf{k}}_{3}-{\mathbf{k}}_{3}^{\prime}$). However, the FWM signal without the EIT window (not satisfying the two-photon Doppler-free configuration [8]) can be neglected. When the two coupling laser fields *E*_{2} (connecting transition |1> to |2>) and ${E}_{4}$ (connecting the transition |1> to |4>) are added, two SWM processes will occur [11]. First, without the strong coupling field ${E}_{4}$, a simple SWM1 process (${E}_{\text{S}1}$) at frequency ${\omega}_{1}$ is generated from the probe beam (*E*_{1}), the coupling field (${E}_{2}$), and two pump fields (${E}_{3}$ and ${E}_{3}^{\prime}$), via the perturbation chain (I): ${\rho}_{00}^{(0)}\stackrel{{G}_{1}}{\to}{\rho}_{10}^{(1)}\stackrel{{G}_{2}}{\to}{\rho}_{20}^{(2)}\stackrel{{G}_{2}^{*}}{\to}{\rho}_{10}^{(3)}\stackrel{{({G}_{3}^{\prime})}^{*}}{\to}{\rho}_{30}^{(4)}\stackrel{{G}_{3}}{\to}{\rho}_{10}^{(5)}$ (satisfying ${\mathbf{k}}_{\text{S}1}={\mathbf{k}}_{1}+{\mathbf{k}}_{2}-{\mathbf{k}}_{2}+{\mathbf{k}}_{3}-{\mathbf{k}}_{3}^{\prime}$) [11]. When the power of ${E}_{2}$ is strong enough, it will start to dress the energy level $|1\u3009$ to create the primarily-dressed states $|+\u3009$ and $|-\u3009$, as shown in Fig. 1(c). This dressed SWM1 process can be described via the perturbation chain (II): ${\rho}_{00}^{(0)}\stackrel{{\omega}_{1}}{\to}{\rho}_{\pm 0}^{(1)}\stackrel{{\omega}_{2}}{\to}{\rho}_{20}^{(2)}\stackrel{-{\omega}_{2}}{\to}{\rho}_{\pm 0}^{(3)}\stackrel{-{\omega}_{3}}{\to}{\rho}_{30}^{(4)}\stackrel{{\omega}_{3}}{\to}{\rho}_{\pm 0}^{(5)}$. Such self-dressing effect, i.e. one of the participating fields for generating SWM dresses the involved energy level $|1\u3009$, which then modifies the SWM process itself, is unique for such multi-wave mixing processes in multi-level systems. Similarly, for another SWM process (with fields ${E}_{1}$, ${E}_{4}$, ${E}_{3}$ and ${E}_{3}^{\prime}$), without the strong coupling field ${E}_{2}$, it will generate a signal field ${E}_{\text{S}2}$ at frequency ${\omega}_{1}$ via the perturbation chain (III): ${\rho}_{00}^{(0)}\stackrel{{G}_{1}}{\to}{\rho}_{10}^{(1)}\stackrel{{G}_{4}}{\to}{\rho}_{40}^{(2)}\stackrel{{G}_{4}^{*}}{\to}{\rho}_{10}^{(3)}\stackrel{{({G}_{3}^{\prime})}^{*}}{\to}{\rho}_{30}^{(4)}\stackrel{{G}_{3}}{\to}{\rho}_{10}^{(5)}$ (satisfying ${\mathbf{k}}_{\text{S}2}={\mathbf{k}}_{1}+{\mathbf{k}}_{4}-{\mathbf{k}}_{4}+{\mathbf{k}}_{3}-{\mathbf{k}}_{3}^{\prime}$). When the power of ${E}_{4}$ is strong enough, it will start to dress the energy level $|1\u3009$ to create the primarily-dressed states $|+\u3009$ and $|-\u3009$, as shown in Fig. 1(d). This dressed SWM2 process can be described via the perturbation chain (IV): ${\rho}_{00}^{(0)}\stackrel{{\omega}_{1}}{\to}{\rho}_{\pm 0}^{(1)}\stackrel{{\omega}_{4}}{\to}{\rho}_{40}^{(2)}\stackrel{-{\omega}_{4}}{\to}{\rho}_{\pm 0}^{(3)}\stackrel{-{\omega}_{3}}{\to}{\rho}_{30}^{(4)}\stackrel{{\omega}_{3}}{\to}{\rho}_{\pm 0}^{(5)}$.

Next, when both coupling fields (${E}_{2}$ and ${E}_{4}$) are on at the same time, they can dress the energy level |1> together. For the SWM1 process (${E}_{\text{S}1}$), ${E}_{2}$ first produces the primarily-dressed states $|\pm \u3009$, then ${E}_{4}$ produces the secondarily-dressed states $|\pm \pm \u3009$ at a proper frequency detuning (i.e. either tuned to the upper or lower dressed state, $|+>$ or $|->$), as shown in Fig. 1(e) via the perturbation chain (V): ${\rho}_{00}^{(0)}\stackrel{{\omega}_{1}}{\to}{\rho}_{\pm \pm 0}^{(1)}\stackrel{{\omega}_{2}}{\to}{\rho}_{20}^{(2)}\stackrel{-{\omega}_{2}}{\to}{\rho}_{\pm \pm 0}^{(3)}\stackrel{-{\omega}_{3}}{\to}{\rho}_{30}^{(4)}\stackrel{{\omega}_{3}}{\to}{\rho}_{\pm \pm 0}^{(5)}$. This generates the secondary AT splitting for the SWM1 signal. The situation for the SWM2 (${E}_{\text{S}2}$) process is similar.

The two primarily-dressed states induced by ${E}_{2}$ can be written as $|\pm >=\mathrm{sin}{\theta}_{1\pm}|1>+\mathrm{cos}{\theta}_{1\pm}|2>$ (Fig. 1(c)). When ${E}_{4}$ only couples to the dressed state $|+>$, the secondarily-dressed states are then given by $|+\pm >=\mathrm{sin}{\theta}_{2\pm}|+>+\mathrm{cos}{\theta}_{2\pm}|4>$ (Fig. 1(e)), where $\mathrm{tan}{\theta}_{1\pm}=-{a}_{1\pm}/{G}_{2}$, $\mathrm{tan}{\theta}_{2\pm}=-{a}_{2\pm}/{G}_{4}$, ${a}_{1\pm}={\Delta}_{2}-{\lambda}_{\pm}^{(1)}$ and ${a}_{2\pm}={\Delta}_{4}-{\lambda}_{+}^{(1)}-{\lambda}_{+\pm}^{(1)}$. One can obtain the eigenvalues ${\lambda}_{\pm}^{(1)}=({\Delta}_{2}\pm \sqrt{{\Delta}_{2}^{2}+4{\left|{G}_{2}\right|}^{2}})/2$ (measured from level $|1>$) of $|\pm >$, and ${\lambda}_{+\pm}^{(1)}=({\Delta}_{4}^{\prime}\pm \sqrt{{{\Delta}^{\prime}}_{4}^{2}+4{\left|{G}_{4}\right|}^{2}})/2$ (measured from level $|+>$) of $|+\pm >$, where ${\Delta}_{4}^{\prime}={\Delta}_{4}-{\lambda}_{+}^{(1)}$. When ${E}_{4}$ only couples to the dressed state $|->$, the secondarily-dressed states are given by $|-\pm >=\mathrm{sin}{\theta}_{2\pm}|->+\mathrm{cos}{\theta}_{2\pm}|4>$, where ${a}_{2\pm}={\Delta}_{4}-{\lambda}_{-}^{(1)}-{\lambda}_{-\pm}^{(1)}$, and other parameters are the same as before. Then, we obtain the same eigenvalues ${\lambda}_{\pm}^{(1)}$, and ${\lambda}_{-\pm}^{(1)}=({\Delta}_{4}^{\prime}\pm \sqrt{{{\Delta}^{\prime}}_{4}^{2}+4{\left|{G}_{4}\right|}^{2}})/2$ (measured from level $|->$) of $|-\pm >$, where ${\Delta}_{4}^{\prime}={\Delta}_{4}-{\lambda}_{-}^{(1)}$. Similarly, when ${E}_{4}$ induces the two primarily-dressed states, and ${E}_{2}$ acts as the external-dressing field, one can get the following corresponding eigenvalues: , ${\lambda}_{+\pm}^{(2)}=({\Delta}_{2}^{\prime}\pm \sqrt{{{\Delta}^{\prime}}_{2}^{2}+4{\left|{G}_{2}\right|}^{2}})/2$ (${\Delta}_{2}^{\prime}={\Delta}_{2}-{\lambda}_{+}^{(2)}$), and ${\lambda}_{-\pm}^{(2)}=({\Delta}_{2}^{\prime}\pm \sqrt{{{\Delta}^{\prime}}_{2}^{2}+4{\left|{G}_{2}\right|}^{2}})/2$ (${\Delta}_{2}^{\prime}={\Delta}_{2}-{\lambda}_{-}^{(2)}$).

In general for arbitrary strengths of the fields ${E}_{1}$, ${E}_{3}$, ${E}_{3}^{\prime}$, ${E}_{2}$ and ${E}_{4}$, one needs to solve the coupled density-matrix equations to obtain ${\rho}_{10}^{(5)}$ for the SWM processes, which we have done in simulating the experimental results later on. For simplicity, we have solved the coupled equations with perturbation chain (I) to obtain the nonlinear density-matrix element for the multi-dressed SWM processes (including self-dressing and external-dressing) as: ${\rho}_{10}^{(5)}=i{G}_{\text{S}1}/(ABCDE)$, where, $A={d}_{1}+{G}_{1}^{2}/{\Gamma}_{1}+{G}_{1}^{2}/{\Gamma}_{0}+{G}_{2}^{2}/{d}_{2}+{G}_{4}^{2}/{d}_{4}$, $B={d}_{2}+{G}_{1}{}^{2}/{d}_{2}^{\prime}$, $C={d}_{1}+{G}_{1}^{2}/{\Gamma}_{1}+{G}_{2}^{2}/{d}_{2}+{G}_{3}^{2}/{d}_{3}+{G}_{4}^{2}/{d}_{4}$, $D={d}_{3}+{G}_{1}{}^{2}/{d}_{\prime}^{3}$, $E={d}_{1}+{G}_{1}^{2}/{\Gamma}_{1}+{G}_{1}^{2}/{\Gamma}_{0}+{G}_{3}^{2}/{d}_{3}+{G}_{4}^{2}/{d}_{4}$,${d}_{1}={\Gamma}_{10}+i{\Delta}_{1}$, ${d}_{2}={\Gamma}_{20}+i({\Delta}_{1}+{\Delta}_{2})$, ${d}_{2}^{\prime}=i{\Delta}_{2}+{\Gamma}_{21}$, ${d}_{3}={\Gamma}_{30}+i({\Delta}_{1}-{\Delta}_{3})$, ${d}_{\prime}^{3}={\Gamma}_{31}-i{\Delta}_{3}$, ${d}_{4}={\Gamma}_{40}+i({\Delta}_{1}+{\Delta}_{4})$ with ${\Delta}_{i}={\Omega}_{i}-{\omega}_{i}$ and ${\Gamma}_{ij}$ being the transverse relaxation rate between states $|i\u3009$ and $|j\u3009$. For the SWM1 signal (due to the weak probe field), the expression can be simplified as: ${\rho}_{\text{S}1}^{(5)}=i{G}_{\text{S}1}/[({d}_{1}+{G}_{2}^{2}/{d}_{2}){d}_{2}({d}_{1}+{G}_{2}^{2}/{d}_{2}+{G}_{3}^{2}/{d}_{3}){d}_{3}({d}_{1}+{G}_{3}^{2}/{d}_{3})]$. Similarly, for the SWM2 signal, the expression is simplified as: ${\rho}_{\text{S}2}^{(5)}=i{G}_{\text{S}2}/[({d}_{1}+{G}_{4}^{2}/{d}_{4}){d}_{4}({d}_{1}+{G}_{3}^{2}/{d}_{3}+{G}_{4}^{2}/{d}_{4}){d}_{3}({d}_{1}+{G}_{3}^{2}/{d}_{3})]$, where ${G}_{\text{S}2}={G}_{1}{G}_{3}^{\prime}{}^{*}{G}_{3}{G}_{4}{G}_{4}^{*}$.

There exist two ladder-type EIT windows in Fig. 1(b), i.e., the $|0>-|1>-|2>$ EIT1 window satisfying ${\Delta}_{1}+{\Delta}_{2}=0$ (induced by the coupling field ${E}_{2}$) and the $|0>-|1>-|4>$ EIT2 window satisfying ${\Delta}_{1}+{\Delta}_{4}=0$ (induced by the coupling field ${E}_{4}$). The EIT1 and EIT2 windows contain the SWM1 signal (${E}_{\text{S}1}$) and the SWM2 signal (${E}_{\text{S}2}$), respectively. Next, we will consider the AT splitting of the SWM signals within the EIT windows (Figs. 2 -4 ).

## 3. Autler-Townes Splitting

When the external-dressing field ${E}_{4}$ is blocked, we get the SWM1 signal within the EIT1 window (which is an inverted-Y system) [15]. Figure 2 (a1, b1, c1) presents the SWM1 signal intensity versus the probe field detuning (${\Delta}_{1}$) for different field powers of ${P}_{1}$, and ${P}_{3}$ with the same frequency detuning of ${\Delta}_{2}=-50\text{MHz}$. Obviously, the SWM1 signal shows two peaks due to multi-dressing effects (Fig. 1(c)). With the power increases, the intensity of the SWM1 signal increases accordingly, while the left peak height is always greater than the height of the right peak. Meanwhile, the increments of the AT splitting separations ${\Delta}_{i}={\lambda}_{+}^{(1)}-{\lambda}_{-}^{(1)}\approx 2\sqrt{{\left|{G}_{j}\right|}^{2}+{\left|{G}_{i0}\right|}^{2}}$ ($i=$a, b, c corresponds to $j=$1, 2, 3, respectively, and ${\left|{G}_{\text{a0}}\right|}^{2}={\left|{G}_{20}\right|}^{2}+{\left|{G}_{30}\right|}^{2}$, ${\left|{G}_{\text{b0}}\right|}^{2}={\left|{G}_{10}\right|}^{2}+{\left|{G}_{30}\right|}^{2}$, ${\left|{G}_{\text{c0}}\right|}^{2}={\left|{G}_{10}\right|}^{2}+{\left|{G}_{20}\right|}^{2}$), increase obviously with increased powers (Rabi frequencies) ${P}_{1}$ (${G}_{1}$), ${P}_{2}$ (${G}_{2}$) and ${P}_{3}$ (${G}_{3}$), respectively and fixed ${P}_{2}$&${P}_{3}$ (${G}_{20}\&{G}_{30}$), ${P}_{1}$&${P}_{3}$ (${G}_{10}\&{G}_{30}$) and ${P}_{1}$&${P}_{2}$ (${G}_{10}\&{G}_{20}$), respectively. The two peaks of the double-peak SWM1 signal (Figs. 2(a1), (b1) and (c1)) correspond, from left to right, to the primarily-dressed states $|+\u3009$ and $|-\u3009$, respectively (Fig. 1(c)). Moreover, the experimentally measured (peak separation) results in Figs. 2(a1), (b1) and (c1) are in good agreement with our theoretical calculations (solid curves), as shown in Figs. 2(a2), (b2) and (c2), respectively.

When the external-dressing field ${E}_{2}$ is blocked, we get the SWM2 signal in the EIT2 window. Figure 3 presents the SWM2 signal intensity versus the probe field detuning (${\Delta}_{1}$) for different powers of ${P}_{1}$, ${P}_{3}$ and ${P}_{4}$ with the same frequency detuning of ${\Delta}_{4}=0\text{MHz}$. Figure 3(a) depicts the measured EIT windows induced by the self-dressing field ${E}_{4}$ versus ${\Delta}_{1}$ (satisfying ${\Delta}_{1}+{\Delta}_{4}=0$). Such EIT window increases with ${P}_{4}$ power increasing. The SWM2 signal has three peaks. In general, when the power increases, the intensity of the SWM2 signal also increases accordingly. However, the states of AT splittings change differently for different power changes. If only ${P}_{1}$ power increases, the right peak first increases and then decreases, while the height of the middle peak is always larger than the height of either the left or right peak. If only ${P}_{3}$ power increases, the right peak always increases, while the height of the middle peak is always larger than the height of either the left or right peak. If only ${P}_{4}$ power increases, the height of the right peak increases monotonously to approach the height of the middle peak. The two primarily-dressed states $|+>$and $|->$ are induced by ${E}_{4}$. When ${E}_{1}$ and ${E}_{3}$ couples to the dressed state $|->$, the secondarily-dressed states $|-+\u3009$ and $|--\u3009$ appear. The three peaks in the SWM2 signal (Figs. 3(b1), (c1) and (d1)) correspond, from left to right, to the primarily-dressed state $|+>$, the secondarily-dressed states $|-+\u3009$ and $|--\u3009$, respectively (Fig. 1(d)). Based on our theoretical analysis, the two primarily-dressed states dressed by ${E}_{4}$ can be written as $|\pm >=\mathrm{sin}{\alpha}_{1\pm}|1>+\mathrm{cos}{\alpha}_{1\pm}|4>$, and the secondarily-dressed states dressed by ${E}_{3}$are given by $|-\pm >=\mathrm{sin}{\alpha}_{2\pm}|->+\mathrm{cos}{\alpha}_{2\pm}|3>$, where $\mathrm{tan}{\alpha}_{1\pm}=-{b}_{1\pm}/{G}_{4}$, $\mathrm{tan}{\alpha}_{2\pm}=-{b}_{2\pm}/\sqrt{{\left|{G}_{1}\right|}^{2}+{\left|{G}_{3}\right|}^{2}}$, ${b}_{1\pm}={\Delta}_{4}-{\lambda}_{\pm}^{(3)}$, and ${b}_{2\pm}={\Delta}_{3}-{\lambda}_{-}^{(3)}-{\lambda}_{-\pm}^{(3)}$. One can obtain the eigenvalues ${\lambda}_{\pm}^{(3)}=({\Delta}_{4}\pm \sqrt{{\Delta}_{4}^{2}+4{\left|{G}_{4}\right|}^{2}})/2$ (measured from level $|1>$) of $|\pm >$, and ${\lambda}_{-\pm}^{(3)}=({\Delta}_{3}^{\prime}\pm \sqrt{{{\Delta}^{\prime}}_{3}^{2}+4({\left|{G}_{1}\right|}^{2}+{\left|{G}_{3}\right|}^{2})})/2$ (measured from level $|->$) of $|-\pm >$, where ${\Delta}_{3}^{\prime}=-{\Delta}_{3}-{\lambda}_{-}^{(3)}$.

Meanwhile, the increments of the AT splitting separations ${\text{\Delta}}_{\text{b}}\text{=}{\lambda}_{\text{-+}}^{\text{(3)}}\text{-}{\lambda}_{\text{--}}^{\text{(3)}}\text{=2}\sqrt{{\left|{G}_{\text{1}}\right|}^{\text{2}}\text{+}{\left|{G}_{\text{30}}\right|}^{\text{2}}}$, ${\Delta}_{\text{c}}={\lambda}_{-+}^{(3)}-{\lambda}_{--}^{(3)}=2\sqrt{{\left|{G}_{3}\right|}^{2}+{\left|{G}_{10}\right|}^{2}}$, and ${\Delta}_{\text{d}}={\lambda}_{+}^{(3)}-{\lambda}_{-+}^{(3)}=2{G}_{\text{4}}-\sqrt{{\left|{G}_{\text{10}}\right|}^{\text{2}}\text{+}{\left|{G}_{\text{30}}\right|}^{\text{2}}}\approx 2{G}_{\text{4}}$ (${G}_{\text{4}}>>{G}_{\text{10,30}}$) increase obviously with increased powers (Rabi frequencies) of ${P}_{1}$ (${G}_{1}$), ${P}_{3}$ (${G}_{3}$) and ${P}_{4}$ (${G}_{4}$), respectively and fixed ${P}_{3}$&${P}_{4}$ (${G}_{30}\&{G}_{40}$), ${P}_{1}$&${P}_{3}$ (${G}_{10}\&{G}_{30}$) and ${P}_{1}$&${P}_{4}$ (${G}_{10}\&{G}_{40}$), respectively. When the self-dressing ${P}_{4}$ power changes, the state of the AT splitting obviously has an essential distinction from the former two states as the powers of ${P}_{1}$ and ${P}_{3}$ change. The experimentally measured results in Figs. 3(b1), (c1) and (d1) are in good agreement with our theoretical calculations shown in Figs. 3(b2), (c2) and (d2), respectively.

After studying the self-dressing AT splitting of the individual SWM1 or SWM2 signal, we will now consider the cross-dressing AT splitting between the two SWM signals. Figures 4(a1) - 4(a3) present the interplay between the two SWM signals versus the probe field detuning (${\Delta}_{1}$) for different external-dressing field detunings (${\Delta}_{2}$) with ${\Delta}_{4}=0\text{MHz}$. Here, we consider the case with ${G}_{2}<{G}_{4}$. The upper-curve in each figure is the probe transmission with two ladder-type EIT windows and the lower-curve gives the measured SWM signals. In Fig. 4(a1), the left EIT window ($|0>-|1>-|4>$ satisfying ${\Delta}_{1}+{\Delta}_{4}=0$) is induced by the coupling field ${E}_{4}$,which contains the SWM2 signal (${E}_{\text{S}2}$), and the right one ($|0>-|1>-|2>$ satisfying ${\Delta}_{1}+{\Delta}_{2}=0$, Δ_{2} = -15MHZ) is induced by the coupling field ${E}_{2}$, which contains the SWM1 signal (${E}_{\text{S}1}$). Since the right EIT window (${\Delta}_{2}=-150\text{MHz}$) is quite far from the left EIT window, these two SWM signals have little effect on each other (Fig. 4(a1)). When the frequency of ${E}_{2}$ is tuned to move the right EIT window ($|0>-|1>-|2>$) towards the left one, the two EIT windows overlap at ${\Delta}_{2}=-15\text{MHz}$, which leads to the triple AT splitting for SWM2, i.e., the right peak of SWM2 signal is further split into two peaks (Fig. 4(a2), satisfying ${\Delta}_{2}={\lambda}_{--}^{(3)}$). It is the coupling field ${E}_{2}$ that couples the secondarily-dressed state $|--\u3009$ dressed by ${E}_{3}$ and splits it into two triply-dressed states $|--+>$ and $|--->$. The four peaks of the SWM2 signal in Fig. 4(a2) correspond, from left to right, to the primarily-dressed state $|+>$, the secondarily-dressed states $|-+\u3009$, the triply-dressed states $|--+>$ and $|--->$, respectively (Fig. 1(e)). We can write the triply-dressed states as $|--\pm >=\mathrm{sin}{\alpha}_{3\pm}|--\u3009+\mathrm{cos}{\alpha}_{3\pm}|2>$, where $\mathrm{tan}{\alpha}_{3\pm}=-{b}_{3\pm}/{G}_{2}$, ${b}_{3\pm}={\Delta}_{2}-{\lambda}_{-}^{(3)}-{\lambda}_{--}^{(3)}-{\lambda}_{--\pm}^{(3)}$, and the other parameters are the same as before. One can obtain the eigenvalues ${\lambda}_{--\pm}^{(3)}=({\Delta}_{2}^{\prime}\pm \sqrt{{{\Delta}^{\prime}}_{2}^{2}+4{\left|{G}_{2}\right|}^{2}})/2$ (measured from level $|--\u3009$) for the dressed states $|--\pm >$, where ${\Delta}_{2}^{\prime}={\Delta}_{2}-{\lambda}_{-}^{(3)}-{\lambda}_{--}^{(3)}$. With the right EIT window ($|0>-|1>-|2>$) continuously moving to the left, the SWM1 signal splits the left peak of the SWM2 signal when ${\Delta}_{2}=15\text{MHz}$, as shown in Fig. 4(a3). Corresponding to the moving SWM1 signal in Fig. 4(a1), Fig. 4(b1) presents the measured self-dressing double-peak SWM1 signal versus ${\Delta}_{1}$ with ${\Delta}_{2}=0\text{MHz}$. Corresponding to the fixed SWM2 signal in Figs. 4(a2) and 4(a3), Figs. 4(b3) and 4(b5) show the measured SWM2 signals, whose right and left peaks are split by the SWM1 signal, versus ${\Delta}_{1}$ with an increasing external-dressing ${P}_{2}$ power, respectively. Figures 4(b2), 4(b4) and 4(b6) give the corresponding power dependences of the AT-splitting separations. The corresponding separations are determined by ${\Delta}_{b1}={\lambda}_{+}^{(1)}-{\lambda}_{-}^{(1)}\approx 2{G}_{2}$, ${\Delta}_{b3}={\lambda}_{--+}^{(3)}-{\lambda}_{---}^{(3)}\approx 2{G}_{2}$, and ${\Delta}_{b5}\approx 2{G}_{2}$, respectively.

## 4. Conclusion

In summary, we have observed the self- and externally-dressed AT splittings of the SWM processes within EIT windows in a five-level atomic system. Such AT splitting demonstrates the interactions between two coexisting SWM processes. The controlled multi-channel splitting signals in nonlinear optical processes can find potential applications in optical communication and quantum information processing, such as wavelength-demultiplexer [18–20].

## Acknowledgments

This work was supported by NSFC (10974151, 61078002, 61078020), NCET (08-0431), RFDP (20100201120031), 2009xjtujc08, XJJ20100100, XJJ20100151.

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