1. Introduction

Nonlinear polarization is an important physics quantity that reflects the interaction between optics and media, and it can be expanded in series as $P(t)={\epsilon }_0\chi E={\chi }^{(1)}E(t)+{\chi }^{(3)}{E}^3(t)+{\chi }^{(5)}{E}^5(t)+{\chi }^{(7)}{E}^7(t)\mathrm{...}$ in an isotropic medium. Multi-wave mixing (MWM) represents different odd-order nonlinearity polarizations χ(i). For example, four-wave mixing (FWM) is represented by ${\chi }^{(3)}$ and eight-wave mixing (EWM) is represented by ${\chi }^{(7)}$ and is produced by seven light fields; it can be used to transfer information on the light field to the medium. Such MWM processes can make the energy transfer between different waves [1, 2], can be applied for atom quantum memories [3] and unique wavelength creation [4], and can be correlated with many photon pairs [5]. Along different lines, highly excited Rydberg atoms, owing to their intriguing properties such as long lifetime, large dipole moment and strong interactions at long distances have been extensively investigated for quantum information applications [6]. However, many studies related to Rydberg atoms have primarily been conducted in magneto-optical traps [7, 8], which dramatically limits the practical applications of these atoms because of the complexity of the system. In recent years, Rydberg atoms have been investigated in thermal vapor using the electromagnetically induced transparency (EIT) technique, which has attracted the attention of many researchers to demonstrate non-destructive detection [9], GHz Rabi-flopping [10], Rydberg-Rydberg interaction [11], and have direct application on microwave sensing [12]. Because EIT is a quantum coherent effect that has dual functions, including its depiction by the imaginary density matrix element [13], which can theoretically represent absorption and provide a relatively ideal environment to suppress the Doppler Effect. The other function is the real part of the density matrix element, which describes dispersion; therefore, EIT can be used as a tool to apply for slow-light group velocity matching and enhance the intensity of MWM [14]. Consequently, Rydberg MWM processes can be investigated using the dual functions of EIT. In addition, inspired by the demonstration of Rydberg four-wave mixing (FWM) in a degenerate-energy-level system in thermal vapour, which revives signals beyond the frozen-gas limit [15, 16], and six-wave mixing (SWM) is used to convert millimeter waves to an optical field [17]. Our group has generated co-occurring SWM and EWM in a non-degenerate Rydberg thermal atomic ensemble [18, 19]. For these two MWM processes, EWM presents obvious advantages because of the multiple EIT windows. For example, EWM has a much narrower linewidth than SWM and can possess certain advantages in microwave sensing [20]. However, directly extracting and controlling EWM is always difficult. In this paper, this problem will be solved and the corresponding parametric control mechanism is established in circularly polarized probe field. Meanwhile, the spatial images of MWM are obtained and analyzed in these processes to study the nonlinear dispersion property that Rydberg excitation induces, which can provide insights for detailed study of the underlying Rydberg MWM spatial transmission properties and characteristics.

In this letter, we experimentally present the nonlinear spectra and spatial images showing the polarized-dressing Rydberg EWM process, which is enhanced by co-occurring EIT windows in a five-level atomic system. Quarter-wave plate (QWP) are used to change the polarizations of the probe field to directly observe the intensity evolution of Rydberg (37D_5/2) MWM and the corresponding spatial images. Then, high-contrast Rydberg EWM is extracted and evolved according to the power, detuning and polarization of the dressing field. In the research process, intriguing vortex-style images and chaos-like phenomena related to the nonlinear phase variation are observed.

2. Experimental scheme

The experiment was implemented in a thermal ⁸⁵Rb vapour cell where ground atomic density is about 1.0 × 10¹² cm⁻³. The 1cm long cell with natural abundance Rb vapour is wrapped with μ-metal sheets and heated to 65°C. Five laser beams from four commercial external cavity diode lasers (ECDLs) were used to establish the energy-level structure system in Fig. 1(a). Firstly, the weak probe beam E₁ (with a wavelength of 780 nm, power 1mw, and a diameter of 0.8 mm) drives 5S_1/2(F = 3) to 5P_3/2(F = 3). Beams E₃ (780 nm, 10mw, 1 mm) and E₃′ (780 nm, 23mw, 1 mm) derived from the same ECDL split through a polarization beam splitter (PBS) that connects the transition between 5P_3/2 (F = 3) and 5S_1/2(F = 2). With these three beams injected, one FWM that satisfies the phase-matching condition k_F = k₁ + k₃−k₃′ can be generated in the Λ-type three-level subsystem |0〉↔|1〉↔|3〉. Then the high-power coupling beam E₂ (480 nm, 100mw, 1 mm) produced by frequency doubling a 960nm ECDL is added to drive 5P_3/2(F = 3) to the 37D_5/2 Rydberg state. As a result, one EIT-assisted SWM E_S1 (satisfying phase-matching condition k_S1 = k₁ + k₃−k₃′ + k₂−k₂) can occur. The EIT window for SWM is generated by |0〉↔|1〉↔|2〉 transition. In the following, the dressing field E₄ (776 nm, 35mw, 0.9 mm) participates in driving the transition of 5P_3/2(F = 3) to 5D_5/2(F = 2) to generate the other SWM E_S2 assisted by |0〉↔|1〉↔|4〉 EIT window (satisfying phase matching condition k_S2 = k₁ + k₃−k₃′ + k₄−k₄) in the |0〉↔|1〉↔|3〉↔|4〉↔|1〉 transition. At last, an EWM process (k_EWM = k₁ + k₃−k₃′ + k₂−k₂ + k₄−k₄) with two overlapped EIT windows should be obtained in the process |0〉↔|1〉↔|3〉↔|4〉↔|1〉↔|2〉↔|1〉, where both k₂ and k₄ are used twice in the same propagation direction.

 

Fig. 1 (colour online) (a) K-type five-level system of ⁸⁵Rb atoms, where F = 3 (|0〉) and F = 2 (|3〉) which are two hyperfine state of the ground state 5S_1/2, and a first excited state 5P_3/2, F = 3(|1〉), a low-lying excited state 5D_5/2, F = 2 (|4〉), and a highly excited Rydberg state 37D_5/2 (|2〉) are involved. (b) Spatial arrangement of laser beams in the experiment and EWM phase matching condition diagram, where E_i on behalf of different laser field, E_M is the MWM field. (c) Zeeman sublevels with various transition pathways.

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All laser beams were spatially aligned as shown in Fig. 1(b). The probe beam E₁ (frequency ω₁, wave vector k₁, Rabi-frequency Ω₁ and frequency detuning Δ₁) can be modulated by a QWP before entering the atomic medium. Here, ${\Delta }_i={\omega }_i-{\omega }_{ij}$ is the frequency detuning, ${\omega }_i$ is the frequency of laser E_i, ${\omega }_{ij}$ is the resonant transition frequency between |i〉 and |j〉, Ω_i is the Rabi frequency that corresponds to optical field E_i and can be described as ${\mu }_{ij}{E}_i/\hslash$, ${\mu }_{ij}$ is the transition dipole moment, and E_i is the energy of the laser field. Two pump beams E₃ (ω₃, k₃, Ω₃, Δ₃) and E₃′ (k₃′, Ω₃′,) with an angle of 0.3° between them propagate in the opposite direction of E₁. Two dressing fields E₂ (ω₂, k₂, Ω₂, Δ₂) and E₄ (ω₄, k₄, Ω₄, Δ₄) are injected into the cell in the opposite direction of E₁ and overlap in the central point of the sample. These MWM signals all emit in the opposite direction of E₃′ (see Fig. 1(b)) and are identified by their corresponding EIT windows and detected by an avalanche photodiode detector (APD). The generated MWM signals are extracted through a PBS cube, the horizontal component of the generated MWM signals is detected by the APD, and the vertical component is received by a charge coupled device (CCD) camera. The probe transmission is detected by the other APD. When QWP is changed, the polarization of probe field will vary from linear polarization to circular one so that the selection rule and CG coefficient are also modified. Zeeman sublevels with various transition pathways and corresponding Clebsch-Gordan (CG) coefficient are shown in Fig. 1(c).

3. Basic theory

Theoretically, the imaginary part of the first-order density matrix element ${\rho }_{0_{M+q}{0}_M}{}^{\left(1\right)}$ proportionally determines the absorption of the probe beam and intensity of the probe transmission signal. The corresponding matrix elements can be obtained by solving the coupled density-matrix equations [21] and density-matrix element ${\rho }_{1_{M+q}{0}_M}{}^{\left(1\right)}$ with two EIT processes, which is described as follows:

Generally, the expression of density matrix elements that correspond to MWM signals can also be depicted by solving the density matrix equations [21]. Because the polarization of the probe field for the classical FWM signal is changed, the third-order density matrix element can be obtained via pathway${\rho }_{0_M{0}_M}^{(0)}\stackrel{{\Omega }_{1_{M+q}}^{0,\pm }}{\to }{\rho }_{1_{M\text{+}q}{0}_M}^{(1)}\stackrel{{\Omega }_{3_{M+q}}}{\to }{\rho }_{2_{M+q}{0}_M}^{(2)}\stackrel{{{\Omega }^{\prime }}_{3_{M+q}}}{\to }{\rho }_{1_{M+q}{0}_M}^{(3)}$:

For Rydberg interaction, we have calculated the Rydberg excitation density via the optical Bloch equation (OBE) using the mean-field model and obtain corresponding excitation density $N_2=C{N}_1^{0.2}{\left(\left|{\Omega }_{2_{M+q}}\right|/n^{11}\right)}^{0.4}$, their detailed deduction process can be found in [22], where N₁ is the density of atoms at level |1〉 (with the EIT and optical pumping effect considered) and given by $N_1=\frac{1}{2}{N}_0\left(\frac{{\Omega }_{1_{M+q}}^{0,\pm }*{\Omega }_{1_{M+q}}^{0,\pm }}{\mathrm{Re}[d_{1_{M+q}}+|{\Omega }_{2_{M+q}}{|}^2/d_{2_{M+q}}+|{\Omega }_{4_{M+q}}{|}^2/d_{4_{M+q}}]}+\frac{{\Omega }_{3_{M+q}}^2}{\mathrm{Re}[d_{3_{M+q}{1}_{M+q}}]}\right)$; $d_{3_{M+q}{1}_{M+q}}={\Gamma }_{1_{M+q}{3}_{M+q}}+i{\Delta }_3$; N₀ is the density of ground-state atoms; and C is a constant mainly determined by the coefficient of the vdW interaction and results from the numerical integration outside the given sphere and excitation efficiency between |0〉 and |1〉. We utilized these calculation results to modify the fifth-order density matrix element which determine the SWM signals considering interaction of Rydberg, meanwhile via perturbation chains${\rho }_{0_M{0}_M}^{(0)}\stackrel{{\Omega }_{1_{M+q}}^{0,\pm }}{\to }{\rho }_{1_{M+q}{0}_M}^{(1)}\stackrel{{\Omega }_{3_{M+q}}}{\to }{\rho }_{2_{M+q}{0}_M}^{(2)}\stackrel{-{\Omega }_{3_{M+q}}}{\to }{\rho }_{1_{M+q}{0}_M}^{(3)}\stackrel{{\Omega }_{2_{M+q}}}{\to }{\rho }_{2_{M+q}{0}_M}^{(4)1}\stackrel{-{\Omega }_{2_{M+q}}}{\to }{\rho }_{1_{M+q}{0}_M}^{(5)1}$, such fifth-order density matrix element can be depicted as

4. Results and Discussion

First, the high-contrast 37D_5/2 Rydberg EWM signal is observed with a circularly polarized probe field. The MWM spectrum is the strongest with the detuning fixed at Δ₁ = −400 MHz. Figures 2(a) and 2(b) show the probe transmission signals and corresponding MWM signals versus Δ₂ at different rotation angles θ of the QWP with E₄ on and off, respectively. The values in the horizontal axis in Fig. 2 denote the values of θ. Figures 2(a1) and 2(a2) are the cases with all beams on, and each peak in Fig. 2(a1) shows the probe transmission signals (at Δ₂ ≈-Δ₁) induced by the dressing field E₂ according to Eq. (1), which determines the intensity of the probe transmission. The EIT window caused by E₄ (hiding in the baseline) can interact with the EIT of E₂. The heights of the EIT peaks are maximum at θ = 0, gradually reach the lowest point at θ = 45°, and finally return to the original height when θ = 90°. This phenomenon can be explained by Eq. (1) because different CG coefficients determine the different transitions between Zeeman sublevels, which can lead to different Rabi frequencies. For example, ${\left(\left|{\Omega }_{1_{M\text{+}q}}^{\pm }\right|\right)}^2/{\left(\left|{\Omega }_{1_M}^0\right|\right)}^2=5$ in Fig. 1(c) illustrates that the dressing effects of ${\Omega }_{1_{M+q}}^{\pm }$ are greater than that of ${\Omega }_{1_M}^0$, and factor ${\Omega }_{1_{M+q}}^{\pm }\text{/}{\Gamma }_{0_{M+q}{0}_M}$ also plays an important role such that the height of the EIT peaks are lowest at θ = 45°. The dashed line in Fig. 2(a1) represents the transparent degree of E₄ obtained by the actual experiment offset, which has the same evolution regularity as the peaks. Figure 2(b1) shows the identical probe transmission spectra as Fig. 2(a1) except for blocking E₄, where the peaks of EIT are created by E₂ and interactions do not occur with the EIT of E₄. By comparing Figs. 2(b1) with 2(a1), the peaks in Fig. 2(a1) are smaller than those in Fig. 2(b1) at the identical polarization of E₁, which is mainly caused by the cascade relation between the dressing effect of E₂ and E₄, which is represented by the respective terms ${\left|{\Omega }_{2_{M+q}}^{}\right|}^{0.4}/d_{2_{M+q}}^{}$ and ${\left|{\Omega }_{4_{M+q}}^{}\right|}^2/d_{4_{M+q}}^{}$ in Eq. (1).

 

Fig. 2 EIT and MWM spectra with changes in the polarization of the probe field versus Δ₂. (a) EIT (a1) and MWM (a2) intensity spectra, which vary with the θ of QWP. (b) Blocking of E₄, EIT (b1) and SWM (b2) intensity spectra, which vary with the θ of QWP. (c) EIT (c1) and SWM (c2) intensity spectra changes with the power of E₁ when θ = 45°. (d) Identical to (c) except the power of E₂ is changed. (e) Images of the probe transmission at θ = 0° (e1), 35° (e2), 45° (e3), 55° (e4) and 65° (e5). (f) SWM images that correspond to (e). Corresponding Rabi frequency excluding (c) and (d), with Ω₁ = 2π × 54 MHz, Ω₂ = 2π × 7.6 MHz, Ω₃ = 2π × 142 MHz, and Ω₃′ = 2π × 224 MHz, and the ground atom density is 1.0 × 10¹² cm⁻³. For the color coding each distribution has been normalized by the actual maximum intensity.

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The corresponding MWM signals with and without E₄ are shown in Figs. 2(a2) and 2(b2), respectively. Because the frequency of the E₂ field is scanned, the FWM signal is in the baseline of observed signals. Each peak in Fig. 2(a2) is a Rydberg MWM signal, which consists of Rydberg SWM and EWM, whereas Fig. 2(b2) shows the pure Rydberg SWM signal. An analysis and comparison of Figs. 2(a2) and 2(b2) shows that the MWM (SWM + EWM) peak is smaller than SWM at θ = 0° and 90°. However, the suppression dip of the pure SWM appears at θ = 45°, whereas the MWM (SWM + EWM) signal shows a peak.

In other words, the high-contrast EWM signal can be observed in a circularly polarized probe field and suppressed SWM occur in the linearly polarized case, which represents an interesting phenomenon that can help us to directly extract EWM from mixed MWM signals. The contrast ratio of EWM compared with other MWM (FWM + SWM in the baseline) is approximately 96%, which is obtained based on the difference between the highest point of the MWM spectrum and the baseline (FWM + SWM). Such an EWM is generated by a seven-photon process assisted by two co-occurring EIT windows. The linewidth of the EWM (<30 MHz) can be described by Δω_E = Δω₁₀(Δω₂₁/Δϖ₂₁)(Δω₄₁/Δϖ₄₁), where Δω_ij is the linewidth of the EIT window (the measured linewidth of the Rydberg EIT is approximately 30 MHz in the experiment [18]). Δω_ij suppresses the absorption of the field that excites the transition |i>↔|j>; and Δϖ_ij is the linewidth of the absorption spectrum without the dressing effect. According to this description formula, one can see more EIT windows render the linewidth of Rydberg EWM signals much narrower. The direct observation of the EWM can be attributed to the stronger dressing effect from the circularly polarized E₁ according to Eq. (3). When θ = 45°, the Rydberg SWM is suppressed to generate a dip, and the term ${\left|{\Omega }_{1_{M+q}}^{\pm }\right|}^2/{\Gamma }_{0_{M+q}{0}_M}$ behaves as a constant in Eq. (3). Although EWM is also suppressed by ${\left|{\Omega }_{1_{M+q}}^{\pm }\right|}^2/{\Gamma }_{0_{M+q}{0}_M}$ as shown in Eq. (8), the peak of EWM remains observable in the experiment. When E₁ is linearly polarized, the dressing effect of E₁ is weaker; therefore, the suppression effect of E₄ is easily reflected, and only suppressed SWM signals are observed. To sufficiently certify this process, we present Figs. 2(c) and 2(d) to show the probe transmission signals and SWM signals under circularly polarized E₁ change with power of P₁ and P₂, respectively. With a fixed value of P₂ = 100 mW and increasing P₁, the SWM changes from a small peak to a dip and the depth of the dip increases. When P₁ is small, the term ${\Omega }_{1_{M+q}}^{\pm }\text{/}{\Gamma }_{0_{M+q}{0}_M}$ is also sufficiently small; therefore, the peaks of SWM initially increase. When P₁ further increases, the term ${\Omega }_{1_{M+q}}^{\pm }\text{/}{\Gamma }_{0_{M+q}{0}_M}$ begins to suppress SWM. When P₁ = 0.5 mW and P₂ changes and the suppression dip for SWM varies, which further demonstrates that the suppression dip is mainly caused by E₁.

We also detect images of the probe and MWM with the same condition in Fig. 2(b) to visually investigate the nonlinear dispersion property that Rydberg excitation induces. Rydberg EIT can cause modifications on the refractive index and nonlinear phase shift due to the interparticle interactions in the nonlinear processes [26]. Nonlinear dispersion related to spatial location can cause spatial focusing (or defocusing), shift, and splitting [27] so that the spatial effect of corresponding images can visually advocate the change of nonlinear dispersion. Details for the theoretical derivation of Δn_r and $\Delta {\Phi }_R$ have been discussed in basic theory part. The spatial focusing, splitting and shift phenomena influenced by $\Delta {\Phi }_R$ can be used to analyze the nonlinear dispersion property induced by Rydberg excitation. Figures 2(e) and 2(f) show the probe and MWM images versus Δ₂ with the angle of the QWP at 0°, 35°, 45°, 55° and 65°, respectively. The images at these five certain θ values can demonstrate the evolution of the MWM spatial transmission well. The case of θ = 90° is almost same with θ = 0°. From the shape evolution of probe images as shown in Fig. 2(e), the spatial variation phenomena cannot be observed in these results, which illustrates that the probe transmission images cannot reflect the spatial characters. The SWM images that correspond to Fig. 2(e) are shown in Fig. 2(f). In these figures, different types of images are obtained with different polarized probe fields, which demonstrate that different types of images are caused primarily by the polarized nonlinear phase ${\varphi }_{0,\pm }$ in Eq. (3). When the polarization of the probe field is near the circular polarization as shown in Fig. 2(f2), the spatial splitting in the multi-photon coherent SWM images is observed in the y direction; i.e., the polarized probe field overlaps with vertically polarized E₂ in the y direction. When the scanning of E₂ approaches 0 MHz, the energy level of Rydberg begins to play a role and make the image focusing in Fig. 2(f4), which is consistent with the theory that the attractive atomic interaction causes self-focusing nonlinearities reported in [24]. In particular, interesting vortex-like images are observed in Fig. 2(f5) because E₁, E₃, and E₃′ are superposed in a certain polarized field match and the spatial patterns are formed. This spatial pattern is caused by the multi-beam superposition, which makes the spatial polygon patterns form and induces a new phase factor [25]. The image pattern looks like a circular shape, which can be understood by the phase factor $e^{i\phi }$ in Eq. (3). In Fig. 2(f5), when the scanning frequency of E₂ approaches resonance, the Rydberg interaction can make ΔΦ_R vary to change $\exp \left(i\Delta {\Phi }_R\right)$, and the vortex-like image gradually becomes focused because the factor $\exp \left[i\left(\Delta {\Phi }_R+\phi \right)\right]$ in Eq. (3) varies. Consequently, the evolution of the SWM images can be attributed to two aspects of the nonlinear phase: the polarized nonlinear phase changes the SWM transmission image style and the Rydberg interaction makes the SWM signal focus.

Figure 3 shows the evolution of the Rydberg EWM signal as a function of the power of the external field E₄ at different powers of the probe field with a circular polarization. By analysing Fig. 2(c2), we set the power of E₁ to 100 μW, 500 μW and 1 mW, which makes the Rydberg SWM exhibit a peak, a half dip and a half peak, and a dip, respectively. When the power of E₄ modulates, the variation of EWM is clearly observed. The expressions Eqs. (3) and (8) determine the intensity of the SWM and EWM signals, respectively. The term ${\left|{\Omega }_{1_{M\text{+}q}}^{\pm }\right|}^2/{\Gamma }_{0_{M+q}{0}_M}$ suppresses the MWM (FWM + SWM + EWM) signals, and ${\left|{\Omega }_{4_{M+q}}\right|}^2/\left[{\Gamma }_{4_{M+q}{0}_M}+i({\Delta }_1+{\Delta }_4)\right]$ plays the dual role of modulating the EWM and suppressing the SWM signals. The field E₄ plays the vital role of suppressing and enhancing EWM under the two-photon resonant condition ${\Delta }_1+{\Delta }_4=0$. Figure 3(a1) shows the SWM signals, which unexpectedly decrease with increases of E₄ because ${\left|{\Omega }_{1_{M\text{+}q}}^{\pm }\right|}^2/{\Gamma }_{0_{M+q}{0}_M}<<{\left|{\Omega }_{4_{M+q}}\right|}^2/\left[{\Gamma }_{4_{M+q}{0}_M}+i({\Delta }_1+{\Delta }_4)\right]$ in Eq. (8), which is majorly suppressed by E₄ with a smaller E₁ power. This result illustrates that under a small power of the probe field, the enhancement effect of E₄ is smaller than the suppression effect in Eq. (8).

 

Fig. 3 MWM signals with different powers of E₄ by scanning Δ₂ when Δ₁ = −400 MHz. Signal evolution with fixed E₄ power by setting the E₁ power to 100 μW (Ω₁ = 2π × 10 MHz) (a1), 500 μW (Ω₁ = 2π × 54 MHz) (a2), and 1 mW (Ω₁ = 2π × 108 MHz) (a3). Corresponding dependence curves are shown in (b). Corresponding images of the probe (c) and SWM (d) at different powers of E₁ (Ω₂ = 2π × 7.6 MHz; Ω₃ = 2π × 142 MHz; Ω₃′ = 2π × 224 MHz; and maximum Ω₄ = 2π × 140 MHz). The color coding is same with Fig. 2.

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Figure 3(a2) shows that the EWM signals increase with P₄ from 0 mW to 4 mW and the suppression effect of E₄ increases; subsequently, the EWM signal disappears when E₄ increases from 5 mW to 8 mW. This phenomenon can be easily understood by the competition relation between ${\left|{\Omega }_{1_{M\text{+}q}}^{\pm }\right|}^2/{\Gamma }_{0_{M+q}{0}_M}$ and ${\left|{\Omega }_{4_{M+q}}\right|}^2/\left[{\Gamma }_{4_{M+q}{0}_M}+i({\Delta }_1+{\Delta }_4)\right]$in Eq. (8). Initially, ${\left|{\Omega }_{1_{M\text{+}q}}^{\pm }\right|}^2/{\Gamma }_{0_{M+q}{0}_M}>{\left|{\Omega }_{4_{M+q}}\right|}^2/\left[{\Gamma }_{4_{M+q}{0}_M}+i({\Delta }_1+{\Delta }_4)\right]$, and the enhanced EWM is observed. When the power of E₄ is increased to the maximum, the term ${\left|{\Omega }_{4_{M+q}}\right|}^2/\left[{\Gamma }_{4_{M+q}{0}_M}+i({\Delta }_1+{\Delta }_4)\right]$ becomes larger than ${\left|{\Omega }_{1_{M\text{+}q}}^{\pm }\right|}^2/{\Gamma }_{0_{M+q}{0}_M}$, which leads to the suppressed SWM. In Fig. 3(a3), when P₄ increases, the suppression dip gradually becomes shallow; then, the EWM peak is observed when ${\left|{\Omega }_{1_{M\text{+}q}}^{\pm }\right|}^2/{\Gamma }_{0_{M+q}{0}_M}>>{\left|G_{4_{M+q}}\right|}^2/\left[{\Gamma }_{4_{M+q}{0}_M}+i({\Delta }_1+{\Delta }_4)\right]$ is satisfied in Eq. (8). This regularity is clearly observed in the dependence curves in Fig. 3(b), where the dots are the experimental results and the solid curves are the theoretical simulations according to Eq. (8). The theoretical subtracting offset is basically consistent with the results of the experiment. Moreover, a chaos-like phenomenon appears in the spectrum signals when P₄ is strong as shown in Fig. 3(a). The chaos-like phenomenon in these signals represents the baseline oscillation, and the background of the spectrum varies from peace to noise as shown in the upper part of Fig. 3(a), which is caused directly by the interaction between different orders of MWM and can be attributed to the intrinsic Kerr nonlinearity enhancement in the medium, where dynamic instability and the route to chaos occur [28].

The circularly polarized dressed spatial images are shown in Figs. 3(c) and 3(d). Figure 3(c) shows the probe images with different probe field powers and reflects the intensity of the doubly dressing EIT windows. Figure 3(d) shows the EWM images as the power of the probe field increases from bottom to top. In these processes, the circular polarization of the probe field is fixed. When the power of the probe field increases from 400 μW to 800 μW, the spatial shift and focusing of the images are observed clearly at approximately Δ₂ = 0 MHz in Fig. 3(d). Based on an analysis of Fig. 2, the polarized nonlinear phase and optical field superposition make the image styles changed. The focusing is attributable to the phase modulation caused by Rydberg interactions. The spatial shift phenomenon, which cannot be observed in the SWM images, is a result of EWM. A comparison of Figs. 3(d3) and 3(d1) shows that the obvious spatial shift is related to EWM. The spatial shift can be described by the nonlinear phase shift equation ${\varphi }_2\text{=}2k_{2}z{n}_2^{}{I}_2{e}^{-{(\varsigma -{\varsigma }_2)}^2/2}/n_0{I}_{F,S}$. Based on the equation, the Rydberg interaction leads to a change in n₂, and FWM and SWM are suppressed; moreover, as I_F,S becomes small, ${\varphi }_2$becomes strong. These results advocate that EWM is more sensitive to Rydberg-Rydberg interaction [19].

In Fig. 4, we investigate the EWM signals as a function of detuning Δ₄. Figure 4(a) shows the evolution of the Rydberg MWM signal when Δ₂ is scanned at discrete Δ₄. The circularly polarized probe field is appropriately selected so that the Rydberg SWM becomes a pure suppression dip. When |Δ₄| = 100 MHz, a large detuning of E₄ has a limited effect on the Rydberg SWM because the dressing effect is weaker. When |Δ₄| is tuned to 50 MHz, a spectrum with a half dip and a half peak appears, which illustrates that E₄ begins to enhance the SWM to generate the observable EWM. When Δ₄ = 0, the two-photon resonant condition is satisfied; therefore, the stronger EWM is observed. Figure 4(a) clearly shows the evolution from the SWM dip to the EWM peak. Figure 4(b) shows the simulated results obtained with the expression of ${\rho }_{1_{M+q}{0}_M}^{(7)}$and ${\rho }_{1_{M+q}{0}_M}^{(5)}$, and the values are scaled according to experimental data. Such trend of simulation results are basically accord with experimental observations. This evolution process is easily controlled in the experiment and can be effectively applied to all-optical switches. Based on this idea as an example of application, we demonstrate different degrees of switching in which the E₄ field is turned ON and OFF using a manual switch. As shown in Fig. 4(c), with E₄ ON and OFF, the signals can be switched between SWM dip and EWM peak, and the intensity increases as the power of E₁ increases.

 

Fig. 4 (a) Variation of MWM signals with Δ₄ when Δ₂ is scanned and Δ₁ is fixed at −400 MHz. (b) Theoretical simulation that corresponds to (a). (c) Switch between SWM and EWM by controlling the power of E₁ by scanning Δ₂. Corresponding Rabi frequency at Ω₁ = 2π × 54 MHz; Ω₂ = 2π × 7.6 MHz; Ω₃ = 2π × 142 MHz; Ω₃′ = 2π × 224 MHz; and Ω₄ = 2π × 100 MHz.

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Finally, to directly control the EWM with polarization, variations of the EWM with the polarization of the dressing field E₄ are shown in Fig. 5. Figure 5(a1) shows the EWM signals that vary with the polarization of E₄ in the circularly polarized probe field. In this case, with the scanned Δ₂, the dips on the SWM signals are shown at the top right corner of the figure, and each peak represents pure EWM created in the |0〉↔|1〉↔|3〉↔|1〉↔|2〉↔|1〉↔|4〉↔|1〉 excitation process. When the angle of E₄ QWP changes from 0° to 45°, the polarization of E₄ changes from linearly polarized to circularly polarized, which decreases the size of the EWM peaks. This phenomenon can be explained as the dressing effect of the circularly polarized E₄, which is stronger than the linearly polarized E₄. The corresponding CG coefficient is calculated and labelled in Fig. 5(c), and ${\left(\left|{\Omega }_{4_{M+1}}^{\text{+}}\right|\right)}^2/{\left(\left|{\Omega }_{4_M}^0\right|\right)}^2=3$ is obtained. As shown in Fig. 5(a1), the height of the EWM signals decreases and subsequently increases because the E₄ polarization is adjusted from circular into linear when the angle of the QWP changes from 45° to 90°. To avoid the effect of detuning Δ₁ on this regularity, Fig. 5(a2) shows the case in Fig. 5(a1) but with Δ₁ = −700 MHz. Compared with Fig. 5(a1), the same regularity is presented, although each EWM peak is smaller because of the larger detuning Δ₁. Figure 5(b) shows the dependence curves that correspond to Fig. 5(a1) (black) and Fig. 5(a2) (red), where the dots are experimental observations and solid curves are the theoretic simulations. Here, solid curves are simulated by the overall factors of Eq. (8), and the offset in the experimental observations has been subtracted. The theory is consistent with the experiment.

 

Fig. 5 (a1) EWM signals versus Δ₂ at different polarizations of the dressing field E₄ when the probe field is circularly polarized and Δ₁ is fixed at −400 MHz. (a2) Identical to (a1) but with Δ₁ = −700 MHz. (b) Dependence curves with respect to the angle of QWP added on E₄. (c) Corresponding Zeeman-level diagram between 5P_3/2 and 5D_5/2. The Rabi-frequencies of each field are identical to those in Fig. 4.

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5. Conclusion

In summary, we directly detect the pure Rydberg EWM and present a series of detailed experiment results on multi-parameter control mechanism. To obtain such Rydberg high order processes in the experiment is quite challenging. Nonlinearity is small for high-order process, the characters of nonlinear process in our work are technically obtained by electromagnetically induced transparency (EIT) changes with the control fields; EIT enhances the nonlinearities together with polarized filed dressing make high contrast EWM spectrum are extracted directly and then be controlled by the power, detuning, and polarization of the dressing fields. Coexisting SWM and EWM are modulated in different style, in which one is suppression dip, the other is enhancement peak. Such method overcomes the problem that Rydberg EWM always mixes with Rydberg SWM so that relatively strong Rydberg EWM spectrum can be obtained in lab and controlled directly, this controllable EWM process with an ultra-narrow linewidth is sensitive to Rydberg-Rydberg interactions and has the potential to become an effectively non-destructive detection method for investigating the underlying Rydberg property [29] and could be applied in quantum long-distance communication [30]. In addition, the images of the probe and MWM are presented to reflect the nonlinear response of Rydberg–Rydberg interaction. The common cases of focusing and phase shifting are observed, which can reflect Rydberg nonlinear phase variation and establish the relation between refractive index of Rydberg medium and optical field. This study will not only exploit a new method of investigating the phase control of Rydberg nonlinear processes and spatial transmission characteristics, but also provide the basis for the advanced design that is highly dependent on the refractive index materials; more importantly has research value in the field of optical information processing.