1. Introduction

Efficient four-wave mixing (FWM) processes enhanced by atomic coherence in multilevel atomic systems [1–4] are of great current interest. Recently, destructive and constructive interferences in a two-level atomic system [5] and competition via atomic coherence in a four-level atomic system [6] with two coexisting FWM processes were studied. Also, the interactions of doubly dressed states and the corresponding effects of atomic systems have attracted many researchers in recent years [7,8]. The interaction of double-dark state and splitting of a dark state in a four-level atomic system were studied theoretically in an electromagnetically induced transparency (EIT) system by Lukin et al. [7]. The triple-peak absorption spectrum, which was observed later in the N-type cold atomic system by Zhu et al., verified the existence of the secondarily dressed states [8]. Recently, we had theoretically investigated three types of doubly dressed schemes in a five-level atomic system [9] and observed three-peak Autler-Townes (AT) splitting of the secondary dressing FWM signal [10]. In addition, we reported the evolution of suppression and enhancement of FWM signal by controlling an additional laser field [11].

As two or more laser beams pass through an atomic medium, the cross-phase modulation (XPM), as well as modified self-phase modulation (SPM), can potentially affect the propagation and spatial patterns of the incident laser beams. Laser beam self-focusing [12] and pattern formation [13] have been extensively investigated with two laser beams propagating in atomic vapors. Recently, we have observed spatial shift [14] and spatial splitting [15–17] of the FWM beams generated in multi-level atomic systems, which can be well controlled by additional dressing laser beams via XPM. Studies on such spatial shift and splitting of the laser beams can be very useful in understanding the formation and interactions of spatial solitons [16] in the Kerr nonlinear systems and signal processing applications, such as spatial image storage [18], entangled spatial images [19], soliton pair generation [20], and influences of higher-order (such as fifth-order) nonlinearities [21].

In this paper, we first report our experimental studies of the interaction of four coexisting FWM processes in a two-level atomic system by blocking different laser beams. Next, we investigate the various suppression/enhancement of the degenerate-FWM (DFWM) signals and two dispersion centers, which are caused by the cascade dressing interaction of two dressing fields. The experimental results clearly show the evolutions of the enhancement and suppression, from pure enhancement to partial enhancement/suppression, then to pure suppression, further to partial enhancement/suppression, and finally to enhancement, which are in good agreement with the theoretical calculations. In addition, we also observe the spatial splitting in the x and y directions of DFWM signal due to different spatially alignment of the probe and coupling beams.

2. Theoretical model and experimental scheme

The two relevant experimental systems are shown in Figs. 1(a) and 1(b). Three energy levels from sodium atom in heat pipe oven are involved in the experimental schemes. The pulse laser beams are aligned spatially as shown in Fig. 1(c). In the Fig. 1(a), energy levels |0〉 ($3S_{1/2}$) and |1〉 ($3P_{3/2}$) form a two-level atomic system. Coupling field ${\mathit{E}}_1$ (with wave vector ${\mathbf{k}}_1$ and the Rabi frequency $G_1$) together with ${\mathit{E}}_1^{\prime }$ (${\mathbf{k}}_1^{\prime }$ and $G_1^{\prime }$) (connecting the transition between |0〉 and |1〉) having a small angle ($\sim {0.3}^{\circ }$) propagates in the opposite direction of the probe field ${\mathit{E}}_3$ (${\mathbf{k}}_3$ and $G_3$) (also connecting the transition between |0〉 and |1〉). These three laser beams come from the same near-transform-limited dye laser (with a 10 Hz repetition rate, 5 ns pulse width, and $0.04$ ${\text{cm}}^{-1}$ linewidth) with the same frequency detuning ${\Delta }_1={\omega }_{10}-{\omega }_1$, where ${\omega }_{10}$ is the transition frequency between |0〉 and |1〉. The coupling fields ${\mathit{E}}_1$ and ${\mathit{E}}_1^{\prime }$ induce a population grating between states |0〉 to |1〉, which is probed by ${\mathit{E}}_3$. This generates a DFWM process (Fig. 1(a)) satisfying the phase-matching condition of ${\mathbf{k}}_{\text{F}1}={\mathbf{k}}_1-{\mathbf{k}}_1^{\prime }+{\mathbf{k}}_3$. Then, two additional coupling fields ${\mathit{E}}_2$ (${\mathbf{k}}_2$, $G_2$) and ${\mathit{E}}_2^{\prime }$ (${\mathbf{k}}_2^{\prime }$, $G_2^{\prime }$) are applied as scanning fields connecting the transition from |0〉 to |1〉 with the same frequency detuning ${\Delta }_2={\omega }_{10}-{\omega }_2$; the two additional coupling fields are from another similar dye laser set at ${\omega }_2$ to dress the energy level |1〉. The fields ${\mathit{E}}_2$, ${\mathit{E}}_2^{\prime }$, and ${\mathit{E}}_3$ produce a non-degenerate FWM (NDFWM) signal ${\mathbf{k}}_{\text{F}2}$ (satisfying ${\mathbf{k}}_{\text{F}2}={\mathbf{k}}_2-{\mathbf{k}}_2^{\prime }+{\mathbf{k}}_3$). When the five laser beams are all on, there also exist other two FWM processes ${\mathbf{k}}_{\text{F}3}$ (satisfying ${\mathbf{k}}_{\text{F}3}={\mathbf{k}}_2-{\mathbf{k}}_1^{\prime }+{\mathbf{k}}_3$) and ${\mathbf{k}}_{\text{F}4}$ (satisfying ${\mathbf{k}}_{\text{F}4}={\mathbf{k}}_1-{\mathbf{k}}_2^{\prime }+{\mathbf{k}}_3$) in the same directions as ${\mathit{E}}_{\text{F}1}$ and ${\mathit{E}}_{\text{F2}}$, respectively.

 

Fig. 1 (a) and (b) The diagram of relevant Na energy levels. (c) The scheme of the experiment. Inset gives the spatial alignments of the incident beams.

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Under the experimental condition, ${\mathit{E}}_1$ (or ${\mathit{E}}_1^{\prime }$) with detuning ${\Delta }_1$ depletes two groups of atoms with different velocities at the same time, such as negative velocities group and positive velocities group. At ${\Delta }_1<0$, the positive velocities group will see ${\mathit{E}}_1$ (or ${\mathit{E}}_1^{\prime }$) with detuning ${\Delta }_1+k_{1}v$ and ${\mathit{E}}_3$ with detuning ${\Delta }_1-k_{3}v$. The frequency of the DFWM ${\mathit{E}}_{\text{F}1}$in this case will be ${\omega }_f=\left({\omega }_1-kv\right)-\left({\omega }_1-kv\right)+\left({\omega }_1+k_{3}v\right)={\omega }_1+k_{3}v$ due to the conservation of energy. Correspondingly, at ${\Delta }_1>0$, negative velocities group will see ${\mathit{E}}_1$ (or ${\mathit{E}}_1^{\prime }$) with detuning ${\Delta }_1-k_{1}v$ and ${\mathit{E}}_3$ with detuning ${\Delta }_1+k_{3}v$. The frequency of ${\mathit{E}}_{\text{F}1}$ will be ${\omega }_f=\left({\omega }_1+kv\right)-\left({\omega }_1+kv\right)+\left({\omega }_1-k_{3}v\right)={\omega }_1-k_{3}v$. Such changing implies that a group of atoms with certain velocities can satisfy the condition ${{\Delta }^{\prime }}_1={\Delta }_1\pm k_{1}v$, where ${{\Delta }^{\prime }}_1$ is the detuning of ${\mathit{E}}_1$ (or ${\mathit{E}}_1^{\prime }$) based on both saturation excitation and atomic coherence effect. As a result, the self-dressing field ${\mathit{E}}_1$ (or ${\mathit{E}}_1^{\prime }$) can be considered as the outer dressing field and separates the level |0〉 into two dressing states $|G_1\pm >$, as shown in Fig. 2 . In addition, the Doppler effect and the power broadening effect on the weak FWM signals need to be considered.

 

Fig. 2 (Color online) The interplay and mutual suppression/enhancement between two coexisting FWM signals (${\mathit{E}}_{F1}$ and ${\mathit{E}}_{F3}$). (a1) the upper curves: pure DFWM signal ${\mathit{E}}_{F1}$ (with both ${\mathit{E}}_2$and ${\mathit{E}}_2^{\prime }$ blocked) (squares), singly dressed DFWM signal ${\mathit{E}}_{F1}$ (with ${\mathit{E}}_2$ blocked) (triangles), coexisting singly dressed DFWM signal ${\mathit{E}}_{F1}$ and FWM signal ${\mathit{E}}_{F3}$ (with ${\mathit{E}}_2^{\prime }$ blocked) (reverse triangles), and coexisting dressed DFWM signal ${\mathit{E}}_{F1}$ and FWM signal ${\mathit{E}}_{F3}$ (circles);lower curves: pure FWM signal ${\mathit{E}}_{F3}$ (with both ${\mathit{E}}_1$and ${\mathit{E}}_2^{\prime }$ blocked) (left triangle), singly-dressed FWM signal (with ${\mathit{E}}_1$ blocked) (right triangle); ${\Delta }_1=0$. The inserted plot: corresponding to the dressed-state picture. (b1) the condition are the same to that in (a1) except ${\Delta }_1=-25.9\text{\hspace{0.17em} GHz}$. The inserted plot: corresponding to the dressed-state picture. (a2) and (b2) theoretical plots corresponding to the experimental parameters of (a1) and (b1), respectively.

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When ${\mathit{E}}_1$, ${\mathit{E}}_1^{\prime }$, ${\mathit{E}}_2$, ${\mathit{E}}_2^{\prime }$ and ${\mathit{E}}_3$ are open, the DFWM process ${\mathit{E}}_{F1}$and NDFWM processes ${\mathit{E}}_{F\text{2}}$, ${\mathit{E}}_{F3}$ and ${\mathit{E}}_{F4}$ are generated simultaneously, and there exists interplay among these four FWM signals in the two-level atomic system. These generated FWM signals have the frequencies ${\omega }_{F1}={\omega }_{F2}={\omega }_1$, ${\omega }_{F3}={\omega }_2$, and ${\omega }_{F4}=2{\omega }_1-{\omega }_2$. They are split into two equal components by a $50\%$ beam splitter before being detected. One is captured by the CCD camera, and the other is detected by photomultiplier tubes ($D_1$ or $D_2$) and a fast gated integrator (gate width of 50 ns). Also, they are monitored by digital acquisition card.

In order to interpret the following experimental results, we perform the theoretical calculation on the four coexisting FWM processes. First, we consider four FWM processes to be or perturbed by corresponding laser beams. In two-level configuration, there exist the transition paths to generate FWM signals. They can be described by perturbation chains (F1) ${\rho }_{00}^{(0)}\stackrel{({\mathit{E}}_1)*}{\to }{\rho }_{10}^{(1)}\stackrel{({\mathit{E}}_1\text{'})*}{\to }{\rho }_{00}^{(2)}\stackrel{{\mathit{E}}_3}{\to }{\rho }_{10}^{(3)}$, (F2) ${\rho }_{00}^{(0)}\stackrel{({\mathit{E}}_2\text{'})*}{\to }{\rho }_{10}^{(1)}\stackrel{{\mathit{E}}_2}{\to }{\rho }_{00}^{(2)}$ $\stackrel{{\mathit{E}}_3}{\to }{\rho }_{10}^{(3)}$, (F3) ${\rho }_{00}^{(0)}\stackrel{{\mathit{E}}_2}{\to }{\rho }_{10}^{(1)}\stackrel{({\mathit{E}}_1\text{'})*}{\to }{\rho }_{00}^{(2)}\stackrel{{\mathit{E}}_3}{\to }{\rho }_{10}^{(3)}$, and (F4) ${\rho }_{00}^{(0)}\stackrel{{\mathit{E}}_1}{\to }{\rho }_{10}^{(1)}$ $\stackrel{({\mathit{E}}_2\text{'})*}{\to }{\rho }_{00}^{(2)}\stackrel{{\mathit{E}}_3}{\to }{\rho }_{10}^{(3)}$, respectively. For the DFWM signal ${\mathit{E}}_{\text{F}1}$, in fact, this DFWM generation process can be viewed as a series of transitions: the first step is from |0〉 to |1〉 with absorption of a coupling photon ${\mathit{E}}_1$, and the final state of this process can be dressed by the dressing field ${\mathit{E}}_2$ (or ${\mathit{E}}_2^{\prime }$). The second step is the transition from |1〉 to |0〉, and the final state cannot be dressed by any field. The third step is the transition from |0〉 to |1〉 with the emission of a probe photon ${\mathit{E}}_3$, and the final state of this process can be dressed by ${\mathit{E}}_2$ (or ${\mathit{E}}_2^{\prime }$). Then, the last transition is from |1〉 to |0〉, which emits a FWM photon at frequency ${\omega }_1$. Thus, we can obtain the dressed perturbation chain ${\rho }_{00}^{(0)}\stackrel{({\mathit{E}}_1)*}{\to }{\rho }_{G2\pm 0}^{(1)}\stackrel{({\mathit{E}}_1\text{'})*}{\to }$ ${\rho }_{00}^{(2)}\stackrel{{\mathit{E}}_3}{\to }{\rho }_{G2\pm 0}^{(3)}$. Similarly, we can obtain the other dressed perturbation chains as (DF2) ${\rho }_{00}^{(0)}\stackrel{{\mathit{E}}_2}{\to }$ ${\rho }_{G1\pm 0}^{(1)}\stackrel{({\mathit{E}}_2\text{'})*}{\to }{\rho }_{00}^{(2)}\stackrel{{\mathit{E}}_3}{\to }{\rho }_{G1\pm 0}^{(3)}$, (DF3) ${\rho }_{00}^{(0)}\stackrel{{\mathit{E}}_2}{\to }{\rho }_{G2\pm 0}^{(1)}\stackrel{({\mathit{E}}_1\text{'})*}{\to }{\rho }_{00}^{(2)}\stackrel{{\mathit{E}}_3}{\to }{\rho }_{G2\pm 0}^{(3)}$, and (DF4) ${\rho }_{00}^{(0)}\stackrel{{\mathit{E}}_1}{\to }$ ${\rho }_{G1\pm 0}^{(1)}\stackrel{({\mathit{E}}_2\text{'})*}{\to }$ ${\rho }_{00}^{(2)}\stackrel{{\mathit{E}}_3}{\to }{\rho }_{G1\pm 0}^{(3)}$, respectively. The expressions of the corresponding density matrix elements related to the four FWM processes are ${\rho }_{F1}^{(3)}=-i{G}_3{G}_1{(G_1^{\prime })}^{*}/({\Gamma }_{00}{B}_1^2)$, ${\rho }_{F2}^{(3)}=-i{G}_3{G}_2{(G_2^{\prime })}^{*}/({\Gamma }_{00}{d}_3{B}_2)$, ${\rho }_{F3}^{(3)}=-i{G}_2{G}_3{(G_1^{\prime })}^{*}/(d_4{B}_3^2)$, ${\rho }_{F4}^{(3)}=-i{G}_1{G}_3{(G_2^{\prime })}^{*}/(d_5{d}_6{B}_1)$, respectively, where$d_1={\Gamma }_{10}+i{\Delta }_1$, $d_2={\Gamma }_{00}+i({\Delta }_1/m-{\Delta }_2)$, $d_3={\Gamma }_{10}+i{\Delta }_2$, $d_4={\Gamma }_{00}+i({\Delta }_2-{\Delta }_1)$, $d_5={\Gamma }_{10}+i(2{\Delta }_1-{\Delta }_2)$, $d_6=[{\Gamma }_{00}+i({\Delta }_1-{\Delta }_2)]$, $A_1=|G_2{|}^2/d_2$, $A_2=G_1^2/{\Gamma }_{00}$, $A_3=|G_2{|}^2/{\Gamma }_{00}$, $B_1=d_1+A_1$, $B_2=d_1+A_2$, $B_3=d_3+A_3$. Here $G_i=-{\mu }_i{\mathit{E}}_i/\hslash$ (i = 1, 2, 3) is the Rabi frequency; ${\Gamma }_{10}$, ${\Gamma }_{20}$, and ${\Gamma }_{00}$ are the transverse relaxation rates, and ${\Delta }_i$ ($i=1,2$) is the detuning factor.

The experiments are carried out in a vapor cell containing sodium. The cell, 18-cm long, is heated up to a temperature of about ${230}^{\circ }\text{C}$ and crossed by linearly polarized laser beams which interact with the atoms. In the two-level atomic system, the coupling fields ${\mathit{E}}_1$ and ${\mathit{E}}_1^{\prime }$ (with diameter of 0.8 mm and power of 9 μW), and the probe field ${\mathit{E}}_3$ (with diameter of 0.8 mm and power of 3 μW) are tuned to the line center (589.0 nm) of the $|0〉$ to $|1〉$ transition, which generate the DFWM signal ${\mathit{E}}_{\text{F}1}$ at frequency ${\omega }_1$. The coupling fields ${\mathit{E}}_2$ and ${\mathit{E}}_2^{\prime }$ (with diameter of 1.1 mm and powers of 20 μW and 100 μW, respectively) are scanned simultaneously around the $|0〉$ to $|1〉$ transition to dress the DFWM process ${\mathit{E}}_{F1}$.

3. Cascade dressing interaction

We first investigate the interaction of four coexisting FWM signals in the two-level atomic system by blocking different laser beams. Firstly, by blocking ${\mathit{E}}_2$ (or ${\mathit{E}}_1$), the DFWM signal ${\mathit{E}}_{\text{F}1}$ (or the FWM signal ${\mathit{E}}_{F3}$) is suppressed by the coupling field ${\mathit{E}}_2^{\prime }$ as can be seen from the upper triangle points [or the right triangle points in Fig. 2(a1)], compared to the pure DFWM signal ${\mathit{E}}_{\text{F}1}{}^{}$ (or the FWM signal ${\mathit{E}}_{F3}$). Next, when laser beams ${\mathit{E}}_1$, ${\mathit{E}}_1^{\prime }$, ${\mathit{E}}_2$ and ${\mathit{E}}_3$ are turned on, two coexisting FWM processes (${\mathit{E}}_{\text{F}1}$ and ${\mathit{E}}_{F3}$) couple to each other (the lower triangle points), and the intensities of total FWM signals are increased, as can be attributed to the combination of two FWM signal processes (${\mathit{E}}_{\text{F}1}$ and ${\mathit{E}}_{F3}$). Finally, when all the five laser beams are turned on, the DFWM signal ${\mathit{E}}_{\text{F}1}$ and the FWM signal ${\mathit{E}}_{F3}$ are both greatly suppressed by corresponding dressing fields. So the intensities of total FWM signals are extremely decreased, as shown in the circles points in Fig. 2(a1).

These effects can be explained effectively by the dressed-state picture. The dressing field ${\mathit{E}}_2^{\prime }$ couples the transition $|0>$ to $|1>$ and creates the dressed states $|G_2\pm >$, which leads to single-photon transition $|0>\to |1>$ off-resonance [the inserted plot in Fig. 2(a1)]. At exact single-photon resonance with ${\Delta }_1=0$, the DFWM signal ${\mathit{E}}_{\text{F}1}$ intensity is greatly suppressed by the means of scanning the dressing field ${\mathit{E}}_2^{\prime }$ across the resonance (${\Delta }_2=0$), as the upper triangle points in Fig. 2(a1) shows. At the same time, the FWM signal ${\mathit{E}}_{F3}$ experiences similar process [the right triangle points in Fig. 2(a1)].

Furthermore, an appropriate ${\Delta }_1$ value at which ${\mathit{E}}_{\text{F}1}$ is either enhanced or suppressed is chosen in the investigation. In this case, compared to the pure DFWM signal ${\mathit{E}}_{\text{F}1}$ [square points in Fig. 2(b1)], the dressed DFWM signal ${\mathit{E}}_{\text{F}1}$ is enhanced [the upper and low triangle points in Fig. 2(b1)]. However, the dressed FWM signal ${\mathit{E}}_{F3}$ is suppressed due to the destructive interference [right triangle points in Fig. 2(b1)] compared to the pure signal [left triangle points in Fig. 2(b1)]. The upper triangle in Fig. 2(b1) combines the two FWM processes (${\mathit{E}}_{\text{F}1}$ and ${\mathit{E}}_{F3}$), which are dressed by laser beams ${\mathit{E}}_1{}^{\prime }$ and ${\mathit{E}}_2{}^{\prime }$, respectively. After calculating ${\rho }_{F1}^{(3)}$ and ${\rho }_{F3}^{(3)}$ under the above experiment conditions, good agreements are obtained between the theoretical calculations and the experimental results as shown in Figs. 2(a2) and 2(b2), respectively.

After that, we investigate the evolutions of the interaction between these two coexisting FWM signals by the means of setting different frequency detuning ${\Delta }_1$ values, where the fixed spectra corresponds to the suppression and enhancement of DFWM signal ${\mathit{E}}_{\text{F}1}$, and the shifting spectra corresponds to the FWM signal ${\mathit{E}}_{F3}$ as shown in Figs. 3 (a1)–3(a7). It is obvious in Figs. 3(a1)–3(a3) that, as the frequency detuning ${\Delta }_1$ varies from ${\Delta }_1<0$ to zero from up to down, the DFWM signal ${\mathit{E}}_{\text{F}1}$ shows the evolution from enhancement to partial enhancement/suppression, and then to suppression. At the same time, the FWM signal ${\mathit{E}}_{F3}$ varies from intense to weak (when two FWM signals ${\mathit{E}}_{\text{F}1}$ and ${\mathit{E}}_{F3}$ overlap) and shifts from left side to right side, which satisfies the two-photon resonant condition (${\Delta }_1-{\Delta }_2=0$). When ${\Delta }_1$ changes further to be positive, a symmetric process is observed [i.e., suppression in Fig. 3(a5), partial suppression/enhancement in Fig. 3(a6), and pure enhancement in Fig. 3(a7)]. It should be noted here that FWM signal ${\mathit{E}}_{F3}$ still shifts from left side to right side. Figure 3(a4) shows the weakened FWM signal due to the strong effect of the Doppler absorption. Especially, the DFWM signal ${\mathit{E}}_{\text{F}1}$ at a large one-photon detuning is extremely weak when $G_2=0$. However, the strong dressing field can cause the resonant excitation of one of the dressed states if the enhanced condition ${\Delta }_1-{\Delta }_2\pm G_2=0$ is satisfied. In such case, the DFWM signal ${\mathit{E}}_{\text{F}1}$ is strongly enhanced [Fig. 3(a1)], mainly due to the one-photon resonance ($|0>\to |G_2+|$) [the insert plot in Fig. 2(b1)]. As ${\Delta }_1=0$, the intensity of the DFWM signal ${\mathit{E}}_{\text{F}1}$ is greatly suppressed [Fig. 3(a3)], similar to the case of the upper triangle curves in Fig. 2(a1). Also, we can observe the FWM signal ${\mathit{E}}_{F3}$ is suppressed due to the destructive interference. In addition, Figs. 3(b1)–3(b7) show the interaction of another two coexisting the FWM processes (${\mathit{E}}_{F2}$ and ${\mathit{E}}_{F4}$), in which the fixed spectra corresponds to the FWM signal ${\mathit{E}}_{F2}$, and the shifting spectra corresponds to FWM signal ${\mathit{E}}_{F4}$ for different frequency detuning ${\Delta }_1$ values.

 

Fig. 3 (a) and (b) Measured evolution of the four FWM signals [(${\mathit{E}}_{F1}$and ${\mathit{E}}_{F3}$) and (${\mathit{E}}_{F2}$ and ${\mathit{E}}_{F4}$), respectively] versus ${\Delta }_2$ for different ${\Delta }_1$ values. (a1)-(a7) and (b1)-(b7): ${\Delta }_1=$-139.1, −103.87, −29.5, 0, 29.8, 100.1, 155.7GHz, respectively.

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Now, we concentrate on the cascade dressing interaction and the two dispersion centers of FWM images with two dressing fields ${\mathit{E}}_1^{\prime }$ and ${\mathit{E}}_2^{\prime }$ in the two-level atomic system. In order to investigate the cascade dressing interaction, the power of the coupling field ${\mathit{E}}_1^{\prime }$ is set at $80\mu \text{W}$. So the DFWM signal ${\mathit{E}}_{\text{F}1}$ shows a spectrum of the AT splitting due to self-dressed effect [10] induced by beam ${\mathbf{k}}_1^{\prime }$ when ${\Delta }_1$ is scanned and the dressing field ${\mathit{E}}_2^{\prime }$ is off, as shown in the dashed curve of Fig. 4(a) . When the beam ${\mathbf{k}}_2^{\prime }$ is on, the DFWM signal ${\mathit{E}}_{\text{F}1}$ is dressed by both ${\mathit{E}}_1^{\prime }$ and ${\mathit{E}}_2^{\prime }$, and therefore shows the cascade dressing interaction, as shown in Figs. 4(a) and 4(c). Specifically, by discretely choosing different detuning values within ${\Delta }_1<0$ and scanning ${\Delta }_2$, the DFWM signal ${\mathit{E}}_{\text{F}1}$shows the evolution of the successively occurring pure enhancement, partial suppression/enhancement, pure suppression, partial enhancement/suppression and enhancement processes, as shown in the left side of Fig. 4(a). When ${\Delta }_1$ changes to be positive, a symmetric process occurs, in the right side of Fig. 4(a), which is well described by the theoretical curves [Fig. 4(b)].

 

Fig. 4 (Color online) (a) Measured suppression and enhancement of DFWM signal ${\mathit{E}}_{F1}$ versus ${\Delta }_2$ for different ${\Delta }_1$values in the two-level system. ${\Delta }_1=$-69.1, −55.5, −38.7, −19.2, 0, 14.7, 28.8, 42.2 and 57.3 GHz, respectively. The dashed curve is the double-peak DFWM signal ${\mathit{E}}_{F1}$ versus ${\Delta }_1$. (b) Theoretical plots corresponding to the experimental parameters in (a). (c) The same measures to (a) with the same condition, except that the laser beams ${\mathit{E}}_1$, ${\mathit{E}}_1^{\prime }$ overlap in the middle of heat oven. (d1)-(d9) the dressed-state pictures of the suppression or enhancement of the DFWM signal. The states $|G_1\pm >$ (dashed lines) and the states $|G_1+\pm >$ or $|G_1-\pm >$ (dot-dashed lines), respectively.

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In order to explain this phenomenon, the dressed-state picture is adopted, as shown in Fig. 4(d). First, the DFWM signal ${\mathit{E}}_{\text{F}1}$is dressed by both fields ${\mathit{E}}_2^{\prime }$ and ${\mathit{E}}_1^{\prime }$. The corresponding expression of the modified density matrix element of DFWM ${\mathit{E}}_{\text{F}1}$ process is ${{\rho }^{\prime }}_{{\text{E}}_{F1}}^{(3)}=-i{G}_3{G}_1{(G_1^{\prime })}^{*}/(B_4{B}_5^2)$, where $d_6={\Gamma }_{00}+A_5$, $d_7={\Gamma }_{10}-i{\Delta }_2$, $A_4=G_1^2/B_1$, $A_5=G_2^2/d_7$, $A_6=G_1^2/d_6$, $B_4={\Gamma }_{00}+A_4$ and $B_5=B_1+A_6$. Next, the inner dressing field ${\mathit{E}}_1^{\prime }$ dresses the state $|0>$ to create two new dressing states $|G_1\pm >$, and then the strong dressing field ${\mathit{E}}_2^{\prime }$ creates new states $|G_1+\pm >$ or $|G_1-\pm >$ around states $|G_1\pm >$, as scanning the frequency detuning ${\Delta }_2^{}$. As a result of this dressing scheme, the DFWM signal ${\mathit{E}}_{F1}$ is extremely small when $G_2=0$ and ${\Delta }_1$ is set far away from the resonance point at both ${\Delta }_1<0$ and ${\Delta }_1>0$, respectively. Another result is that the strong fields can cause resonant excitation of one of the dressed state (i.e., $|G_1-->$ or $|G_1++>$ [Fig. 4(d1) and Fig. 4(d9), respectively], which can lead to the enhancement of FWM signals. Specifically, if the condition ${\Delta }_1-{\Delta }_2=-G_2^{}$ and ${\Delta }_1-{\Delta }_2=G_2^{}$ [corresponding to the dressed states shown in Figs. 4(d1) and 4(d9)] is satisfied, the DFWM signal ${\mathit{E}}_{F1}$ is obviously enhanced, as shown in the curves of Fig. 4(a1) and 4(a9), respectively. As ${\Delta }_1$ changes to be near the resonance point, we can get a partial enhancement/suppression of DFWM signal ${\mathit{E}}_{F1}$. The first and second transition states satisfy suppression condition ${\Delta }_1-{\Delta }_2=0$ and enhancement condition ${\Delta }_1-{\Delta }_2=-G_2^{}$ (Fig. 4(d2)), [enhancement condition ${\Delta }_1-{\Delta }_2=G_2^{}$ and suppression condition ${\Delta }_1-{\Delta }_2=0$, as shown in Fig. 4(d4)], as leads to the first suppression and next enhancement (the curve of Fig. 4(a2)) [or the first enhancement and next suppression, as shown in the curve of Fig. 4(a4)]. When ${\Delta }_1$ reaches the point ${\Delta }_1-{\Delta }_2=0$, the suppression effect gets dominant due to the dressed states $|G_1-->$ [Fig. 4(d3)], so the DFWM signal ${\mathit{E}}_{F1}$ is purely suppressed, as shown in the curve of Fig. 4(a3). For the point ${\Delta }_1^{}={\Delta }_2^{}=0$ between the two resonance points, the curve of Fig. 4(a5) shows a pair of suppressed peaks. In fact, they are induced by the outer dressing field ${\mathit{E}}_2^{\prime }$ which can largely weaken the suppression effect of the inner dressing field ${\mathit{E}}_1^{\prime }$ on DFWM signal. Furthermore, the other cascade dressing field ${\mathit{E}}_2^{\prime }$ splits such suppressed peak into a pair of suppressed peaks, as shown in the curve of Fig. 4(a5). Figure 4(a) shows the various suppression/enhancement of the DFWM signal ${\mathit{E}}_{F1}$ and its two dispersion centers, which is caused by the cascade dressing interaction of the two dressing fields ${\mathit{E}}_1$ and ${\mathit{E}}_2^{\prime }$.

In addition, the spatial splitting in x-direction of the FWM signal beams induced by additional dressing laser beams is observed simultaneously as shown in Fig. 5(a) . It is observed that, the number of the splitting spots increases when the FWM intensity is suppressed. To understand these phenomena, we need to consider the cross-phase modulation (XPM) on the FWM signals. As described in our previous investigation [17], the spatial splitting of the FWM beam can be controlled by the intensities of the involved laser beams, the cross-Kerr nonlinear coefficients and the atomic density, according to the nonlinear phase shift $\varphi =2k_{\text{F}1}{n}_{2}z{I}_1{e}^{-r^2/2}/(n_0{I}_{F1}^{\prime })$. Here the additional transverse propagation wave-vector is $\delta k_r=\partial \varphi /\partial r$. The change of phase (ϕ) distribution in the laser propagating equations determines the spatial splitting of the laser beams. In theoretical calculation, we can obtain the intensity of the ${\mathit{E}}_{\text{F}1}$ beam by $I_{F1}^{\prime }\propto |{{\rho }^{\prime }}_{{\text{E}}_{F1}}^{(3)}{|}^2=|-i{G}_3{F}_a^{}{|}^2$ [17], with $F_a=G_1{(G_1^{\prime })}^{*}/(B_4{B}_5^2)$ and the nonlinear cross-Kerr refractive index $n_2^{}\propto \mathrm{Re}(-i\mu G_{F1}{F}_a^2/h)$. When ${\Delta }_2$ is scanned in the experiment, the intensity $I_1$ of the laser beam ${\mathit{E}}_1^{\prime }$ and $n_2$ almost stays constant for different detuning ${\Delta }_2$. So ϕ is primarily determined by $I_{F1}^{\prime }$. When the suppression condition (${\Delta }_1-{\Delta }_2=0$) is satisfied, and the intensity of FWM signal $I_{F1}^{\prime }$ reaches its minimum, the spatial splitting will become stronger as shown in Fig. 5(a) (the suppression positions located at ${\Delta }_2\approx -12\text{GHz}$). While in the enhancement condition with the larger $I_{F1}^{\prime }$, ϕ is decreased, and therefore the splitting is weakened correspondingly, as shown in Fig. 5(a), where the enhancement condition is located at ${\Delta }_2\approx -28.3\text{GHz}$.

 

Fig. 5 (Color online) (a) DFWM signal ${\mathit{E}}_{F1}$ images when ${\Delta }_1=$-69.1, −55.5, −38.7GHz. (b) DFWM signal images when ${\Delta }_1=$-55.5, −38.7, −19.2GHz.

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Specially, we observed the y-direction spatial splitting images of the DFWM signal ${\mathit{E}}_{\text{F}1}$ [Fig. 5(b)] by carefully arranging laser beams ${\mathbf{k}}_1$ and${\mathbf{k}}_1^{\prime }$. In the experiment, the beams ${\mathit{E}}_1$ and ${\mathit{E}}_1^{\prime }$ are deliberately aligned in y-z plane with an angle θ ($\sim {0.05}^{\circ }$) to induce a grating in the same plane with the fringe spacing $\Lambda ={\lambda }_1/\theta$. Because θ is far less than the angle of ${\mathit{E}}_1$ and ${\mathit{E}}_1^{\prime }$ in the x-z plane, Λ is big enough for observing the splitting caused by the induced grating. Furthermore when ${\mathit{E}}_1$and ${\mathit{E}}_1^{\prime }$ are set in the middle of the oven, ${\mathit{E}}_{\text{F}1}$and ${\mathit{E}}_1^{\prime }$ overlap in y-direction due to the phase matching condition. As a result, the splitting of ${\mathit{E}}_{\text{F}1}$ in x-direction due to the nonlinear cross-Kerr effect from ${\mathit{E}}_1^{\prime }$ disappears simultaneously. Because Λ remains nearly the same for the changeless θ and ${\lambda }_1$, a larger spot of the ${\mathit{E}}_{\text{F}1}$ beam with larger intensity will be split to more parts. In Fig. 5(b), the field ${\mathit{E}}_3$ is stronger than that in Fig. 5(a), which leads to stronger FWM signals passing through the grating in y-direction. So we can easily obtain the splitting in y-direction. Moreover, in the enhanced position, the profile of the FWM signal become larger, and more split parts induced by the grating can be obtained. Here, Figs. 5(b1)-5(b3) show the experimental spots corresponding to the curves in Figs. 4(c1)-4(c3). However, the effects of suppression and enhancement of the DFWM signal ${\mathit{E}}_{\text{F}1}$ are much worse due to the special spatial alignment of the laser beams, as shown in Fig. 4(c) compared to that in Fig. 4(a). In Fig. 4(a), the${\mathit{E}}_1$, ${\mathit{E}}_1^{\prime }$ and ${\mathit{E}}_2$, ${\mathit{E}}_2^{\prime }$ are all set at the back of the heat oven. But in Fig. 4(c), only ${\mathit{E}}_1$and ${\mathit{E}}_1^{\prime }$ are deliberately moved to the middle of the oven to demonstrate the splitting of ${\mathit{E}}_{\text{F}1}$ in the y-direction. So the dressing effect on ${\mathit{E}}_{\text{F}1}$ by ${\mathit{E}}_2^{\prime }$ in Fig. 4(c) appears worse than that in Fig. 4(a).

In order to verify the cascade dressing interaction and two dispersion centers of FWM image. The dresses field ${\mathit{E}}_2^{\prime }$ is tuned to the line center (568.8 nm) of the $|1>$ to $|2>$ ($4D_{3/2,5/2}$) transition, and a ladder type three-level atomic system forms, as shown in Fig. 1(b). With the dressed perturbation chains, we can obtain $I_{{\text{E}}_{F1}}^{″}\propto |{\rho }_{{\text{E}}_{F1}}^{″(3)}{|}^2$, where ${\rho }_{{\text{E}}_{F1}}^{″(3)}=-i{G}_3{G}_1{(G_1^{\prime })}^{*}/(B_6{B}_7^2)$, with $d_8={\Gamma }_{00}+i({\Delta }_1+{\Delta }_2)$, $d_9={\Gamma }_{21}+i{\Delta }_2$,$d_{10}={\Gamma }_{11}+A_9$, $d_{11}=d_1+A_7$, $A_7=G_2^2/d_8$, $A_8=G_1^2/d_{11}$, $A_9=G_2^2/d_9$, $A_{10}=G_1^2/d_{10}$, $B_6={\Gamma }_{00}+A_8$ and $B_7=d_1+A_7+A_{10}$.

We repeated above the experiment with the same experimental conditions [the data points in Fig. 4(a)], and obtained the results as shown in Fig. 6(a) . Comparing the results in Fig. 6(a) with those in Fig. 4(a), we can obtain the similar observations of suppression and enhancement of the DFWM signal ${\mathit{E}}_{\text{F}1}$, except the shapes of partial suppressions/ enhancement. For instance, the curve of Fig. 4(a2) shows first suppression, and next enhancement, is different from the curve of Fig. 6(a2), which shows first enhancement, and next suppression. The reason for this contrast is the difference of levels structure. As ${\Delta }_1$ is set near the resonance point (${\Delta }_1=-55.8\text{GHz}$), the new dressed-state $|G_1-->$ moves from upper to lower as ${\Delta }_2$ is scanned from positive to zero [Fig. 4(d2)]. So the DFWM signal ${\mathit{E}}_{\text{F}1}$ is first enhanced as the condition ${\Delta }_1+{\Delta }_2=-G_2^{}$ is satisfied, and then suppressed (the suppression condition ${\Delta }_1+{\Delta }_2=0$), as shown in the curve of Fig. 6(a2). Simultaneously, we obtain the corresponding suppressions and enhancements of x direction spatial splitting of DFWM signal ${\mathit{E}}_{\text{F}1}$ images, as shown in Fig. 6(c). Here Figs. 6(c1)–6(c9) show the experimental spots corresponding to the curves in Figs. 6(a1)–6(a9).

 

Fig. 6 (a) Measured suppression and enhancement of DFWM signal ${\mathit{E}}_{F1}$ versus ${\Delta }_2$ for different ${\Delta }_1$ values. ${\Delta }_1=$-70.5, −55.8, −32.7, −15.4, 0, 16.5, 32.5, 48.2 and 72.1 GHz, respectively. The dashed curve is the double-peak DFWM signal ${\mathit{E}}_{F1}$ versus ${\Delta }_1$. (b) Theoretical plots corresponding to the experimental parameters in (a). (c1)-(c9) DFWM signal ${\mathit{E}}_{F1}$ images. The condition here is the same as that in (a).

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4. Conclusion

In conclusion, we have experimentally observed the suppression and enhancement of the spatial FWM signal by the controlled cascade interaction of additional dressing fields, and the corresponding controlled spatial splitting of FWM signal caused by the enhanced cross-Kerr nonlinearity due to atomic coherence in two- and three-level atomic systems. In addition, we report the interplay between the two coexisting FWM signals, which can be tuned to overlap or separate by varying frequency detunings. Such controllable FWM processes can have important applications in wavelength conversion for spatial signal processing and optical communication.