1. Introduction

Rydberg atoms and dipolar molecules attract more and more attention due to their potential applications in quantum computing and scalable quantum information processing [1, 2], etc. It is worth mentioning that the quantum logic gate is designed by employing the sensitivity of highly excited state energy to the interaction between neighboring Rydberg atoms [3, 4]. With the development of modern laser cooling and trapping techniques, ultracold Rydberg gases and plasmas have been experimentally created, in which the Rydberg interaction is very strong that has been accurately calculated [5–9]. The mutual interactions between Rydberg atoms, include van der Waals and dipole-dipole interactions, can shift the energy levels and prevent more than one atom in sizable spatial domain from being excited to the Rydberg state by a resonant laser field, i.e., the van der Waals [10–12] or resonant dipole-dipole blockade [13–16], which makes the Rydberg atomic system a promising candidate to produce quantum logic gates.

As an atomic coherence phenomena, electromagnetically induced transparency (EIT) has been investigated intensively in last two decades [17, 18], because the absorption of a probe beam in a medium can be significantly suppressed [19] in the EIT window. Along with suppressed linear absorption, large nonlinear susceptibility has been obtained under EIT [20, 21], and many nonlinear phenomena have been effectively enhanced [22, 23]. Especially, the coexisting four-wave mixing (FWM) and six-wave mixing (SWM) [24, 25] and even higher-order nonlinear processes [26] have been experimentally demonstrated. Recently, it is encouraging that many atomic coherence phenomena, such as coherent population trapping [27], stimulated Raman adiabatic passage [28], EIT [29] and even the anti-blockade by dressing effect [30, 31], have been demonstrated in Rydberg atomic assemble. The FWM involving Rydberg states in the diamond-type atomic system has also been experimentally demonstrated in ultracold atoms [32] and thermal vapor [33], respectively. However, the influence of the Rydberg blockade effect on the FWM process was not explicitly discussed and investigation in higher order nonlinear phenomena was not involved.

In this paper, we first theoretically investigate the FWM and SWM generation in the atomic systems with Rydberg levels. In both diatomic and quadratomic systems, we not only demonstrate the modulation of EIT and electromagnetically induced absorption (EIA) of the probe transmission, but also demonstrate enhancement, suppression and avoided crossing of the dressed multi-wave mixing (MWM) signals through the interplays between the dressing effect and primary as well as secondary Rydberg blockades, which can be controlled by the external electric field intensity and atomic internuclear distance. In addition, we also demonstrate the anti-blockade effect in FWM process.

The paper is organized as follows. In Section 2, we present the basic theory about the MWM and Rydberg blockade. In Section 3, the interactions between the dressing effect and primary blockade in the diatomic system are investigated. In Section 4, we use the primary and secondary blockades in the quadratomic system to modulate the MWM processes. In Section 5, we present the anti-blockade in FWM process.

2. Basic theory

The atomic system and relevant beam geometry are shown in Figs. 1(a) and 1(b), respectively. This five-level system can be experimentally constructed by the energy levels 5S_1/2 (F = 3, |0>), 5S_1/2 (F = 2, |3>), 5P_1/2 (|1>), 5D_3/2 (|2>), and 70S_1/2 (or 69D_3/2, |4>) of ⁸⁵Rb. The resonant wavelengths (frequencies) in this system are 795 nm (Ω₁≈Ω₃ = 377.1 THz) from both of 5S_1/2 (F = 3) and 5S_1/2 (F = 2) to 5P_1/2, 762.1 nm (Ω₂ = 393.3 THz) for 5P_1/2 to 5D_3/2, and 474.2 nm (Ω₄ = 632.18 THz) for 5P_1/2 to 70S_1/2, respectively. If |4> is 69D_3/2, the resonant frequency will be Ω₄ = 632.2 THz. The probe beam E₁ (frequency ω₁, wave vector k₁, Rabi frequency G₁ and frequency detuning Δ₁) connects the transition |0>−|1>; E₂ (ω₂, k₂, G₂ and Δ₂) and ${E^{\prime }}_2$ (ω₂, ${k^{\prime }}_2$, ${G^{\prime }}_2$ and Δ₂) drive |1>−|2>; E₃ (ω₃, k₃, G₃ and Δ₃) and ${E^{\prime }}_3$ (ω₃, ${k^{\prime }}_3$, ${G^{\prime }}_3$ and Δ₃) connect |1>−|3>; E₄ (ω₄, k₄, G₄ and Δ₄) drives |1>−|4>. The Rabi frequency G_i is defined as G_i = μ_ijE_i/ћ, where μ_ij is the transition dipole moment between |i> and |j>. The frequency detuning Δ_i is defined as Δ_i = Ω _i −ω_i.

 

Fig. 1 (a) The diagram of ⁸⁵Rb energy levels with different coupling schemes in the five-level system. (b) The beam geometry diagram, in which the elliptic sample is designed so that atom density is uniform and the statistical variation of internuclear distance R can be neglected. (c) and (d) The diatomic systems consisting of 3 × 3 and 4 × 4 levels, respectively. The lines correspond to different light beams. Solid lines: E₁; long dashed lines: E₂ (${E^{\prime }}_2$); short dashed lines: E₄; dash-dotted lines: E₃ (${E^{\prime }}_3$). (e) The energy level shift curves for different symmetries of 70s-70s (dashed lines), 70p-70p (dotted lines) and 69d-69d (solid lines) of rubidium. (f) and (g) The diagrams of dressed FWM process in a Y-type four-level diatomic system and dressed SWM process in a reverse-Y type four-level diatomic system, with Rydberg blockade, respectively.

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In such beam geometry configuration, the two-photon Doppler-free condition will be satisfied in the two ladder-type subsystems |0>−|1>−|2> and |0>−|1>−|4> with two EIT windows and different dressed MWM signals occurring. When all the beams except E₃ and ${E^{\prime }}_3$ are on, a FWM signal E_F dressed by strong beam E₄ satisfying the phase-matching condition k_F = k₁ + k₂−${k^{\prime }}_2$ can be generated in the |0>−|1>−|2> subsystem. While if all the beams except E₂ and ${E^{\prime }}_2$ are on, a SWM signal E_S satisfying k_S = k₁ + k₃−${k^{\prime }}_3$ + k₄−k₄ will be generated in the |0>−|1>−|3>−|4> subsystem.

There is considerable interaction between two atoms with Rydberg levels, which can form a diatomic system. First, for two single atoms in the subsystem |0>−|1>−|2>, 3 × 3 = 9 energy levels will be generated in the corresponding diatomic system, i.e., |00>, |01> (|10>), |11>, |02> (|20>), |12> (|21>) and |22> with eigen-energies being 0, ћΩ₁, 2ћΩ₁, ћΩ₂, ћ(Ω₁ + Ω₂) and 2ћΩ₂, respectively, as shown in Fig. 1(c). Next, when a four-level subsystem |0>−|1>−|3>−|4> is involved, we can obtain 4 × 4 = 16 states as shown in Fig. 1(d).

In general, the intensity of the generated MWM is proportional to the amplitude of the corresponding density matrix element ρ₁₀ (ρ₁₀₀₀ for diatomic system), which can be obtained by solving the density-matrix equations with perturbation method. First, when only E₁, E₂ and ${E^{\prime }}_2$ are on, the FWM process (denoted as F1) generating E_F happens in the |00>−|01>−|20> ladder-type EIT window formed by the diatomic levels. Without considering the dressing effect, by the perturbation chain (I) ${\rho }_{0000}^{(0)}\stackrel{{\omega }_1}{\to }{\rho }_{1000}^{(1)}\stackrel{{\omega }_2}{\to }{\rho }_{2000}^{(2)}\stackrel{-{\omega }_2}{\to }$ ${\rho }_{1000}^{(3)}$, we can obtain the undressed third-order density element ${\rho }_{1000}^{(3)}=G_F/d_{10}^2{d}_{20}$ for E_F, with $G_F=-i{G}_1{G}_2{({G^{\prime }}_2)}^{*}\exp (i{k}_F\cdot r)$, d₁₀ = Γ₁₀ + iΔ₁, d₂₀ = Γ₂₀ + i(Δ₁ + Δ₂) and Γ_ij being the transverse relaxation rate between |i> and |j>. Then, when all the beams except E₂ and ${E^{\prime }}_2$ are on, the SWM process S1 generating E_S happens in the |00>−|10>−|40> ladder-type EIT window. By the undressed perturbation chain (II) ${\rho }_{0000}^{(0)}\stackrel{{\omega }_1}{\to }{\rho }_{1000}^{(1)}\stackrel{-{\omega }_3}{\to }{\rho }_{3000}^{(2)}\stackrel{{\omega }_3}{\to }$ ${\rho }_{1000}^{(3)}\stackrel{{\omega }_4}{\to }{\rho }_{4000}^{(4)}\stackrel{-{\omega }_4}{\to }{\rho }_{1000}^{(5)}$, we can obtain the undressed fifth-order density element ${\rho }_{1000}^{(5)}=G_S/(d_{10}^3{d}_{30}{d}_{40})$ for E_S, with $G_S=i{G}_1{G}_3{({G^{\prime }}_3)}^{*}{G}_4{(G_4)}^{*}\exp (i{k}_S\cdot r)$ and d₃₀ = Γ₃₀ + i(Δ₁−Δ₃). Even though E_F and E_S propagate in approximately the same direction, we can separate them in the experiment by selecting different EIT windows.

When E₁, E₂, ${E^{\prime }}_2$ and E₄ are all on, E_F will be dressed and we rewrite it as $E_F^D$. Autler-Townes splitting of self-dressing and external-dressing will occur when energy-levels are dressed, the splitting condition as Fig. 2 . (d1-d9). Level |1> will be split into two dressed levels |G_{4 ±} >, and the corresponding diatomic level |10> will split into |G_{4 ±} 0>. An illustration of such level-splitting can be found in Fig. 2 (d1). As a result, the dressed perturbation chain of FWM process F1 changes into (III) ${\rho }_{0000}^{(0)}\stackrel{{\omega }_1}{\to }{\rho }_{G_4{}_{\pm }000}^{(1)}\stackrel{{\omega }_2}{\to }{\rho }_{2000}^{(2)}\stackrel{-{\omega }_2}{\to }{\rho }_{G_4{}_{\pm }000}^{(3)}$, and we can obtain ${\rho }_F^{(3)}=G_F/[d_{20}{(d_{10}+|G_4{|}^2/d_{40})}^2]$, with d₄₀ = Γ₄₀ + i(Δ₁ + Δ₄). Because E₄ does not participate in F1 process directly, we call it as external-dressing field. When E₂ and ${E^{\prime }}_2$ are strong sufficiently, they will also bring dressing effect, that |10> will be first split into |G_{2 ±} 0> by E₂ (${E^{\prime }}_2$), and then split into third energy levels |G₂₊G_{4 ±} 0> or |G_2-G_{4 ±} 0>. As a result, the dressed perturbation chain responsible for the FWM process F1 generating the doubly-dressed signal $E_F^{DD}$ changes into (IV) ${\rho }_{0000}^{(0)}\stackrel{{\omega }_1}{\to }{\rho }_{G_2{}_{\pm }{G}_4{}_{\pm }000}^{(1)}\stackrel{{\omega }_2}{\to }{\rho }_{2000}^{(2)}$ $\stackrel{-{\omega }_2}{\to }{\rho }_{G_2{}_{\pm }{G}_4{}_{\pm }000}^{(3)}$, and we can obtain

 

Fig. 2 (a) The theoretical calculations of the intensity of (a1) the probe transmission and (a2) the doubly-dressed FWM signal $E_F^{DD}$ versus Δ₁ at discrete internuclear distances R = 77000, 78800, 80200, 82000 and 84000 a.u. (Δ₄ is fixed at Δ₄ = −70 MHz). (b) The theoretical calculations of the intensity of (b1) the probe transmission and (b2) $E_F^{DD}$versus R at discrete probe detunings Δ₁ = −50, −25, −11, −5, 0, 5, 11, 25, and 50 MHz (Δ₄ is fixed at Δ₄ = −60 MHz). (c) The theoretical calculations of the intensity of $E_F^{DD}$ versus (c1) probe detuning Δ₁ (horizontal axis) and dressing detuning Δ₄ (vertical axis), (c2) Δ₁ and R, and (c3) Δ₄ and R. (d) The energy levels corresponding to (b). (d1) to (d9) correspond to the nine curves from left to right in (b1) and (b2). The beam connecting |10>−|20> split |10> into |G_{2 ±} 0>, and |G₂₊0> will be split into |G₂₊G_{4 ±} 0> by the beam connecting |10>−|40>.

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The above equations obviously show that the MWM signal intensities are intensively determined by the frequency detunings of the involved fields. In the case of Rydberg atoms, the atom-atom interaction will bring new characteristics to MWM signal by affecting the frequency detunings. Due to the long-range interaction potential between Rydberg atoms with high principal quantum number n, there will be an energy level shift from the unperturbed energy level. Such energy shift could be called as blockade effect. To distinguish from the secondary blockade which will be introduced in the quadratomic system in Section 4, we call this effect in the diatomic system as the primary blockade. Since the potential depends on the atomic internuclear distance R between two atoms, the energy shift also has such dependence and can be approximated by the polynomial of 1/R, which is different for Rydberg states with different angular quantum number. For instance, ΔE_ns(R)≈−C₆/R⁶−C₈/R⁸−C₁₀/R¹⁰ is for ns-ns interaction, ΔE_np(R)≈−C₅/R⁵−C₆/R⁶−C₈/R⁸ for np-np interaction, and ΔE_np(R)≈−C₅/R⁵−C₆/R⁶−C₇/R⁷ for nd-nd interaction [5]. In Fig. 1(e), these energy level shift curves around n = 70 in Rubidium are given. The internuclear distance R can be experimentally controlled by changing the atomic intensity, i.e., the cooling conditions in magneto-optical trap (MOT) in ultracold gas or the temperature in hot atom vapor.

When the influence of blockade effect on the dressed FWM process is investigated, level |4> is set as 70S_1/2, and the corresponding energy shift of level |40> in the diatomic system can be written as ΔE_70s(R). With this energy level shift, the resonant frequency Ω₄ will change into ${{\Omega }^{\prime }}_4={\Omega }_4+\Delta E_{70s}(R)/\hslash$. As a result, the modified frequency detuning can be written as ${{\Delta }^{\prime }}_4={\Delta }_4+\Delta E_{70s}(R)/\hslash$. Therefore, the matrix density element responsible for the singly-dressed FWM is modified into ${\rho }_F^{(3)}=G_F/[d_{20}{(d_{10}+|G_4{|}^2/{d^{\prime }}_{40})}^2]$ with ${d^{\prime }}_{40}={\Gamma }_{40}+i[{\Delta }_1+{\Delta }_4\text{+}\Delta E_{70s}(R)/\hslash ]$. And for doubly-dressed FWM, we have

3. Primary blockade of MWM in the diatomic system

3.1 Doubly-dressed FWM

It is well known that the probe transmission and the enhancement as well as suppression of the FWM signal can be controlled by the dressing effect of light field which modifies the unperturbed levels significantly, and affected by the detuning of dressing field strongly. As the Rydberg blockade could shift energy levels, it can modulate the probe transmission and FWM signal via the modified dressing field detuning. In order to better explain the following theoretical calculations, we resort to the diatomic doubly-dressed energy level diagram in Fig. 2(d). Due to the doubly-dressing effect, |10> is totally split into three or two levels as shown in Fig. 2(d). In the region with Δ₁<0, when Δ₄ + ΔE_70s(R)/ħ is around |G₂₊0>, two secondarily split levels |G₂₊G_{4 ±} 0> could be created from |G₂₊0> by the external-dressing effect as shown in Figs. 2(d1)-(d4). Symmetrically, in the region with Δ₁>0, when Δ₄ + ΔE_70s(R)/ħ is around |G_2-0>, |G_2-G_{4 ±} 0> could be obtained as shown in Figs. 2(d6)-(d9). When R is adjusted to make Δ₄ + ΔE_70s(R)/ħ resonant with the midpoint of |G_{2 ±} 0>, the first and secondary dressing effects can together split |10> into |G₂G_{4 ±} 0>, as shown in Fig. 2(d5).

First of all, the effect of Rydberg blockade can be revealed when Δ₁ is scanned continuously at different R. Under the doubly-dressing effect, for a given R set at discrete values orderly from 77000 to 84000 a.u., which correspond to discrete Δ₄ + ΔE_70s(R)/ħ values from 100 to −100 MHz, the probe transmission always shows double EIT windows as shown in Fig. 2(a1), which are in the subsystems |00>−|10>−|20> (appearing at Δ₁ + Δ₂ = 0) and |00>−|10>−|40> (appearing at Δ₁ + Δ₂ + ΔE_70s(R)/ħ = 0), respectively. When Δ₂ = 0 is fixed and R is changed, the EIT window |00>−|10>−|20> is fixed at Δ₁ + Δ₂ = 0 but the EIT window |00>−|10>−|40> moves. Especially, when R = 80200 a.u. (Δ₄ + ΔE_70s(R)/ħ = 0), the two EIT windows overlap as shown in Fig. 2(a1), and a double-peak FWM signal is obtained (Fig. 2(a2)), in which the two peaks correspond to the resonance of the FWM signal to |G₂G₄₊0> and |G₂G_4-0>, as shown in Fig. 2(d5). When R≠80200 a.u. (Δ₄ + ΔE_70s(R)/ħ≠0) and Δ₁ is scanned from negative to positive, the FWM signal presents three peaks, which correspond to the resonances to three dressed states, respectively. Firstly, resonance to the levels |G_{2 ±} 0> leads to the two primary Autler-Townes (AT) splitting peaks. Then, as shown in Fig. 2(a1), when R is changed to move the |00>−|10>−|40> EIT window onto the left FWM peak (Δ₄ + ΔE_70s(R)/ħ>0), secondary AT splitting occurs and the left peak in the primary splitting is broken into two peaks, which correspond to the resonance of FWM signal to the two secondarily dressed states |G₂₊G₄₊0> and |G₂₊G_4-0>, respectively (Fig. 2(a2)). Symmetrically, in the region withΔ₄ + ΔE_70s(R)/ħ<0 where Δ₁ is scanned, the three peaks structure correspond to |G₂₊0>, |G_2-G₄₊0> and |G_2-G_4-0>, respectively.

The effect of Rydberg blockade can be also revealed when R is scanned at different Δ₁. In Fig. 2(b1), the baseline height of each curve at fixed Δ₁ represents the probe transmission without the dressing effects of E₄, while the peak and dip on each baseline represent EIT and EIA, respectively. On the other hand, the baseline height of each curve at fixed Δ₁ in Fig. 2(b2) represents the FWM signal without the dressing effects of E₄, while the peak and dip on each baseline respectively represent the enhancement and suppression of $E_F^{DD}$, which respectively correspond to EIA or EIT of the probe transmission.

Compared to the scanning of Δ₁ in each curve in Fig. 2(a), the scanning of R in Fig. 2(b) can directly reveal the positions of dressed states without the classical absorption dip in probe transmission and two-photon Lorentz lineshape in FWM signal. At different Δ₁, R can be always controlled to change in a region to guarantee that probe field has two-photon resonance or dressed state resonance. When Δ₂ = 0 is fixed, and R is scanned with Δ₁ = 0, on the one hand, the probe transmission shows pure-EIT (Fig. 2(b1)) at the two-photon resonance point Δ₄ + ΔE_70s(R)/ħ = 0. On the other hand, a pure-suppression dip of $E_F^{DD}$ is correspondingly obtained (Fig. 2(b2)) because the probe field E₁ could not resonate with |G₂G_{4 ±} 0>, as shown in Fig. 2(d5), which weakens the wave mixing nonlinearity significantly. So we call the condition Δ₁ + Δ₄ + ΔE_70s(R)/ħ = 0 as suppression condition. Second, with Δ₁ = −5 MHz, the probe transmission shows first an EIA dip and next a EIT peak (Fig. 2(b1)). Correspondingly, $E_F^{DD}$ is first enhanced and next suppressed (Fig. 2(b2)). The reason for the first EIA and corresponding enhancement of $E_F^{DD}$ is that E₁ first resonates with the dressed state |G₂₊G_4-0> when R is scanned from small to large (as a result ΔE_70s(R) of |40> changes from large to small), which enhances the probe absorption and nonlinearity. Thus, we call ${\Delta }_1+{\lambda }_{G_{\text{2+}}{G}_{\text{4-}}}=0$ as enhancement condition, in which ${\lambda }_{G_{\text{2+}}{G}_{\text{4-}}}={{{\Delta }^{\prime }}^{\prime }}_4/2+\sqrt{{({{{\Delta }^{\prime }}^{\prime }}_4/2)}^2+|G_4{|}^2}+G_2$ is the difference between the eigen-frequency of |G₂₊G_4-0> and that of undressed |10>, where ${{{\Delta }^{\prime }}^{\prime }}_4={\Delta }_4$ + ΔE_70s(R)/ħ–G₂ represents the detuning of E₄ from |G₂₊0>. While the reason for the next EIT and corresponding suppression of $E_F^{DD}$ is that two-photon resonance occurs so as to satisfy the suppression condition (Fig. 2(d4)). Third, when Δ₁ = −11 MHz, a pure-EIT of probe field and corresponding pure-suppression of $E_F^{DD}$ occur, as E₁ could not resonate with |G₂₊G_{4 ±} 0>, as shown in Fig. 2(d3). Fourth, when Δ₁ = −25 MHz, the probe transmission shows first EIT with corresponding suppression of $E_F^{DD}$ because the suppression condition is satisfied. While the reason for the following EIA in probe transmission and corresponding enhancement of $E_F^{DD}$ is that E₁ resonates with |G₂₊G₄₊0> and the enhancement condition ${\Delta }_1+{\lambda }_{G_{\text{2+}}{G}_{\text{4+}}}=0$ is satisfied, with ${\lambda }_{G_{\text{2+}}{G}_{\text{4+}}}={{{\Delta }^{\prime }}^{\prime }}_4/2-\sqrt{{({{{\Delta }^{\prime }}^{\prime }}_4/2)}^2+|G_4{|}^2}+G_2$. Finally, when Δ₁ = −50 MHz, far away from the resonance point (Fig. 2(d1)), the pure-EIA of probe transmission as well as the pure-enhancement of $E_F^{DD}$ are obtained because E₁ could only resonant with |G₂₊G₄₊0>. When R is scanned in the Δ₁>0 region and the values of Δ₁ are chosen symmetrically to those negative ones above, we can obtain that the behavior of probe transmission (FWM) is mirror symmetrical to those obtained above with respect to Δ₁ = 0, as shown in Fig. 2(b). It should be noted that the EIA of probe transmission or enhancement of FWM signal in the Δ₁>0 region is due to E₁ or $E_F^{DD}$ resonance to |G_2-G_{4 ±} 0>, as shown in Figs. 2(d6)-(d9).

From a global perspective (Fig. 2(b2), dashed line), we can see that the global profile constructed by the baseline of each FWM signal curve at different Δ₁ has two peaks, revealing the AT splitting due to the self-dressing effect. The variation in each curve, i.e., transition between enhancement and suppression, shows three symmetric centers at Δ₁ = 0, $-$11 and 11 MHz due to the interaction between E₂ (${E^{\prime }}_2$) and E₄. The pure suppression at Δ₁ = 0 is induced by the self-dressing effect, while pure suppressions at Δ₁ = ± 11 MHz are caused by the external-dressing effect. Correspondingly, the global profile (dashed line) of the probe transmission in Fig. 2(b1) shows an EIT window within an absorption dip. It also has three symmetric centers, which are the same as those of the FWM signal.

Furthermore, Figs. 2(c1) and 2(c2) present the intensity of the doubly-dressed FWM signal $E_F^{DD}$ versus Δ₁ and Δ₄, Δ₁ and R, respectively. They are the typical avoided-crossing plots in the diatomic system, which reflect the twice AT splittings (along the horizontal axis) due to E₂ (${E^{\prime }}_2$) and E₄, and the transition between enhancement and suppression (along the vertical axis) due to the interactions among the dressing fields. Figure 2(c3) presents the intensity of $E_F^{DD}$ versus Δ₄ and R, which reflects the interaction between the dressing effect and primary Rydberg blockade. The trajectory of the main bright line has a rightward deflection, because the energy level shift ΔE_70s(R) decreases with increasing R, as shown in Fig. 1(e).

3.2 Singly-dressed SWM

The singly-dressed SWM process S1 when power of E₃ is weak in the diatomic reverse Y-type four-level system mentioned in Section 2 can be also controlled by the blockade effect, as shown in Fig. 3 . Firstly, when Δ₁ is scanned for a given R, the probe transmission always shows an EIT window on each curve (Fig. 3(a1)) at Δ₁ + Δ₄ + ΔE_69d(R)/ħ = 0. As R is changed from 58000 to 68300 a.u. (Δ₄ + ΔE_69d(R)/ħ correspondingly changes from $-$70 to 70 MHz), the EIT window |00>−|10>−|40> moves from right to left. Meanwhile, $E_S^D$ presents double peaks at the positions where E₁ resonates with |G_{4 ±} 0> as shown in Fig. 3(a2), due to AT splitting.

 

Fig. 3 (a) The theoretical calculations of the intensity of (a1) the probe transmission and (a2) the singly-dressed SWM signal $E_S^D$ versus ${\Delta }_1$ at discrete internuclear distances R = 58000, 59500, 62000, 65000 and 68300 a.u. (${\Delta }_4$ is fixed at ${\Delta }_4$ = 100 MHz). (b) The theoretical calculations of the intensity of (b1) the probe transmission, (b2) the enhancement and suppression of $E_S^D$ without considering ${d^{\prime }}_{40}$, and (b3) the complete $E_S^D$ versus R at discrete probe detunings ${\Delta }_1$ = −80, −20, 0, 20, and 80 MHz (${\Delta }_4$ is fixed at ${\Delta }_4$ = 90 MHz). (c) The theoretical calculations of the intensity of $E_S^D$ versus (c1) ${\Delta }_1$ (horizontal axis) and dressing detuning ${\Delta }_4$ (vertical axis), (c2) ${\Delta }_1$ and R, and (c3) ${\Delta }_4$ and R. (d) The energy levels corresponding to (b). (d1) to (d5) correspond to the curves from left to right in (b).

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Next, when R is scanned with fixed Δ₁ and Δ₃, as shown in Fig. 3(b3), the behavior of $E_S^D$ will appear as a mixture effect of a single-peak line shape due to classical two-photon resonance emission, and the enhancement peak (suppression dip) due to dressing effect, which corresponds to the terms ${d^{\prime }}_{40}$ and $d_{10}+|G_4{|}^2/{d^{\prime }}_{40}$ in Eq. (4), respectively. To emphasize the interaction between the light field dressing effect and Rydberg blockade, we also calculate by eliminating the term ${d^{\prime }}_{40}$. The results of the two cases are shown in Figs. 3(b3) and 3(b2), respectively.

As shown in Fig. 3(b1), when R is scanned from small to large, for the curves from left to right with Δ₁ from $-8\text{0}$ to 80 MHz, the probe transmission shows the evolution from pure-EIA, to first EIA and then EIT, to pure-EIT, to first EIT and then EIA, finally to pure-EIA. Correspondingly, the SWM signal obtained in complete expression shows the evolution from single peak, to two unequal-height peaks, to two equal-height peaks, again to unequal-height two peaks, and finally to single peak (Fig. 3(b3)). Eliminating the dressing and blockade effect, we can find that the SWM signal shows the evolution from pure-enhancement, to first enhancement and then suppression, to pure-suppression, to first suppression and then enhancement, and finally to pure-enhancement (Fig. 3(b2)). It is obvious that every enhancement and suppression in Fig. 3(b2) correspond to EIA and EIT in Fig. 3(b1), respectively, which is similar to the case of doubly-dressed FWM $E_F^{DD}$ obtained in Fig. 2.

In order to better understand the phenomena mentioned above, we resort to the singly-dressed energy diagrams in the diatomic system (Fig. 3(d)). Due to the self-dressing effect of E₄, the energy level |10> is split into two dressed states |G_{4 ±} 0>, as shown in Fig. 3(d1)-(d5). We should note here that E₄ participates in the generation of E_S directly, so E₄ also offers self-dressing effect, that’s to say there is no external-dressing effect for SWM process. When R is scanned at Δ₁ = 0, the pure-EIT of probe transmission and pure-suppression of the SWM signal are obtained, because two-photon resonance occurs soas to satisfy the suppression condition Δ₁ + Δ₄ + ΔE_69d(R)/ħ = 0, where probe field could not resonate with either of the two dressed energy levels |G_{4 ±} 0>, as shown in Fig. 3(d3). In the region Δ₁<0 (Δ₁ = −20 MHz), the reason for the first EIA and then EIT of probe transmission in Fig. 3(b1), and corresponding first enhancement and then suppression of $E_S^D$ in Fig. 3(b2), is that E₁ first resonates with |G₄₊0> (Fig. 3(d2)), i.e., the enhancement condition Δ₁ = −${{\Delta }^{\prime }}_4/2+\sqrt{{({{\Delta }^{\prime }}_4/2)}^2+|G_4{|}^2}$ is satisfied, with ${{\Delta }^{\prime }}_4$ = Δ₄ + ΔE_69d(R)/ħ; and then two-photon resonance occurs, i.e., the suppression condition ${\Delta }_1\text{+}{{\Delta }^{\prime }}_4\text{=}0$ is satisfied. When Δ₁ changes to be positive (Δ₁ = 20 MHz), the reason for the first EIT and then EIA of probe transmission, and first suppression and then enhancement of $E_S^D$ is that two-photon resonance first occurs; and then E₁ resonates with the dressed state |G_4-0>, i.e., the enhancement condition Δ₁ = −${{\Delta }^{\prime }}_4/2-\sqrt{{({{\Delta }^{\prime }}_4/2)}^2+|G_4{|}^2}$ is satisfied, as shown in Fig. 3(d4). When Δ₁ is set far away from resonance point (Δ₁ = ± 80 MHz), the pure-EIA and corresponding pure-enhancement of E_S are obtained because E₁ can only resonate with one of |G_{4 ±} 0>, as shown in Figs. 3(d1) and 3(d5), respectively. The global profile composed of the baseline of each curve in Fig. 3(b1) (dashed line) shows a simple single-photon absorption dip, and that in Fig. 3(b2) shows a two photon resonance (Δ₁−Δ₃ = 0) emission peak. This is different from the case of FWM because the SWM is singly dressed.

Furthermore, Figs. 3(c1) and 3(c2) present $E_S^D$ calculated by the reduced expression, versus Δ₁ and Δ₄, Δ₁ and R, respectively. Similar to Figs. 2(c1)-(c2), avoided-crossing plots are obtained, in which the AT splitting due to dressing field E₄ can be seen along the horizontal axis, and the transition between enhancement and suppression of $E_S^D$ is obtained along the vertical axis. Figure 3(c3) presents the intensity of $E_S^D$ versusΔ₁ and R calculated by reduced expression, which reflects the interaction between the dressing effect and Rydberg blockade. The trajectory of the main bright line has a leftward deflection with increasing R, as the value of the negative energy level shift ΔE_69d(R) increases, as shown in Fig. 1(e).

4. Primary and secondary blockades of MWM in the quadratomic system

4.1 Rydberg interaction in the quadratomic system

As mentioned above, when two atoms form a diatomic structure with strong Rydberg interaction in it, the diatomic energy levels will be shifted from the ones without potential perturbation. Furthermore, if the Rydberg energy level of a diatomic system has strong probability of dipole transition to that of another diatomic system, for instance, |40s40s> and |40p40p>, the perturbation will make the eigen-frequency shift again. We can call this effect as the secondary blockade to distinguish from the primary blockade in the diatomic system. Specifically, without the transition perturbation, the eigen-energy of |40p40p> (|40s40s>) is E(40p40p) + ΔE(40p40p) (E(40s40s) + ΔE(40s40s)), where E(40p40p) (E(40s40s)) is the eigen-energy without any blockade, and ΔE(40p40p) (ΔE(40s40s)) is the energy level shift due to primary blockade. So the energy gap between |40s40s> and |40p40p> without secondary blockade is Δ = E(40p40p)−E(40s40s) + ΔE(40p40p)−ΔE(40s40s), as shown in Fig. 4(a) . Because E(40p40p) (E(40s40s)) can be controlled by the external electric field intensity ε due to a consequence of the enormous Stark shifts, and ΔE(40p40p) (ΔE(40s40s)) by R, the energy gap ΔE by both ε and R. When ε = 0 is fixed, the energy level shift curves versus R of |40s40s> and |40p40p> are plotted in Fig. 4(b), from which ΔE(R) can be obtained. The dipole matrix element between these two diatomic states is . Take the dipole transition interaction into account and adopt the rotating wave approximation, the Hamiltonian of the subsystem composed of |40p40p> and |40s40s> is

 

Fig. 4 (a) The left figure presents the energy levels of a quadratomic system in atom states; μ₁ is the dipole matrix element between |40s> and |40p>. The right figure illustrates the corresponding energy levels in pair sates, in which Δ is the energy gap between |40s40s> and |40p40p>. (b) The energy level shift curves of |40s40s> and |40p40p> when only the primary blockade is considered, with the external electric field intensity ε = 0. Solid line: |40s40s>; dashed lines: |40p40p>. The only line ascending with increasing R of |40p40p> is selected in calculating Δ. (c) The eigen-frequencies of the two levels E₊(R) (the upper curve) and E₋(R) (the lower curve) with both the primary and secondary Rydberg blockades considered, versus R with ε fixed, when the eigen-frequency of |40s40s> without any blockade is set as zero. The neighborhoods of points A, B and C correspond to the regions where only resonant dipole-dipole interaction works, both resonant dipole-dipole and van der Waals interactions work, and only van der Waals interaction works, respectively. (d) The same quantity in (c) but versus ε with R fixed.

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Firstly, if |ΔΕ(ε, R)/2|>>$\left|{\mu }_1^2/R^3\right|$, Eq. (6) yields into

Then, when ε is tuned to make ΔE(ε, R) approach zero, i.e., in the case of |ΔΕ(ε,R)/2|<<$\left|{\mu }_1^2/R^3\right|$, the system exhibits a pair state resonance, and

If ε is fixed and ΔE(ε, R) only changes with R. When R is fixed, ΔE(ε, R) only has a linear dependence on ε. In the following, we will investigate the behavior of MWM signals in the two cases.

4.2 Atomic internuclear distance modulation on MWM

As described above, the energy level shift caused by the primary and secondary Rydberg blockades can be effectively controlled by the atomic internuclear distance R, and therefore the FWM and SWM signals can be also modulated due to the dependence of modified dressing filed detuning on the energy level shift. What’s more, by tuning the external electric field intensity, we can drive the Rydberg interaction into different regimes, in which the interaction has different R-dependence. Figures 5(a) and 5(b) present the evolution of $E_F^{DD}$ and $E_S^D$ when R is scanned with different Δ₁ in each curve, in different regimes of Rydberg interaction, respectively. The density matrix elements for the FWM and SWM processes are obtained as the following:

 

Fig. 5 (a) The intensity of the doubly-dressed FWM signal $E_F^{DD}$ in the quadratomic system, with energy level shift Δ_|ss>+ versus R at discrete probe detunings Δ₁ = −100, −50, −20, −10, 0, 10, 20, 50, and 100 MHz with the conditions (a1) |ΔΕ(ε, R)/2|>>$\left|{\mu }_1^2/R^3\right|$, (a2) |ΔΕ(ε, R)/2| at intermediate value compared with $\left|{\mu }_1^2/R^3\right|$, and (a3) |ΔΕ(ε, R)/2|<<$\left|{\mu }_1^2/R^3\right|$. (b) The intensity of the singly-dressed SWM signal $E_S^D$ in the quadratomic system, with the same condition as that in (a), except that Δ₁ = −100, −20, 0, 20, and 100 MHz from the left curve to right one in each row.

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As shown in Figs. 5(a1)-(a3), the peaks and dips in the curves within the same column broaden from top to bottom. The explanation is that the Rydberg interactions in Figs. 5(a1)-(a3) are in the regimes illustrated by the neighborhoods of the points C, B and A in Fig. 4(c), respectively. It is obvious that ΔΕ_|ss>+ decreases with increasing R more slowly in the neighborhood of A (B) than in that of B (C). So the characteristic signal of two-photon or dressed state resonance with the same frequency width in the same column is mapped into larger scanning length of R in the curve in Fig. 5(a3) (Fig. 5(a2)), than that in Fig. 5(a2) (Fig. 5(a1)), which appears with the rightward moving of peak and dip.

The SWM signal evolutions in Figs. 5(b1)-(b3) are similar to those in Figs. 5(a1)-(a3), except that each row has only one symmetry center and the global profile is a single-photon emission peak (dashed line). This is because the SWM is singly dressed, while the FWM is doubly dressed. Also, the enhancement peak in each curve here is due to the resonance with |G₄₊0> or |G_4-0>. What’s more, we can find that the transition between enhancement and suppression for the SWM process is just opposite to that in Fig. 3(b2). The reason is that as R increases, the energy level shift in the diatomic system (ΔE_69d(R)) increases while that in the quadratomic system (ΔΕ_|ss>+ (R)) decreases.

4.3 External electric field intensity modulation on MWM

Since the energy level shift can be controlled by ε, the dressed FWM and SWM processes can be also controlled by ε. As discussed above, ΔE(ε) and E _± are deduced with fixed R, and the latter one is plotted with ε in Fig. 4(d). Figures 6 (a1) and 6(a2) are obtained by substituting the energy level shift into Eq. (10.1), respectively, when ε is scanned with different Δ₁ in each curve. With ε scanned from small to large, ΔE_{|ss> ±} (ε) show declining tendency, as shown in Fig. 4(d). As a result, from the left curve to right one, the FWM signal shows the evolution from pure-enhancement, to first suppression and then enhancement, to pure-suppression, to first enhancement and then suppression, to pure-suppression, to first suppression and then enhancement, to pure-suppression, to first enhancement and then suppression, finally to pure-enhancement. Similar to the analyses in Fig. 5, the enhancement here is due to the dressed state resonance, and suppression due to two-photon resonance. Nevertheless, the enhancement conditions for Δ₁<0 and Δ₁>0 are different. With Δ₁<0, the enhancement condition is ${\Delta }_1+{\lambda }_G{{}_{{{}_2}_{+}}}_{G_{4\pm }}=0$, with ${\lambda }_G{{}_{{{}_2}_{+}}}_{G_{4\pm }}={{\Delta }^{″}}_4/2\pm \sqrt{{\left(\Delta {″}_4/2\right)}^2+{\left|G_4\right|}^2}+G_2$ being the eigen-frequency difference between |G₂₊G_{4 ±} 0> and |10>, where ${{{\Delta }^{\prime }}^{\prime }}_4={\Delta }_4$ + ΔE_{|ss> ±} (ε)/ħ–G₂ (ΔE_|ss>+(ε) for Fig. 6(a1) and ΔE_|ss>-(ε) for Fig. 6(a2)); With Δ₁>0, the enhancement condition is ${\Delta }_1+{\lambda }_G{{}_{{{}_2}_{-}}}_{G_{4\pm }}=0$, with ${\lambda }_G{{}_{{{}_2}_{-}}}_{G_{4\pm }}={{\Delta }^{″}}_4/2\pm \sqrt{{\left(\Delta {″}_4/2\right)}^2+{\left|G_4\right|}^2}-G_2$ with ${{{\Delta }^{\prime }}^{\prime }}_4={\Delta }_4$ + ΔE_{|ss> ±} (ε)/ħ + G₂. The suppression condition is always Δ₁ + Δ₄ + ΔE_{|ss> ±} (ε)/ħ = 0. The three symmetry centers and two-peak structure in global profile also appear in Figs. 6(a1) and 6(a2) (dashed lines).

 

Fig. 6 (a) The intensity of the doubly-dressed FWM signal $E_F^{DD}$ in the quadratomic system, versus ε at discrete probe detunings Δ₁ = −100, −50, −20, −10, 0, 10, 20, 50, and 100 MHz, considering both primary and secondary Rydberg interactions for (a1) the upper energy level shift Δ_|ss>+(ε), and (a2) the lower energy level shift Δ_|ss>−(ε); R is fixed at R = 9000 a.u. (b) The intensity of the singly-dressed SWM signal $E_S^D$ in the quadratomic system, with the same condition as that in (a), except that Δ₁ = −100, −20, 0, 20, and 100 MHz, from the left curve to right one in each row.

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Figures 6(b1) and 6(b2) are plotted by substituting ΔE_{|ss> ±} (ε) into Eq. (9.2) without the term Γ₄₀ + i(Δ₁ + Δ₄ + ΔE_{|ss> ±} (ε)/ħ). From the left curve to right one, the SWM signal shows the evolution from pure-enhancement, to first suppression and then enhancement, to pure-suppression, to first enhancement and then suppression, finally to pure-enhancement. In such case, the enhancement condition is ${\Delta }_1+{\lambda }_{G_{4+}}\text{=}0$ for Δ₁<0 and ${\Delta }_1+{\lambda }_{G_4{}_{-}}\text{=}0$ for Δ₁>0, where ${\lambda }_{G_{\text{4}\pm }}={{\Delta }^{\prime }}_4/2\pm \sqrt{{({{\Delta }^{\prime }}_4/2)}^2+|G_4{|}^2}$ with ${{\Delta }^{\prime }}_4={\Delta }_4$ + ΔE_{|ss> ±} (ε)/ħ. The suppression condition is always ${\Delta }_1+{{\Delta }^{\prime }}_4=0$, with ΔE_|ss>+(ε) for Fig. 6(b1) and ΔE_|ss>-(ε) for Fig. 6(b2).

What’s more, we can find an interesting phenomenon with ε scanned. As shown in Figs. 6(a1) and 6(b1), from the left curve to right one with Δ₁ increasing, the peak and dip are gradually broadened. To explain such broadening, on the one hand, all the curves in Figs. 6(a1) and 6(b1) are plotted with ΔE_|ss>+(ε), i.e., the upper curve in Fig. 4(d), in which the decreasing rate of ΔE_|ss>+(ε) becomes slowed down with increasing ε. On the other hand, in a curve with a certain Δ₁, compared with the curve at its left side with smaller Δ₁, ε must be scanned in a region where smaller ΔE_|ss>+(ε) can be reached to guarantee the satisfaction of the enhancement and suppression conditions. Therefore, the same frequency width for an enhancement peak (suppression dip), to raise (fall) and fall (raise), is mapped into larger scanning range in ε axis in a certain curve than in its left one in the same row, which appears with the broadening of peak (dip). In this way, the gradually narrowing of the peak/dip in curves from left to right in Figs. 6(a2) and 6(b2) can be also explained, considering the gradually accelerated decreasing rate of ΔE_|ss>-(ε) with increasing ε.

5. Anti-blockade in FWM process

Genuine control of MWM signal requires that the Rydberg blockade effect can be both revealed and eliminated (anti-blockade effect). Now, we study how to eliminate the primary blockade effect via controlling the dressing field, thus resulting in the anti-blockade effect. For an undressed FWM process F2 generating signal ${E^{\prime }}_F$ in the three-level system composed of |00>, |10> and |40>, as shown in Fig. 7 (b1), which includes an additional laser beam ${E^{\prime }}_4$ (ω₄,${k^{\prime }}_4$,${G^{\prime }}_4$ and Δ₄) connecting |40>−|10>. Compared with the system in Fig. 1(a), the density matrix element is ${\rho }_{\text{10}00}^{\text{(3)}}\text{=}-i{G}_1{G}_4{\left({G^{\prime }}_4\right)}^{\ast }\text{/}{d}_{10}^2{d}_{40}$. If |4> is set as 70S_1/2 and only the primary Rydberg blockade in the diatomic system is considered, the energy level shift of |40> could be written as ΔE_70s(R), and correspondingly the term d₄₀ will change into ${d^{\prime }}_{40}\text{=}{\Gamma }_{40}\text{+}i\left({\Delta }_4\text{+}{\Delta }_1\text{+}\Delta E_{\text{70}s}\text{(}R\text{)/}\hslash \right)$. When R = 10⁶ a.u., ΔE_70s(R) is so small that it can be neglected. In such case, with a certain Δ₄, the signal ${E^{\prime }}_F$ will have an emission peak at Δ₁ + Δ₄ = 0, i.e., the position where E₁ and E₄ have two-photon resonance (Fig. 7(b1)), as shown in the top curve in Fig. 7(a1). When R decreases, ΔE_70s(R) becomes noticeable. As a result, at the position Δ₁ + Δ₄ = 0, the two-photon resonance will occur at the position Δ₁ + Δ₄ + ΔE_70s(R)/ħ = 0 (Fig. 7(b2)). The second to fifth curves from top to bottom in Fig. 7(a1) show the right-to-left shift of the emission peak with decreasing R, i.e., increasing ΔE_70s(R), which reflects the blockade effect. Next, as shown in Fig. 7(b3), we add a strong field E₅ (ω₅, k₅, G₅ and Δ₅) connecting an additional level |20> with |40>, and |40> is split into |G_{5 ±} 0> with the eigen-frequencies as${\lambda }_{G_{\text{5}\pm }}={\Delta }_5/2\pm \sqrt{{({\Delta }_5/2)}^2+|G_5{|}^2}$. So the detuning of E₄ is modified into ${{\Delta }^{\prime }}_{\text{4}}\text{=}{\Delta }_{\text{4}}\text{+}\Delta E_{\text{70}s}\text{(}R\text{)/}\hslash \text{+}{\lambda }_{G_{\text{5}\pm }}$. As a result, two peaks occur in Figs. 7(a2) and 7(a3), when the two-photon resonance conditions ${\Delta }_{\text{1}}\text{+}{\Delta }_{\text{4}}\text{+}\Delta E_{\text{70}s}\text{(}R\text{)/}\hslash \text{+}{\lambda }_{G_{\text{5}\pm }}\text{=0}$ are satisfied. As shown in Fig. 7(a2), in the second to fifth curves from top to bottom, one of the two peaks reoccurs at Δ₁ + Δ₄ = 0, i.e., the blockade effect is eliminated with Δ₅ = 0, −13, −22 and −42 MHz, respectively. It is because that Δ₅ is properly chosen for each curves with different R to ensure $\Delta E_{\text{70}s}\text{(}R\text{)}/\hslash \text{+}{\lambda }_{G_{\text{5-}}}\text{=0}$. Similarly, as shown in Fig. 7(a3), one of peaks reoccurs at Δ₁ + Δ₄ = 0, in the second to fifth curves from top to bottom, with different dressing field Rabi frequencies G₅ = 15, 23, 32 and 41 MHz, respectively. It is also because the condition $\Delta E_{\text{70}s}\text{(}R\text{)}/\hslash \text{+}{\lambda }_{G_{5-}}\text{=0}$ is satisfied in each curve due to the adoption of appropriate G₅, which can be controlled by changing the power of E₅ in experiment.

 

Fig. 7 (a1) The intensity of the undressed FWM signal ${E^{\prime }}_F$ in a ladder-type system, versus the probe detuning Δ₁ at discrete atomic internuclear distances R = 1000000, 125000, 115000, 106000 and 98000 a.u. from top to bottom. For each curve in (a2) and (a3), R is the same as that of the corresponding curve in (a1). (a2) The intensity of the singly-dressed FWM signal, versus Δ₁ at discrete dressing detunings Δ₁ = 0, −13, −22, and −44 MHz, corresponding to the second to the fifth curves from top to bottom, respectively. The top curve is the same as the top one in (a1). (a3) The intensity of the singly-dressed FWM signal versus Δ₁ at discrete Rabi frequencies of dressing field G₅ = 15, 23, 32 and 41 MHz, corresponding to the second to fifth curves from top to bottom. The top curve is the same as the top one in (a1). Δ₄ = 0. (b) The energy levels corresponding to (a).

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In a word, for a given internuclear distance R, namely a certain energy level shift, we can move one of the two dressed states to the position of undressed level without blockade effect, via changing frequency detuning or power of the dressing field.

6. Singly-dressed FWM with multiple Rydberg levels

We consider the case that the atomic states |1>, |2> and |4> in Fig. 1(a) are all Rydberg states, and the corresponding states |10>, |20> and |40> in the diatomic system have energy level shift ΔE₁(R), ΔE₂(R) and ΔE₄(R), respectively. For experimental demonstration, the configuration requires E₂ and E₄ to be microwave fields. The density matrix element related to $E_F^D$ could be written as

In calculation, we set |1>, |2> and |4> as 40P_1/2, 60S_1/2 and 70S_1/2, respectively. Then ΔE₁(R)/ħ, ΔE₂(R)/ħ and ΔE₄(R)/ħ can be obtained after substituting the real parameters into the R-dependent expressions of ΔE_40p(R), ΔE_60s(R) and ΔE_70s(R), respectively. They all present a declining trend as R increases. The calculated intensity of $E_F^D$ and probe transmission versus R at different Δ₁ are plotted in Figs. 8(a) and 8(b), respectively.

 

Fig. 8 The theoretical calculations of (a) singly-dressed FWM signal $E_F^D$ and (b) the probe transmission versus R at discrete probe detunings Δ₁ = 5, 10, 15, 20, 25, 30, 35 and 40 MHz from top to bottom. (c) The energy diagrams corresponding to (a).

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As shown in Fig. 8(b), there are two dips and one wide region between them in each curve, which correspond to the two EIA and one broadened EIT due to strong dressing effect, respectively. Meanwhile, three peaks are obtained in each curve in Fig. 8(a), in which the two broad ones (at two sides) correspond to the two EIA dips in Fig. 8(b). An intuitive understanding of these three peaks can be obtained from Eq. (10), in which three zero points related to three resonance peaks exist in the denominator, i.e., the middle peak corresponds to ${d^{\prime }}_{20}$ and the peaks at two sides correspond to ${d^{\prime }}_{10}+{\left|G_4\right|}^2/{d^{\prime }}_{40}$.

A more detailed analysis can be obtained with the help of singly-dressed energy-level diagrams. As shown in Fig. 8(c), |10> is split into |G_{4 ±} 0> by E₄. The reason for the first EIA of probe transmission and corresponding peak of FWM signal curve is that E₁ resonates with |G_4-0> (Fig. 8(c1)), thus the enhancement condition ${\Delta }_1\text{+}{\lambda }_{G_{4-}}\text{=}0$is satisfied with ${\lambda }_G{}_{{{}_4}_{-}}={{\Delta }^{\prime }}_4/2-\sqrt{{\left(\Delta {\prime }_4/2\right)}^2+{\left|G_2\right|}^2}$ being the frequency detuning between |10> and |G_4-0>, in which ${{\Delta }^{\prime }}_4={\Delta }_4\text{+}\Delta E_4\left(R\right)/\hslash$. As R increases, the second EIA of the probe transmission with corresponding peak of FWM signal is obtained when E₁ resonates with |G₄₊0> (Fig. 8(c3)), and the condition ${\Delta }_1\text{+}{\lambda }_{G_{4+}}\text{=}0$ is satisfied with ${\lambda }_G{}_{{{}_4}_{+}}={{\Delta }^{\prime }}_4/2+\sqrt{{\left(\Delta {\prime }_4/2\right)}^2+{\left|G_2\right|}^2}$. Moreover, the middle peak of FWM signal is because of the two-photon resonance, i.e., Δ₁ + Δ₂ + ΔE₂(R)/ħ = 0, as shown in Fig. 8(c2). With Δ₁ increasing (from top to bottom in Figs. 8(a) and 8(b)), we can find that the positions of the EIA dips of probe transmission, and three peaks of FWM signal all move toward the right direction along the R axis, because ΔE_40p(R), ΔE_60s(R) and ΔE_70s(R) must be smaller to ensure E₁ with larger Δ₁ get two-photon and dressed state resonances.

7. Conclusion

In summary, we have first proposed a scheme to control the MWM signal in Rydberg atoms by the interaction between the dressing effect and Rydberg blockade. In the diatomic system, the primary blockade can be employed to modulate the enhancement, suppression and avoided crossing of doubly-dressed FWM and singly-dressed SWM signals, by controlling the atomic internuclear distance. In the quadratomic system, secondary blockade occurs and besides the internuclear distance, the external electric field intensity can be also exploited to effectively control the enhancement and suppression of MWM signals. Moreover, we have also demonstrated the anti-blockade effect, i.e., the elimination of primary blockade effect in MWM process, by the counteraction between Rydberg blockade and dressing effect of light field. Such investigation can have potential applications in the quantum computing with Rydberg atom as the carrier of qubit.