## Abstract

By selectively blocking specific laser beams, we investigate coexisting seven distinguishable dressed odd-order multi-wave mixing (MWM) signals in a K-type five-level atomic system. We demonstrate that the enhancement and suppression of dressed four-wave mixing (FWM) signal can be directly detected by scanning the dressing field instead of the probe field. We also study the temporal and spatial interference between two FWM signals. Surprisingly, the pure-suppression of six-wave mixing signal has been shifted far away from resonance by atomic velocity component. Moreover, the interactions among six MWM signals have been studied.

©2012 Optical Society of America

## 1. Introduction

Recently a lot of attention has been concentrated on the four-wave mixing (FWM) [1–5] and six-wave-mixing (SWM) [6-8] under atomic coherence. And electromagnetically induced transparency (EIT) [9, 10] is an beneficial tool to investigate these multi-wave mixing (MWM) processes since the weak generated signals can be allowed to transmit through the resonant atomic medium with little absorption. It also plays an important role in lasing without inversion [11], quantum communications [12], slow light [13], photon controlling and information storage [14–16]. Furthermore, the enhancement and suppression of FWM also attracted the attention of many researchers, which has been experimentally studied and the generated FWM signals can be selectively enhanced and suppressed [17]. Besides, the doubly-dressed states in cold atoms were observed, in which triple-photon absorption spectrum exhibits a constructive interference between transition paths of two closely spaced, doubly-dressed-states [18, 19]. In addition, the generated FWM and SWM signals can be made to coexist and interfere with each other not only in the frequency domain but also spatially, using phase control [20].

In this paper, we show seven coexisting distinguishable multi-wave mixing signals (including three FWM and four SWM signals) by selectively blocking different laser beams in a K-type five-level atomic system. Also by blocking several certain laser beams respectively, the interactions among six MWM signals have been studied. In addition, when scanning the frequency detuning of external-dressing, self-dressing and probe fields respectively in the dressed FWM process, we first analyze the corresponding relationship and differentia between the experimental results of different scanning methods, and demonstrate that scanning the dressing field can be used as a technique to directly observe the dressing effects of FWM process. Also, we first observe the enhancement and suppression of SWM signal at large detuning, due to the atomic velocity component and optical pumping effect. Moreover, we demonstrate the temporal and spatial interferences between two FWM signals.

## 2. Basic theory and experimental scheme

The experiments are performed in a five-level atomic system as shown in Fig. 1(a)
where the five energy levels are $5{S}_{1/2}(F=3)(|0\u3009)$, $5{P}_{3/2}(|1\u3009)$, $5{D}_{5/2}(|2\u3009)$, $5{S}_{1/2}(F=2)(|3\u3009)$ and $5{D}_{3/2}(|4\u3009)$ in ^{85}Rb. The resonant frequencies are ${\Omega}_{1}$, ${\Omega}_{2}$, ${\Omega}_{3}$ and ${\Omega}_{4}$ for transitions$|0\u3009$ to $|1\u3009$, $|1\u3009$ to $|2\u3009$,$|1\u3009$ to $|3\u3009$ and $|1\u3009$ to $|4\u3009$ respectively. The lower transition $|0\u3009$ to $|1\u3009$ is driven and probed by a weak laser beam ${E}_{1}$, and the two strong coupling beams ${E}_{2}$ and ${{E}^{\prime}}_{2}$ connect with the transition $|1\u3009$ to $|2\u3009$. Two pumping beams ${E}_{3}$ and ${{E}^{\prime}}_{3}$ are applied to drive the transition $|1\u3009$ to $|3\u3009$ and two additional strong coupling beams ${E}_{4}$ and ${{E}^{\prime}}_{4}$ drive the transition $|1\u3009$ to $|4\u3009$. In the experimental setup, the two coupling beams ${E}_{2}$ (frequency ${\omega}_{2}$, wave vector ${k}_{\text{2}}$, Rabi frequency ${G}_{2}$ and frequency detuning ${\Delta}_{2}$, where ${\Delta}_{i}={\Omega}_{i}-{\omega}_{i}$) and ${{E}^{\prime}}_{2}$ (${\omega}_{2}$, ${{k}^{\prime}}_{2}$, ${{G}^{\prime}}_{2}$ and ${\Delta}_{2}$) with vertical polarization (wavelength 775.978nm) are from the same external cavity diode laser (ECDL) . The other two vertically polarized coupling beams ${E}_{4}$ (${\omega}_{4}$, ${k}_{\text{4}}$, ${G}_{4}$ and ${\Delta}_{4}$) and ${{E}^{\prime}}_{4}$ (${\omega}_{4}$, ${{k}^{\prime}}_{4}$, ${{G}^{\prime}}_{4}$ and ${\Delta}_{4}$) are from a tapered-amplifier diode laser with the same wavelength 776.157nm. Two pumping beams ${E}_{3}$(${\omega}_{3}$, ${k}_{\text{3}}$, ${G}_{3}$ and ${\Delta}_{3}$) and ${{E}^{\prime}}_{3}$ (${\omega}_{3}$, ${{k}^{\prime}}_{3}$, ${{G}^{\prime}}_{3}$ and ${\Delta}_{3}$) with equal power are split from a LD beam and have polarizations vertical with each other via a polarization beam splitter. The probe beam ${E}_{1}$ (${\omega}_{1}$, ${k}_{\text{1}}$, ${G}_{1}$ and ${\Delta}_{1}$) is generated by a ECDL (Toptica DL100L) with horizontally polarization. These laser beams are spatially designed in a square-box pattern as shown in Fig. 1(b), in which the laser beams ${E}_{2}$, ${{E}^{\prime}}_{2}$, ${E}_{3}$, ${{E}^{\prime}}_{3}$, ${E}_{4}$ and ${{E}^{\prime}}_{4}$ propagate through the Rubidium vapor cell (50mm long) with a temperature of 60 °C in the same direction with small angles (about 0.3°) between one another, and the probe beam ${E}_{1}$ propagates in the opposite direction with a small angle from the other beams.

In such beam geometric configuration, the two-photon Doppler-free conditions will be satisfied for the two ladder-type subsystems both $|0\u3009\to |1\u3009\to |2\u3009$ and $|0\u3009\to |1\u3009\to |4\u3009$, thus two EIT windows appear. When all the seven laser beams (${E}_{1}$, ${E}_{2}$, ${{E}^{\prime}}_{2}$, ${E}_{3}$, ${{E}^{\prime}}_{3}$, ${E}_{4}$, ${{E}^{\prime}}_{4}$) are turned on, three FWM processes ${E}_{F2}^{}$ (satisfying the phase-matching condition ${k}_{F2}={k}_{\text{1}}+{k}_{2}-{{k}^{\prime}}_{2}$), ${E}_{F4}^{}$ (satisfying ${k}_{F4}={k}_{\text{1}}+{k}_{4}-{{k}^{\prime}}_{4}$) and ${E}_{F3}^{}$ (satisfying ${k}_{F3}={k}_{\text{1}}-{{k}^{\prime}}_{3}+{k}_{3}$), and four SWM processes ${E}_{S2}^{}$ (satisfying ${k}_{S2}={k}_{\text{1}}-{{k}^{\prime}}_{3}+{k}_{3}+{k}_{2}-{k}_{2}$), ${{E}^{\prime}}_{S2}$ (satisfying ${{k}^{\prime}}_{S2}={k}_{\text{1}}-{{k}^{\prime}}_{3}+{k}_{3}+{{k}^{\prime}}_{2}-{{k}^{\prime}}_{2}$), ${E}_{S4}^{}$ (satisfying ${k}_{S4}={k}_{\text{1}}-{{k}^{\prime}}_{3}+{k}_{3}+{k}_{4}-{k}_{4}$) and ${{E}^{\prime}}_{S4}$ (satisfying ${{k}^{\prime}}_{S4}={k}_{\text{1}}-{{k}^{\prime}}_{3}+{k}_{3}+{{k}^{\prime}}_{4}-{{k}^{\prime}}_{4}$) can occur simultaneously. The propagation direction of all the generated signals with horizontal polarization is determined by the phase-matching conditions, so all the signals propagate along the same direction deviated from probe beam at an angle θ, as shown in Fig. 1(a). The wave-mixing signals are detected by an avalanche photodiode detector, and the probe beam transmission is simultaneously detected by a silicon photodiode.

Generally, the expression of the density-matrix element related to the MWM signals can be obtained by solving the density-matrix equations. For the simple FWM process of ${E}_{F2}^{}$, via the perturbation chain ${\rho}_{00}^{(0)}\stackrel{{\omega}_{1}}{\to}{\rho}_{10}^{(1)}\stackrel{{\omega}_{2}}{\to}{\rho}_{20}^{(2)}\stackrel{-{\omega}_{2}}{\to}{\rho}_{10}^{(3)}$ we can obtain the third-order density element ${\rho}_{F2}^{(3)}={G}_{F2}/({d}_{1}^{2}{d}_{2})$, the amplitude of which determines the intensity of the simple FWM process, where ${G}_{F2}=-i{G}_{1}{G}_{2}{({{G}^{\prime}}_{2})}^{*}\mathrm{exp}(i{k}_{F2}\cdot r)$, ${d}_{1}={\Gamma}_{10}+i{\Delta}_{1}$, ${d}_{2}={\Gamma}_{20}+i({\Delta}_{1}+{\Delta}_{2})$, and ${\Gamma}_{ij}$ is the transverse relaxation rate between states $|i\u3009$ and $|j\u3009$. Similarly, for the simple FWM process of ${E}_{F4}^{}$, we can obtain ${\rho}_{F4}^{(3)}={G}_{F4}/({d}_{1}^{2}{d}_{4})$ via ${\rho}_{00}^{(0)}\stackrel{{\omega}_{1}}{\to}{\rho}_{10}^{(1)}\stackrel{{\omega}_{4}}{\to}{\rho}_{40}^{(2)}\stackrel{-{\omega}_{4}}{\to}{\rho}_{10}^{(3)}$, where ${G}_{F4}=-i{G}_{1}{G}_{4}{({{G}^{\prime}}_{4})}^{*}\mathrm{exp}(i{k}_{F4}\cdot r)$, ${d}_{4}={\Gamma}_{40}+i({\Delta}_{1}+{\Delta}_{4})$. And for the simple FWM process of ${E}_{F3}^{}$, we can obtain ${\rho}_{F3}^{(3)}={G}_{F3}/({d}_{1}^{2}{d}_{4})$ via ${\rho}_{00}^{(0)}\stackrel{{\omega}_{1}}{\to}{\rho}_{10}^{(1)}\stackrel{-{\omega}_{3}}{\to}{\rho}_{30}^{(2)}\stackrel{{\omega}_{3}}{\to}{\rho}_{10}^{(3)}$, where ${G}_{F3}=-i{G}_{1}{G}_{3}{({{G}^{\prime}}_{3})}^{*}\mathrm{exp}(i{k}_{F3}\cdot r)$, ${d}_{3}={\Gamma}_{30}+i({\Delta}_{1}-{\Delta}_{3})$. Additionally, via perturbation chain ${\rho}_{00}^{(0)}\stackrel{{\omega}_{1}}{\to}{\rho}_{10}^{(1)}\stackrel{-{\omega}_{3}}{\to}{\rho}_{30}^{(2)}\stackrel{{\omega}_{3}}{\to}{\rho}_{10}^{(3)}\stackrel{{\omega}_{2}}{\to}{\rho}_{20}^{(4)}\stackrel{-{\omega}_{2}}{\to}{\rho}_{10}^{(5)}$ we can obtain ${\rho}_{S2}^{(5)}={G}_{S2}/({d}_{1}{}^{3}{d}_{2}{d}_{3})$ for the simple SWM process of${E}_{S2}$, where ${G}_{S2}=i{G}_{1}{G}_{2}{G}_{2}^{*}{{G}^{\prime}}_{3}{}^{*}{G}_{3}\mathrm{exp}(i{k}_{S2}\cdot r)$ (or ${{\rho}^{\prime}}_{S2}^{(5)}={{G}^{\prime}}_{S2}/({d}_{1}{}^{3}{d}_{2}{d}_{3})$ for ${{E}^{\prime}}_{S2}$, where ${{G}^{\prime}}_{S2}=i{G}_{1}{{G}^{\prime}}_{2}{{G}^{\prime}}_{2}^{*}{{G}^{\prime}}_{3}{}^{*}{G}_{3}\mathrm{exp}(i{{k}^{\prime}}_{S2}\cdot r)$). And via ${\rho}_{00}^{(0)}\stackrel{{\omega}_{1}}{\to}{\rho}_{10}^{(1)}\stackrel{-{\omega}_{3}}{\to}{\rho}_{30}^{(2)}\stackrel{{\omega}_{3}}{\to}{\rho}_{10}^{(3)}\stackrel{{\omega}_{4}}{\to}{\rho}_{40}^{(4)}\stackrel{-{\omega}_{4}}{\to}{\rho}_{10}^{(5)}$ we obtain ${\rho}_{S4}^{(5)}={G}_{S4}/({d}_{1}{}^{3}{d}_{4}{d}_{3})$ for the simple SWM process of ${E}_{S4}$, where ${G}_{S4}=i{G}_{1}{G}_{4}{G}_{4}^{*}{{G}^{\prime}}_{3}{}^{*}{G}_{3}\mathrm{exp}(i{k}_{S4}\cdot r)$ (or ${{\rho}^{\prime}}_{S4}^{(5)}={{G}^{\prime}}_{S4}/({d}_{1}{}^{3}{d}_{4}{d}_{3})$ for ${{E}^{\prime}}_{S4}$, where ${{G}^{\prime}}_{S4}=i{G}_{1}{{G}^{\prime}}_{4}{{G}^{\prime}}_{4}^{*}{{G}^{\prime}}_{3}{}^{*}{G}_{3}\mathrm{exp}(i{{k}^{\prime}}_{S4}\cdot r)$). Further, the dressing effect on these MWM signals and the interaction among them will be researched in the following section.

In the experiments, these multi-wave mixing signals are researched by selectively blocking different laser beams. When the beams ${E}_{3}$, ${{E}^{\prime}}_{3}$, ${E}_{4}$ and ${{E}^{\prime}}_{4}$ are blocked and ${E}_{1}$,${E}_{2}$ and ${{E}^{\prime}}_{2}$ are turned on (as shown in Fig. 1(c)), only the ${E}_{F2}^{}$ signal with self-dressing effect could be generated in the ladder-type three-level subsystem $|0\u3009\to |1\u3009\to |2\u3009$. When the beams ${E}_{3}$ and ${{E}^{\prime}}_{3}$ are blocked and other beams on (Fig. 1(d)), two FWM signals ${E}_{F2}^{}$ and ${E}_{F4}^{}$ will coexist in the Y-type four-level subsystem, both of them would be perturbed by self-dressing effect and external-dressing effect, and spatiotemporal coherent interference and interaction between them could be observed. When the beams ${{E}^{\prime}}_{2}$ and ${{E}^{\prime}}_{4}^{}$ are blocked only (Fig. 1(e)), two SWM signals ${E}_{S2}^{}$ and ${E}_{S4}$ will coexist and interact with each other.

## 3. Dressed odd-order multi-wave mixing

By individually adjusting the frequency detunings ${\Delta}_{2}$ and ${\Delta}_{4}$, these generated wave-mixing signals can be separated in spectra for the identification, or be overlapped for investigating the interplay among them. Firstly, by detuning the frequency of the participating laser beams and blocking one or two participating laser beams, we can successfully separate two EIT windows and these MWM signals can be identified. Figures 2(a) -2(e) present the measured signals versus the probe detuning ${\Delta}_{1}$ with different laser beams blocked, in which the lower curves are the measured MWM signals while the corresponding probe transmission signals versus ${\Delta}_{1}$ are shown in the upper curves. Figure 2(a) depicts the case when all the participating laser beams are turned on, and Figs. 2(b), 2(c), 2(d) and 2(e) show the measured signals when the laser beams or beam combinations ${{E}^{\prime}}_{2}$, ${E}_{2}$, ${E}_{4}\&{{E}^{\prime}}_{4}$ and ${E}_{3}\&{{E}^{\prime}}_{3}$ are blocked, respectively. In the upper curves, the left EIT window is created by ${E}_{4}$(${{E}^{\prime}}_{4}$) at ${\Delta}_{1}\text{=}-\text{140MHz}$ satisfying ${\Delta}_{1}+{\Delta}_{4}\text{=}0$ and the right one is created by ${E}_{2}$(${{E}^{\prime}}_{2}$) at ${\Delta}_{1}\text{=150MHz}$ satisfying ${\Delta}_{1}+{\Delta}_{2}\text{=}0$. The generated FWM and SWM signals except for ${E}_{F3}^{}$ all fall into these two separate EIT windows, so the linear absorptions of the generated signals are greatly suppressed. Specifically, the MWM signals related to ${E}_{4}$(${{E}^{\prime}}_{4}$) (i.e. ${E}_{F4}^{}$, ${E}_{S4}^{}$ and ${{E}^{\prime}}_{S4}$) fall into the $|0\u3009\to |1\u3009\to |4\u3009$ EIT window while the MWM signals related to ${E}_{2}$(${{E}^{\prime}}_{2}$) (i.e. ${E}_{F2}^{}$, ${E}_{S2}^{}$ and ${{E}^{\prime}}_{S2}$) fall into the $|0\u3009\to |1\u3009\to |2\u3009$ EIT window. The FWM signal ${E}_{F3}^{}$ can be obtained in Figs. 2(a)-(d), and it appears as a Doppler broadened background signal because the Doppler-free condition cannot be satisfied in the FWM process generating ${E}_{F3}^{}$ with opposite propagation direction between ${E}_{1}^{}$ and ${E}_{3}^{}$(${{E}^{\prime}}_{3}^{}$). By comparing Fig. 2(a) with Fig. 2(d), we find the interaction between MWM signals related to ${E}_{4}$(${{E}^{\prime}}_{4}$) and those related to ${E}_{2}$(${{E}^{\prime}}_{2}$) does not exist when the two EIT widows are separated from each other, since the intensity of the MWM signals related to ${E}_{2}$(${{E}^{\prime}}_{2}$) behaves identical in both curves.

Moreover, the interaction between FWM and SWM processes in the same EIT window of ${E}_{2}$(${{E}^{\prime}}_{2}$) can be observed, by comparing the total MWM signal related to ${E}_{2}$(${{E}^{\prime}}_{2}$) (Fig. 2(a)) with the sum of FWM signal ${E}_{F2}^{}$ (Fig. 2(e)), SWM signals ${E}_{S2}$ (Fig. 2(b)) and ${{E}^{\prime}}_{S2}$ (Fig. 2(c)) in amplitude. We can find that the MWM signal is suppressed by 40%, which shows the interaction and competition between FWM and SWM when they coexist. This phenomenon could be explained by the dressing effect of ${E}_{3}^{}$(${{E}^{\prime}}_{3}^{}$) on FWM signal ${E}_{F2}^{}$. The dressed FWM process can be described by ${\rho}_{00}^{(0)}\stackrel{{\omega}_{1}}{\to}{\rho}_{({G}_{3}\pm )0}^{(1)}\stackrel{{\omega}_{2}}{\to}{\rho}_{20}^{(2)}\stackrel{-{\omega}_{2}}{\to}{\rho}_{({G}_{3}\pm )0}^{(3)}$, we can obtain ${\rho}_{F2}^{(3)}={G}_{F2}/[{d}_{2}{({d}_{1}+{\left|{G}_{3}^{b}\right|}^{2}/{d}_{3})}^{2}]$, where ${G}_{3}^{b}\text{=}{G}_{3}+{{G}^{\prime}}_{3}$. And the density-matrix element related to the SWM processes can be obtained as ${\rho}_{S2}^{(5)}={G}_{S2}/({d}_{1}{}^{3}{d}_{2}{d}_{3})$ and ${{\rho}^{\prime}}_{S2}^{(5)}={{G}^{\prime}}_{S2}/({d}_{1}{}^{3}{d}_{2}{d}_{3})$ (for simplicity, the self-dressing effects on ${E}_{F2}^{}$, ${E}_{S2}$ and ${{E}^{\prime}}_{S2}$ are not considered here). Since these MWM processes exist at the same time in the experiment, and the signals are copropagating in the same direction, the total detected MWM signal (Fig. 2(a)) will be proportional to the mod square of ${\rho}_{\text{M}}$, where ${\rho}_{\text{M}}={\rho}_{S2}^{(5)}+{{\rho}^{\prime}}_{S2}^{(5)}+{\rho}_{\text{F2}}^{(3)}$.

Next, we investigate the singly-dressed FWM process in the ladder-type three-level subsystem $|0\u3009\to |1\u3009\to |2\u3009$ (shown as Fig. 1(c)) when only the laser beams ${E}_{1}$, ${E}_{2}$ and ${{E}^{\prime}}_{2}$ are turned on. In this three-level subsystem, only the wave-mixing signal ${E}_{F2}^{}$ is generated, with self-dressing effect of ${E}_{2}$(${{E}^{\prime}}_{2}$). According to the perturbation chain of the self-dressed FWM process ${\rho}_{00}^{(0)}\stackrel{{\omega}_{1}}{\to}{\rho}_{({G}_{2}\pm )0}^{(1)}\stackrel{{\omega}_{2}}{\to}{\rho}_{20}^{(2)}\stackrel{-{\omega}_{2}}{\to}{\rho}_{({G}_{2}\pm )0}^{(3)}$, we obtain ${\rho}_{F2}^{(3)}={G}_{F2}/[{d}_{2}{({d}_{1}+{\left|{G}_{2}^{b}\right|}^{2}/{d}_{2})}^{2}]$ for this singly-dressed FWM process, where ${G}_{2}^{b}\text{=}{G}_{2}+{{G}^{\prime}}_{2}$.

The spectra of the singly-dressed FWM process are shown in Fig. 3 . Figures 3(a1) and (a2) respectively present the intensities of the probe transmission (Fig. 3(a1)) and ${E}_{F2}^{}$ (Fig. 3(a2)) versus ${\Delta}_{2}$ at discrete ${\Delta}_{1}$ values. Figure 3(a3) and (a4) respectively depict the intensities of probe transmission (Fig. 3(a3)) and ${E}_{F2}^{}$ (Fig. 3(a4)) versus ${\Delta}_{1}$ at discrete ${\Delta}_{2}$ values, and the Doppler Broadening of the probe transmission signal in Fig. 3(a3) has been subtracted. Figures 3(b1), (b3), (b4) and (b5) are the theoretical calculations corresponding to Figs. 3(a1)-(a4), and Fig. 3(b2) represents the theoretical enhancement and suppression of ${E}_{F2}^{}$, which respectively depicted as the peak and dip on each baseline of the curves, by the self-dressing effect. Notice the experimentally obtained ${E}_{F2}^{}$ signal when scanning ${\Delta}_{2}$ (Fig. 3(a2)) includes two components: the pure FWM signal when not considering dressing effect, and the modification (enhancement and suppression) of the FWM process which is theoretically shown in Fig. 3(b2). Figure 3(c) shows the singly-dressed energy level diagrams corresponding to the curves at discrete frequency detunings in Fig. 3(a).

When scanning ${\Delta}_{2}$, the FWM signal shows the evolution from pure-enhancement (${\Delta}_{\text{1}}=-80\text{MHz}$), to first enhancement and next suppression (${\Delta}_{\text{1}}=-20\text{MHz}$), to pure-suppression (${\Delta}_{\text{1}}=0$), to first suppression and next enhancement (${\Delta}_{\text{1}}=20\text{MHz}$), to pure-enhancement (${\Delta}_{\text{1}}=80\text{MHz}$), as shown in Fig. 3(b2). And the corresponding probe transmission shows the evolution from pure-EIA, to first EIA and next EIT, to pure-EIT, to first EIT and next EIA, finally to pure-EIA (electromagnetically induced absorption) in series as shown in Fig. 3(a1). The height of each baseline of the curves represents the probe transmission without dressing effect of ${E}_{2}$(${{E}^{\prime}}_{2}$) versus probe detuning ${\Delta}_{1}$, while the peak and dip on each baseline represent EIT and EIA respectively. We can see that every enhancement and suppression correspond to EIA and EIT respectively, and the curves show symmetric behavior.

In order to understand the phenomena mentioned above, we resort to the singly-dressed energy level diagrams in Fig. 3(c). With the self-dressing effect of ${E}_{2}$(${{E}^{\prime}}_{2}$), the energy level $|1\u3009$ will be split into two dressed states $|{G}_{2}\pm \u3009$, as shown in Figs. 3(c1)-3(c5). When ${\Delta}_{2}$ is scanned at ${\Delta}_{1}=0$, on the one hand, EIT is obtained in Fig. 3(a1) at the point ${\Delta}_{2}=0$ where the suppression condition ${\Delta}_{1}+{\Delta}_{2}=0$ is satisfied. On the other hand, a pure-suppression of ${E}_{F2}^{}$ is gotten in Fig. 3(b2) because the probe field ${E}_{1}$ could not resonate with either of the two dressed energy levels $|{G}_{2}\pm \u3009$, as shown in Fig. 3(c3). In the region with ${\Delta}_{1}<0$, when ${\Delta}_{2}$ is scanned, the probe transmission shows EIA firstly and EIT afterwards in Fig. 3(a1) at ${\Delta}_{1}\text{=}-20\text{MHz}$. Correspondingly, ${E}_{F2}^{}$ is first enhanced when the EIA is gotten and next suppressed when the EIT is obtained, shown in Fig. 3(b2) at ${\Delta}_{1}\text{=}-20\text{MHz}$. The reason for the first EIA and the corresponding enhancement of ${E}_{F2}^{}$ is that the probe field ${E}_{1}$ resonates with the dressed state $|{G}_{2}+\u3009$ at first, thus the enhancement condition ${\Delta}_{1}+({\Delta}_{2}+\sqrt{{\Delta}_{2}^{2}+4{\left|{G}_{2}^{b}\right|}^{2}})/2=0$ is satisfied. While the reason for the next EIT and the corresponding suppression of ${E}_{F2}^{}$ is that two-photon resonance occurs so as to satisfy the suppression condition ${\Delta}_{1}+{\Delta}_{2}=0$ (see Fig. 3(c2)). When ${\Delta}_{1}$ changes to be positive, the curves at ${\Delta}_{1}\text{=2}0\text{MHz}$ show symmetric evolution behavior with the curves at ${\Delta}_{1}\text{=}-20\text{MHz}$, i.e., EIT as well as a suppression of ${E}_{F2}^{}$ are obtained due to the two-photon resonance which matched the suppression condition ${\Delta}_{1}+{\Delta}_{2}=0$ firstly; and then EIA as well as an enhancement of ${E}_{F2}^{}$ are obtained when ${E}_{1}$ is in resonance with $|{G}_{2}-\u3009$ satisfying the enhancement condition ${\Delta}_{1}+({\Delta}_{2}-\sqrt{{\Delta}_{2}^{2}+4{\left|{G}_{2}^{b}\right|}^{2}})/2=0$, as depicted in Fig. 3(c4). When ${\Delta}_{1}$ is far away from resonance point (${\Delta}_{1}\text{=}\pm 80\text{MHz}$), the pure-EIA as well as the pure-enhancement of ${E}_{F2}^{}$ are obtained because the probe field can only resonate with one of the two dressed states $|{G}_{2}\pm \u3009$ (as shown in Figs. 3(c1) and 3(c5)).

On the other hand, when ${\Delta}_{2}$ is set at discrete values orderly from positive to negative and ${\Delta}_{1}$ is scanned, the probe transmission shows an EIT window on each curve in Fig. 3(a3) satisfying ${\Delta}_{1}+{\Delta}_{2}\text{=}0$. Also, the FWM signal ${E}_{F2}^{}$ presents double peaks (Fig. 3(a4)), due to Autler-Townes (AT) splitting. The two peaks are obtained when ${E}_{1}$ resonates with $|{G}_{4}+\u3009$ and $|{G}_{4}-\u3009$, respectively. The theoretical calculations (Fig. 3(b)) are in good agreement with the experimental results (Fig. 3(a)).

Moreover, when we compare the results of these two kinds of scanning method (i.e. scanning ${\Delta}_{2}$ at discrete ${\Delta}_{1}$ values, and scanning ${\Delta}_{1}$ at discrete ${\Delta}_{2}$ values), an interesting corresponding relationship between them could be discovered, as expressed with the dash lines in Fig. 3(a) and 3(b). Referring to the dressed energy level diagrams in Figs. 3(c), one can easily find out that the curves in the same column which are connected by dash lines correspond to the same dressed energy level diagram in Fig. 3(c), although these curves are obtained by scanning different fields. In other words, the positions of enhancement points and suppression points of FWM signal in the probe frequency detuning (${\Delta}_{1}$) domain correspond with the positions in the dressing frequency detuning (${\Delta}_{2}$) domain, satisfying same enhancement/suppression conditions. Take the curves obtained when scanning ${\Delta}_{2}$ at ${\Delta}_{1}\text{=}-20\text{MHz}$ in Figs. 3(b1)-3(b2), and curves obtained when scanning ${\Delta}_{1}$ at ${\Delta}_{2}=20\text{MHz}$ in Figs. 3(b4)-3(b5) for example. These four curves, although gotten by scanning different field, correspond to the same energy state depicted in Fig. 3(c2) and reveal some common features. When ${E}_{1}$ resonates with the dressed state $|{G}_{2}+\u3009$, a dip of probe transmission (EIA) is gotten both in Fig. 3(b1) and 3(b4), and a peak (enhancement point) of ${E}_{F2}^{}$ appears correspondingly both in Fig. 3(b2) and 3(b5), as the left dash line expresses; when two-photon resonance occurs at the point ${\Delta}_{1}+{\Delta}_{2}=0$, a peak (EIT) is gotten both in Fig. 3(b1) and 3(b4), and a dip (suppression point) of ${E}_{F2}^{}$ appears correspondingly both in Fig. 3(b2) and 3(b5), as the right dash line expresses. Additionally, we notice that when scanning the probe detuning, two enhancement points (i.e. the two peaks of AT splitting) and one suppression point could be obtained, while when scanning the dressing detuning, only one enhancement point and one suppression point could be gotten at most. The reason is that the two splitting states $|{G}_{2}\pm \u3009$ could not move across the original position of $|1\u3009$ and therefore only one of them can resonate with ${E}_{1}$ when scanning the dressing detuning.

Furthermore, the spectra of the doubly-dressed FWM process of ${E}_{F2}^{}$ in the Y-type four-level subsystem are investigated as shown in Fig. 4 , with the laser beams ${E}_{3}$, ${{E}^{\prime}}_{3}$ and ${{E}^{\prime}}_{4}$ blocked and ${E}_{1}$, ${E}_{2}$, ${{E}^{\prime}}_{2}$ and ${E}_{4}$ turned on. Since ${E}_{4}$ is turned on, the FWM signal ${E}_{F2}^{}$ is dressed by ${E}_{4}$ (external-dressing effect) as well as ${E}_{2}$(${{E}^{\prime}}_{2}$) (self-dressing effect). According to the perturbation chain of the doubly-dressed FWM process: ${\rho}_{00}^{(0)}\stackrel{{\omega}_{1}}{\to}{\rho}_{({G}_{2}\pm {G}_{4}\pm )0}^{(1)}\stackrel{{\omega}_{2}}{\to}{\rho}_{20}^{(2)}\stackrel{-{\omega}_{2}}{\to}{\rho}_{({G}_{2}\pm {G}_{4}\pm )0}^{(3)}$, we can obtain ${\rho}_{F2}^{(3)}={G}_{F2}/[{({d}_{1}+{\left|{G}_{2}^{b}\right|}^{2}/{d}_{2}+{\left|{G}_{4}\right|}^{2}/{d}_{4})}^{2}{d}_{2}]$ for the doubly-dressed FWM process.

Firstly, the probe transmission and FWM signal ${E}_{F2}^{}$ versus probe detuning ${\Delta}_{1}$ and self-dressing detuning ${\Delta}_{2}$ are investigated (Figs. 4(a)-4(c)). Figures 4(a1) and 4(a2) present the intensities of the probe transmission (Fig. 4(a1)) and ${E}_{F2}^{}$ (Fig. 4(a2)), respectively, versus ${\Delta}_{2}$ at discrete ${\Delta}_{1}$ values. Figures 4(a3) and (a4) depict the intensities of probe transmission (Fig. 4(a3)) and ${E}_{F2}^{}$ (Fig. 4(a4)) versus${\Delta}_{1}$, respectively, at discrete ${\Delta}_{2}$ values (with fixed ${\Delta}_{4}$ at ${\Delta}_{4}=0$). Notice the Doppler Broadening of the probe transmission signal in Fig. 3(a3) has been subtracted. Figures 4(b1), 4(b3), 4(b4) and 4(b5) are the theoretical calculations corresponding to Fig. 4(a1)-(a4), while Fig. 4(b2) represents the theoretical enhancement and suppression of ${E}_{F2}^{}$, respectively expressed by the peak and dip on each baseline of the curves. Figure 4(c) show the doubly-dressed energy level diagrams corresponding to the curves at discrete detuning values in Fig. 4(a).

When ${\Delta}_{1}$ is set at discrete values orderly from negative to positive and ${\Delta}_{2}$ is scanned, the experimentally obtained ${E}_{F2}^{}$ signal is shown in Fig. 4(a2), including two components: the pure FWM signal when not considering dressing effect, and the modification (enhancement and suppression) of the FWM process which is theoretically shown in Fig. 4(b2). The profile of all the baselines in Fig. 4(b2) reveals AT splitting of ${E}_{4}$, and the transition of enhancement and suppression in each curve is induced by the interaction between ${E}_{2}$(${{E}^{\prime}}_{2}$) and ${E}_{4}$, showing the evolution from pure-enhancement (${\Delta}_{\text{1}}=-80\text{MHz}$), to first enhancement and next suppression (${\Delta}_{\text{1}}=-40\text{MHz}$), to pure-suppression (${\Delta}_{\text{1}}=-20\text{MHz}$), to first suppression and next enhancement (${\Delta}_{\text{1}}=-10\text{MHz}$), to pure-suppression (${\Delta}_{\text{1}}=0$), to first enhancement and next suppression (${\Delta}_{\text{1}}=10\text{MHz}$), to pure-suppression (${\Delta}_{\text{1}}=20\text{MHz}$), to first suppression and next enhancement (${\Delta}_{\text{1}}=40\text{MHz}$), finally to pure-enhancement (${\Delta}_{\text{1}}=80\text{MHz}$). Correspondingly, the probe transmission shows the evolution from pure-EIA, to first EIA and next EIT, to pure-EIT, to first EIT and next EIA, to pure-EIT, to first EIA and next EIT, to pure-EIT, to first EIT and next EIA, finally to pure-EIA in series as shown in Fig. 4(a1). The height of the baseline of each curve represents the probe transmission without dressing field ${E}_{2}$(${{E}^{\prime}}_{2}$) versus probe detuning ${\Delta}_{1}$. The profile of these baselines reveals an EIT window induced by external-dressing field ${E}_{4}$ at ${\Delta}_{\text{1}}\text{=}-{\Delta}_{4}$. While the peak and dip on each baseline of the curves represent EIT and EIA induced by self-dressing fields ${E}_{2}$ and ${{E}^{\prime}}_{2}$. We can see that every enhancement and suppression correspond to EIA and EIT respectively, which is similar to the singly-dressing case observed in Fig. 3.

Such variations in the probe transmission and the transition of enhancement and suppression of ${E}_{F2}^{}$ are caused by the interaction of the dressing fields ${E}_{2}$(${{E}^{\prime}}_{2}$) and ${E}_{4}$. Because of the doubly-dressing effect, the energy level $|1\u3009$ is totally split into three dressed states (shown in Fig. 4(c)). Firstly under the external-dressing effect of ${E}_{4}$, the energy level $|1\u3009$ will be broken into two primarily dressed states $|{G}_{4}\pm \u3009$. Then in the region with ${\Delta}_{1}<0$, when ${\Delta}_{2}$ is scanned around $|{G}_{4}+\u3009$, two secondarily dressed states $|{G}_{4}+{G}_{2}\pm \u3009$ could be created from $|{G}_{4}+\u3009$ by the self-dressing effect of ${E}_{2}$(${{E}^{\prime}}_{2}$), as shown in Figs. 4(c1)-4(c4). Symmetrically, in the region with ${\Delta}_{1}>0$, when ${\Delta}_{2}$ is scanned around $|{G}_{4}-\u3009$, two secondarily dressed states $|{G}_{4}-{G}_{2}\pm \u3009$ could be created from $|{G}_{4}-\u3009$, as shown in Figs. 4(c6)-(c9). Since the phenomena and analysis method are similar with those in the singly-dressing case, here we only give the enhancement and suppression conditions as ${\Delta}_{1}+({{\Delta}^{\prime}}_{2}\pm \sqrt{{{\Delta}^{\prime}}_{2}^{2}+4{\left|{G}_{2}^{b}\right|}^{2}})/2+{G}_{4}=0$ and ${\Delta}_{1}+{\Delta}_{2}=0$ for ${\Delta}_{1}<0$, where ${{\Delta}^{\prime}}_{2}={\Delta}_{2}-{G}_{4}$ represents the detuning of ${E}_{2}$(${{E}^{\prime}}_{2}$) from $|{G}_{4}+\u3009$, and ${\Delta}_{1}+({{\Delta}^{\prime}}_{2}\pm \sqrt{{{\Delta}^{\prime}}_{2}^{2}+4{\left|{G}_{2}^{b}\right|}^{2}})/2-{G}_{4}=0$ and ${\Delta}_{1}+{\Delta}_{2}=0$ for ${\Delta}_{1}>0$, where ${{\Delta}^{\prime}}_{2}={\Delta}_{2}+{G}_{4}$ represents the detuning of ${E}_{2}$(${{E}^{\prime}}_{2}$) from $|{G}_{4}-\u3009$. When the enhancement condition is satisfied, the probe field ${E}_{1}$ resonates with one of the secondarily dressed states, leading to an EIA of probe transmission and enhancement of ${E}_{F2}^{}$. When the suppression condition is satisfied, two-photon resonance occurs, leading to an EIT and suppression of ${E}_{F2}^{}$. Especially, we notice the enhancement and suppression of ${E}_{F2}^{}$ in Fig. 4(b2) show symmetric behavior with three symmetric centers at ${\Delta}_{\text{1}}=0,-20,\text{and}\text{20}\text{MHz}$, all of which are pure-suppression. The pure-suppression at ${\Delta}_{\text{1}}=0\text{MHz}$ is induced by primary dressing effect of ${E}_{4}$, while pure-suppressions at ${\Delta}_{1}\text{=}\pm 20\text{MHz}$ are caused by the secondary dressing effect of ${E}_{2}$(${{E}^{\prime}}_{2}$).

On the other hand, when ${\Delta}_{2}$ is set at discrete values orderly from positive to negative and ${\Delta}_{1}$ is scanned, the intensity of probe transmission shows double EIT windows on each curve in Fig. 4(a3), which are EIT windows $|0\u3009\to |1\u3009\to |2\u3009$ (appearing at ${\Delta}_{1}=-{\Delta}_{2}$) and $|0\u3009\to |1\u3009\to |4\u3009$ (appearing at ${\Delta}_{1}=-{\Delta}_{4}$), respectively. As ${\Delta}_{4}$ is fixed at ${\Delta}_{4}=0$ and ${\Delta}_{2}$ is set at discrete values from positive to negative, the EIT window $|0\u3009\to |1\u3009\to |4\u3009$ is fixed at ${\Delta}_{1}=-{\Delta}_{4}=0$ and the EIT window $|0\u3009\to |1\u3009\to |2\u3009$ moves from negative to positive. Especially, when ${\Delta}_{2}$ is set at ${\Delta}_{2}=0$, the two EIT windows overlap as shown in Fig. 4(a3), and a double-peak FWM signal is obtained because both ${E}_{2}$(${{E}^{\prime}}_{2}$) and ${E}_{4}$ dress the energy level $|1\u3009$ simultaneously into two dressed states $|+\u3009$ and $|-\u3009$, as shown in Fig. 4(a4). When ${\Delta}_{2}$ is set at ${\Delta}_{2}\ne 0$, in the process of scanning ${\Delta}_{1}$, the FWM signal ${E}_{F2}^{}$ presents three peaks, corresponding to the three dressed state respectively. Firstly, ${E}_{4}$ dresses $|1\u3009$ into two primarily dressed state $|{G}_{4}+\u3009$ and $|{G}_{4}-\u3009$, corresponding to primary AT splitting. Then when the frequency of ${E}_{2}^{}$ and ${{E}^{\prime}}_{2}$ is tuned so as to move the $|0\u3009\to |1\u3009\to |2\u3009$ EIT window into the left FWM peak (${\Delta}_{2}>0$), secondary AT splitting occurs and the left peak splits into two peaks, respectively corresponding to secondarily dressed states $|{G}_{4}+{G}_{2}+\u3009$ and $|{G}_{4}+{G}_{2}-\u3009$. Symmetrically, in the region with ${\Delta}_{2}<0$, the three peaks corresponding to $|{G}_{4}+\u3009$, $|{G}_{4}-{G}_{2}+\u3009$ and $|{G}_{4}-{G}_{2}-\u3009$ respectively. The theoretical calculations (Fig. 4(b)) are in good agreement with the experimental results (Fig. 4(a)).

When comparing the results of these two kinds of scanning method (i.e. scanning ${\Delta}_{2}$ at discrete ${\Delta}_{1}$ values, and scanning ${\Delta}_{1}$ at discrete ${\Delta}_{2}$ values), the corresponding relationship could also be discovered, as expressed with the dash lines in Figs. 4(a) and 4(b). By referring to the energy level diagrams in Figs. 4(c), one can easily find out that positions of enhancement points and suppression points of FWM signal in the probe frequency detuning (${\Delta}_{1}$) domain correspond with the positions in the self-dressing frequency detuning (${\Delta}_{2}$) domain, satisfying same enhancement/suppression conditions. Take the curves at ${\Delta}_{1}=-40\text{MHz}$ in Figs. 4(b1)-(b2) and the curves at ${\Delta}_{2}=40\text{MHz}$ in Figs. 4(b4)-4(b5) for example. These four curves, although gotten by scanning different fields, correspond to the same energy state depicted in Fig. 4(c2) and reveal some common features. When ${E}_{1}$ resonates with the dressed state $|{G}_{4}+{G}_{2}+\u3009$, a dip of probe transmission (EIA) is gotten both in Fig. 4(b1) and 4(b4), and a peak (enhancement point) of ${E}_{F2}^{}$ signal appears both in Fig. 4(b2) and 4(b5), as the left dash line expresses; when two-photon resonance (${\Delta}_{1}+{\Delta}_{2}=0$) occurs, a peak (EIT) is gotten both in Fig. 4(b1) and 4(b4), and a dip (suppression point) appears both in Fig. 4(b2) and (b5), as the right dash line expresses. Especially, the position of the pure-suppression at ${\Delta}_{1}\text{=}0$ in Fig. 4(b2) corresponds to the center of primary AT splitting at ${\Delta}_{2}\text{=}0$ in Fig. 4(b5); and the positions of the pure-suppression at ${\Delta}_{1}\text{=}\pm 20\text{MHz}$ in Fig. 4(b2) correspond to the center of secondary AT splitting at ${\Delta}_{2}\text{=}\pm 20\text{MHz}$ in Fig. 4(b5). We also notice that when ${\Delta}_{1}$ scanned, three enhancement points and two suppression points could be obtained, while when scanning ${\Delta}_{2}$ only one enhancement point and one suppression point could be gotten at most.

Next, we investigate the probe transmission and the enhancement and suppression of ${E}_{F2}^{}$ versus the probe detuning ${\Delta}_{1}$ and external-dressing detuning ${\Delta}_{4}$ (Figs. 4(d)-4(e)). Figures 4(d1) and 4(d2) respectively present the intensities of the probe transmission (Fig. 4(d1)) and the enhancement and suppression of ${E}_{F2}^{}$ (Fig. 4(d2)) versus ${\Delta}_{4}$ at discrete ${\Delta}_{1}$ values. While Fig. 4(d3) and (d4) depict the intensities of probe transmission (Fig. 4(d3)) and ${E}_{F2}^{}$ (Fig. 4(d4)) versus ${\Delta}_{1}$ at discrete ${\Delta}_{4}$ values (with fixed ${\Delta}_{2}$ at ${\Delta}_{2}=0$). Figures 4(e1)-4(e4) are the theoretical calculations corresponding to Figs. 4(d1)-4(d4). Figure 4(f) shows the doubly-dressed energy level diagrams corresponding to the curves at discrete detuning values in Fig. 4(d). Similar to the above discussion of Figs. 4(a)-(c), the signal ${E}_{F2}^{}$ is dressed by both ${E}_{2}$(${{E}^{\prime}}_{2}$) and ${E}_{4}$. Firstly, under the self-dressing effect of ${E}_{2}$(${{E}^{\prime}}_{2}$), the state $|1\u3009$ will be broken into two primarily dressed states $|{G}_{2}\pm \u3009$. Then in the region with ${\Delta}_{1}<0$, when ${\Delta}_{4}$ is scanned around $|{G}_{2}+\u3009$, two secondarily dressed states $|{G}_{2}+{G}_{4}\pm \u3009$ could be created from $|{G}_{2}+\u3009$ by the external-dressing field ${E}_{4}$, as shown in Fig. 4(f1)-4(f4). Symmetrically, in the region with ${\Delta}_{1}>0$, when ${\Delta}_{4}$ is scanned around $|{G}_{2}-\u3009$, two secondarily dressed states $|{G}_{2}-{G}_{4}\pm \u3009$ could be created from $|{G}_{2}-\u3009$, as shown in Fig. 4(f6)-4(f9). Here we only give the enhancement and suppression conditions as ${\Delta}_{1}+({{\Delta}^{\prime}}_{4}\pm \sqrt{{{\Delta}^{\prime}}_{4}^{2}+4{\left|{G}_{4}\right|}^{2}})/2+{G}_{2}=0$ and ${\Delta}_{1}+{\Delta}_{4}=0$ for ${\Delta}_{1}<0$, where ${{\Delta}^{\prime}}_{4}={\Delta}_{4}-{G}_{2}$ represents the detuning of ${E}_{4}$ from $|{G}_{2}+\u3009$, and ${\Delta}_{1}+({{\Delta}^{\prime}}_{4}\pm \sqrt{{{\Delta}^{\prime}}_{4}^{2}+4{\left|{G}_{4}\right|}^{2}})/2-{G}_{2}=0$ and ${\Delta}_{1}+{\Delta}_{4}=0$ for ${\Delta}_{1}>0$, where ${{\Delta}^{\prime}}_{4}={\Delta}_{4}+{G}_{2}$ represents the detuning of ${E}_{4}$ from $|{G}_{2}-\u3009$. Unlike scanning self-dressing detuning (Fig. 4(a2)), by scanning the external-dressing detuning the enhancement and suppression of ${E}_{F2}^{}$ could be detected directly, excluding the pure FWM component (Fig. 4(d2)). We can see that the enhancement and suppression of ${E}_{F2}^{}$ in Fig. 4(d2) shows similar evolution with that in Fig. 4(b2). The profile of all the baselines, which has two peaks, reveals AT splitting of ${E}_{2}$(${{E}^{\prime}}_{2}$), and the transition of enhancement and suppression in each curve is induced by the interaction between ${E}_{2}$(${{E}^{\prime}}_{2}$) and ${E}_{4}$ with three symmetric centers at ${\Delta}_{\text{1}}=0,-20,\text{and}\text{20}\text{MHz}$, all of which are pure-suppression. The pure-suppression at ${\Delta}_{\text{1}}=0\text{MHz}$ is induced by primary dressing effect of ${E}_{2}$(${{E}^{\prime}}_{2}$), while pure-suppressions at ${\Delta}_{1}\text{=}\pm 20\text{MHz}$ are caused by the secondary dressing effect of ${E}_{4}$. On the other hand, when ${\Delta}_{1}$ is scanned, the FWM signal ${E}_{F2}^{}$ in Fig. 4(d4) also presents three peaks, corresponding to the three dressed state respectively. Moreover, the corresponding relationship between scanning probe detuning and scanning external-dressing detuning is similar with above, as expressed by the dash lines in Fig. 4(d) and 4(e). It is obvious that the theoretical calculations (Fig. 4(e)) are in good agreement with the experimental results (Fig. 4(d)).

Comparing with the singly-dressed FWM process in Fig. 3, we notice the doubly-dressed FWM process, although derives from the former, shows more complexities since one more dressing field is considered. When scanning probe detuning, doubly-dressed FWM signal shows three peaks resulting from two orders of AT splitting (Fig. 4(a4) and 4(d4)); whereas singly-dressed FWM signal shows only two peaks resulting from of AT splitting of self-dressing effect (Fig. 3(a4)). When scanning the dressing detuning, only one symmetric center appears in singly-dressing case (Fig. 3(b2)), whereas three symmetric centers appear in doubly-dressing case respectively at ${\Delta}_{\text{1}}=0,-20,\text{and}\text{20}\text{MHz}$ (Fig. 4(b2) and 4(e2)), all of which reveals pure-suppression. The symmetric center at ${\Delta}_{1}\text{=}0$ is caused by the primary dressing effect, while the two symmetric centers at ${\Delta}_{1}\text{=}\pm 20\text{MHz}$ are due to the secondary dressing effect.

Synthetically, based on the analysis above, we find the methods of scanning the probe detuning (Fig. 3(a3)-3(a4), Fig. 4(a3)-4(a4), Fig. 4(d3)-4(d4)), scanning self-dressing detuning (Fig. 3(a1)-3(a2), Fig. 4(a1)-4(a2)) and scanning external-dressing detuning (Fig. 4(d1)-4(d2)) individually show some different features and advantages on research the FWM process. When scanning the probe detuning, the obtained FWM signal includes two components: the pure FWM signal when not considering dressing effects, and the modification (revealing AT splitting) of the FWM process. When scanning self-dressing detuning, the obtained signal also includes two components: the pure FWM signal and the modification (revealing the transition between enhancement and suppression) of the FWM process. While by scanning external-dressing detuning, the enhancement and suppression could be detected directly, excluding the pure FWM component. On the other hand, by scanning the probe detuning, all enhancement points and suppression points could be observed corresponding to the peaks and dips of AT splitting. In singly-dressing case, there are two enhancement points and one suppression point (Fig. 3(a3)-3(a4)), and in doubly-dressing case, three enhancement points and two suppression points (Fig. 4(a3)-4(a4)), etc. In contrast, by scanning dressing detuning, at most one enhancement point and one suppression point could be gotten in the spectra. Furthermore, the positions of the enhancement and suppression points when scanning dressing detuning match with the positions of corresponding points when scanning probe detuning, as the dash lines express in Fig. 3 and Fig. 4.

After that, we demonstrate a new type of phase-controlled, spatiotemporal coherent interference between two FWM processes (${E}_{F2}^{}$ and ${E}_{F4}^{}$) in a four-level, Y-type subsystem when the laser beams ${E}_{3}$ and ${{E}^{\prime}}_{3}$ are turned off (shown as Fig. 1(d)). With a specially designed spatial configuration for the laser beams with phase matching and an appropriate optical delay introduced in one of the coupling laser beams, we can have a controllable phase difference between the two FWM processes in the subsystem. When this relative phase is varied, temporal and spatial interferences can be observed. The interference in the time domain is in the femtosecond time scale, corresponding to the optical transition frequency excited by the delayed laser beam. In the experiment, the beam ${{E}^{\prime}}_{2}$ is delayed by an amount $\tau $ using a computer-controlled stage. The CCD and an avalanche photodiode (APD) are set on the propagation path of the two FWM signals to measure them. By changing the frequency detuning ${\Delta}_{4}$, the $|0\u3009-|1\u3009-|4\u3009$ EIT window can be shifted toward the $|0\u3009-|1\u3009-|2\u3009$ EIT window. When the two EIT windows overlap with each other, temporal and spatial interferences of two FWM signal ${E}_{F2}^{}$ and ${E}_{F4}^{}$ can be observed, as shown in Fig. 5 .

The coexisting ${E}_{F1}^{}$ and ${E}_{F2}^{}$ signals give the total detected intensity as:$I(\tau ,r)\propto |{\chi}_{F2}^{(3)}{|}^{2}+|{\chi}_{F4}^{(3)}{|}^{2}+2\left|{\chi}_{F2}^{(3)}\right|\left|{\chi}_{F4}^{(3)}\right|\mathrm{cos}({\phi}_{2}-{\phi}_{4}+\phi )$, where ${\chi}_{F2}^{(3)}=\frac{N{\mu}_{1}^{2}}{\hslash {\epsilon}_{0}{G}_{1}}{\rho}_{F2}^{(3)}\times \frac{{\mu}_{2}^{2}}{{\hslash}^{2}{G}_{2}{{G}^{\prime}}_{2}^{*}}$ $=-i{\mu}_{1}^{2}{\mu}_{2}^{2}N/\left\{{\epsilon}_{0}{\hslash}^{3}{({d}_{1}+{G}_{2}^{2}/{d}_{2}+{G}_{4}^{2}/{d}_{4})}^{2}{d}_{2}\right\}=\left|{\chi}_{F2}^{(3)}\right|\mathrm{exp}(i{\phi}_{2})$, ${\chi}_{F4}^{(3)}=\frac{N{\mu}_{1}^{2}}{\hslash {\epsilon}_{0}{G}_{1}}{\rho}_{F4}^{(3)}\times \frac{{\mu}_{4}^{2}}{{\hslash}^{2}{G}_{4}{{G}^{\prime}}_{4}^{*}}$ $=-i{\mu}_{1}^{2}{\mu}_{4}^{2}N/\left\{{\epsilon}_{0}{\hslash}^{3}{({d}_{1}+{G}_{2}^{2}/{d}_{2}+{G}_{4}^{2}/{d}_{4})}^{2}{d}_{4}\right\}=\left|{\chi}_{F4}^{(3)}\right|\mathrm{exp}(i{\phi}_{4})$, and $\phi =\Delta k\cdot r-{\omega}_{2}\tau $ with the frequency of spatial interference $\Delta k={k}_{F2}-{k}_{F4}=({k}_{2}-{{k}^{\prime}}_{2})-({k}_{4}-{{k}^{\prime}}_{4})$. ${\mu}_{1}$,${\mu}_{2}$, and ${\mu}_{4}$ are the dipole moments of the transitions $|0\u3009\to |1\u3009$, $|1\u3009\to |2\u3009$, and $|1\u3009\to |4\u3009$, respectively. From the expression of $I(\tau ,r)$, we can see that the total signal has an ultrafast time oscillation with a period of $2\pi /{\omega}_{2}$ and spatial interference with a period of $2\pi /\Delta k$, which forms a spatiotemporal interferogram. Fig. 5(a) shows a three-dimensional interferogram pattern, and Fig. 5(b) shows the temporal interference with a much longer time delay in beam ${{E}^{\prime}}_{2}$ while Fig. 5(c) shows its projections on time. Figure 5(c) depicts a typical temporal interferogram with the temporal oscillation period of $2\pi /{\omega}_{2}=\text{2}.6\text{fs}$ corresponding to the $|1\u3009$ to $|2\u3009$ transition frequency of ${\Omega}_{2}=2.4{\text{fs}}^{-1}$ in ${}^{\text{85}}\text{R}\text{b}$.This gives a technique for precision measurement of atomic transition frequency in optical wavelength range. The solid curve in Fig. 5(c) is the theoretical calculation from the full density-matrix equations. It is easy to see that the theoretical results fit well with the experimentally measured results.

Now, we concentrate on the SWM process when ${\Delta}_{3}$ is at large detuning, with ${{E}^{\prime}}_{2}$ and ${{E}^{\prime}}_{4}$ blocked (shown as Fig. 1(e)). When take the atomic velocity component and the dressing effect of ${E}_{3}$(${{E}^{\prime}}_{3}$) into consideration, the enhancement and suppression of the SWM signal would be shifted far away from resonance, as shown in Fig. 6 .

When considering Doppler effect the atom moving towards the probe laser beam with velocity $v$, the frequency of ${E}_{1}$ is changed to ${\omega}_{1}+{\omega}_{1}v/c$, and the frequencies of ${E}_{2}$, ${E}_{3}$ and ${E}_{4}$ are changed to ${\omega}_{2}-{\omega}_{2}v/c$, ${\omega}_{3}-{\omega}_{3}v/c$, ${\omega}_{4}-{\omega}_{4}v/c$ under our experimental geometry configuration, therefore their detunings are changed to ${\Delta}_{1}-{\omega}_{1}v/c$, ${\Delta}_{2}+{\omega}_{2}v/c$, ${\Delta}_{3}+{\omega}_{3}v/c$, ${\Delta}_{4}+{\omega}_{4}v/c$. Noticing that in such beam geometric configuration, the two-photon Doppler-free condition will not be satisfied for the $\Lambda $-type three-level subsystem $|0\u3009\to |1\u3009\to |3\u3009$, the atomic velocity component ${\omega}_{3}v/c$ will behaves dominant. The density-matrix element of SWM signal ${E}_{S4}^{}$ can be obtained as ${\rho}_{S4}^{(5)}={G}_{S4}/[{{d}^{\prime}}_{1}{{d}^{\prime}}_{3}{{d}^{\prime}}_{4}{({{d}^{\prime}}_{1}+{\left|{G}_{4}\right|}^{2}/{{d}^{\prime}}_{4}\text{+}{\left|{G}_{3}^{b}\right|}^{2}/{{d}^{\prime}}_{3})}^{2}]$ via the self-dressed perturbation chain ${\rho}_{00}^{(0)}\stackrel{{\omega}_{1}}{\to}{\rho}_{({G}_{4}\pm {G}_{3}\pm )0}^{(1)}\stackrel{-{\omega}_{3}}{\to}{\rho}_{30}^{(2)}\stackrel{{\omega}_{3}}{\to}{\rho}_{({G}_{4}\pm {G}_{3}\pm )0}^{(3)}\stackrel{{\omega}_{4}}{\to}{\rho}_{40}^{(4)}\stackrel{-{\omega}_{4}}{\to}{\rho}_{10}^{(5)}$, where ${{d}^{\prime}}_{1}={d}_{1}-i{\omega}_{1}v/c+{\gamma}_{1}$, ${{d}^{\prime}}_{3}={d}_{3}-i{\omega}_{3}v/c-i{\omega}_{1}v/c+{\gamma}_{1}-{\gamma}_{3}$, ${{d}^{\prime}}_{4}={d}_{4}+i{\omega}_{4}v/c-i{\omega}_{1}v/c+{\gamma}_{1}+{\gamma}_{4}$, and ${\gamma}_{1}$, ${\gamma}_{3}$, ${\gamma}_{4}$ are the half linewidth of laser beams ${E}_{1}$, ${E}_{3}$, ${E}_{4}$ respectively. When the SWM signal ${E}_{S4}^{}$ is externally dressed by ${E}_{2}$ (defined as ${E}_{S4}^{D}$), the solved expression is ${\rho}_{S4}^{(5)}={G}_{S4}/[{{d}^{\prime}}_{3}{{d}^{\prime}}_{4}({{d}^{\prime}}_{1}+{\left|{G}_{2}\right|}^{2}/{{d}^{\prime}}_{2}){({{d}^{\prime}}_{1}+{\left|{G}_{2}\right|}^{2}/{{d}^{\prime}}_{2}+{\left|{G}_{4}\right|}^{2}/{{d}^{\prime}}_{4}\text{+}{\left|{G}_{3}^{b}\right|}^{2}/{{d}^{\prime}}_{3})}^{2}]$ via the dressed perturbation chain: ${\rho}_{00}^{(0)}\stackrel{{\omega}_{1}}{\to}{\rho}_{({G}_{4}\pm {G}_{3}\pm {G}_{2}\pm )0}^{(1)}\stackrel{-{\omega}_{3}}{\to}{\rho}_{30}^{(2)}\stackrel{{\omega}_{3}}{\to}{\rho}_{({G}_{4}\pm {G}_{3}\pm {G}_{2}\pm )0}^{(3)}\stackrel{{\omega}_{4}}{\to}{\rho}_{40}^{(4)}\stackrel{-{\omega}_{4}}{\to}{\rho}_{({G}_{2}\pm )0}^{(5)}$, where ${{d}^{\prime}}_{2}={d}_{2}+i{\omega}_{2}v/c-i{\omega}_{1}v/c+{\gamma}_{1}+{\gamma}_{2}$ and ${\gamma}_{2}$ are the half linewidth of laser beam ${E}_{2}$. Similarly, the density-matrix element of SWM signal${E}_{S2}^{}$ can be obtained as ${\rho}_{S2}^{(5)}={G}_{S2}/[{{d}^{\prime}}_{1}{{d}^{\prime}}_{2}{{d}^{\prime}}_{3}{({{d}^{\prime}}_{1}+{\left|{G}_{2}\right|}^{2}/{{d}^{\prime}}_{2}\text{+}{\left|{G}_{3}^{b}\right|}^{2}/{{d}^{\prime}}_{3})}^{2}]$ via the self-dressed perturbation chain: ${\rho}_{00}^{(0)}\stackrel{{\omega}_{1}}{\to}{\rho}_{({G}_{2}\pm {G}_{3}\pm )0}^{(1)}\stackrel{-{\omega}_{3}}{\to}{\rho}_{30}^{(2)}\stackrel{{\omega}_{3}}{\to}{\rho}_{({G}_{2}\pm {G}_{3}\pm )0}^{(3)}\stackrel{{\omega}_{2}}{\to}{\rho}_{20}^{(4)}\stackrel{-{\omega}_{2}}{\to}{\rho}_{10}^{(5)}$. When the SWM signal ${E}_{S2}^{}$ is externally dressed by ${E}_{4}$ (defined as ${E}_{S2}^{D}$), the solved expression is ${\rho}_{S2}^{(5)}={G}_{S2}/[{{d}^{\prime}}_{2}{{d}^{\prime}}_{3}({{d}^{\prime}}_{1}+{\left|{G}_{4}\right|}^{2}/{{d}^{\prime}}_{4}){({{d}^{\prime}}_{1}+{\left|{G}_{2}\right|}^{2}/{{d}^{\prime}}_{2}+{\left|{G}_{4}\right|}^{2}/{{d}^{\prime}}_{4}\text{+}{\left|{G}_{3}^{b}\right|}^{2}/{{d}^{\prime}}_{3})}^{2}]$ via the dressed perturbation chain: ${\rho}_{00}^{(0)}\stackrel{{\omega}_{1}}{\to}{\rho}_{({G}_{2}\pm {G}_{3}\pm {G}_{4}\pm )0}^{(1)}\stackrel{-{\omega}_{3}}{\to}{\rho}_{30}^{(2)}\stackrel{{\omega}_{3}}{\to}{\rho}_{({G}_{2}\pm {G}_{3}\pm {G}_{4}\pm )0}^{(3)}\stackrel{{\omega}_{2}}{\to}{\rho}_{20}^{(4)}\stackrel{-{\omega}_{2}}{\to}{\rho}_{({G}_{4}\pm )0}^{(5)}$. From the expressions of ${E}_{S4}^{D}$ and ${E}_{S2}^{D}$ signals, one can see that the two SWM processes are closely connected by mutual dressing effect.

In Fig. 6(a), we present the probe transmission (Fig. 6(a1)) and the measured SWM signal (Fig. 6(a2)) by scanning ${\Delta}_{2}$ at different designated ${\Delta}_{1}$ values, with ${G}_{2}<<{G}_{4}$. In Fig. 6(a1), The profile of each baseline represents the probe transmission without dressing field ${E}_{2}$ versus probe detuning ${\Delta}_{1}$, which reveals an EIT window ($-{\text{80MHz<\Delta}}_{\text{1}}\text{<80MHz}$) induced by ${E}_{4}$, and the peak on each baseline is the EIT induced by ${E}_{2}$. In Fig. 6(a2), the profile of each baseline represents the intensity variation of the triple-peak SWM signal ${E}_{S4}^{}$ versus ${\Delta}_{1}$. The peak and dip on each baseline include the dressed SWM signal ${E}_{S2}^{D}$ and the enhancement and suppression of ${E}_{S4}$ induced by ${E}_{2}$. Considering ${G}_{2}<<{G}_{4}$, we can deduce the signal of ${E}_{S2}^{D}$ is quite small. Therefore, the peak and dip on each baseline mainly represent the enhancement and suppression of SWM signal ${E}_{S4}$ induced by ${E}_{2}$.

One can see that the curves in Fig. 6(a2) shows pure-suppression at ${\Delta}_{1}=0\text{MHz}$ and ${\Delta}_{1}=-2\text{50MHz}$. This two pure-suppressions can be explained by the triple-dressing effect of ${E}_{2}$, ${E}_{3}$(${{E}^{\prime}}_{3}$) and ${E}_{4}$. The enhancement and suppression of the SWM is caused by the triply-dressing fields. Firstly, due to the self-dressing effect of ${E}_{4}$, the state $|1\u3009$ would be split into two dressed states $|{G}_{4}\pm \u3009$. Next, the dressing field ${E}_{3}$(${{E}^{\prime}}_{3}$) split the state $|{G}_{4}\text{+}\u3009$ into $|{G}_{4}\text{+}{G}_{3}\pm \u3009$. Finally, when ${\Delta}_{2}$ is scanned, ${E}_{2}$ will further split $|{G}_{4}\text{+}{G}_{3}\pm \u3009$ into two dressed states $|{G}_{4}\text{+}{G}_{3}+{G}_{2}\pm \u3009$ or $|{G}_{4}\text{+}{G}_{3}-{G}_{2}\pm \u3009$; or split $|{G}_{4}-\u3009$ into two dressed states $|{G}_{4}-{G}_{2}\pm \u3009$, as shown in Fig. 6(b). When two-photon resonance occurs at the original states $|{G}_{4}\text{+}{G}_{3}\text{+}\u3009$ (Fig. 6(b2)), $|{G}_{4}\text{+}{G}_{3}-\u3009$ (Fig. 6(b3)) or $|{G}_{4}-\u3009$ (Fig. 6(b4)), the pure-suppressions of SWM can be obtained. When Doppler effect being considered, the dominant atomic velocity component ${\omega}_{3}v/c$ moving the $|{G}_{4}\text{+}{G}_{3}\text{+}\u3009$ state far away from the resonance, the pure-suppression on the left induced by triply-dressing effect will be shifted to large detuning. In the experiment, only two pure-suppressions can be obtained, of which the left one is caused by two-photon resonance at original $|{G}_{4}\text{+}{G}_{3}\text{+}\u3009$ state, and the right one is related to the original $|{G}_{4}\text{+}{G}_{3}-\u3009$ state. This inconsistence is because when the frequency of ${E}_{3}$(${{E}^{\prime}}_{3}$) is at large detuning (${\Delta}_{3}>>0$), the enhancement and suppression of SWM signal is no more symmetrical. Specifically, due to the optical pumping effect corresponding to the transition from $|3\u3009$ to $|1\u3009$ by ${E}_{3}$ and ${{E}^{\prime}}_{3}$, the suppression will be intensified with ${\Delta}_{1}<0$ (especially when ${\Delta}_{1}+{\Delta}_{3}=0$) as shown in Fig. 6(b); but with ${\Delta}_{1}>0$, the inexistence of such effect makes the suppression caused by the two photon resonance at original $|{G}_{4}-\u3009$ state unobtainable.

In this way, we first demonstrate that the enhancement and suppression signal can be observed out of the EIT window (${\text{-80MHz<\Delta}}_{\text{1}}\text{<+80MHz}$) through the Doppler frequency shift led by atomic velocity component and optical pumping. From the figures one can see that even ${\Delta}_{1}$ is at large detuning (Fig. 6(b1) and 6(b5)), the enhancement and suppression of SWM will still exists in the region with ${\Delta}_{1}<0$.

Finally, the interaction of the six wave-mixing signals is studied. When all seven laser beams are turned on, two FWM signals (${E}_{F2}$ and ${E}_{F4}$) and four SWM signals (${E}_{S2}^{}$, ${{E}^{\prime}}_{S2}$, ${E}_{S4}^{}$ and ${{E}^{\prime}}_{S4}$) can be generated simultaneously and interact with each other (considering the FWM signal ${E}_{F3}$ is so weak that it could be negligible). When the $|0\u3009\to |1\u3009\to |4\u3009$ EIT window and the $|0\u3009\to |1\u3009\to |2\u3009$ EIT window are tuned separated, the interaction of MWM processes related to the same EIT window has been displayed in Fig. 2, by scanning probe detuning under different blocking conditions. Here, we overlap the two EIT windows experimentally, therefore the interaction between these two groups of wave-mixing signals (those related to ${E}_{2}$(${{E}^{\prime}}_{2}$) and those related to ${E}_{4}$(${{E}^{\prime}}_{4}$)) will be studied, by scanning the dressing field detuning ${\Delta}_{2}$ at discrete probe detuning ${\Delta}_{1}$ values (as shown in Fig. 7 ).

Generally, due to the mutual dressings of the two ladder subsystems, we can obtain the density elements ${\rho}_{F2}^{(3)}={G}_{F2}/[{d}_{2}{({d}_{1}+{\left|{G}_{4}^{b}\right|}^{2}/{d}_{4})}^{2}]$ (where ${G}_{4}^{b}\text{=}{G}_{4}+{{G}^{\prime}}_{4}$) for the external-dressed FWM process of ${E}_{F2}$, ${\rho}_{S2}^{(5)}={G}_{S2}/[{d}_{3}{d}_{2}{({d}_{1}+{\left|{G}_{4}^{b}\right|}^{2}/{d}_{4})}^{3}]$ (or ${{\rho}^{\prime}}_{S2}^{(5)}={{G}^{\prime}}_{S2}/[{d}_{3}{d}_{2}{({d}_{1}+{\left|{G}_{4}^{b}\right|}^{2}/{d}_{4})}^{3}]$) for the external-dressed SWM process of ${E}_{S2}$ (or ${{E}^{\prime}}_{S2}$); and ${\rho}_{F4}^{(3)}={G}_{F4}/[{d}_{4}{({d}_{1}+{\left|{G}_{2}^{b}\right|}^{2}/{d}_{2})}^{2}]$ for the external-dressed FWM process ${E}_{F4}$, ${\rho}_{S4}^{(5)}={G}_{S4}/[{d}_{3}{d}_{4}{({d}_{1}+{\left|{G}_{2}^{b}\right|}^{2}/{d}_{2})}^{3}]$ (or ${{\rho}^{\prime}}_{S4}^{(5)}={{G}^{\prime}}_{S4}/[{d}_{3}{d}_{4}{({d}_{1}+{\left|{G}_{2}^{b}\right|}^{2}/{d}_{2})}^{3}]$) for the external-dressed SWM process of ${E}_{S4}$ (or ${{E}^{\prime}}_{S4}$), when not considering the self-dressing effects. The total detected MWM signal (Fig. 7(a)) will be proportional to the mod square of ${{\rho}^{\prime}}_{\text{M}}$, where ${{\rho}^{\prime}}_{\text{M}}=({\rho}_{S2}^{(5)}+{{\rho}^{\prime}}_{S2}^{(5)}+{\rho}_{F2}^{(3)})+({\rho}_{S4}^{(5)}+{{\rho}^{\prime}}_{S4}^{(5)}+{\rho}_{F4}^{(3)})$.

The measured total MWM signal when all the laser beams are turned on is depicted in Fig. 7(a). The global profile of the baselines of each curve, which mainly includes the self-dressed ${E}_{F4}$, ${E}_{S4}$ and ${{E}^{\prime}}_{S4}$ signals, exhibits AT splitting induced by ${E}_{4}$(${{E}^{\prime}}_{4}$). The peak on each profile is mainly composed of the doubly-dressed ${E}_{F2}$, ${E}_{S2}$, ${{E}^{\prime}}_{S2}$ signals and the enhancement and suppression of ${E}_{F4}$, ${E}_{S4}$ and ${{E}^{\prime}}_{S4}$ induced by ${E}_{2}$(${{E}^{\prime}}_{2}$). To understand the interaction of these six generated signals deeply, we divide it into two parts: the interaction of FWM signals ${E}_{F2}$ and ${E}_{F4}$ (shown as Fig. 7(b)), and the interaction of SWM signals ${E}_{S2}$, ${{E}^{\prime}}_{S2}$ and ${E}_{S4}$, ${{E}^{\prime}}_{S4}$. Since ${{E}^{\prime}}_{S2}$ and ${E}_{S2}$ share similar characteristics (so do ${E}_{S4}$ and ${{E}^{\prime}}_{S4}$), the interaction between ${{E}^{\prime}}_{S2}$, ${E}_{S2}$ and ${E}_{S4}$, ${{E}^{\prime}}_{S4}$ can be studied by only investigating the interaction between ${E}_{S2}$ and ${E}_{S4}$ (shown as Fig. 7(c)) [21]. Therefore, by blocking different laser beams and scanning ${\Delta}_{2}$ at discrete ${\Delta}_{1}$ values, the interaction of these six wave-mixing signals can be observed directly, separated into the interplay between two FWM signals, two SWM signals and the interplay between FWM and SWM signals.

First we investigate the interplay between the two FWM signals ${E}_{F2}$ and ${E}_{F4}$ in the Y-type subsystem (Fig. 1(d)) by blocking the laser beams ${E}_{3}$ and ${{E}^{\prime}}_{3}$. The interplay between these two FWM signals will occur when we overlap the two separated EIT windows, as shown in Fig. 7(b). Figure 7(b1) shows the measured FWM signal versus ${\Delta}_{2}$ at discrete ${\Delta}_{1}$ values, which including the information of both ${E}_{F2}$ and ${E}_{F4}$, with mutual dressings. In Fig. 7(b1), the global profile of baselines of all the curves represents the intensity variation of ${E}_{F4}$ at designated probe detuning values, and the peak and dip on each baseline include two components: the doubly-dressed ${E}_{F2}$ signal and the enhancement and suppression of ${E}_{F4}$ induced by ${E}_{2}$(${{E}^{\prime}}_{2}$). These two components could be individually detected by additionally blocking ${{E}^{\prime}}_{2}$ or ${{E}^{\prime}}_{4}$, as shown in Fig. 7(b2) and 7(b3) separately. When blocking ${{E}^{\prime}}_{2}$, the information related to ${E}_{F4}$ could be extracted since ${E}_{F2}$ is turned off (Fig. 7(b2)). The global profile of all the baselines in Fig. 7(b2) reveals AT splitting of ${E}_{F4}$, and the peak and dip of each curve represent the enhancement and suppression of ${E}_{F4}$ induced by ${E}_{2}$, which show similar evolution to the curves in Fig. 4(d2). On the other hand, when turning on ${{E}^{\prime}}_{2}$ and blocking ${{E}^{\prime}}_{4}$, the doubly-dressed ${E}_{F2}$ signal could be obtained in Fig. 7(b3), which is similar to Fig. 4(a2). It is quite obvious that the measured total FWM signal (Fig. 7(b1)) is approximate to the sum of the enhancement and suppression of ${E}_{F4}$ which mainly behaves dips (Fig. 7(b2)), and the dressed FWM signal ${E}_{F2}$, which mainly behave peaks (Fig. 7(b3)).

Next we investigate the interplay between two SWM signals ${E}_{S2}$ and ${E}_{S4}$ in Fig. 7(c) [21], with ${{E}^{\prime}}_{2}$ and ${{E}^{\prime}}_{4}$ blocked (shown as Fig. 1(e)). When all the five laser beams (${E}_{1}$, ${E}_{2}$, ${E}_{4}$, ${E}_{3}$ and ${{E}^{\prime}}_{3}$) are turned on, two external-dressed SWM signals ${E}_{S2}^{D}$ and ${E}_{S4}^{D}$ will form simultaneously, as shown in Fig. 7(c1). The global baseline variation profile shows the intensity variation of the SWM signal ${E}_{S4}^{}$ revealing AT splitting. The peak and dip on each baseline include the SWM signal ${E}_{S2}^{D}$ and ${E}_{S4}^{D}-{E}_{S4}^{}$ which represents the enhancement and suppression of ${E}_{S4}^{}$ caused by ${E}_{2}$. When the beam ${E}_{4}$ is also blocked, only the measured SWM signal ${E}_{S2}^{}$ remains, as shown in Fig. 7(c3). By subtracting the SWM signal ${E}_{S2}^{}$ (Fig. 7(c3)) and the height of each baseline from the total signal (Fig. 7(c1)), the sum of the signals ${E}_{S4}^{D}-{E}_{S4}^{}$ and ${E}_{S2}^{D}-{E}_{S2}^{}$ revealing the pure dressing effect can be obtained, as shown in Fig. 7(c2). Here, ${E}_{S2}^{D}-{E}_{S2}^{}$ expresses the enhancement or suppression of the ${E}_{S2}^{}$ caused by the ${E}_{4}$. On the one hand, we can see from the curve (c2) that when two-photon resonance occurs at ${\Delta}_{1}\text{=}-20\text{MHz}$ and ${\Delta}_{1}\text{=}25\text{MHz}$, the depth of the dip is approximately maximum, meaning that the suppression is most significant. On the other hand, when ${E}_{1}$ resonates with $|{G}_{4}+{G}_{2}+\u3009$ and $|{G}_{4}-{G}_{2}-\u3009$, the generated SWM signals are enhanced as shown by the small peaks.

When the FWM signals and SWM signals coexist in Fig. 7(a) with all seven beams on, the interaction of these generated wave-mixing signals can be obtained. Theoretically, the intensity of the measured total MWM signal in Fig. 7(a) can be described as sum of the FWM signal intensity (Fig. 7(b1)), the SWM signal intensity (Fig. 7(c1)) and the intensity of the SWM signals relate to ${{E}^{\prime}}_{S2}$ and ${{E}^{\prime}}_{S4}$ which is similar to Fig. 7(c1). From the experimental result, one can see that the generated signal with all laser beams tuned on in (a) is approximate to the sum of the FWM intensity in (b1) and the SWM intensity in (c1), but behaves FWM dominant. Because the SWM signals are too weak to be distinguished when compared with the FWM. We also notice that when the FWM signals and SWM signals coexist and interplay with each other, the enhancement and suppression effect of FWM will be weakened by the interaction of six MWM signal.

## 4. Conclusion

In summary, we distinguish seven coexisting multi-wave mixing signals in a K-type five-level atomic system by selectively blocking different laser beams. And the interactions among these MWM signals have been studied by investigating the interaction between two FWM signals, between two SWM signals, and the interaction between FWM and SWM signals. We also report our experimental results on the dressed FWM process by scanning the frequency detuning of the probe field, the self-dressing field and the external-dressing field respectively, proving the corresponding relationship between different scanning methods. Especially, by scanning external-dressing detuning, the enhancements and suppressions of FWM can be detected directly. In addition, we successfully demonstrate the temporal interference between two FWM signals with a femtosecond time scale. Moreover, when ${\Delta}_{3}$ is far away from resonance, we first observe the enhancement and suppression of SWM signal at large detuning, which is moved out of the EIT window through the Doppler frequency shift led by atomic velocity component.

## Acknowledgments

This work was supported by 973 Program (2012CB921804), NSFC (10974151, 61078002, 61078020, 11104214, 61108017, 11104216), NCET (08-0431), RFDP (20110201110006, 20110201120005, 20100201120031).

## References and links

**1. **P. R. Hemmer, D. P. Katz, J. Donoghue, M. Cronin-Golomb, M. S. Shahriar, and P. Kumar, “Efficient low-intensity optical phase conjugation based on coherent population trapping in sodium,” Opt. Lett. **20**(9), 982–984 (1995). [CrossRef] [PubMed]

**2. **Y. Q. Li and M. Xiao, “Enhancement of nondegenerate four-wave mixing based on electromagnetically induced transparency in rubidium atoms,” Opt. Lett. **21**(14), 1064–1066 (1996). [CrossRef] [PubMed]

**3. **M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. **82**(26), 5229–5232 (1999). [CrossRef]

**4. **D. A. Braje, V. Balić, S. Goda, G. Y. Yin, and S. E. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. **93**(18), 183601 (2004). [CrossRef] [PubMed]

**5. **H. Kang, G. Hernandez, and Y. Zhu, “Resonant four-wave mixing with slow light,” Phys. Rev. A **70**(6), 061804 (2004). [CrossRef]

**6. **Z. C. Zuo, J. Sun, X. Liu, Q. Jiang, G. S. Fu, L. A. Wu, and P. M. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+1)-level system with phase-conjugate geometry,” Phys. Rev. Lett. **97**(19), 193904 (2006). [CrossRef] [PubMed]

**7. **H. Ma and C. B. de Araujo, “Interference between third- and fifth-order polarizations in semiconductor doped glasses,” Phys. Rev. Lett. **71**(22), 3649–3652 (1993). [CrossRef] [PubMed]

**8. **D. J. Ulness, J. C. Kirkwood, and A. C. Albrecht, “Competitive events in fifth order time resolved coherent Raman scattering: Direct versus sequential processes,” J. Chem. Phys. **108**(10), 3897–3902 (1998). [CrossRef]

**9. **S. E. Harris, “Electromagnetically induced transparency,” Phys. Today **50**(7), 36 (1997). [CrossRef]

**10. **M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. **77**(2), 633–673 (2005). [CrossRef]

**11. **A. Imamoğlu and S. E. Harris, “Lasers without inversion: interference of dressed lifetime-broadened states,” Opt. Lett. **14**(24), 1344–1346 (1989). [CrossRef] [PubMed]

**12. **L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature **414**(6862), 413–418 (2001). [CrossRef] [PubMed]

**13. **L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature **397**(6720), 594–598 (1999). [CrossRef]

**14. **M. D. Lukin and A. Imamoğlu, “Controlling photons using electromagnetically induced transparency,” Nature **413**(6853), 273–276 (2001). [CrossRef] [PubMed]

**15. **C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature **409**(6819), 490–493 (2001). [CrossRef] [PubMed]

**16. **D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. **86**(5), 783–786 (2001). [CrossRef] [PubMed]

**17. **C. B. Li, H. B. Zheng, Y. P. Zhang, Z. Q. Nie, J. P. Song, and M. Xiao, “Observation of enhancement and suppression in four-wave mixing processes,” Appl. Phys. Lett. **95**(4), 041103 (2009). [CrossRef]

**18. **M. Yan, E. G. Rickey, and Y. F. Zhu, “Observation of doubly dressed states in cold atoms,” Phys. Rev. A **64**(1), 013412 (2001). [CrossRef]

**19. **M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A **60**(4), 3225–3228 (1999). [CrossRef]

**20. **B. Anderson, Y. P. Zhang, U. Khadka, and M. Xiao, “Spatial interference between four- and six-wave mixing signals,” Opt. Lett. **33**(18), 2029–2031 (2008). [CrossRef] [PubMed]

**21. **Z. Wang, Y. Zhang, H. Chen, Z. Wu, Y. Fu, and H. Zheng, “Enhancement and suppression of two coexisting six-wave-mixing processes,” Phys. Rev. A **84**(1), 013804 (2011). [CrossRef]