1. Introduction

The coherent superposition of atomic states forms the base for a great deal of interesting phenomena in nonlinear laser spectroscopy. One of these phenomena resulting from the quantum interference between dressed states [1,2] is electromagnetically induced transparency (EIT) [3–7]. Under EIT condition, several higher-order nonlinear optical processes including four-wave mixing (FWM) [8–10] are allowed to occur in multi-level atomic systems, since the weak generated signals can be allowed to transmit through the resonant atomic medium with little absorption. Meanwhile, the fluorescence due to spontaneous emission can also generate within the EIT windows [11,12] and the competition between amplified spontaneous emission and four-wave-mixing process has been studied [13].

Furthermore, the suppression and enhancement of FWM, which are respectively corresponding to EIT and EIA (electromagnetically induced absorption) of probe transmission, also attracted the attention of many researchers [2,14,15]. By altering the frequency detunings of incident laser fields, the switch between dark state (EIT of probe transmission and suppression of FWM) and bright state (EIA of probe transmission and enhancement of FWM) is obtainable [2,13,15]. It is also reported recently by manipulating the phase difference between the two circularly polarized components of a single coherent field, the EIT-EIA switch could be realized [16].

In this paper, we first report the switch between dark state and bright state by manipulating the deflection angle of certain coupling beams in a Y-type or cascade atomic rubidium system. Such phenomenon is dramatically astonished in comparison with previous works, where only the signal’s linewidth changes by altering the angle between beams [17,18]. We have offered a mechanism based on the angle-modulation on the Rabi frequency, which is capable to explain the aforementioned switch. Simultaneously, the FWM signal due to atomic coherence and the fluorescence signals due to spontaneous emission are studied in company with probe transmission signal. By manipulation the deflection angle, the generated FWM and fluorescence processes can also transform from suppression to enhencement along with the EIT-EIA switch in the probe transmission spectrum. Such angle-modulated switch could have potential applications in optical communication and quantum information processing. Moreover, in the fluorescence signal, the peak of two-photon fluorescence due to classical emission and the dip of single-photon fluorescence due to dressing effect are distinguished, both in separate spectral curves and in the global profile of spectrum. Furthermore, the experimental results with two different ground-state hyperfine levels (GSHL) $5S_{1/2}F\text{=}2$ and $5S_{1/2}F\text{=3}$ of ${}^{85}\text{Rb}$ are compared for the first time.

2. Basic theory and experimental scheme

2.1 Experimental setup

The experiment is carried out in a rubidium vapor cell, whose energy levels of $5S_{1/2}$ ($|0〉$), $5P_{3/2}$ ($|1〉$), $5D_{5/2}$ ($|2〉$) and $5D_{3/2}$ ($|3〉$) form a four-level Y-type atomic system, as shown in Fig. 1(a). The resonant frequencies are ${\Omega }_1,$ ${\Omega }_2$ and ${\Omega }_3$ for transitions$|0〉$ to $|1〉,$ $|1〉$ to $|2〉$ and $|1〉$ to $|3〉$ respectively. The temperature of the atomic vapor cell is set at ${60}^{\circ }C.$ A weak probe beam $E_1$ (with frequency ${\omega }_1,$ wave vector $k_1,$ Rabi frequency $G_1$ and frequency detuning ${\Delta }_1,$ where ${\Delta }_i={\Omega }_i-{\omega }_i$) from an external cavity diode laser (ECDL), is horizontally polarized and probes the lower transition $|0〉$ to $|1〉.$ Two coupling laser beams $E_2$ (${\omega }_2,$ $k_2,$ $G_2$ and ${\Delta }_2$) and ${E^{\prime }}_2$ (${\omega }_2,$ ${k^{\prime }}_2,$ ${G^{\prime }}_2$ and ${\Delta }_2$) splitting from a cw Ti:sapphire laser with vertical polarization, drive the upper transition $|1〉$ to $|2〉.$ Another two coupling laser beams $E_3$ (${\omega }_3,$ $k_3,$ $G_3$ and ${\Delta }_3$) and ${E^{\prime }}_3$ (${\omega }_3,$ ${k^{\prime }}_3,$ ${G^{\prime }}_3$ and ${\Delta }_3$) splitting from an ECDL with vertical polarization, drive the upper transition $|1〉$ to $|3〉.$ Using this experimental setup, we will study three kinds of signals simultaneously: the transmission of probe beam, the four-wave mixing signals $E_{F1}^{}$ and $E_{F2}^{},$ and the fluorescence signals $R_0^{},$ $R_1^{}$ and $R_2^{}$ (shown in Fig. 1(a)). Especially, we mainly focus on the control of signal patterns through varying the direction of incident beams.

 

Fig. 1 (a) Relevant four-level Y-type atomic system with one probe field $E_1,$ two coupling fields $E_2$ and ${E^{\prime }}_2,$ and another two coupling fields $E_3$ and ${E^{\prime }}_3.$ $E_{F1}$ and $E_{F2}$ are the generated FWM signals. $R_0,$ $R_1$ and $R_2$ are the generated fluorescence signals. (b) Normal phase-matching spatial beam geometry. (c)-(e) The abnormal propagation configurations for the ladder type subsystem and Y-type system, with the deflection angles $\alpha$ and $\beta .$ The dash lines in (c)-(e) represent the direction of beams when the deflection angles equal $0^{\circ }.$

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In normal experimental configuration, the five laser beams are spatially designed in a square-box pattern as shown in Fig. 1(b), in which the coupling beams $E_2,$ ${E^{\prime }}_2,$ $E_3$ and ${E^{\prime }}_3$ propagate through the Rb vapor cell in the same direction with small angles (about $0.3{}^{\circ }$) between one another, and the probe beam $E_1$ propagates in the opposite direction of $E_2.$ In such beam geometric configuration, the two-photon Doppler-free conditions will be satisfied for the two ladder-type subsystems $|0〉-|1〉-|2〉$ and $|0〉-|1〉-|3〉,$ thus two EIT windows appear in the probe transmission spectrum. Also, two FWM processes $E_{F1}^{}$ (generated by $E_1,$ $E_2$ and ${E^{\prime }}_2$) and $E_{F2}^{}$ (generated by $E_1,$ $E_3$ and ${E^{\prime }}_3$) can occur simultaneously within the two EIT windows, both propagating in the direction of $k_{\text{F}}$ (at the lower right corner of Fig. 1(b)) satisfying the phase-matching condition $k_{F1}=k_{\text{1}}+k_2-{k^{\prime }}_2$ or $k_{F2}=k_{\text{1}}-{k^{\prime }}_3+k_3.$ In our experiments, we used a silicon photodiode to monitor the transmitted probe spectrum, and an avalanche photodiode detector to measure the generated FWM signals.

In addition to FWM signals induced by atomic coherence, three fluorescence signals due to spontaneous emission are studied simultaneously: the decay of photons from $|1〉$ to $|0〉$ generate single-photon fluorescence signal $R_0^{},$ and the decay of photons from $|2〉$ or $|3〉$ to $|1〉$ separately generate two-photon fluorescence signals $R_1^{}$ and $R_2^{},$ as shown in Fig. 1(a). These non-directional fluorescence signals are collected by a photodiode located at the side of the vapor cell. Similar to FWM signals, the two-photon fluorescence $R_1^{}$ and $R_2^{}$ also fall into the EIT windows and form the Doppler-free sharp peaks in frequency domain.

In our experiments, we especially focus on the angle-modulated switch on the probe transmission signal, FWM signals and fluorescence signals. When certain coupling beams are deflected with a small angle from their “normal” directions, the behaviors of the detected signals will change significantly: EIT peak in the probe transmission spectrum would switch to EIA dip; the suppression of FWM signal would alter to enhancement; and the pattern of fluorescence signals would also change correspondingly. We use the symbol $\alpha$ to represent the deflection of the coupling beams $E_2$ and ${E^{\prime }}_2$ from their normal directions (as shown in Fig. 1(c)-1(d)), and use the symbol $\beta$ to represent the deflection of the coupling beams $E_3$ and ${E^{\prime }}_3$ from their normal directions (as shown in Fig. 1(e)). By altering the deflection angle $\alpha$ or $\beta$ in different conditions, we can observe the switches of signals’ pattern we stated above. In the following we term such deflected spatial geometry (Fig. 1(c)-1(e)) the “abnormal” propagation configuration, to distinguish it from the “normal” spatial geometry shown in Fig. 1(b).

2.2 Basic theory

Generally, the behaviors of detected signals can be described by density matrix elements with different orders. Specially, the probe transmission signal can be described by the opposite of the imaginary part of first-order density matrix element ${\rho }_{10}^{(1)}$ (the superscript of the notation represents the order of density matrix element, or perturbation order), the intensity of FWM signals can be described by the third-order one ${\rho }_{10}^{(3)},$ and the intensity of fluorescence signals are related to the even-order ones ${\rho }_{11}^{(2)},$ ${\rho }_{22}^{(4)}$ and ${\rho }_{33}^{(4)},$ which are the various diagonal elements of the density matrix. The expressions of these elements can be obtained by solving the coupled density-matrix equations [12,19].

Via the Liouville pathway (perturbation chain) ${\rho }_{00}^{(0)}\stackrel{E_1}{\to }{\rho }_{10}^{(1)},$ the element ${\rho }_{10}^{(1)}$ can be written as:

For the fluorescence signals, the intensity of single-photon fluorescence ($R_0^{}$) and two-photon fluorescence ($R_1^{}$ and $R_2^{}$) are separately proportional to the square of the module of second-order matrix element (${\rho }_{11}^{(2)}$) and fourth-order matrix elements (${\rho }_{22}^{(4)}$ and ${\rho }_{33}^{(4)}$), since the square of the module of diagonal elements represent the density of particles in corresponding states. First of all, with only probe beam $E_1^{}$ turned on, the single-photon fluorescence signal $R_0^{}$ generates, the process of which is described by the pathway ${\rho }_{00}^{(0)}\stackrel{E_1}{\to }{\rho }_{10}^{(1)}\stackrel{{(E_1)}^{*}}{\to }{\rho }_{11}^{(2)}$. Guided by the pathway, we can easily obtain the expression of ${\rho }_{11}^{(2)}$ from the density-matrix equations, as:

Next, in the two ladder type subsystems $|0〉-|1〉-|2〉$ and $|0〉-|1〉-|\text{3}〉,$ the two-photon fluorescence signals $R_1^{}$ and $R_{\text{2}}^{}$ generate separately. In $|0〉-|1〉-|2〉$ subsystem, the generation of $R_1^{}$ can be described by ${\rho }_{00}^{(0)}\stackrel{E_1}{\to }{\rho }_{10}^{(1)}\stackrel{E_2}{\to }{\rho }_{20}^{(2)}\stackrel{{(E_1)}^{*}}{\to }{\rho }_{21}^{(3)}\stackrel{{(E_2)}^{*}}{\to }{\rho }_{22}^{(4)}.$ Step by step guided by the pathway, we can get the expressions of related elements: ${\rho }_{1\text{0}}^{(1)}=-i{G}_1/d_1$ via ${\rho }_{00}^{(0)}\stackrel{E_1}{\to }{\rho }_{10}^{(1)},$ ${\rho }_{20}^{(2)}=-G_1{G}_2\text{/(}{d}_1{d}_2\text{)}$ via ${\rho }_{10}^{(1)}\stackrel{E_2}{\to }{\rho }_{20}^{(2)},$ ${\rho }_{21}^{(3)}=i{G}_1^2{G}_2\text{/(}{d}_1{d}_2{d}_{\text{4}}\text{)}$ ($d_4={\Gamma }_{21}+i{\Delta }_2$) via ${\rho }_{20}^{(2)}\stackrel{{(E_1)}^{*}}{\to }{\rho }_{21}^{(3)},$ and finally:

When the coupling beams are deflected with a small angle $\alpha$ or $\beta ,$ because of phase matching conditions, the coupling strength (Rabi frequency) becomes a function of the angles. As is known, Rabi frequency $G_i$ is defined as $G_i={\mu }_i{E}_i\cos \theta \text{/}\hslash$ where $E_i$ represents the electric field, ${\mu }_i$ is the dipole moment of transition the light field excites, and $\theta$ represents the angle between the polarization of the light and the transition dipole moment. Now, when the additional deflection angle $\alpha$ between $E_2$ and the opposite direction of $E_1$ is introduced in, the orientation of electric field $E_2$ changes, and the Rabi frequency $G_2$ should be modified as $G_2={\mu }_2{E}_2\cos \text{(}\theta \pm \alpha \text{)/}\hslash .$ Similarly, when the beam $E_3$ is deflected with the angle $\beta ,$ the Rabi frequency $G_3$ should be modified as $G_3={\mu }_3{E}_3\cos \text{(}\theta \pm \beta \text{)/}\hslash .$ In a word, by manipulating the deflection angles, we can control the coupling strength, and thereby control the switch between dark state and bright state. Although the angles $\alpha$ or $\beta$ are relatively small, we will find the signals are strikingly sensitive to their alterations.

3. Observation of angle modulation in ladder type subsystem

We have deduced the expressions of related density matrix elements and discussed the mechanism of angle-modulated switch. In the following sections, we will present the experimental results of angle modulation in ladder type subsystem (this section), in Y-type system (Sec. 4), and the direct observation of angle-modulated suppression-enhancement switch of FWM (Sec. 5).

We first consider the angle modulation in the $|0〉-|1〉-|2〉$ ladder type subsystem when three beams $E_1,$ $E_2$ and ${E^{\prime }}_2$ are turned on (as shown in Fig. 1(c)). We separately show the results with the two ground-state hyperfine levels (GSHL) of ${}^{85}\text{Rb}$: $5S_{1/2}F\text{=3}$ in Fig. 2 and $5S_{1/2}F\text{=}2$ in Fig. 3. In both cases the angle modulation effect can be observed clearly, but some discrepancies can also be observed in the results with two different ground states.

 

Fig. 2 (a)-(c) Measured signals versus ${\Delta }_2$ at discrete probe detuning ${\Delta }_1$ and discrete deflection angle $\alpha \text{,}$ the ground-state is $5S_{1/2}F\text{=3}\text{.}$ The top curves ((a1)-(a4)) are probe transmission; the middle curves ((b1)-(b4)) are FWM signal; and the bottom curves ((c1)-(c4)) are fluorescence signal. The experimental parameters are $P_1=8\text{mW,}$ $P_2=10\text{mW}$ and ${P^{\prime }}_2=10\text{mW}\text{.}$ (d1)-(d3) Calculated probe transmission versus $\alpha$ at three typical detunings.

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Fig. 3 (a)-(c) Measured signals versus ${\Delta }_2$ at discrete ${\Delta }_1$ and $\alpha \text{,}$ the ground state is $5S_{1/2}F\text{=2}\text{.}$ The experimental parameters are the same as Fig. 2. (d1)-(d2) Magnified sub-graphs for $5S_{1/2}F\text{=3}$ and $F\text{=2}$ of ${}^{85}\text{Rb}.$ (e) Realistic energy level diagram showing the hyperfine levels of each driven state, where the FWM transitions with the least number of decay channels are presented.

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In Figs. 2(a)-2(c), the probe transmission (Figs. 2(a1)-2(a4)), FWM (Figs. 2(b1)-2(b4)) and fluorescence signals (Figs. 2(c1)-2(c4)) are presented simultaneously, obtained by scanning ${\Delta }_2$ with probe detuning ${\Delta }_1$ and deflection angle $\alpha$ separately set at typical values. The obtained signals are arranged in a three-dimensional box, so that the variation of the curves versus both ${\Delta }_1$ and $\alpha$ is explicitly displayed. When the beams $E_2$ and ${E^{\prime }}_2$ propagate from their normal direction without deflection ($\alpha =0{}^{\circ }$), in the probe transmission spectrum we can see EIT (peaks higher than the baseline) appearing in the center area within the Doppler absorption background, EIA (small dips lower than the base lines) emerging at large probe detunings, and partial-EIT-partial-EIA appearing in the transitional areas (as shown in Fig. 2(a2)), this is just the same as the results in previous work [2]. However, when the coupling beams are deflected from normal directions ($\alpha \ne 0{}^{\circ },$ as the geometry shown in Fig. 1(c)), switches between EIA and EIT can be observed at each ${\Delta }_1$ point. For example, in the case of $\alpha =-0.08{}^{\circ }$ (Fig. 2(a1)), strong EIT peak appears at negative ${\Delta }_1$ points, and EIA dip appears at positive ${\Delta }_1$ points. On the contrary, when $\alpha$ is set at positive values (Figs. 2(a3)-2(a4)), obvious EIA dip appears at negative ${\Delta }_1$ points, while EIT peak emerges at positive probe detunings. In sum, when $\alpha \ne 0{}^{\circ },$ the symmetrical pattern of probe transmission versus ${\Delta }_1$ is broken. We also present the simulation of such EIT-EIA switch, as shown in Figs. 2(d1)-2(d3), which is in agreement with the experimental results.

As is known, the suppression of FWM is obtained in EIT window, and the enhancement of FWM is in company with EIA. Thus the switch between the suppression and enhancement of FWM will appear along with the EIT-EIA switch. In Figs. 2(b1)-2(b4), the suppression-enhancement switch of FWM is reflected in the variation in signal’s intensity with different angles. For instance, when $\alpha$ is set at $-0.08{}^{\circ }$ (Fig. 2(b1)) or $0{}^{\circ }$ (Fig. 2(b2)), the FWM signal reaches its maximum at ${\Delta }_1=-300\text{MHz;}$ and when $\alpha$ is set at $0.08{}^{\circ }$ (Fig. 2(b3)), it reaches the maximum around ${\Delta }_1=-450\text{MHz}\text{.}$ Admittedly, there’re some other factors which could also lead to the variation in the intensity of FWM. For instance, the FWM generally weakens with $\alpha$ increasing, because the effective overlap cross section of the beams generating FWM decreases. Therefore analyzing the variation in the intensity of FWM is not an ideal way to observe the suppression-enhancement switch. In Sec. 5, we will observe the switch in FWM directly using another method.

The fluorescence signals (Figs. 2(c1)-2(c4)) is composed of two components: the single-photon fluorescence $R_0^{}$ related to matrix element ${\rho }_{11}^{(2)}$ and the two-photon fluorescence $R_1^{}$ related to matrix element ${\rho }_{22}^{(4)}.$ Basically, the obtained fluorescence signal appears as a dip containing a sharp peak on each base line (details can be clearly seen in the amplified sub-figure in Fig. 3(d1)). The dip represents the suppression of $R_0^{}$ induced by the dressing effect of $E_2$(${E^{\prime }}_2$), corresponding with EIT in probe transmission spectrum. The peak within the dip is the emission of fluorescence $R_1^{},$ which is corresponding with EIA according to Eqs. (1b) and (5b). Therefore, in the process of altering the angle $\alpha ,$ $R_0^{}$ will get stronger suppression when EIT appears, and $R_{\text{1}}^{}$ will be enhanced in the presence of EIA (more clear details will be shown in Fig. 4).

 

Fig. 4 (a) and (c) Measured probe transmission (top curves), FWM signal $E_{F1}$ (middle curves), and fluorescence signal (bottom curves) versus ${\Delta }_2$ at discrete ${\Delta }_3,$ with fixed ${\Delta }_1\text{=}0\text{MHz}$ and ${E^{\prime }}_3$ blocked. For (a1)-(a3) $\alpha =0.04{}^{\circ },$ and for (c1)-(c3) $\alpha =0.16{}^{\circ }.$ The other parameters are $P_1=7.8\text{mW,}$ $P_2=6.9\text{mW,}$${P^{\prime }}_2=15.9\text{mW}$ and $P_3=46.0\text{mW}\text{.}$ (b) and (d) The calculated curves corresponding to (a) and (c) respectively. (e) The corresponding dressed state diagrams with typical ${\Delta }_3$ values.

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Now we turn to the results for the other ground state: $5S_{1/2}F\text{=}2$ of ${}^{85}\text{Rb}$ (Figs. 3(a)-3(c)). The phenomenon of angle-modulation for $F\text{=}2$ is similar with $F\text{=}3,$ except for some discrepancies. To show the details clearly, we magnified two typical sub-graphs from Figs. 2(a)-(c) and Figs. 3(a)-3(c) respectively, as shown in Figs. 3(d1)-3(d2). In the case of $F\text{=}3$ (Fig. 2(a)), when ${\Delta }_1$ is set at negative points (for example ${\Delta }_1=-300\text{MHz}$), the probe transmission signal can change from strong EIT ($\alpha \text{=}-0.08{}^{\circ }$), to weak EIT ($\alpha \text{=}0{}^{\circ }$), then to weak EIA ($\alpha \text{=}0.08{}^{\circ }$), and finally to strong EIA ($\alpha \text{=}0.16{}^{\circ }$). But when ${\Delta }_1$ is set at positive points, the switch process is not as striking as above. By contrast, in the case of $F\text{=}2$ (Fig. 3(a)), the striking EIA-EIT switch happens in positive probe detuning region. Besides, the strongest FWM generation also appears in positive probe detuning region for $F\text{=}2$ (Fig. 3(b)), which is different from the case of $F\text{=}3$ where the strongest FWM appears in negative detuning region (Fig. 2(b)). Moreover, the fluorescence signals for the two ground states are also different. Comparing the fluorescence signal for $F\text{=}2$ with that for $F\text{=}3,$ we find the suppression dip of fluorescence disappears for $F\text{=}2$ ground state (Fig. 3(d2)). These discrepancies could be explained with the assistance of the realistic energy level diagram in Fig. 3(e). For the ground state $F\text{=}3,$ the higher frequency transition $F\text{=3}\to F^{\prime }\text{=4}$ is closed; in other words the pathways involving $F^{\prime }\text{=4}$ have the fewest decay channels. Therefore the FWM generation and the switch for $F\text{=}3$ are strongest in the negative-detuned region where the $F^{\prime }\text{=4}$ level lies [20]. On the contrary, for the ground state $F\text{=}2,$ the lower frequency transition $F\text{=2}\to F^{\prime }\text{=1}$ is closed. That’s why for $F\text{=}2$ the strongest FWM and switch appear in positive-detuned region.

In following sections, the experiments are all performed with ground state $F\text{=}3$ of ${}^{85}\text{Rb}.$

4. Observation of angle modulation in Y-type subsystem

In this section, we emphasize on the angle modulation in the Y-type system where the doubly dressing effect should be considered. First in Fig. 4, with $E_1,$ $E_2,$ ${E^{\prime }}_2$ and $E_3$ turned on and ${E^{\prime }}_3$ blocked (as the geometry shown in Fig. 1(d)), we study the signals by scanning ${\Delta }_2$ at different ${\Delta }_3$ points, with the angle $\alpha$ set at two typical values $\alpha =0.04{}^{\circ }$ (Fig. 4(a)) and $\alpha =0.16{}^{\circ }$ (Fig. 4(c)). The theoretical calculations corresponding to Figs. 4(a) and 4(c) are presented in Figs. 4(b) and (d).

Under the doubly dressing condition, two EIT would form simultaneously in the probe transmission spectrum: the $|0〉-|1〉-|\text{2}〉$ EIT and the $|0〉-|1〉-|\text{3}〉$ EIT. In the case of $\alpha =0.04{}^{\circ },$ the global profile of the transmitted probe signal (Fig. 4(a1)) versus ${\Delta }_3$ reaches its summit at ${\Delta }_3=-{\Delta }_1=0,$ representing the $|0〉-|1〉-|\text{3}〉$ EIT window; on the other hand, the peak on each curve versus ${\Delta }_2$ is the $|0〉-|1〉-|\text{2}〉$ EIT window, satisfying ${\Delta }_1\text{+}{\Delta }_{\text{2}}=0.$ When $\alpha$ is adjusted to $\alpha =0.16{}^{\circ },$ the $|0〉-|1〉-|\text{2}〉$ EIT peaks can be observed totally switch to EIA dips, as shown in Fig. 4(c1). Notice the $|0〉-|1〉-|\text{3}〉$ EIT profile remains the same under the two angles, because changing the direction of $E_2$ and ${E^{\prime }}_2$ will not influence the dressing effect of $E_3.$

In Fig. 4(a1) and 4(c1), when comparing the curves at ${\Delta }_{\text{3}}=0\text{MHz,}$ ${\Delta }_{\text{3}}=\pm \text{3}0\text{MHz}$ and ${\Delta }_{\text{3}}=\pm \text{6}0\text{MHz,}$ one can discover that both the $|0〉-|1〉-|\text{2}〉$ EIT in Fig. 4(a1) and the $|0〉-|1〉-|\text{2}〉$ EIA in Fig. 4(c1) reach their minimum amplitude at ${\Delta }_{\text{3}}=0\text{MHz,}$ matching the condition ${\Delta }_{\text{1}}+{\Delta }_2={\Delta }_{\text{1}}+{\Delta }_{\text{3}}=0.$ This is due to the strong sequential-cascade-dressing interaction between the two ladder type subsystems $|0〉-|1〉-|\text{2}〉$ and $|0〉-|1〉-|\text{3}〉,$ according to the doubly dressed term $d_1+{\left|G_2\right|}^2/d_2+{\left|G_3\right|}^2/d_3$ in Eq. (1c). Such interaction can be illustrated with the dressed state diagrams in Fig. 4(e). Figures 4(e1)-4(e5) separately present the diagrams of dressed states with ${\Delta }_3$ gradually altering from negative to positive. Due to the dressing effect of $E_{\text{3}},$ the energy level $|1〉$ would be split into two dressed states $|+〉$ and $|-〉,$ the positions of which altering along with ${\Delta }_3.$ As we know, the larger the relative frequency of a field to the transition it drives, the weaker the dressing effect is. When ${\Delta }_{\text{3}}=0\text{MHz}$ (Fig. 4(e3)), the relative frequency of $E_2$(${E^{\prime }}_2$) to the transition $|+〉\to |2〉$ or $|-〉\to |2〉$ is large, therefore the dressing effect of $E_2$(${E^{\prime }}_2$) is relatively weak and the $|0〉-|1〉-|\text{2}〉$ EIT/EIA is small; with $\left|{\Delta }_{\text{3}}\right|$ increasing, the relative frequency of $E_2$(${E^{\prime }}_2$) to one of the two transitions $|+〉\to |2〉$ and $|-〉\to |2〉$ gets smaller, therefore the dressing effect of $E_2$(${E^{\prime }}_2$) becomes larger and the $|0〉-|1〉-|\text{2}〉$ EIT/EIA becomes stronger.

For the FWM signal $E_{F1}^{}$ generated by $E_1,$ $E_2$ and ${E^{\prime }}_2$ shown in Fig. 4(a2), we can see its intensity is much weaker at resonant point (${\Delta }_3\approx -{\Delta }_1=0$) than at detuned ${\Delta }_3,$ for the suppression of the external dressing field $E_3$ on $E_{F1}^{}$ is strongest around ${\Delta }_3=-{\Delta }_1.$ When $\alpha$ is adjusted to $0.16{}^{\circ }$ (Fig. 4(c2)), $E_{F1}^{}$ is greatly strengthened at each ${\Delta }_3$ point. This is for the reason that the original suppression effect on $E_{F1}^{}$ induced by the self-dressing fields $E_2$(${E^{\prime }}_2$) transforms to enhancement effect when $\alpha =0.16{}^{\circ },$ corresponding to the switch from EIT to EIA of the transmitted probe field.

The fluorescence signal in Figs. 4(a3) and 4(c3) includes the doubly dressed single-photon fluorescence $R_0^{},$ and the two-photon fluorescence $R_1^{}.$ For $R_0^{},$ on the one hand it is suppressed by $E_3$ to its minimum around ${\Delta }_3\approx -{\Delta }_1=0,$ corresponding to the $|0〉-|1〉-|\text{3}〉$ EIT profile in Figs. 4(a1) and (c1); on the other hand it is also suppressed by $E_2$(${E^{\prime }}_2$), shown as the dip on each curve, corresponding to the $|0〉-|1〉-|\text{2}〉$ EIT peak. The sharp peak within each dip represents the emission of the two-photon fluorescence $R_1^{}.$ It is obvious in Fig. 4(a3) that the suppression dip of $R_0^{}$ induced by $E_2$(${E^{\prime }}_2$) gets much shallower when ${\Delta }_3$ approaches the resonant point, in agreement with the weakened $|0〉-|1〉-|\text{2}〉$ EIT peaks around ${\Delta }_3=0$ in Fig. 4(a1). Under the condition of $\alpha =0.16{}^{\circ }$ (Fig. 4(c3)), the suppression dip on each curve become shallower compared with those in Fig. 4(a3). This corresponds to the behavior of probe transmission signal which changes from EIT ($\alpha =0.04{}^{\circ }$) to EIA ($\alpha =0.16{}^{\circ }$). To make the facts above more evident, we present the corresponding theoretical calculated results in Figs. 4(b) and 4(d), which are in good agreement with the experimental results in Figs. 4(a) and 4(c).

Next, the generated signals under a specific abnormal configuration ($\alpha =0.1\text{6}{}^{\circ }$) are present in Fig. 5, where we will put emphasis on the variation of fluorescence signals in particular. Figures 5(a) and (c) present the measured signals by scanning ${\Delta }_2$ at different ${\Delta }_3$ points with $\alpha =0.1\text{6}{}^{\circ },$ in which the probe detuning ${\Delta }_{\text{1}}$ is set at ${\Delta }_1\text{=}0\text{MHz}$ for Fig. 5(a) and ${\Delta }_1\text{=}-150\text{MHz}$ for Fig. 5(c). Similar with the case in Fig. 4(c1), the abnormal EIA dip instead of normal EIT peak appears around ${\Delta }_1+{\Delta }_3=0$ in the transmitted probe spectrum, due to the modulation of angle $\alpha .$ However, we notice the EIA dip in Figs. 5(a1) and 5(c1) emerges only in a small region around ${\Delta }_1+{\Delta }_3=0,$ whereas the EIA in Fig. 4(c1) appears in a extensive region, this is the result of the smaller probe field power $P_1$ in Fig. 5 compared with Fig. 4. For the FWM signal in Figs. 5(a2) and 5(c2), it’s obvious that the intensity is greatly larger at the detuning point ${\Delta }_1\text{=}-150\text{MHz}$ (Fig. 5(c2)) than at ${\Delta }_1\text{=}0\text{MHz}$ (Fig. 5(a2)), due to the different numbers of decay channels between transitions, which has already been discussed in Sec. 3.

 

Fig. 5 (a) and (c) Measured probe transmission (top curves), FWM signal $E_{F1}$ (middle curves), and fluorescence signals (bottom curves) versus ${\Delta }_2$ at discrete ${\Delta }_3,$ with $\alpha =0.1\text{6}{}^{\circ },$ ${E^{\prime }}_3$ blocked and ${\Delta }_1$ fixed at ${\Delta }_1\text{=}0\text{MHz}$ for (a) and ${\Delta }_1\text{=}-150\text{MHz}$ for (c). The other parameters are $P_1=\text{4mW,}$ $P_2=\text{12}\text{.6mW,}$${P^{\prime }}_2=\text{6}.\text{3mW}$ and $P_3=40\text{mW}\text{.}$ (b) and (d) Calculated fluorescence signals corresponding to (a3) and (c3) separately.

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When we turn to the fluorescence signals in Figs. 5(a3) and 5(c3), we find all three types of fluorescence ($R_0^{},$ $R_1^{}$ and $R_2^{}$) arise in the spectrum under such experimental condition. To discriminate them clearly, we present the corresponding calculated fluorescence signals as $R_0^{}$ ($\text{~}{\left|{\rho }_{\text{11}}\right|}^{\text{2}}$), $R_1^{}$ ($\text{~}{\left|{\rho }_{\text{22}}\right|}^{\text{2}}$), $R_2^{}$ ($\text{~}{\left|{\rho }_{\text{33}}\right|}^{\text{2}}$) and the total fluorescence signal ($\text{~}{\left|{\rho }_{\text{11}}\right|}^{\text{2}}\text{+}{\left|{\rho }_{\text{22}}\right|}^{\text{2}}\text{+}{\left|{\rho }_{\text{33}}\right|}^{\text{2}}$) separately in Figs. 5(b) and 5(d), among which the calculated total fluorescence signal in Figs. 5(b4) and 5(d4) is the simulation of the experimental detected fluorescence signal in Figs. 5(a3) and 5(c3). When ${\Delta }_1\text{=}0\text{MHz,}$ the calculated single-photon fluorescence $R_0^{}$ is shown in Fig. 5(b1), where we can see each curve reveals as a dip resulting from the suppression effect of $E_2$(${E^{\prime }}_2$). Besides, the global profile of the curves also reveals as a big dip (shown as the dash line), for $R_0^{}$ is also suppressed by $E_3.$ Figs. 5(b2) and 5(b3) show the two-photon fluorescence $R_1^{}$ and $R_2^{},$ respectively. Under the method of scanning ${\Delta }_2$ at different ${\Delta }_3$ points, the fluorescence signal $R_1^{}$ reveals as an emission peak on each spectral line (Fig. 5(b2)); while the fluorescence signal $R_2^{}$ reveals as an emission profile composed of a series of horizontal lines at each ${\Delta }_3$ point (Fig. 5(b3)). Therefore, when we turn to the total fluorescence signal in Fig. 5(b4), it is obvious that its intensity versus ${\Delta }_{\text{2}}$ (the curve at each ${\Delta }_3$ point) reveals as a dip (the suppression induced by $E_2$(${E^{\prime }}_2$) on $R_0^{}$) containing a sharp peak ($R_1^{}$), and fluorescence intensity versus ${\Delta }_3$ (the global profile of curves at different ${\Delta }_3$ points) also behaves as a dip (the suppression induced by $E_{\text{3}}$ on $R_0^{}$) containing a peak ($R_{\text{2}}^{}$).

When ${\Delta }_1$ is tuned away from resonance, the amplitudes of both the suppression dips and the emission peaks change, as shown in Fig. 5(d). First, the suppression dips of $R_0^{}$ both in the profile and in each curve become shallower in Fig. 5(d1) compared with those in Fig. 5(b1), since the dressing effect weakens with detuned probe field. On the other hand, both $R_1^{}$ peak (Fig. 5(d2)) and $R_{\text{2}}^{}$ peak (Fig. 5(d3)) get stronger, compared with those in Figs. 5(b2) and (b3). According to Eqs. (5b) and (6b), the two-photon fluorescence signals are under suppression around ${\Delta }_1\text{=}0\text{MHz}$, and such suppression effects weaken when $\left|{\Delta }_1\right|$ increases. This is why $R_1^{}$ and $R_{\text{2}}^{}$ get strong with detuned ${\Delta }_1.$

Next, with all the five beams turned on, we investigate the function of deflection angle $\alpha$ in the interplay of two ladder type subsystems $|0〉-|1〉-|2〉$ and $|0〉-|1〉-|3〉,$ as shown in Fig. 6. Here, the probe detuning ${\Delta }_1$ is scanned with ${\Delta }_2$ set at four different values, and we present the experimental results with two different angles: $\alpha \text{=}0{}^{\circ }$ for Figs. 6(a1)-6(a4); and $\alpha \text{=}0.1\text{2}{}^{\circ }$ for Figs. 6(b1)-6(b4).

 

Fig. 6 Measured probe transmission (top curves), FWM (middle curves) and fluorescence (bottom curves) versus ${\Delta }_1$ when all the beams are turned on. The coupling detuning ${\Delta }_2$ is set at $80\text{MHz}$ ((a1) and (b1)), $60\text{MHz}$ ((a2) and (b2)), $40\text{MHz}$ ((a3) and (b3)), and $-150\text{MHz}$ ((a4) and (b4)); ${\Delta }_3$ is fixed at ${\Delta }_3\text{=}60\text{MHz}\text{.}$ $\alpha$ is set at $\alpha =0{}^{\circ }$ for (a1)-(a4), and $\alpha =0.1\text{2}{}^{\circ }$ for (b1)-(b4). The other experimental parameters are $P_1=8.2\text{mW,}$ $P_2=18.3\text{mW,}$ ${P^{\prime }}_2=9.6\text{mW,}$ $P_3=29.0\text{mW,}$ and ${P^{\prime }}_3=25.0\text{mW}\text{.}$

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In the former case (Figs. 6(a1)-6(a4)), two EIT windows separately related to the ladder-type subsystems $|0〉-|1〉-|2〉$ and $|0〉-|1〉-|3〉$ appear in the probe transmission spectrum (the top curves). Simultaneously, two FWM signals $E_{F1}^{}$ and $E_{F\text{2}}^{}$ (the middle curves) are generated within the two EIT windows. In the fluorescence signals (the bottom curves), the background curve revealing the emission profile of single-photon fluorescence $R_{\text{0}}^{},$ two dips appear at ${\Delta }_1\text{=}-{\Delta }_2$ and ${\Delta }_1\text{=}-{\Delta }_{\text{3}}$ upon the emission profile representing the suppression induced by $E_2({E^{\prime }}_2)$ and $E_{\text{3}}({E^{\prime }}_{\text{3}})$, respectively. Here by scanning ${\Delta }_1,$ the hypothesis that $R_{\text{0}}^{}$ could be regard as the counterpart of the probe transmission signal is reconfirmed: the emission profile of $R_{\text{0}}^{}$ is corresponding to the absorption background of the probe transmission spectrum, and the two suppression dips of $R_{\text{0}}^{}$ are corresponding to the two EIT windows. Besides, within the two suppression dips of $R_{\text{0}}^{}$ the two-photon fluorescence signals $R_{\text{1}}^{}$ and $R_{\text{2}}^{}$ are generated as small peaks, respectively (although $R_{\text{2}}^{}$ signal at ${\Delta }_1\text{=}-{\Delta }_{\text{3}}$ is unobvious).

As ${\Delta }_3$ is fixed at $60\text{MHz}$ and ${\Delta }_2$ is changed from $80\text{MHz}$ (Fig. 6(a1)), to $60\text{MHz}$ (Fig. 6(a2)), $40\text{MHz}$ (Fig. 6(a3)) and finally to $-150\text{MHz}$ (Fig. 6(a4)), the measured signals related to $|0〉-|1〉-|3〉$ subsystem ($|0〉-|1〉-|3〉$ EIT, $E_{F\text{2}}^{}$ and $R_{\text{2}}^{}$) are always fixed at ${\Delta }_1=-{\Delta }_3=-60\text{MHz,}$ and the characteristic signals related to $|0〉-|1〉-|2〉$ subsystem ($|0〉-|1〉-|2〉$ EIT, $E_{F\text{1}}^{}$ and $R_{\text{1}}^{}$) will shift from left to right. The two groups of signals partially overlap when ${\Delta }_2\text{=}80\text{MHz}$ (Fig. 6(a1)) and ${\Delta }_2\text{=}40\text{MHz}$ (Fig. 6(a3)), completely overlap when ${\Delta }_2\text{=}60\text{MHz}$ (Fig. 6(a2)), and finally separate when ${\Delta }_2\text{=}-150\text{MHz}$ (Fig. 6(a4)). When the two groups of signals completely or partially overlap, they will interact with each other. For example, when the two FWM signals completely overlap, the intensity of the total FWM signal is suppressed to its minimum (Fig. 6(a2)), resulting from the strongest mutually dressing effect of $|0〉-|1〉-|2〉$ and $|0〉-|1〉-|3〉$ subsystems.

Then, under the abnormal propagation configuration where $E_2$ is deflected with $\alpha \text{=}0.1\text{2}{}^{\circ }$ (Figs. 6(b1)-6(b4)), the EIT peak of the $|0〉-|1〉-|2〉$ subsystem transforms to an EIA dip. Similar to the case of $\alpha \text{=}0{}^{\circ },$ the EIA of $|0〉-|1〉-|2〉$ and EIT of $|0〉-|1〉-|3〉$ partially overlap (Figs. 6(b1) and 6(b3)), completely overlap (Fig. 6(b2)), and finally separate (Fig. 6(b4)). The two groups of characteristic signals still interact with each other. But the interaction behaves differently now. For example, for the FWM in Figs. 6(b1)-6(b3), the sum of $E_{F\text{1}}^{}$ and $E_{F\text{2}}^{}$ reaches the maximum amplitude when they completely overlap in Fig. 6(b2), which is different from the case in Figs. 6(a1)-6(a3) where the sum of $E_{F\text{1}}^{}$ and $E_{F\text{2}}^{}$ reaches the minimum amplitude when completely overlapping. This is because the FWM signal $E_{F\text{2}}^{}$ get enhancement instead of suppression in the case of $\alpha \text{=}0.1\text{2}{}^{\circ }.$ With respect to the fluorescence signal, the two-photon emission peak of $R_{\text{1}}^{}$ is strengthened in the condition of $\alpha \text{=}0.1\text{2}{}^{\circ },$ corresponding to the $|0〉-|1〉-|2〉$ EIA.

5. Observation of angle modulated suppression-enhancement switch of FWM

In above sections the EIT-EIA switch modulated by angle $\alpha$ has been thoroughly discussed. The angle-modulated suppression-enhancement switch of FWM, on the other hand, only reflects in the variation of signal’s amplitude. In this section, we will modulate our spatial geometry so that the switch of FWM could be observed directly. In Fig. 7 and Fig. 8, we adopt the geometry shown in Fig. 1(e), where the external dressing field $E_3$ instead of self-dressing field $E_{\text{2}}$ is deflected with the angle $\beta$. The corresponding detuning ${\Delta }_{\text{3}}$ is scanned here.

 

Fig. 7 Measured probe transmission ((a1) and (b1)), FWM signal $E_{F1}^{}$ with enhancement and suppression ((a2) and (b2)), and fluorescence signal ((a3) and (b3)) versus ${\Delta }_3$ at discrete ${\Delta }_1,$ with fixed ${\Delta }_2\text{=}-100\text{MHz}$ and ${E^{\prime }}_3$ blocked. The deflection angle $\beta$ is $\beta =0{}^{\circ }$ for (a1)-(a3), and $\beta =0.1\text{2}{}^{\circ }$ for (b1)-(b3). The other experimental parameters are $P_1=4.5\text{mW,}$ $P_2=12.9\text{mW,}$ ${P^{\prime }}_2=8.2\text{mW}$ and $P_3=29.0\text{mW}\text{.}$

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Fig. 8 Measured probe transmission ((a1) and (b1)), the enhancement and suppression of FWM signal $E_{F1}^{}$ ((a2) and (b2)), and fluorescence ((a3) and (b3)) versus ${\Delta }_3$ with $\beta =-0.{\text{04}}^{\circ },$ ${\text{0}}^{\circ },$ $0.{\text{04}}^{\circ },$ $0.{\text{08}}^{\circ },$ $0.{\text{12}}^{\circ }$ and $0.{\text{16}}^{\circ }$ from top to bottom. (a) and (b) are separately the signals obtained at left peak and right peak of the FWM double-peak profile. The other parameters are $P_1=\text{3}\text{.6mW,}$ $P_2=\text{30}.\text{6mW,}$ ${P^{\prime }}_2=\text{5}\text{.4mW}$ and $P_3=\text{14}.0\text{mW}\text{.}$

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Figure 7 shows concerning signals with two angle: $\beta =0{}^{\circ }$ (normal) for (a1)-(a3), and $\beta =0.1\text{2}{}^{\circ }$ (abnormal) for (b1)-(b3). The behaviors of the probe transmission and the fluorescence signals are similar with above figures, so we mainly focus on the FWM. In Figs. 7(a2) and 7(b2), the curves at discrete ${\Delta }_1$ points form a double-peak profile (dash line), representing the AT-splitting of FWM signal $E_{F1}^{}$ induced by self dressing effect of $E_2$(${E^{\prime }}_2$). The peak or dip on each baseline, on the other hand, means $E_{F1}^{}$ is enhanced or suppressed by the external dressing field $E_3$. By manipulating the deflection angle $\beta$, we can observe the switch between enhancement and suppression directly. For instance, under the normal case (Fig. 7(a2)), the FWM signal undergoes suppression at the two global peaks (${\Delta }_1\text{=7}0\text{MHz}$ and ${\Delta }_1\text{=115MHz}$) and the global valley (${\Delta }_1\text{=100MHz}$), slight enhancement at two edges of the double-peak (${\Delta }_1\text{=4}0\text{MHz}$ and ${\Delta }_1\text{=160MHz}$), and partial-suppression-partial-enhancement at other points. Now, with $\beta$ altered to $0.1\text{2}{}^{\circ }$(Fig. 7(b2)), the original suppression dips around the left global peak are all replaced by pure enhancement, and around the right global peak the suppression also diminishes obviously. Such suppression-enhancement switch of FWM is in concord with the EIT-EIA switch in Sec. 3 and 4.

Besides, we notice that unlike the strong two-photon fluorescence peak ($R_1^{}$) in Fig. 4 and Fig. 5, in Figs. 7(a3) and 7(b3) the two-photon fluorescence peak ($R_{\text{2}}^{}$) is rather weak within each dip of single-photon fluorescence. This difference results from the discrepancy of spontaneous transition probability between the transitions $|2〉\to |\text{1}〉$ and $|\text{3}〉\to |\text{1}〉.$ Theoretical calculation shows that the photons in the excited state $|3〉$($5D_{3/2}$) are more likely to transit to $5P_{\text{1}/2}$ rather than $|1〉$($5P_{3/2}$), while for the excited state $|2〉$($5D_{5/2}$) the transition $|2〉\text{(}5D_{5/2}\text{)}\to |\text{1}〉\text{(}5P_{3/2}\text{)}$ is dominant [21], which results in $R_1^{}$ much stronger than $R_{\text{2}}^{}.$

In Fig. 8, we continuously change $\beta$ from $-0.{\text{04}}^{\circ }$ to $0.{\text{16}}^{\circ }$ with ${\Delta }_3$ scanned, so that the evolution of the signals versus the deflection angle can be observed more clearly. Figures 8(a) and (b) separately depict the signals at the left and right peaks (corresponding to ${\Delta }_1\text{=7}0\text{MHz}$ and ${\Delta }_1\text{=115MHz}$ in Figs. 7(a) and 7(b)) of the FWM double-peak profile. When the angle $\beta$ changes from negative to positive, the height of EIT for probe transmission signal increases from small to large in the beginning, and then decreases to small again, as shown in Figs. 8(a1) and 8(b1) from top to bottom. For fluorescence signal, we can see the suppression (dip) of it also changes from small to big, then to small with increasing $\beta ,$ as shown in Figs. 8(a3) and 8(b3), corresponding to the variation of probe transmission signal. The enhancement and suppression of the FWM signal $E_{F1}^{}$ are shown in Figs. 8(a2) and 8(b2). At the left peak of the AT splitting double-peak structure (Fig. 8(a2)), the dressing effect on $E_{F1}^{}$ evolutes from pure suppression, to partial-suppression-partial-enhancement, then to pure enhancement with $\beta$ increasing. Especially, when $\beta =0{}^{\circ },$ the suppression dip gets deepest,corresponding to the case in Fig. 7(a2); when $\beta =0.1\text{2}{}^{\circ },$ $E_{F1}^{}$ undergoes strong enhancement, corresponding to the case in Fig. 7(b2). At the right peak (Fig. 8(b2)), the FWM signal mainly shows the pattern of left-suppression and right-enhancement. Although the switch from suppression to enhancement in Fig. 8(b2) is not as prominent as that in Fig. 8(a2), the tendency could also be seen.

6. Conclusion

We have investigated the four-wave mixing (FWM), fluorescence and the probe transmission simultaneously in the atomic rubidium system. By manipulating the deflection angle of certain coupling beams, the switch between dark state (EIT of probe transmission and suppression of the nonlinear optical processes) and bright state (EIA of probe transmission and enhancement of the nonlinear optical processes) is obtained. We have separately investigated such angle-modulated switch in ladder-type atomic system with singly-dressing effect and Y-type system with doubly-dressing effect. Such angle-modulated switch could have potential applications in optical communication and quantum information processing. Moreover, in the ladder-type system, we have observed and analyzed similarities and discrepancies between the two ground-state hyperfine levels F = 2 and F = 3 of Rb 85.